Designing heteroclinic and excitable networks in phase space using two populations of coupled cells

We give a constructive method for realizing an arbitrary directed graph (with no one-cycles) as a heteroclinic or an excitable dynamic network in the phase space of a system of coupled cells of two types. In each case, the system is expressed as a system of first order differential equations. One of the cell types (the $p$-cells) interacts by mutual inhibition and classifies which vertex (state) we are currently close to, while the other cell type (the $y$-cells) excites the $p$-cells selectively and becomes active only when there is a transition between vertices. We exhibit open sets of parameter values such that these dynamical networks exist and demonstrate via numerical simulation that they can be attractors for suitably chosen parameters

Almost complete and equable heteroclinic networks

Heteroclinic connections are trajectories that link invariant sets for an autonomous dynamical flow: these connections can robustly form networks between equilibria, for systems with flow-invariant spaces. In this paper we examine the relation between the heteroclinic network as a flow-invariant set and directed graphs of possible connections between nodes. We consider realizations of a large class of transitive digraphs as robust heteroclinic networks and show that although robust realizations are typically not complete (i.e. not all unstable manifolds of nodes are part of the network), they can be almost complete (i.e. complete up to a set of zero measure within the unstable manifold) and equable (i.e. all sets of connections from a node have the same dimension). We show there are almost complete and equable realizations that can be closed by adding a number of extra nodes and connections. We discuss some examples and describe a sense in which an equable almost complete network embedding is an optimal description of stochastically perturbed motion on the network

Heteroclinic Dynamics in Network Dynamical Systems with Higher-Order Interactions

Heteroclinic structures organize global features of dynamical systems. We analyze whether heteroclinic structures can arise in network dynamics with higher-order interactions which describe the nonlinear interactions between three or more units. We find that while commonly analyzed model equations such as network dynamics on undirected hypergraphs may be useful to describe local dynamics such as cluster synchronization, they give rise to obstructions that allow to design heteroclinic structures in phase space. By contrast, directed hypergraphs break the homogeneity and lead to vector fields that support heteroclinic structures.Comment: 38 pages, 4 Figure

Mathematical frameworks for oscillatory network dynamics in neuroscience

The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances where this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear—for example, heteroclinic network attractors. In this review we present a set of mathemat- ical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical frame- work for further successful applications of mathematics to understanding network dynamics in neuroscience

From coupled networks of systems to networks of states in phase space

This is the author accepted manuscript. The final version is available from American Institute of Mathematical Sciences (AIMS) via the DOI in this record.Dynamical systems on graphs can show a wide range of behaviours beyond simple synchronization - even simple globally coupled structures can exhibit attractors with intermittent and slow switching between patterns of synchrony. Such attractors, called heteroclinic networks, can be well described as networks in phase space and in this paper we review some results and examples of how these robust attractors can be characterised from the synchrony properties as well how coupled systems can be designed to exhibit given but arbitrary network attractors in phase space