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.

中文翻译：

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有界域上神经场的动力学：Dirichlet边界条件的接口方法。

连续神经场方程模型在皮质表面相互作用的神经元的大规模时空动力学模型。他们已经在有界和无界域上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。相对很少有人关注将神经活动强加于边界上或其在诱导图案化状态中的作用。在这里，我们通过研究具有Dirichlet边界条件的Amari类型（位于一维和二维有界域上）的神经场模型来纠正这种不平衡。Amari模型具有Heaviside非线性，该非线性允许使用接口动力学描述神经场的局部解。我们展示了如何通过导出封装边界效应的接口的法向速度规则来概括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了解在没有边界条件和存在边界条件的情况下空间扩展模式如何发展。模式形成的理论结果显示与完整的神经场模型的仿真非常吻合。此外，引入了界面动力学的数值方案，并用于探讨狄利克雷边界条件可以限制迷宫结构增长的方式。界面动力学中局部状态的线性稳定性分析用于了解在没有边界条件和存在边界条件的情况下空间扩展模式如何发展。模式形成的理论结果显示与完整的神经场模型的仿真非常吻合。此外，引入了界面动力学的数值方案，并用于探讨狄利克雷边界条件可以限制迷宫结构增长的方式。界面动力学中局部状态的线性稳定性分析用于了解在没有边界条件和存在边界条件的情况下空间扩展模式如何发展。模式形成的理论结果显示与全神经场模型的仿真非常吻合。此外，引入了界面动力学的数值方案，并用于探讨狄利克雷边界条件可以限制迷宫结构增长的方式。