Fast Voltage Dynamics of Voltage–Conductance Models for Neural Networks

We present the conductance limit of the voltage–conductance model with random firing voltage when conductance dynamics are slower than the voltage dynamics. The result of the limiting procedure is a transport/Fokker–Planck equation for conductance variable with a non-linear drift which depends on the total firing rate. We analyze the asymptotic behavior of the limit equation under two possible rescalings which relate the voltage scale, the conductance scale and the firing rate. We provide the sufficient framework in which the limiting procedure can be rigorously justified. Moreover, we also suggest a sufficient condition on the parameters and firing distribution in the limiting conductance equation under which we are able to obtain a unique stationary state and its asymptotic stability. Finally, we provide several numerical illustrations supporting the analytic results.