Long-Term Inactivation of Sodium Channels as a Mechanism of Adaptation in CA1 Pyramidal Neurons.

Many hippocampal CA1 pyramidal cells function as place cells, increasing their firing rate when a specific place field is traversed. The dependence of CA1 place cell firing on position within the place field is asymmetric. We investigated the source of this asymmetry by injecting triangular depolarizing current ramps to approximate the spatially tuned, temporally diffuse depolarizing synaptic input received by these neurons while traversing a place field. Ramps were applied to CA1 pyramidal neurons from male rats in vitro (slice electrophysiology) and in silico (multicompartmental NEURON model). Under control conditions, CA1 neurons fired more action potentials at higher frequencies on the up-ramp versus the down-ramp. This effect was more pronounced for dendritic compared with somatic ramps. We incorporated a four-state Markov scheme for NaV1.6 channels into our model and calibrated the spatial dependence of long-term inactivation according to the literature; this spatial dependence was sufficient to explain the difference in dendritic versus somatic ramps. Long-term inactivation reduced the firing frequency by decreasing open-state occupancy, and reduced spike amplitude during trains by decreasing occupancy in the closed state, which comprises the available pool. PKC activator phorbol-dibutyrate, known to reduce NaV long-term inactivation, removed spike amplitude attenuation in vitro more visibly in dendrites and greatly reduced adaptation, consistent with our hypothesized mechanism. Intracellular application of a peptide inducing long-term NaV inactivation elicited spike amplitude attenuation during spike trains in the soma and greatly enhanced adaptation. Our synergistic experimental/computational approach shows that long-term inactivation of NaV1.6 is a key mechanism of adaptation in CA1 pyramidal cells.SIGNIFICANCE STATEMENT The hippocampus plays an important role in certain types of memory, in part through context-specific firing of "place cells"; these cells were first identified in rodents as being particularly active when an animal is in a specific location in an environment, called the place field of that neuron. In this in vitro/in silico study, we found that long-term inactivation of sodium channels causes adaptation in the firing rate that could potentially skew the firing of CA1 hippocampal pyramidal neurons earlier within a place field. A computational model of the sodium channel revealed differential regulation of spike frequency and amplitude by long-term inactivation, which may be a general mechanism for spike frequency adaptation in the CNS.

Interneuronal network model of theta-nested fast oscillations predicts differential effects of heterogeneity, gap junctions and short term depression for hyperpolarizing versus shunting inhibition.

Theta and gamma oscillations in the hippocampus have been hypothesized to play a role in the encoding and retrieval of memories. Recently, it was shown that an intrinsic fast gamma mechanism in medial entorhinal cortex can be recruited by optogenetic stimulation at theta frequencies, which can persist with fast excitatory synaptic transmission blocked, suggesting a contribution of interneuronal network gamma (ING). We calibrated the passive and active properties of a 100-neuron model network to capture the range of passive properties and frequency/current relationships of experimentally recorded PV+ neurons in the medial entorhinal cortex (mEC). The strength and probabilities of chemical and electrical synapses were also calibrated using paired recordings, as were the kinetics and short-term depression (STD) of the chemical synapses. Gap junctions that contribute a noticeable fraction of the input resistance were required for synchrony with hyperpolarizing inhibition; these networks exhibited theta-nested high frequency oscillations similar to the putative ING observed experimentally in the optogenetically-driven PV-ChR2 mice. With STD included in the model, the network desynchronized at frequencies above ~200 Hz, so for sufficiently strong drive, fast oscillations were only observed before the peak of the theta. Because hyperpolarizing synapses provide a synchronizing drive that contributes to robustness in the presence of heterogeneity, synchronization decreases as the hyperpolarizing inhibition becomes weaker. In contrast, networks with shunting inhibition required non-physiological levels of gap junctions to synchronize using conduction delays within the measured range.

Inactivation mode of sodium channels defines the different maximal firing rates of conventional versus atypical midbrain dopamine neurons.

