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Synchronous rhythms represent a core mechanism for sculpting temporal coordination of neural activity in the brainwide network. This review focuses on oscillations in the cerebral cortex that occur during cognition, in alert behaving conditions. Over the last two decades, experimental and modeling work has made great strides in elucidating the detailed cellular and circuit basis of these rhythms, particularly gamma and theta rhythms. The underlying physiological mechanisms are diverse (ranging from resonance and pacemaker properties of single cells, to multiple scenarios for population synchronization and wave propagation), but also exhibit unifying principles. A major conceptual advance was the realization that synaptic inhibition plays a fundamental role in rhythmogenesis, either in an interneuronal network or in a recipropocal excitatory-inhibitory loop. Computational functions of synchronous oscillations in cognition are still a matter of debate among systems neuroscientists, in part because the notion of regular oscillation seems to contradict the common observation that spiking discharges of individual neurons in the cortex are highly stochastic and far from being clock-like. However, recent findings have led to a framework that goes beyond the conventional theory of coupled oscillators, and reconciles the apparent dichotomy between irregular single neuron activity and field potential oscillations. From this perspective, a plethora of studies will be reviewed on the involvement of long-distance neuronal coherence in cognitive functions such as multisensory integration, working memory and selective attention. Finally, implications of abnormal neural synchronization are discussed as they relate to mental disorders like schizophrenia and autism.
In 1958, Frédéric Bremer published a seminal review in this journal on ‘theoretical and experimental data pertaining the nature, origin, synchrony and functional significance of brain waves’ . Half a century later, the study of cortical rhythms has become an area of converging interests across many disciplines in Neuroscience [145, 143]. Network oscillations are attractive for cellular neurophysiologists interested in understanding network behavior in terms of the underlying biophysical mechanisms. Synchronous oscillations can be studied in detail using in vivo and in vitro preparations, and increasingly in combination with powerful genetic tools. Therefore, for neurophysiologists and theorists alike, the study of synchronous rhythms offers an excellent venue to investigate how collective network dynamics emerge from the interplay between cellular biophysics and synaptic circuits. On the other hand, for systems neurophysiologists working with alert animals, who have long recorded single neurons with tremendous success in uncovering neural correlates of behavior, understanding coordinated neural population patterns in a circuit represents a new challenge. Moreover, cognition involves a large network of brain structures, therefore it is critical to elucidate neuronal interactions between different brain regions, increasingly with the help of multi-electrode recordings and imaging techniques. Coherent oscillations, more generally neuronal synchrony, may play a role in well timed coordination and communication between neural populations simultaneously engaged in a cognitive process.
There are many commonalities among network oscillations in various nervous systems. Indeed, it has been suggested that cortical circuits are similar to central pattern generators . However, whereas central pattern generators are well described in terms of coupled oscillators , it is questionable that cortical rhythms can be conceptualized in that framework. Typically, brain rhythms of the waking brain are reflected by small-amplitude scalp electroencephalogram (EEG) signals. They occur intermittently by short episodes in time [378, 711], and synchronization is subtle and typically confined to restricted neural populations [238, 630]. As a matter of fact, awake behaving states were traditionally called ‘desynchronized EEG’ in contrast to large-amplitude slow oscillations observed in quiet (non-rapid eye movement, non-REM) sleep [908, 909, 913, 236]. It is an interesting and, by no means obvious, question as to how desynchronization or asynchrony is achieved in a neural network [2, 345, 1029, 243, 997, 1016, 123, 126, 241, 538, 232]. Presumably, cortical networks in wakefulness are predominantly asynchronous, at the same time weak oscillations and selective coherence are present. At the single neuron level, spike discharges of cortical cells are highly irregular, characterized by some measures as approximate Poisson processes [884, 862], hence far from behaving as clock-like oscillators. Synaptic inputs to cortical neurons measured in vivo also display large stochastic fluctuations . Such irregular neural firing patterns may be computationally desirable to maximize information content of spike trains for neural coding of sensory information , generating stochastic choice behavior in decision making [1034, 226, 487], and other brain functions. At first sight, the strongly stochastic nature of neuronal spike discharges is at odds with the popular description of network rhythms in terms of synchronization of clock-like neural oscillators.
Unlike a well defined function such as respiration or walking for a rhythmic motor pattern generation system , it has been elusive to conclusively demonstrate that specific brain processes causally depend on certain cortical oscillations [877, 876, 861]. Instead of focusing on rhythmicity per se, the function of synchronous oscillations should be better appraised in the more general perspective of correlation between spike times of neurons. Neural correlation implies synchronization on some timescale, which can occur with or without oscillations. High variability of spike discharges itself depends on correlations in the inputs [917, 420], since the variability of inputs that converge onto a cortical neuron would be largely averaged out through summation over a large number of uncorrelated neurons . In the cortex, pairwise neural correlation of trial-by-trial response fluctuations (noise correlation) is typically small [339, 1097, 568, 40, 187]; recent work indicates that the population average of noise correlations is approximately zero in some dynamical states of the cortex [262, 801, 433]. But even weak neural correlations can have strong impact on a network’s state , and may be important for various computations ranging from sensory coding, to attentional gain modulation, to planning and goal choices [376, 890, 992, 838, 728, 1016, 851, 36, 233, 221, 642, 771, 294, 835, 979, 941]. Perhaps no recent findings better highlighted the importance of correlated spike times than the discovery of spike-timing dependent plasticity (STDP) for synaptic learning [1, 344, 643, 82, 1093, 81, 213, 151, 879]. STDP depends on the relative timing of the pre- and post- synaptic spikes as well as the firing rates [643, 880, 879, 540]; if the spike discharges of a pair of neurons were uncorrelated Poisson processes, there would be no net modification of synaptic connections between the two [505, 891, 826]. In the hippocampus, for instance, STDP represents an attractive mechanism for sequence learning in spatial navigation . As an animal moves along a linear track, location- and direction-selective place cells in the hippocampus are activated sequentially. If two neural pools fire one after another within tens of milliseconds, according to the STDP rule, synaptic connections from those firing earlier to those firing later should be strengthened, whereas synaptic connections in the opposite direction are weakened. An observable consequence is that the shape of the spatial tuning curve of hippocampal place cells becomes asymmetric (skewed) with experience, a prediction that was confirmed experimentally . This example illustrates how a learning mechanism specifically depends on temporal correlations of neural firing activity. Correlated activity (coherent rhythms in particular) also plays an important role in circuit pattern formation of the developing brain [289, 1082, 88].
Synchronous oscillations are a simple and appealing way to produce or enhance temporal correlations between neurons , which are critical for STDP. Conversely, STDP may be a driving force to determine neuronal synchronization patterns in a network [464, 157, 731, 941, 319], or to detect specific temporal patterns in the input . Interestingly, a recurrent network endowed with STDP at local synapses could self-organize into complex oscillatory patterns, perhaps at the border between synchronization and desychronization, with irregular yet correlated spatiotemporal dynamics . One of the main objectives of this review is to marshal experimental data and theoretical work that are beginning to resolve the apparent dichotomy between stochasticity and synchrony of cortical dynamics in cognition.
Bremer’s 1958 review focused on the ‘alpha rhythm’, large-amplitude EEG sinusoidal (~ 10 Hz) oscillations that spontaneously occur in an awake and relaxed state [110, 108, 20, 910]. By that time, it was already known that other EEG rhythms could be observed in certain behavioral states , or be induced by sensory stimulus . However, these macroscopic EEG signals, when recorded from the scalp, have small-amplitudes and are spatially poorly localized. Better measurements could be obtained invasively using subdural electrocorticogram (ECoG) with the scalp removed and electrodes directly placed on the cortical (pial) surface, so that the signals are not smeared spatially by passing through the skull; or mesoscopic local field potential (LFP) with electrodes inserted deep into the brain [733, 732]. Using ECoG or LFP recordings in animals, a variety of brain rhythms have been found across various behavioral states during wakefulness, which are briefly described below.
This review will focus on these synchronous cortical oscillations directly associated with cognitive behavior.
The last two decades have seen the accumulation of a wealth of our knowledge about brain rhythms during behavior, thanks to a confluence of experimental advances. First, in human studies, magnetoencephalogram (MEG), and iEEG recordings using as subjects epileptic patients (who are undergoing iEEG for localization of the seizure onset regions before surgery), provide measurements with a higher spatiotemporal resolution and much better signal-to-noise ratio than the scalp EEG. Second, an increasing number of physiologists record simultaneously LFP and single units from animals performing cognitive tasks, it is thus possible to examine the timing of spiking activity of individual cells relative to LFP oscillations during behavior. Third, new techniques make it possible to record in vivo from identified neuron types (pyramidal cells or different subclasses of GABAergic interneurons), yielding valuable information about how they relate to each other and to a population rhythm. Fourth, network oscillations similar to behaviorally relevant rhythms such as gamma and theta can be reproduced in cortical slices in vitro, where the circuit mechanisms underlying rhythm generation can be thoroughly analyzed. At the same time, it has been increasingly recognized that the membrane dynamics of single neurons are determined by a diverse set of voltage-gated and calcium-gated ion channels. As a result, some subclass of cortical cells display intrinsic oscillations and are potentially capable of serving as pacemakers of network rhythms. The history of research on synchronous neural oscillations is an excellent case of progressive advances from macroscopic (average) description of neural populations to a microscopic understanding of circuit dynamics of interconnected neurons endowed with all their glorious electrophysiological details. Fifth, with the new targeted optogenetic manipuation that allows one to rapidly and selectively activate or de-activate a subtype of nerve cells [227, 613, 411], it is now possible to establish causal links between an observed neural activity pattern and its underlying circuit mechanism(s). Sixth and finally, mathematical theory and ever more sophisticated biophysically-based neuronal network modeling provide valuable tools for exploring the dynamic behavior of such recurrent neural circuits, sometimes with surprising results.
What are the cellular mechanisms of synchronous cortical rhythms? In his 1958 review, Bremer articulated the idea that ‘the cortical waves are the synchronized pulsations of cortical neurons’, and proposed that synchronization of large-amplitude alpha oscillations across the two hemispheres was produced by intracortical excitatory connections. The idea of synchronization by excitatory connections is intuitively appealing, as one would expect that mutual excitation between principal neurons should bring them into firing together. This idea gained support from experimental and computational studies on epilepsy, where blockade of synaptic inhibition in a cortical network led to extremely synchronous neural firing patterns, resembling epileptic discharges [969, 968]. In the 1990’s, however, modeling studies have begin to question the generality of this mechanism, with the finding that whether excitation synchronizes coupled neurons depends on the membrane properties of the constituent cells and the time courses of synaptic currents [415, 1000, 527]. On the other hand, it has been known for a long time [21, 1060, 1061, 303, 318] that synchronous rhythms can be generated by an interplay between excitation and inhibition, as fast recurrent excitation followed by slower feedback inhibition readily gives rise to oscillations. In this scenario, inhibition plays an important role in rhythmogenesis. More surprisingly, in the early 1990’s Wang and Rinzel found, in modeling studies, that mutually inhibitory synaptic interactions can synchronize a population of GABAergic neurons, therefore coherent oscillations can emerge in a purely inhibitory neuronal network [1036, 1037]. This ‘interneuronal network scenario’ was initially revealed in a model of thalamic oscillations [1036, 1037], but has proven to be a general principle [1000, 1035, 275, 970]. In particular, theoretical and experimental work found that this mechanism is capable of generating synchronous gamma (~ 40 Hz) oscillations in a hippocampal network of inhibitory neurons in vitro [1053, 1035, 970]. Both the interneuronal network and the excitatory- inhibitory loop scenarios critically depend on synaptic inhibition, which thus does not merely control the overall network excitability but plays an essential role in temporally sculpting cortical network activity patterns during behavior.
In the first part of this review, I will cover a rich variety of single neuron properties (Section II), basic synaptic circuit mechanisms and network architectures (Sections III-IV) underlying the generation of cortical network oscillations associated with cognition. Then, I will marshal the evidence showing that the theory of coupled oscillators is not adequate for capturing the salient characteristics of cortical dynamics of the waking brain. I will present an alternative theoretical framework for describing weakly synchronous population rhythms in which single cells fire action potentials irregularly and intermittently in time (Section V). This is followed by Section VI, which is devoted to recent work on behaving animals and human that explore functional implications of synchrony and coherent rhythms, with an emphasis on long-distance neural communication and top-down signaling. In Section VII, I will briefly discuss possible implications for abnormal neural synchronization associated with mental disorders. Finally, I will summarize some general insights that have been gained from experimental and theoretical studies (Section VIII).
Single neurons in the cortex are endowed with a rich repertoire of active ion channels that are distributed across the somatodendritic membrane [597, 623, 899, 879]. As a result, diverse cell types display a variety of subthreshold membrane response dynamics and spike firing patterns. Synchronization of a circuit depends critically on the intrinsic membrane properties of its constituent cells. In a recurrent network, precisely how single neurons adjust the timing of action potentials in response to phasic synaptic inputs from other neurons greatly affects whether the network ultimately becomes synchronized or not. Moreover, the selection of a frequency band at which coherent oscillations occur is shaped by pacemaker or resonance properties of individual cells. Thanks to many studies over the last two decades, with close two-way interactions between experiments and theory, we now know a lot about the interplay between single neuron dynamics and network synchronization. In fact, this area represents an example par excellence of why detailed single cell properties do matter with regard to collective network dynamics. In this Section, I shall review important aspects of cellular electroresponsiveness relevant to network oscillations.
All single neurons are oscillators, insofar as they respond to a constant input current I by repetitive firing of action potentials at a certain frequency f (period T0 = 1/f). In the classic Hodgkin-Huxley model of action potential , this periodic behavior is mathematically described as a stable oscillatory state. Since action potentials are stereotypical events, it makes intuitive sense to focus on the timing of action potential, or the phase (between 0 to 360 degrees), relative to the oscillatory cycle. In this view, the issue of network synchronization can be formulated in terms of how synaptic interactions affect the phases of action potential among connected neurons. This ‘phase reduction theory’, pioneered by Kopell, Ermentrout and others, is well founded under the ‘weak coupling condition’, when synaptic interactions are not too strong to greatly alter the intrinsic frequency of neural oscillators [539, 524, 276, 280, 1065, 999, 413, 525]. Even when synaptic interactions are not weak, the phase reduction theory still provides useful insights. For a sinusoidal oscillation, sin(2πft), which arises from a linear system, phase shift induced by a brief pulse perturbation is the same regardless of when the pulse is applied. In contrast, neuronal oscillators are highly nonlinear, characterized by a stereotypical fast event (the action potential) followed by a slow refractory and recovery period. For such ‘relaxation type’ oscillators [25, 922], the phase response highly depends on the timing of the input pulse. This is quantified by a ‘phase response curve’ (PRC), where the induced phase shift of spike is plotted as a function of the phase at which a brief depolarizing current pulse is applied [359, 1065, 811, 279, 210].
The classic Hodgkin-Huxley model exhibits a biphasic ‘type II’ PRC (Fig. 2A) [415, 119]: a depolarizing pulse induces a phase advance (making the spike to fire earlier), when it is applied some time before the next spike is expected to occur, but it actually delays the phase of the next spike when applied soon after the spike discharge. This is because during that time the neuron is recovering from afterhyperpolarization, as the delayed rectifier potassium current decays away and the fast sodium current de-inactivates. Both decay processes are slowed momentarily by the brief depolarization, therefore it takes longer for the next spike to be generated.
Intuitively, synaptic excitation is expected to synchronize coupled neurons, if neurons that happen to fire earlier in an oscillatory cycle advance the spike times of other (‘later firing’) neurons, which in turn delay the spike times of ‘earlier firing’ neurons, so that all neurons will tend to fire more synchronously in the next oscillatory cycle and the network will be brought in phase. This is precisely what happens for type II neurons coupled by fast (‘instantaneous’) excitatory synapses (Fig. 2B). Assume that initially cell #1 fires ahead of cell #2. Then synaptic excitation from cell #1 advances the next spike in cell #2. When cell #2 fires, cell #1 is still in the recovery phase, so synaptic excitation from cell #2 actually delays the next spike in cell #1. Therefore, interactions in both directions tend to reduce the phase difference between the two cells, eventually leading to spike-to-spike synchrony. However, this intuitive reasoning is no longer sufficient if synaptic interactions are not brief instantaneous perturbations, but have significant time courses of their own (see below).
Unlike the classic Hodgkin-Huxley model, a neuronal PRC may not display a significant ‘negative lobe’ [275, 119, 658]. That is, the spike phase is always advanced regardless of the stimulus timing, even though the amount of induced phase shift depends on the timing of the pulse perturbation. Such a ‘type I’ PRC is shown in Fig. 2C (upper panel), using a model modified from the classic Hodgkin-Huxley model with some parameter variations . The parameter changes reduce the potassium conductance and speed up the sodium and potassium current kinetics. As a result, the recovery from hyperpolarization is fast, and the delay portion of the PRC essentially disappears. Therefore, type I and type II PRCs can be produced by the same set of ion channels but with quantitatively different properties.
For type I neurons, a cell that fires earlier is phase-advanced by another cell that fires later, which acts against reducing the relative phase difference between the two. Hence it is not straightforward to ascertain intuitively synchronization by mutual excitation to type I neurons. As it turns out, a mathematical analysis using the phase reduction theory, which will be discussed in Section III, found that type I neurons tend to be much more easily synchronized by mutual inhibition than excitation [415, 998, 275]. This theoretical finding shows that single cell properties can qualitatively alter network behavior in a completely counter-intuitive way.
Phase response properties have been routinely characterized for neurons in central pattern generator systems [772, 638]. More recently, in vitro slice studies have directly assessed PRCs of neurons in the mammalian cerebral cortex, using current perturbations that mimic excitatory or inhibitory postsynaptic potential (EPSP or IPSP) [802, 719, 978, 936]. It was generally found that for a given cell, the classification of its PRC into type I or type II is consistent for a range of intrinsic oscillation frequencies, and not sensitive to the exact amplitude of PSP perturbations [719, 936]. Distinct neuron types exhibit differential preponderance for phase response properties. Nevertheless, neurons within a neural population are heterogeneous and exhibit both type I and type II PRCs , possibly as a result of quantitative differences in the expression of ion channels across neurons in a population.
The precise shape of PRC of a neuron depends on its repertoire and somatodendritic distribution of various ion channels. In general, regenerative currents (such as a persistent sodium current INaP that provides a positive feedback for membrane depolarization) tend to shift leftward a PRC and eradicate the negative lobe, thereby favoring type I behavior [766, 362]. On the other hand, restorative currents (such as a voltage-activated M-type potassium current IM that acts as a negative feedback for voltage changes) tend to accentuate the negative lobe [1044, 206, 277, 362]. This is because a current like IM is activated by depolarization during an action potential, and decays afterwards as the membrane recovers from afterhyperpolarization. A brief depolarization soon after a spike would counter this decay process, hence delaying the timing of the next spike. The hyperpolarization activated cation current IH de-activates with depolarization, hence has a similar effect as a depolarization-activated outward current. IH is thus expected to accentuate the negative lobe and favors type II PRC behavior, a prediction that was confirmed with pharmacologically blockade of IH in pyramidal neurons . Another important issue is how PRC depends on the location of synaptic inputs in a spatially complex neuron [362, 799]. Goldberg et al.  addressed this question using both modeling and patch-clamp recording in vitro. They found that dendritic PRC (obtained using a current pulse applied onto the dendrite, 35 to 250 μm from the soma) is a linearly filtered version of somatic PRC (obtained using a current pulse applied onto the soma); the filter depends on the passive and active properties of the dendrite. As a result, somatic and dendritic PRCs can in principle show disparate behaviors. For example, because of an increasing gradient of expression of the H ion channel conductance from soma to distal dendrite , dendrites of a pyramidal cell can display type II PRC, whereas its somatic PRC can be type I.
A calcium-activated potassium current (IKCa) and, to a lesser degree, IM and IH, underlie spike-frequency adaptation, which is a common characteristic of pyramidal neurons [668, 619, 620, 650, 293, 1027, 793] and certain subclasses of inhibitory cells [499, 500]. As it turns out, a large IM (which has a low voltage threshold and is significant at resting membrane potential), but not IKCa (which depends on calcium influx through the opening of high-threshold calcium channels during action potentials), can switch a neuron from type I to type II [277, 782]. Conversely, a reduction of IM (e.g. by cholinergic modulation) can transform a pyramidal cell from type II to type I . Hence, type I versus type II PRC behavior depends on low-threshold potassium currents, but is not necessarily correlated with a neuron’s (primarily IKCa dependent) spike-frequency adaptation. Indeed, in vitro pyramidal cells display strong spike-frequency adaptation, yet show mostly type I behavior; when they do display type II PRCs the negative lobe is fairly shallow [802, 395, 978, 362]. Also, ‘low-threshold spiking’ (LTS) interneurons display spike-frequency adaptation, whereas fast spiking (FS) interneurons do not [668, 499, 500, 645]. At the same time, the observed tendency appears to be type I behavior for LTS cells and type II behavior for FS cells . Such differential PRC properties of different neural types have important implications for their distinct roles in generating network rhythms (see Section III).
