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1.  Effect of phase response curve skew on synchronization with and without conduction delays 
A central problem in cortical processing including sensory binding and attentional gating is how neurons can synchronize their responses with zero or near-zero time lag. For a spontaneously firing neuron, an input from another neuron can delay or advance the next spike by different amounts depending upon the timing of the input relative to the previous spike. This information constitutes the phase response curve (PRC). We present a simple graphical method for determining the effect of PRC shape on synchronization tendencies and illustrate it using type 1 PRCs, which consist entirely of advances (delays) in response to excitation (inhibition). We obtained the following generic solutions for type 1 PRCs, which include the pulse-coupled leaky integrate and fire model. For pairs with mutual excitation, exact synchrony can be stable for strong coupling because of the stabilizing effect of the causal limit region of the PRC in which an input triggers a spike immediately upon arrival. However, synchrony is unstable for short delays, because delayed inputs arrive during a refractory period and cannot trigger an immediate spike. Right skew destabilizes antiphase and enables modes with time lags that grow as the conduction delay is increased. Therefore, right skew favors near synchrony at short conduction delays and a gradual transition between synchrony and antiphase for pairs coupled by mutual excitation. For pairs with mutual inhibition, zero time lag synchrony is stable for conduction delays ranging from zero to a substantial fraction of the period for pairs. However, for right skew there is a preferred antiphase mode at short delays. In contrast to mutual excitation, left skew destabilizes antiphase for mutual inhibition so that synchrony dominates at short delays as well. These pairwise synchronization tendencies constrain the synchronization properties of neurons embedded in larger networks.
PMCID: PMC3858834  PMID: 24376399
synchrony; synchronization; pulsatile coupling; phase locking; phase resetting
2.  Effects of conduction delays on the existence and stability of one to one phase locking between two pulse-coupled oscillators 
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.
PMCID: PMC3130804  PMID: 21344300
3.  Responses of a bursting pacemaker to excitation reveal spatial segregation between bursting and spiking mechanisms 
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.
PMCID: PMC3160527  PMID: 21360137
6.  Implications of Cellular Models of Dopamine Neurons for Schizophrenia 
Midbrain dopamine neurons are pacemakers in vitro, but in vivo they fire less regularly and occasionally in bursts that can lead to a temporary cessation in firing produced by depolarization block. The therapeutic efficacy of antipsychotic drugs used to treat the positive symptoms of schizophrenia has been attributed to their ability to induce depolarization block within a subpopulation dopamine neurons. We summarize the results of experiments characterizing the physiological mechanisms underlying the ability of these neurons to enter depolarization block in vitro, and our computational simulations of those experiments. We suggest that the inactivation of voltage-dependent Na+ channels, and in particular the slower component of this inactivation, is critical in controlling entry into depolarization block. In addition, an ether-a-go-related gene (ERG) K+ current also appears to be involved by delaying entry into and speeding recovery from depolarization block. Since many antipsychotic drugs share the ability to block this current, ERG channels may contribute to the therapeutic effects of these drugs.
PMCID: PMC4351765  PMID: 24560140
bursting; depolarization block; antipsychotic drugs; substantia nigra; ventral tegmental area; pacemaker; ether-a-go-go-related gene (ERG) potassium current; Kv11
7.  Effect of Heterogeneity and Noise on Cross Frequency Phase-Phase and Phase-Amplitude Coupling 
Network (Bristol, England)  2014;25(0):38-62.
