Clustering dynamics in delayed neural networks
Describing the time evolution of neural networks is a challenging tasks due to the large number of nonlinear network elements and connections in the system, and also due to the fact that the transmission of biological signals involves elaborate processes. In this talk, a modeling framework is considered where biological signal transmission is not modeled in detail but the time it takes to transmit the signal is accounted for using time delays. This approach, while retaining the essential infinite dimensionality of the dynamics, allows us to develop parsimonious models that are scalable for large networks. However, the obtained large, complex delayed dynamical systems cannot be analyzed by standard methods developed for "simple" delayed systems. To solve this problem, I present a mathematical technique that allows us to analyze the rhythmic patterns of neural networks by exploiting the graph connectivity of the underlying network. Using this method, the clustering behavior of neural networks of realistic size can be analyzed and the stability of the arising spatiotemporal patters can be evaluated. It will be demonstrated that neural systems may exploit time delays to change the arising spatiotemporal patterns when encoding/decoding information or performing computation.