Stabilization of unstable rigid rotation of spiral waves in excitable media

Control of spiral waves in optogenetically modified cardiac tissue by periodic optical stimulation

Resonant drift of nonlinear spiral waves occurs in various types of excitable media under periodic stimulation. Recently a novel methodology of optogenetics has emerged, which allows to affect excitability of cardiac cells and tissues by optical stimuli. In this paper we study if resonant drift of spiral waves in the heart can be induced by a homogeneous weak periodic optical stimulation of cardiac tissue. We use a two-variable and a detailed model of cardiac tissue and add description of depolarizing and hyperpolarizing optogenetic ionic currents. We show that weak periodic optical stimulation at a frequency equal to the natural rotation frequency of the spiral wave induces resonant drift for both depolarizing and hyperpolarizing optogenetic currents. We quantify these effects and study how the speed of the drift and its direction depend on the initial conditions. We also derive analytical formulas based on the response function theory which correctly predict the drift velocity and its trajectory. We conclude that optogenetic methodology can be used for control of spiral waves in cardiac tissue and discuss its possible applications.

Resonant drift of synchronized spiral waves in excitable media

Many methods have been employed to investigate the drift behavior of spiral waves in an effort to understand and control their dynamics. Here, we propose a joint method to control the dynamics of spiral waves by applying both a rotating external field and periodic forcing. First, spirals in different regimes (including rigidly rotating, meandering, and drifting spirals) are synchronized by suitable rotating external fields. Then, under weak periodic forcing at frequencies equal to those of the external fields, the synchronized spirals undergo a directional and simple drift. Dependence of the drift velocity of the synchronized spiral on the initial rotational phase of the rotating external field and on the initial phase of the periodic forcing is studied in detail. We hope that our theoretical results can be observed in experiments, such as in the Belousov-Zhabotinsky reaction.

Signal-Coupled Subthreshold Hopf-Type Systems Show a Sharpened Collective Response

Astounding properties of biological sensors can often be mapped onto a dynamical system below the occurrence of a bifurcation. For mammalian hearing, a Hopf bifurcation description has been shown to work across a whole range of scales, from individual hair bundles to whole regions of the cochlea. We reveal here the origin of this scale invariance, from a general level, applicable to all dynamics in the vicinity of a Hopf bifurcation (embracing, e.g., neuronal Hodgkin-Huxley equations). When subject to natural "signal coupling," ensembles of Hopf systems below the bifurcation threshold exhibit a collective Hopf bifurcation. This collective Hopf bifurcation occurs at parameter values substantially below where the average of the individual systems would bifurcate, with a frequency profile that is sharpened if compared to the individual systems.

Regulation of Spatiotemporal Patterns by Biological Variability: General Principles and Applications to Dictyostelium discoideum

Spatiotemporal patterns often emerge from local interactions in a self-organizing fashion. In biology, the resulting patterns are also subject to the influence of the systematic differences between the system’s constituents (biological variability). This regulation of spatiotemporal patterns by biological variability is the topic of our review. We discuss several examples of correlations between cell properties and the self-organized spatiotemporal patterns, together with their relevance for biology. Our guiding, illustrative example will be spiral waves of cAMP in a colony of Dictyostelium discoideum cells. Analogous processes take place in diverse situations (such as cardiac tissue, where spiral waves occur in potentially fatal ventricular fibrillation) so a deeper understanding of this additional layer of self-organized pattern formation would be beneficial to a wide range of applications. One of the most striking differences between pattern-forming systems in physics or chemistry and those in biology is the potential importance of variability. In the former, system components are essentially identical with random fluctuations determining the details of the self-organization process and the resulting patterns. In biology, due to variability, the properties of potentially very few cells can have a driving influence on the resulting asymptotic collective state of the colony. Variability is one means of implementing a few-element control on the collective mode. Regulatory architectures, parameters of signaling cascades, and properties of structure formation processes can be "reverse-engineered" from observed spatiotemporal patterns, as different types of regulation and forms of interactions between the constituents can lead to markedly different correlations. The power of this biology-inspired view of pattern formation lies in building a bridge between two scales: the patterns as a collective state of a very large number of cells on the one hand, and the internal parameters of the single cells on the other.

