Anomalous Collective Dynamics Induced by Pulse-Coupling?
– from asymmetric frequency locking to chaos

Diemut Regel, Hinrich Arnoldt and Marc Timme

Chair for Network Dynamics, TU Dresden, 01069 Dresden and Max Planck Institute for Dynamics and Self-Organization, 37077 Goettingen

Standard dynamical systems’ theory focuses on invariant sets and their stability, as defined through ordinary differential equations or iterated maps. In such systems, the influence between two variables is usually a direct function of these variables, as for instance mediated by the right hand side of an ordinary differential equation. Such state-dependent coupling (SDC) is in stark contrast to pulse-coupling (PC) where interactions, e.g. between neurons via chemical synapses, explicitly depend on times of events generated by a variable and only those times directly impact the dynamics of other variables. While some system features, such as simple stability properties, are often closely related between systems with SDC and PC, several striking differences exist. Here we analyze three examples. First, symmetrically coupled single-variable oscillators may interchange their phase order, in contrast to classic results by Golubitsky, Stewart and others [1,2] on balanced equivalence relations. Second, asymmetric n:m frequency locking emerges even for identical, symmetrically coupled oscillators [2]. Third, a system of only two pulse-coupled single-variable oscillators may exhibit chaos [3], as indicated by irregular, nonperiodic motion and positive Lyapunov exponents. We explain the mechanisms underlying these phenomena and discuss consequences for modeling and analysis of spiking neural circuits.

[1] I. Stewart, M. Golubitsky, and M. Pivato, SIAM J. Appl. Dyn. Syst. 2, 609 (2003).

[2] M. Golubitsky, K. Josić, and E. Shea-Brown, J. Nonlinear. Sci. 16, 201 (2006).

[3] H. Kielblock, C. Kirst, and M. Timme, Chaos 21, 025113 (2011).

[4] D. Regel et al., in prep.