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Limit cycle oscillators that are coupled in a pulsatile manner are referred to as pulse coupled oscillators. In these oscillators, the interactions take the form of brief pulses such that the effect of one input dies out before the next is received. A phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike in an oscillatory neuron depending upon where in the cycle the input is applied. PRCs can be used to predict phase locking in networks of pulse coupled oscillators. In some studies of pulse coupled oscillators, a specific form is assumed for the interactions between oscillators, but a more general approach is to formulate the problem assuming a PRC that is generated using a perturbation that approximates the input received in the real biological network. In general, this approach requires that circuit architecture and a specific firing pattern be assumed. This allows the construction of discrete maps from one event to the next. The fixed points of these maps correspond to periodic firing modes and are easier to locate and analyze for stability compared to locating and analyzing periodic modes in the original network directly. Alternatively, maps based on the PRC have been constructed that do not presuppose a firing order. Specific circuits that have been analyzed under the assumption of pulsatile coupling include one to one lockings in a periodically forced oscillator or an oscillator forced at a fixed delay after a threshold event, two bidirectionally coupled oscillators with and without delays, a unidirectional N-ring of oscillators, and N all-to-all networks.

We review the principal assumptions underlying the application of phase-response curves (PRCs) to synchronization in neuronal networks. The PRC measures how much a given synaptic input perturbs spike timing in a neural oscillator. Among other applications, PRCs make explicit predictions about whether a given network of interconnected neurons will synchronize, as is often observed in cortical structures. Regarding the assumptions of the PRC theory, we conclude: (i) The assumption of noise-tolerant cellular oscillations at or near the network frequency holds in some but not all cases. (ii) Reduced models for PRC-based analysis can be formally related to more realistic models. (iii) Spike-rate adaptation limits PRC-based analysis but does not invalidate it. (iv) The dependence of PRCs on synaptic location emphasizes the importance of improving methods of synaptic stimulation. (v) New methods can distinguish between oscillations that derive from mutual connections and those arising from common drive. (vi) It is helpful to assume linear summation of effects of synaptic inputs; experiments with trains of inputs call this assumption into question. (vii) Relatively subtle changes in network structure can invalidate PRC-based predictions. (viii) Heterogeneity in the preferred frequencies of component neurons does not invalidate PRC analysis, but can annihilate synchronous activity.

The infinitesimal phase response curve (PRC) of a neural oscillator to a weak input is a powerful predictor of network dynamics; however, many networks have strong coupling and require direct measurement of the PRC for strong inputs under the assumption of pulsatile coupling. We incorporate measured noise levels in firing time maps constructed from PRCs to predict phase-locked modes of activity, phase difference, and locking strength in 78 heterogeneous hybrid networks of 2 neurons constructed using the dynamic clamp. We show that noise may either destroy or stabilize a phase-locked mode of activity.

Networks of model neurons were constructed and their activity was predicted using an iterated map based solely on the phase resetting curves (PRCs). The predictions were quite accurate provided that the resetting to simultaneous inputs was calculated using the sum of the simultaneously active conductances, obviating the need for weak coupling assumptions. Fully synchronous activity was observed only when the slope of the PRC at a phase of zero, corresponding to spike initiation, was positive. A novel stability criterion was developed and tested for all to all networks of identical, identically connected neurons. When the PRC generated using N-1 simultaneously active inputs becomes too steep, the fully synchronous mode loses stability in a network of N model neurons. Therefore, the stability of synchrony can be lost by increasing the slope of this PRC either by increasing the network size or the strength of the individual synapses. Existence and stability criteria were also developed and tested for the splay mode in which neurons fire sequentially. Finally, N/M synchronous sub-clusters of M neurons were predicted using the intersection of parameters that supported both between cluster splay and within cluster synchrony. Surprisingly, the splay mode between clusters could enforce synchrony on sub-clusters that were incapable of synchronizing themselves. These results can be used to gain insights into the activity of networks of biological neurons whose PRCs can be measured.

