Self-organization of the excitable field in a reaction-diffusion system

我们研究了一个称为Bonhoeffer-van der Pol(BVP)型方程的可激发反应扩散系统的动力学和斑图的形成。这组方程已用于模拟脉冲沿神经公理和Belousov-Zhabotinsky反应的传播。此外，它不仅允许时间相关的解，而且还允许空间周期的定常解，其中激发域构成周期晶格。非平衡开放系统最大的特征之一就是存在时间有序和空间局域模式。两者都源于李亚普诺夫泛函的缺失。在这里，我将总结所获得的结果，着重于概念方面。反应扩散系统中的空间周期结构已被扩散不稳定性解释。然而，我们已经证明了BVP方程在特定的极限下归结为一个变分系统。同时含有短程和长程界面的Lyapounov泛函与嵌段共聚熔体和近晶液晶等平衡中间相的Lyapounov泛函基本相同。此外，简化的系统与视皮层模式形成的模型方程密切相关。这些都是以前没有认识到的BVP方程的重要性质。我想强调的另一个结果是远离平衡的动力多稳定性的作用。我们证明了BVP方程有多种解，这些解可以共存。例如，在受激域的呼吸运动中会发生正相振荡和反相振荡。我们还分析了传播脉冲和静止区域的共存区域。我们认为这些多稳性与生物系统的信息传输和自组织有关。然而，进一步的调查有待于未来的研究。

We have studied the dynamcs and the pattern formation in an excitable reaction-diffusion system so called Bonhoeffer-van der Pol(BVP)type equation. This set of equations has been used for modeling pulse propagation along the nerve axiom and Belousov-Zhabotinsky reaction. Furthermore, it admits not only the time-dependent solutions but also a spatially periodic steady solution where excited domains constitute a periodic lattice. It is also known that an excited domain undergoes a breathing motion by changing the parameter.One of the most characteristic features of nonequilibrium open systems is the existence of temporal order and of spatially localized pattern. Both are originated from the absence of any Lyapounov functionals. Here I shall summarize the results obtained emphasizing the conceptual aspects.A spatially periodic structure in a reaction-diffusion system has been interpreted by the diffusional instability. However, we have shown that BVP equation in a particular limit reduces to a variational system. The Lyapounov functional which contains both a short range and a long range intemdons is found to be essentially the same as the one for equilibrium mesophases such as block copolymer melts and smectic liquid crystals. Furthermore, the reduced system is closely related to the model equation for pattern formation in visual cortex. These are important properties of BVP equation unrecognized previously.The other result that I would like to stress is the role of dynamical multistability far from equilibrium. We have shown that BVP equation exhibits various solutions which can coexist. For instance, An enphase and an anti-phase oscillations occur in the breathing motion of excited domains. We have also analyzed the coexistence region of a propagating pulse and a motionless domain. We expect that these multistabilitis are relevant to information transportation and self-organization in biological system. However further investigation is left for a future study.