Numerical Approximation for Nonlinear Noisy Leaky Integrate-and-Fire Neuronal Model

Dipty Sharma, Paramjeet Singh, Ravi P. Agarwal and Mehmet Emir Koksal
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Dipty Sharma: School of Mathematics, Thapar Institute of Engineering & Technology, Patiala 147004, India
Paramjeet Singh: School of Mathematics, Thapar Institute of Engineering & Technology, Patiala 147004, India
Ravi P. Agarwal: Department of Mathematics, Texas A& M University-Kingsville, Kingsville, TX 78363, USA
Mehmet Emir Koksal: Department of Mathematics, Ondokuz Mayis University, Atakum, Samsun 55139, Turkey

Mathematics, 2019, vol. 7, issue 4, 1-15

Abstract: We consider a noisy leaky integrate-and-fire (NLIF) neuron model. The resulting nonlinear time-dependent partial differential equation (PDE) is a Fokker-Planck Equation (FPE) which describes the evolution of the probability density. The finite element method (FEM) has been proposed to solve the governing PDE. In the realistic neural network, the irregular space is always determined. Thus, FEM can be used to tackle those situations whereas other numerical schemes are restricted to the problems with only a finite regular space. The stability of the proposed scheme is also discussed. A comparison with the existing Weighted Essentially Non-Oscillatory (WENO) finite difference approximation is also provided. The numerical results reveal that FEM may be a better scheme for the solution of such types of model problems. The numerical scheme also reduces computational time in comparison with time required by other schemes.

Keywords: neuronal variability; Fokker-Planck-Kolmogorov equations; Galerkin finite element method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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