Spatiotemporal chaos on reaction diffusion systems

Emerging Behavior and Spatiotemporal Chaos in Reaction-Diffusion Models:
GPU-accelerated simulations in a web browser over the internet.


We present here a set of interactive programs to study and analyze several models of excitable media in tissue. As the waves they produce propagate through the media, the models exhibit complex spatiotemporal dynamics that cannot be appreciated from an analysis of the underlying equations or even verbal descriptions. Here, we allow users to perform in real time simulations of these models and to watch the patterns develop and change over time as the simulated dynamical waves propagate. The parameters governing the model's behavior can be changed on the fly to alter the dynamics. In addition, users can apply perturbations and periodic pacing, that change the patterns locally an globaly and watch the response.

One of the main advantages of these programs is that the models are implemented using WebGL, which allows the simulations to be run over the Internet, independent of computer architecture and operating system. WebGL utilizes available hardware, including graphics cards, to improve performance, thereby allowing simulations to run as a parallel implementation on the computer's GPU and to respond rapidly to user interactions. In fact, some smart phones are capable of running these simulations. Our implementations also feature very large domains, (256x256,512x512 and 1024x1024) in 2d and one 3D, that provide sufficient space for complex and interesting patterns to form and evolve.

We have implementated three generic models of excitable media, the FitzHugh-Nagumo model, the Barkley model (which is a simplification of the FitzHugh-Nagumo model), and the Gray-Scott model, which among other things includes an unusual backfiring feature. In addition, we have implemented two models specific to excitable cardiac electrophysiology that develop complex spatiotemporal patterns, the Karma model and the BuenoOrovio–Cherry–Fenton model. Altogether, these models provide a fascinating portrait of complex spatiotemporal dynamics in excitable media.

WebGL Programs

These are programs that run in parallel on the computer's graphic cards (GPU) via a web-browser. There is no need for compiler or a special operating system, the new standards on HTML5 allows for Mozilla or Google Chrome to run shader programs like WebGL in parallel on the computer's GPU over the web. In these programs, click inside the tissue to initiate or induce an excitation(s). The parameters of the models can be changed in the windows and various activities can be produced. Allthe options in the codes are relatively easy to use and are self explanatory. However, we have added some movies that help to illustrate some of the effects, uses and dynamics that can be achieved with these programs.

Note: You will need Google Chrome or Mozilla to play the WebGL programs. With mozilla in some cases it may be needed to give permision for webGL to run, which is very simple as shown here.

FitzHugh-Nagumo model [1]

        ∂U/∂t = F(U, V) = Du∇2U + (U(1 - U)(U - a) - V),
        ∂V/∂t = H(U, V) = (bU - V - δ)ε
    Here is a link to a couple of our Java Applets that are usefull for learning and underestanding the dynamics of a single FHN cell. FHN applet 1 and FHN applet 2
  • Large wave length (512x512) ( (1024x1024 grid). This model can produce spiral waves with very diverse tip trajectories[6], that can be studied by varying for example ε from 0.015 to 0.00002.
  • Small wave length (512x512) ( (1024x1024 grid). By rescaling the time betwen the U and V variables with ε, it is possible to change the wavelenght of the system. The simulations in this link produce spiral waves that have faster rotation and smaller wavelengths.
  • Movie one
  • Movie two
  • Three dimensional simulation Dynamics of scroll waves in the negative tension regime, this results in scroll wave breakup[7]. The parameters used in this simulation produce distabilization of the initial vortex filament, leading to the creation of multiple filaments. The parameters of the model can be modified dynamically in the code to prevent breakup and keep a rotating spiral wave stable.

Barkley model [2]

    ∂U/∂t = Du∇2U + U(1 - U)(U - Uth ) ,
    ∂V/∂t = (U - V)ε ,     Uth = (V - b)/a .