The ability of spiking neurons to synchronize their activity in a network depends on the response behavior of these neurons as quantified by the phase response curve (PRC) and on coupling properties. The PRC characterizes the effects of transient inputs on spike timing and can be measured experimentally. Here we use the adaptive exponential integrate-and-fire (aEIF) neuron model to determine how subthreshold and spike-triggered slow adaptation currents shape the PRC. Based on that, we predict how synchrony and phase locked states of coupled neurons change in presence of synaptic delays and unequal coupling strengths. We find that increased subthreshold adaptation currents cause a transition of the PRC from only phase advances to phase advances and delays in response to excitatory perturbations. Increased spike-triggered adaptation currents on the other hand predominantly skew the PRC to the right. Both adaptation induced changes of the PRC are modulated by spike frequency, being more prominent at lower frequencies. Applying phase reduction theory, we show that subthreshold adaptation stabilizes synchrony for pairs of coupled excitatory neurons, while spike-triggered adaptation causes locking with a small phase difference, as long as synaptic heterogeneities are negligible. For inhibitory pairs synchrony is stable and robust against conduction delays, and adaptation can mediate bistability of in-phase and anti-phase locking. We further demonstrate that stable synchrony and bistable in/anti-phase locking of pairs carry over to synchronization and clustering of larger networks. The effects of adaptation in aEIF neurons on PRCs and network dynamics qualitatively reflect those of biophysical adaptation currents in detailed Hodgkin-Huxley-based neurons, which underscores the utility of the aEIF model for investigating the dynamical behavior of networks. Our results suggest neuronal spike frequency adaptation as a mechanism synchronizing low frequency oscillations in local excitatory networks, but indicate that inhibition rather than excitation generates coherent rhythms at higher frequencies.

Author Summary

Synchronization of neuronal spiking in the brain is related to cognitive functions, such as perception, attention, and memory. It is therefore important to determine which properties of neurons influence their collective behavior in a network and to understand how. A prominent feature of many cortical neurons is spike frequency adaptation, which is caused by slow transmembrane currents. We investigated how these adaptation currents affect the synchronization tendency of coupled model neurons. Using the efficient adaptive exponential integrate-and-fire (aEIF) model and a biophysically detailed neuron model for validation, we found that increased adaptation currents promote synchronization of coupled excitatory neurons at lower spike frequencies, as long as the conduction delays between the neurons are negligible. Inhibitory neurons on the other hand synchronize in presence of conduction delays, with or without adaptation currents. Our results emphasize the utility of the aEIF model for computational studies of neuronal network dynamics. We conclude that adaptation currents provide a mechanism to generate low frequency oscillations in local populations of excitatory neurons, while faster rhythms seem to be caused by inhibition rather than excitation.

Gamma oscillations can synchronize with near zero phase lag over multiple cortical regions and between hemispheres, and between two distal sites in hippocampal slices. How synchronization can take place over long distances in a stable manner is considered an open question. The phase resetting curve (PRC) keeps track of how much an input advances or delays the next spike, depending upon where in the cycle it is received. We use PRCs under the assumption of pulsatile coupling to derive existence and stability criteria for 1:1 phase-locking that arises via bidirectional pulse coupling of two limit cycle oscillators with a conduction delay of any duration for any 1:1 firing pattern. The coupling can be strong as long as the effect of one input dissipates before the next input is received. We show the form that the generic synchronous and anti-phase solutions take in a system of two identical, identically pulse-coupled oscillators with identical delays. The stability criterion has a simple form that depends only on the slopes of the PRCs at the phases at which inputs are received and on the number of cycles required to complete the delayed feedback loop. The number of cycles required to complete the delayed feedback loop depends upon both the value of the delay and the firing pattern. We successfully tested the predictions of our methods on networks of model neurons. The criteria can easily be extended to include the effect of an input on the cycle after the one in which it is received.

