On designing heteroclinic networks from graphs

Abstract

Robust heteroclinic networks are invariant sets that can appear as attractors in symmetrically coupled or otherwise constrained dynamical systems. These networks may have a very complicated structure that is poorly understood and determined to a large extent by the constraints and dimension of the system. As these networks are of great interest as dynamical models of biological and cognitive processes, it is useful to understand how particular graphs can be realised as robust heteroclinic networks that are attracting. This paper presents two methods of realizing arbitrarily complex directed graphs as robust heteroclinic networks for flows generated by ODEs---we say the ODEs {\em realise} the graphs as heteroclinic networks between equilibria that represent the vertices. Suppose we have a directed graph on $n_v$ vertices with $n_e$ edges. The "simplex realisation" embeds the graph as an invariant set of a flow on an $(n_v-1)$-simplex. This method realises the graph as long as it is one- and two-cycle free. The "cylinder realisation" embeds a graph as an invariant set of a flow on a $(n_e+1)$-dimensional space. This method realises the graph as long as it is one-cycle free. In both cases we find the graph as an invariant set within an attractor, and discuss some illustrative examples, including the influence of noise and parameters on the dynamics. In particular we show that the resulting heteroclinic network may or may not display "memory" of the vertices visited.Comment: 31 pages, 16 figure

Heteroclinic network dynamics on joining coupled cell networks

“…Using the same arguments for the local and global construction of heteroclinic orbits presented in Sections 5.2 and 5.3 of [2], that allow to conclude the support by the dynamics of N 1 of a robust attracting simple heteroclinic cycle involving the two equilibria p and q in ∆ 1 with heteroclinic connections in S 12) in the restriction of the join equations (4.9) to the synchrony subspace P 1 × ∆ 2 (∆ 1 × P 2 ) the invariant hyperplanes H p and H q (H p and H q ) are attractors. …”

“…For this case, we prove the existence of coupled cell dynamics for the join network supporting a robust simple heteroclinic network with four partially synchronous equilibria. The method followed here can be used iteratively and can be generalised to other coupled cell networks with dynamics realising heteroclinic networks as, for example, in Ashwin et al [12] and Field [15].…”

Sequences by Metastable Attractors: Interweaving Dynamical Systems and Experimental Data

“…In general, one may say that MS are quasistationary states with an attractive input channel and a repelling output channel, the simplest examples are hyperbolic saddles that may be connected along their stable and unstable manifolds, thus forming a heteroclinic orbit (HO) [2]. In case of dispersive saddles with one-dimensional unstable manifolds, a HO may assume the form of a stable heteroclinic sequence (SHS) [3,4] or a heteroclinic network [5]. In the following, we will investigate a set of several of such connected dispersive saddles and call it SHS in accordance to Rabinovich et al [4].…”

“…For instance, in a previous series of studies, sequences of MS have been identified in encephalographic data during cognitive tasks [17] and in early human auditory processing [22]. Since the brain processes stimuli under experimental conditions from a starting state at rest and returns to a resting state, sequences of MS in a heteroclinic network represent HOs [5]. Studies on the dimensionality of the system dynamics of these MS have revealed that the data close to MS can be described analytically by low-dimensional dynamical systems [22], whereas the transitions between MS are high-dimensional.…”

“…These components are well-known to reflect neural processing mechanisms [22,25]. Summarizing, brain signals may exhibit HO, processing information in steps of low-dimensional self-organized MS [5].…”