Regularization of ill-posed point neuron models

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 Arzel\`{a}-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 Arzel\`{a}-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.