How Chaotic is the Balanced State?

Sven Jahnke

doi:10.3389/neuro.10.013.2009

Abstract

Large sparse circuits of spiking neurons exhibit a balanced state of highly irregular activity under a wide range of conditions. It occurs likewise in sparsely connected random networks that receive excitatory external inputs and recurrent inhibition as well as in networks with mixed recurrent inhibition and excitation. Here we analytically investigate this irregular dynamics in finite networks keeping track of all individual spike times and the identities of individual neurons. For delayed, purely inhibitory interactions we show that the irregular dynamics is not chaotic but stable. Moreover, we demonstrate that after long transients the dynamics converges towards periodic orbits and that every generic periodic orbit of these dynamical systems is stable. We investigate the collective irregular dynamics upon increasing the time scale of synaptic responses and upon iteratively replacing inhibitory by excitatory interactions. Whereas for small and moderate time scales as well as for few excitatory interactions, the dynamics stays stable, there is a smooth transition to chaos if the synaptic response becomes sufficiently slow (even in purely inhibitory networks) or the number of excitatory interactions becomes too large. These results indicate that chaotic and stable dynamics are equally capable of generating the irregular neuronal activity. More generally, chaos apparently is not essential for generating the high irregularity of balanced activity, and we suggest that a mechanism different from chaos and stochasticity significantly contributes to irregular activity in cortical circuits.

Highlights

Most neurons in the brain communicate by emitting and receiving electrical pulses, called action potentials or spikes, via chemically operating synaptic connections
It is characterized by individual neurons that display largely fluctuating membrane potentials and highly variable inter-spike-intervals (ISIs) as well as by low correlations between the neurons (v.Vreeswijk and Sompolinsky, 1996, 1998; Brunel, 2000; Vogels and Abbott, 2005; Kumar et al, 2007). This dynamical state seemed to be in contradiction to cortical anatomy, where each neuron receives a huge number of synapses, typically 103–104 (Braitenberg and Schüz, 1998): One might expect that a large number of uncorrelated, or weakly correlated synaptic inputs to one neuron, given the central limit theorem, sums up to a regular total input signal with only small relative fluctuations, excluding the emergence of irregular dynamics
Irregular spiking activity that robustly arises in balanced state models constitutes a generic feature of cortical dynamics

Summary

Introduction

Most neurons in the brain communicate by emitting and receiving electrical pulses, called action potentials or spikes, via chemically operating synaptic connections. 0. The neuronal dynamics is smooth except at times when events, namely sendings or receivings of spikes happen.

Results

Conclusion