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Transition to chaos in neuronal networks with balanced inhibition and excitation
Jonathan Kadmon1,2 and Haim Sompolinsky1,2,3
1Racah Institute of Physics and 2 Edmond and Lily Safra Center for Neuroscience
The Hebrew University of Jerusalem, Israel
3Center for Brain Science, Harvard University, Cambridge, MA

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 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 non-linearity are known to exhibit chaotic dynamics when the synaptic gain (i.e., connection variance) is sufficiently large. In the limit of infinitely large network, there is a sharp transition from a fixed point to chaos, as the synaptic gain reaches a critical value. Near 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 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, in line with known constraints in many biological systems. Using Dynamic Mean Field Theory, we study the conditions for the existence a transition from a state with static firing rates, to chaotic rate fluctuations and characterize the properties of systems near this transition. We show that dilute excitatory-inhibitory 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 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.