Our goal is to understand how nearly synchronous modes arise in heterogenous networks of neurons. In heterogenous networks, instead of exact synchrony, nearly synchronous modes arise, which include both 1:1 and 2:2 phase-locked modes. Existence and stability criteria for 2:2 phase-locked modes in reciprocally coupled two neuron circuits were derived based on the open loop phase resetting curve (PRC) without the assumption of weak coupling. The PRC for each component neuron was generated using the change in synaptic conductance produced by a presynaptic action potential as the perturbation. Separate derivations were required for modes in which the firing order is preserved and for those in which it alternates. Networks composed of two model neurons coupled by reciprocal inhibition were examined to test the predictions. The parameter regimes in which both types of nearly synchronous modes are exhibited were accurately predicted both qualitatively and quantitatively provided that the synaptic time constant is short with respect to the period and that the effect of second order resetting is considered. In contrast, PRC methods based on weak coupling could not predict 2:2 modes and did not predict the 1:1 modes with the level of accuracy achieved by the strong coupling methods. The strong coupling prediction methods provide insight into what manipulations promote near-synchrony in a two neuron network and may also have predictive value for larger networks, which can also manifest changes in firing order. We also identify a novel route by which synchrony is lost in mildly heterogenous networks.

Central pattern generators (CPGs) frequently include bursting neurons that serve as pacemakers for rhythm generation. Phase resetting curves (PRCs) can provide insight into mechanisms underlying phase locking in such circuits. PRCs were constructed for a pacemaker bursting complex in the pyloric circuit in the stomatogastric ganglion of the lobster and crab. This complex is comprised of the Anterior Burster (AB) neuron and two Pyloric Dilator (PD) neurons that are all electrically coupled. Artificial excitatory synaptic conductance pulses of different strengths and durations were injected into one of the AB or PD somata using the Dynamic Clamp. Previously, we characterized the inhibitory PRCs by assuming a single slow process that enabled synaptic inputs to trigger switches between an up state in which spiking occurs and a down state in which it does not. Excitation produced five different PRC shapes, which could not be explained with such a simple model. A separate dendritic compartment was required to separate the mechanism that generates the up and down phases of the bursting envelope (1) from synaptic inputs applied at the soma, (2) from axonal spike generation and (3) from a slow process with a slower time scale than burst generation. This study reveals that due to the nonlinear properties and compartmentalization of ionic channels, the response to excitation is more complex than inhibition.

The phase-resetting curve (PRC) of a neural oscillator describes the effect of a perturbation on its periodic motion and is therefore useful to study how the neuron responds to stimuli and whether it phase locks to other neurons in a network. Combining theory, computer simulations and electrophysiological experiments we present a simple method for estimating the PRC of real neurons. This allows us to simplify the complex dynamics of a single neuron to a phase model. We also illustrate how to infer the existence of coherent network activity from the estimated PRC.

How stable synchrony in neuronal networks is sustained in the presence of conduction delays is an open question. The Dynamic Clamp was used to measure phase resetting curves (PRCs) for entorhinal cortical cells, and then to construct networks of two such neurons. PRCs were in general Type I (all advances or all delays) or weakly type II with a small region at early phases with the opposite type of resetting. We used previously developed theoretical methods based on PRCs under the assumption of pulsatile coupling to predict the delays that synchronize these hybrid circuits. For excitatory coupling, synchrony was predicted and observed only with no delay and for delays greater than half a network period that cause each neuron to receive an input late in its firing cycle and almost immediately fire an action potential. Synchronization for these long delays was surprisingly tight and robust to the noise and heterogeneity inherent in a biological system. In contrast to excitatory coupling, inhibitory coupling led to antiphase for no delay, very short delays and delays close to a network period, but to near-synchrony for a wide range of relatively short delays. PRC-based methods show that conduction delays can stabilize synchrony in several ways, including neutralizing a discontinuity introduced by strong inhibition, favoring synchrony in the case of noisy bistability, and avoiding an initial destabilizing region of a weakly type II PRC. PRCs can identify optimal conduction delays favoring synchronization at a given frequency, and also predict robustness to noise and heterogeneity.