Cross-frequency coupling is hypothesized to play a functional role in neural computation. We apply phase resetting theory to two types of cross-frequency coupling that can occur when a slower oscillator periodically forces one or more oscillators: phase-phase coupling, in which the two oscillations are phase-locked, and phase-amplitude coupling, in which the amplitude of the driven oscillation is modulated. Our first result is that the shape of the phase resetting curve predicts the tightness of locking to a pulsatile forcing periodic input at any ratio of forced to intrinsic period; the tightness of the locking decreases as the ratio increases. Theoretical expressions were obtained for the probability density of the phases for a population of heterogeneous oscillators or a noisy single oscillator. Results were confirmed using two types of simulated networks and experiments on hippocampal CA1 neurons. Theoretical expressions were also obtained and confirmed for the probability density of N spike times within a single cycle of low frequency forcing. The second result is a suggested mechanism for phase-amplitude coupling in which progressive desynchronization leads to decreasing amplitude during a low frequency forcing cycle. Network simulations confirmed the theoretical viability of this mechanism, and that it generalizes to more diffuse input.
PMCID: PMC3972019  PMID: 24571097
8.  A Mathematical Model of a Midbrain Dopamine Neuron Identifies Two Slow Variables Likely Responsible for Bursts Evoked by SK Channel Antagonists and Terminated by Depolarization Block 
Midbrain dopamine neurons exhibit a novel type of bursting that we call “inverted square wave bursting” when exposed to Ca2+-activated small conductance (SK) K+ channel blockers in vitro. This type of bursting has three phases: hyperpolarized silence, spiking, and depolarization block. We find that two slow variables are required for this type of bursting, and we show that the three-dimensional bifurcation diagram for inverted square wave bursting is a folded surface with upper (depolarized) and lower (hyperpolarized) branches. The activation of the L-type Ca2+ channel largely supports the separation between these branches. Spiking is initiated at a saddle node on an invariant circle bifurcation at the folded edge of the lower branch and the trajectory spirals around the unstable fixed points on the upper branch. Spiking is terminated at a supercritical Hopf bifurcation, but the trajectory remains on the upper branch until it hits a saddle node on the upper folded edge and drops to the lower branch. The two slow variables contribute as follows. A second, slow component of sodium channel inactivation is largely responsible for the initiation and termination of spiking. The slow activation of the ether-a-go-go-related (ERG) K+ current is largely responsible for termination of the depolarized plateau. The mechanisms and slow processes identified herein may contribute to bursting as well as entry into and recovery from the depolarization block to different degrees in different subpopulations of dopamine neurons in vivo.
PMCID: PMC4385104  PMID: 25852980
10.  Functional characterization of ether-a-go-go-related gene potassium channels in midbrain dopamine neurons: implications for a role in depolarization block 
The European journal of neuroscience  2012;36(7):2906-2916.
Bursting activity by midbrain dopamine neurons reflects the complex interplay between their intrinsic pacemaker activity and synaptic inputs. Although the precise mechanism responsible for the generation and modulation of bursting in vivo has yet to be established, several ion channels have been implicated in the process. Previous studies with nonselective blockers suggested that ether-a-go-go-related gene (ERG) K+ channels are functionally significant. Here, electrophysiology with selective chemical and peptide ERG channel blockers (E-4031 and rBeKm-1) and computational methods were used to define the contribution made by ERG channels to the firing properties of midbrain dopamine neurons in vivo and in vitro. Selective ERG channel blockade increased the frequency of spontaneous activity as well as the response to depolarizing current pulses without altering spike frequency adaptation. ERG channel block also accelerated entry into depolarization inactivation during bursts elicited by virtual NMDA receptors generated with the dynamic clamp, and significantly prolonged the duration of the sustained depolarization inactivation that followed pharmacologically evoked bursts. In vivo, somatic ERG blockade was associated with an increase in bursting activity attributed to a reduction in doublet firing. Taken together, these results show that dopamine neuron ERG K+ channels play a prominent role in limiting excitability and in minimizing depolarization inactivation. As the therapeutic actions of antipsychotic drugs are associated with depolarization inactivation of dopamine neurons and blockade of cardiac ERG channels is a prominent side effect of these drugs, ERG channels in the central nervous system may represent a novel target for antipsychotic drug development.
PMCID: PMC4042402  PMID: 22780096
KV11; hERG; KCNH2; bursting; antipsychotic drugs
11.  Slow Noise in the Period of a Biological Oscillator Underlies Gradual Trends and Abrupt Transitions in Phasic Relationships in Hybrid Neural Networks 
PLoS Computational Biology  2014;10(5):e1003622.