Resonance refers to the phenomenon of band-passed neuronal response, with maximal response at a preferred input frequency. In studies using sinusoidal input currents of varying frequencies, or the ‘impedance amplitude profile’ protocol that sweeps through many frequencies in a single stimulus [784, 455, 454, 453], many neuron types were found to display resonance, namely a peak in their impedance (the ratio of the subthreshold membrane potential response and the input current intensity) at a particular (resonance) stimulus frequency. Resonance is intuitively expected when single neurons are damped oscillators hence display an intrinsic preferred frequency [1025, 552]. However, resonance does not necessarily require damped oscillation of single neurons [453, 805]. It readily takes place when there are two opposing processes, one counteracts membrane potential Vm changes at low frequencies (with a time constant τ1), the other does the same at high frequencies (with a time constant τ2)). The combination of the two gives rise to a peaked response at a frequency fR intermediate between 1/(2πτ1) and 1/(2πτ2) .
Membrane passive properties (e.g. the leak conductance) naturally act as a low-pass filter (τ2 = τm ~ 10 ms), whereas voltage-gated active channels give rise to high-pass filtering. For example, an IH opposes voltage changes in a subthreshold voltage range: a hyperpolarization of Vm is counteracted by an increased inward IH, whereas when Vm is depolarized IH inactivation has the same effect as activation of an outward current. IH has a time constant on the order of a hundred milliseconds, thus behaves as a high-pass filter. The interplay with the leak conductance as a low-pass filter results in a resonance at fR ~ 5–10 Hz. An H-current dependent mechanism underlies resonance observed in layer II stellate cells of entorhinal cortex, and combined with membrane noise gives rise to subthreshold oscillations in these cells [519, 242, 396, 298, 397, 356]. Interestingly, IH-dependent resonance is most prominent when the inputs are located at distal dendrites [991, 191, 192, 637], consistent with an increasing gradient of IH expression from soma to distal dendrite of pyramidal neurons . Another example is a voltage-gated potassium current, like IM or a low-threshold slowly inactivating potassium current IKS, which counter-acts membrane potential changes in a depolarized range. Together with suppressed responses at high frequencies by the leak conductance, such a current produces resonance when neurons are excited to significantly above the resting membrane potential but below firing threshold [453, 445]. A single neuron may display more than one resonance (bandpass) filters that are spatially segregated, for instance an IH dependent mechanism in the dendrite and an IM dependent mechanism in the soma .
Typically, resonance by a depolarization-activated K+ current is greatly amplified by the presence of a persistent sodium current INaP, which is a positive feedback operating in the same voltage range as IM or IKS [1025, 601, 394, 455, 454, 1047, 453, 463, 445, 705]. The resonance frequency fR critically depends on the activation time constant of the slow potassium current , often in the theta (4–8 Hz) [1025, 394, 1046, 445] or gamma (~ 40 Hz) frequency range .
How does subthreshold resonance relate to neuronal firing of action potentials? It has been shown that, in hippocampus, neurons display similar frequency preference for membrane potential responses and spike discharges at low firing rates (when Vm stays below firing threshold most of the time) . Moreover, resonance frequency is in the theta frequency range for pyramidal cells, and in the gamma frequency range for fast spiking interneurons, suggesting differential roles of these two major cell types in theta- versus gamma-frequency network oscillations . In a systematical modeling study of neuronal resonance, Richardson et al  considered two model neurons with resonance frequency at fR, one based on IH and the other on IKS and INaP. They used input currents composed of three components: a baseline that controls the overall average spike rate r0 of the model neuron, a weak sinusoidal wave at frequency f and a background noise term mimicking stochastic synaptic bombardments in vivo . They found that, interestingly, subthreshold resonance may or may not be reflected in the spike firing activity, depending on the degree of neural firing variability. When noise is weak and firing is regular, the maximal modulation by the sinusoidal input occurs at f r0, so the preferred frequency is the firing rate itself. By contrast, when noise is strong and firing irregular, the firing response is best modulated by the subthreshold resonance frequency at f fR. This is true even when the mean firing rate r0 is much higher than fR, e.g. r0 = 20 Hz and fR = 5 Hz (Fig. 3). Although somewhat surprising, this result may be explained by observing that, with a stronger noise, a lower mean input is needed to produce the same mean firing rate r0, because spikes are more often triggered by stochastic input fluctuations. Therefore on average the membrane potential is lower where subthreshold resonance plays a more important role. In other words, noise is important for unmasking a subthreshold resonance in the spike firing patterns. This insight is likely to have implications for understanding the relationship between highly irregular neural discharges and coherent network rhythms (Section V).
Strong resonance is usually a manifestation of membrane potential oscillations, because even a transient (‘damped’) oscillation endows a neuron with an intrinsic preferred frequency and an sinusoidal input with matched frequency evokes maximal responses [453, 805]. A neuron becomes a pacemaker if it displays sustained intrinsic oscillations [599, 15, 600, 106, 1038]. In the neocortex, membrane potential oscillations below spike threshold were first reported at theta frequency in layer II entorhinal cortical neurons by Alonso and Llinás in 1989  and at gamma frequency in neocortical neurons by Llinás et al. in 1991 . In both cases the intrinsic rhythmicity was shown to be TTX sensitive, suggesting that a low threshold Na+ current plays a critical role. These findings motivated a computational model, published in 1993, proposing that the observed intrinsic membrane potential oscillations arise from the interplay between a low-threshold persistent Na+ current INaP and a slow low-threshold K+ current (IKS) . Oscillation naturally occurs as a result of the interaction between a positive feedback (between Vm depolarization and INaP activation) and a slower negative feedback (depolarization induced IKS activation). The activation time constant of IKS controls the recovery phase of the oscillation cycle hence largely determines the rhythmic frequency which can be in the gamma or theta frequency range . The same model was analyzed in the study of resonance by Richardson et al. .
Notably, the model of  predicts a different relationship between the oscillation frequency and the input drive in these two cases. In the gamma frequency regime, subthreshold oscillation frequency increases with the input drive (see Section II-C). In the theta frequency regime, the frequency remains roughly constant (exhibiting a plateau) as long as the oscillation is subthreshold, and starts to increase with the applied current intensity when spike firing becomes significant. The predicted plateau phenomenon was observed in stellate cells of entorhinal cortex (Fig. 5F in .)
Among neural types that exhibit subthreshold membrane oscillations, of special interest are GABAergic neurons in the nucleus basalis  and medial septum  (Fig. 4A). The nucleus basalis and medial septum are the major parts of the basal forebrain that provide the neuromodulatory transmitter acetylcholine to the neocortex and hippocampus, respectively [833, 909, 910, 669]. In addition to cholinergic cells, these structures also contain GABAergic cells with axonal projections to the cortical mantle. GABAergic cells in the medial septum are believed to act as a pacemaker for theta rhythm in the hippocampus, and those in the nucleus basalis may play a similar role for theta rhythm in the neocortex. As shown in Fig. 4A, medial septum non-cholinergic (putative GABAergic) cells display ‘mixed-mode oscillations’ in which periodic repetition of clusters of spikes are interspersed in time with epochs of subthreshold oscillations. The inter-cluster oscillation is in the theta frequency range, whereas both subthreshold membrane potential oscillations and intra-cluster spike firing rate are in the gamma frequency range. Such mixed-mode oscillations are especially interesting because in the entorhinal cortex and hippocampus theta rhythm is often temporally nested with faster gamma-frequency oscillations [923, 146, 888, 104, 176, 148, 472, 358, 181]. Cluster firing was first reported experimentally in 1939 by Arvanitaki ; it was modeled and analyzed mathematically in the 1980’s by Rinzel . A biophysical basis of such inter-nested gamma and theta mixed-mode oscillation was described by the aforementioned ionic channel model [1025, 1031] (Fig. 4B). The mechanism relies on two assumptions. First, a low-threshold K+ current, like a slow A-type current, should be transient and inactivates slowly. Second, the neuronal spike afterhyperpolarization is strong, so that the average membrane potential is lower during spike discharges than during subthreshold epochs. Thus, when the neuron fires a cluster of spikes, the de-inactivation of IKS slowly builds up during spike afterhyperpolarizations, so the amplitude of IKS increases gradually and eventually becomes large enough to terminate a spiking episode. When the cell does not fire spikes in a subsequent subthreshold epoch, the membrane potential is relatively depolarized and IKS slowly decreases due to inactivation, until the cell is sufficiently recovered and can start to fire again. The subthreshold oscillations are produced by the interplay between a Na+ current and the low-threshold activation of IKS; the same oscillatory wave modulates repetitive action potentials during clustered spike firing, hence the frequency is similar for the two. In this scenario, the theta frequency of intrinsic oscillations in septal GABAergic cells is largely controlled by a single current (IKS). This model prediction has not yet been tested experimentally.
Interestingly, in recent years such subthreshold oscillations have been observed in many cell types, including neocortical neurons , mitral cells in the olfactory bulb [231, 48], magnocellular neuron in hypothalamus , mesencephalic trigeminal neurons , dorsal column nuclei neurons in culture , hippocampal interneurons in the lacunosum-moleculare layer . Fig. 4C shows an example of olfactory mitral cell. Similar to the model (Fig. 4B, lower right), in this cell the frequency of subthreshold oscillation and the rate of intra-cluster spike discharges remain similar, when they both vary with increased input current drive (hence mean Vm depolarization). In all these diverse cell types, Vm oscillations were shown to be blocked by TTX but not by calcium channel blockers, confirming the model prediction that subthreshold oscillations are independent of calcium channels. Moreover, in some studies, voltage-gated K+ currents have been shown to play a critical role. Membrane potential oscillations were abolished by TEA (15 mM) in neocortical cells (3–15 Hz at 34°)  and hypothalamic magnocellular neurons (10–70 Hz at 32°) , and by 4-AP in mitral cells of the olfactory bulb (20–40 Hz at 30°)  and hippocampal lacunosum-moleculare interneurons (7 Hz at 32°) .
If a K+ current that opposes INaP does not inactivate, for example of an M-type, clustered spiking interspersed with subthreshold oscillatory episodes is still possible, but no longer displays regular periodicity [394, 6, 271]. In this case, spikes are triggered by noise, on top of the membrane potential oscillatory wave . Also, whereas some cell types are able to exhibit purely subthreshold oscillations [13, 519, 231, 798], in other neurons subthreshold membrane oscillations always occur interspersed with spike clusters [16, 858, 755, 101]. This difference presumably reflects whether the spike threshold is significantly above, or overlaps with, the voltage range for activation of the low-threshold ion channels responsible for generating membrane potential oscillations . In either case, with increasing amplitude of input current, the number of spikes per cluster increases, and spike firing becomes dominant over subthreshold oscillatory episodes.
Taken together, there is a convergence of evidence in support of a general ion channel mechanism for subthreshold rhythmogenesis. This mechanism involves a voltage-gated Na+ current, and the oscillation frequency is controlled by the activation kinetics of K+ currents. There exist a wide diversity of voltage-gated K+ currents with disparate kinetic properties and operating in different voltage ranges [893, 859, 178]. This offers a powerful means for generating membrane oscillations at distinct frequencies, by a selective expression of a subset of these currents in distinct cell types.
Like pacemaker neurons in central pattern generators , subtypes of cortical cells fire bursts of spikes (brief clusters of spikes) rhythmically at preferred frequencies . Bursting neurons have the potential to serve as pacemakers for synchronous network oscillations , and bursts of spikes may underlie or enhance neuronal resonance [435, 956]. In the cortex, two types of rhythmic bursting neurons have been observed. Hippocampal pyramidal cells  and layer 5 neocortical pyramidal neurons  show a preponderance for slow rhythmic bursting at 4–15 Hz. High frequency (> 300 Hz) action potential burst firing depends on rapid regenerative processes mediated by voltage-gated calcium and sodium channels [494, 1072, 875, 968, 1058, 506]. Calcium-activated potassium currents (with time constants ~ 100 ms) terminate the burst, control the time course of after-burst recovery, hence help set the oscillation frequency (~ 10 Hz) [10, 875, 968, 369, 159]. Interestingly, bursting occurs more readily, when its underlying inward and outward channels are located in the dendrite, separated from the perisomatic region where action potentials are produced [968, 773, 629, 506]. Furthermore, phasic burst firing in layer 5 pyramidal neurons  (and in dopaminergic neurons [481, 583, 228]) depends on the voltage-gated NMDA receptors for glutamate mediated synaptic transmission.
In the neocortex, another class of pyramidal neurons (called ‘chattering cells’) exhibit fast rhythmic bursting in the gamma frequency range, with intraburst spike rates of 300–500 Hz (Fig. 5A) [373, 904, 307, 122, 730, 211, 154, 220]. Because the gamma cycle is only tens of milliseconds, potassium currents such as IM and IKCa are likely to be too slow to set the periodicity of fast bursting in chattering cells. A compartmental model  suggests that the fast rhythmic bursting in chattering neurons can be generated by a Ca2+-independent ionic mechanism (Fig. 5B). Instead, it relies on a voltage-gated persistent Na+ current (INaP ) that is electrotonically separated from the action potential initiation zone. In one scenario, perisomatic action potentials propagate back to the dendritic sites, where a Na+-dependent after-depolarization is produced, which in turn triggers more spikes in the perisomatic region. Alternatively, the Na+ channels underlying after-depolarization may be near the soma, whereas the spike generation zone is in the initial segment of the axon. In either case, the ‘ping-pong’ iterative interplay between spikes and after-depolarization underlies a burst of spikes, which is terminated by the activation of a K+ current. The de-activation of the K+ current during hyperpolarization leads to the recovery and eventually to the start of a new burst . It has also been proposed that after-depolarization depends on a calcium- activated cation current . The sodium-dependent and calcium-independent model was supported by an in vitro experiment which showed that the spike afterdepolarization and the generation of spike bursts were suppressed by phenytoin which blocked persistent Na+ channels, but not sensitive to blocking calcium transmembrane influx or intracellular chelation of free Ca2+ [122, 211]. Interestingly, a TTX-sensitive sodium channel in interplay with a Kv3 channel in the dendrites were also found to underlie oscillatory bursting in a sensory neuron in electric fish [984, 574, 247, 985], suggesting that this mechanism may be versatile in vertebrates. Moreover, calcium-dependent processes can modulate, rather than generate, chattering behavior. In chattering neuron models, slow K+ currents like IM and IKCa tend to suppress burst firing by virtue of reduced cellular excitability, and a reduction of these currents by neuromodulators like acytelcholine can transform a cell from tonic spiking to fast rhythmic bursting [1028, 972]. Since arousal and attention are associated with cholinergic activation of the cortical system [908, 909, 669, 913], this result suggest a cholinergic mechanism for promoting neocortical gamma oscillations in attentive states.
To summarize, putative pacemaker neurons for the gamma and theta rhythms of the waking brain seem to critically rely on Na+ and K+ currents, which give rise to either subthreshold membrane oscillations or rhythmic bursting. Subthreshold oscillations and repetitive bursting may have different implications for synchronization of coupled neurons. Subthreshold oscillations could subserve a signal carrier for phase-locking and resonance, by virtue of the cell’s sensitivity to small but precisely timed inputs [13, 106, 177, 1048, 217]. On the other hand, bursts may provide a reliable signal for the rhythmicity to be transmitted across probabilistic and unreliable synapses between neurons [590, 1028, 659].
A generic cortical circuit consists of two major cell types: excitatory principal neurons and inhibitory interneurons. It follows that three types of synchronization mechanisms by chemical synapses are conceivable: recurrent excitation between principal neurons, mutual inhibition between interneurons, and feedback inhibition through the excitatory-inhibitory loop. More recently, electrical synapses by gap junctions have also been proposed to contribute to neural synchrony. Here I will describe each of these mechanisms in turn.
For any type of synaptic coupling, a general mathematical approach for studying synchrony of neural oscillators is provided by the theory of phase-coupled oscillators [539, 524, 276, 415]. The idea is that, if synaptic interactions are relatively weak, their effect is the synaptic current from the presynaptic neuron into the postsynaptic neuron, averaged over the oscillation period. Specifically, consider two oscillatory neurons with an intrinsic period of T0 (frequency f = 1/T0), described by their respective phases 1 and 2 (defined between 0 and 2π, with the spike time defined at zero phase by convention). The synaptic current from cell #2 to cell #1 is Isyn(1, 2) = gsyns2(2)(V (1) − Vsyn) where s2 is a ‘gating variable’ for the synaptic current and activated by spike firing of cell #2, gsyn is the maximum synaptic conductance and Vsyn is the synaptic reversal potential. The interaction function H(1, 2) is given by the average of the product of Isyn(1, 2) and Z(1) (a single neuron’s phase response curve, see Section II-A) over an oscillatory period. Mathematically, we have [276, 1000, 415, 279]
where Cm is the membrane capacitance.
The dynamical equations for describing the two interacting neurons are given by
Subtracting these two equations yields the following equation for the phase difference Φ = 1 − 2:
where Hodd(Φ) is the odd part of the H function, which determines the phase relationship. Any steady state phase locking at Φss is given by solving Hodd(Φ) = 0, which means that at Φ = Φss, dΦ/dt is zero hence the relative phase no longer changes over time. The stability of a steady state phase locking is determined by the slope of Hodd through the steady state. Consider a steady state Φss at which point the slope of Hodd is negative. Combined with the fact that Hodd is zero at Φss, we know that dΦ/dt = Hodd is positive for Φ slightly smaller than Φss, and negative otherwise. As a result, if Φ is perturbed to be below (respectively, above) Φss, then it is predicted to increase (respectively decrease) over time; in either case the system evolves back to Φss. In other words, coupling acts to attenuate any small perturbation away from the steady state and Φss is stable. Conversely, a steady state is dynamically unstable if the slope of Hodd at Φss is positive.
The phase reduction theory thus provides a general framework for predicting the synchronization properties of neurons that behave as oscillators and are coupled by relatively weak synaptic interactions. In general, type II neurons are synchronized by sufficiently fast synapses (as illustrated in II-A, Fig. 2B), relative to the oscillation period [415, 1000]. For a higher frequency rhythm, the period is shorter, and faster synaptic kinetics are required to produce synchrony. Moreover, the stronger the ‘negative lobe’ of the type II PRC, the higher the likelihood for synchrony by mutual excitation. This is why the presence of voltage-gated K+ currents encourages synchrony by excitatory connections between pyramidal neurons . The same qualitative explanation applies to synchronization of slow rhythmic firing of epileptic discharges (characterized by large after-hyperpolarizations) in a disinhibited pyramidal cell population [964, 968, 773].
As shown above, a key insight from theoretical analysis is that in order to determine network synchrony, interactions between spiking neurons cannot be treated as instantaneous, quantitative synaptic time courses matter a great deal. Furthermore, the required synaptic kinetics is different for different oscillation frequencies. Thus, AMPA receptor mediated glutamatergic transmission, with a decay time constant τsyn in the millisecond range, may be suited for synchronizing slow rhythms at a few hertz [415, 718]. For example, in hippocampal slices, activation of cholinergic muscarinic receptors by carbachol can induce theta frequency oscillation that is independent of synaptic inhibition . Experiments and modeling showed that AMPA receptor mediated excitatory synapses (time constant of a few milliseconds) are suffcient for synchronizing 5–10 Hz rhythmic bursting (with a period ~ 100 ms) in the carbachol induced rhythm in vitro . This rhythm, however, is quite different from theta oscillations in behaving animals during spatial navigation, as it will be discussed later (Section IV-F).
On the other hand, an excitatory-excitatory mechanism does not seem to be suitable for robustly synchronizing faster rhythms (with a much shorter period of tens of milliseconds), such as gamma (~ 40 Hz) oscillations. This is the case even when single neurons display intrinsic rhythmic bursting, as shown in a model of coupled chattering cells . Thus, synaptic excitation mediated by ionotropic glutamate receptors, alone, is likely to be insufficient for generating coherent network oscillations at high frequencies.
With slow synapses (relative to an oscillation period), inhibition rather than excitation generates neural synchrony [1036, 1037, 1000, 415]. For type I neurons, even fast excitation does not lead to zero-phase synchrony [415, 275]. Indeed, it was found that type I neurons typically show anti-phase behavior with excitatory coupling [415, 275, 1035]. On the other hand, reciprocal inhibition turned out to be an effective mechanism for neuronal synchrony. This is illustrated in Fig. 2C (lower panel), where the function Hodd(Φ) is shown for a pair of type I model neurons characterized by the PRC given in Fig. 2C (upper panel). With kinetics of GABAA receptor mediated synaptic currents (τsyn 10 ms), the anti-phase state is unstable while the zero-phase state is stable, hence the pair of neurons are perfectly synchronized.
Wang and Buzsaki  found that, in this network model of GABAergic interneurons, spike firing (oscillation) frequency of individual neurons can be varied in a broad range, e.g. by changing the mean external drive or the strength of synaptic interactions among neurons in the network; yet the synchrony is high only in the gamma frequency band . Therefore, networks of inhibitory interneurons provide a mechanism for coherent brain oscillations, in particularly gamma rhythm. Whittington, Traub and collaborators [1053, 970] were the first to report experimentally synchronous ~ 40 Hz oscillations in rat hippocampus in vitro, when fast excitatory glutamate synaptic transmissions mediated by AMPA and NMDA receptors were blocked . In the experiment, when metabotropic glutamate receptors were activated pharmacologically, hippocampal slices exhibited oscillatory activity patterns. Oscillatory IPSPs in the 40 Hz frequency range were observed in simultaneously recorded pyramidal cells. These IPSPs presumably originated from the firing activities of GABAergic interneurons. Drugs that slowed down the kinetics of GABAA receptor mediated synaptic currents led to lower oscillation frequency, confirming a model prediction and lending support of an interneuronal network mechanism for gamma oscillations [1053, 970, 1035].
