Transition to Chaos in Random Neuronal Networks

Abstract

Firing patterns in the central nervous system often exhibit strong temporal irregularity and considerable heterogeneity in time-averaged response properties. Previous studies suggested that these properties are the outcome of the intrinsic chaotic dynamics of the neural circuits. Indeed, simplified rate-based neuronal networks with synaptic connections drawn from Gaussian distribution and sigmoidal nonlinearity are known to exhibit chaotic dynamics when the synaptic gain (i.e., connection variance) is sufficiently large. In the limit of an infinitely large network, there is a sharp transition from a fixed point to chaos, as the synaptic gain reaches a critical value. Near the onset, chaotic fluctuations are slow, analogous to the ubiquitous, slow irregular fluctuations observed in the firing rates of many cortical circuits. However, the existence of a transition from a fixed point to chaos in neuronal circuit models with more realistic architectures and firing dynamics has not been established. In this work, we investigate rate-based dynamics of neuronal circuits composed of several subpopulations with randomly diluted connections. Nonzero connections are either positive for excitatory neurons or negative for inhibitory ones, while single neuron output is strictly positive with output rates rising as a power law above threshold, in line with known constraints in many biological systems. Using dynamic mean field theory, we find the phase diagram depicting the regimes of stable fixed-point, unstable-dynamic, and chaotic-rate fluctuations. We focus on the latter and characterize the properties of systems near this transition. We show that dilute excitatoryinhibitory architectures exhibit the same onset to chaos as the single population with Gaussian connectivity. In these architectures, the large mean excitatory and inhibitory inputs dynamically balance each other, amplifying the effect of the residual fluctuations. Importantly, the existence of a transition to chaos and its critical properties depend on the shape of the single-neuron nonlinear input-output transfer function, near firing threshold. In particular, for nonlinear transfer functions with a sharp rise near threshold, the transition to chaos disappears in the limit of a large network; instead, the system exhibits chaotic fluctuations even for small synaptic gain. Finally, we investigate transition to chaos in network models with spiking dynamics. We show that when synaptic time constants are slow relative to the mean inverse firing rates, the network undergoes a transition from fast spiking fluctuations with constant rates to a state where the firing rates exhibit chaotic fluctuations, similar to the transition predicted by rate-based dynamics. Systems with finite synaptic time constants and firing rates exhibit a smooth transition from a regime dominated by stationary firing rates to a regime of slow rate fluctuations. This smooth crossover obeys scaling properties, similar to crossover phenomena in statistical mechanics. The theoretical results are sup...

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Is chaos making a difference? Synchronization transitions in chaotic and nonchaotic neuronal networks

“…It is generally accepted that the inherent instability of chaos in 4 nonlinear systems dynamics, facilitates the extraordinary ability of neural systems to 5 respond quickly to changes in their external inputs [3], to make transitions from one 6 pattern of behavior to another when the environment is altered [4], and to create a rich neurons [3,[13][14][15][16][17][18][19][20][21]. The first type of chaotic dynamics in neural systems is typically 14 accompanied by microscopic chaotic dynamics at the level of individual oscillators.…”

“…Synchronous chaos has been demonstrated in networks of 18 both biophysical and non-biophysical neurons [3,13,15,17,[22][23][24], where neurons display 19 synchronous chaotic firing-rate fluctuations. The latter cases, the chaotic behavior is a 20 result of network connectivity, since isolated neurons do not display chaotic dynamics or 21 burst firing. More recently, a totally different mechanism showed that asynchronous 22 chaos, where neurons exhibit asynchronous chaotic firing-rate fluctuations, emerge 23 generically from balanced networks with multiple time scales synaptic dynamics [20].…”

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