Controlling the Time Evolution of Paths in a Hierarchical Heteroclinic Network

A heteroclinic network is a sequence of trajectories connecting saddles in a topological network. Although this particular nonlinear dynamics looks exceptional, it is frequently found in ordinary differential equations under certain constraints like symmetries or delay. Accordingly it is predicted in models of coupled phase oscillators, pulse-coupled oscillators and models of winnerless competition. The applications range from social systems to ecological systems, fluid mechanics, chemostats, computation to neuronal networks. In relation to neuronal systems, heteroclinic cycles (and, more generally, heteroclinic sequences) in models of winnerless competition provide mechanisms for generating transient dynamics, which can be sensitive to the very input and robust against perturbations at the same time. The transient feature may correspond to cognitive processes in the brain and episodes in ecological or social systems that are inherently transient themselves. More specifically, for neuronal networks heteroclinic dynamics has been proposed as a mechanism of binding between different information modalities in the brain. An explicit hierarchy in time scales is implemented in the so-called chunking dynamics of the brain, where chunking refers to the phenomenon that the brain uses to perform information processing of long sequences by splitting them into shorter information items, called chunks.
In contrast to former descriptions, the hierarchy in time scales which we consider in this contribution, is merely implemented via the choice of rates in the generalized Lotka-Volterra equations (GLV). From the physics point of view, our main interest is in a possible modulation of fast oscillations by slow oscillations generated via heteroclinic dynamics. The two levels of hierarchy which we consider here refer both to the structure (that is a heteroclinic cycle of heteroclinic cycles) and to time scales. The elementary items correspond to the dominance of single species in 1-species saddles. The short time scales refer to fast oscillations between different saddles within one small heteroclinic cycle (SHC) (analogous to one "chunk"), while the slow time scale is generated by a large heteroclinic cycle (LHC) between three SHCs. If we choose the rates of the predation matrix in a way that certain conditions on the eigenvalues of the Jacobian are satisfied, our GLV-dynamics allows already for a tuning of time scales over orders of magnitude. Moreover, by changing a single parameter, the levels of hierarchy can be reduced to one or none. The parameter can be either the death rate of species or the strength of additive noise. This way we demonstrate how to control the path of the species trajectories through a desired heteroclinic network.
In general, the model has applications to systems in which slow oscillations modulate fast oscillations with rather sudden transitions between the temporary winners. We also indicate generalizations towards more than two levels in the hierarchical structure and outline the impact of the hierarchical structure in time on spatial pattern formation when the local heteroclinic dynamics is coupled on a spatial grid.