The Finite Element Method and the Fokker-Planck Equation

An application to Neuroscience

The interest in stochastic processes has increased remarkably in the last few years, in part motivated by the investigation of the constructive role of noise in many biological systems. A quantitative description of these phenomena often requires the solution of complicated Fokker-Planck equations. We apply an efficient approach from computational engineering, the finite-element method, to numerically solve the Fokker-Planck equation in two dimensions. This approach permits us to find the solution to complicated stochastic problems. In particular, with this method we have studied the stochastic synchronization of neuronal oscillators, a phenomenon that has attracted considerable attention in neuroscience recently. More specifically, we have shown that resonators (type II neural oscillators) respond and synchronize more reliably when provided correlated stochastic inputs than do integrators (type I neural oscillators) [PDF]. This result is consistent with recent experimental and computational work.

Movie: Stationary probability density of finding two uncoupled neural oscillators driven by correlated noise at any given combination of their phases. The phase of one oscillator is given by the horizontal axis and the phase of the other oscillator is given by the vertical axis. The movie shows the evolution of the stationary probability density as a function of the input correlation, r. The probability of the synchronous state (diagonal of the plots) increases as r increases and is larger for two type II oscillators than for two type I oscillators. The details of the calculations are described in R.F.Galán et al, Physical Review E, 76, 056110 [PDF].