Author Summary

Individual oscillators, such as pendulum-based clocks and fireflies, can spontaneously organize into a coherent, synchronized entity with a common frequency. Neurons can oscillate under some circumstances, and can synchronize their firing both within and across brain regions. Synchronized assemblies of neurons are thought to underlie cognitive functions such as recognition, recall, perception and attention. Pathological synchrony can lead to epilepsy, tremor and other dynamical diseases, and synchronization is altered in most mental disorders. Biological neurons synchronize despite conduction delays, heterogeneous circuit composition, and noise. In biological experiments, we built simple networks in which two living neurons could interact via a computer in real time. The computer precisely controlled the nature of the connectivity and the length of the communication delays. We characterized the synchronization tendencies of individual, isolated oscillators by measuring how much a single input delivered by the computer transiently shortened or lengthened the cycle period of the oscillation. We then used this information to correctly predict the strong dependence of the coordination pattern of the firing of the component neurons on the length of the communication delays. Upon this foundation, we can begin to build a theory of the basic principles of synchronization in more complex brain circuits.

Cerebellar Purkinje cells display complex intrinsic dynamics. They fire spontaneously, exhibit bistability, and via mutual network interactions are involved in the generation of high frequency oscillations and travelling waves of activity. To probe the dynamical properties of Purkinje cells we measured their phase response curves (PRCs). PRCs quantify the change in spike phase caused by a stimulus as a function of its temporal position within the interspike interval, and are widely used to predict neuronal responses to more complex stimulus patterns. Significant variability in the interspike interval during spontaneous firing can lead to PRCs with a low signal-to-noise ratio, requiring averaging over thousands of trials. We show using electrophysiological experiments and simulations that the PRC calculated in the traditional way by sampling the interspike interval with brief current pulses is biased. We introduce a corrected approach for calculating PRCs which eliminates this bias. Using our new approach, we show that Purkinje cell PRCs change qualitatively depending on the firing frequency of the cell. At high firing rates, Purkinje cells exhibit single-peaked, or monophasic PRCs. Surprisingly, at low firing rates, Purkinje cell PRCs are largely independent of phase, resembling PRCs of ideal non-leaky integrate-and-fire neurons. These results indicate that Purkinje cells can act as perfect integrators at low firing rates, and that the integration mode of Purkinje cells depends on their firing rate.

Author Summary

By observing how brief current pulses injected at different times between spikes change the phase of spiking of a neuron (and thus obtaining the so-called phase response curve), it should be possible to predict a full spike train in response to more complex stimulation patterns. When we applied this traditional protocol to obtain phase response curves in cerebellar Purkinje cells in the presence of noise, we observed a triangular region devoid of data points near the end of the spiking cycle. This “Bermuda Triangle” revealed a flaw in the classical method for constructing phase response curves. We developed a new approach to eliminate this flaw and used it to construct phase response curves of Purkinje cells over a range of spiking rates. Surprisingly, at low firing rates, phase changes were independent of the phase of the injected current pulses, implying that the Purkinje cell is a perfect integrator under these conditions. This mechanism has not yet been described in other cell types and may be crucial for the information processing capabilities of these neurons.

New tools for analysis of oscillatory networks using phase response theory (PRT) under the assumption of pulsatile coupling have been developed steadily since the 1980s, but none have yet allowed for analysis of mixed systems containing nonoscillatory elements. This caveat has excluded the application of PRT to most real systems, which are often mixed. We show that a recently developed tool, the functional phase resetting curve (fPRC), provides a serendipitous benefit: it allows incorporation of nonoscillatory elements into systems of oscillators where PRT can be applied. We validate this method in a model system of neural oscillators and a biological system, the pyloric network of crustacean decapods.

The response of an oscillator to perturbations is described by its phase-response curve (PRC), which is related to the type of bifurcation leading from rest to tonic spiking. In a recent experimental study, we have shown that the type of PRC in cortical pyramidal neurons can be switched by cholinergic neuromodulation from type II (biphasic) to type I (monophasic). We explored how intrinsic mechanisms affected by acetylcholine influence the PRC using three different types of neuronal models: a theta neuron, single-compartment neurons and a multi-compartment neuron. In all of these models a decrease in the amount of a spike-frequency adaptation current was a necessary and sufficient condition for the shape of the PRC to change from biphasic (type II) to purely positive (type I).