Two subpopulations of midbrain dopamine (DA) neurons are known to have different dynamic firing ranges in vitro that correspond to distinct projection targets: the originally identified conventional DA neurons project to the dorsal striatum and the lateral shell of the nucleus accumbens, whereas an atypical DA population with higher maximum firing frequencies projects to prefrontal regions and other limbic regions including the medial shell of nucleus accumbens. Using a computational model, we show that previously identified differences in biophysical properties do not fully account for the larger dynamic range of the atypical population and predict that the major difference is that originally identified conventional cells have larger occupancy of voltage-gated sodium channels in a long-term inactivated state that recovers slowly; stronger sodium and potassium conductances during action potential firing are also predicted for the conventional compared to the atypical DA population. These differences in sodium channel gating imply that longer intervals between spikes are required in the conventional population for full recovery from long-term inactivation induced by the preceding spike, hence the lower maximum frequency. These same differences can also change the bifurcation structure to account for distinct modes of entry into depolarization block: abrupt versus gradual. The model predicted that in cells that have entered depolarization block, it is much more likely that an additional depolarization can evoke an action potential in conventional DA population. New experiments comparing lateral to medial shell projecting neurons confirmed this model prediction, with implications for differential synaptic integration in the two populations.

The Transcription Factor Shox2 Shapes Neuron Firing Properties and Suppresses Seizures by Regulation of Key Ion Channels in Thalamocortical Neurons.

Thalamocortical neurons (TCNs) play a critical role in the maintenance of thalamocortical oscillations, dysregulation of which can result in certain types of seizures. Precise control over firing rates of TCNs is foundational to these oscillations, yet the transcriptional mechanisms that constrain these firing rates remain elusive. We hypothesized that Shox2 is a transcriptional regulator of ion channels important for TCN function and that loss of Shox2 alters firing frequency and activity, ultimately perturbing thalamocortical oscillations into an epilepsy-prone state. In this study, we used RNA sequencing and quantitative PCR of control and Shox2 knockout mice to determine Shox2-affected genes and revealed a network of ion channel genes important for neuronal firing properties. Protein regulation was confirmed by Western blotting, and electrophysiological recordings showed that Shox2 KO impacted the firing properties of a subpopulation of TCNs. Computational modeling showed that disruption of these conductances in a manner similar to Shox2's effects modulated frequency of oscillations and could convert sleep spindles to near spike and wave activity, which are a hallmark for absence epilepsy. Finally, Shox2 KO mice were more susceptible to pilocarpine-induced seizures. Overall, these results reveal Shox2 as a transcription factor important for TCN function in adult mouse thalamus.

Intrinsic Mechanisms of Frequency Selectivity in the Proximal Dendrites of CA1 Pyramidal Neurons.

Gamma oscillations are thought to play a role in learning and memory. Two distinct bands, slow (25-50 Hz) and fast (65-100 Hz) gamma, have been identified in area CA1 of the rodent hippocampus. Slow gamma is phase locked to activity in area CA3 and presumably driven by the Schaffer collaterals (SCs). We used a combination of computational modeling and in vitro electrophysiology in hippocampal slices of male rats to test whether CA1 neurons responded to SC stimulation selectively at slow gamma frequencies and to identify the mechanisms involved. Both approaches demonstrated that, in response to temporally precise input at SCs, CA1 pyramidal neurons fire preferentially in the slow gamma range regardless of whether the input is at fast or slow gamma frequencies, suggesting frequency selectivity in CA1 output with respect to CA3 input. In addition, phase locking, assessed by the vector strength, was more precise for slow gamma than fast gamma input. This frequency selectivity was greatly attenuated when the slow Ca2+-dependent K+ (SK) current was removed from the model or blocked in vitro with apamin. Perfusion of slices with BaCl2 to block A-type K+ channels tightened this frequency selectivity. Both the broad-spectrum cholinergic agonist carbachol and the muscarinic agonist oxotremorine-M greatly attenuated the selectivity. The more precise firing at slower frequencies persisted throughout all of the pharmacological manipulations conducted. We propose that these intrinsic mechanisms provide a means by which CA1 phase locks to CA3 at different gamma frequencies preferentially in vivo as physiological conditions change with behavioral demands.SIGNIFICANCE STATEMENT Gamma frequency activity, one of multiple bands of synchronous activity, has been suggested to underlie various aspects of hippocampal function. Multisite recordings within the rat hippocampal formation indicate that different behavioral tasks are associated with synchronized activity between areas CA3 and CA1 at two different gamma bands: slow and fast gamma. In this study, we examine the intrinsic mechanisms that may allow CA1 to selectively "listen" to CA3 at slow compared with fast gamma and suggest mechanisms that gate this selectivity. Identifying the intrinsic mechanisms underlying differential gamma preference may help to explain the distinct types of CA3-CA1 synchronization observed in vivo under different behavioral conditions.

