In artificial neural networks, the diffusion phenomenon of electrons exists inevitably, due to the electromagnetic field of neural networks is heterogeneous. In this paper, we study the spatio-temporal dynamical behaviors of a reaction-diffusion neural network with leakage delay. By analyzing the corresponding characteristic equation, the sufficient and necessary conditions of Turing instability are obtained and the existence of Turing, Hopf, and Turing-Hopf bifurcations is also established. Furthermore, the truncated normal form up to third order is derived to understand and classify the spatio-temporal dynamics close to the Turing-Hopf bifurcation point. By numerical simulations, we find a pair of spatially inhomogeneous periodic solutions and illustrate the effects of time delays and spatial diffusion on the spatio-temporal dynamics of the model.

在人工神经网络中，由于神经网络的电磁场是异质的，因此不可避免地存在电子的扩散现象。在本文中，我们研究了具有泄漏延迟的反应扩散神经网络的时空动力学行为。通过分析相应的特征方程，获得了图灵不稳定性的充要条件，并建立了图灵，霍普夫和图灵-霍普夫分支的存在性。此外，推导了高达三阶的截断正态形式，以了解和分类接近Turing-Hopf分叉点的时空动力学。通过数值模拟