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

1
School of Mathematics, Thapar Institute of Engineering & Technology, Patiala 147004, India
2
Department of Mathematics, Texas A& M University-Kingsville, Kingsville, TX 78363, USA
3
Department of Mathematics, Ondokuz Mayis University, Atakum, Samsun 55139, Turkey
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(4), 363; https://doi.org/10.3390/math7040363
Received: 1 March 2019 / Revised: 14 April 2019 / Accepted: 15 April 2019 / Published: 21 April 2019
(This article belongs to the Section Mathematics and Computers Science)
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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. View Full-Text
Keywords: neuronal variability; Fokker-Planck-Kolmogorov equations; Galerkin finite element method neuronal variability; Fokker-Planck-Kolmogorov equations; Galerkin finite element method
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Sharma, D.; Singh, P.; Agarwal, R.P.; Koksal, M.E. Numerical Approximation for Nonlinear Noisy Leaky Integrate-and-Fire Neuronal Model. Mathematics 2019, 7, 363.

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