In order to study the ability of coupled neural oscillators to synchronize in the presence of intrinsic as opposed to synaptic noise, we constructed hybrid circuits consisting of one biological and one computational model neuron with reciprocal synaptic inhibition using the dynamic clamp. Uncoupled, both neurons fired periodic trains of action potentials. Most coupled circuits exhibited qualitative changes between one-to-one phase-locking with fairly constant phasic relationships and phase slipping with a constant progression in the phasic relationships across cycles. The phase resetting curve (PRC) and intrinsic periods were measured for both neurons, and used to construct a map of the firing intervals for both the coupled and externally forced (PRC measurement) conditions. For the coupled network, a stable fixed point of the map predicted phase locking, and its absence produced phase slipping. Repetitive application of the map was used to calibrate different noise models to simultaneously fit the noise level in the measurement of the PRC and the dynamics of the hybrid circuit experiments. Only a noise model that added history-dependent variability to the intrinsic period could fit both data sets with the same parameter values, as well as capture bifurcations in the fixed points of the map that cause switching between slipping and locking. We conclude that the biological neurons in our study have slowly-fluctuating stochastic dynamics that confer history dependence on the period. Theoretical results to date on the behavior of ensembles of noisy biological oscillators may require re-evaluation to account for transitions induced by slow noise dynamics.
Author Summary
Many biological phenomena exhibit synchronized oscillations in the presence of noise and heterogeneity. These include brain rhythms that underlie cognition and spinal rhythms that underlie rhythmic motor activity like breathing and locomotion. A two oscillator system was constructed in which most of the circuit was implemented in a computer model, and was therefore completely known and under the control of the investigators. The one biological component was an oscillator in which an apparently novel manifestation of biological noise was identified, dynamical noise in the period of the oscillator itself. This study quantifies how much noise and heterogeneity this simple two oscillator system can tolerate before desynchronizing. More complicated systems of oscillators may follow similar principles.
PMCID: PMC4022488  PMID: 24830924
13.  Pacemaker rate and depolarization block in nigral dopamine neurons: a somatic sodium channel balancing act 
Midbrain dopamine (DA) neurons are slow intrinsic pacemakers that undergo depolarization (DP) block upon moderate stimulation. Understanding DP block is important because it has been correlated with the clinical efficacy of chronic antipsychotic drug treatment. Here we describe how voltage-gated sodium (NaV) channels regulate DP block and pacemaker activity in DA neurons of the substantia nigra using rat brain slices. The distribution, density and gating of NaV currents were manipulated by blocking native channels with tetrodotoxin and by creating virtual channels and anti-channels with dynamic clamp. Although action potentials initiate in the axon initial segment (AIS) and NaV channels are distributed in multiple dendrites, selective reduction of NaV channel activity in the soma was sufficient to decrease pacemaker frequency and increase susceptibility to DP block. Conversely, increasing somatic NaV current density raised pacemaker frequency and lowered susceptibility to DP block. Finally, when NaV currents were restricted to the soma, pacemaker activity occurred at abnormally high rates due to excessive local subthreshold NaV current. Together with computational simulations, these data show that both the slow pacemaker rate and the sensitivity to DP block that characterizes DA neurons result from the low density of somatic NaV channels. More generally, we conclude that the somatodendritic distribution of NaV channels is a major determinant of repetitive spiking frequency.
PMCID: PMC3494994  PMID: 23077037
14.  Phase response theory extended to nonoscillatory network components 
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.
