Layer and frequency dependencies of phase response properties of pyramidal neurons in rat motor cortex

Optimal phase-selective entrainment of electrochemical oscillators with different phase response curves

We investigate the entrainment of electrochemical oscillators with different phase response curves (PRCs) using a global signal: the goal is to achieve the desired phase configuration using a minimum-power waveform. Establishing the desired phase relationships in a highly nonlinear networked system exhibiting significant heterogeneities, such as different conditions or parameters for the oscillators, presents a considerable challenge because different units respond differently to the common global entraining signal. In this work, we apply an optimal phase-selective entrainment technique in both a kinetic model and experiments involving electrochemical oscillators in achieving phase synchronized states. We estimate the PRCs of the oscillators at different circuit potentials and external resistance, and entrain pairs and small sets of four oscillators in various phase configurations. We show that for small PRC variations, phase assignment can be achieved using an averaged PRC in the control design. However, when the PRCs are sufficiently different, individual PRCs are needed to entrain the system with the expected phase relationships. The results show that oscillator assemblies with heterogeneous PRCs can be effectively entrained to desired phase configurations in practical settings. These findings open new avenues to applications in biological and engineered oscillator systems where synchronization patterns are essential for system performance.

Optimal phase-selective entrainment of heterogeneous oscillator ensembles

We develop a framework to design optimal entrainment signals that entrain an ensemble of heterogeneous nonlinear oscillators, described by phase models, at desired phases. We explicitly take into account heterogeneity in both oscillation frequency and the type of oscillators characterized by different Phase Response Curves. The central idea is to leverage the Fourier series representation of periodic functions to decode a phase-selective entrainment task into a quadratic program. We demonstrate our approach using a variety of phase models, where we entrain the oscillators into distinct phase patterns. Also, we show how the generalizability gained from our formulation enables us to meet a wide range of design objectives and constraints, such as minimum-power, fast entrainment, and charge-balanced controls.

Resurgent Na+ currents promote ultrafast spiking in projection neurons that drive fine motor control

The underlying mechanisms that promote precise spiking in upper motor neurons controlling fine motor skills are not well understood. Here we report that projection neurons in the adult zebra finch song nucleus RA display robust high-frequency firing, ultra-narrow spike waveforms, superfast Na⁺ current inactivation kinetics, and large resurgent Na⁺ currents (I_NaR). These properties of songbird pallial motor neurons closely resemble those of specialized large pyramidal neurons in mammalian primary motor cortex. They emerge during the early phases of song development in males, but not females, coinciding with a complete switch of Na+ channel subunit expression from Navβ3 to Navβ4. Dynamic clamping and dialysis of Navβ4's C-terminal peptide into juvenile RA neurons provide evidence that Navβ4, and its associated I_NaR, promote neuronal excitability. We thus propose that I_NaR modulates the excitability of upper motor neurons that are required for the execution of fine motor skills.

Firing rate-dependent phase responses of Purkinje cells support transient oscillations

Both spike rate and timing can transmit information in the brain. Phase response curves (PRCs) quantify how a neuron transforms input to output by spike timing. PRCs exhibit strong firing-rate adaptation, but its mechanism and relevance for network output are poorly understood. Using our Purkinje cell (PC) model, we demonstrate that the rate adaptation is caused by rate-dependent subthreshold membrane potentials efficiently regulating the activation of Na+ channels. Then, we use a realistic PC network model to examine how rate-dependent responses synchronize spikes in the scenario of reciprocal inhibition-caused high-frequency oscillations. The changes in PRC cause oscillations and spike correlations only at high firing rates. The causal role of the PRC is confirmed using a simpler coupled oscillator network model. This mechanism enables transient oscillations between fast-spiking neurons that thereby form PC assemblies. Our work demonstrates that rate adaptation of PRCs can spatio-temporally organize the PC input to cerebellar nuclei.

The effects of chloride dynamics on substantia nigra pars reticulata responses to pallidal and striatal inputs

As a rodent basal ganglia (BG) output nucleus, the substantia nigra pars reticulata (SNr) is well positioned to impact behavior. SNr neurons receive GABAergic inputs from the striatum (direct pathway) and globus pallidus (GPe, indirect pathway). Dominant theories of action selection rely on these pathways' inhibitory actions. Yet, experimental results on SNr responses to these inputs are limited and include excitatory effects. Our study combines experimental and computational work to characterize, explain, and make predictions about these pathways. We observe diverse SNr responses to stimulation of SNr-projecting striatal and GPe neurons, including biphasic and excitatory effects, which our modeling shows can be explained by intracellular chloride processing. Our work predicts that ongoing GPe activity could tune the SNr operating mode, including its responses in decision-making scenarios, and GPe output may modulate synchrony and low-frequency oscillations of SNr neurons, which we confirm using optogenetic stimulation of GPe terminals within the SNr.

How to correctly quantify neuronal phase-response curves from noisy recordings

At the level of individual neurons, various coding properties can be inferred from the input-output relationship of a cell. For small inputs, this relation is captured by the phase-response curve (PRC), which measures the effect of a small perturbation on the timing of the subsequent spike. Experimentally, however, an accurate experimental estimation of PRCs is challenging. Despite elaborate measurement efforts, experimental PRC estimates often cannot be related to those from modeling studies. In particular, experimental PRCs rarely resemble the characteristic theoretical PRC expected close to spike initiation, which is indicative of the underlying spike-onset bifurcation. Here, we show for conductance-based model neurons that the correspondence between theoretical and measured phase-response curve is lost when the stimuli used for the estimation are too large. In this case, the derived phase-response curve is distorted beyond recognition and takes on a generic shape that reflects the measurement protocol and masks the spike-onset bifurcation. We discuss how to identify appropriate stimulus strengths for perturbation and noise-stimulation methods, which permit to estimate PRCs that reliably reflect the spike-onset bifurcation - a task that is particularly difficult if a lower bound for the stimulus amplitude is dictated by prominent intrinsic neuronal noise.