Synchronization of globus pallidus (GP) neurons and cortically-entrained oscillations between GP and other basal ganglia nuclei are key features of the pathophysiology of Parkinson's disease. Phase response curves (PRCs), which tabulate the effects of phasic inputs within a neuron's spike cycle on output spike timing, are efficient tools for predicting the emergence of synchronization in neuronal networks and entrainment to periodic input. In this study we apply physiologically realistic synaptic conductance inputs to a full morphological GP neuron model to determine the phase response properties of the soma and different regions of the dendritic tree. We find that perisomatic excitatory inputs delivered throughout the inter-spike interval advance the phase of the spontaneous spike cycle yielding a type I PRC. In contrast, we demonstrate that distal dendritic excitatory inputs can either delay or advance the next spike depending on whether they occur early or late in the spike cycle. We find this latter pattern of responses, summarized by a biphasic (type II) PRC, was a consequence of dendritic activation of the small conductance calcium-activated potassium current, SK. We also evaluate the spike-frequency dependence of somatic and dendritic PRC shapes, and we demonstrate the robustness of our results to variations of conductance densities, distributions, and kinetic parameters. We conclude that the distal dendrite of GP neurons embodies a distinct dynamical subsystem that could promote synchronization of pallidal networks to excitatory inputs. These results highlight the need to consider different effects of perisomatic and dendritic inputs in the control of network behavior.

Spatiotemporal pattern formation in neuronal networks depends on the interplay between cellular and network synchronization properties. The neuronal phase response curve (PRC) is an experimentally obtainable measure that characterizes the cellular response to small perturbations, and can serve as an indicator of cellular propensity for synchronization. Two broad classes of PRCs have been identified for neurons: Type I, in which small excitatory perturbations induce only advances in firing, and Type II, in which small excitatory perturbations can induce both advances and delays in firing. Interestingly, neuronal PRCs are usually attenuated with increased spiking frequency, and Type II PRCs typically exhibit a greater attenuation of the phase delay region than of the phase advance region. We found that this phenomenon arises from an interplay between the time constants of active ionic currents and the interspike interval. As a result, excitatory networks consisting of neurons with Type I PRCs responded very differently to frequency modulation compared to excitatory networks composed of neurons with Type II PRCs. Specifically, increased frequency induced a sharp decrease in synchrony of networks of Type II neurons, while frequency increases only minimally affected synchrony in networks of Type I neurons. These results are demonstrated in networks in which both types of neurons were modeled generically with the Morris-Lecar model, as well as in networks consisting of Hodgkin-Huxley-based model cortical pyramidal cells in which simulated effects of acetylcholine changed PRC type. These results are robust to different network structures, synaptic strengths and modes of driving neuronal activity, and they indicate that Type I and Type II excitatory networks may display two distinct modes of processing information.

Author Summary

Synchronization of the firing of neurons in the brain is related to many cognitive functions, such as recognizing faces, discriminating odors, and coordinating movement. It is therefore important to understand what properties of neuronal networks promote synchrony of neural firing. One measure that is often used to determine the contribution of individual neurons to network synchrony is called the phase response curve (PRC). PRCs describe how the timing of neuronal firing changes depending on when input, such as a synaptic signal, is received by the neuron. A characteristic of PRCs that has previously not been well understood is that they change dramatically as the neuron's firing frequency is modulated. This effect carries potential significance, since cognitive functions are often associated with specific frequencies of network activity in the brain. We showed computationally that the frequency dependence of PRCs can be explained by the relative timing of ionic membrane currents with respect to the time between spike firings. Our simulations also showed that the frequency dependence of neuronal PRCs leads to frequency-dependent changes in network synchronization that can be different for different neuron types. These results further our understanding of how synchronization is generated in the brain to support various cognitive functions.