At first sight it seems paradoxical that reciprocal inhibition can bring coupled neurons to fire spikes synchronously, since spiking in one neuron triggers a synaptic current that hyperpolarizes the membrane potentials of target neurons. In fact, ‘half-center’ oscillators consisting of two reciprocally inhibitory neurons (or neural pools) in anti-phase represent a standard circuit mechanism for central pattern generators that, by innervating antagonistic sets of muscles, produce rhythmic motor patterns such as walking and swimming [856, 347, 638]. As depicted in a classic model , consider a pair of neurons endowed with some slow fatigue process and coupled reciprocally by fast synaptic inhibition. At any moment one (neuron #1) is active and effectively suppresses the other (neuron #2). When this inhibition wanes over time and eventually falls below a critical level, for example due to spike rate adaptation of neuron #1 or short-term depression of synapses, neuron #2 is ‘released’ from inhibition, becomes active and suppresses neuron #1 in turn. In the ‘release scenario’, coupled inhibitory neurons tend to fire out-of-phase with each other. There is, however, a fundamentally different mode of behavior for coupled inhibitory neurons, called the ‘escape scenario’ . In this scenario, coupled by slower synapses, GABAergic neurons can simultaneously fire action potentials, undergo inhibition together and ‘escape’ from it synchronously. Whether an inhibitory neural network displays release or escape type behavior depends on the details of active neural properties and synaptic kinetics [882, 638], and a dynamic clamp experiment suggests that the same circuit can be switched from one to the other mode by neuromodulation .
It is worth emphasizing again that different synaptic subtypes are suitable for synchronizing circuit rhythms at different frequency ranges. Thus, whereas GABAB receptor mediated inhibitory synapses (τsyn 100 ms) may play a role in synchronizing slow rhythms at a few hertz, GABAA receptor dependent inhibition (τsyn 5–10 ms) may be adequate for synchronizing fast rhythms in the gamma frequency band. Of course, synaptic currents cannot be too slow relative to the oscillation cycle, otherwise interactions between neurons would become roughly tonic and phasic information critical for synchronization would be lost [1035, 1049]. For type I neurons, synchrony can be realized even with relatively fast inhibitory synapses . Synaptic latency (including axonal delay) [123, 621] and rise-time [1000, 1049, 33], as well as the decay time, contribute to determining the synchronous firing patterns. These theoretical results are supported by the observation that, during development of the rodent brain, the frequency and synchrony of gamma oscillations reach adult levels in a time course that parallels with that of cellular properties of basket cells and kinetic time constants of their synaptic connections [547, 250].
Just as it is difficult to synchronize an array of clocks with a wide range of intrinsic periods, neural synchrony is more difficult to achieve in the presence of heterogeneity, which introduces a variety of intrinsic oscillation frequencies among individual neurons [539, 1035]. This is an important issue because the oscillation frequency of a neuron varies considerably with the input current intensity, with type I neurons typically showing a much wider range of firing frequencies than type II neurons [811, 937]. For example, the frequency-current relationship of a fast-spiking interneuron has a slope of about 400 Hz/nA [668, 730, 542]. In a ‘regular spiking’ pyramidal cell that displays spike-frequency adaptation, the f-I slope is similar (about 350 Hz/nA) for the initial spike firing and ~ 150 Hz/nA in the adapted steady state [650, 730]. This implies that a small variation in the net input current leads to a large change in the oscillation frequency. In vivo, excitatory synaptic currents evoked by sensory stimulation were estimated to be in the range of 0.5–1.0 nA in cortical neurons [11, 23, 405]. Given such input variations, and the high heterogeneity of both cellular and synaptic properties in interneurons , how can synchronous oscillations be maintained robustly? Modeling studies suggest that synchrony can be more robustly realized in spite of heterogeneity when GABAA receptor mediated synaptic inhibition is strong [947, 53, 1008, 54]. Furthermore, network oscillations may be enhanced by subclasses of specialized neurons with distinct frequency preference due to their resonance or pacemaker biophysical properties (Section II). Finally, as we shall see in Section V, sparsely synchronized oscillations, characterized by irregular firing of individual cells, are more robust in the presence of heterogeneities.
Back in the early 1970’s, Wilson and Cowan  showed that, in a population firing rate model, network oscillation emerges naturally from reciprocal interactions between excitatory and inhibitory neural pools, under the condition that recurrent connections are strong and inhibition is slower or delayed relative to excitation. Such a model was proposed for gamma oscillations in the olfactory bulb [300, 303] and in the hippocampus [443, 578]. The mechanism is easy to understand as follows. An oscillatory cycle begins when fast excitation drives up neural firing in a positive feedback, until slower inhibition is recruited to eventually bring down population activity. As the excitatory drive to interneurons wanes, the network recovers from inhibition and the next cycle starts anew, leading to rhythmic behavior [318, 976, 1028, 414, 747].
In contrast to ‘rate synchrony’ at the level of population activities, ‘spike-to-spike synchrony’ refers to precise phase-locking of spike times among neurons in a network oscillation. In the scenario in which a coherent oscillation is generated among a population of GABAergic cells, pyramidal neurons are presumably synchronized by inhibitory inputs to produce a population rhythm. Consistent with this scenario, an early modeling study by Lytton and Sejnowski found that IPSPs impinging on the somatic region were more effective than EPSPs on dendrites at rhythmically entraining the spike discharges of a pyramidal cell in the gamma frequency band, suggesting that perisoma-targeting basket cells play a role in phase locking pyramidal neurons . This idea was supported experimentally by Cobb et al. , who showed that in hippocampal slices, spiking activity of pyramidal cells could be entrained with a high temporal precision by a single basket cell. This is demonstrated again recently using optogenetic method to activate a subtype of genetically targeted neurons in the intact brain. Two groups reported that activating channorodopsin-2 (a cation channel) in fast spiking interneurons led to entrainment of pyramidal cells preferentially at gamma frequencies in mouse frontal  and somatosensory  cortex. When photoactivation was administrated at different frequencies in fast spiking interneurons but not pyramidal cells, the LFP power at the stimulus frequency exhibited a pronounced peak in the gamma frequency band (Fig. 7) .
Entrainment involves unidirectional inhibitory-to-excitatory coupling, whereas in the hippocampus  and neocortex  interneurons and pyramidal cells are reciprocally connected. An alternative to the interneuronal network model is that spike-to-spike synchrony emerges from the interplay between excitatory principal neurons and inhibitory interneurons. In models of interacting excitatory and inhibitory neural oscillators, two regimes were identified [1062, 96, 414, 62]. In the first regime, interneurons are sufficiently driven by external inputs and show a phase advance relative to excitatory neurons. The oscillation frequency is comparable to single neuron firing rates. In the other, recurrent excitatory inputs provide a major drive to interneurons, which lag behind excitatory cells but the phase delay is never larger than a quarter of the oscillation cycle (π/2 radians, or 90 degrees). Although mutual excitation between principal neurons cannot by itself produce synchrony, it enhances oscillation generated by the feedback inhibition through amplifying excitatory neural population activity. A mathematical analysis found that the second regime is less sensitive to heterogeneity hence more robust than the first regime .
In a local cortical circuit of the intact brain, both the interneuronal network mechanism and the excitatory-inhibitory feedback loop mechanism are present, and one or the other may be more predominant depending on the relative strengths of the two [127, 414, 96, 949]. To some extent these two mechanisms may be redundant to ensure the robustness of a synchronous rhythm. However, they are not likely to behave exactly the same way. For example, synchrony by the excitatory-inhibitory feedback loop mechanism is actually less prominent in the presence, than in the absence, of the inhibitory-to-inhibitory coupling between GABAergic neurons [413, 939, 414]. This is because I-to-I coupling reduces the effective time constant of the inhibitory neural population, an accelerated dynamics of interneurons runs counter to a significant delay of feedback inhibition that is required in the excitatory-inhibitory loop scenario for rhythmogenesis . More importantly, in cortical circuits where neuronal activity is highly irregular, the interneuronal network and the excitatory-inhibitory loop give rise to synchronous oscillations in different frequency bands, which potentially are predominant in different brain states or serving different functions (see Section V).
Short-term synaptic plasticity induces temporal filtering of presynaptic inputs, with the characteristic frequency dependence determined by the relative strength and time constants of facilitation and depression [1098, 942, 944]. At a synaptic connection exhibiting both facilitation and depression, the release probability of transmitters (hence the synaptic response per presynaptic spike) could be a non-monotonic function of the presynaptic firing frequency: an increased firing rate induces facilitation leading to a larger release probability; this release probability reaches a maximum at a preferred input frequency, then decreases at higher frequencies where depression becomes dominant [644, 659]. This by itself does not constitute a ‘synaptic resonance’, because the synaptic response per unit time, which is the product of the release probability and the presynaptic firing rate, is still a monotonic function of the firing rate. Some synapses in cortical neurons also display a ‘notch’ (a dip) of synaptic response in a narrow range of presynaptic firing frequencies . Synaptic filtering may contribute to frequency- dependent neural responses, often in interplay with presynaptic burst firing [659, 296, 466, 696]. Its role in synchronous oscillations awaits to be fully elucidated in the future.
As indicated above, combining fast positive feedback and slow negative feedback offers a general recipes for rhythmogenesis. Delayed negative feedback  is a versatile mechanism that can be realized, in diverse forms and on multiple timescales, in neural circuits without synaptic inhibition. A common negative feedback process at the single cell level is spike-rate adaptation of pyramidal neurons, originating from depolarization induced activations of voltage-gated and calcium-gated potassium currents (IM and IKCa), which in turn reduce the neuronal excitability. In a population of interacting pyramidal cells, such a negative feedback (with a kinetic time constant of ~ 100 ms) in interplay with fast recurrent synaptic excitation can give rise to network oscillations at ~ 10 Hz [1029, 999, 321, 929]. Another slow negative feedback for generating rhythmic patterns is short-term synaptic depression of recurrent synapses onto excitatory neurons [1029, 928, 603, 929, 827, 508], or short-term facilitation of excitatory synapses onto inhibitory interneurons . In these cases, the biophysical time constants of facilitation and depression, ranging from several hundreds of milliseconds to a few seconds, set the scale of network oscillation periodicity.
The ~ 10Hz oscillations observed in layer 5 of the neocortex appears to be generated by such a mechanism, in which the positive feedback relies on intrinsic voltage-gated inward currents as well as synaptic excitation, and the negative feedback is independent of the GABAA receptor mediated synaptic inhibition [875, 292]. The independence from synaptic inhibition was recently confirmed by an in vivo study with anesthetized rats, in which patch-clamp intracellular recording revealed that 10–15 Hz membrane oscillations were prominent in layer 5 but not layer 2/3 neurons of the primary visual cortex and that these oscillations disappeared when the voltage of the recorded cell was near 0 mV (the reversal potential of excitatory postsynaptic currents) .
A qualitatively similar scenario underlies the generation of slow (< 1 Hz) oscillations observed in quiet sleep [679, 912, 190, 916, 553, 839, 915, 905, 195, 610, 779, 776]. In this case, negative feedbacks operate on longer timescales (a few seconds) that are mediated by activity-dependent K+ conductances in single neurons [839, 183, 752] and synaptic depression [190, 68, 440]. According to this scenario, the Up state is maintained through synaptic interactions between pyramidal cells; this excitation wanes over time due to synaptic depression or/and reduced neural spike discharges, and when it falls below a threshold level the network undergoes a transition to the Down state, which is quasi-stable with the help of an inwardly rectifying K+ channel . In the Down state neurons are hyperpolarized and largely silent, the activity-dependent negative feedback processes decay away, and the network recovers leading to the onset of the next Up state . The dynamics of Up and Down states vary greatly: the Up states can be longer than Down states [198, 190, 916, 954] or otherwise [912, 911]; transitions between Up and Down states are sometimes nearly periodic [912, 751] but often highly stochastic [198, 916, 553, 225]. In an in vitro preparation, slow (< 1 Hz) oscillations spontaneously occur in ferret cortical slices that resemble slow oscillations observed in vivo, offering an opportunity to identify its underlying cellular and circuit mechanisms . Experiments  and computational modeling  suggest that the transition from the Up state to the Down state is induced, at least in part, by the opening of a Na+-activated K+ current (IKNa), known to exist in these neurons. Here, the negative feedback is activated by the buildup of the intracelluar Na+ concentration through sodium spikes, and the recovery is due to the extrusion of Na+ ions by a Na+/K+ pump (operating on a timescale of ~ 10 s) . A network model  of interconnected pyramidal and interneuron populations, endowed with this cellular mechanism, reproduced salient in vivo and in vitro observations, including a gradual increase in the input resistance of pyramidal cells over the course of an Up state  and a maintained balance between synaptic excitation and inhibition in spite of changes in the firing rate [869, 405, 268, 240]. A more recent study showed that the same model exhibits beta and gamma frequency oscillations in the Up state similar to those observed in vivo in awake animals. This result is in line with the increasing evidence that Up states of slow oscillations during non-REM sleep resemble closely to the activated state of wakefulness in characteristics ranging from EEG or LFP patterns and spatial correlation length, to membrane fluctuations of single cells [914, 234, 239, 830, 831] (reviewed in [240, 404]).
Electrical synapses are gap junction mediated connections between neurons. In the mammalian brain, neuronal gap junctions are prevalent during postnatal development, and remain present in adulthood at least in certain classes of neurons, such as parvalbumin-expressing GABAergic cells [333, 680, 322]. A direct demonstration of gap junctions requires intracellular recording simultaneously from two neurons, by depolarizing or hyperpolarizing one neuron with current injection and measuring the resulting change in the membrane potential of another neuron. Such experiments became possible only recently with the advance of new techniques, like infrared-differential interference contrast optics that allows the experimenter to visually identify recorded neurons. Remarkably, it was found that gap junctions almost exclusively connect GABAergic interneurons within the same class. For example, perisomatic-targeting, parvalbumin-expressing FS cells are connected among themselves by gap junctions; so are dendrite-targeting, somatostatin-expressing LTS cells, but electrical synapses are rare between mixed-type (FS-LTS) cell pairs [331, 350]. So far, at least five such distinct networks of electrically coupled GABAergic interneurons have been identified (for a review see ).
Compared to chemical synapses, electrical coupling has properties that seem especially desirable for subserving neural synchronization: it is fast and usually bidirectional (at least between neurons of the same type), and acts to equalize the membrane potentials of the connected neurons. This mechanism of intra-population coordination may be especially important for some cell types, such as LTS interneurons which exhibit a paucity of inhibitory synapses among themselves . Electrical synapses have been suggested to promote oscillatory activity [331, 350] and contribute to coincidence detection in inhibitory neurons . Computational modeling studies generally found that sufficiently strong electrical synapses encourage synchrony [971, 175, 966, 767, 351, 631, 927], but can also have desynchronizing effects depending on the interplay with inhibitory synapses and the intrinsic membrane properties of the constituent cells [867, 729, 767]. A pair of FS interneurons or LTS neurons coupled by gap junctions showed similar phase-locking behavior, despite their differences in the biophysical properties. Interestingly, FS and LTS neurons are synchronized preferentially in the gamma and beta frequency bands, respectively . It should be noted that in the cortex gap junctions are restricted to neighboring neurons, within 50 μm . Therefore, electrical coupling is spatially very localized, like diffusion in physical and chemical systems.
To experimentally assess the role of electrical synapses in brain rhythms, one must be able to induce a disruption of gap junction communications and examine its impact on a network oscillation. Presently there are no drugs that block gap junctions with a high degree of selectivity, but genetic manipulations have proven a powerful tool. Connexin 36 is the predominant gap junction protein expressed in neurons of the mammalian brain . Both in vitro  and in vivo  studies showed that in mice, connexin 36 knockout only produced a reduction of the field potential gamma power (or the degree of network synchronization), without affecting the oscillation frequency. Given that electrical synapses are much faster than chemical synapses, they may be especially important for synchronizing the ultrafast (100–200 Hz) sharp wave ripples. An in vitro experiment supports a critical involvement of gap junctions in the generation of sharp wave ripples . Electrical coupling between Purkinje cells, provided that it is sufficiently strong, has also been proposed as a mechanism for sustaining ultrafast oscillations in the cerebellum [622, 681]. On the other hand, connexin 36 knockout had no effect on ultrafast (> 100 Hz) field ripple oscillations in mice in vitro  or in vivo . Traub and colleagues proposed that other connexin subtypes, forming electrical synapses between the axons of excitatory cells, underlie fast oscillations in the hippocampus [971, 849, 967, 211, 409]. This idea arose from a network model in which single neurons were described with elaborated dendritic and axonal arbors , which is rarely done in theoretical studies of large-scale cortical networks. Unlike gap junctions at the dendrite, those located between axons make it possible for a spike in one neuron to directly trigger a spike in another neuron, which was found to be necessary to generate ultrafast oscillations in a sparsely connected model network . Other modeling studies suggest that chemical synapses may be sufficient to generate > 100 Hz fast oscillations in an interneuronal network [127, 342, 125, 222] (see Section V).
Mechanisms for synchronizing neurons discussed above rely on oscillatory synaptic inputs, which are produced in a recurrent network. Synchronization between neural oscillators can also be realized by shared but random inputs in a network where neurons do not directly interact with each other [770, 940, 714, 898, 329, 330]. In this scenario, common (correlated) noise gives rise to phase synchronization of otherwise uncoupled oscillators, when two conditions are satisfied . First, single neurons are intrinsic oscillators with roughly the same frequency. For this to be realized in spite of input heterogeneity, this scenario is better suited and more robust for type II cells (which display a relatively flat current-frequency curve [811, 937]) than for type I cells (which display a steep current-frequency curve [1035, 658, 639]). Second, neurons receive partially correlated fluctuating synaptic inputs, that can be broad-band (i.e. they do not have to be rhythmic) but should be fast relative to the period of the intrinsic neuronal oscillation. Under these conditions, brief (compared to the intrinsic oscillation period) and correlated fluctuations in the inputs act as a common pulse to reset the oscillation phases of neurons in a network. This can be understood by considering a population of uncoupled neural oscillators, which is initially asynchronous. This means that spike times of neurons are uniformly distributed on the oscillation cycle. The response of each neuron to a common pulse is given by the phase response curve (which may be type I or type II) (Section II-A). This brief input would speed up those that fire later than the others, and either slow down (for type II cells) or do not significant alter (for type I cells) the spiking phase of those cells that are ‘ahead’, thereby leading to population synchrony . As expected intuitively, an intermediate level of noise is optimal: too little noise would not be sufficient to induce phase synchronization, whereas too much noise would overwhelm and destroy intrinsic neuronal oscillations.
Synchronous fast oscillations are prominent in the olfactory system, and play a major role in odor stimulus processing [303, 584, 556, 1043, 308, 441, 555, 501, 117, 847, 79, 503, 161, 461]. Numerous studies have shown that coherent rhythmicity in the rat olfactory bulb depends on the reciprocal, dendrodendritic synaptic interactions between excitatory principal mitral cells and inhibitory granule interneurons [791, 700, 734, 218, 544, 545], with possibly additional contribution of electrical coupling [306, 683]. Many models for gamma oscillations in the olfactory bulb are based on a feedback inhibition mechanism [303, 584, 67, 66, 61, 109]. Galán et al.  noted, however, that so far there is no direct proof that inhibitory synaptic potentials are synchronized in mitral cells. They proposed to apply the scenario of correlation-induced stochastic synchrony to the olfactory bulb, where shared noisy synaptic inputs could synchronize mitral neurons endowed with subthreshold oscillations (their membrane oscillation frequency as well as spiking rate are in the gamma band and relatively insensitive to input strength (Fig. 4C)) [231, 640, 48]. What are the sources of common synaptic noise and, more generally, whether this scenario or the excitatory-inhibitory loop scenario is responsible for the rhythmogenesis in the olfactory bulb, remain to be determined in future experiments.
In summary, two-way interactions between theory and experimentation have proven extremely fruitful in uncovering fundamental processes through which neural networks (cortical circuits in particular) are synchronized. Mutual excitation between principal cells, in conjunction with some negative feedback process other than synaptic inhibition, is suitable for generating certain forms of slow oscillations. On the other hand, mutual inhibition between interneurons and reciprocal loop between excitatory and inhibitory cells provide two general mechanisms for rhythmogenesis, especially fast cortical oscillations. In addition, gap junctions contribute to population synchrony. Finally, correlated noise may also lead to spike-to-spike synchrony of otherwise uncoupled neural oscillators. These diverse core mechanisms may be deployed differentially, or in any combinations, in different cortical circuits.
Studies of a couple of neurons have yielded valuable information about the synchro nization mechanisms (Section III). To go beyond neural pairs toward predicting the behavior of large populations of neurons, a key step is to have a firm grasp of how network connectivity properties sculpt activity patterns. Recent advances in neural anatomy and imaging have opened a new era of experimental studies of detailed ‘wiring diagrams’ of a functional circuit [173, 586, 860]. In computational studies of synchronous oscillations, different types of network architectures have been considered, which will be reviewed below.
