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Neural Computation

September 1, 2003, Vol. 15, No. 9, Pages 2129-2146
(doi: 10.1162/089976603322297322)
© 2003 Massachusetts Institute of Technology
A Simple and Stable Numerical Solution for the Population Density Equation
Article PDF (430.46 KB)

A population density description of large populations of neurons has generated considerable interest recently. The evolution in time of the population density is determined by a partial differential equation (PDE). Most of the algorithms proposed to solve this PDE have used finite difference schemes. Here, I use the method of characteristics to reduce the PDE to a set of ordinary differential equations, which are easy to solve. The method is applied to leaky-integrate-and-fire neurons and produces an algorithm that is efficient and yields a stable and manifestly nonnegative density. Contrary to algorithms based directly on finite difference schemes, this algorithm is insensitive to large density gradients, which may occur during evolution of the density.