Non-stationary dynamics of interspike intervals in neuronal populations

We study the joint dynamics of membrane potential and time since the last spike in a population of integrate-and-fire neurons using a population density framework. This leads to a two-dimensional Fokker-Planck equation that captures the evolution of the full neuronal state, along with a one-dimensional hierarchy of equations for the moments of the inter-spike interval (ISI). The formalism allows us to characterize the time-dependent ISI distribution, even when the population is far from stationarity, such as under time-varying external input or during network oscillations. By performing a perturbative expansion around the stationary state, we also derive an analytic expression for the linear response of the ISI distribution to weak input modulations.