CELLULAR AUTOMATA MODELS OF RING DYNAMICS

1996 ◽  
Vol 07 (06) ◽  
pp. 863-871 ◽  
Author(s):  
JANKO GRAVNER
Keyword(s):  
Phase Transition ◽  
Cellular Automata ◽  
Excitation Level ◽  
Excitable Media ◽  
Convergence To Equilibrium ◽  
Visual Feature ◽  
Positive Probability ◽  
Fixed Probability ◽  
Ring Dynamics

This paper describes three models arising from the theory of excitable media, whose primary visual feature are expanding rings of excitation. Rigorous mathematical results and experimental/computational issues are both addressed. We start with the much-studied Greenberg–Hastings model (GHM) in which the rings are very short-lived, but they do have a transient percolation property. By contrast, in the model we call annihilating nested rings (ANR), excitation centers only gradually lose strength, i.e., each time they become inactive (and then stay so forever) with a fixed probability; we show how the long-term global connectivity properties of the set of excited sites undergo a phase transition. Second part of the paper is devoted to digital boiling (DB) in which new rings spontaneously appear at rested sites with a positive probability. We focus on such (related) issues as convergence to equilibrium, equilibrium excitation level and success of the basic coupling.

Recurrent ring dynamics in two-dimensional excitable cellular automata

1999 ◽  
Vol 36 (2) ◽  
pp. 492-511 ◽  
Author(s):  
Janko Gravner
Keyword(s):  
Excitable Media ◽  
Two Dimensional ◽  
External Stimulation ◽  
Asymptotic Shape ◽  
Ring Dynamics ◽  
Initial Placement ◽  
Random Growth ◽  
A Site ◽  
Continuous Nucleation

TheGreenberg–Hastings model(GHM) is a simple cellular automaton which emulates two properties of excitable media: excitation by contact and a refractory period. We study two ways in which external stimulation can makeringdynamics in the GHM recurrent. The first scheme involves the initial placement of excitation centres which gradually lose strength, i.e. each time they become inactive (and then stay inactive forever) with probability 1 −pf. In this case, the density of excited sites must go to 0; however, their long–term connectivity structure undergoes a phase transition aspfincreases from 0 to 1. The second proposed rule utilizes continuous nucleation in that new rings are started at every rested site with probabilityps. We show that, for smallps, these dynamics make a site excited about everyps−1/3time units. This result yields some information about the asymptotic shape of a closely related random growth model.

Recurrent ring dynamics in two-dimensional excitable cellular automata

1999 ◽  
Vol 36 (02) ◽  
pp. 492-511 ◽  
Author(s):  
Janko Gravner
Keyword(s):  
Excitable Media ◽  
Two Dimensional ◽  
External Stimulation ◽  
Asymptotic Shape ◽  
Ring Dynamics ◽  
Initial Placement ◽  
Random Growth ◽  
A Site ◽  
Continuous Nucleation

The Greenberg–Hastings model (GHM) is a simple cellular automaton which emulates two properties of excitable media: excitation by contact and a refractory period. We study two ways in which external stimulation can make ring dynamics in the GHM recurrent. The first scheme involves the initial placement of excitation centres which gradually lose strength, i.e. each time they become inactive (and then stay inactive forever) with probability 1 − p f. In this case, the density of excited sites must go to 0; however, their long–term connectivity structure undergoes a phase transition as p f increases from 0 to 1. The second proposed rule utilizes continuous nucleation in that new rings are started at every rested site with probability p s . We show that, for small p s , these dynamics make a site excited about every p s −1/3 time units. This result yields some information about the asymptotic shape of a closely related random growth model.

scholarly journals ON DIVERSITY OF CONFIGURATIONS GENERATED BY EXCITABLE CELLULAR AUTOMATA WITH DYNAMICAL EXCITATION INTERVALS

