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Techniques for obtaining analytical solutions for the somatic shunt cable model. (English) Zbl 0622.92010
Mathematical expressions are obtained for the Green’s function corresponding to an instantaneous pulse of current injected at a single point along the dendritic cable in a somatic shunt neuron model. The convergence of the Green’s function is determined from an estimate of its truncation error. The Green’s function, when used in a convolution formula, enables one to compute the voltage response at any specified point for an arbitrary synaptic input at a given location.
Examples of synaptic input considered are (1) a current step, injected at the soma, and (2) a smooth current time course of the form \(\alpha^ 2Te^{-\alpha T}\) injected at a given location along the dendritic cable. Alternative representations for the Green’s function are given to enable both the small and the large time behavior of the voltage to be analyzed. The treatment of synaptic input as a conductance change is also discussed. The Volterra integral equation is solved analytically by employing a Neumann expansion to obtain the voltage response.

MSC:
92Cxx Physiological, cellular and medical topics
45D05 Volterra integral equations
78A70 Biological applications of optics and electromagnetic theory
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