A Convection-Siffusion Type Boundary Value Problem to Determine the Expected Time for the Generation of Action Potentials in Nerve Cells by Random Synaptic Inputs in the Dendrites

According to Lange and Miura [1], the determination of the expected time for the generation of action potentials in nerve cells by random synaptic inputs in the dendrites can be modeled as general boundary value problem for the linear second order differential difference equation with prescribed boundary condition. This biological problem motivates the study of boundary value problems for singularly perturbed differential difference equation with delay. In this paper, a boundary value problem for a second-order singularly perturbed delay differential equation which models the above biological problem is considered on [0,2]. The solution of this problem exhibits an initial layer at 0 and an interior layer at 1. A numerical method composed of a classical finite difference scheme applied on a piecewise-uniform Shishkin mesh is suggested to solve the problem. The biological implications are studied. [1] C.G.Lange and R.M.Miura, \textit{ Singular perturbation analysis of oundary-value problems for differential-difference equations.V. small shifts with layer behavior }, SIAM J.Appl.Math, Vol.54 1 249--272 (1994).