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Prediction implies the estimation of future states of dynamical systems on the basis of time series. Unavoidable uncertainty in making predictions stems from errors and fluctuations associated with making measurements, and also from the complexity of the dynamics themselves. To be predicted, the time series have to contain some kind of repeatability, which can be exploited in the course of forecasting. In particular, even irregular time series are often characterized by the repeatability that implies fuzzy recurrences of the states of the system under study. Recently, the recurrence quantification analysis was used in order to assess numerically the horizon of predictability of chaotic fluctuations of the phytoplankton abundance in the Naroch Lakes system consisting of three reservoirs, Lake Naroch, Lake Myastro and Lake Batorino (Medvinsky et al., 2015. Chaos far away from the edge of chaos: A recurrence quantification analysis of plankton time series. Ecol. Complex., 23, 61–67). Here, we present the results of the analysis of the dynamics of bacterioplankton populations, which inhabit the Naroch Lakes. We demonstrate that the dynamics are chaotic. The horizons of predictability of the bacterioplankton dynamics are shown to be equal to 4.8 months for Small Stretch of Lake Naroch, 4.6 months for Large Stretch of Lake Naroch, 4.7 months for Lake Myastro, and 3.4 months for Lake Batorino. Chaoticity of fluctuations in population abundance can be either an immanent feature of the dynamics or be related to environmental influences. In order to evaluate the action of changes in the environment on plankton dynamics, we assessed numerically the extent to which chaotic fluctuations of bacterioplankton and phytoplankton abundances in the Naroch Lakes were synchronized with temperature oscillations. With the use of the analysis of phase relations between bacterioplankton and temperature time series we show that the chaotic bacterioplankton oscillations are synchronized with water temperature oscillations, while chaotic fluctuations of the phytoplankton abundance are not synchronized with the temperature oscillations in Lake Naroch and Lake Myastro in contrast to the phytoplankton fluctuations in Lake Batorino, the smallest of the Naroch Lakes, where phytoplankton fluctuations are phase-locked by the temperature oscillations. We conclude that temperature is the factor that has significant impact on predictability of the bacterioplankton fluctuations, while dynamics and predictability of phytoplankton dynamics can apparently be controlled not only by the temperature but also by trophic interactions and nutrient supply.

It was found via the mathematical simulation of the effect of fibroblasts on the electrical activity of the sinoatrial node cells that the interaction with fibroblasts through gap junctions increases the oscillation frequency of pacemakers. Under the effect of fibroblasts, the oscillation amplitude of a true pacemaker decreases until the end of the spontaneous activity. The oscillation amplitude of a latent pacemaker decreases considerably less.

It was previously found that the exploratory activity of adult Wistar rats with their vibrissae cut in the period from 9 to 20 postnatal days was characterized by lower intragroup variability in comparison with control rats [3]. The present study has shown that the earlier limitation of species-specific afferentation (whisker trimming on postnatal days 2-9) does not induce such changes. We conclude that high plasticity of the brain during the early postnatal period provides better adaptation to the deficit of sensory information.