Some basic insights into neural synchronization were obtained in studies of models where neurons are coupled globally (every neuron is connected to every other neuron). When perfect (zero phase) synchrony is predicted for a pair of neurons, the same often holds true for an all-to-tall network. For example, Fig. 6A shows synchronous gamma oscillations of a population of type I neurons coupled globally by synaptic inhibition, with the behavior of a pair of the same neurons described in Fig. 2C. On the other hand, when a pair of neurons display phase shift relative to each other, the activity pattern of an all-to-all coupled population of such neurons could be totally asynchronous, or exhibit a variety of complex patterns. Perhaps the most commonly observed pattern is clustering, where a network breaks into a small number of fixed clusters of neurons [367, 526, 1040, 1035, 1049, 366, 413, 53, 5]. Neurons within a cluster fire together, and different clusters take turn to fire over time, hence single neurons fire intermittently (only once in a few oscillation cycle). The network frequency is equal to the number of clusters times the firing frequency of single cells. In a clustering state, single cells still discharge spikes in a fairly regular fashion. Moreover, clustering states usually occur in a restricted range of noise .
In local cortical circuits, synaptic connections are sparse [892, 902]; the probability for a pair of pyramidal cells to be connected in either direction is about 0.1–0.2 [892, 573]. Coherent oscillations require a minimum connectedness. Consider a network in which any neuron make synaptic contacts on average with Msyn neurons. Modeling work revealed that, in a randomly connected interneuronal network model, synchrony depends on Msyn in a highly nonlinear fashion. Obviously, if Msyn is zero, neurons are not connected to each other and fire independently. It turns out that, if Msyn is below a threshold Mcrit, the network remains completely asynchronous. Coherent oscillation emerges as a qualitatively new type of collective network behavior when Msyn is above Mthr (Fig. 6B). Importantly, this threshold does not depend on the network size, thus small and very large networks can have the same threshold connectedness for synchrony. Intuitively, synchronization requires a sufficient amount of network connectedness, e.g. there should be a large connected neural cluster of rougly the same size as the entire network. For random graphs, this only requires that the number of links per node be larger than a threshold value, independent of the network size [273, 12]. Interestingly, synchronization of a neural rhythm also depends on the biophysics of synaptic interactions and intrinsic cellular dynamics. This is strikingly demonstrated in a network model of mutually inhibitory neurons where, given similar synaptic coupling time constants, the threshold connectedness Mthr is about 50 with conductance-based Hodgkin-Huxley type (type I) model interneurons , but is several thousands with ‘leaky integrate-and-fire’ (LIF) model neurons . By comparison, In the hippocampus CA1, the axon arbor of a single basket cell contacts, on average, about 60 other basket neurons within a spatial span of approximately 500 μm . Hence a model with Hodgkin-Huxley type neurons but not LIF neurons would predict that gamma synchrony can be realized with realistic sparse connectivity in a population of fast spiking interneurons. These findings underscored the importance of detailed physiological properties of single neurons in determining collective network behavior. The conclusion about the small minimum number of synapses per neuron required for synchrony in a random network also holds for two population models of excitatory and inhibitory neurons [1040, 96].
Connectivity in a local cortical network is not completely random [892, 149, 699, 554]. In particular, synaptic connections depend on the spatial distance between connected neurons, which can greatly affect global synchrony , or generate propagating waves (Section IV-D).
Consider, again, a population of mutually inhibitory interneurons. Even if interneurons and their connections are distributed homogeneously, high-density local connectivity alone may not give rise to global coherence in large networks. The reason is that with an increasing network size, the average ‘path length’ (the minimum distance between two neurons through intermediate connected neurons) and the associated synaptic delays become excessively high. As suggested by Watts and Strogatz (, however, the path length can be dramatically decreased by a few randomly placed ‘short-cuts’ that connect distant parts of the network. Can a small number of ‘long-range’ interneurons serve this exact role? To assess this possibility, we simulated a network model of 4,000 inhibitory cells , which corresponds to approximately all basket cells in the dorsal hippocampus of the rat . We considered how re-wiring from local to long-range connections affects the synchronization behavior of the network. With a local architecture, the connection probability between two neurons decreases with their distance, according to a Gaussian distribution. When the Gaussian distribution is narrow (characteristic range 20 neurons), the network is asynchronous and the population firing rate is essentially constant in time (Fig. 8A). Next, while fixing the average number of connections per neuron, a fraction p of the existing connections are re-wired according to a probability that falls off as an inverse power law with the distance. The power distribution makes it much more likely to have long-range connections than a Gaussian or exponential distribution (Fig. 8C). The network synchrony becomes significant with p > 0.1 (Fig. 8D). As illustrated in Fig. 8B, the network with p = 0.25 exhibits highly synchronous network oscillations at 150 Hz, in the frequency range of sharp wave ripples (100–200 Hz). Therefore, wide-range network synchrony could be achieved with a small fraction of the local connectivity was removed and replaced by long-range connections. In fact, increasing p above 0.4 only enhances the network synchrony modestly. On the other hand, the ‘wire length’ of the network (the total length of inter-neuronal connections) obviously increases with p (Fig. 8D). Therefore, there is a trade-off between the need of long-range connections for network synchronization, and minimizing the cost for these long-range wires (Fig. 8D). Hence, an optimal design for network synchronization with a minimal wire-cost can be achieved by a division of labor between a large subpopulation of short-range interneurons and a small subpopulation of long-range interneurons cells in the hippocampus. Another modeling study also found that spiking coherence between neural pairs could be maximal in the small-world network topology .
Complex networks have been the subject of active research in recent years [1042, 12], and various types of network connectivity schemes have begun to be applied to Neuroscience: small-world networks (with predominantly local connections and rare long-distance connections) [546, 823, 149, 806, 1085], networks endowed with ‘hubs’ (a few nodes with an unusually large number of connections with the rest of the network)  or a community structure , networks with dynamic motifs (elementary building blocks from which more complex dynamic behavior can be assembled) [1095, 464]. For example, even a few percent of neurons with an exceptionally large number of connections (‘hubs’) can dramatically alter a circuit’s dynamical behavior .
The proposal of GABAergic ‘hub neurons’ has gained some empirical support from a recent work . Monitoring multiple neurons by calcium imaging of spontaneous activity and assessing functional connectivity by pair-wise correlation analysis of the observed calcium activity, Bonifazi et al found that hippocampal CA3 networks in the developing rats and mice exhibit a scale-free topology. The distribution of the number of output links per neuron decays as a power law, with an exponent of ~ 1.2, and ‘hub neurons’ (defined as the top 40% highest-connected) are a subpopulation of GABAergic cells with widespread axonal arbors. Stimulation of hub neurons, but not those with low connectivity, affected synchronous network dynamics, demonstrating the functional importance of these hub neurons . The simple models discussed above were designed for a local area; in a large-scale cortical network, the spatial structure of coherent oscillations is likely to be much more complex. In awake states, the correlation between LFPs decays with the distance between the recording electrodes, with a characteristic spatial length of ~ 3–5 millimeters within a cortical area [134, 238]. On the other hand, long-distance synchronization among selective neural populations in disparate cortical regions has been reported during cognitive tasks (see Section VI). A future challenge is to elucidate the wiring topology and statistics that underlie brainwide synchronization.
Sensory processing, decision making, motor action all engage selective neural populations. Localized neural activity either remains spatially confined in time, or propagates as a wave among neural pools that are spatially separated but engaged in the same computation or behavioral state. Wave propagation has been observed on multiple spatial scales. MEG studies of human subjects revealed gamma oscillatory episodes that started in the frontal cortex and propagated front-to-back across the cortical surface (Fig. 9A) [804, 598]. Slow sleep oscillations measured with high-density EEG recordings also appear to start in the frontal cortex and travel in an antero-posterior direction [652, 654]; slow waves primarily involve areas in the pre-frontal cortex [214, 709] that overlap with the default network in the human resting state . In alert monkeys performing an instructed delayed reaching task, LFP beta oscillations, evoked by the instruction cue in the motor and premotor cortical areas, are not synchronous globally but propagate as waves whose latencies and amplitudes provide information about the visual target to be reached . In vivo multi-electrode EEG recording or voltage-sensitive dye imaging has also revealed stimulus-induced waves in the olfactory bulb [301, 743, 264], turtle visual cortex  and rat somatosensory cortex . Numerous studies have also demonstrated wave propagation of evoked responses in cortical slices in vitro [935, 189, 1079, 1076] (for movie demos, see http://www9.georgetown.edu/faculty/wuj/propagationwave.html). Wave propagation naturally occurs because connections between neurons are spatially confined. Within a cortical area, the probability of synaptic contact decreases with the physical distance between a pair of neurons for both excitatory pyramidal cells and inhibitory interneurons [353, 874, 902]. Therefore, spiking activity spreads from a group of neurons to its neighbors and onto the rest of the network as a propagating wave. Wave propagation can also take place in an electrically coupled neural network, when the distance-dependence of electrical coupling is taken into account , just as expected for ‘excitable media’, diffusion-coupled spatially extended chemical or biological systems .
A key characteristic of brain waves is propagation speed . A ‘wavefront’ is an activity packet that sweeps across a network, such as a brief burst of spikes, a response evoked by sensory stimulus, or the event of switching from an inactive (Down) state to an active (Up) state. For a wavefront, the propagation speed is determined by cellular and synaptic properties that control the surge of activity, but not what happens afterwards such as the slow afterhyperpolarization [278, 459, 815, 282]. In vivo (using anesthetized animals) [383, 115, 762, 410] and in vitro [1075, 839] studies estimated propagation speeds to be on the order of ~ 10 mm/sec in the rodent cortex. A similar wave speed was observed in a human study of binocular rivalry. As a subject experienced an alternation from one to another perceived pattern, functional brain imaging showed that the dominance of the new pattern emerged locally and propagated like a traveling wave as it rendered the old pattern invisible . Much faster propagating waves, at ~ 100 mm/sec, have been reported in ferret piriform cortex , in rodent cortex (using voltage-sensitive dye imaging) in vivo (Fig. 9B)  and in vitro [189, 74], and in the motor cortex of behaving monkeys . Wave speed appears to be sensitive to modest changes of synaptic inhibition that sculpts cortical local circuits, as it is known that ~ 10–20 % suppression of GABAA receptor mediated inhibition can increase wave speed by tenfold [172, 162, 839, 183, 448]. Moreover, the inclusion of a small amount of long-distance connections, in addition to predominantly local connections, reminiscent of ‘small-world’ type architecture with a mixture of mostly local and sparsely long-distance connections (Section III-C), significantly increases the signal propagation speed .
In contrast to smooth wavefront propagation, another, ‘lurching’ type of waves propagates in a discontinuous manner [368, 812, 170]. Imagine a spatially structured inhibitory circuit of GABAergic neurons that are endowed with a ‘postinhibitory rebound’ property (spike discharges triggered by a long-lasting hyperpolarization) [908, 597, 196, 47, 1039, 1026, 237]. A group of cells fire a burst of spikes, inhibiting for a period of time their neighboring cells. The latter, when synaptic inhibition has waned, escape from it and discharge in turn, leading to another episode of activity that lurches forward across the neural population. Because the network is silent during periods when synaptic inhibition is strong, the activity pattern is discontinuous in a lurching wave. Lurching waves may also take place in an excitatory network, provided that synaptic transmission has a sufficiently long latency [113, 365].
Wave fronts should be distinguished from traveling waves, in which neurons fire rhythmically and the oscillation phase varies as a function of time and spatial location . For a plane wave, the phase is linear with the spatial position x and time t, like = 2π(ft − (f/υ)x), where f is the oscillation frequency, υ is the propagation velocity, and λ = υ/f is the wavelength. In two or three dimensional nerve tissues, more complex waves such as spiral wave (Fig. 9C) can take place [686, 448, 508]. Traveling waves have been observed in a number of studies (reviewed in [279, 1076]), including MEG gamma oscillations , EEG slow sleep oscillations , evoked responses in the cortex [32, 781, 302], and hippocampal theta oscillations . Traveling waves during theta oscillations in CA1 propagate along the septotemporal axis of the hippocampus, with υ 80 – 100 mm/sec and λ 10 – 15 mm. Given that the length of the septotemporal axis of CA1 is L ~ 10 mm, the total phase difference is L/λ, between 240 and 360 degrees, i.e. two thirds to a full theta cycle . Interestingly, in all existing studies, the total phase shift is often ~ π/2, never larger than 2π. In other words, the observed spatial variation in peak amplitude is less than the full oscillation cycle, regardless of oscillation frequency (e.g. theta or gamma), the physical size of the neural system or species examined . Modeling studies showed that a phase wave depends on the spatial extent and connectivity delay (including the axonal delay and synaptic latency). In the same model, a gradual increase in the connection delay can lead to an abrupt transition from global synchronous oscillation (when delays are short) to phase waves (when delays are long) .
So far I have discussed networks consisting of either one neural population or two (excitatory and inhibitory) neural populations. However, neural circuits in the cerebral cortex are more complex. In particular, given the wide diversity of inhibitory cell subtypes [305, 645], what are their relative roles in rhythmogenesis? In this regard, the best studied case is theta and gamma rhythms in the hippocampus. Early pioneering in vivo studies found that, in the rat hippocampus, putative pyramidal cells (p) show overall maximal firing rate close to the trough of the LFP theta (recorded in the pyramidal layer of CA1), and fast-spiking interneurons in the pyramidal layer (int(p)) preceded pyramidal cells by about 60° [208, 209]. In vitro and modeling studies suggested that different interneurons play differential roles in theta oscillations [354, 89, 821]. More recently, using juxtacellular recording that allowed investigators to identify the cell type of the recorded neuron, Somogyi, Klausberger, and their collaborators were able to characterize theta phase preferences of diverse hippocampal interneurons in vivo of rats under urethane anesthesia (Fig. 10A) [512, 513, 514, 889, 511]. Interestingly, two types of phase relationships were observed. Basket and axon-axonic interneurons, which control the spiking output of pyramidal cells are roughly out of phase with the pyramidal cells; whereas bistratified and oriens-lacunosum-moleculare (O-LM) interneurons, which gate synaptic inputs to pyramidal dendrites, are approximately in phase with the pyramidal cells. More precisely, maximal firing occured at 20° for pyramidal cells, 271° for parvalbumin basket cells, 185° for axo-axonic cells, 1° for bistratified cells and 19° for O-LM cells (0° is at the trough). Phase relationships between several cell types during hippocampal theta have been analyzed in modeling studies [341, 406, 744], but remain only partially understood. Note that the phase difference between fast spiking neurons and pyramidal cells appears to be much smaller in the awake than anesthetized condition, therefore the phase relationships between different cell types depend on the animal’s behavioral state.
The juxtacellular recording technique applied to gamma rhythm revealed that, in CA1, dendrite-targeting bistratified cells (rather than perisoma-targeting basket cells) exhibited the strongest modulation by field oscillations [982, 511]. This does not necessarily imply, however, that bistratified cells play a more important role than basket cells in the rhythmogenesis. One reason is that neural activity in CA1 might reflect oscillatory inputs from CA3 as a ‘source area’ for gamma rhythm [291, 209]. In cholinergically induced gamma oscillations in CA3 slices, basket and axoaxonal cells were found to display the largest rhythmic modulation among diverse interneurons . Moreover, spike firing of these interneuron subtypes and pyramidal cells displayed different preferred phases of the gamma cycle [408, 407] (Fig. 10B). These phase relationships provide clues with unprecedented detail about the organization and temporal dynamics of the underlying complex circuit.
Whereas local circuits are capable of generating gamma oscillations [104, 291, 53, 209, 632, 35], the mechanism of the hippocampal theta rhythm appears to involve larger systems. Lesioning the fornix-fimbria abolished hippocampal theta, while rhythmic firings persisted in septal cells [763, 1012, 22], leading to the hypothesis that the hippocampal theta rhythm depends on the afferent inputs from the medial septum [87, 918, 1006]. The medial septum supplies acetylcholine to the hippocampus, this modulation of excitability is believed to be important for promoting theta rhythm in the hippocampus . In in vitro hippocampal slices, cholinergic activation alone, e.g. by the muscarinic receptor agonist carbachol, can induce oscillations at theta frequency [618, 965, 290]. This rhythmic phenomenon in vitro differs from theta rhythm in vivo in that it is independent of GABAergic synaptic transmission [618, 965] and becomes similar to epileptic discharges with a high dose of carbachol . Pyramidal cells fire bursts of spikes regularly in carbachol induced theta in vitro , whereas they fire sparsely in behaving animals . Interestingly, a recent study  found that the intact isolated hippocampus in toto (but not 500 μm thick slices) displayed spontaneous, atropine (a muscarinic receptor antagonist) resistant, theta (~ 5 Hz) oscillations, while pyramidal cells fired spikes sparsely (less than 0.5 Hz). This self-generated theta appeared to originate in CA1; there are multiple theta oscillators along the septotemporal axis in CA1, with faster oscillators entraining slower ones. The circuit mechanism of this in vitro theta rhythm, possibly involving specific type(s) of GABAergic cells , remains to be elucidated in future experiments. Thus, under certain conditions, theta frequency oscillations could emerge within the hippocampal circuit. The fact remains, however, that in vivo the medial septum is necessary for generating theta oscillations in the hippocampus and entorhinal cortex. Perhaps, the propensity of hippocampal networks to oscillate at theta frequency reflects a resonance mechanism (Section II-B) rather than theta rhythmogenesis itself. Indeed, hippocampal pyramidal cells show maximal subthreshold membrane potential changes (resonance) as well as firing preference to sinusoidal inputs in the theta frequency range [577, 769].
Interestingly, theta oscillation observed in anesthetized animals is blocked by atropine, but theta oscillation associated with free movements in behaving animals is insensitive to atropine (hence independent of muscarinic activation) [1002, 531, 918]. If movement-related theta is atropine-resistant, precisely how does it depend on the medial septum? As mentioned in Section II-C, medial septum contains two major types of neurons [105, 384], which are thought to play distinct roles in theta rhythm generation: cholinergic cells acting via muscarinic receptors modulate slowly the excitability of hippocampal neurons [180, 326]; whereas GABAergic cells play a role of pacemakers via fast GABAA receptor mediated synaptic transmission , by selectively innervating GABAergic interneurons but not pyramidal cells in the hippocampus . Septal GABAergic cells display rhythmic firing at theta frequency in vivo [340, 509, 255, 99, 1005], are endowed with a significant H-current thus likely to display resonance in the theta frequency range [1005, 635], and display a temporal lead over the hippocampal local field theta oscillation . It has therefore been hypothesized that septal GABAergic afferents impose theta rhythmicity upon GABAergic cells in the hippocampus , that in turn phasically pace spike firing of pyramidal neurons [1083, 963]. This scenario was supported by several lines of evidence. First, when cholinergic neurons (but not GABAergic neurons) in the medial septum were chemically damaged, the frequency of hippocampal theta was found to remain unchanged, but the power was dramatically reduced [571, 29], consistent with the view that cholinergic cells contribute to theta rhythm by providing a tonic drive to the hippocampus. Second, GABAA receptors in septal GABAergic cells but not cholinergic cells contain α1 subunit , and zolpidem (a preferential GABAA α1 subunit modulator) suppressed theta oscillation both in the hippocampus and septum . Third, local injection of GABA A receptor antagonist in the medial septum eliminated theta rhythmicity in the putative cholinergic cells, but not that in the putative GABAergic cells (as differentiated by broad- versus brief- spikes, respectively) . This last result could be interpreted as evidence that oscillations in cholinergic cells depend on the inhibitory synaptic inputs from GABAergic neurons, which may behave as pacemakers for the hippocampal theta [918, 141] (see Section II-C). There is also evidence that non-cholinergic (putative GABAergic) cells in the nucleus basalis promote LFP oscillations in the neocortex .
Somewhat surprisingly, a modeling study found that synaptic interactions could not synchronize a medial septum network of interconnected GABAergic cells endowed with intrinsic mixed-mode oscillations (as in Fig. 4B); but synchronization was realized in a reciprocal loop between these septal GABAergic cells and a subtype of GABAergic cells in the hippocampus . This result suggests a critical role in theta synchrony of backprojections from the hippocampus to the medial septum. The latter do not originate from pyramidal cells, but a subpopulation of GABAergic, calbindin- and somatostatin-containing cells with horizontally oriented dendrites located in stratum oriens [14, 962, 479]. Septally projecting hippocampal cells fire at the trough of the theta cycle  (like O-LM cell in Fig. 9A). A reciprocal GABA-GABA septohippocampal loop is well established anatomically: septally projecting hippocampal GABAergic cells receive a large proportion (up to 54% on somata and 27% on dendrites) of GABAergic inputs from the medial septum , and hippocamposeptal fibers innervate hippocampally projecting GABAergic, but not cholinergic, neurons in the medial septum . Inhibitory feedback projection from the hippocampus onto GABAergic cells in the medial septum was also physiologically confirmed in an in vitro septohippocampal preparation containing the intact interconnecting fornix/fimbria pathway . Whether the reciprocal loop is critical for theta synchronization can be directly tested by selectively disabling the hippocamposeptal pathway with pharmacological, transgenic or optogenetic manipulations.
