Firing rate distributions in plastic networks of spiking neurons

In recurrent networks of leaky integrate-and-fire (LIF) neurons, mean-field theory has proven successful in describing various statistical properties of neuronal activity at equilibrium, such as firing rate distributions. Mean-field theory has been applied to networks in which either the synaptic weights are homogeneous across synapses and the number of incoming connections of individual neurons is heterogeneous, or vice versa. Here we extend the previous mean-field formalisms to treat networks in which these two sources of structural heterogeneity occur simultaneously, including networks whose synapses are subject to plastic, activity-dependent modulation. The plasticity in our model is mediated by the introduction of one spike trace per neuron: a chemical signal that is released every time the neuron emits a spike and which is degraded over time. The temporal evolution of the trace is controlled by its degradation rate $r_p$ and by the neuron's underlying firing rate $\nu$. When the ratio $\alpha=\nu / r_p$ tends to infinity, the trace can be rescaled to be a reliable estimation of the neuron's firing rate. In this regime, the value of any synaptic weight at equilibrium is a function of the pre- and post-synaptic firing rates, and this relation can be used in the mean-field formalism. The solution to the mean-field equations specifies the firing rate and synaptic weight distributions at equilibrium. These equations are exact in the limit of reliable traces but they already provide accurate results when the degradation rate lies within a reasonable range, as we show by comparison with simulations of the full neuronal dynamics in networks composed of excitatory and inhibitory LIF neurons. Overall, this work offers a way to explore and better understand the way in which plasticity shapes both activity and structure in neuronal networks.