Effect of phase response curve skew on synchronization with and without conduction delays

Constructive role of shot noise in the collective dynamics of neural networks

Finite-size effects may significantly influence the collective dynamics of large populations of neurons. Recently, we have shown that in globally coupled networks these effects can be interpreted as additional common noise term, the so-called shot noise, to the macroscopic dynamics unfolding in the thermodynamic limit. Here, we continue to explore the role of the shot noise in the collective dynamics of globally coupled neural networks. Namely, we study the noise-induced switching between different macroscopic regimes. We show that shot noise can turn attractors of the infinitely large network into metastable states whose lifetimes smoothly depend on the system parameters. A surprising effect is that the shot noise modifies the region where a certain macroscopic regime exists compared to the thermodynamic limit. This may be interpreted as a constructive role of the shot noise since a certain macroscopic state appears in a parameter region where it does not exist in an infinite network.

Role of Interaction Delays in the Synchronization of Inhibitory Networks

Neural oscillations provide a means for efficient and flexible communication among different brain areas. Understanding the mechanisms of the generation of brain oscillations is crucial to determine principles of communication and information transfer in the brain circuits. It is well known that the inhibitory neurons play a major role in the generation of oscillations in the gamma range, in pure inhibitory networks, or in the networks composed of excitatory and inhibitory neurons. In this study, we explore the impact of different parameters and, in particular, the delay in the transmission of the signals between the neurons, on the dynamics of inhibitory networks. We show that increasing delay in a reasonable range increases the synchrony and stabilizes the oscillations. Unstable gamma oscillations characterized by a highly variable amplitude of oscillations can be observed in an intermediate range of delays. We show that in this range of delays, other experimentally observed phenomena such as sparse firing, variable amplitude and period, and the correlation between the instantaneous amplitude and period could be observed. The results broaden our understanding of the mechanism of the generation of the gamma oscillations in the inhibitory networks, known as the ING (interneuron-gamma) mechanism.

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.

Analyzing the competition of gamma rhythms with delayed pulse-coupled oscillators in phase representation

Networks of neurons can generate oscillatory activity as result of various types of coupling that lead to synchronization. A prominent type of oscillatory activity is gamma (30-80 Hz) rhythms, which may play an important role in neuronal information processing. Two mechanisms have mainly been proposed for their generation: (1) interneuron network gamma (ING) and (2) pyramidal-interneuron network gamma (PING). In vitro and in vivo experiments have shown that both mechanisms can exist in the same cortical circuits. This raises the questions: How do ING and PING interact when both can in principle occur? Are the network dynamics a superposition, or do ING and PING interact in a nonlinear way and if so, how? In this article, we first generalize the phase representation for nonlinear one-dimensional pulse coupled oscillators as introduced by Mirollo and Strogatz to type II oscillators whose phase response curve (PRC) has zero crossings. We then give a full theoretical analysis for the regular gamma-like oscillations of simple networks consisting of two neural oscillators, an "E neuron" mimicking a synchronized group of pyramidal cells, and an "I neuron" representing such a group of interneurons. Motivated by experimental findings, we choose the E neuron to have a type I PRC [leaky integrate-and-fire (LIF) neuron], while the I neuron has either a type I or type II PRC (LIF or "sine" neuron). The phase representation allows us to define in a simple manner scenarios of interaction between the two neurons, which are independent of the types and the details of the neuron models. The presence of delay in the couplings leads to an increased number of scenarios relevant for gamma-like oscillatory patterns. We analytically derive the set of such scenarios and describe their occurrence in terms of parameter values such as synaptic connectivity and drive to the E and I neurons. The networks can be tuned to oscillate in an ING or PING mode. We focus particularly on the transition region where both rhythms compete to govern the network dynamics and compare with oscillations in reduced networks, which can only generate either ING or PING. Our analytically derived oscillation frequency diagrams indicate that except for small coexistence regions, the networks generate ING if the oscillation frequency of the reduced ING network exceeds that of the reduced PING network, and vice versa. For networks with the LIF I neuron, the network oscillation frequency slightly exceeds the frequencies of corresponding reduced networks, while it lies between them for networks with the sine I neuron. In networks oscillating in ING (PING) mode, the oscillation frequency responds faster to changes in the drive to the I (E) neuron than to changes in the drive to the E (I) neuron. This finding suggests a method to analyze which mechanism governs an observed network oscillation. Notably, also when the network operates in ING mode, the E neuron can spike before the I neuron such that relative spike times of the pyramidal cells and the interneurons alone are not conclusive for distinguishing ING and PING.

Effect of Phase Response Curve Shape and Synaptic Driving Force on Synchronization of Coupled Neuronal Oscillators

The role of the phase response curve (PRC) shape on the synchrony of synaptically coupled oscillating neurons is examined. If the PRC is independent of the phase, because of the synaptic form of the coupling, synchrony is found to be stable for both excitatory and inhibitory coupling at all rates, whereas the antisynchrony becomes stable at low rates. A faster synaptic rise helps extend the stability of antisynchrony to higher rates. If the PRC is not constant but has a profile like that of a leaky integrate-and-fire model, then, in contrast to the earlier reports that did not include the voltage effects, mutual excitation could lead to stable synchrony provided the synaptic reversal potential is below the voltage level the neuron would have reached in the absence of the interaction and threshold reset. This level is controlled by the applied current and the leakage parameters. Such synchrony is contingent on significant phase response (that would result, for example, by a sharp PRC jump) occurring during the synaptic rising phase. The rising phase, however, does not contribute significantly if it occurs before the voltage spike reaches its peak. Then a stable near-synchronous state can still exist between type 1 PRC neurons if the PRC shows a left skewness in its shape. These results are examined comprehensively using perfect integrate-and-fire, leaky integrate-and-fire, and skewed PRC shapes under the assumption of the weakly coupled oscillator theory applied to synaptically coupled neuron models.

