Pattern formation in excitable media

Meron, Ehud

doi:10.1016/0370-1573(92)90098-k

Cited by 409 publications

(320 citation statements)

References 167 publications

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“…Models of this type have been studied in various physical and chemical contexts [1,2,[29][30][31][32]. The specific form chosen here is…”

Section: Reaction-diffusion Modelmentioning

confidence: 99%

“…(23) the dynamics of a propagating front involve two degrees of freedom: a translational degree of freedom, x f , determining the front position, and an order parameter, v f , determining the direction of propagation. The latter has not been appreciated enough since most works to date [1,2] have focused on conditions far from the front bifurcation. In that case the two stationary states, v f = ± 4 √ 2(η c − η)/3q −1/2 , representing fronts propagating in opposite directions, are highly stable.…”

Section: Solution Of the Free Boundary Problemmentioning

confidence: 99%

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Order parameter equations for front transitions: Planar and circular fronts

et al. 1997

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Near a parity breaking front bifurcation, small perturbations may reverse the propagation direction of fronts. Often this results in nonsteady asymptotic motion such as breathing and domain breakup. Exploiting the time scale differences of an activator-inhibitor model and the proximity to the front bifurcation, we derive equations of motion for planar and circular fronts. The equations involve a translational degree of freedom and an order parameter describing transitions between left and right propagating fronts. Perturbations, such as a space dependent advective field or uniform curvature (axisymmetric spots), couple these two degrees of freedom. In both cases this leads to a transition from stationary to oscillating fronts as the parity breaking bifurcation is approached. For axisymmetric spots, two additional dynamic behaviors are found: rebound and collapse.

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“…Models of this type have been studied in various physical and chemical contexts [1,2,[29][30][31][32]. The specific form chosen here is…”

Section: Reaction-diffusion Modelmentioning

confidence: 99%

Section: Solution Of the Free Boundary Problemmentioning

confidence: 99%

Order parameter equations for front transitions: Planar and circular fronts

et al. 1997

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“…medium (Meron 1992). At certain conditions, such waves can form vortices which take on the shape of spiral waves rotating around a small core area.…”

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confidence: 99%

Negative Tension of Scroll Wave Filaments and Turbulence in Three-Dimensional Excitable Media and Application in Cardiac Dynamics

2012

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Scroll waves are vortices that occur in three-dimensional excitable media. Scroll waves have been observed in a variety of systems including cardiac tissue, where they are associated with cardiac arrhythmias. The disorganization of scroll waves into chaotic behavior is thought to be the mechanism of ventricular fibrillation, whose lethality is widely known. One possible mechanism for this process of scroll wave instability is negative filament tension. It was discovered in 1987 in a simple two variables model of an excitable medium. Since that time, negative filament tension of scroll waves and the resulting complex, often turbulent dynamics was studied in many generic models of excitable media as well as in physiologically realistic models of cardiac tissue. In this article, we review the work in this area from the first simulations in FitzHugh-Nagumo type models to recent studies involving detailed ionic models of cardiac tissue. We discuss the relation of negative filament tension and tissue excitability and the effects of discreteness in the tissue on the filament tension. Finally, we consider the application of the negative tension mechanism to computational cardiology, where it may be regarded as a fundamental mechanism that explains differences in the onset of arrhythmias in thin and thick tissue.

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“…The characteristic feature of excitable systems [Meron, 1992] is that perturbations exceeding a certain threshold can induce a temporary large deviation from the equilibrium state. The plankton bloom produced artificially by iron fertilization experiments has been shown to be consistent with excitable system behaviour [Pitchford and Brindley, 1999].…”

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Ocean fertilization experiments may initiate a large scale phytoplankton bloom

Neufeld

Haynes

Garçon

et al. 2002

Geophysical Research Letters

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[1] Oceanic plankton plays an important role in the marine food chain and through its significant contribution to the global carbon cycle can also influence the climate. Plankton bloom is a sudden rapid increase of the population. It occurs naturally in the North Atlantic as a result of seasonal changes. Ocean fertilization experiments have shown that supply of iron, an important trace element, can trigger a phytoplankton bloom in oceanic regions with low natural phytoplankton density. Here we use a simple mathematical model of the combined effects of stirring by ocean eddies and plankton evolution to consider the impact of a transient local perturbation, e.g., in the form of iron enrichment as in recent 'ocean fertilization' experiments. The model not only explains aspects of the bloom observed in such experiments but predicts the unexpected outcome of a large scale bloom that in its extent could be comparable to the spring bloom in the North Atlantic. [Truscott and Brindley, 1994a] suggested that plankton blooms might be explained by the excitability of the dynamical system describing biological interactions in the plankton ecosystem. The characteristic feature of excitable systems [Meron, 1992] is that perturbations exceeding a certain threshold can induce a temporary large deviation from the equilibrium state. The plankton bloom produced artificially by iron fertilization experiments has been shown to be consistent with excitable system behaviour [Pitchford and Brindley, 1999]. However, previous work has considered only the spatially homogeneous situation, and need not be relevant to the iron fertilization experiments, in which localised perturbations produce a localised bloom that is subsequently strongly affected by stirring by mesoscale ocean eddies.[3] The role of stirring has been recognised and investigated in the context of the SOIREE experiment [Boyd et al., 2000;Abraham et al., 2000] and the spatial characteristics of the bloom filament was found to be similar to that seen in tracer release experiments [Sundermeyer and Price, 1998], in having a roughly constant width and exponentially increasing length. However, the biological evolution in this analysis was modelled by a simple linear growth rate, which cannot be adequate for predicting the long term behavior of the bloom when nonlinearities, e.g. interactions with other species, become important.[4] Here we consider a, perhaps minimal, model that unifies the two approaches described above by taking into account spatial distribution, stirring and non-linear interactions between biological components. The model equations arewhere P, Z represent the phytoplankton and zooplankton concentration (in nitrogen currency) and F is the deviation of the iron concentration from the original homogeneous background. All three components are advected by a two-dimensional flow, v(r, t), obtained from a so-called seeded eddy model [Abraham, 1998], and diffusive terms represent the effect of small-scale mixing (k = 4 m 2 s À1 ). The biological terms on the right-han...

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Pattern formation in excitable media

Cited by 409 publications

References 167 publications

Order parameter equations for front transitions: Planar and circular fronts

Order parameter equations for front transitions: Planar and circular fronts

Negative Tension of Scroll Wave Filaments and Turbulence in Three-Dimensional Excitable Media and Application in Cardiac Dynamics

Ocean fertilization experiments may initiate a large scale phytoplankton bloom

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