具有指数时间核的神经场动力学
Dynamics of neural fields with exponential temporal kernel
AI总结：
本文研究了带指数时间核的标准神经场方程，通过动态分岔分析发现其不允许静态分岔，但能产生行波等时空非恒定解，并给出了显式分岔条件。
摘要：
我们考虑带有指数时间核的标准神经场方程。我们分析了平衡解的时间无关（静态）和时间相关（动态）分岔以及涌现的时空波形模式。我们证明指数时间核不允许静态分岔，如鞍结分岔、音叉分岔，特别是静态Turing分岔。然而，指数时间核具有考虑神经元过去活动有限记忆的重要特性，这是Green函数所不具备的。通过动态分岔分析，我们给出了显式分岔条件。Hopf分岔导致时间非恒定但空间恒定的解，而Turing-Hopf分岔产生空间和时间均非恒定的解，特别是行波。分岔参数为指数时间核的系数、神经信号的传输速度、突触的时间延迟率以及兴奋性与抑制性突触权重的比值。
英文摘要：
We consider the standard neural field equation with an exponential temporal kernel. We analyze the time-independent (static) and time-dependent (dynamic) bifurcations of the equilibrium solution and the emerging spatiotemporal wave patterns. We show that an exponential temporal kernel does not allow static bifurcations such as saddle-node, pitchfork, and in particular, static Turing bifurcations. However, the exponential temporal kernel possesses the important property that it takes into account the finite memory of past activities of neurons, which Green's function does not. Through a dynamic bifurcation analysis, we give explicit bifurcation conditions. Hopf bifurcations lead to temporally non-constant, but spatially constant solutions, but Turing-Hopf bifurcations generate spatially and temporally non-constant solutions, in particular, traveling waves. Bifurcation parameters are the coefficient of the exponential temporal kernel, the transmission speed of neural signals, the time delay rate of synapses, and the ratio of excitatory to inhibitory synaptic weights.
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