Model Of Excitable Media Research Articles

An efficient high-order compact approach for spiral wave dynamics by the FHN model

We simulate spiral waves through the famous Fitzhugh–Nagumo (FHN) model of excitable media by employing zero flux and periodic boundary conditions. This model has been solved numerically by modifying a recently developed implicit high-order compact (HOC) finite difference scheme. We also propose a novel approach for implementing the HOC scheme on boundaries of the domain for the periodic boundary conditions. Henceforth, we investigate the effects of the zero flux and periodic boundary conditions on the patterns in excitable media and the resulting dynamics. The introduction of periodic boundary conditions is seen to trigger the breakup of the spiral waves as opposed to the ones found by imposing no-flux boundary conditions. Our computed grid independent solutions are found to be extremely close to well-known numerical results. In the process, we also probe the unconditional stability of the HOC discretization for the FHN model.

Meander pattern of spiral wave and the spatial distribution of its cycle length.

One of the most interesting dynamics of rotating spiral waves in an excitable medium is meandering. The tip of a meandering spiral wave moves along a complex trajectory, which often takes the shape of an epitrochoid or hypotrochoid with inward or outward petals. The cycle lengths (CLs) of a meandering spiral wave are not constant; rather, they depend on the meandering dynamics. In this paper, we show that the CLs take two mean values, outside T^{out} and inside T^{in} the meandering trajectory with a ratio of T^{in}/T^{out}=(n+1)/n for the inward and (n-1)/n for the outward petals, where n is the number of petals in the tip trajectory. We illustrate this using four models of excitable media and prove this result. These formulas are shown to be suitable for a meandering spiral wave in an anatomical model of the heart. We also show that the effective periods of overdrive pacing of meandering spiral waves depend on the electrode location relative to the tip trajectory. Overall, our approach can be used to study the meandering pattern from the CL data; it should work for any type of dynamics that produces complex tip trajectories of the spiral wave, for example, for a drift due to heterogeneity.

Динамика активности в виртуальных сетях: сравнение модели распространения эпидемии и модели возбудимой среды

Динамика активности в виртуальных сетях: сравнение модели распространения эпидемии и модели возбудимой среды

Numerical solutions of reaction-diffusion equations: Application to neural and cardiac models

We describe the implementation of the explicit Euler, Crank-Nicolson, and implicit alternating direction methods for solving partial differential equations and apply these methods to obtain numerical solutions of three excitable-media models used to study neurons and cardiomyocyte dynamics. We discuss the implementation, accuracy, speed, and stability of these numerical methods.

Excitable waves and direction-sensing in Dictyostelium discoideum: steps towards a chemotaxis model

In recent years, there have been significant advances in our understanding of the mechanisms underlying chemically directed motility by eukaryotic cells such as Dictyostelium. In particular, the local excitation and global inhibition (LEGI) model has proven capable of providing a framework for quantitatively explaining many experiments that present Dictyostelium cells with tailored chemical stimuli and monitor their subsequent polarization. In their natural setting, cells generate their own directional signals via the detection and secretion of cyclic adenosine monophosphate (cAMP). Here, we couple the LEGI approach to an excitable medium model of the cAMP wave-field that is propagated by the cells and investigate the possibility for this class of models to enable accurate chemotaxis to the cAMP waveforms expected in vivo. Our results indicate that the ultra-sensitive version of the model does an excellent job in providing natural wave rectification, thereby providing a compelling solution to the ‘back-of-the-wave paradox’ during cellular aggregation.

Virtual network as excitable medium

We simulated the spread of an activity in a virtual group using the model of excitable medium. We assumed that the structure of the virtual group corresponds to a scale- free network. In our simulation, the network consists of 100 nodes, the average degree of the nodes is 1.98. We considered the propagation of excitation both in a homogeneous and an inhomogeneous excitable medium. The simulation showed that the initial conditions have a little effect on the behaviour of the model. In inhomogeneous medium, fraction of the excited nodes increases, when permanent excited elements (‘active’ centres) appear in the network. The fraction of the excited nodes increases, when we increase the number of the permanent excited elements. Locations of the active centres do not affect at the level of excitation. External source of activator increases the fraction of the excited nodes in the scale-free network with distribution of parameters.

