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Finite volume and asymptotic methods for stochastic neuron models with correlated inputs

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Abstract

We consider a pair of stochastic integrate and fire neurons receiving correlated stochastic inputs. The evolution of this system can be described by the corresponding Fokker–Planck equation with non-trivial boundary conditions resulting from the refractory period and firing threshold. We propose a finite volume method that is orders of magnitude faster than the Monte Carlo methods traditionally used to model such systems. The resulting numerical approximations are proved to be accurate, nonnegative and integrate to 1. We also approximate the transient evolution of the system using an Ornstein–Uhlenbeck process, and use the result to examine the properties of the joint output of cell pairs. The results suggests that the joint output of a cell pair is most sensitive to changes in input variance, and less sensitive to changes in input mean and correlation.

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Authors and Affiliations

  1. Department of Mathematics, University of Houston, Houston, USA

    Robert Rosenbaum, Fabien Marpeau, Jianfu Ma, Aditya Barua & Krešimir Josić

  2. Department of Biology and Biochemistry, University of Houston, Houston, USA

    Krešimir Josić

Authors
  1. Robert Rosenbaum
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  2. Fabien Marpeau
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  3. Jianfu Ma
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  4. Aditya Barua
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  5. Krešimir Josić
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Corresponding author

Correspondence to Robert Rosenbaum.

Additional information

R. Rosenbaum and F. Marpeau contributed equally to this work.

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Rosenbaum, R., Marpeau, F., Ma, J. et al. Finite volume and asymptotic methods for stochastic neuron models with correlated inputs. J. Math. Biol. 65, 1–34 (2012). https://doi.org/10.1007/s00285-011-0451-3

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  • DOI: https://doi.org/10.1007/s00285-011-0451-3

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