Gray-Scott model [3]

    ∂U/∂t = Du∇2U + (U2V - aU)/ε
    ∂V/∂t = Dv∇2V - U2V +(b-cV)
  • (256x256) ( (512x512 grid). This model has a large diversiy of patterns, from selfreplicating spots (reminiscent of cellular division), to spiral waves, to pulsating activations and labyrinth patterns as show in the movie below. Transitions between these regimes can be obtained for example by varyng the diffusion coefficient Du from 1. to 200.
  • Movie one.
  • Movie two.

Karma model [4]

    dE/dt = F(E, n) = [A - (n/nb)M][1 - tanh(E - 3)]E2/2 - E,
    dn/dt = H(E, n) = ε [Θ(E - 1) - n].
    where A = 1.5415.
  • (512x512) This model has a regime where breakup of spiral waves is due to a period doubling bifurcation. This induces oscillations of the wavelength that lead to breakup and complex dynamics[8][9]. For example, by varying the parameter a and lowering below 1.0, spirals will be stable with no oscillations. By slowly increasing a above 1.5 will produce oscillations on the wavelength and eventual breakup of spiral waves leading to multiple spiral waves.

BuenoOrovio-Cherry-Fenton model [5]

    du/dt = -(Jfi + Jso + Jsi ) ,
    dv/dt = [1 - H(u - Vv )](v∞ - v)/τv - - H(u - Vv )v/τv ,
    dw/dt = [1 - H(u - Vw )](w∞ - w)/τw- - H(u - Vw )w/τw ,        
    ds/dt = ((1 + tanh[ks (u - Us )])/2 - s)/τs ,
  • (512x512) This is a minimal model that describes the electrical activity of human ventricular cardiac tissue. The dynamics of the spiral waves and their stability are a function of model parameters as shown on the figure below[10]. For a survey of many existing mathematical models of cariac cells see ref [11].
Main Bibliography: [1] FitzHugh-Nagumo model in Scholarpedia
[2]Dwight Barkley A model for fast computer simulation of waves in excitable media Physica D 49 (1991) 61-70.
[3] P. Gray and S.K. Scott, Autocatalytic reactions in the isothermal, continuous stirred tank reactor: Oscillations and instabilities in the system A+ 2B --> 3B; B --> C     Chem. Engng. Sci. 39, 1087 (1984).
[4] Alain Karma Spiral Breakup in Model Equations of Action Potential Propagation in Cardiac Tissue Phys.Rev.Lett. 71, 1103 (1993)
[5] Bueno-Orovio A., Cherry EM, Fenton FH. Minimal Model for Human Ventricular Action Potential in Tissue. Journal of Theoretical Biology 2008; 253 544-560.
Supplemental Bibliography:
[6]A. T. Winfree, Varieties of spiral wave behavior in excitable media , Chaos 1, 303–333 1991.
[7] V. N. Biktashev, A. V. Holden, and H. Zhang, ‘‘Tension of organizing filaments of scoll waves,’’ Philos. Trans. R. Soc. London, Ser. A 347, 611–630 1994.
[8] M. R. Guevara, G. Ward, A. Shrier, and L. Glass, ‘‘Electrical alternans and period doubling bifurcations,’’ Comput. Cardiol. 167–170 1984.
[9] Watanabe M, Fenton F, Evans S, Hastings H, Karma A. Mechanisms for Discordant Alternans. Journal of Cardiovascular Electrophysiology 2001; 12: 196-206.
[10] Bartocci E, et al. “Teaching cardiac electrophysiology modeling to undergraduate students: Lab exercises and GPU programming for the study of arrhythmias and spiral wave dynamics” Advances in Physiology Education, 35: 427-437. 2011.
[11] Models of Cardiac Cell in Scholarpedia.
By: Evgeny Demidov, demidov@ipm.sci-nnov.ru
      Elizabeth M. Cherry, excsma@rit.edu and
      Flavio H. Fenton, flavio.fenton@physics.gatech.edu
Part of this work was performed under NSF grant # 1341190 Collaborative Research: Dynamical Systems Program.