Numerical Methods for Solving Kinetic Equations of Neuronal Net- work Dynamics

Recently developed kinetic theory for neuronal network dynamics has been demonstrated to be a powerful theoretical framework for investigating coarse-grained dynamical properties of neuronal networks. The kinetic equations are a system of (1+1)dimensional nonlinear partial differential equations (PDE) on a bounded domain with the following features: (i) the boundary conditions are nonlinear and they are themselves a functional of the present solution; (ii) the PDEs themselves are self-consistently specified by parameters which are functions of the boundary values of the solution; and (iii) the PDEs can be stiff in space and time. Numerical methods are presented here for efficiently and accurately solving these kinetic 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 fixedpoint iteration can only achieve a linear convergence for the self-consistent parameters. The practicability and efficiency of our numerical methods for solving the kinetic equations are illustrated with numerical examples. It is further demonstrated that the kinetic equations can very well capture the coarse-grained dynamical properties of integrate-and-fire neuronal networks. Corresponding author: Aaditya V. Rangan, Courant Institute of Mathematical Sciences, New York University, New York, NY 10012. Phone: 212-998-3303, Fax: 212-995-4121, email: [email protected].