Zúñiga-Galindo W A, Zambrano-Luna B A
School of Mathematical & Statistical Sciences, University of Texas Rio Grande Valley, One West University Blvd., Brownsville, TX 78520, USA.
Entropy (Basel). 2023 Jun 16;25(6):949. doi: 10.3390/e25060949.
This work aims to study the interplay between the Wilson-Cowan model and connection matrices. These matrices describe cortical neural wiring, while Wilson-Cowan equations provide a dynamical description of neural interaction. We formulate Wilson-Cowan equations on locally compact Abelian groups. We show that the Cauchy problem is well posed. We then select a type of group that allows us to incorporate the experimental information provided by the connection matrices. We argue that the classical Wilson-Cowan model is incompatible with the small-world property. A necessary condition to have this property is that the Wilson-Cowan equations be formulated on a compact group. We propose a -adic version of the Wilson-Cowan model, a hierarchical version in which the neurons are organized into an infinite rooted tree. We present several numerical simulations showing that the -adic version matches the predictions of the classical version in relevant experiments. The -adic version allows the incorporation of the connection matrices into the Wilson-Cowan model. We present several numerical simulations using a neural network model that incorporates a -adic approximation of the connection matrix of the cat cortex.
这项工作旨在研究威尔逊 - 考恩模型与连接矩阵之间的相互作用。这些矩阵描述了皮质神经布线，而威尔逊 - 考恩方程提供了神经相互作用的动态描述。我们在局部紧阿贝尔群上建立威尔逊 - 考恩方程。我们证明了柯西问题是适定的。然后，我们选择一种类型的群，使我们能够纳入连接矩阵提供的实验信息。我们认为经典的威尔逊 - 考恩模型与小世界性质不兼容。具有此性质的一个必要条件是威尔逊 - 考恩方程要在紧群上建立。我们提出了威尔逊 - 考恩模型的 - 进数版本，这是一个层次版本，其中神经元被组织成一棵无限根树。我们给出了几个数值模拟，表明 - 进数版本在相关实验中与经典版本的预测相匹配。 - 进数版本允许将连接矩阵纳入威尔逊 - 考恩模型。我们使用一个神经网络模型进行了几个数值模拟，该模型纳入了猫皮质连接矩阵的 - 进数近似。
