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Numerical study of singularly perturbed differential-difference equation arising in the modeling of neuronal variability. (English) Zbl 1238.65063
Summary: We present a numerical study of a class of boundary value problems of singularly perturbed differential difference equations (SPDDE) which arise in computational neuroscience in particular in the modeling of neuronal variability. The mathematical modeling of the determination of the expected time for the generation of action potential in the nerve cells by random synaptic inputs in dendrites includes a general boundary-value problem for singularly perturbed differential difference equation with shifts. The problem considered in this paper exhibit turning point behavior which add to the complexity in the construction of numerical approximation to the solution of the problem as well as in obtaining theoretical estimates on the solution. Exponentially fitted finite difference scheme based on Il’in-Allen-Southwell fitting is used on a specially designed mesh. Some numerical examples are given to validate convergence and computational efficiency of the proposed numerical scheme. Effect of the shifts on the layer structure is illuminated for the considered examples.
MSC:
65L03Functional-differential equations (numerical methods)
92C20Neural biology
65L11Singularly perturbed problems for ODE (numerical methods)
34B99Boundary value problems for ODE
39A99Difference equations
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