Pattern formation is abundant in nature—from the rich ornaments of sea shells and the diversity of animal coat patterns to the myriad of fractal structures in biology and pattern-forming colonies of bacteria. Particularly fascinating are patterns changing with time, spatiotemporal patterns, like propagating waves and aggregation streams. Bacteria form large branched and nested aggregation-like patterns to immobilize themselves against water flow. The individual amoeba in Dictyostelium discoideum colonies initiates a transition to a collective multicellular state via a quorum-sensing form of communication—a cAMP signal propagating through the community in the form of spiral waves—and the subsequent chemotactic response of the cells leads to branch-like aggregation streams. The theoretical principle underlying most of these spatial and spatiotemporal patterns is self-organization, in which local interactions lead to patterns as large-scale collective”modes” of the system. Over more than half a century, these patterns have been classified and analyzed according to a”physics paradigm,” investigating such questions as how parameters regulate the transitions among patterns, which (types of) interactions lead to such large-scale patterns, and whether there are "critical parameter values" marking the sharp, spontaneous onset of patterns. A fundamental discovery has been that simple local interaction rules can lead to complex large-scale patterns. The specific pattern "layouts" (i.e., their spatial arrangement and their geometric constraints) have received less attention. However, there is a major difference between patterns in physics and chemistry on the one hand and patterns in biology on the other: in biology, patterns often have an important functional role for the biological system and can be considered to be under evolutionary selection. From this perspective, we can expect that individual biological elements exert some control on the emerging patterns. Here we explore spiral wave patterns as a prominent example to illustrate the regulation of spatiotemporal patterns by biological variability. We propose a new approach to studying spatiotemporal data in biology: analyzing the correlation between the spatial distribution of the constituents’ properties and the features of the spatiotemporal pattern. This general concept is illustrated by simulated patterns and experimental data of a model organism of biological pattern formation, the slime mold Dictyostelium discoideum. We introduce patterns starting from Turing (stripe and spot) patterns, together with target waves and spiral waves. The biological relevance of these patterns is illustrated by snapshots from real and theoretical biological systems. The principles of spiral wave formation are first explored in a stylized cellular automaton model and then reproduced in a model of Dictyostelium signaling. The shaping of spatiotemporal patterns by biological variability (i.e., by a spatial distribution of cell-to-cell differences) is demonstrated, focusing on two Dictyostelium models. Building up on this foundation, we then discuss in more detail how the nonlinearities in biological models translate the distribution of cell properties into pattern events, leaving characteristic geometric signatures.

Feedback control of flow alignment in sheared liquid crystals

Based on a continuum theory, we investigate the manipulation of the nonequilibrium behavior of a sheared liquid crystal via closed-loop feedback control. Our goal is to stabilize a specific dynamical state, that is, the stationary "flow alignment," under conditions where the uncontrolled system displays oscillatory director dynamics with in-plane symmetry. To this end we employ time-delayed feedback control (TDFC), where the equation of motion for the ith component q(i)(t) of the order parameter tensor is supplemented by a control term involving the difference q(i)(t)-q(i)(t-τ). In this diagonal scheme, τ is the delay time. We demonstrate that the TDFC method successfully stabilizes flow alignment for suitable values of the control strength K and τ; these values are determined by solving an exact eigenvalue equation. Moreover, our results show that only small values of K are needed when the system is sheared from an isotropic equilibrium state, contrary to the case where the equilibrium state is nematic.

Transient behavior in systems with time-delayed feedback

We investigate the transient times for the onset of control of steady states by time-delayed feedback. The optimization of control by minimizing the transient time before control becomes effective is discussed analytically and numerically, and the competing influences of local and global features are elaborated. We derive an algebraic scaling of the transient time and confirm our findings by numerical simulations in dependence on feedback gain and time delay.

