We study the effect of additive noise on integro-differential neural field equations. In particular, we analyze an Amari-type model driven by a Q-Wiener process, and focus on noise-induced transitions and escape. We argue that proving a sharp Kramers' law for neural fields poses substantial difficulties, but that one may transfer techniques from stochastic partial differential equations to establish a large deviation principle (LDP). Then we demonstrate that an efficient finite-dimensional approximation of the stochastic neural field equation can be achieved using a Galerkin method and that the resulting finite-dimensional rate function for the LDP can have a multiscale structure in certain cases. These results form the starting point for an efficient practical computation of the LDP. Our approach also provides the technical basis for further rigorous study of noise-induced transitions in neural fields based on Galerkin approximations.Mathematics Subject Classification (2000): 60F10, 60H15, 65M60, 92C20.

中文翻译：

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非局部随机神经场的大偏差。

我们研究了加性噪声对积分微分神经场方程的影响。特别是，我们分析了由 Q-Wiener 过程驱动的 Amari 型模型，并重点关注噪声引起的转变和逃逸。我们认为，证明神经场的尖锐克莱默定律带来了很大的困难，但人们可以将随机偏微分方程的技术转移到建立大偏差原理（LDP）。然后我们证明了使用伽辽金方法可以实现随机神经场方程的有效有限维近似，并且所得的 LDP 有限维速率函数在某些情况下可以具有多尺度结构。这些结果构成了 LDP 的高效实际计算的起点。我们的方法还为基于伽辽金近似的神经场噪声引起的转变的进一步严格研究提供了技术基础。数学学科分类（2000）：60F10、60H15、65M60、92C20。