This paper aims at presenting an efficient and accurate numerical method for treating both deterministic- and stochastic-type neural field equations (NFEs) in the presence of external stimuli input (or without it). The devised numerical integration means belongs to the class of Galerkin-type spectral approximations. The particular effort is focused on an efficient practical implementation of the solution technique because of the partial integro-differential fashion of the NFEs in use, which are to be integrated, numerically. Our method is implemented in Matlab. Its practical performance and efficiency are investigated on three variants of an NFE model with external stimuli inputs. We study both the deterministic case of the mentioned model and its stochastic counterpart to observe important differences in the solution behavior. First, we observe only stable one-bump solutions in the deterministic neural field scenario, which, in general, will be preserved in our stochastic NFE scenario if the level of random disturbance is sufficiently small. Second, if the area of the external stimuli is large enough and exceeds the size of the bump, considerably, the stochastic neural field solution’s behavior may change dramatically and expose also two- and three-bump patterns. In addition, we show that strong random disturbances, which may occur in neural fields, fully alter the behavior of the deterministic NFE solution and allow for multi-bump (and even periodic-type) solutions to appear in all variants of the stochastic NFE model studied in this paper.

本文旨在提出一种有效且准确的数值方法，用于处理确定型和随机型神经场方程（NFE）在存在外部刺激输入（或没有外部刺激输入）的情况下。所设计的数值积分方法属于伽勒金型谱逼近类。由于正在使用的NFE的局部整数微分方式，需要特别努力，以解决方案技术的有效实际实施为重点，这些方式将在数值上进行积分。我们的方法是在Matlab中实现的。在具有外部刺激输入的NFE模型的三个变体上研究了其实际性能和效率。我们研究上述模型的确定性情况及其随机对应物，以观察解决方案行为中的重要差异。首先，在确定性神经场场景中，我们仅观察到稳定的一次碰撞解，通常，如果随机干扰水平足够小，将在我们的随机NFE场景中保留。其次，如果外部刺激的面积足够大并且超过凸起的大小，则随机神经场解的行为可能会发生巨大变化，并同时暴露出两个和三个凸起的模式。此外，我们表明，可能在神经场中发生的强烈随机干扰会完全改变确定性NFE解决方案的行为，并允许在随机NFE模型的所有变体中出现多峰（甚至周期型）解决方案本文进行了研究。随机神经场解决方案的行为可能会发生巨大变化，并且还会暴露出两个和三个碰撞模式。此外，我们表明，可能在神经场中发生的强烈随机干扰会完全改变确定性NFE解决方案的行为，并允许在随机NFE模型的所有变体中出现多峰（甚至周期型）解决方案本文进行了研究。随机神经场解决方案的行为可能会发生巨大变化，并且还会暴露出两个和三个凹凸模式。此外，我们表明，可能在神经场中发生的强烈随机干扰会完全改变确定性NFE解决方案的行为，并允许在随机NFE模型的所有变体中出现多峰（甚至周期型）解决方案本文进行了研究。