It is well known that integer-order neural networks with diffusion have rich spatial and temporal dynamical behaviors, including Turing pattern and Hopf bifurcation. Recently, some studies indicate that fractional calculus can depict the memory and hereditary attributes of neural networks more accurately. In this paper, we mainly investigate the Turing pattern in a delayed reaction–diffusion neural network with Caputo-type fractional derivative. In particular, we find that this fractional neural network can form steadily spatial patterns even if its first-derivative counterpart cannot develop any steady pattern, which implies that temporal fractional derivative contributes to pattern formation. Numerical simulations show that both fractional derivative and time delay have influence on the shape of Turing patterns.

众所周知，具有扩散的整数阶神经网络具有丰富的时空动力学行为，包括图灵模式和 Hopf 分岔。最近，一些研究表明分数阶微积分可以更准确地描述神经网络的记忆和遗传属性。在本文中，我们主要研究具有 Caputo 型分数导数的延迟反应扩散神经网络中的图灵模式。特别是，我们发现即使它的一阶导数对应物不能发展任何稳定的模式，这个分数神经网络也可以形成稳定的空间模式，这意味着时间分数导数有助于模式的形成。数值模拟表明，分数导数和时间延迟都对图灵图的形状有影响。