Role of the Axon Initial Segment in the Control of Spontaneous Frequency of Nigral Dopaminergic Neurons In Vivo.

The spontaneous tonic discharge activity of nigral dopamine neurons plays a fundamental role in dopaminergic signaling. To investigate the role of neuronal morphology and architecture with respect to spontaneous activity in this population, we visualized the 3D structure of the axon initial segment (AIS) along with the entire somatodendritic domain of adult male mouse dopaminergic neurons, previously recorded in vivo We observed a positive correlation of the firing rate with both proximity and size of the AIS. Computational modeling showed that the size of the AIS, but not its position within the somatodendritic domain, is the major causal determinant of the tonic firing rate in the intact model, by virtue of the higher intrinsic frequency of the isolated AIS. Further mechanistic analysis of the relationship between neuronal morphology and firing rate showed that dopaminergic neurons function as a coupled oscillator whose frequency of discharge results from a compromise between AIS and somatodendritic oscillators. Thus, morphology plays a critical role in setting the basal tonic firing rate, which in turn could control striatal dopaminergic signaling that mediates motivation and movement.SIGNIFICANCE STATEMENT The frequency at which nigral dopamine neurons discharge action potentials sets baseline dopamine levels in the brain, which enables activity in motor, cognitive, and motivational systems. Here, we demonstrate that the size of the axon initial segment, a subcellular compartment responsible for initiating action potentials, is a key determinant of the firing rate in these neurons. The axon initial segment and all the molecular components that underlie its critical function may provide a novel target for the regulation of dopamine levels in the brain.

Morphological and Biophysical Determinants of the Intracellular and Extracellular Waveforms in Nigral Dopaminergic Neurons: A Computational Study.

Action potentials (APs) in nigral dopaminergic neurons often exhibit two separate components: the first reflecting spike initiation in the dendritically located axon initial segment (AIS) and the second the subsequent dendro-somatic spike. These components are separated by a notch in the ascending phase of the somatic extracellular waveform and in the temporal derivative of the somatic intracellular waveform. Still, considerable variability exists in the presence and magnitude of the notch across neurons. To systematically address the contribution of AIS, dendritic and somatic compartments to shaping the two-component APs, we modeled APs of previously in vivo electrophysiologically characterized and 3D-reconstructed male mouse and rat dopaminergic neurons. A parsimonious two-domain model, with high (AIS) and lower (dendro-somatic) Na+ conductance, reproduced the notch in the temporal derivatives, but not in the extracellular APs, regardless of morphology. The notch was only revealed when somatic active currents were reduced, constraining the model to three domains. Thus, an initial AIS spike is followed by an actively generated spike by the axon-bearing dendrite (ABD), in turn followed mostly passively by the soma. The transition from being a source compartment for the AIS spike to a source compartment for the ABD spike satisfactorily explains the extracellular somatic notch. Larger AISs and thinner ABD (but not soma-to-AIS distance) accentuate the AIS component. We conclude that variability in AIS size and ABD caliber explains variability in AP extracellular waveform and separation of AIS and dendro-somatic components, given the presence of at least three functional domains with distinct excitability characteristics.SIGNIFICANCE STATEMENT Midbrain dopamine neurons make an important contribution to circuits mediating motivation and movement. Understanding the basic rules that govern the electrical activity of single dopaminergic neurons is therefore essential to reveal how they ultimately contribute to movement and motivation as well as what goes wrong in associated disorders. Our computational study focuses on the generation and propagation of action potentials and shows that different morphologies and excitability characteristics of the cell body, dendrites and proximal axon can explain the diversity of action potentials shapes in this population. These compartments likely make differential contributions both to normal dopaminergic signaling and could potentially underlie pathological dopaminergic signaling implicated in addiction, schizophrenia, Parkinson's disease, and other disorders.

Phase response theory explains cluster formation in sparsely but strongly connected inhibitory neural networks and effects of jitter due to sparse connectivity.