Phase response curves (PRCs) have been widely used to study synchronization in neural circuits comprised of pacemaking neurons. They describe how the timing of the next spike in a given spontaneously firing neuron is affected by the phase at which an input from another neuron is received. Here we study two reciprocally coupled clusters of pulse coupled oscillatory neurons. The neurons within each cluster are presumed to be identical and identically pulse coupled, but not necessarily identical to those in the other cluster. We investigate a two cluster solution in which all oscillators are synchronized within each cluster, but in which the two clusters are phase locked at nonzero phase with each other. Intuitively, one might expect this solution to be stable only when synchrony within each isolated cluster is stable, but this is not the case. We prove rigorously the stability of the two cluster solution and show how reciprocal coupling can stabilize synchrony within clusters that cannot synchronize in isolation. These stability results for the two cluster solution suggest a mechanism by which reciprocal coupling between brain regions can induce local synchronization via the network feedback loop.

Spike generation in cortical neurons depends on the interplay between diverse intrinsic conductances. The phase response curve (PRC) is a measure of the spike time shift caused by perturbations of the membrane potential as a function of the phase of the spike cycle of a neuron. Near the rheobase, purely positive (type I) phase-response curves are associated with an onset of repetitive firing through a saddle-node bifurcation, whereas biphasic (type II) phase-response curves point towards a transition based on a Hopf-Andronov bifurcation. In recordings from layer 2/3 pyramidal neurons in cortical slices, cholinergic action, consistent with down-regulation of slow voltage-dependent potassium currents such as the M-current, switched the PRC from type II to type I. This is the first report showing that cholinergic neuromodulation may cause a qualitative switch in the PRCs type implying a change in the fundamental dynamical mechanism of spike generation.

We demonstrate that two key theoretical objects used widely in Computational Neuroscience, the phase-resetting curve (PRC) from dynamics and the spike triggered average (STA) from statistical analysis, are closely related when neurons fire in a nearly regular manner and the stimulus is sufficiently small. We prove that the STA due to injected noisy current is proportional to the derivative of the PRC. We compare these analytic results to numerical calculations for the Hodgkin-Huxley neuron and we apply the method to neurons in the olfactory bulb of mice. This observation allows us to relate the stimulus-response properties of a neuron to its dynamics, bridging the gap between dynamical and information theoretic approaches to understanding brain computations and facilitating the interpretation of changes in channels and other cellular properties as influencing the representation of stimuli.

Fast-spiking (FS) cells in the neocortex are interconnected both by inhibitory chemical synapses and by electrical synapses, or gap-junctions. Synchronized firing of FS neurons is important in the generation of gamma oscillations, at frequencies between 30 and 80 Hz. To understand how these synaptic interactions control synchronization, artificial synaptic conductances were injected in FS cells, and the synaptic phase-resetting function (SPRF), describing how the compound synaptic input perturbs the phase of gamma-frequency spiking as a function of the phase at which it is applied, was measured. GABAergic and gap junctional conductances made distinct contributions to the SPRF, which had a surprisingly simple piecewise linear form, with a sharp midcycle break between phase delay and advance. Analysis of the SPRF showed how the intrinsic biophysical properties of FS neurons and their interconnections allow entrainment of firing over a wide gamma frequency band, whose upper and lower frequency limits are controlled by electrical synapses and GABAergic inhibition respectively.

Author Summary

Oscillations of the electrical field in the brain at 30–80 Hz (gamma oscillations) reflect coordinated firing of neurons during cognitive, sensory, and motor activity, and are thought to be a key phenomenon in the organization of neural processing in the cortex. Synchronous firing of a particular type of neuron, the inhibitory fast-spiking (FS) cell, imposes the gamma rhythm on other cells in the network. FS cells are highly interconnected by both gap junctions and chemical inhibition. In this study, we probed FS cells with a synthetic conductance stimulus which mimics the electrical effect of these complex connections in a controlled way, and directly measured how the timing of their firing should be affected by nearby FS neighbours. We were able to fit a mathematically simple but accurate model to these measurements, the “synaptic phase-resetting function”, which predicts how FS neurons synchronize at different frequencies, noise levels, and synaptic connection strengths. This model gives us deeper insight into how the FS cells synchronize so effectively at gamma oscillations, and will be a building-block in large-scale simulations of the FS cell network aimed at understanding the onset and stability of patterns of gamma oscillation in the cortex.