Diverse action of repeated corticosterone treatment on synaptic transmission, neuronal plasticity, and morphology in superficial and deep layers of the rat motor cortex

One of the adverse effects of prolonged stress in rats is impaired performance of skilled reaching and walking tasks. The mechanisms that lead to these abnormalities are incompletely understood. Therefore, we compared the effects of twice daily repeated corticosterone injections for 7 days on miniature excitatory postsynaptic currents (mEPSCs), as well as on synaptic plasticity and morphology of layers II/III and V pyramidal neurons of the primary motor cortex (M1) of male Wistar rats. Corticosterone treatment resulted in increased frequency, but not amplitude, of mEPSCs in layer II/III neurons accompanied by increased complexity of the apical part of their dendritic tree, with no changes in the density of dendritic spines. The frequency and amplitude of mEPSCs as well as the parameters characterizing the complexity of the dendritic tree were not changed in layer V cells; however, their dendritic spine density was increased. While corticosterone treatment resulted in an increase in the amplitude of field potentials evoked in intralaminar connections within layer II/III, it did not influence field responses in layer V intralaminar connections, as well as the extent of chemically induced layer V long-term potentiation (chemLTP) by the application of tetraethylammonium (TEA, 25 mM). However, chemLTP induction in layer II/III was impaired in slices prepared from corticosterone-treated animals. These data indicate that repeated 7-day administration of exogenous corticosterone induces structural and functional plasticity in the M1, which occurs mainly in layer II/III pyramidal neurons. These findings shed light on potential sites of action and mechanisms underlying stress-induced impairment of motor functions.

Neurons as oscillators

Regularly spiking neurons can be described as oscillators. In this article we review some of the insights gained from this conceptualization and their relevance for systems neuroscience. First, we explain how a regularly spiking neuron can be viewed as an oscillator and how the phase-response curve (PRC) describes the response of the neuron's spike times to small perturbations. We then discuss the meaning of the PRC for a single neuron's spiking behavior and review the PRCs measured from a variety of neurons in a range of spiking regimes. Next, we show how the PRC can be related to a number of common measures used to quantify neuronal firing, such as the spike-triggered average and the peristimulus histogram. We further show that the response of a neuron to correlated inputs depends on the shape of the PRC. We then explain how the PRC of single neurons can be used to predict neural network behavior. Given the PRC, conduction delays, and the waveform and time course of the synaptic potentials, it is possible to predict neural population behavior such as synchronization. The PRC also allows us to quantify the robustness of the synchronization to heterogeneity and noise. We finally ask how to combine the measured PRCs and the predictions based on PRC to further the understanding of systems neuroscience. As an example, we discuss how the change of the PRC by the neuromodulator acetylcholine could lead to a destabilization of cortical network dynamics. Although all of these studies are grounded in mathematical abstractions that do not strictly hold in biology, they provide good estimates for the emergence of the brain's network activity from the properties of individual neurons. The study of neurons as oscillators can provide testable hypotheses and mechanistic explanations for systems neuroscience.

Phase-Dependent Modulation of Oscillatory Phase and Synchrony by Long-Lasting Depolarizing Inputs in Central Neurons

Oscillatory neural activities have been implicated in various types of information processing in the CNS. The procerebral (PC) lobe of the land mollusk Limax valentianus shows an ongoing oscillatory local field potential (LFP). Olfactory input increases both the frequency and spatial synchrony of the LFP oscillation by a nitric oxide (NO)-mediated mechanism, but how NO modulates the activity in a specific manner has been unclear. In the present study, we used electrical stimulation and NO uncaging to systematically analyze the response of the LFP oscillation and found phase-dependent effects on phase shifting and synchrony. The neurons that presumably release NO in the PC lobe preferentially fired at phases in which NO has a synchronizing effect, suggesting that the timing of NO release is regulated to induce a stereotyped response to natural sensory stimuli. The phase-response curve (PRC) describes the timing dependence of responses of an oscillatory system to external input. PRCs are usually constructed by recording the temporal shifts of the neural activity in response to brief electrical pulses. However, NO evokes a long-lasting depolarization persisting for several cycles of oscillation. The phase-response relationship obtained by NO stimulation was approximately the integral of the PRC. A similar relationship was also shown for regular firing of mouse cerebellar Purkinje cells receiving step depolarization, suggesting the generality of the results to oscillatory neural systems with highly distinct properties. These results indicate novel dynamic effects of long-lasting inputs on network oscillation and synchrony, which are based on simple and ubiquitous mechanisms.

Phase Response Synchronization in Neuronal Population with Time-Varying Coupling Strength

We present the dynamic model of global coupled neuronal population subject to external stimulus by the use of phase sensitivity function. We investigate the effect of time-varying coupling strength on the synchronized phase response of neural population subjected to external harmonic stimulus. For a time-periodic coupling strength, we found that the stimulus with increasing intensity or frequency can reinforce the phase response synchronization in neuronal population of the weakly coupled neural oscillators, and the neuronal population with stronger coupling strength has good adaptability to stimulus. When we consider the dynamics of coupling strength, we found that a strong stimulus can quickly cause the synchronization in the neuronal population, the degree of synchronization grows with the increasing stimulus intensity, and the period of synchronized oscillation induced by external stimulation is related to stimulus frequency.

Estimation of the phase response curve from Parkinsonian tremor

Phase response curves (PRCs), characterizing the response of an oscillator to weak external perturbation, have been estimated from a broad range of biological oscillators, including single neurons in vivo. PRC estimates, in turn, provide an intuitive insight into how oscillatory systems become entrained and how they can be desynchronized. Here, we explore the application of PRC theory to the case of Parkinsonian tremor. Initial attempts to establish a causal effect of subthreshold transcranial magnetic stimulation applied to primary motor cortex on the filtered tremor phase were unsuccessful. We explored the possible explanations of this and demonstrate that assumptions made when estimating the PRC in a traditional setting, such as a single neuron, are not arbitrary when applied to the case of tremor PRC estimation. We go on to extract the PRC of Parkinsonian tremor using an iterative method that requires varying the definition of the tremor cycle and estimating the PRC at multiple peristimulus time samples. Justification for this method is supported by estimates of PRC from simulated single neuron data. We provide an approach to estimating confidence limits for tremor PRC and discuss the interpretational caveats introduced by tremor harmonics and the intrinsic variability of the tremor's period.