One of the open issues is the synaptic mechanism that can account for the observed relationship between theta oscillations in the entorhinal cortex and hippocampus. The atropine-resistant form of theta oscillation in the hippocampus depends on the integrity of afferent inputs from the entorhinal cortex [492, 141], therefore movement-related hippocampal theta involves the entorhinal cortex and the medial septum in a interconnected large circuitry. There are multiple generators for field theta oscillations in the entorhinal cortex and hippocampus [522, 702, 695, 691]. CA1 pyramidal neurons fire with the lowest probability at a phase of the theta cycle where the direct synaptic excitation from the entorhinal cortex into their distal dendrites is maximal [492, 208, 141]. Simultaneous recording from the two structures in awake behaving rats revealed that the peak of the population activity in the entorhinal cortex is aligned with the timing of dendritic excitation of hippocampal target neurons, but the firing of the latter is typically offset by a half theta cycle . Note that, in anesthetized animals, perisoma-targeting interneurons show maximal firing ~ 100–180 degrees out of phase with pyramidal cells  (Fig. 9A), hence at the same preferred theta phase as the entorhinal inputs onto the CA1 distal dendrites. This output-controlling inhibition could veto direct entorhinal excitation, thereby rendering the activities of the hippocampus and entorhinal cortex relatively independent on some timescale. However, in non-anesthetized animals, putative perisoma-targeting interneurons fire roughly 60° ahead of pyramidal cells, not 180° out of phase . Furthermore, even if the activity of perisomsa-targeting interneurons is phasically aligned with the synaptic excitation from the entorhinal cortex, it should delay firing of pyramidal cells by a few milliseconds, but not a half theta cycle (~ 60 ms). It will be a challenge to understand, using both experiments and modeling, how perisoma-targeting and dendrite-targeting inhibitory neural types in interplay with pyramidal cells give rise to the observed theta phase relationships, as well as their computational implications.
In summary, network architecture is a predominant determinant of spatiotemporal neural activity patterns, the richness and complexity have only beginning to be revealed and promise to be a fertile ground in the coming years with the accelerated pace of knowledge accumulation about neural circuit wiring properties. Yet, connectivity alone does not predict behavior. As illustrated by examples above, synchrony depends on a combination of the required network properties (such as the minimal connectedness) and the dynamics of constituent single neurons and synapses.
In the cortex, especially in awake behaving conditions, even when a synchronous oscillation is detected in LFP, it is often not readily discernable in highly irregular spike trains of single units [706, 468]. To resolve this puzzle, a necessary step is to understand the physical basis of EEG and LFP signals and how they relate to the electrical activity of single cells. There is a reasonable consensus that LFP is a weighted spatial average of the flow of currents along the dendrosomatic axis of pyramidal neurons [201, 202, 690, 144]. The spatial extent over which neural signals are integrated by LFP has been estimated to be ~ 500 μm [532, 595, 75], based on a model of the columnar structure for some stimulus selectivity. Recent work used more direct analysis. Optical imaging from the cat visual cortex revealed that LFP could be quantitatively accounted for by membrane potentials, filtered by a Gaussian kernel with a width of σ ~ 100 μm. It follows that more than 95% of the LFP signals originated within a radius of 250 μm (2.5σ) of the recording electrode tip . Another study, using LFP and MUA recordings, obtained similar estimates in the monkey visual cortex: LFP sums signals spatially with a radius of ~ 120 μm in layer 4B, and 200 μm in superficial and deep layers . Therefore, LFP appears to reflect neural signals within a narrower spatial extent than previously thought.
The relationship between LFP and spiking activity remains a topic of active research. A recent monkey physiological study found that a linear filter operation on the activity of a few neurons can account for a significant fraction of the LFP time course ; and a study combining LFP and single-unit recording from epileptic human subjects found that broad-band (2–150 Hz) LFP power is positively correlated with the single- unit firing rate . Yet, it is generally held that LFP primarily reflects synaptic inputs rather than spiking outputs [164, 690, 37]. For example, in anesthetized monkeys, LFP in the visual cortex was not significantly affected when spike activity was reduced by pharmacologically induced hyperpolarization . There is also evidence that LFP in a visual cortical area shows similar stimulus-feature selectivity as the spiking output of upstream neurons that project to the recorded site . However, this does not mean that LFP exclusively reflects afferent inputs from outside of the recorded brain area, because cortical networks are endowed with an abundance of intrinsic connections, therefore a large fraction of synaptic connections onto a cortical neuron originate within the local circuit [253, 85]. A recent study showed that a recurrent cortical network model can reproduce many features of LFP observed in a monkey experiment, suggesting precisely how spike rates relate to EEG power in different frequency bands . Furthermore, voltage-dependent membrane changes other than synaptic events, including spike afterhyperpolarization, also contribute to the generation of LFP signals [144, 492].
Isolated cortical neurons fire regularly under certain conditions, e.g. when driven by a constant current injection in vitro. In the intact brain, however, cortical cells are bombarded by highly fluctuating synaptic inputs, and typically fire seemingly random spike trains. A simple statistic of variability is Fano factor, defined as the ratio of the variance over the mean of spike count (in a proper time window). Another measure is the coefficient of variation (CV) of inter-spike intervals (ISIs). Fano factor and CV are both zero for periodic responses, and one for Poisson processes. Variability of spiking responses is higher in the cortex than in the periphery . In cat and monkey primary visual cortex, Fano factor ~ 1 for neurons recorded from anesthetised [958, 957, 884, 153] and awake [430, 1014, 883, 742] animals (but see [495, 392]). Fano factor or CV is also close to 1 for spiking activity of single-units recorded in extrastriate cortical areas of alert behaving monkeys, including the middle temporal area MT [883, 116, 863, 39], V4 [664, 688], prefrontal cortex [184, 864], and motor cortex . A recent study compared CV of neural activity in several cortical areas, and found that the average CV of single units is ~ 1.0 for MT, ~ 0.96 for the lateral intraparietal association area (LIP), and ~ 0.87 for motor-like area 5 in the parietal cortex . In all three areas examined, ISI histograms are well fitted by gamma distributions without repetitive peaks, hence neural activity is typically far from being clock-like. However, a high (~ 1) Fano factor or CV is not necessarily inconsistent with the presence of synchronous oscillations. Although these are useful measures of neural variability, they do not probe temporal structures and are not sufficient to determine whether a spike train is consistent with a Poisson process [497, 17]. In particular, importantly, a significant fraction of measured variability is likely due to the variability of neural firing rates (rather than Poissonian spiking with a fixed firing rate) .
Nevertheless, it is notable that, even when simultaneously recorded LFP exhibits coherent fast oscillations, spike trains of cortical neurons are still highly irregular (Fig. 11). In the neocortex and hippocampus, gamma oscillations observed in in vivo are typically associated with stochastic and sparse spiking activity of individual cells [263, 377, 104, 238, 208, 312, 759, 209, 185, 181]. Csicsvari et al.  examined spiking activity of hippocampal pyramidal neurons and interneurons that were associated with field oscillations. During sharp-wave ripples, CA1 cells fire at much lower rates (~ 5 Hz for pyramidal cells; ~ 30 Hz for interneurons) than LFP oscillations (100–200 Hz). Similarly, during inter-nested theta (3–7 Hz) and gamma (30–70 Hz) rhythms, pyramidal cells and interneurons fire stochastically at lower rates compared to gamma (~ 1.5 Hz for pyramidal cells, 15 Hz for interneurons) . During both sharp-wave ripples and theta oscillations, the interspike intervals are broadly distributed with CV ~1 for pyramidal cells and ~0.7 for interneurons . Caution is warranted in interpreting the reported CV values, since a high CV might, at least partly, be due to the burstiness of spike discharges . Moreover, distinct cell types display different degrees of modulation by a field rhythm (Fig. 9); a minority of neurons might conceivably behave as a pacemaker  but are not clearly reflected in the population averages. Also, the aforementioned statistics are global averages, and may be different for any neuron at specific times, for example when a hippocampal cell fires at a high rate and may display prominent oscillations as the rat traverses across its place field (Fig. 1A). On the other hand, it is also true that theta rhythm in the rat hippocampus represents perhaps the most prominent rhythm observed in the mammalian cortex during behavior. In the neocortex, field oscillations associated with cognitive tasks are generally much weaker. There is a dearth of work in which the variability of single-unit spike trains and field oscillations are analyzed at the same time. Such studies in the future would be highly desirable, particularly in behaving animals.
Assuming that LFP monitors subthreshold membrane changes, can LFP oscillations in a cortical circuit coexist with predominantly random spiking output of constituent single cells?
To reconcile irregular neural firing with a LFP rhythm, one possibility is that the majority of neurons in a network fire irregularly and asynchronously so that they are largely averaged out in LFP, and field oscillations are generated by a small subset of neurons endowed with pacemaker properties (like chattering cells in the neocortex). This is unlikely. In a study with in vivo intracellular recordings , fast rhythmic bursting (chattering) neurons were identified by their responses to constant current injection. Interestingly, it was found that, when these cells responded to visual stimuli, hyperpolarization of the membrane potential did not reduce the relative power of visually evoked gamma oscillations in the voltage response, suggesting that gamma oscillations triggered by visual stimulation are predominantly produced by synaptic inputs, not intrinsic rhythmic bursting of single cells.
In an alternative scenario, most neurons participate in the generation of field oscillations, but the rhythmic modulation of neural firing probability is small, as illustrated in Fig. 12B for hippocampal pyramidal cells during gamma oscillations. A framework to describe such network oscillatory dynamics has been developed, taking a diametrically opposite view from that of coupled oscillators. The idea is to start with a network in which neurons predominantly behave as ‘Geiger counters’, and examine how a coherent rhythm might emerge from a dynamical destabilization of the asynchronous network state [124, 123, 127, 342, 126, 125]. Brunel and Hakim  and Brunel  assumed that isolated neurons are driven by a large amount of background synaptic noise and generate Poisson-like spike trains, and considered populations of such neurons randomly connected by delayed synaptic interactions (while neglecting synaptic current time courses). It is intuitive to see why oscillations occur in such a network of inhibitory cells: neurons fire spikes together, generating IPSPs that occur only after a delay τ. The shared feedback inhibition hyperpolarizes neurons synchronously. When this inhibition wanes over time, another collective firing episode can occur again, leading to a population oscillation with the period roughly equals to 2τ . Given that monosynaptic transmission latency is typically only a few milliseconds, the frequency of oscillations based on synaptic delay in a local circuit is typically very high (e.g. 200 Hz with τ = 2.5 ms). This conceptual framework can be generalized by taking into account realistic synaptic kinetics and applied to physiologically realistic rhythms [127, 342]. In addition, delays are longer for long-distance synaptic interactions, and delayed inhibitory feedback has been proposed as a mechanism of rhythmogenesis in systems ranging from the electric fish [248, 249, 589, 641] to the basal ganglia-thalamoc-cortical system of the mammalian brain .
Brunel and Hakim  analyzed the network behavior as a function of the synaptic coupling strength. When the coupling strength is zero, neurons do not interact with each other, the completely asynchronous network is characterized by a constant (r0) population firing rate over time. When the coupling strength is gradually increased beyond a certain threshold level, the asynchronous state is no longer stable because small perturbations are amplified by strong synaptic interactions, and a network rhythm emerges. The transition from asynchrony to collective rhythmicity can be described mathematically by the theory of bifurcations in nonlinear dynamical systems . Biologically, this illustrates how graded changes in network property (like the coupling strength) can yield qualitatively different behavior in a recurrent neural circuit. Because the oscillatory component is significantly smaller than the tonic component and noise, spike output of single cells remains highly irregular; and the mean firing rate of neurons (r0) can be essentially uncorrelated with the network oscillation frequency (f). Therefore, a measure of LFP (based on an averaged synaptic current) displays small oscillations superimposed on a large baseline component, and the presence of field oscillation coexists with Poisson-like spike discharges of single cells.
If single neurons discharge spikes irregularly with no intrinsic periodicity, what determine the frequency f of a network rhythm? Under the condition that the oscillatory component is small, a systematic answer is provided by a linear analysis . Consider a population of mutually inhibitory neurons. Assume that an oscillatory population firing rate is given by r(t) r0 + r1 sin(2πft), where f is the oscillation frequency and the oscillation amplitude r1 is small compared to r0 (Fig. 12A). The recurrent network synaptic current produced by this neural firing is given by Isyn(t) = I0 + I1 sin(2πft + Φsyn(f)), where the synaptic phase shift Φsyn(f) is determined by the synaptic filtering properties such as the latency, rise time and decay time. On the other hand, given a sinusoidal input current (the sum of noisy background drive and −Isyn(t), where the minus sign originates from the inhibitory nature of the inputs and yields an additional phase shift of π), the spiking response of a neuron is determined by its intrinsic membrane properties. The rate of spiking output r(t) is a tonic component plus a sinusoidal wave, characterized by an amplitude r1(f) and a phase shift Φcell(f) with respect to the input current, or r(t) r0 + r1 sin(2πft + Φsyn(f) + Φcell(f) + π). Because this r(t) is the same as the one that gives rise to Isyn(t), self-consistency implies that 
Solving this equation determines the frequency f. It is remarkable that this equation (hence the network oscillation frequency f) is independent of the neuronal firing rate r0, demonstrating a salient characteristic of such sparsely synchronized rhythms. The analytically predicted network oscillation frequency has been checked and shown to be fairly accurate in direct simulations of spiking neural network models [127, 342]. This theoretical framework can be extended to two populations of pyramidal cells and inhibitory interneurons [123, 127, 342]. An example is shown in Fig. 13, where the model network displays gamma oscillations while single cells fire sparsely with wide distributions of firing rates across neural populations.
This analysis shows that even though single-neuron activity is highly irregular, the frequency of a collective network rhythm is determined by both the synaptic properties (Φsyn(f)) and single cell’s membrane properties (Φcell(f)). The amplitude factor r1(f) and the phase shift Φcell(f) can be analyzed by studying a single neuron in response to an injected current with three components (a baseline I0, a small sinusoidal wave, and a noise filtered by a synaptic time constant τsyn). This insight inspired a number of studies using different types of single neuron models. It turns out that, with realistic synaptic filtering time constants, for a leaky integrate-and-fire (LIF) neuron model, both r1 and Φcell are approximately insensitive to the frequency f . On the other hand, for a Hodgkin-Huxley type I neuron , r1(f) decays like 1/f, and Φcell shows an increased lag with larger f, independent of τsyn [297, 342]. This is captured by a simpler, ‘exponential LIF’ model (taking into account the nonlinear action potential upstroke) [297, 342].
Motivated by these theoretical predictions, an in vitro experiment has been carried out, where single cortical pyramidal cells were driven by the same type of noisy current as in modeling studies . It was found that, as predicted theoretically, in pyramidal cells the phase delay Φcell(f) increases with f, and the phase shift curve is insensitive to the noise filtering time constant. Moreover, at low frequency f ~ 10 Hz, pyramidal neurons actually show a phase advance with respective to the current drive . This phase advance may result from a potassium current [668, 650, 293, 1027] or an H-type hyperpolarization activated cation current , which produces a preferred spike timing at the upstroke (phase-advanced relative to the peak) of a sinusoidal input current [492, 623, 625, 95, 506, 321, 338, 715]. At high input frequencies, such slow ionic currents are less effective, and the cellular phase shift becomes a lag due to the passive membrane RC property [297, 342, 125]. These results highlight the point that single neuron properties do matter with regard to determining collective network synchrony, even in the cases where single neuron behavior is highly irregular. Since different single neuron models (e.g. the linear LIF or a Hodgkin-Huxley type neuron) are characterized by different Φcell(f), different choices of single cell models can lead to divergent conclusions about a network oscillation’s frequency.
In a network of interconnected pyramidal cells and interneurons, a synchronous rhythm can predominantly arise from an interneuronal (I-I loop) mechanism or an excitatory-inhibitory (E-I) loop mechanism (Fig. 14A). The latter is akin to that proposed in previous firing-rate models [1060, 303, 578, 318, 976]; what is new is a framework for quantitatively predicting the oscillation frequency and the phase relationships in terms of cellular and synaptic properties. Interestingly, it can be shown that the oscillation frequency produced by the E-I loop is generally slower than that by the I-I loop [127, 342]. Thus, given typical synaptic time constants of AMPA receptor mediated excitation and GABAA receptor mediated inhibition, it is expected that 100–200 Hz ultrafast rhythms could arise from an I-I scenario, with the frequency largely controlled by the synaptic transmission latency and rise time; and ~40 Hz gamma rhythm primarily depend on an E-I scenario, which is more sensitive to synaptic decay times (Fig. 14B). This is in stark contrast to networks of coupled oscillators for which the interneuronal circuit mechanism preferentially synchronizes in the gamma frequency range (Fig. 6).
Generally, the network dynamics involves a mixture of both the I-I loop and E-I loop scenarios. One such a situation is when the synaptic excitation and inhibition balance is the same in pyramidal cells and interneurons. In a model, when the excitation/inhibition ratio is gradually increased (for both E and I neural populations at the same time), the network oscillation frequency can either decrease, or first increase then show an abrupt drop and then decrease continuously , illustrating complex behaviors that can occur even in a rather simple system of two randomly connected neural populations.
As a rule, stronger recurrent synaptic interactions greatly enhance the network’s propensity to exhibit synchronous oscillations (Fig. 14A). This result sheds insight onto why gamma-band oscillations naturally occur in CA3 [291, 632] which is endowed with more excitatory local collaterals than CA1 in the hippocampus , or superficial layers 2/3 of the necortex , which are endowed with a higher abundance of local horizontal connections than in deep layers 5/6 (in the cat V1, 21.6%, 0.9% and 4.6% of all excitatory-excitatory synapses are horizontal synaptic connections between pyramidal cells in layer 2/3, layer 5 and layer 6, respectively ). Furthermore, as we shall discuss in Section VI, cognitive functions such as selective attention involve highly recurrent areas like the prefrontal cortex prone to fast oscillations, which may explain why beta and gamma rhythms are often enhanced throughout the brain with the engagement of ‘cognitive-type’ cortical circuitry.
There is increasing experimental evidence that fast rhythms in the cerebral cortex display salient characteristics of sparsely synchronized oscillations.
As mentioned earlier, data from Csicsvari et al.  and others showed that sharp-wave ripples appear to represent an example of sparsely synchronized oscillation, with both pyramidal cells and interneurons showing a high degree of irregularity and firing intermittently at much lower firing rates than field oscillations (100–200 Hz), and pyramidal cells typically firing ahead of fast spiking interneurons during the oscillation cycle [140, 147, 208, 148, 561].
Fast (~200 Hz) rhythm has also been observed, first by Adrian [7, 460, 171], in the cerebellum. The cerebellum is a structure critical for fine motor coordination, and it has been hypothesized  that precise timing required for motor control involves millisecond-scale synchrony between Purkinje cells [73, 261, 868]. A recent study, using tetrode recordings from multiple single- units in vivo, found that precise synchrony between Purkinje neurons is directly related to fast field oscillations of the rat cerebellum in vivo . This cerebellar rhythm shared characteristics of a sparsely synchronized oscillation. The field oscillation frequency was ~150–250 Hz, whereas an average firing rate of individual Purkinje cells was about 38 Hz. Neuronal spike discharges did not occur at multiple intervals of the field oscillatory period. Cross-correlation of neighboring Purkinje cells displayed a peak at ~ 1–5 ms, but quickly (within 2 cycles) decayed to a large baseline, as expected from highly stochastic neural dynamics.
Current-source density analysis suggested that the fast oscillations were generated in the layers containing soma and proximal dendrites of Purkinje cells. The precise synchrony did not appear to be caused by common excitation from parallel fiber inputs, because the spatial extent of coherence was similar in the direction ‘on beam’ (along the parallel fibers) or ‘off beam’ (parasagittally). Moreover, blocking AMPA receptor mediated excitation by GYKI 52466 actually enhanced the oscillation power. On the other hand, blocking GABAA receptors by picrotoxin reduced the oscillation power by about 40%, suggesting that synaptic inhibition contributes to in vivo ~200 Hz cerebellar oscillations. This conclusion was in contrast to another study reporting that gap junctions played a critical role in ~ 100 Hz oscillations induced by nicotine receptor activation in in vitro cerebellar slices . De Solages et al. replicated salient in vivo experimental observations in an inhibitory neural network model, and Equation (5) predicted approximately the network oscillation frequency. Interestingly, the generation of ~200 Hz oscillations requires fast time constants of inhibitory postsynaptic currents, consistent with the measurements in vitro from the mature cerebellum (> 3 fold quicker compared to those in the developing cerebellum). Moreover, using a two-compartment neuron model for Purkinje cells, with recurrent synaptic inhibition near the soma relatively isolated from a large dendrite, produced significantly higher oscillation frequencies than when Purkinje cells were described by a one-compartment model. These results illustrate, again, how detailed synaptic and cellular properties together determine the frequency of a collective network oscillation with irregular neuronal discharges.
