Kinematics of Wave Patterns in Excitable Media

Author(s):  
A. S. Mikhailov
1979 ◽  
Vol 34 (5) ◽  
pp. 565-570 ◽  
Author(s):  
. E. Rössler ◽  
C. Kahlert

Abstract Two-dimensional excitable media, for example the excitable version of the Belousov-Zhabotin-sky reaction, are capable of forming spiral-shaped self-sustaining rotating wave patterns (rotors). In order to explain Winfree's experimental observation of an irregular "meandering" of the rotor's core region, we present a numerical simulation of a continuous, two-variable excitable medium in two space dimensions. Two phenomena occur: 1) an irregular motion of the rotor's core; 2) a non-stationary peak inside the core region. Thus, "meandering" is obtained, together with a new phenomenon, the "peak". A sufficient condition for both phenomena is that the underlying (local) system be stiff, that is, admit two time scales for its approximate description. In this case, (1) the "reaching distance" of diffusion is small as compared with the core's radius, and (2) rotation symmetry of the core implies a gradation in local frequencies (increase toward the center), supposing radial decoupling. We propose that both constraints act together to induce a spontaneous breakdown of rotation symmetry under an increase of stiffness. Immediately (or soon) thereafter, the core is no longer synchronized by the wave circling around it; instead, excitation from the surrounding wave penetrates the core along a new path from time to time, causing a non-repetitive short-cut.


2018 ◽  
Vol 119 (6) ◽  
pp. 2212-2228 ◽  
Author(s):  
Michael E. Rule ◽  
Carlos Vargas-Irwin ◽  
John P. Donoghue ◽  
Wilson Truccolo

Previous studies on the origin and properties of spatial patterns in motor cortex β-local field potential (β-LFP) oscillations have focused on planar traveling waves. However, it is unclear 1) whether β-LFP waves are limited to plane waves, or even 2) whether they are propagating waves of excito-excitatory activity, i.e., primarily traveling waves in excitable media; they could reflect, instead, reorganization in the relative phases of transient oscillations at different spatial sites. We addressed these two problems in β-LFPs recorded via microelectrode arrays implanted in three adjacent motor cortex areas of nonhuman primates during steady-state movement preparation. Our findings are fourfold: 1) β-LFP wave patterns emerged as transient events, despite stable firing rates of single neurons concurrently recorded during the same periods. 2) β-LFP waves showed a richer variety of spatial dynamics, including rotating and complex waves. 3) β-LFP wave patterns showed no characteristic wavelength, presenting instead a range of scales with global zero-lag phase synchrony as a limiting case, features surprising for purely excito-excitatory waves but consistent with waves in coupled oscillator systems. 4) Furthermore, excito-excitatory traveling waves induced by optogenetic stimulation in motor cortex showed, in contrast, a characteristic wavelength and reduced phase synchrony. Overall, β-LFP wave statistics differed from those of induced traveling waves in excitable media recorded under the same microelectrode array setup. Our findings suggest phase reorganization in neural coupled oscillators contribute significantly to the origin of transient β-LFP spatial dynamics during preparatory steady states and outline important constraints for spatially extended models of β-LFP dynamics in motor cortex. NEW & NOTEWORTHY We show that a rich variety of transient β-local field potential (β-LFP) wave patterns emerge in motor cortex during preparatory steady states, despite stable neuronal firing rates. Furthermore, unlike optogenetically induced traveling waves, β-LFP waves showed no characteristic wavelength, presenting instead a range of scales with global phase synchrony as a limiting case. Overall, our statistical analyses suggest that transient phase reorganization in neural coupled oscillators, beyond purely excito-excitatory traveling waves, contribute significantly to the origin of motor cortex β-LFP wave patterns.


1996 ◽  
Vol 6 (11) ◽  
pp. 1417-1434 ◽  
Author(s):  
Joceline Lega ◽  
Jean-Marc Vince
Keyword(s):  

1997 ◽  
Vol 36 (04/05) ◽  
pp. 290-293
Author(s):  
L. Glass ◽  
T. Nomura

Abstract:Excitable media, such as nerve, heart and the Belousov-Zhabo- tinsky reaction, exhibit a large excursion from equilibrium in response to a small but finite perturbation. Assuming a one-dimensional ring geometry of sufficient length, excitable media support a periodic wave of circulation. As in the periodic stimulation of oscillations in ordinary differential equations, the effects of periodic stimuli of the periodically circulating wave can be described by a one-dimensional Poincaré map. Depending on the period and intensity of the stimulus as well as its initial phase, either entrainment or termination of the original circulating wave is observed. These phenomena are directly related to clinical observations concerning periodic stimulation of a class of cardiac arrhythmias caused by reentrant wave propagation in the human heart.


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