Emergence and combinatorial accumulation of jittering regimes in spiking oscillators with delayed feedback

Interaction via pulses is common in many natural systems, especially neuronal. In this article we study one of the simplest possible systems with pulse interaction: a phase oscillator with delayed pulsatile feedback. When the oscillator reaches a specific state, it emits a pulse, which returns after propagating through a delay line. The impact of an incoming pulse is described by the oscillator's phase reset curve (PRC). In such a system we discover an unexpected phenomenon: for a sufficiently steep slope of the PRC, a periodic regular spiking solution bifurcates with several multipliers crossing the unit circle at the same parameter value. The number of such critical multipliers increases linearly with the delay and thus may be arbitrary large. This bifurcation is accompanied by the emergence of numerous "jittering" regimes with nonequal interspike intervals (ISIs). Each of these regimes corresponds to a periodic solution of the system with a period roughly proportional to the delay. The number of different "jittering" solutions emerging at the bifurcation point increases exponentially with the delay. We describe the combinatorial mechanism that underlies the emergence of such a variety of solutions. In particular, we show how a periodic solution exhibiting several distinct ISIs can imply the existence of multiple other solutions obtained by rearranging of these ISIs. We show that the theoretical results for phase oscillators accurately predict the behavior of an experimentally implemented electronic oscillator with pulsatile feedback.

Dynamics of on-off neural firing patterns and stochastic effects near a sub-critical Hopf bifurcation

On-off firing patterns, in which repetition of clusters of spikes are interspersed with epochs of subthreshold oscillations or quiescent states, have been observed in various nervous systems, but the dynamics of this event remain unclear. Here, we report that on-off firing patterns observed in three experimental models (rat sciatic nerve subject to chronic constrictive injury, rat CA1 pyramidal neuron, and rabbit blood pressure baroreceptor) appeared as an alternation between quiescent state and burst containing multiple period-1 spikes over time. Burst and quiescent state had various durations. The interspike interval (ISI) series of on-off firing pattern was suggested as stochastic using nonlinear prediction and autocorrelation function. The resting state was changed to a period-1 firing pattern via on-off firing pattern as the potassium concentration, static pressure, or depolarization current was changed. During the changing process, the burst duration of on-off firing pattern increased and the duration of the quiescent state decreased. Bistability of a limit cycle corresponding to period-1 firing and a focus corresponding to resting state was simulated near a sub-critical Hopf bifurcation point in the deterministic Morris-Lecar (ML) model. In the stochastic ML model, noise-induced transitions between the coexisting regimes formed an on-off firing pattern, which closely matched that observed in the experiment. In addition, noise-induced exponential change in the escape rate from the focus, and noise-induced coherence resonance were identified. The distinctions between the on-off firing pattern and stochastic firing patterns generated near three other types of bifurcations of equilibrium points, as well as other viewpoints on the dynamics of on-off firing pattern, are discussed. The results not only identify the on-off firing pattern as noise-induced stochastic firing pattern near a sub-critical Hopf bifurcation point, but also offer practical indicators to discriminate bifurcation types and neural excitability types.

On the firing rate dependency of the phase response curve of rat Purkinje neurons in vitro

Synchronous spiking during cerebellar tasks has been observed across Purkinje cells: however, little is known about the intrinsic cellular mechanisms responsible for its initiation, cessation and stability. The Phase Response Curve (PRC), a simple input-output characterization of single cells, can provide insights into individual and collective properties of neurons and networks, by quantifying the impact of an infinitesimal depolarizing current pulse on the time of occurrence of subsequent action potentials, while a neuron is firing tonically. Recently, the PRC theory applied to cerebellar Purkinje cells revealed that these behave as phase-independent integrators at low firing rates, and switch to a phase-dependent mode at high rates. Given the implications for computation and information processing in the cerebellum and the possible role of synchrony in the communication with its post-synaptic targets, we further explored the firing rate dependency of the PRC in Purkinje cells. We isolated key factors for the experimental estimation of the PRC and developed a closed-loop approach to reliably compute the PRC across diverse firing rates in the same cell. Our results show unambiguously that the PRC of individual Purkinje cells is firing rate dependent and that it smoothly transitions from phase independent integrator to a phase dependent mode. Using computational models we show that neither channel noise nor a realistic cell morphology are responsible for the rate dependent shift in the phase response curve.

Changes of Firing Rate Induced by Changes of Phase Response Curve in Bifurcation Transitions

We study dynamical mechanisms responsible for changes of the firing rate during four different bifurcation transitions in the two-dimensional Hindmarsh-Rose (2DHR) neuron model: the saddle node on an invariant circle (SNIC) bifurcation to the supercritical Andronov-Hopf (AH) one, the SNIC bifurcation to the saddle-separatrix loop (SSL) one, the AH bifurcation to the subcritical AH (SAH) one, and the SSL bifurcation to the AH one. For this purpose, we study slopes of the firing rate curve with respect to not only an external input current but also temperature that can be interpreted as a timescale in the 2DHR neuron model. These slopes are mathematically formulated with phase response curves (PRCs), expanding the firing rate with perturbations of the temperature and external input current on the one-dimensional space of the phase [Formula: see text] in the 2DHR oscillator. By analyzing the two different slopes of the firing rate curve with respect to the temperature and external input current, we find that during changes of the firing rate in all of the bifurcation transitions, the calculated slope with respect to the temperature also changes. This is largely dependent on changes in the PRC size that is also related to the slope with respect to the external input current. Furthermore, we find phase transition–like switches of the firing rate with a possible increase of the temperature during the SSL-to-AH bifurcation transition.