In vivo gamma rhythm in the olfactory bulb [264, 496, 816], hippocampus [104, 208, 209], neocortex [914, 238, 423, 783], all displays features of a sparsely synchronized oscillation, with single neurons firing irregularly at lower rates than LFP oscillations. In in vitro neocortical or hippocampal slices, when gamma rhythm is induced by muscarinic acetylcholine receptor activation, pyramidal neurons fire sparsely in less than 5% of the cycles [291, 132]. Intracellular voltage measurements and voltage-clamp recordings of synaptic currents provided a wealth of evidence that strongly supports of the excitatory-inhibitory loop mechanism for beta and gamma oscillations. First, spiking probability of single neurons displays a peaked distribution of phases with respect to the field gamma cycle, with pyramidal cells leading perisomatic-targeting interneurons by a few (ranging from 2 to 8) milliseconds [209, 408, 632, 423, 746]. This means that inhibitory neurons lag pyramidal cells by a significant phase difference (In vivo, ~ 60° ; in vitro, ~ 55°  or ~ 23° [408, 746]). This phase lag is qualitatively consistent with the feedback inhibition model proposed by Freeman  and Leung .
Second, synaptic excitation is not tonic, as expected for a predominantly interneuronal network mechanism. During spontaneous gamma oscillations in the hippocampus  and odor evoked beta oscillations in the piriform cortex , in vivo whole-cell voltage-clamp measurements from neurons in revealed that EPSCs precede IPSCs by a few milliseconds. Both in vivo and in vitro, excitatory and inhibitory synaptic conductances in a single neuron fluctuate together on a rapid timescale, leading to considerable cycle-by-cycle variations of the gamma cycle duration .
Third, in vitro carbachol induced fast oscillations were abolished by blockade of either fast excitatory AMPA receptors or inhibitory GABAA receptors in the hippocampus [291, 632] and neocortex [132, 155]. A reduction by genetic manipulation of glutamatergic synaptic transmission in parvalbumin-expressing fast spiking interneurons led to a reduced gamma oscillations . It might be argued that, in these experiments, blocking AMPA receptors deprived a cortical network of its main excitatory drive (which was not compensated by other means) necessary to induce synchronous activity in an interneuronal network. Nevertheless, these studies lend strong support of a feedback loop mechanism for self-sustained gamma rhythm in hippocampus and neocortex. In recent optogenetic studies, gamma oscillations were induced by light stimulation of fast spiking interneurons, but they were largely eliminated by blocking AMPA and NMDA receptors , and optically stimulating pyramidal cells enhanced gamma oscillations , suggesting that gamma rhythomogenesis depended on excitatory drive of interneurons by pyramidal cells, in a feedback loop.
LFP beta (~ 15–30 Hz) rhythm is also consistent with the sparsely synchronized oscillation behavior, as evident in single neuron activity recorded from the motor cortex in behaving monkeys (see, for example. Fig. 3 in ). The mechanisms of beta-band oscillations have been much less studied but may be generated by a similar circuit mechanism as gamma rhythm. For example, two Hodgkin-Huxley type models of interacting pyramidal neurons and interneurons, with a number of commonalities but also some differences, exhibit network oscillations either in the gamma band  or beta band [183, 185]. The timescale of beta rhythmicity is comparable with the spike afterhyperpolarization lasting for 30–100 ms that has been revealed by intracellular recording of motor cortex neurons in awake monkeys . The slower frequency may be determined by a mixture of cellular and synaptic properties, including voltage- and calcium-gated potassium currents in pyramidal cells and the strength of recurrent excitatory connections [1052, 1054, 817], perhaps also involving specific subtypes of interneurons [758, 682, 667].
It is presently not fully understood what are the circuit properties required to quantitatively explain the observed relative phase lag between excitatory and inhibitory cells during gamma and beta rhythms . A critical step forward is to not just look at the phase relationship of spike firing of different neural populations, but also the phase relationship between the spiking activity of a neural population and the synaptic current produced in target neural populations. In a remarkable paper, again studying in vitro cholinergically induced gamma oscillations, Oren et al.  measured phase distributions of excitatory and inhibitory synaptic currents in the pyramidal cells and three different types of interneurons. It will be interesting to see, in future work, whether the observed phase relationships of firing activity and synaptic currents can be quantitatively accounted for by the theory of sparsely synchronized oscillations [124, 127, 342, 662].
Even when fast oscillations are generated by reciprocal interactions between pyramidal cells and interneurons, inhibitory synaptic inputs may be especially suited for mediating precise synchrony in a cortical circuit [616, 177, 756, 778, 666, 934, 701], and subclasses of interneurons that target soma or dendrites of pyramidal cells produce entrainment in a frequency- and phase-specific manner . This is consistent with the finding in vivo that inhibitory synaptic potentials contained more power in the gamma frequency range than excitatory synaptic potentials (Fig. 15C), and barrages of IPSPs but not EPSPs showed marked synchrony between simultaneously recorded pyramidal neurons [423, 405, 35, 404]. In this view, high frequency LFP fluctuations are likely to reflect more synaptic inhibition than excitation.
To conclude, the theory of sparsely synchronized oscillations provides a promising new framework, that goes beyond the idea of coupled oscillators, for describing fast rhythms in the cerebral cortex. These two theories make distinct predictions. In particular, according to the model of coupled oscillators the interneuronal mechanism generates synchrony preferentially in the gamma frequency range; in contrast, the theory of sparsely synchronized oscillations predicts that gamma rhythms primarily arises from interactions between excitatory and inhibitory neurons, whereas mutual inhibition between interneurons naturally gives rise to ultrafast (> 100 Hz) oscillations. Importantly, the theory of sparsely synchronized oscillations makes it possible to reconcile population oscillations with highly stochastic firing of single neurons.
The previous sections covered the basic physiological mechanisms of brain oscillations observed in behaving conditions, and a framework for describing cortical rhythms with irregular firing of single neurons as commonly seen in alert behaving states. We are thus in a position to discuss how cortical synchronization may contribute to cognitive functions. Most of the work summarized here involves descending top-down signaling in the neocortex of monkeys and humans.
Recent years have seen a surge of research on oscillations and their synchronization across cortical areas during a variety of cognitive tasks. While these studies hold great promises, they also brought into the fore many methodological issues that remain to be resolved. As pointed out by Uhlhaas et al. : ‘In contrast to conventional analyses of ERP [evoked response potential], e.g., there are currently no general guidances and standards regarding the analyses of task-related oscillations and their synchronization. Analyses of oscillations are dependent upon several parameters, such as the choice of size of the width of the analysis window, the assessment of pre-task baseline activity, the methods to assess spectral densities, phase locking, and coherence. Last but not least, even details such as the definition of reference electrodes in EEG recordings need to be standardized to guarantee replicability and comparison of results’.
Without diverting into technical details, it is worth briefly pointing out here a number of issues, some of which are not only about measurement or analysis but fundamentally depend on our understanding of the nature of neural dynamics (e.g. neurons are viewed as regular oscillators or irregularly firing devices) in the cerebral cortex.
First, caution must be exercised with interpreting scalp EEGs [1088, 314]. For example, certain forms of induced gamma band responses from scalp EEG, characterized by transient enhancement of broad band (30–100 Hz) power ~200–300 ms after stimulus onset , may actually be an artifact related to eye muscles that produce miniature saccades  (see [676, 1089] for a debate; among the contentious issues is the choice of the reference point for EEG data). MEG, intracranial EEG and LFP are less prone to such artifacts.
Second, unlike clock-like rhythm (with a narrowly peaked power spectrum) observed in vitro under certain conditions (Fig. 15A), the power spectrum of LFP recorded in vivo is typically dominated by a power-law (~ 1/fα) over a wide range of frequencies [148, 576, 431, 185]. A rhythm, if present, is manifested by the existence of a small peak superimposed on this broad power spectrum (Fig. 15B). A common practice is to filter out this power decay (‘normalizing the power spectrum’) prior spectral analysis [689, 878, 873], often assuming α = 1 (‘1/f noise’). This, however, could lead to artifacts. For example, frequency scaling of LFP power spectrum may depend on the behavioral state, with α varying between 1 and 2 [70, 69].
The power law of LFP spectrum is partly due to low-pass filtering by the extracellular medium [70, 71], but also reflects fluctuations of neuronal signals on multiple timescales. Indeed, during episodes of spontaneous network activity (‘Up state’) [240, 404] or sensory responses , excitatory and inhibitory synaptic potentials in a single neuron display fluctuations on a range of timescales (Fig. 15C); and such synaptic currents across the dendrosomatic membrane are a major component of the physiological substrate of LFP. Further, it has been suggested that, in some cases, an enhanced high-frequency power might not reflect synchronous rhythms but a decrease in the low frequency power and a concomitant broadband power increase . Therefore, in order to assess synchronous oscillations, it is important to understand better LFP broadband power structure and its dependence on behavioral conditions.
Third, since the voltage produced by volume conducting currents decays inversely with distance from the point sources, LFPs in neighboring structures might be synchronous merely as a result of volume conduction [732, 955] (but see ). Current source density (CSD) analysis [722, 299, 690, 266, 764] avoids this problem. On the other hand, rhythms are often not detectable in the spike trains of individual cells. For these reasons, spike-field coherence has become a measure of choice for assessing relative timing of neural activity to a population rhythm, or synchronization between brain regions (with LFP recorded in one area, and spiking activity in another). Statistically reliable estimation of spike-field coherence, however, remains a difficult problem. For example, results of multitaper spectral analysis are sensitive to the number of tapers used and to a suitable smoothing frequency window. The values of these parameters are often chosen ad hoc; a disciplined approach is lacking . Observed coherence (defined between 0 and 1) is usually quite small: in attention studies, with LFP and spiking activity recorded from the same area, spike-field coherence is typically on the order of 0.1 for multi-unit spike trains  and single-unit spike trains , whereas spike-spike coherence of multi-units is ~ 0.05 . This underscores the notion that oscillatory synchrony, if present, is weak between neurons.
Fourth, a peak in the coherence spectrum of two neural signals is readily interpretable when both signals are rhythmic (each with a peaked power spectrum) (Fig. 15D). However, it can happen that one of the signals is rhythmic while the other is not. This is illustrated by a study with awake cats . It was found that LFP coherence between the basolateral amygdala and the rhinal cortical areas was peaked in the gamma (35–45 Hz) frequency range; the LFP of the amygdala showed a peak in the same gamma band, but those from the perihinal and entorhinal cortices did not.
Moreover, in principle, it is conceivable that a coherence spectrum exhibits a peak, in the absence of a rhythm. Fig. 15E shows an example in which, while the LFP power spectrum is devoid of a clear peak, spike-field (between LFP and a multi-unit spike train) and spike-spike (between spike trains of two multi-units recorded from different electrodes) coherence spectra display a peak in the gamma band . Thus, a peaked coherence spectrum does not necessarily imply the presence of a rhythm.
Fifth, LFP signals may be distorted because the circuit of a metal microelectrode operates as a voltage divider with a frequency-dependent gain . This results in frequency-dependent phase shifts and amplitude attenuation when the electrode impedance is not negligible relative to the effective input impedance of the head-stage amplifier . Consequently, caution is warranted in the analysis of frequency-dependent phase shifts (such as the relative phase between LFP and spike discharges) and interpreting them as evidence of a ‘phase code’.
Sixth, synchronous events occur episodically, typically lasting only for ~ 200 – 500 ms [378, 711, 678, 285, 349, 760] (Fig. 15D). Several methods have been developed, including multitapering [759, 1069, 1091, 560], wavelet transforms [932, 566, 549, 550, 561], empirical mode decomposition using short-term Hilbert transform , ‘matching pursuit’ using Gabor functions . But the relative advantages and shortcomings of these methods are not fully understood. At the very least, it should be insisted that reported oscilatory/synchronous episodes must last much longer than the time window used in such analyses.
In response to long-lasting inputs, short oscillation episodes may reoccur intermittently in a slow temporal pattern [378, 711, 1043, 67, 788, 557]. Modeling studies discussed above are not adequate to account for recurrence of transient episodes of neural activity, which remain to be better understood theoretically (but see ).
Seventh, coherence spectrum reflects both correlation of power amplitudes and phase synchronization. To avoid this confound, elaborated methods such as phase-locking statistics [543, 1004] and power-correlation as a function of the relative phase  have been developed to more directly assess phase coherence of two neural signals. However, such methods are justified if the two signals are strongly oscillatory with a nearly identical rhythmic frequency (so the phase is well defined in the same way for both signals). By contrast, the applicability of such methods is not entirely clear if the two signals have broad band power spectra, as is generally the case. At a minimum, such studies need to ensure that each of the two signals does display a peaked power spectrum (hence a rhythm is present), and that the peaks of the two spectra have a considerable overlapping frequency range. Lack of such information would render it difficult to evaluate the meaning and implications of reported phase coherence.
Eighth, as there are often two or more inter-nested rhythms [548, 907, 143, 819], analysis techniques have been proposed to describe cross-frequency coupling (e.g. to examine how the gamma amplitude depends on the theta phase). A common practice is to compute band-passed signals at two different frequencies f1 and f2 (e.g. theta and gamma), and assess the statistical dependence between the two amplitude time series, the two phase time series, or one amplitude and the other phase time series [150, 748, 229, 216, 1084]. Clearly, many of the aforementioned concerns also apply to the analysis of cross- frequency coupling.
Ninth, long-distance communication involves interactions between multiple cortical areas. Analysis methods have been developed to interrogate the directionality of the flow of signals between two or multiple recorded areas, such as the Granger causality analysis [371, 118, 244, 560, 93, 381] and its frequency-domain representation [244, 38]. It is currently unclear how directionality inferred from such methods can be mapped onto the actual anatomical and functional connectivity in the brain [288, 402]. A fusion of several techniques may help make progress on this important issue.
Tenth, even when there is no rhythm (in the sense that an LFP power spectrum does not exhibit a well defined peak), it is a common practice to divide an LFP power spectrum into different frequency bands [112, 1017, 1018, 704, 57, 204, 80]. This is often done with an arbitrary choice of passbands, but methods such as the empirical mode decomposition [447, 585] have been developed that decompose a time-series into narrow-band components by identifying the physical time scales intrinsic to the data. This approach has been used, for instance, in investigations of the physiological basis of functional magnetic resonance imaging, by examining the correlation between blood oxygenation level-dependent (BOLD) responses and LFP power in different frequency bands. Correlations were found especially strong between BOLD and LFP gamma power in the neocortex [606, 604, 704, 725, 727, 636, 1013, 361, 605, 1090] and theta power in the hippocampus .
Similarly, a coherence spectrum may not show a peak at all, but one can still consider the coherence in different frequency bands, and seek whether the coherence at a particular frequency is enhanced or decreased in a comparison between two stimulation or behavioral conditions. For example, a recent study  examined LFP coherence between the lateral intraparietal area (LIP) in the parietal cortex and the frontal eye field (FEF) in the frontal cortex of behaving monkeys, during visual search which is either based on bottom-up attention (when a salient target ‘pops up’) or top-down attention (looking for a target that is similar to distractors). The coherence spectrum did not display a clear peak under either of the two conditions. The difference between the two coherence spectra, however, showed that top-down attention was associated with a differential enhancement in the beta frequency range and a suppression in the gamma frequency range.
It stands to reason that LFP powers or coherence powers in different frequency bands provide different information about the recorded signals and its neuronal underpinning. Nevertheless, confusion often arises if band-passed powers are identified with various rhythms, whereas there is no explicit assessment as to whether a well defined rhythm or peaked coherence spectrum actually exists. Conceptually, will we be able to achieve a fundamental understanding of the cortical dynamics, by dividing a broad power spectrum into frequency bands and theorizing the band-passed powers in terms of oscillations? Probably not. An adequate framework is presently lacking to explain these diverse types of network coherence.
For the sake of clarity, below I will review studies of neural synchronization in cognitive behaviors without dwelling on methodological issues. However, it is worth keeping in mind the potential pitfalls discussed above, which call forth a concerted effort to set the standards for the field.
Phase coding is a key, yet still controversial, concept in studies of the role of neural oscillations in local circuit functions and in long-distance communication between brain areas [144, 143, 850, 313, 951, 786, 955], as well as in memory functions [594, 591, 425, 488, 592]. In general, the phase coding hypothesis posits that neurons represent information in terms of spike times with respect to the phases of ongoing network fluctuations, rather than only by spike rates. In a simple situation, network fluctuations (e.g. measured by LFP) are rhythmic with a well defined periodicity T. If neuronal spike times (defined in the phase variable = 2πt/T mod(2π)) are not uniformly distributed between 0 and 2π, but narrowly concentrated around a preferred phase of the LFP oscillation, then spike phases may convey information (e.g. about a sensory stimulus or behavioral metrics), above and beyond average spike counts.
An extensively studied example of phase coding is the neural representation of space in the hippocampus and entorhinal cortex. As introduced at the beginning of this article, during exploration in a familiar environment, a pyramidal ‘place cell’ in the hippocampus is activated when the animal passes through its ‘place field’ , hence it shows a bell-shaped activity profile in space and (given a constant velocity) over time (Fig. 16A). Furthermore, spike firing exhibits phase precession, shifting to earlier phases of the theta cycle as the animal moves across the place field (Fig. 1A). Pooled-trial phase precession ranges over 360 degrees (a full theta cycle) [740, 881, 418, 456, 1099, 661, 1081, 660]; there is a large trial-to-trial variability and single-trial phase precession range is limited to ~ 180° . The firing rate of a place cell is a bell-shaped function of spatial location, hence is roughly the same for the animal to be at different locations of equal distance to the center of the place field. In contrast, because of the phase precession phenomenon, the preferred phase of neural firing relative to the theta cycle is a linear function of the animal’s location across the place field, thus can potentially disambiguate locations that spike counts alone cannot. Indeed, analyses suggest that position reconstruction from activity of hippocampal place cells can be improved by taking into account the contributions of phase coding [881, 473, 456, 865].
Phase precession of hippocampal place cells has been proposed to result from recurrent network dynamics [977, 1021, 674, 945]. However, neurons in the medial entorhinal cortex layer II, which provide the main inputs to the hippocampus, also display hippocampus-independent phase precession . Another, ‘somadendritic interference model’ for phase precession [492, 625, 338] assumes the interplay of two oscillatory inputs onto pyramidal cells: a pacemaker input (e.g. somatic inhibition driven by the medial septum) and a dendritic excitation (e.g. from the entorhinal cortex) that is spatially dependent hence increases as the animal moves toward the center of the place field. A somewhat different, ‘dual oscillator interference model’ assumes the linear addition of two independent oscillatory influences onto pyramidal cells [740, 575]. Consider a place cell that receives the sum of two sinusoidal inputs I = A(sin(2πf0t) + sin(2πfit)), where f0 represents an external input (and reflected by LFP), fi is determined by some intrinsic mechanism and is higher than f0. If the difference between fi and f0 is small, the two waves move in and out of phase in an interference pattern, with the beating frequency proportional to (fi − f0). Assume further that the neuron fires spikes when the total input exceeds a threshold, then spike timing naturally displays phase precession with respect to the LFP theta wave at f0. Since the change in the firing phase Δ ~ fbt, and the location of the rat in the place field Δx vt (where v is the running speed), Δ (fb/v)Δx grows linearly with the spatial location. Moreover, if the intrinsic frequency fi increases with v, such that fb κv, then Δ κΔx, i.e. the linear relationship between the spike phase and the spatial location does not depend on the running speed. This prediction has been confirmed experimentally .
The membrane potential oscillation during phase precession is differentially predicted by the three models: it exhibits a beating pattern in the dual oscillator interference model, a ramp-like depolarization with increasing theta oscillation amplitudes in the soma-dendritic interference model, and a ramp-like depolarization with a constant theta oscillation amplitude in the recurrent network model (Fig. 16B). In a tour de force, Harvey et al. tested these models by intracellular recording from hippocampal cells in mice performing virtual navigation . It was revealed that the membrane potential of pyramidal neurons exhibits the time course predicted by the ‘soma-dendrite interference model’ but not other models (Fig. 16C). Furthermore, the membrane potential undergoes phase precession with respect to LFP, whereas spiking always occurs at the peak of the transmembrane voltage oscillations (Fig. 16D–F), in agreement with the ‘soma-dendritic interference model’. It remains to be seen whether the recurrent network model can be modified to account for these data.
The dual oscillator interference model has also been applied to the entorhinal cortex, where layer II pyramidal cells display multiple firing fields as predicted by the model (Fig. 15B, top trace). Indeed, these ‘grid cells’ fire whenever the rat is on any vertex of a triangular lattice spanning the whole surface of the environment [400, 843, 703]. In recent years, the dual oscillator interference model has been proposed to account for phase precession as well as the spatial pattern itself of grid cell activity [737, 136, 135, 426]. The basic insight is that given animal’s running speed v, a temporal interference pattern is converted into a spatial interference pattern (a grid) of firing activity, with the characteristic spatial length [136, 135, 86]. The model predicts that the grid scale decreases with the frequency of theta modulation of grid cells, but does not depend on the animal’s running speed, which was supported by experimental measurements from behaving rats . Furthermore, the grid scale increases from dorsal to ventral medial entorhinal cortex . Assuming that the frequency fi reflects intrinsic membrane oscillations of stellate cells in layer II of the entorhinal cortex [13, 1047, 272], the oscillatory interference model predicts a decreasing gradient of the oscillation frequency along the dorso-ventral axis. Consistent with this prediction, an in vitro study found that stellate cells near the dorsal border have slightly higher frequencies (by ~ 2 Hz) of subthreshold membrane oscillations than those near the ventral border of entorhinal cortex [357, 426]. It remains to be seen whether this modest variation in the oscillation frequency is sufficient to account for the observed grid scale gradient from dorsal to ventral medial entorhinal cortex. One potential weakness of this scenario is its extreme sensitivity to noise: computer model simulations found that the grid structure is completely lost when a noise as small as 0.003 Hz was added to the intrinsic frequency fi ; and it was estimated that a biologically constrained oscillatory interference scenario predicts a stable grid pattern only for less than 1 sec, far less than what is observed experimentally . Understanding whether the model can be rendered robustly in spite of neural variability will critically test the oscillatory interference model.