Influence of the medium's dimensionality on defect-mediated turbulence.

Spatiotemporal chaos in oscillatory and excitable media is often characterized by the presence of phase singularities called defects. Understanding such defect-mediated turbulence and its dependence on the dimensionality of a given system is an important challenge in nonlinear dynamics. This is especially true in the context of ventricular fibrillation in the heart, where the importance of the thickness of the ventricular wall is contentious. Here, we study defect-mediated turbulence arising in two different regimes in a conceptual model of excitable media and investigate how the statistical character of the turbulence changes if the thickness of the medium is changed from (quasi-) two- dimensional to three dimensional. We find that the thickness of the medium does not have a significant influence in, far from onset, fully developed turbulence while there is a clear transition if the system is close to a spiral instability. We provide clear evidence that the observed transition and change in the mechanism that drives the turbulent behavior is purely a consequence of the dimensionality of the medium. Using filament tracking, we further show that the statistical properties in the three-dimensional medium are different from those in turbulent regimes arising from filament instabilities like the negative line tension instability. Simulations also show that the presence of this unique three-dimensional turbulent dynamics is not model specific.

Regenerative Hair Waves in Aging Mice and Extra-Follicular Modulators Follistatin, Dkk1, and Sfrp4

Hair cycling is modulated by factors both intrinsic and extrinsic to hair follicles. Cycling defects lead to conditions such as aging associated alopecia. Recently we demonstrated that mouse skin exhibits regenerative hair waves, reflecting a coordinated regenerative behavior in follicle populations. Here, we use this model to explore the regenerative behavior of aging mouse skin. Old mice (>18 months) tracked over several months show that with progressing age hair waves slow down, wave propagation becomes restricted, and hair cycle domains fragment into smaller domains. Transplanting aged donor mouse skin to a young host can restore donor cycling within a 3mm range of the interface, suggesting that changes are due to extra-cellular factors. Therefore, hair stem cells in aged skin can be re-activated. Molecular studies show that extra-follicular modulators Bmp2, Dkk1, and Sfrp4 increase in early anagen. Further, we identify follistatin as an extra-follicular modulator which is highly expressed in late telogen and early anagen. Indeed follistatin induces hair wave propagation and its level decreases in aging mice. We present an excitable medium model to simulate the cycling behavior in aging mice and illustrate how the inter-organ macro-environment can regulate the aging process by integrating both “activator” and “inhibitor” signals.

Influences of periodic mechanical deformation on pinned spiral waves.

In a generic model of excitable media, we study the behavior of spiral waves interacting with obstacles and their dynamics under the influences of simple periodic mechanical deformation (PMD). Depending on the characteristics of the obstacles, i.e., size and excitability, the rotation of a pinned spiral wave shows different scenarios, e.g., embedding into or anchoring on an obstacle. Three different drift phenomena induced by PMD are observed: scattering on small partial-excitable obstacles, meander-induced unpinning on big partial-excitable obstacles, and drifting around small unexcitable obstacles. Their underlying mechanisms are discussed. The dependence of the threshold amplitude of PMD on the characteristics of the obstacles to successfully remove pinned spiral waves on big partial-excitable obstacles is studied.

Spiral–pacemaker interactions in a mathematical model of excitable medium

Interactions of a spiral wave with a pacemaker is studied in a mathematical model of two dimensional excitable medium. Faster pacemakers emitting target waves can abolish spirals by driving them to the border of the medium. Our study shows that a slower pacemaker can modify spiral wave behavior by changing the motion of the spiral core. We analyze the dynamics of the spiral wave near the spiral core and away from the core as a function of size and period of the pacemaker. The pacemaker can cause the spiral wave to drift towards it, and either speed up or slow down the reentrant activity. Furthermore, the drift induced by the pacemaker can result in irregular or quasiperiodic dynamics even at sites away from the pacemaker. These results highlight the influence of pacemakers on complex spiral wave dynamics.