We consider and analyze the influence of spike-timing dependent plasticity (STDP) on homeostatic states in synaptically coupled neuronal oscillators. In contrast to conventional models of STDP in which spike-timing affects weights of synaptic connections, we consider a model of STDP in which the time lags between pre- and/or post-synaptic spikes change internal state of pre- and/or post-synaptic neurons respectively. The analysis reveals that STDP processes of this type, modeled by a single ordinary differential equation, may ensure efficient, yet coarse, phase-locking of spikes in the system to a given reference phase. Precision of the phase locking, i.e. the amplitude of relative phase deviations from the reference, depends on the values of natural frequencies of oscillators and, additionally, on parameters of the STDP law. These deviations can be optimized by appropriate tuning of gains (i.e. sensitivity to spike-timing mismatches) of the STDP mechanism. However, as we demonstrate, such deviations can not be made arbitrarily small neither by mere tuning of STDP gains nor by adjusting synaptic weights. Thus if accurate phase-locking in the system is required then an additional tuning mechanism is generally needed. We found that adding a very simple adaptation dynamics in the form of slow fluctuations of the base line in the STDP mechanism enables accurate phase tuning in the system with arbitrary high precision. Adaptation operating at a slow time scale may be associated with extracellular matter such as matrix and glia. Thus the findings may suggest a possible role of the latter in regulating synaptic transmission in neuronal circuits.

A temperature independent period and temperature entrainment are two defining features of circadian oscillators. A default model of distributed temperature compensation satisfies these basic facts yet is not easily reconciled with other properties of circadian clocks, such as many mutants with altered but temperature compensated periods. The default model also suggests that the shape of the circadian limit cycle and the associated phase response curves (PRC) will vary since the average concentrations of clock proteins change with temperature. We propose an alternative class of models where the twin properties of a fixed period and entrainment are structural and arise from an underlying adaptive system that buffers temperature changes. These models are distinguished by a PRC whose shape is temperature independent and orbits whose extrema are temperature independent. They are readily evolved by local, hill climbing, optimization of gene networks for a common quality measure of biological clocks, phase anticipation. Interestingly a standard realization of the Goodwin model for temperature compensation displays properties of adaptive rather than distributed temperature compensation.

Author Summary

Circadian clocks are biological oscillators which evolved to couple the internal rhythm of animals, plants and even some bacteria to the alternation of light and day. Circadian oscillators are temperature compensated, i.e. they keep a 24-h period irrespective of the temperature of the organism. This is surprising, since many biochemical parameters, including average concentration of clock proteins, vary with temperature. From dynamical system theory, we therefore expect changes in both period and relative lengths of features in the phase response curve which are not seen. We couple mathematical modelling and computational evolution of gene networks to formulate a novel explanation for temperature compensation that accords better with experimental facts than alternatives. Our model has deep mathematical connections with the process of biochemical adaptation, by which cells respond to temporal gradients of signals rather than their absolute value.

For the purpose of elucidating the neural coding process based on the neural excitability mechanism, researchers have recently investigated the relationship between neural dynamics and the spike triggered stimulus ensemble (STE). Ermentrout et al. analytically derived the relational equation between the phase response curve (PRC) and the spike triggered average (STA). The STA is the first cumulant of the STE. However, in order to understand the neural function as the encoder more explicitly, it is necessary to elucidate the relationship between the PRC and higher-order cumulants of the STE. In this paper, we give a general formulation to relate the PRC and the nth moment of the STE. By using this formulation, we derive a relational equation between the PRC and the spike triggered covariance (STC), which is the covariance of the STE. We show the effectiveness of the relational equation through numerical simulations and use the equation to identify the feature space of the rat hippocampal CA1 pyramidal neurons from their PRCs. Our result suggests that the hippocampal CA1 pyramidal neurons oscillating in the theta frequency range are commonly sensitive to inputs composed of theta and gamma frequency components.