The Competing Benefits of Noise and Heterogeneity in Neural Coding

Noise and heterogeneity are both known to benefit neural coding. Stochastic resonance describes how noise, in the form of random fluctuations in a neuron's membrane voltage, can improve neural representations of an input signal. Neuronal heterogeneity refers to variation in any one of a number of neuron parameters and is also known to increase the information content of a population. We explore the interaction between noise and heterogeneity and find that their benefits to neural coding are not independent. Specifically, a neuronal population better represents an input signal when either noise or heterogeneity is added, but adding both does not always improve representation further. To explain this phenomenon, we propose that noise and heterogeneity operate using two shared mechanisms: (1) temporally desynchronizing the firing of neurons in the population and (2) linearizing the response of a population to a stimulus. We first characterize the effects of noise and heterogeneity on the information content of populations of either leaky integrate-and-fire or FitzHugh-Nagumo neurons. We then examine how the mechanisms of desynchronization and linearization produce these effects, and find that they work to distribute information equally across all neurons in the population in terms of both signal timing (desynchronization) and signal amplitude (linearization). Without noise or heterogeneity, all neurons encode the same aspects of the input signal; adding noise or heterogeneity allows neurons to encode complementary aspects of the input signal, thereby increasing information content. The simulations detailed in this letter highlight the importance of heterogeneity and noise in population coding, demonstrate their complex interactions in terms of the information content of neurons, and explain these effects in terms of underlying mechanisms.

Predicting the responses of repetitively firing neurons to current noise

We used phase resetting methods to predict firing patterns of rat subthalamic nucleus (STN) neurons when their rhythmic firing was densely perturbed by noise. We applied sequences of contiguous brief (0.5–2 ms) current pulses with amplitudes drawn from a Gaussian distribution (10–100 pA standard deviation) to autonomously firing STN neurons in slices. Current noise sequences increased the variability of spike times with little or no effect on the average firing rate. We measured the infinitesimal phase resetting curve (PRC) for each neuron using a noise-based method. A phase model consisting of only a firing rate and PRC was very accurate at predicting spike timing, accounting for more than 80% of spike time variance and reliably reproducing the spike-to-spike pattern of irregular firing. An approximation for the evolution of phase was used to predict the effect of firing rate and noise parameters on spike timing variability. It quantitatively predicted changes in variability of interspike intervals with variation in noise amplitude, pulse duration and firing rate over the normal range of STN spontaneous rates. When constant current was used to drive the cells to higher rates, the PRC was altered in size and shape and accurate predictions of the effects of noise relied on incorporating these changes into the prediction. Application of rate-neutral changes in conductance showed that changes in PRC shape arise from conductance changes known to accompany rate increases in STN neurons, rather than the rate increases themselves. Our results show that firing patterns of densely perturbed oscillators cannot readily be distinguished from those of neurons randomly excited to fire from the rest state. The spike timing of repetitively firing neurons may be quantitatively predicted from the input and their PRCs, even when they are so densely perturbed that they no longer fire rhythmically.

Most neurons receive thousands of synaptic inputs per second. Each of these may be individually weak but collectively they shape the temporal pattern of firing by the postsynaptic neuron. If the postsynaptic neuron fires repetitively, its synaptic inputs need not directly trigger action potentials, but may instead control the timing of action potentials that would occur anyway. The phase resetting curve encapsulates the influence of an input on the timing of the next action potential, depending on its time of arrival. We measured the phase resetting curves of neurons in the subthalamic nucleus and used them to accurately predict the timing of action potentials in a phase model subjected to complex input patterns. A simple approximation to the phase model accurately predicted the changes in firing pattern evoked by dense patterns of noise pulses varying in amplitude and pulse duration, and by changes in firing rate. We also showed that the phase resetting curve changes systematically with changes in total neuron conductance, and doing so predicts corresponding changes in firing pattern. Our results indicate that the phase model may accurately represent the temporal integration of complex patterns of input to repetitively firing neurons.

Dense neuron clustering explains connectivity statistics in cortical microcircuits

Local cortical circuits appear highly non-random, but the underlying connectivity rule remains elusive. Here, we analyze experimental data observed in layer 5 of rat neocortex and suggest a model for connectivity from which emerge essential observed non-random features of both wiring and weighting. These features include lognormal distributions of synaptic connection strength, anatomical clustering, and strong correlations between clustering and connection strength. Our model predicts that cortical microcircuits contain large groups of densely connected neurons which we call clusters. We show that such a cluster contains about one fifth of all excitatory neurons of a circuit which are very densely connected with stronger than average synapses. We demonstrate that such clustering plays an important role in the network dynamics, namely, it creates bistable neural spiking in small cortical circuits. Furthermore, introducing local clustering in large-scale networks leads to the emergence of various patterns of persistent local activity in an ongoing network activity. Thus, our results may bridge a gap between anatomical structure and persistent activity observed during working memory and other cognitive processes.

Impact of neuronal properties on network coding: roles of spike initiation dynamics and robust synchrony transfer

Neural networks are more than the sum of their parts, but the properties of those parts are nonetheless important. For instance, neuronal properties affect the degree to which neurons receiving common input will spike synchronously, and whether that synchrony will propagate through the network. Stimulus-evoked synchrony can help or hinder network coding depending on the type of code. In this Perspective, we describe how spike initiation dynamics influence neuronal input-output properties, how those properties affect synchronization, and how synchronization affects network coding. We propose that synchronous and asynchronous spiking can be used to multiplex temporal (synchrony) and rate coding and discuss how pyramidal neurons would be well suited for that task.