Indirect evidence suggests the increased production of reactive oxygen species (ROS) in migraine pathophysiology. In the current study we measured lipid peroxidation product in the rat cortex, trigeminal ganglia and meninges after the induction of cortical spreading depression (CSD), a phenomenon known to be associated with migraine aura, and tested nociceptive firing triggered by ROS in trigeminal nerves ex vivo. Application of KCl to dura mater in anesthetized rats induced several waves of CSD recorded by an extracellular electrode in the cortex. Following CSD, samples of cortex (affected regions were identified with blood oxygen level-dependent (BOLD) functional magnetic resonance imaging (fMRI)), meninges from left and right hemispheres and trigeminal ganglia were taken for biochemical analysis. We found that CSD increased the level of the lipid peroxidation product malondialdehyde (MDA) in the ipsilateral cerebral cortex and meninges, but also in both ipsi- and contralateral trigeminal ganglia. In order to test the pro-nociceptive action of ROS, we applied the mild oxidant hydrogen peroxide to isolated rat hemiskull preparations including preserved trigeminal innervations. Application of hydrogen peroxide to meninges transiently enhanced electrical spiking activity of trigeminal nerves showing a pro-nociceptive action of ROS. In the presence of hydrogen peroxide trigeminal nerves still responded to capsaicin by burst of spiking activity indicating integrity of neuronal structures. The action of hydrogen peroxide was mediated by TRPA1 receptors as it was abolished by the specific TRPA1 antagonist TCS-5861528. Using dorsal root ganglion sensory neurons as test system we found that hydrogen peroxide promoted the release of the migraine mediator calcitonin gene-related peptide (CGRP), which we previously identified as a trigger of delayed sensitization of trigeminal neurons. Our data suggest that, after CSD, oxidative stress spreads downstream within the trigeminal nociceptive system and could be involved in the coupling of CSD with the activation of trigeminovascular system in migraine pathology. © 2013 IBRO.

Temporal lobe epilepsy (TLE) is one of the most common forms of epilepsy, characterized by hippocampal sclerosis and memory deficits. It is well-documented that intrinsic neuronal oscillations and provided by them communications between brain structures are of importance for cognition. Epilepsy disturbs these brain rhythms and presumably therefore affects memory. Here we review studies on cellular and systemic levels devoted to the TLE-induced disturbance of theta oscillations in the septohippocampal system. Special attention is paid to the role of damage of septal and hippocampal GABAergic cells in theta activity abnormalities. We also compare differences between native (in vivo) theta oscillations with those obtained in in vitro preparations of hippocampus and medial septum and find that in vitro they resemble epileptiform activity in some respects.

Changes in the interactions of the hippocampus and medial septal area in the theta range (4–8 Hz) during the generation of convulsive activity in electrical kindling were studied in conscious guinea pigs. Overall electrical activity (EEG)was analyzed by wavelet transformation. Decreases in the coherence of theta oscillations recorded in both structures were seen during generation of convulsive discharges. Phase analysis showed that the phase-leading structure at the initial stage of epileptogenesis was the medial septal area; however, after formation of the pathological focus, the medial septal area lagged the hippocampus in phase. These data may contribute to our understanding of the mechanisms of development of temporal epilepsy.

Changes in rhythmic activity can serve as early biomarkers of pathological alterations, but it remains unclear how different types of rhythmic activity are altered during neurodegenerative processes. Glutamatergic neurotoxicity, evoked by kainic acid (KA), causes hyperexcitation and acute seizures that result in delayed brain damage. We employed wide frequency range (0.1-300Hz) local field potential recordings in guinea pigs to study the oscillatory activity of the hippocampus, entorhinal cortex, medial septum, and amygdala in healthy animals for three months after KA introduction. To clarify whether the activation of endocannabinoid (eCB) system can influence toxic KA action, AM404, an eCB reuptake inhibitor, and URB597, an inhibitor of fatty acid amide hydrolase, were applied. The cannabinoid CB1 receptor antagonist AM251 was also tested. Coadministration of AM404 or URB597 with KA reduced acute behavioral seizures, but electrographic seizures were still registered. During the three months following KA injection, various trends in the oscillatory activities were observed, including an increase in activity power at all frequency bands in the hippocampus and a progressive long-term decrease in the medial septum. In the KA- and KA/AM251-treated animals, disturbances of the oscillatory activities were accompanied by cell loss in the dorsal hippocampus and mossy fiber sprouting in the dentate gyrus. Injections of AM404 or URB597 softened alterations in electrical activity of the brain and prevented hippocampal neuron loss and synaptic reorganization. Our results demonstrate the protective potential of the eCB system during excitotoxic influences.