We show how to predict whether a neural network will exhibit global synchrony (a one-cluster state) or a two-cluster state based on the assumption of pulsatile coupling and critically dependent upon the phase response curve (PRC) generated by the appropriate perturbation from a partner cluster. Our results hold for a monotonically increasing (meaning longer delays as the phase increases) PRC, which likely characterizes inhibitory fast-spiking basket and cortical low-threshold-spiking interneurons in response to strong inhibition. Conduction delays stabilize synchrony for this PRC shape, whereas they destroy two-cluster states, the former by avoiding a destabilizing discontinuity and the latter by approaching it. With conduction delays, stronger coupling strength can promote a one-cluster state, so the weak coupling limit is not applicable here. We show how jitter can destabilize global synchrony but not a two-cluster state. Local stability of global synchrony in an all-to-all network does not guarantee that global synchrony can be observed in an appropriately scaled sparsely connected network; the basin of attraction can be inferred from the PRC and must be sufficiently large. Two-cluster synchrony is not obviously different from one-cluster synchrony in the presence of noise and may be the actual substrate for oscillations observed in the local field potential (LFP) and the electroencephalogram (EEG) in situations where global synchrony is not possible. Transitions between cluster states may change the frequency of the rhythms observed in the LFP or EEG. Transitions between cluster states within an inhibitory subnetwork may allow more effective recruitment of pyramidal neurons into the network rhythm. NEW & NOTEWORTHY We show that jitter induced by sparse connectivity can destabilize global synchrony but not a two-cluster state with two smaller clusters firing alternately. On the other hand, conduction delays stabilize synchrony and destroy two-cluster states. These results hold if each cluster exhibits a phase response curve similar to one that characterizes fast-spiking basket and cortical low-threshold-spiking cells for strong inhibition. Either a two-cluster or a one-cluster state might provide the oscillatory substrate for neural computations.

Calcium dynamics control K-ATP channel-mediated bursting in substantia nigra dopamine neurons: a combined experimental and modeling study.

Burst firing in medial substantia nigra (mSN) dopamine (DA) neurons has been selectively linked to novelty-induced exploration behavior in mice. Burst firing in mSN DA neurons, in contrast to lateral SN DA neurons, requires functional ATP-sensitive potassium (K-ATP) channels both in vitro and in vivo. However, the precise role of K-ATP channels in promoting burst firing is unknown. We show experimentally that L-type calcium channel activity in mSN DA neurons enhances open probability of K-ATP channels. We then generate a mathematical model to study the role of Ca2+ dynamics driving K-ATP channel function in mSN DA neurons during bursting. In our model, Ca2+ influx leads to local accumulation of ADP due to Ca-ATPase activity, which in turn activates K-ATP channels. If K-ATP channel activation reaches levels sufficient to terminate spiking, rhythmic bursting occurs. The model explains the experimental observation that, in vitro, coapplication of NMDA and a selective K-ATP channel opener, NN414, is required to elicit bursting as follows. Simulated NMDA receptor activation increases the firing rate and the rate of Ca2+ influx, which increases the activation of K-ATP. The model suggests that additional sources of hyperpolarization, such as GABAergic synaptic input, are recruited in vivo for burst termination or rebound burst discharge. The model predicts that NN414 increases the sensitivity of the K-ATP channel to ADP, promoting burst firing in vitro, and that that high levels of Ca2+ buffering, as might be expected in the calbindin-positive SN DA neuron subpopulation, promote rhythmic bursting pattern, consistent with experimental observations in vivo. NEW & NOTEWORTHY Recently identified distinct subpopulations of midbrain dopamine neurons exhibit differences in their two primary activity patterns in vivo: tonic (single spike) firing and phasic bursting. This study elucidates the biophysical basis of bursts specific to dopamine neurons in the medial substantia nigra, enabled by ATP-sensitive K+ channels and necessary for novelty-induced exploration. A better understanding of how dopaminergic signaling differs between subpopulations may lead to therapeutic strategies selectively targeted to specific subpopulations.

Resonant Interneurons Can Increase Robustness of Gamma Oscillations.

