Stability of asynchronous firing states in networks with synaptic adaptation

We construct a mean field theory for low-rate asynchronous firing states in networks consisting of excitatory and inhibitory populations of integrate-and-fire neurons with synaptic depression or facilitation. The theory is exact when each neuron receives input from K randomly chosen ones, with 1⪡K⪡N, where N is the total number of neurons. Changes in firing rates produce changes in synaptic strengths and vice-versa, potentially leading to instabilities. We prove that depression of synapses within a population (excitatory or inhibitory) always tends to stabilize the asynchronous state against such fluctuations, while depression acting between populations destabilizes it. Facilitation has the opposite effect.