In the neocortex, the idea of phase coding at <10 Hz is supported by evidence from a number of physiological work on rats, monkeys and humans. In a study of visual responses to natural visual stimuli (movies) in anesthetized monkeys, phase coding was examined in different frequency bands of LFP fluctuations. It was found that spike phases of low-band (1–4 Hz [693, 1051] or 8–12 Hz [504, 750]) passed LFP conveyed a significant amount of additional information about movie stimuli beyond that conveyed by spike counts alone. Such a phase dependence may come about through a conversion from the strength of afferent input into a particular phase of neuronal firing (the ‘rate-to-phase transform’ ). Mclelland and Paulsen examined the ‘rate-to-phase transform’ of single pyramidal cells in rat neocortex in vitro and found that, in the presence of a theta-band membrane oscillation, the level of a tonic excitation could be linearly converted onto a preferred firing phase . In human studies, it was reported that detection of a near-threshold stimulus depends on the stimulus onset time relative to the phase of ongoing 5–10 Hz oscillations in the frontal cortex  or parietal cortex . The likelihood of perception of a faint stimulus depends on neural excitability, which covaries with EEG fluctuations on a fine timescale. As a result, the threshold of conscious perception varies with the phase of theta/alpha band EEG oscillations [274, 995, 657, 137, 1077].
Phase coding has also been proposed for fast gamma-band oscillations . A recent study reported that, in a task when monkey must remember two stimulus items during the mnemonic delay period, spikes of neurons in the prefrontal cortex carried most information about the individual memorized objects at different gamma oscillatory phases according to their order of presentation . In addition, it has been reported that, in visual cortex of awake monkeys, gamma-band phase of neural spiking (in response to visual stimulation) displayed a systematical relationship with the firing rate . Other studies raised questions about the robustness of phase coding at gamma frequencies. Gamma oscillations are very variable in the neocortex; the oscillation frequencies range widely, and the moment-to-moment LFP cycle duration and phases fluctuate greatly over time. For instance, in an in vivo rat experiment, the gamma cycle duration was found to vary considerably (12 to 40 ms), whereas the timing of maximum neural firing relative to the gamma peak remained fixed, implying that the preferred oscillatory phase of neural activity is not reliable . In the aforementioned monkey experiment using movie stimuli, at higher frequencies (> 12 Hz), the phases of band-passed LFP were highly unrepeatable across different trials at fixed sensory input (Fig. 17) , hence phase coding at higher frequencies (including the gamma band) was unreliable (see also ). Search for stimulus coding by EEG gamma phase in human studies has yielded mixed results [975, 518]. Finally, in their in vitro work Mclelland and Paulsen did not observe the ‘rate-to-phase transform’ of single pyramidal cells (the firing phase was independent of the tonic input intensity), if the ongoing rhythm was at gamma rather than theta frequency , in discord with the idea that the input intensity could be encoded by the gamma phase of spiking discharges.
In general, phase noise represents a hindrance for phase modulation as a communication scheme . Quite unlike sinusoidal waves typically analyzed in artificial communication systems, neural circuit oscillations are nonlinear. Nevertheless, whereas stable nonlinear oscillators are resistant to amplitude perturbations, they are sensitive to phase perturbations [359, 710, 922, 465]. Therefore, noise is a considerable concern for the phase coding concept, more so for higher frequency rhythms (e.g. gamma-versus theta-band). Quantifying the variability of LFP phases, the variability and the fraction of spikes phase-locked to LFP fluctuations, will help clarify the concept of phase coding in studies of local processing as well as synchronization between cortical areas discussed below.
Insofar as spike synchrony represents a versatile mechanism for sculpting neural temporal correlations, it is a significant factor in determining synaptic learning by virtue of STDP. In line with the facts that theta phase coding of information is reliable (Section VI-B), and activity induced synaptic plasticity depends on spike timing relative to theta phase [753, 450], numerous studies on behaving humans and animals have implicated theta phase modulation in memory formation and retrieval [515, 424, 60, 425, 488, 518, 432, 156, 24]. A focus of recent research is the interaction between the hippocampus and neocortex, which is functionally important for memory consolidation . Simultaneous recordings from the hippocampus and medial prefrontal cortex in freely behaving rats found that many prefrontal neurons discharge spikes that are phase-locked to the hippocampal theta rhythm [484, 870, 458, 878, 260], presumably driven by monosynaptic connections from the the ventral CA1 [470, 421]; and gamma oscillations generated locally in the neocortex are entrained by the hippocampal theta rhythm . Correlated neural firing and LFP coherence between the hippocampus and frontal cortex were enhanced at the theta frequency during a spatial memory task , correlation analysis of simultaneously recorded single hippocampal and frontal neurons revealed a hippocampofrontal directionality . This theta-entrained communication may play a role in inducing spike-timing dependent plastic changes across these networks. Furthermore, intracranial EEG recording in epilepsy patients [287, 854], rat [1045, 961] and monkey physiology  indicate that gamma-band oscillations and gamma-theta coupling may also contribute to learning and memory processes. In particular, the dialog between hippocampus and neocortex is mediated through the perirhinal and entorhinal cortices. The rhinal cortices are also strongly and reciprocally connected with amygdala; this connection is thought to be important to facilitate learning in emotionally arousing situations. In awake and behaving rats, coherence of gamma oscillations is prominent in LFPs simultaneously recorded from the rhinal cortices and amygdala, and single-unit spiking activity in both structures was modulated by field oscillations, suggesting a role of gamma-band coherence in effective communication between the amygdala and rhinal cortices . There is also evidence for gamma band coupling between the frontal cortex and striatum  and theta-band coupling between the hippocampus and striatum  during action selection, and gamma-band coupling between the amygdala and striatum during learning .
Sleep has long been thought to aid consolidation of newly acquired information in the memory system, and there is accumulating evidence for a beneficial effect of sleep on memory consolidation [919, 100, 1020, 920]. Interestingly, while dreams during REM sleep are often thought of as reactivated memories, recent work suggests that non-REM, slow wave sleep is also involved with memory consolidation. In a spatial task where rats repeatedly ran through a sequence of place fields of several simultaneously recorded hippocampal place cells, it was found that immediately following the experience, these cells repeatedly fired in the precisely same sequential order during slow wave sleep . Furthermore, during slow wave sleep, hippocampal neurons are phase locked to neocortical Up and Down states . Frontal neurons consistently fire within 100 ms after hippocampal neurons and this coupling is much stronger during sharp wave sleep than REM sleep . Coordinated activity in hippocampal and neocortical neurons displayed replay of temporal patterns observed in the previous awake experience, suggesting that slow wave sleep may facilitate simultaneous reactivation of coherent memory traces in the hippocampus and neocortex . In human studies, motor learning increased slow wave activity during subsequent sleep, locally in specific brain regions, and this local increase after learning correlated with improved performance of the task after sleep . In another study, boosting slow oscillations artificially by transcranial electrical field stimulation enhanced hippocampus-dependent memory .
Precisely how slow wave sleep might contribute to memory consolidation remains unclear presently. Compared to wakefulness, slow wave sleep is characterized by a breakdown of effective connectivity in the cortex, as assessed by how the activation of one cortical area (evoked by transcranial magnetic stimulation) is transmitted to the rest of the brain [653, 284]. At the same time, during slow wave sleep the cortex undergoes synchronous Up and Down states, which might be particularly suited for inducing some form of spike-timing dependent plasticity across brain regions. The sign (potentiation versus depression), however, depends on precise timing relationship between neurons [81, 213]. In fact, there is evidence that excitatory synaptic connections in the cortex are overall depressed in sleep and potentiated in wakefulness , suggesting that a primary function of slow wave sleep is homeostatic regulation of synaptic circuitry [959, 283, 655].
A correlation between neural coherence and memory behavior, whenever observed, is typically weak. For example, Jutras et al.  recorded from single units and LFP in hippocampus of monkeys on a recognition memory task. In the experiment, each visual stimulus was presented twice, randomly interleaved with other stimuli. Although the monkeys were not required to produce a behavioral response, memory recall was assessed by monkey’s looking time on a stimulus which is shorter with a repeated item compared to its first presentation. It was found that spike-field coherence increased by ~ 10% from a baseline of ~ 0.1 in the gamma frequency range during encoding of stimuli (at their first presentations) that were subsequently well recognized relative to those stimuli that were poorly recognized (at their repeat presentations). The increased coherence may aid STDP hence learning [486, 1067]. One may reasonably ask: why does this ten percent of ten percent coherence matter, and how about the majority of spikes that are not correlated with field gamma? Stable computation may require that spiking activity of neurons is predominantly asynchronous, and learning through STDP relies on a small fraction of synchronous spikes, but at the present we do not know if that is the case. It will be a challenge, in future research, to develop a quantitative, biophysical, model that can account for memory behavior (good versus poor recognition) based on the observed small change in spike-field coherence.
While wakefulness is not associated with large-amplitude EEG synchronization, subtle synchronous coordination is thought to be important for the integrative brain functions in behaving states. A general computation of this kind is to combine information from different sensory modalities, such as binding of vocalized sounds with vocalizing faces in speech perception . The traditional hierarchical model posits that crossmodal integration is instantiated by convergence of input pathways from different senses to higher-level multisensory areas. Recent experimental evidence, however, points to the possibility that multisensory interplay involves coherent oscillations across cortical areas [490, 687, 1087, 549, 349, 627, 857, 551].
Ghazanfar and colleagues investigated binding of a vocalized sound and its associated mouth movement in monkeys . They recorded LFP and single units from the auditory cortex and the superior temporal sulcus (STS, an association area involved with integrating audiovisual biological motion). While the power spectrum of LFP displays a peak in the gamma band in the auditory cortex when a voice stimulus was presented, and in STS when a face was presented, the inter-areal coherence shows a peak only when both stimuli were presented simultaneously (Fig. 15D), hence gamma band coherence across areas is correlated with multisensory integration. Another study by the same group used visual and auditory inputs that were either looming or receding. It was found that coincident looming visual and auditory stimuli to be particularly salient behaviorally (presumably mimicking an approaching object, potentially a predator) . Gamma power in an auditory cortical area or STS was no different in the unisensory versus multisensory condition. On the other hand, gamma-band coherence between the two areas was significantly larger with congruent visual and auditory looming stimuli, compared to a looming visual or auditory stimulus alone, or to incongruent visual and auditory stimuli (one was looming, the other was receding) . This finding suggests that synchronization may act as a mechanism for rapidly and transiently enhancing functional connections between selective neural populations that encode stimuli of different modalities. The resulting correlated activity of the two cortical areas may impact more effectively on downstream neurons [838, 287, 310, 1068, 549], as a potential mechanism for integration of sensory information across modalities.
As noted at the beginning of this review, gamma oscillation was initially called a ‘hypervigilance rhythm’, because of its correlate with highly alert brain states . Over the years, this idea has received increasing supporting evidence. EEG or MEG recordings in human studies have shown that attention is associated with a reduced EEG alpha rhythm [1073, 452, 845, 780, 946, 983, 152] and increased gamma rhythm [311, 852, 224]. A recent study using ‘dynamic causal models’ of MEG data found that forward and backward functional connections between cortical areas are asymmetric and frequency-dependent; high (gamma) frequencies in higher cortical areas suppressed low (alpha) frequencies in lower areas . Enhanced gamma power associated with attention has been documented for various sensory modalities: audition [328, 952, 482, 785, 886, 388, 203, 223, 285], vision [270, 938, 754, 245, 797, 872], somatosensation [65, 824, 427, 797, 521] and pain [386, 427]. For example, Ray et al.  recorded subdural electrocorticography (ECoG) in epileptic patients, when they were asked to pay attention to either auditory stimuli or vibrotactile stimuli. ECoG in the auditory and somatosensory cortical areas showed an early and a late component of event-related gamma-band activity. The early component was localized in modality specific cortical areas, and not modulated by attention. By contrast, the late component (400 ms after the target stimulus onset) is larger in the somatosensory (respectively auditory) cortex when subjects attended vibrotactile (respectively sound) stimuli. Moreover, at electrode sites in the prefrontal cortex, ECoG gamma activity was negligible when a stimulus was presented in an unattended modality, but large in response to the presentation of an attended stimulus regardless of whether it was auditory or vibrotactile. These observations suggest that gamma-band oscillations in a sensory area are enhanced by selective attention, involving top-down signals from the prefrontal cortex.
In a groundbreaking study, Fries and collaborators  combined neuronal recording with LFP measurements in monkeys performing a task that depended on location-specific visual attention. In this task, monkeys were required to direct attention to one of two visual (moving grating) stimuli which were inside and outside, respectively, the receptive field (RF) of recorded neurons (multi-unit activity) and LFP in the cortical area V4 (an important area for visual object processing) . It was found that attention into the RF, compared to attention outside the RF, reduced LFP power at low frequencies (< 15 Hz) and enhanced gamma power (at 30–90 Hz) (Fig. 1B). Local synchronization was assessed using spike-triggered averaging of LFPs, which was enhanced by attention while multi-unit spike activity remains seemingly stochastic (Fig. 1B). This observation was confirmed with spike-field coherence, which measures phase synchronization between spikes and LFP as a function of frequency. Subsequently, using the same task paradigm. it was reported that large spike-field coherence immediately before the onset of a test stimulus was correlated with faster response times, demonstrating an effect at the behavioral level . In a separate study using a visual search paradigm (where the subject looks for a target stimulus surrounded by distractor stimuli) while recording from monkey’s V4, spike-field coherence displayed a peak in the gamma band when a preferred stimulus was presented in the RF of a recorded neuron and it matched a feature of the target . Because it occurred even when the preferred stimulus is a distractor and the target itself is not in the RF, the finding suggests a role of gamma band coherence in parallel processing of all objects in a cluttered scene. Finally, the same group recently carried out simultaneous recording from neurons in V4 and in FEF (a putative ‘source area’ for top-down attention signal) and found that attention to a stimulus in their joint RFs increased inter-areal coherence especially in the gamma band, with a time shift consistent with the idea that FEF initiates attention induced enhancement of gamma oscillations .
How enhanced synchrony might contribute to selective attention has been investigated in modeling work. In an early study, Niebur et al. [724, 723] proposed that an attentional signal increases synchrony among sensory neurons, and this enhanced temporal correlation is detected by interneurons, which in turn suppress other neurons selective for non-attended stimuli thereby implementing biased competition [698, 230, 803]. Other work argued that competitive advantage by attention is amplified if an attentional signal directly targets interneurons engaged in gamma oscillations [950, 948, 133, 97, 98, 1092, 949]. Assuming that oscillations are generated by an interneuronal mechanism, Tiesinga and collaborators [950, 948] examined the impact of random jitters among (otherwise perfectly clock-like) inhibitory inputs to a pyramidal cell, and showed that less jitter (higher synchrony) leads to a larger slope of the pyramidal cell’s input-output relationship, suggesting that synchrony contributes to gain modulation [665, 974, 973]. This scenario for the multiplicative gain modulation requires that jitters of spike times be small (< 5 ms), thus interneurons must behave almost like regular oscillators. A related but more general mechanism for gain modulation is mediated by correlated excitatory and inhibitory inputs [837, 838]. A fairly consistent observation that has not been taken into account in these models is that the variability of spike trains, measured by the Fano factor, is typically high (~ 1 or larger) and is either unaffected [664, 900] or only slightly decreased  by selective attention. Note that both competition and gain modulation can be realized by mechanisms solely based on rate modulation: e.g. competition through feedback inhibition [803, 30] and gain modulation by balanced inputs  or by a noise-induced input-output power law . Nevertheless, rate modulation may be enhanced by the presence of gamma oscillations , which might lead to improved sensory processing such as signal discrimination .
These modeling studies focused on how a tonic attentional input modulates oscillations intrinsically generated in a sensory circuit. On the other hand, top-down attentional signals may themselves be endowed with rhythmicity. The parietal and prefrontal cortical areas are ‘source areas’ for top-down attentional signals to sensory systems [230, 194, 179, 697], and coherence between these areas and early visual areas is modulated by attention . The parieto-prefrontal circuit is also important for working memory, brain’s ability to temporally hold information ‘online’ . In monkey experiments where subjects were required to remember the location of a stimulus cue across a delay period (a few seconds), neurons in the parietal cortex  and prefrontal cortex  exhibited location-selective persistent activity during the delay (see  for a review). Such persistent neural activity could also serve spatial attention by providing a location-specific and internally sustained top-down attention signal to bias sensory processing. Therefore, working memory and attention control may share a common cortical circuit substrate. There is also evidence that, whereas fast gamma oscillations are involved in the maintenance of relevant information, the slow alpha rhythm is implicated in suppressing irrelevant information in memory [517, 475, 748, 749, 846] and attention [845, 946, 246] tasks. Hence, at a given cortical site, alpha rhythm may be reduced when attention is focused on external stimuli processed by that brain area, but enhanced when sensory information flow to that area is behaviorally irrelevant and should be ignored.
Does activation of the parieto-prefrontal circuit induce similar synchronous oscillations during working memory tasks as in attention tasks? Indeed, MEG, EEG and intracranial EEG recordings in human studies have revealed strong gamma oscillations in parietal, frontal and other cortical areas during working memory tasks [614, 444, 476, 994, 477]. Moreover, in a monkey experiment using a spatial working memory task, LFP and spike-field coherence in the lateral intraparietal (LIP) area showed enhanced gamma oscillations during a mnemonic delay period . A small number of single units in LIP also displayed a peaked power spectrum in the gamma frequency range. In contrast, other studies using the same spatial working memory task, did not find gamma oscillations in the majority of single-units recorded from the posterior parietal cortex area 7a  or dorsolateral prefrontal cortex . These results highlighted the need of simultaneous recording of LFP as well as single-unit activity, in order to resolve the discrepancy between studies and better assess coherent oscillations in an activated parieto-prefrontal network.
Self-sustained persistent activity has been hypothesized to be generated by reverberatory local circuit dynamics [18, 364, 1030]. Interestingly, in biophysically realistic models of working memory [1029, 182, 939], highly recurrent networks naturally give rise to fast coherent oscillations. The amplitude of network gamma oscillations is controlled not only by the strength of recurrent connections, but also the ratio of NMDA and AMPA receptors at the horizontal excitatory connections [1029, 182, 939, 129]. This is because, as discussed in Sections III-C and V-C, oscillations readily occur in an excitatory-inhibitory loop, when synaptic excitation is mediated by AMPA receptors and faster than inhibition.
If the working memory/attention circuit is prone to fast oscillations, attention induced enhancement of gamma oscillations in sensory circuits may simply be a signature of the engagement of such a ‘source area’ for top-down biasing signals. To test this idea, Ardid et al.  considered a two-module model of spiking neurons that consists of a reciprocal loop between a sensory (e.g. MT) network and a working memory (e.g. PFC) network. In this model, feature-selective attentional signal, originating in the working memory circuit, projects top-down to bias stimulus processing in the sensory circuit (Fig. 18A). The model captures biased competition and provides a circuit implementation of the ‘feature-similarity gain principle’, namely the attentional effect on a neuron’s firing rate is a graded ‘gain modulation function’ of the difference between the neuron’s preferred feature and the attended feature [973, 649]. The sensory circuit (in response to a stimulus) and the working memory circuit (when activated during sustained attention), both display sparsely synchronized gamma-band oscillations (Fig. 18B,D) while spiking of sensory neurons is highly irregular (with the Fano factor close to 1) (Fig. 18C,E). Thus, the model reproduces the common observation from monkey experiments that weak network rhythmicity coexists with highly irregular neuronal spike discharges, which is important since how synchrony affects attention depends on the dynamical state of the network. In this model, the activation of the ‘source area’ (which itself is oscillatory) during selective attention results in an enhancement of oscillations in the sensory circuit (Fig. 18D) and inter-areal coherence in the gamma frequency band. In this model, a manipulation that abolishes inter-areal coherence reduced attentional gain modulation by about 10% . Therefore, in a dynamic state where single neurons fire stochastically rather than behaving like oscillators, gamma-band coherence across areas may play a modest but significant role in boosting sensory processing during selective attention.
