13 Continuity Equation and the Fokker-Planck Approach

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)p(u,t). The continuity equation describes the evolution of p(u,t)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)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.