Narrow Results

Analytical and numerical results on the ordering role of external random fluctuations in excitable systems are presented. Our study focuses on a simple model for excitable systems. Regular waves are created and sustained out of noise when the system is forced with random perturbations. Explicit results for the generation and dynamics of rings and targets are presented.

This paper describes three models arising from the theory of excitable media, whose primary visual feature are expanding rings of excitation. Rigorous mathematical results and experimental/computational issues are both addressed. We start with the much-studied Greenberg–Hastings model (GHM) in which the rings are very short-lived, but they do have a transient percolation property. By contrast, in the model we call annihilating nested rings (ANR), excitation centers only gradually lose strength, i.e., each time they become inactive (and then stay so forever) with a fixed probability; we show how the long-term global connectivity properties of the set of excited sites undergo a phase transition. Second part of the paper is devoted to digital boiling (DB) in which new rings spontaneously appear at rested sites with a positive probability. We focus on such (related) issues as convergence to equilibrium, equilibrium excitation level and success of the basic coupling.

Ventricular fibrillation (VF), the major reason behind sudden cardiac death, is turbulent cardiac electrical activity in which rapid, irregular disturbances in spatiotemporal electrical activation of heart make it incapable of any concerted pumping action. We give a brief overview of the simple Panfilov model for ventricular fibrillation, with emphasis on studies that have elucidated the nature of spiral turbulence which is the analog of VF here. The control of such turbulence is briefly touched upon. Preliminary results are presented for the effects of conduction inhomogeneity on spiral breakup, and the transition from functional to anatomical reentry as a function of the size and position of the inhomogeneity.

In this paper, we study the effect of external periodic pulses on spiral dynamics. Resonant entrainment bands were observed on the period T-axis, and T is close to rational multiples of the path curvature period of the spiral tip on the bands. It is also shown that spiral waves are drifted and eliminated by applying the driving method with suitable control parameters, and we reveal the mechanism which forces the spiral wave to periodically shift and rotate. In the domain near the spiral tip, the bidirectional wave excitations are periodically generated by external pulses, and each excitation induces a straight drift of the spiral wave tip. Numerical results show that the parameter range of the external pulse period T, used to successfully eliminate spiral waves, is broaden by appropriately increasing the values of the pulse width and the amplitude. The low-amplitude control scheme is operable in many real systems, and its study is beneficial to understand the forced spiral dynamics.

We consider waves in two-dimensional excitable media. The correct form of the eikonal equation (i.e. the formula that predicts propagation speed as a function of curvature) is a question that has surfaced and resurfaced during the last twenty years or so. Good answers have become available in limiting cases and under certain approximations, notably weak curvatures, low frequencies, and sharp wave fronts. The solution is important in some cardiac pathologies, as well as in our basic understanding of excitable-medium mathematics. After a brief review of curvature effects, particularly in heart tissue, we demonstrate how to obtain a drastically corrected formula that is free of restrictions on frequency and sharpness. (In the original limiting cases the result is unchanged.) Our derivation uses a finite-renormalization method. We illustrate the formula in the context of cardiac tissue. For the sake of definiteness we work in terms of FitzHugh–Nagumo-like models.

Nucleation from a boundary is experimentally and numerically studied in a one-dimensional array in two excitable media consisting of Chua's circuits and the Oregonator model, respectively. Forcing from a boundary with a pulse of constant amplitude and infinite duration gives rise to a periodic wave train propagating through the array. As the amplitude of the pulse increases, wave period evolves to chaos through a period doubling cascade.

We investigate a spontaneous formation of stable multiarmed spirals in two-dimensional excitable media, an effect observed in various biological and chemical systems. A previous study based on FitzHugh–Nagumo-type Pushchino model reported a robust effect of stable two- and three-armed spiral formation from nearby vortices, when the spirals rotate around unexcited cores, i.e. when excitability of the medium is low. In this study, we used a powerful parallel computer cluster to perform an extensive parameter search in two other widely used FitzHugh–Nagumo-type models, as well as in the two-component Oregonator model. We observed formation of stable n-armed spirals, with 2 ≤ n ≤ 10, whenever the excitability of the medium was sufficiently low. Thus, we conclude that the formation and persistence of stable multiarmed spirals (MAS) is not an artifact of one particular model, but, rather, it is an amazing higher-level self-organization property of a generic weakly excitable medium. We also establish quantitatively that such multiarmed spirals serve as high-frequency wave sources — a finding that has a direct relevance to cardiac defibrillation research.

Dynamics of spiral waves in perturbed, e.g. slightly inhomogeneous or subject to a small periodic external force, two-dimensional autowave media can be described asymptotically in terms of Aristotelean dynamics, so that the velocities of the spiral wave drift in space and time are proportional to the forces caused by the perturbation. The forces are defined as a convolution of the perturbation with the spirals Response Functions, which are eigenfunctions of the adjoint linearized problem. In this paper we find numerically the Response Functions of a spiral wave solution in the classic excitable FitzHugh–Nagumo model, and show that they are effectively localized in the vicinity of the spiral core.

