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

几十年来，动物着色、生物组织、植被模式和晶体形成的模式，以及行波(水)波、(激光)脉冲和(电)尖峰等时空模式吸引了不同科学领域的研究人员。NewWave项目提出了一种独特的、史无前例的方法来研究离散的、空间扩展的和延迟耦合的可激发和双稳单元系统中的这种模式，这些系统对神经信号的传播起着重要的作用。例如，可以通过对基本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来获得这样的系统。从研究单个活动单元对其邻域的响应也可以自下而上地获得，随着复杂系统科学和网络科学的出现，耦合单元大系统正受到越来越多的关注。拟议的具体研究涉及克服与混合类型的时延相关的相当大的数学和数值挑战，当(A)考虑离散轨迹之间的有限通信速度时，或(B)在离散系统上施加TW ansatz(共同移动坐标)时。预测的结果为创新离散系统中TW的分叉(质变)分析提供了巨大的潜力，并将通过新的分析和数值手段促进我们对其动态规律的理解，并使开发的方法作为项目的一部分作为开源库提供。这些技术将被用于阐明行波在大脑中的功能作用，更具体地说，将与神经科学的项目合作者一起阐明行波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.