Optimal number of stimulation contacts for coordinated reset neuromodulation

In this computational study we investigate coordinated reset (CR) neuromodulation designed for an effective control of synchronization by multi-site stimulation of neuronal target populations. This method was suggested to effectively counteract pathological neuronal synchrony characteristic for several neurological disorders. We study how many stimulation sites are required for optimal CR-induced desynchronization. We found that a moderate increase of the number of stimulation sites may significantly prolong the post-stimulation desynchronized transient after the stimulation is completely switched off. This can, in turn, reduce the amount of the administered stimulation current for the intermittent ON-OFF CR stimulation protocol, where time intervals with stimulation ON are recurrently followed by time intervals with stimulation OFF. In addition, we found that the optimal number of stimulation sites essentially depends on how strongly the administered current decays within the neuronal tissue with increasing distance from the stimulation site. In particular, for a broad spatial stimulation profile, i.e., for a weak spatial decay rate of the stimulation current, CR stimulation can optimally be delivered via a small number of stimulation sites. Our findings may contribute to an optimization of therapeutic applications of CR neuromodulation.

Spike width and frequency alter stability of phase-locking in electrically coupled neurons

The stability of phase-locked states of electrically coupled type-1 phase response curve neurons is studied using piecewise linear formulations for their voltage profile and phase response curves. We find that at low frequency and/or small spike width, synchrony is stable, and antisynchrony unstable. At high frequency and/or large spike width, these phase-locked states switch their stability. Increasing the ratio of spike width to spike height causes the antisynchronous state to transition into a stable synchronous state. We compute the interaction function and the boundaries of stability of both these phase-locked states, and present analytical expressions for them. We also study the effect of phase response curve skewness on the boundaries of synchrony and antisynchrony.

Neural dynamics and information representation in microcircuits of motor cortex

The brain has to analyze and respond to external events that can change rapidly from time to time, suggesting that information processing by the brain may be essentially dynamic rather than static. The dynamical features of neural computation are of significant importance in motor cortex that governs the process of movement generation and learning. In this paper, we discuss these features based primarily on our recent findings on neural dynamics and information coding in the microcircuit of rat motor cortex. In fact, cortical neurons show a variety of dynamical behavior from rhythmic activity in various frequency bands to highly irregular spike firing. Of particular interest are the similarity and dissimilarity of the neuronal response properties in different layers of motor cortex. By conducting electrophysiological recordings in slice preparation, we report the phase response curves (PRCs) of neurons in different cortical layers to demonstrate their layer-dependent synchronization properties. We then study how motor cortex recruits task-related neurons in different layers for voluntary arm movements by simultaneous juxtacellular and multiunit recordings from behaving rats. The results suggest an interesting difference in the spectrum of functional activity between the superficial and deep layers. Furthermore, the task-related activities recorded from various layers exhibited power law distributions of inter-spike intervals (ISIs), in contrast to a general belief that ISIs obey Poisson or Gamma distributions in cortical neurons. We present a theoretical argument that this power law of in vivo neurons may represent the maximization of the entropy of firing rate with limited energy consumption of spike generation. Though further studies are required to fully clarify the functional implications of this coding principle, it may shed new light on information representations by neurons and circuits in motor cortex.

Origin of intrinsic irregular firing in cortical interneurons

Cortical spike trains are highly irregular both during ongoing, spontaneous activity and when driven at high firing rates. There is uncertainty about the source of this irregularity, ranging from intrinsic noise sources in neurons to collective effects in large-scale cortical networks. Cortical interneurons display highly irregular spike times (coefficient of variation of the interspike intervals >1) in response to dc-current injection in vitro. This is in marked contrast to cortical pyramidal cells, which spike highly irregularly in vivo, but regularly in vitro. We show with in vitro recordings and computational models that this is due to the fast activation kinetics of interneuronal K(+) currents. This explanation holds over a wide parameter range and with Gaussian white, power-law, and Ornstein-Uhlenbeck noise. The intrinsically irregular spiking of interneurons could contribute to the irregularity of the cortical network.

Hippocampal CA1 pyramidal neurons exhibit type 1 phase-response curves and type 1 excitability

Phase-resetting properties of neurons determine their functionality as integrators (type 1) vs. resonators (type 2), as well as their synchronization tendencies. We introduce a novel, bias-correction method to estimate the infinitesimal phase-resetting curve (iPRC) and confirm type 1 excitability in hippocampal pyramidal CA1 neurons in vitro by two independent methods. First, PRCs evoked using depolarizing pulses consisted only of advances, consistent with type 1. Second, the frequency/current (f/I) plots showed no minimum frequency, again consistent with type 1. Type 1 excitability was also confirmed by the absence of a resonant peak in the interspike interval histograms derived from the f/I data. The PRC bias correction assumed that the distribution of noisy phase resetting is truncated, because an input cannot advance a spike to a point in time before the input (the causal limit) and successfully removed the statistical bias for delays in the null PRC in response to zero-magnitude input by computing the phase resetting as the mean of the untruncated distribution. The PRC for depolarization peaked at late phases and decreased to zero by the end of the cycle, whereas delays observed in response to hyperpolarization increased monotonically. The bias correction did not affect this difference in shape, which was due instead to the causal limit obscuring the iPRC for depolarization but not hyperpolarization. Our results suggest that weak periodic hyperpolarizing drive can theoretically entrain CA1 pyramidal neurons at any phase but that strong excitation will preferentially phase-lock them with zero time lag.

Higher-order spike triggered analysis of neural oscillators

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.

When Long-Range Zero-Lag Synchronization is Feasible in Cortical Networks

Many studies have reported long-range synchronization of neuronal activity between brain areas, in particular in the beta and gamma bands with frequencies in the range of 14–30 and 40–80 Hz, respectively. Several studies have reported synchrony with zero phase lag, which is remarkable considering the synaptic and conduction delays inherent in the connections between distant brain areas. This result has led to many speculations about the possible functional role of zero-lag synchrony, such as for neuronal communication, attention, memory, and feature binding. However, recent studies using recordings of single-unit activity and local field potentials report that neuronal synchronization may occur with non-zero phase lags. This raises the questions whether zero-lag synchrony can occur in the brain and, if so, under which conditions. We used analytical methods and computer simulations to investigate which connectivity between neuronal populations allows or prohibits zero-lag synchrony. We did so for a model where two oscillators interact via a relay oscillator. Analytical results and computer simulations were obtained for both type I Mirollo–Strogatz neurons and type II Hodgkin–Huxley neurons. We have investigated the dynamics of the model for various types of synaptic coupling and importantly considered the potential impact of Spike-Timing Dependent Plasticity (STDP) and its learning window. We confirm previous results that zero-lag synchrony can be achieved in this configuration. This is much easier to achieve with Hodgkin–Huxley neurons, which have a biphasic phase response curve, than for type I neurons. STDP facilitates zero-lag synchrony as it adjusts the synaptic strengths such that zero-lag synchrony is feasible for a much larger range of parameters than without STDP.