Curiously, while gamma oscillations are commonly enhanced under conditions that involve cognitive control, top-down signaling itself may be particularly associated with beta-band neural activity. As mentioned in Section I, numerous studies have documented beta oscillations associated with movement behavior [765, 103, 711, 842, 712, 713, 617, 41, 286, 45, 26, 252, 562, 42, 735, 43, 197, 836, 567, 46, 44, 1001] and response inhibition [536, 537, 520, 1094, 925, 774]. But beta power has also been implicated in broader cognitive processes. Descending control in cognitive tasks was examined in several recent monkey physiological experiments, used LFP and single-unit recordings from two cortical areas. In a delayed match-to-sample task, when the subjects had to retain in working memory the identity and spatial location of a visual sample stimulus, in order to decide whether it matches a second stimulus presented a delay later, there was an enhanced coherence at ~ 20 Hz between MT (a sensory area) and LIP (a cognitive area), in LFP or neural spikes, in a way that is specific to the preferred location and feature of the recording sites . A second study used a visual search task, and found that the LFP coherence between LIP and FEF is enhanced at ~ 20 Hz in guided search (which involves top-down signaling from FEF) when distractors resemble the target in some feature (color or orientation of a visual stimulus), compared to ‘pop-up’ (which is a feed-forward process) when the target is dissimilar to the distractors . In a third experiment, subjects performed a reach movement either freely or according to an instructed cue, and spike-field coherence at ~ 15 Hz was enhanced between a parietal (‘reach’) area and a frontal (premotor) area in free choices compared to forced choices . It is noteworthy that in all three experiments, the enhanced coherence was observed in the beta, not gamma, frequency range.
Why would beta oscillations play a special role in top-down signaling? To answer this question, it is essential to consider in more detail reciprocal interactions between cortical areas with laminar structures [611, 612, 787, 167, 943]. In the neocortex, gamma-band oscillations is prominent in superficial layers 2/3 , which is endowed with abundant horizontal connections . In contrast, deep layers 5/6 seem to have a propensity to display lower-frequency beta-band oscillations. In an experiment with monkeys performing an attention task, lamina-localized recording revealed gamma oscillations in superficial layers and beta oscillations in deep layers . In consonance with this observation is the finding from an in vitro somatosensory rat cortical slice preparation with application of kainate . When the superficial and deep layers were separated by a cut through layer 4, they were found to exhibit oscillations in the gamma versus beta frequency ranges, respectively  (see  for a detailed model with multiple cortical layers). Consider two reciprocally connected cortical areas, as schematically illustrated in Fig. 19. Feedforward projections from a ‘lower’ cortical area to a ‘higher’ area originate in the superficial layers, whereas the deep layers of the higher area mediate feedback projections to the lower area as well as to subcortical brain structures [352, 50, 253]. A recent model of the thalamocortical system also proposes that beta rhythm primarily emerges in the deep layers of the cortex . Thus, beta oscillations produced in the deep layers may be especially involved in long-distance signaling along feedback pathways. Top-down signals may encode expectation that guides learning; consistent with this idea was the finding that, in the hippocampus, beta oscillations were prominent especially in the early phase of learning in a novel environment .
Beta oscillations in top-down signaling are especially well documented in the rat olfactory system. Field oscillations occur in both gamma and beta frequency bands in the olfactory bulb [111, 91, 1003, 795, 648, 607] and the piriform cortex [111, 720, 775]. Simultaneous field potential recordings in behaving rats indicated that beta band oscillations originate in the cortex . This is reinforced by a study with urethane-anesthetized rat showing that beta, but not gamma, rhythm was abolished by lesion of the cortifugal pathway (the lateral olfactory tract), hence beta oscillations may be produced within the piriform cortex and (from the deep layers) projected back to the olfactory bulb . Recent two-photon guided microstimulation experiments  demonstrated that feedback projections from the piriform cortex target inhibitory granule cells in the olfactory bulb, with synapses that exhibited short-term facilitation when tested at ~ 20 Hz, suggesting a specific pathway for beta frequency top-down signaling in the olfactory system.
A prominent role of beta-band oscillations in top-down projections would explain why inter-areal coherence associated with selective attention [385, 834], working memory , guided search [726, 138, 139], object recognition , perception [327, 251] or sensori-motor integration [121, 118, 57, 1066] is especially conspicuous in the beta frequency range. It may also account for beta rhythm observed during attention in the basal ganglia  or in the thalamus , as a result of inputs from deep layers of cortical areas upstream. Interestingly, modeling studies showed that synchronization can tolerate longer synaptic delays for beta than gamma oscillation [282, 528, 83], suggesting that long-distance oscillatory synchronization may be more robustly realized at beta frequencies.
Theta band oscillations may also play a role in top-down processing [1017, 1018, 871]. Just as gamma rhythm is commonly associated with working memory and attention, so are field theta oscillations [489, 790, 569, 677, 981] and theta-band inter-areal synchronization . In anesthetized rats, theta frequency oscillations are especially prominent in deep layers of the neocortex . In a study with behaving cats performing a visual discrimination task (stimulus 1, go; stimulus 2, nogo) , it was found that cross-correlation of field potentials from the primary visual cortex and the association area 7 was enhanced on go stimulus trials, in the theta- but not gamma-band. This increased coherence is selective between the infragranular layer of area 7 and supragranular layer of V1 (with area 7 leading V1 by a time lag), suggesting a top-down influence. This and other studies ( reviewed in ) are consistent with the view that long-distance neural synchronization might primarily occur in slower (beta and theta) frequency bands.
If the top-down signaling carries beta-band rhythms, how does this fit with the idea of enhanced gamma-band oscillations during attention and working memory, as discussed above? Feedback projections from a cognitive-type area to a sensory area preferentially synapses onto neurons in the superficial layers 2/3, including pyramidal cells, perisoma-targeting and dendrite-targeting interneurons (Fig. 19) [51, 672]. Therefore, such top-down signals are well suited to modulate gamma oscillations, locally generated in the superficial layers in a context dependent manner.
Synchronization also involves interactions between the neocortex and the thalamus [601, 235, 391, 483, 451, 907, 1007]. In particular, the widespread cortical projections of matrix cells in the dorsal thalamic nuclei  and the intralaminar nonspecific thalamic nuclei  appear to be well suited for synchronizing disparate cortical areas. The relative importance of cortico-cortical versus cortico-thalamic connections remains to be elucidated in the future.
Data reviewed above begin to reveal specific roles of cortical oscillations in integrative brain processes and executive control. To demonstrate that these contributions are critical, one way would be to disrupt oscillations (e.g. by optogenetic methods) and assess the consequences in cognitive behaviors. Another direction is to examine how neural synchronization may be impaired in patients known to exhibit deficits in information integration, working memory and selective attention. Insights gained along this line would not only clarify whether, or in what specific ways, cortical oscillations are truly important for cognition, but also might help us understand the neural circuit basis of cognitive impairments associated with brain disorders. Summarized here briefly, this area of research is still new, but with potentially significant implications in Psychiatry and Neurology [986, 564, 428, 989].
Cognitive deficits are at the core of a wide range of mental pathologies in schizophrenia . Friston argued that key cognitive functions that are impaired in schizophrenia, such as distractibility of attention and loosening of association in thought, could be caused by abnormal functional connections in the brain . A number of theoretical, behavioral and fMRI studies reported impaired connectivity in the schizophrenic brain [438, 558, 439, 1059, 336, 212, 596, 1063]. In physiological studies, there have been an avalanche of recent publications reporting impaired beta-and gamma-band synchronization in schizophrenic patients in a variety of behavioral tasks [541, 570, 894, 895, 534, 926, 986, 926, 174, 587, 398, 295, 987, 55] (reviewed in ). Some of these studies are not free from confounds discussed in Section VI-A but, taken together, a consistent observation is that EEG fluctuations in the beta and gamma bands are abnormal in schizophrenia. For example, while subjects viewed a Gestalt stimulus, phase coherence between distributed cortical areas was reduced in schizophrenic subjects compared to control, in either gamma  or beta  frequency range. A few studies attempted to directly assess the correlation between synchrony and behavior. Spencer and collaborators  found that in response to a Gestalt stimulus, there was a brief (~ 100 ms) burst of gamma-band EEG power phase-locked to the reaction time in the occipital cortex of healthy subjects, but the peaked power was weaker and at lower frequencies in schizophrenic patients. In another study, the behavioral task had two conditions under which the demand for cognitive control was either high or low . It was found that gamma-band power was enhanced under the higher than the low demand condition, in healthy subjects but not in schizophrenic patients (who performed worse than the control group). In yet another experiment, schizophrenic patients did not perform as well as the control group in a perception task (detection of human faces in degraded pictures); at the same time, MEG over parietal cortical sites displayed sensory responses concentrated in the beta/gamma frequency range, that was smaller in schizophrenia than in healthy subjects [953, 989]. Recent studies found abnormal gamma-band EEG signals , and abnormal brain network connectivity as assessed by correlations of beta-band MEG signals recorded in the prefrontal and parietal cortices , during working memory in schizophrenia.
Inter-areal connections may well be aberrant in schizophrenia, but physiologically measured alternation in inter-areal coherence may also arise from impaired dynamics in circumscribed brain structures leading to abnormal brainwide neural signals. For example, a computational study suggests that the lower frequency of EEG oscillations seen in schizophrenic subjects  could be explained by some observed deficits in the inhibitory GABA system of schizophrenic patients in the prefrontal cortex. Specifically, decreased GABA transporter 1 (GAT1) and the 67 kiloDalton isoform of glutamate decarboxylase (GAD67, an enzyme that synthesizes GABA) [579, 580] may lead to a slower GABAA receptor mediated IPSCs, hence lower-frequency oscillations . Moreover, in a rat model of schizophrenia, a reduction of parvalbumin-expressing interneurons in the medial prefrontal cortex and ventral subiculum gave rise to a reduced gamma-band response to a conditioned tone during a latent inhibition paradigm and an impaired behavioral expression of latent inhibition .
On the other hand, schizophrenia is associated with NMDA hypofunction (especially in the prefrontal cortex) [535, 199], which may cause cognitive dysfunctions in multiple ways. First, NMDA receptors are important for synaptic plasticity hence abnormal NMDA signaling is detrimental to learning and memory . Second, working memory may be impaired because persistent activity cannot be stably maintained by fast AMPA receptor mediated excitation alone [1029, 182, 259, 1030, 939, 800, 1023]. Third, NMDA receptor mediated excitatory reverberation may be critical for slow time integration of information underlying decision making computations [1032, 1071, 1034]. Fourth, NMDA hypofunction leads to a selective reduction of the excitatory drive for a subpopulation of interneurons [389, 582, 1024, 72], one of the consequences could be abnormal inhibition-dependent synchronous oscillations [593, 818, 633]. The precise mechanism remains controversial, in part because in the adult rodent frontal cortex, the majority of parvalbumin-expressing fast spiking interneurons, which should play a critical role in gamma oscillations, are devoid of the NMDA receptors . Such changes could occur throughout the cortex, or more selectively in certain cortical regions like the PFC. Since strongly recurrent working memory circuits are particularly prone to fast oscillations, as we discussed above (Section VI-E), abnormal gamma or beta rhythm observed in schizophrenia could be a sign of an impaired working memory circuitry , in line with the well documented working memory and selective attention deficits associated with schizophrenia [363, 379, 52, 258]. In addition, cortical oscillations are strongly influenced by the activation of neuromodulators, and an early lesion study  showed that cortical gamma rhythm in the parietal and frontal cortices depended on dopamine signaling which, as is well known, is severely impaired in the PFC associated with schizophrenia [363, 265, 853].
In summary, extant empirical evidence points to two candidate explanations for reduced beta- and gamma-band synchronization associated with schizophrenia: (a) dysfunctional operations in the prefronto-parietal circuit of critical importance to working memory, attention and decision making, and (b) impairments in long-range connections throughout the brainweb. Obviously, these two are not mutually exclusive, and may both be affected by the disease. Future experiments designed to differentially elucidate these two scenarios would shed important insights onto the brain circuit basis of integration dysfunctions in schizophrenia.
Patients with autism spectrum disorders (ASD) have impaired sociability, communication deficits, and unusual patterns of restricted activities. Individuals with ASD commonly show ‘persistent preoccupation with parts of objects’; they are superior at certain tasks that require attention to local details, but poor at performing tasks based on global meaning. Therefore, a characteristic of ASD appears to be a relative inability to integrate pieces of information into coherent wholes [317, 416, 346, 417]. The ‘weak coherence account’ has recently been examined using fMRI and EEG/MEG techniques. fMRI experiments consistently reported a reduced functional connectivity in ASD subjects [158, 485, 529, 1010, 493]. Of particular interest were the observations of a reduced connectivity in the cortical language system during sentence comprehension , and in the prefronto-parietal system of ‘theory of mind’ when patients were asked to attribute a mental state to animated geometric figures with implied intentions (e.g. coaxing, tricking) . More recently, EEG/MEG studies, with much better time resolution (but poor spatial resolution) than fMRI, have been carried out to assess neural synchronization in ASD subjects [382, 120, 707, 1064, 78, 745, 685] (reviewed in [986, 813, 989]). The results so far are inconclusive, and reported abnormality of gamma-band EEG power in ASD patients was not correlated to behavioral difference with the healthy control group. Among possible reasons are that most published work looked at band-passed EEG powers in different locations rather than inter-areal coherence, and that in these studies subjects were either in the resting state or viewing visual stimuli, but not engaged in processes (such as mentalizing, or communicative intent) that are dysfunctional in ASD. Future studies will undoubtedly address these issues. It will also be interesting, in future work, to combine fMRI with simultaneous EEG/MEG recording in order to investigate whether and how abnormal neural synchrony in the brain system directly correlates with atypical functional connectivity measured by fMRI in ASD subjects.
Interestingly, whereas schizophrenia is associated with NMDA hypofunction, the autistic brain appears to exhibit NMDA hyperfunction. Studies using a rat model of autism (with prenatal exposure to valproic acid) suggest that ASD is associated with an abnormal increase in the NMDA receptor expression, leading to hyper-plasticity and hyperconnectivity between pyramidal cells in local cortical circuits [807, 808, 809, 646]. This would provide an explanation for dysfunctional brain connectivity. In addition, as discussed in Sections III and V (Fig. 14), a tightly maintained balance between synaptic excitation and inhibition is critical to normal neural synchronization. Hence, an abnormal excitation/inhibition ratio (in part due to excessive NMDA receptor expression) associated with heterogeneous subtypes of autism [825, 215] could give rise to aberrant neural synchronization (and potentially the resulting behavioral symptoms) associated with autism spectrum disorders. Therefore, in principle, the same molecular alternation (impaired NMDA signaling) can cause abnormal connectivity assessed by fMRI and abnormal synchrony measured by EEG/MEG, hence the two must be correlated; but they occur through separate processes (synaptic plasticity and ‘online’ excitation/inhibition dynamics, respectively).
Brain dysfunctional connectivity has drawn increasing attention not only in schizophrenia and autism research, but also in the fields of neurodegenerative diseases  and neurological disorders . This in part arose from a growing appreciation that inasmuch as cognitive functions are carried out through coordinated action of multiple interacting brain structures, cognitive deficits associated with brain disorders cannot be fully understood in terms of focal abnormalities confined to certain brain regions. To make progress, we need to know a lot more about the brain connectome [897, 586, 92, 860, 996]. What are the defining features of the brainwide architecture? Can it be described by the small-world network or other types of complex networks [436, 896, 56, 3, 491, 993]? Can we use graph theoretical analysis to assess circuit properties (such as the path length (a measure of effective communication), the clustering coefficient (the degree of clustering of connections), cost efficiency, hub structures (nodes with an exceptionally large number of connections and low clustering)), as a tool to quantify the network basis of mental illness? For example, the brains of both Alzheimer’s disease [901, 219] and schizophrenia [58, 829] show shorter path lengths (indicative of network randomization), but what are the differences in their connectivity properties that can account for the entirely different behavioral symptoms of these two brain diseases? Likewise, abnormal gamma and beta oscillations have been observed in patients with schizophrenia, ASD and other brain disorders, but are these deficits specific in each disorder to serve as a biomaker and phenotype for diagnostic purpose? Future research along these lines is needed to identify distinct types of network dysfunctional connectivity associated with different disorders.
Remarkable progress has been made over the last two decades in our understanding of the physiological basis of synchronous oscillations such as gamma and theta rhythms, which occur in the cerebral cortex of awake behaving mammals. We have learnt that there is a wide diversity of cellular and circuit mechanisms underlying the generation of such rhythms; at the same time some unifying themes and general principles have begun to emerge. Impact of single neuron properties on network oscillations can be systematically characterized according to whether they display type I or type II phase response dynamics, resonance, or pacemaker properties. Intrinsic gamma or theta (subthreshold or bursting) oscillations in individual neurons appear to critically depend on voltage-gated sodium and potassium channels. This is in contrast to, for example, spindle waves during quiet sleep or epileptic discharges where rhythmogenesis of pacemaker neurons relies on calcium channels [964, 908, 968, 236].
As a general rule, for cortical rhythms observed in awake states, circuit synchronization depends on synaptic inhibition, either through mutual inhibition in an interneuronal network, or feedback inhibition in a reciprocal loop between excitatory and inhibitory neural populations. Subclasses of GABAergic cells contribute differentially to different rhythms, and to the timing of pyramidal neuronal activity at different phases of an oscillation cycle. Cortical rhythms can arise from synchronization of coupled oscillators when noise is absent or small, which are well described in the framework of coupled oscillators. Alternatively, they can emerge from strong feedback synaptic interactions in a predominantly asynchronous circuit with highly stochastic spiking activity of individual cells, which are captured by the theory of sparsely synchronized oscillations. These two scenarios should be readily distinguishable when the level of population rhythmicity or coherence is quantitatively assessed at the same time as the variability of single neuron spike activity. Furthermore, importantly, they have contrasting predictions about the circuit mechanisms of different fast cortical rhythms. As we have seen, rhythms produced by an interneuronal mechanism in cortical circuits are ultrafast (> 100 Hz) when single neurons fire irregularly in a sparsely synchronized oscillation, but at ~ 40 Hz when single neurons behave like regular oscillators. With highly irregular firing of single neurons, the excitatory-inhibitory loop mechanism naturally gives rise to gamma-and beta-band oscillations. The propensity is especially high when recurrent connections are strong, a characteristic of ‘cognitive-type’ (working memory and decision making) cortical areas like the prefrontal or posterior parietal cortex. Therefore, gamma and beta oscillations are expected to be naturally enhanced in any behavioral task that engages top-down control signals from these areas. In a laminar cortical circuit, descending signals originate in the deep layers (which are prone to beta rhythm) of a ‘source area’, and project to the superficial layers (where gamma oscillations are prevalent) of a recipient area. This model remains to be tested experimentally, and computational implications of the proposed dynamic interplay between beta and gamma oscillations deserve to be explored in the future.
An increasing number of recent studies have documented long-distance coherence in specific cognitive processes. Under the premise that stochasticity predominates in the cortex, in the sense that single neuron firing is irregular and population activity is prevalently asynchronous, functional implications of weak synchrony embedded in highly stochastic neural activity patterns can be more appropriately appraised in the framework of sparsely synchronized oscillations rather than coupled regular oscillators. Two possible views are conceivable with regard to the role of weak synchrony in long-distance neuronal communication. In the first scenario, the bulk of (asynchronous) spike firing might be somehow blocked, e.g. due to a strict balance between excitation and inhibition . Consequently, as a ‘default mode’ inter-areal communication channel is closed in the absence of synchronization, and transient coherence provides a temporal window of opportunity for effective communication. In an alternative scenario, spike firing (synchronous or not) carries the main flow of information among cortical areas. Some types of computation and synaptic plasticity critically depend on correlations in fluctuating neural activity, and synchronous oscillations provide an appealing means for modulating spatiotemporal neural correlations. This modulation may be modest but important, in the same vein as gain modulation of sensory neurons by selective attention which is typically weak yet functionally significant (e.g. leading to our ability to filter out unattended sensory objects at the behavioral level). While neural firing rates and correlations may co-vary under some conditions [165, 166, 221], they may be independent from each other under other circumstances [114, 312]. If so, enhanced neuronal correlation by weak synchrony could be a viable mechanism to implement flexible and guided communications across the complex brainwide neural network in cognition.
Looking forward, we must be cognizant that weak oscillations are only partial clues of spatiotemporal spiking activity patterns in the cortex that are far from being well understood. The advance of techniques such as high-density multi-electrode recording [142, 560, 771] and two-photon imaging with bulk loading of calcium sensitive dyes [736, 335, 380] has opened up the possibility to monitor simultaneously many single cells; such data will give rise to insights into, and inspire novel mathematical models of, neural circuit dynamics. The challenge is not only technical but also conceptual. Cortical neural circuits are extraordinarily complex, and complex dynamical systems (such as turbulent fluids with a hierarchy of coherent structures embedded in stochastic spatiotemporal processes) cannot be described solely in terms of oscillations or even low-dimensional aperiodic (chaotic) dynamics . New developments lay ahead in our quest of understanding the nature of the coordinated and gated brain dynamic operations underlying flexible cognitive behavior.
I would like to thank Angel Alonso, Nicolas Brunel, Gyuri Buzsáki, Albert Compte, Robert Desimone, Bard Ermentrout, Walter Freeman, Caroline Geisler, Rodolfo Llinás, Eve Marder, David McCormick, John Rinzel, Mavi Sanchez-Vives, Roger Traub for many discussions over the years; Salvador Ardid, Jessica Cardin, Tatiana Engel, Asif Ghazanfar, Chris Harvey, Bilal Heider, Thomas Klausberger, May-Britt Moser and Jianyoung Wu for their kind help with figures. This work was supported by NIH grant 2-R01-MH062349 and the Kavli Foundation.