Biological Cybernetics

, Volume 66, Issue 1, pp 49–60

The path integral for dendritic trees

  • L. F. Abbott
  • E. Farhi
  • Sam Gutmann

DOI: 10.1007/BF00196452

Cite this article as:
Abbott, L.F., Farhi, E. & Gutmann, S. Biol. Cybern. (1991) 66: 49. doi:10.1007/BF00196452


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.

Copyright information

© Springer-Verlag 1991

Authors and Affiliations

  • L. F. Abbott
    • 1
  • E. Farhi
    • 2
  • Sam Gutmann
    • 3
  1. 1.Physics Department and Center for Complex SystemsBrandeis UniversityWalthamUSA
  2. 2.Laboratory for Nuclear Science and Department of Physics, MITCenter for Theoretical PhysicsCambridgeUSA
  3. 3.Department of MathematicsNortheastern UniversityBostonUSA

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