- published: 28 May 2014
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An excitable medium is a nonlinear dynamical system which has the capacity to propagate a wave of some description, and which cannot support the passing of another wave until a certain amount of time has passed (known as the refractory time).
A forest is an example of an excitable medium: if a wildfire burns through the forest, no fire can return to a burnt spot until the vegetation has gone through its refractory period and regrown. In chemistry, oscillating reactions are excitable media, for example the Belousov–Zhabotinsky reaction and the Briggs–Rauscher reaction. Pathological activities in the heart and brain can be modelled as excitable media. A group of spectators at a sporting event are an excitable medium, as can be observed in a Mexican wave (so-called from its initial appearance in the 1986 World Cup in Mexico).
Excitable media can be modelled using both partial differential equations and cellular automata.
Cellular automata provide a simple model to aid in the understanding of excitable media. Perhaps the simplest such model is in. See Greenberg-Hastings cellular automaton for this model.
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The rule: - Cells can be in 10 different states. State 0 is resting. States 1-5 are active, and states 6-9 are refractory. - Count the 8 nearest neighbors of each cell which are in one of the active states. - If the sum is greater or equal to 3 (at least three active neighbors) then cell=1. - States 1 to 9 occur stepwise with no more input. If state=1 then the next state=2. If state=2 then the next state=3, and similarly of all the states up to 9. If state=9 then the next state=0 and the cell is back at rest. Simulation from: https://instruct1.cit.cornell.edu/courses/bionb441/CA/index.html
Particles are displaced double the distance when collisions are detected, this results in the Excitable media like behavior.
Propagating wave-front in Oregonator model of Belousov-Zhabotinsky medium solves Chartres maze. See details in http://arxiv.org/abs/1108.4956
Fig 1b Sub excitable medium Collapsing wave fragments Adamatzky A. Fredkin and Toffoli implemented in Oregonator model of Belousov-Zhabotinsky medium. International Journal of Bifurcation and Chaos, vol. 27, no. 3, 2017.
Fig 1a Sub excitable medium Expanding wave fragments Adamatzky A. Fredkin and Toffoli implemented in Oregonator model of Belousov-Zhabotinsky medium. International Journal of Bifurcation and Chaos, vol. 27, no. 3, 2017.
The dynamics of calcium [Ca2+] wave formation and propagation in smooth muscle cells are modelled on a two dimensional flat surface with periodic boundary conditions and with an initial perturbation: the [Ca2+] concentration is set higher at a small rectangular area on the surface. The local dynamics of the medium are excitable via the spatially constant parameter beta. Beta determines the dynamics: either excitable, subexcitable, nonexcitable, or oscillatory. In this simulation beta = 0.25. The initial perturbation will propagate outwards. The backwards travelling wave is absorbed by a temporary boundary at y = 0. Generated using the Goldbeter et al. (1990) model on a two dimensional spatial domain with diffusion coefficient D=0.12.
The dynamics of calcium [Ca2+] wave formation and propagation in smooth muscle cells are modelled on a two dimensional flat surface with periodic boundary conditions and with an initial perturbation: the [Ca2+] concentration is set higher at a small rectangular area on the surface. The local dynamics of the medium are excitable via the spatially constant parameter beta. Beta determines the dynamics: either excitable, subexcitable, nonexcitable, or oscillatory. In this simulation beta = 0.25 and the initial wavefront has two open ends. These ends will grow in all directions except backwards (where the refractory waveback is). This causes the wave ends to curl and eventually form spiral rotors which send spiral waves rotating outwards. Spiral waves will form when the medium is either osci...
This video is a continuation of the excitable medium cellular automaton shown in the video: Excitable Media http://www.youtube.com/watch?v=-jJPtpxu2N0 The current video shows how a frequent excitation parameter leads to self-organization and sustained pattern emergence in the form of spiral waves, and how this self-organization changes for different refractory periods. For very small refractory periods the dynamics is more turbulent, as the refractory period is increased more self-sustained spiral wave dynamics emerge, for longer refractory periods there occur a travelling wave phase that may be replaced by a spiral wave pattern as excitation proceeds. For a 5 tick refratory period spontaneous phase transitions take place between the non-spiral travelling waves pattern and the spirali...
