Year: 2017
Type: preprint
Abstract: Point neuron models with a Heaviside firing rate function can be ill-posed. That is, the initial-condition-to-solution map might become discontinuous in finite time. If a Lipschitz continuous, but steep, firing rate function is employed, then standard ODE theory implies that such models are well-posed and can thus, approximately, be solved with finite precision arithmetic. We investigate whether the solution of this well-posed model converges to a solution of the ill-posed limit problem as the steepness parameter, of the firing rate function, tends to infinity. Our argument employs the Arzelà-Ascoli theorem and also yields the existence of a solution of the limit problem. However, we only obtain convergence of a subsequence of the regularized solutions. This is consistent with the fact that we show that models with a Heaviside firing rate function can have several solutions. Our analysis assumes that the Lebesgue measure of the time the limit function, provided by the Arzelà-Ascoli theorem, equals the threshold value for firing, is zero. If this assumption does not hold, we argue that the regularized solutions may not converge to a solution of the limit problem with a Heaviside firing function.
Source: arXiv (Cornell University)
Author: Nielsen, Bjørn Fredrik
Institution: Norwegian University of Life Sciences
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Norges Forskningsråd
Award: G7240477248
OpenAlex ID: W2952087646