Gamma oscillations are believed to play a critical role in in information processing, encoding, and retrieval. Inhibitory interneuronal network gamma (ING) oscillations may arise from a coupled oscillator mechanism in which individual neurons oscillate or from a population oscillator in which individual neurons fire sparsely and stochastically. All ING mechanisms, including the one proposed herein, rely on alternating waves of inhibition and windows of opportunity for spiking. The coupled oscillator model implemented with Wang-Buzsáki model neurons is not sufficiently robust to heterogeneity in excitatory drive, and therefore intrinsic frequency, to account for in vitro models of ING. Similarly, in a tightly synchronized regime, the stochastic population oscillator model is often characterized by sparse firing, whereas interneurons both in vivo and in vitro do not fire sparsely during gamma, but rather on average every other cycle. We substituted so-called resonator neural models, which exhibit class 2 excitability and postinhibitory rebound (PIR), for the integrators that are typically used. This results in much greater robustness to heterogeneity that actually increases as the average participation in spikes per cycle approximates physiological levels. Moreover, dynamic clamp experiments that show autapse-induced firing in entorhinal cortical interneurons support the idea that PIR can serve as a network gamma mechanism. Furthermore, parvalbumin-positive (PV(+)) cells were much more likely to display both PIR and autapse-induced firing than GAD2(+) cells, supporting the view that PV(+) fast-firing basket cells are more likely to exhibit class 2 excitability than other types of inhibitory interneurons. SIGNIFICANCE STATEMENT: Gamma oscillations are believed to play a critical role in information processing, encoding, and retrieval. Networks of inhibitory interneurons are thought to be essential for these oscillations. We show that one class of interneurons with an abrupt onset of firing at a threshold frequency may allow more robust synchronization in the presence of noise and heterogeneity. The mechanism for this robustness depends on the intrinsic resonance at this threshold frequency. Moreover, we show experimentally the feasibility of the proposed mechanism and suggest a way to distinguish between this mechanism and another proposed mechanism: that of a stochastic population oscillator independent of the dynamics of individual neurons.

Stochastic slowly adapting ionic currents may provide a decorrelation mechanism for neural oscillators by causing wander in the intrinsic period.

Oscillatory neurons integrate their synaptic inputs in fundamentally different ways than normally quiescent neurons. We show that the oscillation period of invertebrate endogenous pacemaker neurons wanders, producing random fluctuations in the interspike intervals (ISI) on a time scale of seconds to minutes, which decorrelates pairs of neurons in hybrid circuits constructed using the dynamic clamp. The autocorrelation of the ISI sequence remained high for many ISIs, but the autocorrelation of the ΔISI series had on average a single nonzero value, which was negative at a lag of one interval. We reproduced these results using a simple integrate and fire (IF) model with a stochastic population of channels carrying an adaptation current with a stochastic component that was integrated with a slow time scale, suggesting that a similar population of channels underlies the observed wander in the period. Using autoregressive integrated moving average (ARIMA) models, we found that a single integrator and a single moving average with a negative coefficient could simulate both the experimental data and the IF model. Feeding white noise into an integrator with a slow time constant is sufficient to produce the autocorrelation structure of the ISI series. Moreover, the moving average clearly accounted for the autocorrelation structure of the ΔISI series and is biophysically implemented in the IF model using slow stochastic adaptation. The observed autocorrelation structure may be a neural signature of slow stochastic adaptation, and wander generated in this manner may be a general mechanism for limiting episodes of synchronized activity in the nervous system.

Implications of cellular models of dopamine neurons for disease.

This review addresses the present state of single-cell models of the firing pattern of midbrain dopamine neurons and the insights that can be gained from these models into the underlying mechanisms for diseases such as Parkinson's, addiction, and schizophrenia. We will explain the analytical technique of separation of time scales and show how it can produce insights into mechanisms using simplified single-compartment models. We also use morphologically realistic multicompartmental models to address spatially heterogeneous aspects of neural signaling and neural metabolism. Separation of time scale analyses are applied to pacemaking, bursting, and depolarization block in dopamine neurons. Differences in subpopulations with respect to metabolic load are addressed using multicompartmental models.

Feedback control of variability in the cycle period of a central pattern generator.

