Gökçe Aytül, Avitabile Daniele, Coombes Stephen
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK.
J Math Neurosci. 2017 Oct 26;7(1):12. doi: 10.1186/s13408-017-0054-4.
Continuum neural field equations model the large-scale spatio-temporal dynamics of interacting neurons on a cortical surface. They have been extensively studied, both analytically and numerically, on bounded as well as unbounded domains. Neural field models do not require the specification of boundary conditions. Relatively little attention has been paid to the imposition of neural activity on the boundary, or to its role in inducing patterned states. Here we redress this imbalance by studying neural field models of Amari type (posed on one- and two-dimensional bounded domains) with Dirichlet boundary conditions. The Amari model has a Heaviside nonlinearity that allows for a description of localised solutions of the neural field with an interface dynamics. We show how to generalise this reduced but exact description by deriving a normal velocity rule for an interface that encapsulates boundary effects. The linear stability analysis of localised states in the interface dynamics is used to understand how spatially extended patterns may develop in the absence and presence of boundary conditions. Theoretical results for pattern formation are shown to be in excellent agreement with simulations of the full neural field model. Furthermore, a numerical scheme for the interface dynamics is introduced and used to probe the way in which a Dirichlet boundary condition can limit the growth of labyrinthine structures.
连续神经场方程对皮质表面相互作用神经元的大规模时空动态进行建模。它们在有界和无界域上都进行了广泛的分析和数值研究。神经场模型不需要指定边界条件。相对而言，很少有人关注在边界上施加神经活动，或者其在诱导模式状态中的作用。在这里，我们通过研究具有狄利克雷边界条件的阿马里型神经场模型（建立在一维和二维有界域上）来纠正这种不平衡。阿马里模型具有一个阶跃函数非线性，它允许通过界面动力学来描述神经场的局部解。我们展示了如何通过推导一个封装边界效应的界面法向速度规则来推广这种简化但精确的描述。界面动力学中局部状态的线性稳定性分析用于理解在没有和存在边界条件的情况下空间扩展模式是如何发展的。模式形成的理论结果与完整神经场模型的模拟结果显示出极好的一致性。此外，还引入了一种界面动力学的数值方案，并用于探究狄利克雷边界条件限制迷宫状结构生长的方式。