Sleep is essential for nervous system functioning and sleep disorders are associated with several neurodegenerative diseases. However, the macroscale connectivity changes in brain networking during different sleep states are poorly understood. One of the hindering factors is the difficulty to combine functional connectivity investigation methods with spontaneously sleeping animals, which prevents the use of numerous preclinical animal models. Recent studies, however, have implicated that urethane anesthesia can uniquely induce different sleep-like brain states, resembling rapid eye movement (REM) and non-REM (NREM) sleep, in rodents. Therefore, the aim of this study was to assess changes in global connectivity and topology between sleep-like states in urethane anesthetized rats, using blood oxygenation level dependent (BOLD) functional magnetic resonance imaging. We detected significant changes in corticocortical (increased in NREM-like state) and corticothalamic connectivity (increased in REM-like state). Additionally, in graph analysis the modularity, the measure of functional integration in the brain, was higher in NREM-like state than in REM-like state, indicating a decrease in arousal level, as in normal sleep. The fMRI findings were supported by the supplementary electrophysiological measurements. Taken together, our results show that macroscale functional connectivity changes between sleep states can be detected robustly with resting-state fMRI in urethane anesthetized rats. Our findings pave the way for studies in animal models of neurodegenerative diseases where sleep abnormalities are often one of the first markers for the disorder development.

We have studied the dynamics of reentry inside the sinoatrial node (SAN). We have found that reentry is unstable at high intercellular conductance. Rotating reentry induces a slow migrating crescent-shaped functional block near the SAN boundary. Abnormal conduction from atrial tissue into the SAN occurs after decay of the reentry. Acetylcholine increases the lifespan of reentry in the SAN.

The initiation and propagation of electrical pulses in the sinoatrial node under normal conditions and after application of acetylcholine have been simulated. It has been found that normally a single or a few leading centers are formed in the tissue. When acetylcholine is applied, a temporary functional block of conduction may appear; the leading center migrates under these conditions.

The duration of the preautomatic pause as a function of sinoatrial node, the type of pacemaker cells, acetylcholine concentration, the duration of high-frequency stimulation, and the conductivity of gap junctions has been studied. It was found that the preautomatic pause in peripheral pacemakers occurs at a higher concentration of acetylcholine as compared with central pacemakers. The dependence of the duration of the preautomatic pause on the gap junction conductivity is a nonlinear one.

While the sinoatrial node (SAN) is structurally heterogeneous, most computer simulations of electrical activity take into account SAN pacemaker cells only. Our aim was to investigate how fibroblasts affect the SAN activity. We simulated the rabbit sinoatrial node accounting for differences between central and peripheral pacemaker cells, and for fibroblast-myocyte electrical coupling. We have observed that only if fibroblast-myocyte coupling is taken into account, (1) action potential is initiated in the central part of the SAN (within 1.2 mm of the center of simulated tissue); otherwise, leading centers are located on the periphery; (2) few (1 to 6) leading centers initiate action potential in the SAN; otherwise, we observed more than 8 leading centers; (3) acetylcholine superfusion results in a shift of leading centers toward the SAN periphery; and (4) sinus pauses up to 1.9 second follow acetylcholine superfusion. We observed negligible effect of fibroblast-myocyte coupling on the period of SAN activation. We conclude that fibroblast-myocyte coupling may explain action potential initiation and propagation from the center of the SAN observed in experimental studies, while atrial load on the peripheral SAN fails to explain this fact.

We simulated the electric current distribution in the brain during transcranial direct current stimulation (tDCS) using an anatomically accurate human head model. We estimated an effect of common electrode montages on spatial distribution of the electric field during tDCS procedure and analyzed a sensitivity of the technique to variations of electrode size and orientation. We concluded that the used electrode montages are stable with respect to minor changes in electrode size and position, while an assumption of homogeneity and isotropy of the head model results in crucial changes of the electric field distribution. We determined the electrode montages suited to deliver strong effect on hippocampus and cerebellum.

