M. Victoria Carpio-Bernido and Christopher C. Bernido (Research Center for Theoretical Physics, Central Visayan Institute Foundation, Jagna, Bohol 6308, Philippines)

The Bernstein Center Freiburg

Bernstein Seminar

M. Victoria Carpio-Bernido
& Christopher C. Bernido

Research Center for Theoretical Physics
Central Visayan Institute Foundation
Jagna, Bohol 6308, Philippine

 

Applications of Sum-over-All-Histories Approach
to Stochastic Neuronal Models

Abstract:
We study the dynamics of membrane potential fluctuations for a single biological neuron in response to time variations of transmembrane ionic currents. We take as the dynamical variable the relative membrane potential defined as the difference between the membrane and threshold potentials. The conditional probability density for potential fluctuations as a function of time is then obtained as the fundamental solution of the Fokker-Planck equation subject to the appropriate boundary conditions. To facilitate solutions and extensions to more general treatments, we use the path integral approach. Closed forms for the conditional probability density are presented for some voltage and time dependent current modulation coefficients which are analogous to varying drift coefficients for the displacement-dependent diffusion equation.

In considering inter-neuronal interactions, we note that there may be various possible connections with which two distant neurons communicate. To account for all possible neuronal chains connecting two distant neurons, we investigate a sum-over-all-possible-paths formulation, this time with fractional Brownian motion (fBm) for path parametrization. An advantage in using fBm is that it accommodates long-memory properties that may be appropriate in describing biological neuronal connections. Results from this approach are also discussed.

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