13.7 Summary

The momentary state of a population of one-dimensional integrate-and-and fire neurons can be characterized by the membrane potential density $p(u,t)$. The continuity equation describes the evolution of $p(u,t)$ over time. In the special case that neurons in the population receive many inputs that each cause a small change of the membrane potential, the continuity equation has the form of a Fokker-Planck equation. Several populations of integrate-and-fire neurons interact via the population activity $A(t)$ which is identified with the flux across the threshold.

The stationary state of the Fokker-Planck equation and the stability of the stationary solution can be calculated by a mix of analytical and numerical methods, be it for a population of independent or interconnected neurons. The mathematical and numerical methods developed for membrane potential density equations apply to leaky as well as to arbitrary nonlinear one-dimensional integrate-and-fire model. A slow adaptation variable such as in the adaptive exponential integrate-and-fire model can be treated as quasi-stationary in the proximity of the stationary solution. Conductance input can be approximated by an equivalent current-based model.