Phase response curves of subthalamic neurons measured with synaptic input and current injection

Infinitesimal phase response curves (iPRCs) provide a simple description of the response of repetitively firing neurons and may be used to predict responses to any pattern of synaptic input. Their simplicity makes them useful for understanding the dynamics of neurons when certain conditions are met. For example, the sizes of evoked phase shifts should scale linearly with stimulus strength, and the form of the iPRC should remain relatively constant as firing rate varies. We measured the PRCs of rat subthalamic neurons in brain slices using corticosubthalamic excitatory postsynaptic potentials (EPSPs; mediated by both AMPA- and NMDA-type receptors) and injected current pulses and used them to calculate the iPRC. These were relatively insensitive to both the size of the stimulus and the cell's firing rate, suggesting that the iPRC can predict the response of subthalamic nucleus cells to extrinsic inputs. However, the iPRC calculated using EPSPs differed from that obtained using current pulses. EPSPs (normalized for charge) were much more effective at altering the phase of subthalamic neurons than current pulses. The difference was not attributable to the extended time course of NMDA receptor-mediated currents, being unaffected by blockade of NMDA receptors. The iPRC provides a good description of subthalamic neurons' response to input, but iPRCs are best estimated using synaptic inputs rather than somatic current injection.

Identifying type I excitability using dynamics of stochastic neural firing patterns

The stochastic firing patterns are simulated near saddle-node bifurcation on an invariant cycle corresponding to type I excitability in stochastic Morris-Lecar model. In absence of external periodic signal, the stochastic firing manifests continuous distribution in ISI histogram (ISIH), whose amplitude at first increases sharply and then decreases exponentially. In presence of the external periodic signal, stochastic firing patterns appear as two cases of integer multiple firing with multiple discrete peaks in ISIH. One manifests perfect exponential decay in all peaks and the other imperfect exponential decay except a lower first peak. These stochastic firing patterns simulated with or without external periodic signal can be demonstrated in the experiments on rat hippocampal CA1 pyramidal neurons. The exponential decay laws in the multiple peaks are also acquired using probability analysis method. The perfect decay law is determined by the independent characteristic within the firing while the imperfect decay law is from the inhibitory effect. In addition, the stochastic firing patterns corresponding to type I excitability are compared to those of type II excitability. The results not only reveal the dynamics of stochastic firing patterns with or without external signal corresponding to type I excitability, but also provide practical indicators to availably identify type I excitability.

The phase response of the cortical slow oscillation

Cortical slow oscillations occur in the mammalian brain during deep sleep and have been shown to contribute to memory consolidation, an effect that can be enhanced by electrical stimulation. As the precise underlying working mechanisms are not known it is desired to develop and analyze computational models of slow oscillations and to study the response to electrical stimuli. In this paper we employ the conductance based model of Compte et al. (J Neurophysiol 89:2707-2725, 2003) to study the effect of electrical stimulation. The population response to electrical stimulation depends on the timing of the stimulus with respect to the state of the slow oscillation. First, we reproduce the experimental results of electrical stimulation in ferret brain slices by Shu et al. (Nature 423:288-293, 2003) from the conductance based model. We then numerically obtain the phase response curve for the conductance based network model to quantify the network's response to weak stimuli. Our results agree with experiments in vivo and in vitro that show that sensitivity to stimulation is weaker in the up than in the down state. However, we also find that within the up state stimulation leads to a shortening of the up state, or phase advance, whereas during the up-down transition a prolongation of up states is possible, resulting in a phase delay. Finally, we compute the phase response curve for the simple mean-field model by Ngo et al. (EPL Europhys Lett 89:68002, 2010) and find that the qualitative shape of the PRC is preserved, despite its different mechanism for the generation of slow oscillations.

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.

Power-law inter-spike interval distributions infer a conditional maximization of entropy in cortical neurons

The brain is considered to use a relatively small amount of energy for its efficient information processing. Under a severe restriction on the energy consumption, the maximization of mutual information (MMI), which is adequate for designing artificial processing machines, may not suit for the brain. The MMI attempts to send information as accurate as possible and this usually requires a sufficient energy supply for establishing clearly discretized communication bands. Here, we derive an alternative hypothesis for neural code from the neuronal activities recorded juxtacellularly in the sensorimotor cortex of behaving rats. Our hypothesis states that in vivo cortical neurons maximize the entropy of neuronal firing under two constraints, one limiting the energy consumption (as assumed previously) and one restricting the uncertainty in output spike sequences at given firing rate. Thus, the conditional maximization of firing-rate entropy (CMFE) solves a tradeoff between the energy cost and noise in neuronal response. In short, the CMFE sends a rich variety of information through broader communication bands (i.e., widely distributed firing rates) at the cost of accuracy. We demonstrate that the CMFE is reflected in the long-tailed, typically power law, distributions of inter-spike intervals obtained for the majority of recorded neurons. In other words, the power-law tails are more consistent with the CMFE rather than the MMI. Thus, we propose the mathematical principle by which cortical neurons may represent information about synaptic input into their output spike trains.