We address how feedback to a bursting biological pacemaker with intrinsic variability in cycle length can affect that variability. Specifically, we examine a hybrid circuit constructed of an isolated crab anterior burster (AB)/pyloric dilator (PD) pyloric pacemaker receiving virtual feedback via dynamic clamp. This virtual feedback generates artificial synaptic input to PD with timing determined by adjustable phase response dynamics that mimic average burst intervals generated by the lateral pyloric neuron (LP) in the intact pyloric network. Using this system, we measure network period variability dependence on the feedback element's phase response dynamics and find that a constant response interval confers minimum variability. We further find that these optimal dynamics are characteristic of the biological pyloric network. Building upon our previous theoretical work mapping the firing intervals in one cycle onto the firing intervals in the next cycle, we create a theoretical map of the distribution of all firing intervals in one cycle to the distribution of firing intervals in the next cycle. We then obtain an integral equation for a stationary self-consistent distribution of the network periods of the hybrid circuit, which can be solved numerically given the uncoupled pacemaker's distribution of intrinsic periods, the nature of the network's feedback, and the phase resetting characteristics of the pacemaker. The stationary distributions obtained in this manner are strongly predictive of the experimentally observed distributions of hybrid network period. This theoretical framework can provide insight into optimal feedback schemes for minimizing variability to increase reliability or maximizing variability to increase flexibility in central pattern generators driven by pacemakers with feedback.

Slow noise in the period of a biological oscillator underlies gradual trends and abrupt transitions in phasic relationships in hybrid neural networks.

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.

Mathematical analysis of depolarization block mediated by slow inactivation of fast sodium channels in midbrain dopamine neurons.

Dopamine neurons in freely moving rats often fire behaviorally relevant high-frequency bursts, but depolarization block limits the maximum steady firing rate of dopamine neurons in vitro to ∼10 Hz. Using a reduced model that faithfully reproduces the sodium current measured in these neurons, we show that adding an additional slow component of sodium channel inactivation, recently observed in these neurons, qualitatively changes in two different ways how the model enters into depolarization block. First, the slow time course of inactivation allows multiple spikes to be elicited during a strong depolarization prior to entry into depolarization block. Second, depolarization block occurs near or below the spike threshold, which ranges from -45 to -30 mV in vitro, because the additional slow component of inactivation negates the sodium window current. In the absence of the additional slow component of inactivation, this window current produces an N-shaped steady-state current-voltage (I-V) curve that prevents depolarization block in the experimentally observed voltage range near -40 mV. The time constant of recovery from slow inactivation during the interspike interval limits the maximum steady firing rate observed prior to entry into depolarization block. These qualitative features of the entry into depolarization block can be reversed experimentally by replacing the native sodium conductance with a virtual conductance lacking the slow component of inactivation. We show that the activation of NMDA and AMPA receptors can affect bursting and depolarization block in different ways, depending upon their relative contributions to depolarization versus to the total linear/nonlinear conductance.

Effect of heterogeneity and noise on cross frequency phase-phase and phase-amplitude coupling.

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.

Synchrony in Networks of Type 2 Interneurons Is More Robust to Noise with Hyperpolarizing Inhibition Compared to Shunting Inhibition in Both the Stochastic Population Oscillator and the Coupled Oscillator Regimes.

Synchronization in the gamma band (25-150 Hz) is mediated by PV+ inhibitory interneurons, and evidence is accumulating for the essential role of gamma oscillations in cognition. Oscillations can arise in inhibitory networks via synaptic interactions between individual oscillatory neurons (mean-driven) or via strong recurrent inhibition that destabilizes the stationary background firing rate in the fluctuation-driven balanced state, causing an oscillation in the population firing rate. Previous theoretical work focused on model neurons with Hodgkin's Type 1 excitability (integrators) connected by current-based synapses. Here we show that networks comprised of simple Type 2 oscillators (resonators) exhibit a supercritical Hopf bifurcation between synchrony and asynchrony and a gradual transition via cycle skipping from coupled oscillators to stochastic population oscillator (SPO), as previously shown for Type 1. We extended our analysis to homogeneous networks with conductance rather than current based synapses and found that networks with hyperpolarizing inhibitory synapses were more robust to noise than those with shunting synapses, both in the coupled oscillator and SPO regime. Assuming that reversal potentials are uniformly distributed between shunting and hyperpolarized values, as observed in one experimental study, converting synapses to purely hyperpolarizing favored synchrony in all cases, whereas conversion to purely shunting synapses made synchrony less robust except at very high conductance strengths. In mature neurons the synaptic reversal potential is controlled by chloride cotransporters that control the intracellular concentrations of chloride and bicarbonate ions, suggesting these transporters as a potential therapeutic target to enhance gamma synchrony and cognition.

Phase Resetting Curves Determine Stability of Synchrony in One and Two Clusters of Pulse Coupled Oscillators with Delays.

