Talk: Designing hierarchical connection structures in phase space

Abstract

Heteroclinic and excitable dynamics provide a natural mathematical framework for modeling structured intermittent, sequential behavior in complex systems. In many applications, these transitions underlie a specific hierarchy. For example, in neuroscience, a memory may correspond to a sequence of activity, and memory formation corresponds to the modulation of these patterns. In biological or social networks, temporal sequences in collective evolution may change qualitatively due to environmental changes on a higher hierarchical level. In such cases, the system evolves through a sequence of lower-level dynamics—such as local state transitions—before undergoing a qualitative change driven by a higher-level process. In this talk, we present a systematic method to construct vector fields that realize such hierarchical dynamics: given a set of lower-level directed graphs and a top-level digraph encoding transitions between them, we construct a dynamical system where the lower-level connections are heteroclinic and the top-level transitions are excitable with zero threshold. This yields a general framework for modeling systems with multi-scale, hierarchical switching behavior reflecting a prescribed hierarchical structure.

This talk will build on the general setup layed out in Alexander Lohse’s talk “Creating Heteroclinic and Excitable Connections” in the same minisymposium.

Sören von der Gracht

Mathematician researching dynamical systems

Research in network dynamical systems and their applications.