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The path integral for dendritic trees. (English) Zbl 0743.92010
Summary: We construct the path integral for determining the potential on any dendritic tree described by a linear cable equation. This is done by generalizing Brownian motion from a line to a tree. We also construct the path integral for dendritic structures with spatially-varying and/or time-dependent membrane conductivities due, for example, to synaptic inputs.
The path integral allows novel computational techniques to be applied to cable problems. Our analysis leads ultimately to an exact expression for Green’s function on a dendritic tree of arbitrary geometry expressed in terms of a set of simple diagrammatic rules. These rules provide a fast an efficient method for solving complex cable problems.

MSC:
92C20 Neural biology
78A70 Biological applications of optics and electromagnetic theory
60K40 Other physical applications of random processes
60J99 Markov processes
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