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Design of Hierarchical Excitable Networks

Sören von der Gracht, Alexander Lohse

Abstract

We provide a method to systematically construct vector fields for which the dynamics display transitions corresponding to a desired hierarchical connection structure. This structure is given as a finite set of directed graphs $\mathbf{G}_1,\dotsc,\mathbf{G}_N$ (the lower level), together with another digraph $\mathbfΓ$ on $N$ vertices (the top level). The dynamic realizations of $\mathbf{G}_1,\dotsc,\mathbf{G}_N$ are heteroclinic networks and they can be thought of as individual connection patterns on a given set of states. Edges in $\mathbfΓ$ correspond to transitions between these different patterns. In our construction, the connections given through $\mathbfΓ$ are not heteroclinic, but excitable with zero threshold. This describes a dynamical transition between two invariant sets where every $δ$-neighborhood of the first set contains an initial condition with $ω$-limit in the second set. Thus, we prove a theorem that allows the systematic creation of hierarchical networks that are excitable on the top level, and heteroclinic on the lower level. Our results modify and extend the simplex realization method by Ashwin & Postlethwaite.

Design of Hierarchical Excitable Networks

Abstract

We provide a method to systematically construct vector fields for which the dynamics display transitions corresponding to a desired hierarchical connection structure. This structure is given as a finite set of directed graphs (the lower level), together with another digraph on vertices (the top level). The dynamic realizations of are heteroclinic networks and they can be thought of as individual connection patterns on a given set of states. Edges in correspond to transitions between these different patterns. In our construction, the connections given through are not heteroclinic, but excitable with zero threshold. This describes a dynamical transition between two invariant sets where every -neighborhood of the first set contains an initial condition with -limit in the second set. Thus, we prove a theorem that allows the systematic creation of hierarchical networks that are excitable on the top level, and heteroclinic on the lower level. Our results modify and extend the simplex realization method by Ashwin & Postlethwaite.
Paper Structure (24 sections, 1 theorem, 42 equations, 11 figures)

This paper contains 24 sections, 1 theorem, 42 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Excitable and heteroclinic connections
  4. Design---The simplex method
  5. Two level hierarchy---The simplex-simplex method
  6. Construction
  7. The realization result
  8. Robustness
  9. Thresholds of excitable connections
  10. Modification of time-scales
  11. Examples and simulations
  12. Switching substructure
  13. Numerical simulation.
  14. Switching superstructure
  15. Numerical simulation.
  16. ...and 9 more sections

Key Result

Theorem 3.3

Let $\mathbf{G}_1,\dotsc,\mathbf{G}_N$ and $\mathbf{\Gamma}$ be a collection of directed graphs as above, and suppose that they contain no $1$-cycles and no $2$-cycles. Choose the coefficients $a_{jk}$ and $\alpha^j_{ik}$ as in eq:simplex_coeff, according to the adjacency matrices of $\mathbf{\Gamma

Figures (11)

  • Figure 1: Heteroclinic (left) and an excitable (right) connections from $\xi_1$ to $\xi_2$.
  • Figure 2: Heteroclinic connection of depth two from the cycle $[\xi_1 \to \xi_2 \to \xi_3]$ to the equilibrium $\xi$; can also be interpreted as three excitable connections with threshold zero from each equilibrium $\xi_1, \xi_2, \xi_3$ to $\xi$.
  • Figure 3: Sketch of the transition function $b_\varepsilon^j$ used in \ref{['eq:simplex_simplex']}.
  • Figure 4: Digraph on four vertices corresponding to the Kirk-Silber network.
  • Figure 5: Sketch of a hierarchical collection of digraphs. The superstructure and two of the substructures correspond to $3$-cycles, while the third substructure corresponds to the Kirk-Silber network.
  • ...and 6 more figures

Theorems & Definitions (7)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 3.1
  • Remark 3.2
  • Theorem 3.3
  • proof