SPIRAL WAVES IN MEDIA WITH COMPLEX-EXCITABLE DYNAMICS

1999 ◽  
Vol 09 (11) ◽  
pp. 2243-2247 ◽  
Author(s):  
ANDREI GORYACHEV ◽  
RAYMOND KAPRAL
Keyword(s):  
Phase Space ◽  
Temporal Dynamics ◽  
Rotational Symmetry ◽  
Reaction Diffusion ◽  
Structural Features ◽  
Spiral Waves ◽  
Space Trajectories ◽  
Reaction Diffusion Systems ◽  
Oscillatory Dynamics ◽  
Diffusion Systems

The structure of spiral waves is investigated in super-excitable reaction–diffusion systems where the local dynamics exhibits multilooped phase-space trajectories. It is shown that such systems support stable spiral waves with broken rotational symmetry and complex temporal dynamics. The main structural features of such waves, synchronization defect lines, are demonstrated to be similar to those of spiral waves in systems with complex-oscillatory dynamics.

scholarly journals Chiral selection and frequency response of spiral waves in reaction-diffusion systems under a chiral electric field

2014 ◽  
Vol 140 (18) ◽  
pp. 184901 ◽  
Author(s):  
Bing-Wei Li ◽  
Mei-Chun Cai ◽  
Hong Zhang ◽  
Alexander V. Panfilov ◽  
Hans Dierckx
Keyword(s):  
Electric Field ◽  
Frequency Response ◽  
Reaction Diffusion ◽  
Spiral Waves ◽  
Reaction Diffusion Systems ◽  
Chiral Selection ◽  
Diffusion Systems

RELAXATION BEHAVIOR AND PATTERN FORMATION IN REACTION-DIFFUSION SYSTEMS

1994 ◽  
Vol 04 (05) ◽  
pp. 1183-1191 ◽  
Author(s):  
PATRICK HANUSSE ◽  
VICENTE PEREZ-MUÑUZURI ◽  
MONCHO GOMEZ-GESTEIRA
Keyword(s):  
Rotational Symmetry ◽  
Relaxation Oscillation ◽  
Reaction Diffusion ◽  
Relaxation Behavior ◽  
Reaction Diffusion Systems ◽  
Phase Dynamics ◽  
Diffusion Systems ◽  
Universal Definition ◽  
Definition Of ◽  
Mathematical Definitions

The notions of relaxation oscillation and hard excitation have been extensively used and early recognized as important qualitative features of many nonlinear systems. Nevertheless, there seems to exist so far no clear mathematical definitions of these notions. We consider the description of relaxation behavior in oscillating or excitable systems resulting from symmetry breaking of the rotational symmetry of the velocity vector field of the Hopf normal form. From symmetry considerations we detect the first terms responsible for the relaxation character of the phase dynamics in such systems and show that they provide a good general, if not universal, definition of the relaxation properties. We analyze their consequence in the modeling of spatiotemporal patterns such as spiral waves.

scholarly journals Stochastic simulation of the spatio-temporal dynamics of reaction-diffusion systems: the case for the bicoid gradient

2010 ◽  
Vol 7 (1) ◽  
Author(s):  
Paola Lecca ◽  
Adaoha E. C. Ihekwaba ◽  
Lorenzo Dematté ◽  
Corrado Priami
Keyword(s):  
Diffusion Coefficient ◽  
Stochastic Simulation ◽  
Diffusion Coefficients ◽  
Temporal Dynamics ◽  
Concentration Field ◽  
Reaction Diffusion ◽  
Local Concentration ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems ◽  
Spatio Temporal

SummaryReaction-diffusion systems are mathematical models that describe how the concentrations of substances distributed in space change under the influence of local chemical reactions, and diffusion which causes the substances to spread out in space. The classical representation of a reaction-diffusion system is given by semi-linear parabolic partial differential equations, whose solution predicts how diffusion causes the concentration field to change with time. This change is proportional to the diffusion coefficient. If the solute moves in a homogeneous system in thermal equilibrium, the diffusion coefficients are constants that do not depend on the local concentration of solvent and solute. However, in nonhomogeneous and structured media the assumption of constant intracellular diffusion coefficient is not necessarily valid, and, consequently, the diffusion coefficient is a function of the local concentration of solvent and solutes. In this paper we propose a stochastic model of reaction-diffusion systems, in which the diffusion coefficients are function of the local concentration, viscosity and frictional forces. We then describe the software tool Redi (REaction-DIffusion simulator) which we have developed in order to implement this model into a Gillespie-like stochastic simulation algorithm. Finally, we show the ability of our model implemented in the Redi tool to reproduce the observed gradient of the bicoid protein in the Drosophila Melanogaster embryo. With Redi, we were able to simulate with an accuracy of 1% the experimental spatio-temporal dynamics of the bicoid protein, as recorded in time-lapse experiments obtained by direct measurements of transgenic bicoidenhanced green fluorescent protein.

scholarly journals ON BIFURCATIONS OF SPIRAL WAVES IN THE PLANE

1999 ◽  
Vol 09 (08) ◽  
pp. 1501-1516 ◽  
Author(s):  
E. V. NIKOLAEV ◽  
V. N. BIKTASHEV ◽  
A. V. HOLDEN
Keyword(s):  
Period Doubling ◽  
Spiral Wave ◽  
Reaction Diffusion ◽  
Pitchfork Bifurcation ◽  
Spiral Waves ◽  
Model Systems ◽  
Reaction Diffusion Systems ◽  
Double Period ◽  
Diffusion Systems ◽  
Armed Spiral

We describe the simplest bifurcations of spiral waves in reaction–diffusion systems in the plane and present the list of model systems. One-parameter bifurcations of one-armed spiral waves are fold and Hopf bifurcations. Multiarmed spiral waves may additionally undergo a period-doubling pitchfork bifurcation, when two congruent spiral wave solutions, having the "double" period, branch from the original spiral wave at the bifurcation point.

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