# Theory of randomly connected networks with excitation-inhibition balance.

van Vreeswijk, C.A. and Sompolinsky, H. (1995)

Israel J. of Med. Sci. 31:740. (1995)

The source of temporal irregularity in neuronal activity in the brain is a focus of current theoretical and experimental activity. In this work we study the hypothesis that these fluctuations originate from the chaotic dynamics of randomly connected networks, with an approximate balance between excitatory and inhibitory currents. We study a network of simple model neurons consisting of excitatory and inhibitory populations. Each neuron is connected randomly with many excitatory and inhibitory neurons, but the average number of connections per neuron, $K$, is small compared to the total number of neurons, $N$\@. The strength of the synapses is of order $1/\sqrt{K}$\@. The properties of the network have been solved analytically. We show that if the excitatory and inhibitory currents are approximately balanced the network is in a chaotic state. We investigate under which conditions there is a stable state in which the neurons fire at low rates. We show that in these states the firing rates of the individual neurons are distributed over a broad range. The neurons fire irregularly and the interspike interval has a long exponential tail. We show that these results also hold for networks that receive input. The dependence of the network activity on the input is determined. We compare the activity of the network for different inputs. We show analytically that even when two input spike trains have small differences, the precise timing of the neuronal firing is uncorrelated. We also calculate the distribution of equal-time and time-delayed cross-correlations in the network. We find that in this state, the cross-correlations are, on average, of order $K/N$, but their typical value is of order $1/\sqrt{N}$\@.