A fundamental property of circadian clocks is that they entrain to environmental cues. The circadian genes, Period1 and Period2, are involved in entrainment of the mammalian circadian system. To investigate the roles of the Period genes in photic entrainment, we constructed phase response curves (PRC) to light pulses for C57BL/6J wild-type, Per1−/−, Per2−/−, and Per3−/− mice and tested whether the PRCs accurately predict entrainment to non-24 light-dark cycles (T-cycles) and constant light (LL). The PRCs of wild-type and Per3−/− mice are similar in shape and amplitude and have relatively large delay zones and small advance zones, resulting in successful entrainment to T26, but not T21, with similar phase angles. Per1−/− mice have a high-amplitude PRC, resulting in entrainment to a broad range of T-cycles. Per2−/− mice also entrain to a wide range of T-cycles because the advance portion of their PRC is larger than wild-types. Period aftereffects following entrainment to T-cycles were similar among all genotypes. We found that the ratio of the advance portion to the delay portion of the PRC accurately predicts the lengthening of the period of the activity rhythm in LL. Wild-type, Per1−/−, and Per3−/− mice had larger delay zones than advance zones and lengthened (>24hrs) periods in LL, while Per2−/− mice had delay and advance zones that were equal in size and no period lengthening in LL. Together, these results demonstrate that PRCs are powerful tools for predicting and understanding photic entrainment of circadian mutant mice.

Circadian rhythms in physiology and behavior are temporally synchronized to the day:night cycle through the action of light on the circadian clock. In mammals, transduction of the photic signal reaching the circadian oscillator in the suprachiasmatic nucleus (SCN) occurs through the release of glutamate and pituitary adenylate cyclase-activating peptide (PACAP). Our study aimed at clarifying the role played by PACAP in photic resetting and entrainment. We investigated the circadian response to light of PACAP-null mice lacking the 5th exon of the PACAP coding sequence. Specifically, we examined free-running rhythms, entrainment to 12-h light:12-h dark (LD) cycles, the phase response curve (PRC) to single light pulses, entrainment to a 23-h T-cycle, re-entrainment to 6-h phase shifts in LD cycles, and light-induced c-Fos expression. PACAP-null and wild-type mice show similar free-running periods and similar entrainment to 12:12 LD cycles. However, the PRC of PACAP-null mice lacks a phase-advance portion. Surprisingly, despite the absence of phase advance to single light pulses, PACAP-null mice are able to entrain to a 23-h T-cycle, but with a significantly longer phase angle of entrainment than wild types. In addition, PACAP-null mice re-entrain more slowly to a 6-h phase advance of the LD cycle. Nevertheless, induction of c-Fos by light in late night is normal. In all experiments, PACAP-null mice show specific behavioral impairments in response to phase-advancing photic stimuli. These results suggest that PACAP is required for the normal integration of the phase-advancing light signal by the SCN.

Mice lacking CLOCK protein have a relatively subtle circadian phenotype, including a slightly shorter period in constant darkness, differences in phase resetting after 4-hr light pulses in the early and late night, and a variably advanced phase angle of entrainment in a light-dark (LD) cycle (DeBruyne et al., Neuron 50:465–477, 2006). The present series of experiments was conducted to more fully characterize the circadian phenotype of Clock−/− mice under various lighting conditions. A phase-response curve (PRC) to 4-hour light pulses in free-running mice was conducted; the results confirm that Clock−/− mice exhibit very large phase advances after 4 hrs light pulses in the late subjective night, but have relatively normal responses to light at other phases. The abnormal shape of the PRC to light may explain the tendency of CLOCK-deficient mice to begin activity before lights-out when housed in a 12 hrs light: 12 hrs dark lighting schedule. To assess this relationship further, Clock−/− and wild-type control mice were entrained to skeleton lighting cycles (1L:23D, and 1L:10D:1L:12D). Comparing entrainment under the two types of skeleton photoperiods revealed that exposure to 1 hr light in the morning leads to a phase advance of activity onset (expressed the following afternoon) in Clock−/− mice, but not in the controls. Constant light typically causes an intensity-dependent increase in circadian period in mice, but this did not occur in CLOCK-deficient mice. The failure of Clock−/− mice to respond to the period-lengthening effect of constant light likely results from the increased functional impact of light falling in the phase advance zone of the PRC. Collectively, these experiments reveal that alterations in the response of CLOCK-deficient mice to light in several paradigms are likely due to an imbalance in the shape of the PRC to light.