Numerical methods for solving moment equations in kinetic theory of neuronal network dynamics

Aaditya Rangan, David Cai, Louis Tao

Research output: Contribution to journalArticle

Abstract

Recently developed kinetic theory and related closures for neuronal network dynamics have been demonstrated to be a powerful theoretical framework for investigating coarse-grained dynamical properties of neuronal networks. The moment equations arising from the kinetic theory are a system of (1 + 1)-dimensional nonlinear partial differential equations (PDE) on a bounded domain with nonlinear boundary conditions. The PDEs themselves are self-consistently specified by parameters which are functions of the boundary values of the solution. The moment equations can be stiff in space and time. Numerical methods are presented here for efficiently and accurately solving these moment equations. The essential ingredients in our numerical methods include: (i) the system is discretized in time with an implicit Euler method within a spectral deferred correction framework, therefore, the PDEs of the kinetic theory are reduced to a sequence, in time, of boundary value problems (BVPs) with nonlinear boundary conditions; (ii) a set of auxiliary parameters is introduced to recast the original BVP with nonlinear boundary conditions as BVPs with linear boundary conditions - with additional algebraic constraints on the auxiliary parameters; (iii) a careful combination of two Newton's iterates for the nonlinear BVP with linear boundary condition, interlaced with a Newton's iterate for solving the associated algebraic constraints is constructed to achieve quadratic convergence for obtaining the solutions with self-consistent parameters. It is shown that a simple fixed-point iteration can only achieve a linear convergence for the self-consistent parameters. The practicability and efficiency of our numerical methods for solving the moment equations of the kinetic theory are illustrated with numerical examples. It is further demonstrated that the moment equations derived from the kinetic theory of neuronal network dynamics can very well capture the coarse-grained dynamical properties of integrate-and-fire neuronal networks.

Original languageEnglish (US)
Pages (from-to)781-798
Number of pages18
JournalJournal of Computational Physics
Volume221
Issue number2
DOIs
StatePublished - Feb 10 2007

Fingerprint

Kinetic theory
kinetic theory
Numerical methods
boundary value problems
Boundary value problems
Boundary conditions
boundary conditions
moments
pulse detonation engines
newton
ingredients
partial differential equations
closures
Partial differential equations
iteration
Fires

Keywords

  • Divide and conquer
  • Newton's method

ASJC Scopus subject areas

  • Computer Science Applications
  • Physics and Astronomy(all)

Cite this

Numerical methods for solving moment equations in kinetic theory of neuronal network dynamics. / Rangan, Aaditya; Cai, David; Tao, Louis.

In: Journal of Computational Physics, Vol. 221, No. 2, 10.02.2007, p. 781-798.

Research output: Contribution to journalArticle

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