Globally attracting synchrony in a network of oscillators with all-to-all inhibitory pulse coupling

The synchronization tendencies of networks of oscillators have been studied intensely. We assume a network of all-to-all pulse-coupled oscillators in which the effect of a pulse is independent of the number of oscillators that simultaneously emit a pulse and the normalized delay (the phase resetting) is a monotonically increasing function of oscillator phase with the slope everywhere less than 1 and a value greater than 2φ-1, where φ is the normalized phase. Order switching cannot occur; the only possible solutions are globally attracting synchrony and cluster solutions with a fixed firing order. For small conduction delays, we prove the former stable and all other possible attractors nonexistent due to the destabilizing discontinuity of the phase resetting at a phase of 0.

Intrinsic Cellular Properties and Connectivity Density Determine Variable Clustering Patterns in Randomly Connected Inhibitory Neural Networks

The plethora of inhibitory interneurons in the hippocampus and cortex play a pivotal role in generating rhythmic activity by clustering and synchronizing cell firing. Results of our simulations demonstrate that both the intrinsic cellular properties of neurons and the degree of network connectivity affect the characteristics of clustered dynamics exhibited in randomly connected, heterogeneous inhibitory networks. We quantify intrinsic cellular properties by the neuron's current-frequency relation (IF curve) and Phase Response Curve (PRC), a measure of how perturbations given at various phases of a neurons firing cycle affect subsequent spike timing. We analyze network bursting properties of networks of neurons with Type I or Type II properties in both excitability and PRC profile; Type I PRCs strictly show phase advances and IF curves that exhibit frequencies arbitrarily close to zero at firing threshold while Type II PRCs display both phase advances and delays and IF curves that have a non-zero frequency at threshold. Type II neurons whose properties arise with or without an M-type adaptation current are considered. We analyze network dynamics under different levels of cellular heterogeneity and as intrinsic cellular firing frequency and the time scale of decay of synaptic inhibition are varied. Many of the dynamics exhibited by these networks diverge from the predictions of the interneuron network gamma (ING) mechanism, as well as from results in all-to-all connected networks. Our results show that randomly connected networks of Type I neurons synchronize into a single cluster of active neurons while networks of Type II neurons organize into two mutually exclusive clusters segregated by the cells' intrinsic firing frequencies. Networks of Type II neurons containing the adaptation current behave similarly to networks of either Type I or Type II neurons depending on network parameters; however, the adaptation current creates differences in the cluster dynamics compared to those in networks of Type I or Type II neurons. To understand these results, we compute neuronal PRCs calculated with a perturbation matching the profile of the synaptic current in our networks. Differences in profiles of these PRCs across the different neuron types reveal mechanisms underlying the divergent network dynamics.

Cooperation and competition of gamma oscillation mechanisms

Oscillations of neuronal activity in different frequency ranges are thought to reflect important aspects of cortical network dynamics. Here we investigate how various mechanisms that contribute to oscillations in neuronal networks may interact. We focus on networks with inhibitory, excitatory, and electrical synapses, where the subnetwork of inhibitory interneurons alone can generate interneuron gamma (ING) oscillations and the interactions between interneurons and pyramidal cells allow for pyramidal-interneuron gamma (PING) oscillations. What type of oscillation will such a network generate? We find that ING and PING oscillations compete: The mechanism generating the higher oscillation frequency "wins"; it determines the frequency of the network oscillation and suppresses the other mechanism. For type I interneurons, the network oscillation frequency is equal to or slightly above the higher of the ING and PING frequencies in corresponding reduced networks that can generate only either of them; if the interneurons belong to the type II class, it is in between. In contrast to ING and PING, oscillations mediated by gap junctions and oscillations mediated by inhibitory synapses may cooperate or compete, depending on the type (I or II) of interneurons and the strengths of the electrical and chemical synapses. We support our computer simulations by a theoretical model that allows a full theoretical analysis of the main results. Our study suggests experimental approaches to deciding to what extent oscillatory activity in networks of interacting excitatory and inhibitory neurons is dominated by ING or PING oscillations and of which class the participating interneurons are.

Phase-resetting as a tool of information transmission

Models of information transmission in the brain largely rely on firing rate codes. The abundance of oscillatory activity in the brain suggests that information may be also encoded using the phases of ongoing oscillations. Sensory perception, working memory and spatial navigation have been hypothesized to use phase codes, and cross-frequency coordination and phase synchronization between brain areas have been proposed to gate the flow of information. Phase codes generally require the phase of the oscillations to be reset at specific reference points for consistent coding, and coordination between oscillators requires favorable phase resetting characteristics. Recent evidence supports a role for neural oscillations in providing temporal reference windows that allow for correct parsing of phase-coded information.