We introduce a neural field equation with a neuron-dependent Heaviside-type activation function and spatial-dependent delay. The basic object of the study is represented by a Volterra–Hammerstein integral equation involving a discontinuous nonlinearity with respect to the state variable that is both time and space dependent. We replace the discontinuous nonlinearity by its multivalued convexification and obtain the corresponding Volterra–Hammerstein integral inclusion. We investigate the solvability of this inclusion using the properties of upper semicontinuous multivalued mappings with convex closed values. Based on these results, we study the solvability of an initial-prehistory problem for the former neural field equation with the Heaviside-type activation function. The application of multivalued analysis techniques allowed us to avoid some restrictive assumptions standardly used in the investigations of the solutions to neural field equations involving Heaviside-type activation functions.

中文翻译：

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具有神经元相关 Heaviside 型激活函数和空间相关延迟的神经场方程

我们引入了一个神经场方程，它具有依赖于神经元的 Heaviside 型激活函数和依赖于空间的延迟。该研究的基本目标由一个 Volterra-Hammerstein 积分方程表示，该方程涉及与时间和空间相关的状态变量的不连续非线性。我们用它的多值凸化代替了不连续的非线性，并获得了相应的 Volterra-Hammerstein 积分包含。我们使用具有凸闭值的上半连续多值映射的属性来研究这种包含的可解性。基于这些结果，我们研究了具有 Heaviside 型激活函数的前神经场方程的初始史前问题的可解性。