Recurrent networks of non-linear rate units can show a large variety of dynamical regimes depending on the structure of their synaptic connectivity. A classical study  has shown that randomly coupled networks can exhibit a transition from a fixed point to chaotic activity, where firing rates fluctuate in time and across units. More recent analyses have highlighted the potential computational capacities of this novel dynamical regime  .
Most previous studies were conducted on highly simplified random networks, where excitation and inhibition are not segregated. Furthermore, allowing firing rates to be positive or negative quantities, those models generate a globally symmetric dynamics, characterised by a zero mean firing rate both in the static and in the chaotic regime. The transition is then characterised solely in terms of second order statistics, i.e. the firing rate auto-correlation function. To help bridge the gap between the model and a more realistic cortical system, here we investigate the dynamics of a rate network that includes additional biological constraints. As a first step, we introduce a sparsely connected synaptic matrix with segregated inhibition and excitation, obeying Dale's law. Secondly, we restrict firing rates to positive values, by using positively defined activation functions.
Extending the dynamical mean-field theory, we show that the network dynamics can be effectively described through two coupled self-consistent equations for the mean activity and the auto-correlation function. Above the transition, in contrast to the classical model, we find that fluctuations strongly influence the value of the mean firing rate. Indeed, even in an inhibition-dominated network, as synaptic coupling is pushed above the critical value, fluctuations are strong enough to drive an increase in the mean firing rate. Moreover, we show that above the instability two different fluctuating regimes can be distinguished: for moderate values of the synaptic coupling, recurrent inhibition is enough to stabilise fluctuations; for strong couplings, firing rates are stabilised solely by the upper bound imposed on activity. Finally, in order to compare the dynamical features of the rate model with the activity in networks of spiking units, we adapt the mean-field framework to include the basic properties of a network of integrate-and-fire neurons .
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