Previously, our group used phase response curves under a pulsatile coupling assumption to determine the stability of synchrony within a cluster of neural oscillators and between two clusters of oscillators. The interactions of the within and between cluster terms were considered, demonstrating how an alternating firing pattern between clusters could stabilize within cluster synchrony-even in clusters unable to synchronize themselves in isolation. In addition, criteria were derived for synchrony between two pulse coupled oscillators with synaptic delays. In this study, we update our previous work on one and two clusters of coupled oscillators to include delays and demonstrate the validity of the results using a map of the firing intervals based on the phase resetting curve. We use self-connected neurons to represent clusters and derive conditions under which an oscillator can phase-lock itself with a delayed input. Although this analysis only strictly applies to identical neurons receiving identical synapses from the same number of neurons, the principles are general and can be used to understand how to promote or impede synchrony in physiological networks of neurons. Heterogeneity can be interpreted as a form of frozen noise, and approximate synchrony can be sustained despite heterogeneity. The pulse-coupled oscillator model can not only be used to describe biological neuronal networks but also cardiac pacemakers, lasers, fireflies, artificial neural networks, social self-organization, and wireless sensor networks.

Persistent Interruption in Parvalbumin Positive Inhibitory Interneurons: Biophysical and Mathematical Mechanisms.

Persistent activity in principal cells is a putative mechanism for maintaining memory traces during working memory. We recently demonstrated persistent interruption of firing in fast-spiking parvalbumin-expressing interneurons (PV-INs), a phenomenon which could serve as a substrate for persistent activity in principal cells through disinhibition lasting hundreds of milliseconds. Here, we find that hippocampal CA1 PV-INs exhibit type 2 excitability, like striatal and neocortical PV-INs. Modelling and mathematical analysis showed that the slowly inactivating potassium current Kv1 contributes to type 2 excitability, enables the multiple firing regimes observed experimentally in PV-INs, and provides a mechanism for robust persistent interruption of firing. Using a fast/slow separation of times scales approach with the Kv1 inactivation variable as a bifurcation parameter shows that the initial inhibitory stimulus stops repetitive firing by moving the membrane potential trajectory onto a co-existing stable fixed point corresponding to a non-spiking quiescent state. As Kv1 inactivation decays, the trajectory follows the branch of stable fixed points until it crosses a subcritical Hopf bifurcation then spirals out into repetitive firing. In a model describing entorhinal cortical PV-INs without Kv1, interruption of firing could be achieved by taking advantage of the bistability inherent in type 2 excitability based on a subcritical Hopf bifurcation, but the interruption was not robust to noise. Persistent interruption of firing is therefore broadly applicable to PV-INs in different brain regions but is only made robust to noise in the presence of a slow variable.

Persistent Interruption in Parvalbumin-Positive Inhibitory Interneurons: Biophysical and Mathematical Mechanisms.

Persistent activity in excitatory pyramidal cells (PYRs) is a putative mechanism for maintaining memory traces during working memory. We have recently demonstrated persistent interruption of firing in fast-spiking parvalbumin-expressing interneurons (PV-INs), a phenomenon that could serve as a substrate for persistent activity in PYRs through disinhibition lasting hundreds of milliseconds. Here, we find that hippocampal CA1 PV-INs exhibit type 2 excitability, like striatal and neocortical PV-INs. Modeling and mathematical analysis showed that the slowly inactivating potassium current KV1 contributes to type 2 excitability, enables the multiple firing regimes observed experimentally in PV-INs, and provides a mechanism for robust persistent interruption of firing. Using a fast/slow separation of times scales approach with the KV1 inactivation variable as a bifurcation parameter shows that the initial inhibitory stimulus stops repetitive firing by moving the membrane potential trajectory onto a coexisting stable fixed point corresponding to a nonspiking quiescent state. As KV1 inactivation decays, the trajectory follows the branch of stable fixed points until it crosses a subcritical Hopf bifurcation (HB) and then spirals out into repetitive firing. In a model describing entorhinal cortical PV-INs without KV1, interruption of firing could be achieved by taking advantage of the bistability inherent in type 2 excitability based on a subcritical HB, but the interruption was not robust to noise. Persistent interruption of firing is therefore broadly applicable to PV-INs in different brain regions but is only made robust to noise in the presence of a slow variable, KV1 inactivation.