Dynamic pattern formation and collisions in networks of excitable elements.

Spatially extended, excitable systems with resting, activated, and refractory states, and emergent localized propagating patterns, are widespread in nature. Here a unique type of three-state excitable network model is shown to generate such dynamic patterns with rich collective dynamics. It is shown that symmetry breaking leads to the formation of dynamical patterns, leading to a change from local subdiffusive wandering to directed superdiffusive propagation. Furthermore, the model yields a rich repertoire of collision dynamics between localized propagating patterns and between propagating patterns and the refractory wakes of others. This work is particularly motivated by recent experimental studies of neural systems that exhibit localized propagating patterns, exemplifying a far wider class of excitable systems.

[1]  Solitons in the one-dimensional forest fire model. , 2000, Physical review letters.

[2]  Shigeru Kondo,et al.  Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation , 2010, Science.

[3]  P. Bressloff,et al.  Breathers in two-dimensional neural media. , 2005, Physical review letters.

[4]  M. A. Carreira-Perpiñán,et al.  Influence of lateral connections on the structure of cortical maps. , 2004, Journal of neurophysiology.

[5]  H. Willebrand,et al.  Pattern formation in gas discharge systems with high impedance electrodes , 1987 .

[6]  Alan Garfinkel,et al.  Period-doubling bifurcation in an array of coupled stochastically excitable elements subjected to global periodic forcing. , 2009, Physical review letters.

[7]  Y. Astrov,et al.  FORMATION OF CLUSTERS OF LOCALIZED STATES IN A GAS DISCHARGE SYSTEM VIA A SELF-COMPLETION SCENARIO , 1997 .

[8]  Jevin D. West,et al.  Evidence for complex, collective dynamics and emergent, distributed computation in plants , 2004, Proc. Natl. Acad. Sci. USA.

[9]  Paul C. Bressloff,et al.  Two-Dimensional Bumps in Piecewise Smooth Neural Fields with Synaptic Depression , 2011, SIAM J. Appl. Math..

[10]  S. Coombes,et al.  Bumps, breathers, and waves in a neural network with spike frequency adaptation. , 2005, Physical review letters.

[11]  J. Tyson,et al.  A cellular automation model of excitable media including curvature and dispersion. , 1990, Science.

[12]  T. Bonhoeffer,et al.  Mapping Retinotopic Structure in Mouse Visual Cortex with Optical Imaging , 2002, The Journal of Neuroscience.

[13]  Mario Markus,et al.  Two types of performance of an isotropic cellular automaton: stationary (Turing) patterns and spiral waves , 1992 .

[14]  H. Purwins,et al.  EXPERIMENTAL EVIDENCE FOR ZIGZAG INSTABILITY OF SOLITARY STRIPES IN A GAS DISCHARGE SYSTEM , 1997 .

[15]  Andrew Adamatzky,et al.  Glider-based computing in reaction-diffusion hexagonal cellular automata , 2006 .

[16]  B. Hess,et al.  Isotropic cellular automaton for modelling excitable media , 1990, Nature.

[17]  K. Steiglitz,et al.  Energy-Exchange Interactions between Colliding Vector Solitons , 1999 .

[18]  M. E. J. Newman,et al.  Power laws, Pareto distributions and Zipf's law , 2005 .

[19]  A. Grinvald,et al.  Linking spontaneous activity of single cortical neurons and the underlying functional architecture. , 1999, Science.

[20]  Ohira,et al.  Master-equation approach to stochastic neurodynamics. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[21]  Paul C. Bressloff,et al.  Breathing Pulses in an Excitatory Neural Network , 2004, SIAM J. Appl. Dyn. Syst..

[22]  Cees van Leeuwen,et al.  Distributed Dynamical Computation in Neural Circuits with Propagating Coherent Activity Patterns , 2009, PLoS Comput. Biol..

[23]  Orazio Descalzi,et al.  Noise induces partial annihilation of colliding dissipative solitons. , 2009, Physical review letters.

[24]  Feng Qi Han,et al.  Reverberation of Recent Visual Experience in Spontaneous Cortical Waves , 2008, Neuron.

[25]  Drossel,et al.  Self-organized critical forest-fire model. , 1992, Physical review letters.

[26]  Mathias Bode,et al.  Interacting Pulses in Three-Component Reaction-Diffusion Systems on Two-Dimensional Domains , 1997 .

[27]  Irving R Epstein,et al.  Localized patterns in reaction-diffusion systems. , 2007, Chaos.

[28]  Ertl,et al.  Solitary-wave phenomena in an excitable surface reaction. , 1992, Physical review letters.

[29]  A Grinvald,et al.  Coherent spatiotemporal patterns of ongoing activity revealed by real-time optical imaging coupled with single-unit recording in the cat visual cortex. , 1995, Journal of neurophysiology.