Control of spatiotemporal patterns in the Gray-Scott model

This paper studies the effects of a time-delayed feedback control on the appearance and development of spatiotemporal patterns in a reaction-diffusion system. Different types of control schemes are investigated, including single-species, diagonal, and mixed control. This approach helps to unveil different dynamical regimes, which arise from chaotic state or from traveling waves. In the case of spatiotemporal chaos, the control can either stabilize uniform steady states or lead to bistability between a trivial steady state and a propagating traveling wave. Furthermore, when the basic state is a stable traveling pulse, the control is able to advance stationary Turing patterns or yield the above-mentioned bistability regime. In each case, the stability boundary is found in the parameter space of the control strength and the time delay, and numerical simulations suggest that diagonal control fails to control the spatiotemporal chaos.

Controlling the onset of traveling pulses in excitable media by nonlocal spatial coupling and time-delayed feedback

The onset of pulse propagation is studied in a reaction-diffusion (RD) model with control by augmented transmission capability that is provided either along nonlocal spatial coupling or by time-delayed feedback. We show that traveling pulses occur primarily as solutions to the RD equations, while augmented transmission changes excitability. For certain ranges of the parameter settings, defined as weak susceptibility and moderate control, respectively, the hybrid model can be mapped to the original RD model. This results in an effective change in RD parameters controlled by augmented transmission. Outside moderate control parameter settings new patterns are obtained, for example, stepwise propagation due to delay-induced oscillations. Augmented transmission constitutes a signaling system complementary to the classical RD mechanism of pattern formation. Our hybrid model combines the two major signaling systems in the brain, namely, volume transmission and synaptic transmission. Our results provide insights into the spread and control of pathological pulses in the brain.

Perturbation analysis of the Kuramoto phase-diffusion equation subject to quenched frequency disorder

The Kuramoto phase-diffusion equation is a nonlinear partial differential equation which describes the spatiotemporal evolution of a phase variable in an oscillatory reaction-diffusion system. Synchronization manifests itself in a stationary phase gradient where all phases throughout a system evolve with the same velocity, the synchronization frequency. The formation of concentric waves can be explained by local impurities of higher frequency which can entrain their surroundings. Concentric waves in synchronization also occur in heterogeneous systems, where the local frequencies are distributed randomly. We present a perturbation analysis of the synchronization frequency where the perturbation is given by the heterogeneity of natural frequencies in the system. The nonlinearity in the form of dispersion leads to an overall acceleration of the oscillation for which the expected value can be calculated from the second-order perturbation terms. We apply the theory to simple topologies, like a line or sphere, and deduce the dependence of the synchronization frequency on the size and the dimension of the oscillatory medium. We show that our theory can be extended to include rotating waves in a medium with periodic boundary conditions. By changing a system parameter, the synchronized state may become quasidegenerate. We demonstrate how perturbation theory fails at such a critical point.

Failure of feedback as a putative common mechanism of spreading depolarizations in migraine and stroke

The stability of cortical function depends critically on proper regulation. Under conditions of migraine and stroke a breakdown of transmembrane chemical gradients can spread through cortical tissue. A concomitant component of this emergent spatio-temporal pattern is a depolarization of cells detected as slow voltage variations. The propagation velocity of approximately 3 mm/min indicates a contribution of diffusion. We propose a mechanism for spreading depolarizations (SD) that rests upon a nonlocal or noninstantaneous feedback in a reaction-diffusion system. Depending upon the characteristic space and time scales of the feedback, the propagation of cortical SD can be suppressed by shifting the bifurcation line, which separates the parameter regime of pulse propagation from the regime where a local disturbance dies out. The optimization of this feedback is elaborated for different control schemes and ranges of control parameters.

Control and synchronization of spatiotemporal chaos

Chaos control methods for the Ginzburg-Landau equation are presented using homogeneously, inhomogeneously, and locally applied multiple delayed feedback signals. In particular, it is shown that a small number of control cells is sufficient for stabilizing plane waves or for trapping spiral waves, and that successful control is closely connected to synchronization of the dynamics in regions close to the control cells.

Controlling spatiotemporal chaos using multiple delays

A control method for manipulating spatiotemporal chaos is presented using lumped local feedback with several different delay times. As illustrated with the two-dimensional Ginzburg-Landau and the Fitzhugh-Nagumo equation this method can, for example, be used to convert chaotic spiral waves into guided plane waves and for trapping spiral waves.