Martin R, Chappell D J, Chuzhanova N, Crofts J J
School of Science and Technology, Nottingham Trent University, Nottingham, UK.
J Comput Neurosci. 2018 Oct;45(2):133-145. doi: 10.1007/s10827-018-0697-5. Epub 2018 Oct 11.
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.
尽管人类大脑具有高度复杂的特性，但神经场模型通常将皮层视为二维平面的神经元薄片。在此，我们提出一种方法，用于在更类似于通常从神经成像数据获得的皮层几何形状的表面上求解神经场方程。我们的方法包括直接使用配置技术求解偏积分 - 微分方程的积分形式，以及用于确定神经单元之间测地距离的高效数值程序。为了说明我们的方法，我们研究了在周期性方形域、环面的弯曲表面以及大鼠大脑皮层表面上提出的二维神经场方程中的局部活动模式，其中大鼠大脑皮层表面是使用神经成像数据构建的。我们的结果有两方面：首先，我们发现配置技术能够在平坦的周期性域上复制使用更标准的基于傅里叶的方法获得的解，而与底层网格无关。鉴于从现代神经成像数据得出的网格类型具有高度不规则的性质，这一结果尤为重要。其次，通过部署高效的数值方案来计算测地线，我们的方法不仅能够对现实皮层几何形状上的宏观模式形成进行建模，还可以扩展到包括更具生理相关性的皮层结构。重要的是，这种方法提供了一种手段，通过它可以研究皮层几何形状对空间局部神经活动的成核和传播以及其他方面的影响。因此，它有望为癫痫或扩散性抑制等疾病以及工作记忆或注意力等健康认知过程提供基于模型的见解。
