Abbassian A H, Fotouhi M, Heidari M
School of Mathematics, Institute for Research in Fundamental Sciences-IPM, P.O. Box 19395-5746, Tehran, Iran.
Biol Cybern. 2012 Jan;106(1):15-26. doi: 10.1007/s00422-012-0475-9. Epub 2012 Mar 8.
We introduce a modified-firing-rate model based on Hebbian-type changing synaptic connections. The existence and stability of solutions such as rest state, bumps, and traveling waves are shown for this type of model. Three types of kernels, namely exponential, Mexican hat, and periodic synaptic connections, are considered. In the former two cases, the existence of a rest state solution is proved and the conditions for their stability are found. Bump solutions are shown for two kinds of synaptic kernels, and their stability is investigated by constructing a corresponding Evans function that holds for a specific range of values of the kernel coefficient strength (KCS). Applying a similar method, we consider exponential synaptic connections, where traveling wave solutions are shown to exist. Simulation and numerical analysis are presented for all these cases to illustrate the resulting solutions and their stability.
我们引入了一种基于赫布型变化突触连接的修正 firing-rate 模型。针对此类模型，展示了诸如静止状态、 bumps 和行波等解的存在性和稳定性。考虑了三种类型的核，即指数型、墨西哥帽型和周期性突触连接。在前两种情况下，证明了静止状态解的存在性并找到了其稳定性条件。针对两种突触核展示了 bump 解，并通过构建一个适用于核系数强度（KCS）特定值范围的相应埃文斯函数来研究其稳定性。应用类似方法，我们考虑指数突触连接，其中展示了行波解的存在性。针对所有这些情况进行了模拟和数值分析，以说明所得解及其稳定性。
