NEWWAVE: New methods for analysing travelling waves in discrete systems with applications to neuroscience

几十年来，动物着色模式、生物组织、植被模式和晶体的形成，以及时空模式，如行进（水）波、（激光）脉冲和（电）尖峰，吸引了各个科学领域的研究人员。项目NEWWAVE提出了一种独特的和前所未有的方法来研究离散的，空间扩展的和延迟耦合的可兴奋和可兴奋单元系统中的这种模式，这对神经信号传播起着重要作用。这样的系统可以获得，例如，从一个潜在的偏微分方程模型的离散化，从测量数据的离散性，或从介质的微观结构。它们也可以自下而上地获得，从研究单个活动单元对其邻域的响应，耦合单元的大系统随着复杂系统科学和网络科学的出现而越来越受到关注。具体提出的研究涉及克服相当大的数学和数值的挑战与时间延迟的混合型出现时，（A）考虑到离散位点之间的通信的有限速度，或（B）强加一个TW anomaly（共同移动坐标）的离散系统。预计的结果承诺显着的潜力，创新的分歧（质变）分析的TW离散系统，并将通过新的分析和数值手段，并通过使开发的方法作为一个开源库的项目的一部分，我们的理解，他们的动态剧目。这些技术将用于阐明行波在大脑中的功能作用，更具体地说，与神经科学项目合作者一起研究TW在帕金森病中的作用。

Patterns of animal colouration, biological tissues, vegetation patterns and the formation of crystals, as well as spatio-temporalpatterns such as travelling (water) waves, (laser) pulses and (electrical) spikes have captivated researchers in various fields of science for many decades. Project NEWWAVE puts forward a unique and unprecedented approach to studying such patterns in discrete, spatially-extended and delay-coupled systems of excitable and bistable units, which play an important role for neural signal propagation. Such systems can be obtained, for example, from discretisation of an underlying partial differential equation model, from the discreteness of the measured data, or from the microstructure of the medium. They can also be obtained bottom-up, from studying the response of single active unit to its neighbourhood, and large systems of coupled units are gaining more and more attention through the emergence of complex systems science and network science. The specific proposed research involves overcoming considerable mathematical and numerical challenges associated with time delays of mixed-type that appear when either (A) taking into account the finite speed of communication between discrete loci, or (B) imposing a TW ansatz (co-moving coordinates) on a discrete system. The projected results promise significant potential for innovating the bifurcation (qualitative changes) analysis of TWs in discrete systems and will advance our understanding of their dynamic repertoire through new analytical and numerical means, and by making the developed methods available as an open source library as part of the project. The techniques will be used to elucidate the functional role of travelling waves in the brain and, more specifically, the role of TWs in Parkinson's disease together with project collaborators in Neuroscience.