2012 ◽  
Vol 23 (12) ◽  
pp. 1250085 ◽  
Author(s):  
ANDREW ADAMATZKY
Keyword(s):  
Cellular Automata ◽  
Lower Boundary ◽  
Morphological Diversity ◽  
Excitable Media ◽  
Stationary Waves ◽  
Future Studies ◽  
Time Dynamics ◽  
Wide Range ◽  
Localized Excitations ◽  
Upper Boundary

Excitable cellular automata with dynamical excitation interval exhibit a wide range of space-time dynamics based on an interplay between propagating excitation patterns which modify excitability of the automaton cells. Such interactions leads to formation of standing domains of excitation, stationary waves and localized excitations. We analyzed morphological and generative diversities of the functions studied and characterized the functions with highest values of the diversities. Amongst other intriguing discoveries we found that upper boundary of excitation interval more significantly affects morphological diversity of configurations generated than lower boundary of the interval does and there is no match between functions which produce configurations of excitation with highest morphological diversity and configurations of interval boundaries with highest morphological diversity. Potential directions of future studies of excitable media with dynamically changing excitability may focus on relations of the automaton model with living excitable media, e.g. neural tissue and muscles, novel materials with memristive properties and networks of conductive polymers.

A Simple Three-States Cellular Automaton for Modelling Excitable Media

1998 ◽  
Vol 12 (05) ◽  
pp. 601-607 ◽  
Author(s):  
M. Andrecut
Keyword(s):  
Wave Propagation ◽  
Partial Differential Equations ◽  
Cellular Automata ◽  
Differential Equations ◽  
Cellular Automaton ◽  
Cellular Automaton Model ◽  
Excitable Media ◽  
Self Organization ◽  
Partial Differential ◽  
Bz Reaction

Wave propagation in excitable media provides an important example of spatiotemporal self-organization. The Belousov–Zhabotinsky (BZ) reaction and the impulse propagation along nerve axons are two well-known examples of this phenomenon. Excitable media have been modelled by continuous partial differential equations and by discrete cellular automata. Here we describe a simple three-states cellular automaton model based on the properties of excitation and recovery that are essential to excitable media. Our model is able to reproduce the dynamics of patterns observed in excitable media.

scholarly journals Wave onset in central gray matter - its intrinsic optical signal and phase transitions in extracellular polymers

2001 ◽  
Vol 73 (3) ◽  
pp. 351-364 ◽  
Author(s):  
VERA M. FERNANDES-DE-LIMA ◽  
JOÃO E. KOGLER ◽  
JOCELYN BENNATON ◽  
WOLFGANG HANKE
Keyword(s):  
Phase Transition ◽  
Phase Transitions ◽  
Gray Matter ◽  
Action Potentials ◽  
Optical Signal ◽  
Excitable Media ◽  
Extracellular Polymers ◽  
Axon Membrane ◽  
Intrinsic Optical Signal ◽  
Volume Phase

The brain is an excitable media in which excitation waves propagate at several scales of time and space. ''One-dimensional'' action potentials (millisecond scale) along the axon membrane, and spreading depression waves (seconds to minutes) at the three dimensions of the gray matter neuropil (complex of interacting membranes) are examples of excitation waves. In the retina, excitation waves have a prominent intrinsic optical signal (IOS). This optical signal is created by light scatter and has different components at the red and blue end of the spectrum. We could observe the wave onset in the retina, and measure the optical changes at the critical transition from quiescence to propagating wave. The results demonstrated the presence of fluctuations preceding propagation and suggested a phase transition. We have interpreted these results based on an extrapolation from Tasaki's experiments with action potentials and volume phase transitions of polymers. Thus, the scatter of red light appeared to be a volume phase transition in the extracellular matrix that was caused by the interactions between the cellular membrane cell coat and the extracellular sugar and protein complexes. If this hypothesis were correct, then forcing extracellular current flow should create a similar signal in another tissue, provided that this tissue was also transparent to light and with a similarly narrow extracellular space. This control tissue exists and it is the crystalline lens. We performed the experiments and confirmed the optical changes. Phase transitions in the extracellular polymers could be an important part of the long-range correlations found during wave propagation in central nervous tissue.

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