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The path integral for dendritic trees

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Abstract

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 anlaysis leads ultimately to an exact expression for the Green's function on a dendritic tree of arbitrary geometry expressed in terms of a set of simple diagrammatic rules. These rules providing a fast and efficient method for solving complex cable problems.

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Authors and Affiliations

  1. Physics Department and Center for Complex Systems, Brandeis University, 02254, Waltham, MA, USA

    L. F. Abbott

  2. Laboratory for Nuclear Science and Department of Physics, MIT, Center for Theoretical Physics, 02139, Cambridge, MA, USA

    E. Farhi

  3. Department of Mathematics, Northeastern University, 02115, Boston, MA, USA

    Sam Gutmann

Authors
  1. L. F. Abbott
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  2. E. Farhi
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  3. Sam Gutmann
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Additional information

Research supported by Department of Energy Contracts DE-AC02-76ER03230 and DE-AC02-76ER03069 and by National Institute of Mental Health grant MH46742

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Abbott, L.F., Farhi, E. & Gutmann, S. The path integral for dendritic trees. Biol. Cybern. 66, 49–60 (1991). https://doi.org/10.1007/BF00196452

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  • DOI: https://doi.org/10.1007/BF00196452

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