The effect of fluctuations of the transmembrane potential on the generation of the action potential is studied by simulating the rabbit sinoatrial node (SAN) pacemaker. It is shown that the effect of fluctuations is enhanced with an increase in the concentration of acetylcholine and becomes most pronounced at the border of spontaneous activity loss and after it. When applying and washing off acetylcholine, the hysteretic effect is observed.

With the aid of detailed computer simulations, we have estimated distributions of membrane potential and ionic currents in the core region of a sinoatrial node reentry. We observe reduced amplitudes of the measured quantities in the core; the core sizes for potential and currents did not always coincide. Simulations revealed that acetylcholine, when applied in the vicinity of unstable reentry, attracted the reentry to become the core and to stabilize its rotation. Anatomically detailed simulations of sinoatrial node and surrounding atrial tissue revealed that reentry always rotated around small strips of connective tissue. Acetylcholine superfusion over superior part of the sinoatrial node resulted in a drift of reentry in the cranial direction. Under the latter conditions, reentry may coexist with the pacemaker in the caudal part of the sinoatrial node. Copyright © 2016 John WileySons, Ltd.

Mathematical modeling of the electric activity of pairs of true (central) and latent (peripheral) sinoatrial pacemaker cells coupled through gap junctions revealed that attenuation of coupling conductance increases the beat phase shift; below a critical conductance there is no synchronization. The phase difference also depends on the type of interacting cells, being maximal for a “central”-“peripheral” cell pair.

The effect of periodic stimulation of the vagus nerve on the activity of the central cell of the sinoatrial node has been simulated. The regions of synchronization and desynchronization have been revealed, and the phase shift at different stimulation frequencies has been estimated. The positive chronotropic effect has been shown to occur at some frequencies of stimulation.

We have studied the phase dynamics in an oscillatory chemical system (Belousov−Zhabotinsky reaction). The phase distribution and isophase lines in the 2D phase space were reconstructed. The dynamic...

We studied the dynamics of a chemical vortex whose core was disturbed by visible light. The observed sharp change of the shape of the vortex and the unfolding increase in the vortex wavelength proved to be tightly related to a transformation of trigger waves to phase waves. Analytical estimations and direct measurements show that the shape of the vortex is well-fitted by a logarithmic spiral. The results of experiments and of computer simulations are in a good agreement.

We modified the FitzHugh-Nagumo model of an excitable medium so that it describes adequately the dymanics of pulse propagation in the canine myocardium. The modified model is simple enough to be used for intensive 3-dimensional (3D) computations of the whole heart. It simulates the pulse shape and the restitution property of the canine myocardium with good precision.

With the aid of detailed computer simulations, we have estimated distributions of membrane potential and ionic currents in the core region of a sinoatrial node reentry. We observe reduced amplitudes of the measured quantities in the core; the core sizes for potential and currents did not always coincide. Simulations revealed that acetylcholine, when applied in the vicinity of unstable reentry, attracted the reentry to become the core and to stabilize its rotation. Anatomically detailed simulations of sinoatrial node and surrounding atrial tissue revealed that reentry always rotated around small strips of connective tissue. Acetylcholine superfusion over superior part of the sinoatrial node resulted in a drift of reentry in the cranial direction. Under the latter conditions, reentry may coexist with the pacemaker in the caudal part of the sinoatrial node. Copyright © 2016 John Wiley & Sons, Ltd.

The dynamics of phase disturbances in the 1D uniform medium described by FitzHugh-Nagumo equations was studied. It was found that the system synchronizes its oscillations if the initial disturbances are small. Large disturbances resulted in the appearance of phase breaks. Dependent on the system parameters, phase breaks were found to be either stable or unstable; stable breaks led to target pattern-like sources, unstable to extended chaos.

A set of dimensionless parameters describing pulse propagation in the heart, BZ reaction and FHN model is estimated for comparison of the systems. The ranges of the parameters measured allow the BZ reaction to be considered as a rough experimental model of excitable heart tissue. Vulnerability in the BZ reaction is used as an analogue for vulnerability in cardiac tissue.