A SIMPLE SCALAR COUPLED MAP LATTICE MODEL FOR EXCITABLE MEDIA

A simple scalar coupled map lattice (sCML) model for excitable media is derived in this paper. The new model, which has a simple structure, is shown to be closely related to the observed phenomena in excitable media. Properties of the sCML model are also investigated. Illustrative examples show that this kind of model is capable of reproducing the behavior of excitable media and of generating complex spatiotemporal patterns.

Hysteresis and bistability in periodically paced cardiac tissue

Hysteresis in periodically paced cardiac tissue is an important issue due to its relevance to cardiac arrhythmias. In the present paper, the mechanism of hysteresis formation and the related properties are interpreted by numerically investigating the phase I Luo-Rudy model. A formula calculating the width of hysteresis is proposed and well confirmed by numerical simulations. We also find that hysteresis in cardiac tissue shows several characteristics due to couplings among cardiac cells which are absent in a single cell. The influences of the physiological parameters are studied in detail. The model dependence of hysteresis is elucidated by considering a number of well-known models of excitable media. Moreover, the influence of bistability on controlling arrhythmias is revealed.

Traveling waves in a piecewise-linear reaction-diffusion model of excitable medium

One-dimensional autowaves (traveling waves) in excitable medium described by a piecewise-linear reaction-diffusion system have been investigated. Two main types of waves have been considered: a single impulse and a periodic sequence of impulses (wave trains). In a two-component system, oscillations appear due to the presence of the second component in the reaction-diffusion system. In a one-component system, oscillations appear owing to external periodic excitation (forcing). Using semianalytical solutions for the wave profile, the shape and velocity of autowaves have been found. It is shown that the dispersion relation for oscillating sequences of impulses has an anomalous character.

Critical scale of propagation influences dynamics of waves in a model of excitable medium

BackgroundDuration and speed of propagation of the pulse are essential factors for stability of excitation waves. We explore the propagation of excitation waves resulting from periodic stimulation of an excitable cable to determine the minimal stable pulse duration in a rate-dependent modification of a Chernyak-Starobin-Cohen reaction-diffusion model.ResultsVarious pacing rate dependent features of wave propagation were studied computationally and analytically. We demonstrated that the complexity of responses to stimulation and evolution of these responses from stable propagation to propagation block and alternans was determined by the proximity between the minimal level of the recovery variable and the critical excitation threshold for a stable solitary pulse.ConclusionThese results suggest that critical propagation of excitation waves determines conditions for transition to unstable rhythms in a way similar to unstable cardiac rhythms. Established conditions were suitably accurate regardless of rate dependent features and the magnitude of the slopes of restitution curves.

Expanding scroll rings and negative tension turbulence in a model of excitable media

Scroll waves in excitable media, described by the Barkley model, are studied. In the parameter region of weak excitability, negative tension of wave filaments is found. It leads to expansion of scroll rings and instability of wave filaments. A circular filament tends to stretch, bend, loop, and produce an expanding tangle that fills up the volume. The filament does not undergo fragmentation before it touches the boundaries. Statistical properties of such Winfree turbulence of scroll waves are numerically investigated.

