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.
Highlights
The large-scale neural network models in computational neuroscience have become familiar.The classical description of these neural network models is based on the deterministic/stochastic system
One of the most common models is known as the noisy leaky integrate-and-fire (NLIF) neuron model in which the behavior of the whole population of neurons is encoded in a stochastic differential equation (SDE) for the time evolution of membrane potential of a single neuron representative of the network
We construct the numerical approximation of the problem given in (2) and (3) in two ways: First we use finite element method (FEM) for space discretization that provides a system of ordinary differential equations, which is solved by Euler’s backward difference for time
Summary
The large-scale neural network models in computational neuroscience have become familiar. In a latest work [8], it was demonstrated that the problem (2)–(4) can produce a finite time blow-up solution for excitatory networks b > 0 when the initial data is concentrated near sufficient to the threshold voltage. This result was obtained by giving no information about the behavior at the blow-up time. When b ≤ 0 we have that T ∗ = ∞, while for b > 0 there exist classical solutions which blow up at a finite time T ∗ and have diverging mean firing rate as t ↑ T ∗.
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