Error analysis of a specialized numerical method for mathematical models from neuroscience

The exponential Euler method is a nonstandard approximation scheme that was developed specifically for the Hodgkin–Huxley differential equation models that arise in neuroscience and was one of the discretization schemes used in the neural systems package called GENESIS. In this article, we show the scheme is first order accurate, develop a second order accurate extension, and suggest ways the method can be used to compute approximations to certain time-dependent partial differential equations. We also apply the scheme to an integro-differential equation model of neuronal activity and furnish sample computations that illustrate our theoretical results.