Despite the highly convoluted nature of the human brain, neural field models typically treat the cortex as a planar two-dimensional sheet of ne;urons. Here, we present an approach for solving neural field equations on surfaces more akin to the cortical geometries typically obtained from neuroimaging data. Our approach involves solving the integral form of the partial integro-differential equation directly using collocation techniques alongside efficient numerical procedures for determining geodesic distances between neural units. To illustrate our methods, we study localised activity patterns in a two-dimensional neural field equation posed on a periodic square domain, the curved surface of a torus, and the cortical surface of a rat brain, the latter of which is constructed using neuroimaging data. Our results are twofold: Firstly, we find that collocation techniques are able to replicate solutions obtained using more standard Fourier based methods on a flat, periodic domain, independent of the underlying mesh. This result is particularly significant given the highly irregular nature of the type of meshes derived from modern neuroimaging data. And secondly, by deploying efficient numerical schemes to compute geodesics, our approach is not only capable of modelling macroscopic pattern formation on realistic cortical geometries, but can also be extended to include cortical architectures of more physiological relevance. Importantly, such an approach provides a means by which to investigate the influence of cortical geometry upon the nucleation and propagation of spatially localised neural activity and beyond. It thus promises to provide model-based insights into disorders like epilepsy, or spreading depression, as well as healthy cognitive processes like working memory or attention.

中文翻译：

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弯曲几何上的神经场的数值模拟。

尽管人脑具有高度复杂的性质，但神经场模型通常将皮质视为神经元的平面二维薄片。在这里，我们提出了一种解决表面上的神经场方程的方法，该方程更类似于通常从神经影像数据获得的皮质几何形状。我们的方法涉及直接使用搭配技术以及有效的数值程序来确定神经单位之间的测地距离来求解部分积分-微分方程的积分形式。为了说明我们的方法，我们在二维神经场方程中研究了局部活动模式，该方程位于周期平方域，圆环的曲面和大鼠大脑的皮质表面上，后者是使用神经影像数据构建的。我们的结果是双重的：首先，我们发现搭配技术能够在平坦，周期性的域上复制使用更标准的基于傅立叶的方法获得的解决方案，而与底层网格无关。鉴于从现代神经影像数据得出的网格类型高度不规则，因此该结果特别重要。其次，通过部署有效的数值方案来计算测地线，我们的方法不仅能够对现实的皮质几何形状上的宏观图案形成进行建模，而且还可以扩展到包括更多具有生理相关性的皮质结构。重要的是，这种方法提供了一种手段来研究皮质几何形状对空间局部神经活动及其他活动的成核和传播的影响。