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.

中文翻译：

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不适定点神经元模型的正则化。

具有Heaviside放电速率函数的点神经元模型可能会不适定。也就是说，初始条件到解的映射可能会在有限时间内不连续。如果采用Lipschitz连续但陡峭的点火速率函数，则标准ODE理论意味着此类模型具有良好的定位，因此可以使用有限精度算法近似求解。我们研究了当射速函数的陡度参数趋于无穷大时，该适定模型的解是否收敛于不适定极限问题的解。我们的论点采用了Arzelà-Ascoli定理，并且得出了极限问题的一个解。但是，我们只能获得正则化解决方案子序列的收敛性。正如我们所展示的，这与具有重载发射速率函数的模型可以有多种解决方案这一事实是一致的。我们的分析假设由Arzelà-Ascoli定理提供的向量值极限函数v是阈值简单的：也就是说，包含v的一个或多个分量函数等于点火阈值的时间的集合具有零勒贝格测度。如果这个假设不成立，我们认为正规化解可能不会收敛到具有Heaviside激发函数的极限问题的解。