Temperature-modulated synchronization transition in coupled neuronal oscillators

We study two firing properties to characterize the activities of a neuron: frequency-current (f-I) curves and phase response curves (PRCs), with variation in the intrinsic temperature scaling parameter (μ) controlling the opening and closing of ionic channels. We show a peak of the firing frequency for small μ in a class I neuron with the I value immediately after the saddle-node bifurcation, which is entirely different from previous experimental reports as well as model studies. The PRC takes a type II form on a logarithmic f-I curve when μ is small. Then, we analyze the synchronization phenomena in a two-neuron network using the phase-reduction method. We find common μ-dependent transition and bifurcation of synchronizations, regardless of the values of I. Such results give us helpful insight into synchronizations tuned with a sinusoidal-wave temperature modulation on neurons.

Engineering the synchronization of neuron action potentials using global time-delayed feedback stimulation

We experimentally demonstrate the use of continuous, time-delayed, feedback stimulation for controlling the synchronization of neuron action potentials. Phase-based models were experimentally constructed from a single synaptically isolated cultured hippocampal neuron. These models were used to determine the stimulation parameters necessary to produce the desired synchronization behavior in the action potentials of a pair of neurons coupled through a global time-delayed interaction. Measurements made using a dynamic clamp system confirm the generation of the synchronized states predicted by the experimentally constructed phase model. This model was then utilized to extrapolate the feedback stimulation parameters necessary to disrupt the action potential synchronization of a large population of globally interacting neurons.

Synchrony makes neurons fire in sequence, and stimulus properties determine who is ahead

The synchronized activity of cortical neurons often features spike delays of several milliseconds. Usually, these delays are considered too small to play a role in cortical computations. Here, we use simultaneous recordings of spiking activity from up to 12 neurons to show that, in the cat visual cortex, the pairwise delays between neurons form a preferred order of spiking, called firing sequence. This sequence spans up to ∼ 15 ms and is referenced not to external events but to the internal cortical activity (e.g., beta/gamma oscillations). Most importantly, the preferred sequence of firing changed consistently as a function of stimulus properties. During beta/gamma oscillations, the reliability of firing sequences increased and approached that of firing rates. This suggests that, in the visual system, short-lived spatiotemporal patterns of spiking defined by consistent delays in synchronized activity occur with sufficient reliability to complement firing rates as a neuronal code.

Phase-response curves and synchronized neural 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.

Neurophysiological and computational principles of cortical rhythms in cognition

Synchronous rhythms represent a core mechanism for sculpting temporal coordination of neural activity in the brain-wide network. This review focuses on oscillations in the cerebral cortex that occur during cognition, in alert behaving conditions. Over the last two decades, experimental and modeling work has made great strides in elucidating the detailed cellular and circuit basis of these rhythms, particularly gamma and theta rhythms. The underlying physiological mechanisms are diverse (ranging from resonance and pacemaker properties of single cells to multiple scenarios for population synchronization and wave propagation), but also exhibit unifying principles. A major conceptual advance was the realization that synaptic inhibition plays a fundamental role in rhythmogenesis, either in an interneuronal network or in a reciprocal excitatory-inhibitory loop. Computational functions of synchronous oscillations in cognition are still a matter of debate among systems neuroscientists, in part because the notion of regular oscillation seems to contradict the common observation that spiking discharges of individual neurons in the cortex are highly stochastic and far from being clocklike. However, recent findings have led to a framework that goes beyond the conventional theory of coupled oscillators and reconciles the apparent dichotomy between irregular single neuron activity and field potential oscillations. From this perspective, a plethora of studies will be reviewed on the involvement of long-distance neuronal coherence in cognitive functions such as multisensory integration, working memory, and selective attention. Finally, implications of abnormal neural synchronization are discussed as they relate to mental disorders like schizophrenia and autism.

Phase-resetting curve determines how BK currents affect neuronal firing

BK channels are large conductance potassium channels gated by calcium and voltage. Paradoxically, blocking these channels has been shown experimentally to increase or decrease the firing rate of neurons, depending on the neural subtype and brain region. The mechanism for how this current can alter the firing rates of different neurons remains poorly understood. Using phase-resetting curve (PRC) theory, we determine when BK channels increase or decrease the firing rates in neural models. The addition of BK currents always decreases the firing rate when the PRC has only a positive region. When the PRC has a negative region (type II), BK currents can increase the firing rate. The influence of BK channels on firing rate in the presence of other conductances, such as I(m) and I(h), as well as with different amplitudes of depolarizing input, were also investigated. These results provide a formal explanation for the apparently contradictory effects of BK channel antagonists on firing rates.

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.

Synchronization engineering: tuning the phase relationship between dissimilar oscillators using nonlinear feedback

A mild, nonlinear, time-delayed feedback signal was applied to two heterogeneous oscillators in order to synchronize their frequencies with an arbitrary and controllable phase difference. The feedback was designed using phase models constructed from experimental measurements of the intrinsic dynamical properties of the oscillators. The feedback signal produced an interaction function that corresponds to the desired collective behaviour. The synchronized phase difference between the elements can be tuned to any value on the interval 0 and 2pi by shifting the phase of the interaction function using the feedback delay. Numerical simulations were conducted and experiments carried out with electrochemical oscillators.

Bayesian estimation of phase response curves

Phase response curve (PRC) of an oscillatory neuron describes the response of the neuron to external perturbation. The PRC is useful to predict synchronized dynamics of neurons; hence, its measurement from experimental data attracts increasing interest in neural science. This paper introduces a Bayesian method for estimating PRCs from data, which allows for the correlation of errors in explanatory and response variables of the PRC. The method is implemented with a replica exchange Monte Carlo technique; this avoids local minima and enables efficient calculation of posterior averages. A test with artificial data generated by the noisy Morris-Lecar equation shows that the proposed method outperforms conventional regression that ignores errors in the explanatory variable. Experimental data from the pyramidal cells in the rat motor cortex is also analyzed with the method; a case is found where the result with the proposed method is considerably different from that obtained by conventional regression.