Krinski's lab formalized the study of spiral waves in excitable media. Here is a stirred passive medium.
"Vulnerability" is one of the scenario to form spiral wave spontaneously on an excitable medium. This movie shows spiral formation on Belousov-Zhabotinsky reaction generated by vulnerability. I stimulated just behind a precursive chemical wave. Scince the refractry region of the precursive wave inhibits a part of the newborn chemical wave, twin spiral cores are generated.
This is an excitable media cellular automaton model described in Daniel Kaplan and Leon Glass (1995), "Understanding Nonlinear Dynamics", Springer-Verlag, New York. Each cell is in one of three states: Quiescent Excited or Refractory If a cell is quiescent at time t, then it stays quiescent at time t+1 unless one or more of its neighbors is excited at time t, in which case the cell becomes excited at time t+1. If a node is excited at time t, then it becomes refractory at time t+1. If a node is refractory at time t, it becomes quiescent at time t+T where T is the refractory time. Initially, all of the cells are quiescent, except for one chosen at random that is excited. The excitation frequency will lead to the temporal frequency at which a randomly chosen quiescent cell is excited. Thu...
3Comps model nn=50 N =nn+1 ; number of grid points in the x-axis grid=N*N h =0.01; ; Grid size dt =0.01 ; Enter time step ;kinetic parameters k1=-6.5 k2=2.0 k3=1.0 k4=8.5 tau=40.0 ;diffusion constants Difu = 0.00009 Difv = 0.001 Difw = 0.01 ;diversity SIGk1=0.01
Excitation wave-front follows gradients of chemo-attractants emitted by the maze's central chamber. See details in http://arxiv.org/abs/1108.4956
Vibrating Pinscreen Portrait Mosaiced into the pinfield is a photographic portrait of the Exploratorium's founder Frank Oppenheimer. The pins comprising the portrait are one inch in length and are surrounded by pins five eighths of an inch in length that predominate the thirty square inch field. Being of greater mass the one-inch pins lag behind their shorter neighbors when excited at any given frequency. The light reflected off the longer pins' polished heads lags in reaching the viewer, thus revealing the pertinent information. A second condition that makes the revelation possible is that the longer pins comprising the portrait are arranged in specifically calibrated clusters that correspond to relative amounts of light and shade in the photo portrait. This directly translates...
propagating spot in homogeneous media
Fig 1a Sub excitable medium Expanding wave fragments Adamatzky A. Fredkin and Toffoli implemented in Oregonator model of Belousov-Zhabotinsky medium. International Journal of Bifurcation and Chaos, vol. 27, no. 3, 2017.
Fig 1b Sub excitable medium Collapsing wave fragments Adamatzky A. Fredkin and Toffoli implemented in Oregonator model of Belousov-Zhabotinsky medium. International Journal of Bifurcation and Chaos, vol. 27, no. 3, 2017.
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Particles are displaced double the distance when collisions are detected, this results in the Excitable media like behavior.
- By: Jaume Casademunt, Departament dECM, Universitat de Barcelona - Date: 2014-05-07 14:30:00 - Description: At early stages of development, neuronal cultures in vitro spontaneously reach a coherent state of collective firing in a pattern of nearly periodic global bursts. Although understanding the spontaneous activity of neuronal networks is of chief importance in neuroscience, the origin and nature of that pulsation has remained elusive. By combining high-resolution calcium imaging with modelling in silico, we show that this behaviour is controlled by the propagation of waves that nucleate randomly in a set of points that is specific to each culture and is selected by a non-trivial interplay between dynamics and topology. The phenomenon is explained by the noise focusing effec...
- By: Felix Muller, Institut für Physik, Humboldt Universität zu Berlin, Germany - Date: 2010-12-10 12:00:00 - Description: According to the conventional approach neural ensembles are modeled with fixed ionic concentrations in the extracellular environment. However, recent studies on the patho-physiology of tissue from epileptic human patients revealed aberrations in potassium regulation. These findings can be interpreted as neuronal interaction via specific chemical pathways, that influence strongly the behavior of single neurons and large ensembles. Released chemical agents, which diffuse in the extracellular medium is able to lower thresholds of individual excitable units. We address this problem by studying a modified stochastic FitzHugh-Nagumo dynamics. In our model the neur...
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