The Numerical Solution of Neural Field Models Posed on Realistic Cortical Domains

神经场方程（nfe）是描述皮层神经活动的粗粒度动态的非局部和延迟动力系统。在这项工作中，我们将证明nfe可以在任意几何上进行数值求解。我们将以R^3中的2流形为重点开发这些方案，模拟现实的皮层，包括脑回和脑沟。人类连接体项目有大量可用的数据，我们可以将其纳入我们的框架。这包括从许多受试者收集的真实结构连通性估计，以及皮层区域网格。轴突延迟也将被适应，尽管需要更先进的数值技术来开发。该项目的主要目的是开发一种合适的计算机编程语言的数值求解器，以处理任意几何形状、任意连接和包含延迟的任意维度的nfe。利用图灵不稳定性分析和在一些奇异参数极限下的局域状态构造，数值方案将根据理想系统的分析预测进行验证。这项工作最终将提供一种计算机工具（基于个人大脑结构），可以让我们了解临床相关的生理大脑节律，这些节律可以很容易地通过脑电和脑磁图以及功能性磁共振成像记录下来。

Neural field equations (NFEs) are non-local and delayed dynamical systems describing the coarse grained dynamics of cortical neural activity. In this work we will show that NFEs can be numerically solved on arbitrary geometries. We will develop these schemes with a focus on a 2-manifold in R^3, modelling a realistic cortex, including gyri and sulci. The Human Connectome Project has a large collection of data available which we can incorporate into our framework. This includes real structural connectivity estimates collected from a number of subjects, as well as the cortical domain meshes. Axonal delays will also be accommodated, though require more advanced numerical techniques to be developed. This project's main aim is to develop a numerical solver in an appropriate computer programming language that handles NFEs of arbitrary dimension, posed on an arbitrary geometry, with arbitrary connectivity, and the inclusion of delays. The numerical scheme will be validated against analytical predictions from idealised systems, making use of Turing instability analysis and the construction of localised states in some singular parameter limits. This work will ultimately provide an in silico tool (based on individual human brain structure) that can inform our understanding of the clinically relevant physiological brain rhythms that are readily recorded using electro- and magneto-encephalography and functional magnetic resonance imaging.