New methods of identifying the transition rule of a Belousov–Zhabotinskii (BZ) reaction directly from experimental data using cellular automata (CA) models are investigated. The experimental set-up and new techniques for image pre-processing to ensure the identification of representative models are discussed including noise reduction, pixel and color calibration. Two kinds of models, the Greenberg–Hasting model (GHM) and the polynomial CA model are studied in detail. It is shown that the results of identifying a real BZ reacting system are very encouraging and the predicted patterns compare well with the imaged patterns both visually and quantitatively.

We study the dynamics of a reaction–diffusion system composed of two mutually coupled excitable fibers. We focus on the situation in which dynamical properties of the two fibers are not identical because of the parameter difference between the fibers. Using the spatially one-dimensional FitzHugh–Nagumo equations as a model of a single excitable fiber, we show that the system exhibits a rich variety of dynamical behavior, including soliton-like collision between two pulses and recombination of a solitary pulse and synchronized pulses.

The identification problem for excitable media is investigated in this paper. A new scalar coupled map lattice (SCML) model is introduced and the orthogonal least squares algorithm is employed to determinate the structure of the SCML model and to estimate the associated parameters. A simulated pattern and a pattern observed directly from a real Belousov–Zhabotinsky reaction are identified. The identified SCML models are shown to possess almost the same local dynamics as the original systems and are able to provide good long term predictions.

The wavefront profile and the propagation velocity of waves in an experimentally observed Belousov–Zhabotinskii reaction are analyzed and a revised FitzHumgh–Nagumo (FHN) model of these systems is identified. The ratio between the excitation period and the recovery period, for a solitary wave is studied, and included within the model. Averaged traveling velocities at different spatial positions are shown to be consistent under the same experimental conditions. The relationship between the propagation velocity and the curvature of the wavefront are also studied to deduce the diffusion coefficient in the model, which is a function of the curvature of the wavefront and not a constant. The application of the identified model is demonstrated on real experimental data and validated using multistep ahead predictions.

This paper describes the identification of a temperature dependent FitzHugh–Nagumo model directly from experimental observations with controlled inputs. By studying the steady states and the trajectory of the phase of the variables, the stability of the model is analyzed and a rule to generate oscillation waves is proposed. The dependence of the oscillation frequency and propagation speed on the model parameters is then investigated to seek the appropriate control variables, which then become functions of temperature in the identified model. The results show that the proposed approach can provide a good representation of the dynamics of the oscillatory behavior of a Belousov–Zhabotinskii reaction.

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.

The continuous increment in the performance of classical computers has been driven to its limit. New ways are studied to avoid this oncoming bottleneck and many answers can be found. An example is the Belousov–Zhabotinsky (BZ) reaction which includes some fundamental and essential characteristics that attract chemists, biologists, and computer scientists. Interaction of excitation wave-fronts in BZ system, can be interpreted in terms of logical gates and applied in the design of unconventional hardware components. Logic gates and other more complicated components have been already proposed using different topologies and particular characteristics. In this study, the inherent parallelism and simplicity of Cellular Automata (CAs) modeling is combined with an Oregonator model of light-sensitive version of BZ reaction. The resulting parallel and computationally-inexpensive model has the ability to simulate a topology that can be considered as a one-bit full adder digital component towards the design of an Arithmetic Logic Unit (ALU).

Computer simulation is applied to study the role of cellular coupling, dispersion of refractoriness as well as both of them, in the mechanisms underlying cardiac arrhythmias. We first assumed that local ischemia mainly induces cell to cell dispersion in the coupling resistance (case 1), refractory period (case 2) or both (case 3). Numerical experiments, based on the van Capelle and Durrer model, showed that vortices could not be induced in these conditions. In order to be more realistic about coronary circulation we simulated a patchy dispersion of cellular properties, each patch corresponding to the zone irrigated by a small coronary artery. In these conditions, a single activation wave could give rise to abnormal activities. Probabilities of reentry, estimated for the three cases cited above, showed that a severe alteration of the coupling resistance may be an important factor in the genesis of reentry. Moreover, use of isochronal maps revealed that vortices were both stable and sustained with an alteration of coupling alone or along with reductions of action potential duration. Conversely, simulations with reduction of the refractoriness alone induced only transient patterns.

We study the coherence of noise-induced excitations in a modified stochastic Oregonator model for the light-sensitive Belousov-Zhabotinsky (BZ) reaction assuming that the intensity of the applied illumination is a spatio-temporal stochastic field with finite correlation time and correlation length. For a single excitable element, we find coherence resonance (CR) with respect to the correlation time. In the spatially extended medium of diffusively coupled excitable elements, we observe CR for suitable combinations of the correlation time and length of the noise.

During embryonic morphogenesis, a collection of individual neurons turns into a functioning network with unique capabilities. Only recently has this most staggering example of emergent process in the natural world, began to be studied. Here we propose a navigational strategy for neurites growth cones, based on sophisticated chemical signaling. We further propose that the embryonic environment (the neurons and the glia cells) acts as an excitable media in which concentric and spherical chemical waves are formed. Together with the navigation strategy, the chemical waves provide a mechanism for communication, regulation, and control required for the adaptive self-wiring of neurons.