The response of a classical Hodgkin-Huxley neuron to an inhibitory input pulse

A population of uncoupled neurons can often be brought close to synchrony by a single strong inhibitory input pulse affecting all neurons equally. This mechanism is thought to underlie some brain rhythms, in particular gamma frequency (30–80 Hz) oscillations in the hippocampus and neocortex. Here we show that synchronization by an inhibitory input pulse often fails for populations of classical Hodgkin–Huxley neurons. Our reasoning suggests that in general, synchronization by inhibitory input pulses can fail when the transition of the target neurons from rest to spiking involves a Hopf bifurcation, especially when inhibition is shunting, not hyperpolarizing. Surprisingly, synchronization is more likely to fail when the inhibitory pulse is stronger or longer-lasting. These findings have potential implications for the question which neurons participate in brain rhythms, in particular in gamma oscillations.

A comparison of methods to determine neuronal phase-response curves

The phase-response curve (PRC) is an important tool to determine the excitability type of single neurons which reveals consequences for their synchronizing properties. We review five methods to compute the PRC from both model data and experimental data and compare the numerically obtained results from each method. The main difference between the methods lies in the reliability which is influenced by the fluctuations in the spiking data and the number of spikes available for analysis. We discuss the significance of our results and provide guidelines to choose the best method based on the available data.

Phase response curve analysis of a full morphological globus pallidus neuron model reveals distinct perisomatic and dendritic modes of synaptic integration

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 interspike 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.

Fluctuation of gamma-band phase synchronization within the auditory cortex in schizophrenia

OBJECTIVE

To study the phase stability of the 40Hz auditory steady-state response (ASSR) in Sz, and in addition, to investigate inter-hemispheric phase synchronization using ipsilateral and contralateral hemisphere gamma band ASSRs.

METHODS

Whole head magnetoencephalography (MEG) was used to detect ASSR from both hemispheres in Sz patients and their control counterparts. Source localization, spatial and temporal filtering were performed to infer gamma band activity from the neural generators of the ASSR. The response gamma band phase stability relative to a reference signal was quantified using the phase synchronization index (PSI).

RESULTS

Results indicated reduced phase synchronization of the ASSR and the stimulus reference signal in Sz patients compared to control subjects, in addition to reduced inter-hemispheric phase synchronization between contralateral and ipsilateral hemispheric responses in Sz patients.

CONCLUSIONS

Greater intra and inter hemispheric fluctuations of ASSR gamma band phase synchronization in Sz add to previous studies suggesting timing deficiencies within neural populations, possibly caused by impairments of neural network parameters.

SIGNIFICANCE

This study provides experimental support that may aid in understanding the dynamics of neural phase synchrony caused by modifications of underlying neurotransmitter systems, as reflected in disease states such as schizophrenia.

Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold

Information is transmitted in the brain through various kinds of neurons that respond differently to the same signal. Full characteristics including cognitive functions of the brain should ultimately be comprehended by building simulators capable of precisely mirroring spike responses of a variety of neurons. Neuronal modeling that had remained on a qualitative level has recently advanced to a quantitative level, but is still incapable of accurately predicting biological data and requires high computational cost. In this study, we devised a simple, fast computational model that can be tailored to any cortical neuron not only for reproducing but also for predicting a variety of spike responses to greatly fluctuating currents. The key features of this model are a multi-timescale adaptive threshold predictor and a nonresetting leaky integrator. This model is capable of reproducing a rich variety of neuronal spike responses, including regular spiking, intrinsic bursting, fast spiking, and chattering, by adjusting only three adaptive threshold parameters. This model can express a continuous variety of the firing characteristics in a three-dimensional parameter space rather than just those identified in the conventional discrete categorization. Both high flexibility and low computational cost would help to model the real brain function faithfully and examine how network properties may be influenced by the distributed characteristics of component neurons.

Comparison and regulation of neuronal synchronization for various STDP rules

We discuss effects of various experimentally supported STDP learning rules on frequency synchronization of two unidirectional coupled neurons systematically. First, we show that synchronization windows for all STDP rules cannot be enhanced compared to constant connection under the same model. Then, we explore the influence of learning parameters on synchronization window and find optimal parameters that lead to the widest window. Our findings indicate that synchronization strongly depends on the specific shape and the parameters of the STDP update rules. Thus, we give some explanations by analyzing the synchronization mechanisms for various STDP rules finally.

Functional phase response curves: a method for understanding synchronization of adapting neurons

Phase response curves (PRCs) for a single neuron are often used to predict the synchrony of mutually coupled neurons. Previous theoretical work on pulse-coupled oscillators used single-pulse perturbations. We propose an alternate method in which functional PRCs (fPRCs) are generated using a train of pulses applied at a fixed delay after each spike, with the PRC measured when the phasic relationship between the stimulus and the subsequent spike in the neuron has converged. The essential information is the dependence of the recovery time from pulse onset until the next spike as a function of the delay between the previous spike and the onset of the applied pulse. Experimental fPRCs in Aplysia pacemaker neurons were different from single-pulse PRCs, principally due to adaptation. In the biological neuron, convergence to the fully adapted recovery interval was slower at some phases than that at others because the change in the effective intrinsic period due to adaptation changes the effective phase resetting in a way that opposes and slows the effects of adaptation. The fPRCs for two isolated adapting model neurons were used to predict the existence and stability of 1:1 phase-locked network activity when the two neurons were coupled. A stability criterion was derived by linearizing a coupled map based on the fPRC and the existence and stability criteria were successfully tested in two-simulated-neuron networks with reciprocal inhibition or excitation. The fPRC is the first PRC-based tool that can account for adaptation in analyzing networks of neural oscillators.

Asymmetry in pulse-coupled oscillators with delay

We studied the dynamics of synchronization in asymmetrically coupled neural oscillators with time delay. Stability analysis revealed that symmetric excitatory coupling results in synchrony at multiple phase relations. Asymmetry yields two saddle-node bifurcations of the stable states when coupling is asymmetric. By contrast, with inhibitory coupling only in phase or antiphase is stable as long as coupling is symmetric. Otherwise, these stable states shift or even vanish. The reduced bistability range suggests the beneficial role of asymmetric coupling for reliable neural information transfer.