We have simulated 3D propagation of the action potential in the sinoatrial node to study the dynamics of the heart rhythm initiation. We have found that the leading center inside the sinoatrial node is formed by a group of cells, appears spontaneously under normal conditions, and migrates as acetylcholine is applied. The leading center drifts toward the center of the sinoatrial node, if we consider the effect of the surrounding atrial tissue. Abnormal electrical activity, rotating waves of action potential are observed in the sinoatrial node.

The phase jump dynamics in a photosensitive Belousov-Zhabotinsky reaction has been studied. It was found that an initially inhomogeneous phase distribution evolved to a homogeneous one, so that the phase at phase-lag points rises to the value at phase-lead points. A steep phase distribution was shown to become smoother in the course of time, inducing a dephasing wave which restores the system to a uniform phase. Applications of photosensitive chemical reactions to image processing (image storage, inversing, contouring) are discussed

The control of spiral wave parameters by electric current was investigated in the Belousov-Zhabotinsky (BZ) reaction. It was found that the wavelength and the period of spiral waves increase by a factor of up to 3 with electric current (both dc and ac). Using this procedure spiral waves with a period higher than the period of medium bulk oscillations were observed. It was also found that hysteresis phenomena occur in the system considered.

The model of the ferroin-catalyzed Belousov-Zhabotinsky reaction is modified for the temperature range 10-30°C, which is more appropriate for experiments.

Autowaves spreading in a thin layer of the Belousov-Zhabotinsky reaction with ferroin catalyst immobilized in silica gel have been studied. It is found that there is a critical thickness of this layer such that autowaves could not propagate in thinner layers. The results of computer simulations in the two-variable Rovinsky model qualitatively fit the experimental data for the temperature range used.

Most mammals experience cardiac arrest during hypothermia. In contrast, hibernators remain in sinus rhythm even at body temperatures of 0 degrees C.The purpose of this study was to quantify electrical activity and connexin expression in the heart of hibernating Siberian ground squirrel Citellus undulatus.Optical imaging and microelectrode recordings were conducted in Langendorff-perfused hearts and isolated papillary muscles of summer active (SA, n = 19), winter hibernating (WH, n = 21), interbout arousal (IBA, n = 12), and winter active (WA, n = 3) ground squirrels and rabbits (n = 14) at temperatures from +37 degrees C to +3 degrees C.All studied SA and WH hearts maintained spontaneous sinus rhythm, safe propagation through the entire conduction system, and normal pattern of ventricular excitation at all temperatures. However, three of the seven IBA and all rabbit hearts lost excitability at 10 degrees C +/- 1 degrees C and 12 degrees C +/- 1 degrees C, respectively. In WH, SA, and IBA ground squirrels, temperature reduction from 37 degrees C to 3 degrees C resulted in a 10-fold slowing of ventricular conduction velocity and increased excitation threshold. At any temperature, WH ventricles had faster conduction velocity and lower excitation threshold compared with SA and IBA. Immunolabeling demonstrated that connexin43 (Cx43) was significantly up-regulated in WH and WA compared with SA myocardium: Cx43 area density was 12.4 +/- 1.3, 15.0 +/- 3.0 and 8.6 +/- 1.1 microm(2)/1,000 microm(2), respectively. Moreover, Cx45 was expressed in the WH but not in the SA or WA ventricles.Hibernator Citellus undulatus has evolved to maintain safe conduction at extreme hypothermia via up-regulation of Cx43 and Cx45 in order to protect the heart against arrhythmia associated with hypothermia.

We have created a new non-linear computer model to simulate electrical activity of the intestine. Unlike the widely used model by Sarna et al., who used a modified set of Van der Pol oscillators to simulate activity the of gastrointestinal (GI) tract, our model assumes a more realistic approach which involves (i) a description of electrical coupling between cells, and (ii) a description of the two interacting layers of longitudinal muscle (LM) and interstitial cells of Cajal (ICC). The dynamics in each layer are described by a pair of differential equations with variables representing the transmembrane potential and slow currents; the two equations describe the dynamics of the nonlinear membrane in the same spirit as the FitzHugh-Nagumo equations.

