Networks of diffusively time-delay coupled systems: Conditions for synchronization and its relation to the network topology

Synchronization and Partial Synchronization Experiments with Networks of Time-Delay Coupled Hindmarsh–Rose Neurons

“…In this section, we review the theoretical framework presented in Steur et al, 2012;Steur et al, 2014a], which provides qualitative predictions about (partial) synchronization in networks with HR neurons. We represent the HR dynamics by the (more general) system…”

“…Theorem 2 [Steur et al, 2014a]. Suppose that the assumptions of Theorem 1 are satisfied such that k = 2 coupled systems (2), (4) synchronize for any…”

“…Theorem 3 [Steur et al, 2012]. Consider k systems (2) coupled through the transmission delay coupling (3) and let the assumptions of Theorem 1 be satisfied.…”

“…We emphasize (partial) synchronization is understood in this manuscript as the asymptotic match of the states of (some of) the systems. The theoretical results presented in Steur et al, 2012;Steur et al, 2014a] identify a large class of systems that can be synchronized by linear time-delay coupling, (of course, the HR model neuron belongs to this class). The conditions, which are of algebraic nature, are expressed at the level of the systems of the network.…”

“…The focus of this paper is to test the validity of theory using an experimental setup that consists of electronic circuit board realizations of the (mathematical) Hindmarsh-Rose (HR) model neuron [Hindmarsh & Rose, 1984]. Particularly, this manuscript presents results of a series of experiments that are conducted to verify theoretical results about full synchronization , partial synchronization [Steur et al, 2012], and synchronization in relation to the network topology [Steur et al, 2014a]. We emphasize (partial) synchronization is understood in this manuscript as the asymptotic match of the states of (some of) the systems.…”