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Phase transition in one-dimensional excitable media with variable interaction range

Ander Aguirre, Hanbaek Lyu, David Sivakoff

Abstract

We investigate two discrete models of excitable media on a one-dimensional integer lattice $\mathbb{Z}$: the $κ$-color Cyclic Cellular Automaton (CCA) and the $κ$-color Firefly Cellular Automaton (FCA). In both models, sites are assigned uniformly random colors from $\mathbb{Z}/κ\mathbb{Z}$. Neighboring sites with colors within a specified interaction range $r$ tend to synchronize their colors upon a particular local event of 'excitation'. We establish that there are three phases of CCA/FCA on $\mathbb{Z}$ as we vary the interaction range $r$. First, if $r$ is too small (undercoupled), there are too many non-interacting pairs of colors, and the whole graph $\mathbb{Z}$ will be partitioned into non-interacting intervals of sites with no excitation within each interval. If $r$ is within a sweet spot (critical), then we show the system clusters into ever-growing monochromatic intervals. For the critical interaction range $r=\lfloor κ/2 \rfloor$, we show the density of edges of differing colors at time $t$ is $Θ(t^{-1/2})$ and each site excites $Θ(t^{1/2})$ times up to time $t$. Lastly, if $r$ is too large (overcoupled), then neighboring sites can excite each other and such 'defects' will generate waves of excitation at a constant rate so that each site will get excited at least at a linear rate. For the special case of FCA with $r=\lfloor 2/κ\rfloor+1$, we show that every site will become $(κ+1)$-periodic eventually.

Phase transition in one-dimensional excitable media with variable interaction range

Abstract

We investigate two discrete models of excitable media on a one-dimensional integer lattice : the -color Cyclic Cellular Automaton (CCA) and the -color Firefly Cellular Automaton (FCA). In both models, sites are assigned uniformly random colors from . Neighboring sites with colors within a specified interaction range tend to synchronize their colors upon a particular local event of 'excitation'. We establish that there are three phases of CCA/FCA on as we vary the interaction range . First, if is too small (undercoupled), there are too many non-interacting pairs of colors, and the whole graph will be partitioned into non-interacting intervals of sites with no excitation within each interval. If is within a sweet spot (critical), then we show the system clusters into ever-growing monochromatic intervals. For the critical interaction range , we show the density of edges of differing colors at time is and each site excites times up to time . Lastly, if is too large (overcoupled), then neighboring sites can excite each other and such 'defects' will generate waves of excitation at a constant rate so that each site will get excited at least at a linear rate. For the special case of FCA with , we show that every site will become -periodic eventually.

Paper Structure

This paper contains 23 sections, 20 theorems, 104 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Related work and discussion
  3. Fixation vs. Fluctuation.
  4. Clustering and persistence of particle counting walks
  5. Particle flipping and random speeds
  6. Excitation counts and tournamant expansion
  7. Quasi-Stable Periodic Objects in the overcoupled regime
  8. Organization
  9. Arrow Dynamics for undercoupled and critical CCA/FCA
  10. Arrow dynamics for the 3-color critical CCA/FCA
  11. Arrow dynamics for general CCA and FCA
  12. Arrow dynamics for CCA
  13. Arrow dynamics for FCA
  14. CCA and FCA in the undercoupled regime
  15. Critical CCA and FCA
  16. ...and 8 more sections

Key Result

Theorem 1

Let $\xi_{t}$ denote the $\kappa$-color CCA or FCA with interaction range $r$ on the one-dimensional integer lattice $\mathbb{Z}$ for $\kappa\ge 3$, where the initial configuration $\xi_{0}$ is drawn from the uniform product measure $\mathbb{P}$ on ${\mathbb Z}_\kappa^{\mathbb{Z}}$. The following ho

Figures (9)

  • Figure 1: Simulation of CCA and FCA with $\kappa=5$ and $r\in \{1,2,3,4\}$ on 500 nodes in $\mathbb{Z}$ with 50 and 250 iterations, respectively. Time goes from top to bottom. For CCA configurations at every iteration are shown, while for FCA only the ones at times $\kappa t$ are shown.
  • Figure 2: Simulation of CCA and FCA with $\kappa=6$ and $r\in \{1,2,3,4,5\}$ on 500 nodes in $\mathbb{Z}$ with 50 and 250 iterations, respectively. Time goes from top to bottom. For CCA configurations at every iteration are shown, while for FCA only the ones at times $\kappa t$ are shown.
  • Figure 3: Simulation of 3-color critical CCA and FCA ($r=1$) on 50 nodes in $\mathbb{Z}$ with 20 iterations. Time goes from top to bottom where configurations at every iteration are shown for both systems.
  • Figure 4: Simulation of 9-color critical CCA and FCA ($r=4$) on 50 nodes in $\mathbb{Z}$ with 20 iterations. Time goes from top to bottom. For CCA configurations at every iteration are shown, while for FCA only the ones at times $9 t$ are shown.
  • Figure 5: Arrow collisions for CCA with $\kappa=11$ and $r=4$. Pseudo-Lexicographic labeling of arrows ensures proper monotonic ordering.
  • ...and 4 more figures

Theorems & Definitions (46)

  • Theorem 1: Phase transition of CCA/FCA in interaction range
  • Theorem 2: Excitation rate in critical CCA/FCA
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 36 more