Information Transmission Concept Based Model of Wave Propagation in Discrete Excitable Media

A new information transmission concept based model of excitable media with continuous outputs of the model’s cells and variable excitation time is proposed. Continuous character of the outputs instigates infinitesimal inaccuracies in calculations. It generates countless number of the cells’ excitation variants that occur in front of the wave even in the homogenous and isotropic grid. New approach allows obtain many wave propagation patterns observed in real world experiments and known simulation studies. The model suggests a new spiral breakup mechanism based on tensions and gradually deepening clefts that appear in front of the wave caused by uneven propagation speed of curved and planar segments of the wave. The analysis hints that the wave breakdown and daughter wavelet bursting behavior possibly is inherent peculiarity of excitable media with weak ties between the cells, short refractory period and granular structure. The model suggested is located between cellular automaton with discrete outputs and differential equation based models and gives a new tool to simulate wave propagation patterns in applied disciplines. It is also a new line of attack aimed to understand wave bursting, propagation and annihilation processes in isotropic homogenous media.

The causes and processes of the mid-summer population crash of the potato aphids Macrosiphum euphorbiae and Myzus persicae (Hemiptera: Aphididae).

Populations of many phloem-feeding aphid species in temperate regions increase exponentially in early summer and then 'disappear', usually over a time-scale of a few days, in July. To understand these dynamics, empirical investigation of the causes and modelling of the processes underlying population change are required. Numbers of the aphids Myzus persicae(Sulzer) and Macrosiphum euphorbiae (Thomas), monitored over three years in commercial potato fields in the UK, increased to a maximum of 2-2.5 per leaflet on 16 July in 1999 and 2001, and then declined to < 0.25 per leaflet by 26 July. In 2000, aphid numbers remained very low (< 0.25 per leaflet) throughout the season. The onset of the crash in aphid numbers (16-19 July in 1999 and 2001) was consistently associated with changes in the phloem amino acid composition of potato leaflets. Natural enemies, including syrphids, parasitoids, coccinellids, chrysopids and entomopathogenic fungi, increased in abundance throughout the sampling period. The incidence of winged emigrant aphids prior to the crash was low (< 10%). Experimental manipulation during 2001 demonstrated that, during the crash period, the fecundity of aphids (caged on leaves to exclude natural enemies) was depressed by 25-45% relative to earlier in the season, and that presence of natural enemies reduced aphid numbers by up to 68%. Using these data, an excitable medium model was constructed, which provided a robust description of aphid population dynamics in terms of plant development-induced changes in aphid fecundity and temporal change in natural enemy pressure.

Emergent global oscillations in heterogeneous excitable media: the example of pancreatic beta cells.

Using the standard van der Pol-FitzHugh-Nagumo excitable medium model, I demonstrate a generic mechanism, diversity, that provokes the emergence of global oscillations from individually quiescent elements in heterogeneous excitable media. This mechanism may be operating in the mammalian pancreas, where excitable beta cells, quiescent when isolated, are found to oscillate when coupled, despite the absence of a pacemaker region.

Memory in an Excitable Medium: A Mechanism for Spiral Wave Breakup in the Low-Excitability Limit

The electrophysiology of cardiac tissue is altered during acute myocardial ischemia, making the tissue less excitable but, nonetheless, more susceptible to tachyarrythmias which frequently degenerate to fibrillation within several seconds. The transition from tachycardia to fibrillation is associated with the breakup of spiral waves into multiple offspring, and has been linked to steep restitution $(\mathrm{slope}&gt;1)$ of action potential duration (APD). However, restitution curves become so flat during ischemia that this mechanism does not apply. We found that when the response of APD to recent activations is included in a model of excitable media, spiral breakup can occur even when the slope in APD restitutions is $&lt;1$.

Where do dispersion curves end? A basic question in theory of excitable media

We use our recent exactly solvable model of excitable media with nondiffusive control kinetics to study periodic wave trains in an excitable medium. We explicitly find restitution and dispersion curves for such a medium that are protocol independent. We also introduce an approximate stability criterion for periodic waves, which is based on the solitary pulse stability. Using our analytical periodic solutions as the initial conditions in our numerical experiments we demonstrate that this criterion indeed determines the minimal wavelength and propagation velocity below which both a solitary pulse and a periodic wave train quickly die.