We use a 2-D FitzHugh-Nagumo (FHN) bidomain model to simulate activation patterns in cardiac tissue. We analyze the dynamics using vector calculus quantities such as curl and divergence of spatial gradients of the local phase /spl phi/.

Compared to steadily propagating waves (SPW), damped waves (DW), another solution to the nonlinear wave equation, are seldom studied. In cardiac tissue after electrical stimulation in an SPW wake, we observe DW with diminished amplitude and velocity that either gradually decrease as the DW dies, or exhibit a sharp amplitude increase after a delay to become an SPW. The cardiac DW-SPW transition is a key link in understanding defibrillation and stimulation close to the refractory period, and is ideal for a general study of DW dynamics.

Phase Singularity Dynamics.Introduction: Quantitative analysis of complex self-excitatory wave patterns, such as cardiac fibrillation and other high-order reentry, requires the development of new tools for identifying and tracking the most important features of the activation, such as phase singularities. Methods and Results: Image processing operations can be used to detect the phase singularity at the tip of a spiral wave. The phase space behavior of a spatiotemporal sequence of data may be reconstructed using time-series analysis. The phase singularities then are localized efficiently by computing the topologic charge density as the curl of the spatial phase gradient. We analyzed the singularity interaction dynamics of both experimentally observed and numerically simulated instances of quatrefoil reentry and found that the singularity behavior in the experimental preparations can be classified into three categories on the basis of how their separation changes with time. Conclusion: Topologic charge densities can be calculated easily and efficiently to reveal phase singularity behavior. However, the differences between theoretical and experimental observations of singularity separation distances indicate the need for more sophisticated numerical models.

We study the role of the diffusive interaction in plankton dynamics in a patchy environment. We use a minimal reaction-diffusion model of the nutrient---plankton---fish food chain to simulate the diffusive interaction between fish-populated and fish-free habitats. We show that such interaction can give rise to spatiotemporal plankton patterns. The plankton dynamics depend on the fish predation rate and can exhibit both regular and chaotic behavior. We show that limit cycle and chaotic attractor coexist in the system. The entire basin of attraction of the limit cycles is found to be riddled with ``holes'' leading to the competitive chaotic attractors. The chaotic dynamics is typical of a wide range of the fish predation rates.

We have simulated electrical activity of the intestine in a computer model that describes the coupled layers of longitudinal muscle (LM) and interstitial cells of Cajal (ICC). The model suggests that pacemaker activity is due to the ICC layer, while the pulse propagation involves the LM layer that is in the excitatory state. The model describes well the experimentally observed phenomena: frequency change along the intestine, synchronization along short distances and desynchronization for long distances, and the decrease of propagation distance and propagation time along the intestine. We have observed the occurrence of phase interruptions or breaks, which are responsible for the limited values of propagation distance and time.

SEVERAL different types of three-dimensional nonlinear wave structure are expected to occur in chemically excitable media1. In general these are vortex-like structures (Fig. 1, for example) in which the waves adopt complex spatial configurations (curved, knotted, twisted or closed into rings). But until recently only simple vortices and vortex rings have been observed experimentally, for example in the Belousov–Zhabotinsky (B–Z) reaction2–4 and in cardiac tissue5. Here we report a new type of structure, twisted vortices, observed in a B–Z reaction immobilized in agarose gel (Fig. 2). In cross-section, a twisted vortex resembles a rotating spiral with a phase of rotation that changes along the filament. The filament extends through the entire medium, terminating at its free surfaces. The wavelength of a twisted vortex is shorter than that of a simple one and decreases with increasing twist, in accord with theory6,7. The period shows little dependence on twist, contrary to the results of computer experiments6. Twisted vortices decay into simple ones; their twist decays exponentially with a time constant that increases with filament length. These dynamics are consistent with the theory of diffusional untwisting8,9 rather than that of shock waves10.

