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 models with a Heaviside firing rate function can have several solutions, as we show. Our analysis assumes that the vector-valued limit function v, provided by the Arzelà–Ascoli theorem, is threshold simple: That is, the set containing the times when one or more of the component functions of v equal the threshold value for firing, has zero Lebesgue measure. 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.
Highlights
It is important to note that this ill-posed nature of the model is a fundamentally different mathematical property from the possible existence of unstable equilibria, which typically occur if a firing rate function with moderate steepness is used
The existence of a solution matter for point neuron models with a Heaviside firing rate function is summarized in the following theorem
We will first explain why the uniqueness question is a subtle issue for point neuron models with a Heaviside firing rate function
Summary
The limit process β → ∞, using different techniques, is studied in [18, 19] for the stationary solutions of neural field equations It has been observed [20] for the Wilson–Cowan model that this transition is a subtle matter: Using a steep sigmoid firing rate function instead of the Heaviside mapping can lead to significant changes in a Hopf bifurcation point. Smoothening techniques for discontinuous vector fields, which are similar to the regularization method considered in this paper, have been proposed and analyzed for rather general phase spaces [22,23,24] These studies consider qualitative properties of large classes of problems, whereas we focus on a quantitative analysis of a very special system of ODEs. For the sake of easy notation, we will sometimes write (1) in the form τ u (t) = −u(t) + ωSβ u(t) − uθ + q(t), t ∈ (0, T ], (3). For the sake of simplicity, use the same threshold value uθ for all the units in the network; see (4)
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