PMCID: PMC3501682  PMID: 23004844
15.  Synaptic and Intrinsic Determinants of the Phase Resetting Curve for Weak Coupling 
A phase resetting curve (PRC) keeps track of the extent to which a perturbation at a given phase advances or delays the next spike, and can be used to predict phase locking in networks of oscillators. The PRC can be estimated by convolving the waveform of the perturbation with the infinitesimal PRC (iPRC) under the assumption of weak coupling. The iPRC is often defined with respect to an infinitesimal current as zi(ϕ), where ϕ is phase, but can also be defined with respect to an infinitesimal conductance change as zg(ϕ). In this paper, we first show that the two approaches are equivalent. Coupling waveforms corresponding to synapses with different time courses sample zg(ϕ) in predictably different ways. We show that for oscillators with Type I excitability, an anomalous region in zg(ϕ) with opposite sign to that seen otherwise is often observed during an action potential. If the duration of the synaptic perturbation is such that it effectively samples this region, PRCs with both advances and delays can be observed despite Type I excitability. We also show that changing the duration of a perturbation so that it preferentially samples regions of stable or unstable slopes in zg(ϕ) can stabilize or destabilize synchrony in a network with the corresponding dynamics.
PMCID: PMC3059351  PMID: 20700637
16.  Stability of Two Cluster Solutions in Pulse Coupled Networks of Neural Oscillators 
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.
PMCID: PMC3059341  PMID: 20725773
neuronal networks; synchronization; clustering; phase response curves; pulse coupled oscillators
17.  Short Conduction Delays Cause Inhibition Rather than Excitation to Favor Synchrony in Hybrid Neuronal Networks of the Entorhinal Cortex 
PLoS Computational Biology  2012;8(1):e1002306.
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.
PMCID: PMC3252263  PMID: 22241969
18.  Pulse Coupled Oscillators and the Phase Resetting Curve 
Mathematical biosciences  2010;226(2):77-96.
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.
PMCID: PMC3022482  PMID: 20460132
Pulse coupled oscillators; Phase resetting; Phase locking; Synchronization; Splay; Clustering
19.  Phase locking and resetting in human subthalamic neurons 
BMC Neuroscience  2011;12(Suppl 1):P28.
PMCID: PMC3240389
20.  Inclusion of noise in iterated firing time maps based on the phase response curve 
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.
PMCID: PMC2946859  PMID: 20866456
21.  Regulation of Firing Frequency in a Computational Model of a Midbrain Dopaminergic Neuron 
Dopaminergic (DA) neurons of the mammalian midbrain exhibit unusually low firing frequencies in vitro. Furthermore, injection of depolarizing current induces depolarization block before high frequencies are achieved. The maximum steady and transient rates are about 10 and 20 Hz, respectively, despite the ability of these neurons to generate bursts at higher frequencies in vivo. We use a three-compartment model calibrated to reproduce DA neuron responses to several pharmacological manipulations to uncover mechanisms of frequency limitation. The model exhibits a slow oscillatory potential (SOP) dependent on the interplay between the L-type Ca2+ current and the small conductance K+ (SK) current that is unmasked by fast Na+ current block. Contrary to previous theoretical work, the SOP does not pace the steady spiking frequency in our model. The main currents that determine the spontaneous firing frequency are the subthreshold L-type Ca2+ and the A-type K+ currents. The model identifies the channel densities for the fast Na+ and the delayed rectifier K+ currents as critical parameters limiting the maximal steady frequency evoked by a depolarizing pulse. We hypothesize that the low maximal steady frequencies result from a low safety factor for action potential generation. In the model, the rate of Ca2+ accumulation in the distal dendrites controls the transient initial frequency in response to a depolarizing pulse. Similar results are obtained when the same model parameters are used in a multi-compartmental model with a realistic reconstructed morphology, indicating that the salient contributions of the dendritic architecture have been captured by the simpler model.
PMCID: PMC2929809  PMID: 20217204
multicompartmental model; pacemaking; depolarization block
25.  Phase Resetting Curves Determine Synchronization, Phase-locking, and Clustering in Networks of Neural Oscillators 
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.
PMCID: PMC2765798  PMID: 19386918
Network; Synchronization; Oscillator; Rhythm; Phase shift; Synchrony

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