We simulate wave propagation in the whole heart containing a local inhomogeneity whose properties mimic some properties of cardiac tissue during acute phase of infarction. The dynamics of cardiac tissue is described by a FitzHugh-Nagumo (FHN) model. We show that two or several short-period stimulations of the heart lead to the development of a three-dimensional vortex ring, which is a temporal source of high frequency waves. The vortex ring is located near the boundary of the infarction and induces wave patterns which appear as several focal wave sources on the epicard and endocard. We have traced the filament of the vortex and show its dynamics. Continuous stimulation of the heart at high frequency resulted in the Wenckebach effect.

We study vulnerability in the Belousov-Zhabotinsky reaction-using experimental and numerical methods. We show that the width of the vulnerable window (VW) decreases with increasing velocity of wave,propagation. A detailed study of the structure of the VW: revealed a new effect: the development of multiple responses to a single premature stimulus; a single premature stimulation in a homogeneous one-dimensional medium generates two or three propagating waves. We locate this effect in the parametric space and discuss its mechanism.

The evolution of scroll rings with filaments of arbitrary shape has been studied analytically for reaction-diffusion systems with all the state variables having the same diffusion coefficients. Belonging to this class of systems are, in particular, models of the Belousov-Zhabotinskii (BZ) chemical reaction. It has been found that the speed of decrease of an area bounded by the planar scroll filament is given by S = −2 πD ( D is the diffusion coefficient) and is independent of the kind of model used to describe an active medium, time and filament shape; i.e., the integral φν d s  −2 πD , where d s is the differential of the filament length and ν is the drift velocity, always remains valid. Using this integral invariant a number of problems on the drift of closed and unclosed three-dimensional vortices have been solved. The theoretical predictions have been verified in experiments with the BZ reaction.

It is shown that the optimal path in a two-dimensional vector field is deduced by the use of chemical wave in the Belousov-Zhabotinsky reaction (BZ reaction). We also reproduced our experimental result in a numerical simulation based on a two-variable reaction-diffusion equation. The present result provides a simple model for the future application of excitable media to parallel computing, i.e., an excitable medium serves as a self-organized parallel processor.

ROTATING spiral waves have been observed in various excitable media, including heart muscle1, retinae2, cultures of the slime mould Dyctiostelium discoideum3,4 and chemical oscillators such as the Belousov-Zhabotinsky (BZ) reaction5–7. Under certain conditions the spiral wave does not exhibit simple periodic rotation, but quasiperiodic8 (or 'compound'9) rotation, in which the spiral's origin (the tip) meanders10. Recent calculations11 have shown that highly meandering tip motion can impose superstructures on spiral waves. Here we reproduce these patterns experimentally, using the BZ reaction as the excitable medium. We induce high tip meander by applying pulses of electrical current locally at the tip12. Image processing of the patterns reveals a spiral wave of larger wavelength superimposed on the original wave, an effect that can be described in terms of a Doppler shift in the original spiral.

The time delay for transmission of chemical waves via a geometrically asymmetric gap is studied based on the Rovinsky-Zhabotinsky model of the Belousov-Zhabotinsky reaction. It decreases with acidity and increases with the gap width. The time delay is assumed to be the sum of two components: the diffusion process within the gap and the reaction process at the receiving boundary. The latter accounts for the presence of the critical gap width above which no transmission occurs, and consequently, for switching of unidirectional transmission. An equation for the time delay is proposed as a function of the gap width.

Experiments were carried out to study the effect of local disturbances on the vortex dynamics in the light-sensitive Belousov-Zhabotinsky (BZ) reaction. It was found that irradiation with visible light of the core region of a rotating spiral wave (vortex) in an oscillatory medium results in a bifurcation which dramatically changes the shape of the vortex. The bifurcation is seen as an unfolding increase in the vortex wavelength although only a negligible change in the vortex core size is observed. Similar dynamics is expected to occur in oscillatory media of different nature.