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

Delayed feedback control of self-mobile cavity solitons in a wide-aperture laser with a saturable absorber

We investigate the spatiotemporal dynamics of cavity solitons in a broad area vertical-cavity surface-emitting laser with saturable absorption subjected to time-delayed optical feedback. Using a combination of analytical, numerical, and path continuation methods, we analyze the bifurcation structure of stationary and moving cavity solitons and identify two different types of traveling localized solutions, corresponding to slow and fast motion. We show that the delay impacts both stationary and moving solutions either causing drifting and wiggling dynamics of initially stationary cavity solitons or leading to stabilization of intrinsically moving solutions. Finally, we demonstrate that the fast cavity solitons can be associated with a lateral mode-locking regime in a broad-area laser with a single longitudinal mode.

Bifurcation structure of two coupled FHN neurons with delay

This paper presents an investigation of the dynamics of two coupled non-identical FitzHugh-Nagumo neurons with delayed synaptic connection. We consider coupling strength and time delay as bifurcation parameters, and try to classify all possible dynamics which is fairly rich. The neural system exhibits a unique rest point or three ones for the different values of coupling strength by employing the pitchfork bifurcation of non-trivial rest point. The asymptotic stability and possible Hopf bifurcations of the trivial rest point are studied by analyzing the corresponding characteristic equation. Homoclinic, fold, and pitchfork bifurcations of limit cycles are found. The delay-dependent stability regions are illustrated in the parameter plane, through which the double-Hopf bifurcation points can be obtained from the intersection points of two branches of Hopf bifurcation. The dynamical behavior of the system may exhibit one, two, or three different periodic solutions due to pitchfork cycle and torus bifurcations (Neimark-Sacker bifurcation in the Poincare map of a limit cycle), of which detection was impossible without exact and systematic dynamical study. In addition, Hopf, double-Hopf, and torus bifurcations of the non trivial rest points are found. Bifurcation diagrams are obtained numerically or analytically from the mathematical model and the parameter regions of different behaviors are clarified.

Time-delayed feedback control of breathing localized structures in a three-component reaction-diffusion system

The dynamics of a single breathing localized structure in a three-component reaction-diffusion system subjected to time-delayed feedback is investigated. It is shown that variation of the delay time and the feedback strength can lead either to stabilization of the breathing or to delay-induced periodic or quasi-periodic oscillations of the localized structure. A bifurcation analysis of the system in question is provided and an order parameter equation is derived that describes the dynamics of the localized structure in the vicinity of the Andronov-Hopf bifurcation. With the aid of this equation, the boundaries of the stabilization domains as well as the dependence of the oscillation radius on delay parameters can be explicitly derived, providing a robust mechanism to control the behaviour of the breathing localized structure in a straightforward manner.

Bistable dynamics underlying excitability of ion homeostasis in neuron models

When neurons fire action potentials, dissipation of free energy is usually not directly considered, because the change in free energy is often negligible compared to the immense reservoir stored in neural transmembrane ion gradients and the long-term energy requirements are met through chemical energy, i.e., metabolism. However, these gradients can temporarily nearly vanish in neurological diseases, such as migraine and stroke, and in traumatic brain injury from concussions to severe injuries. We study biophysical neuron models based on the Hodgkin-Huxley (HH) formalism extended to include time-dependent ion concentrations inside and outside the cell and metabolic energy-driven pumps. We reveal the basic mechanism of a state of free energy-starvation (FES) with bifurcation analyses showing that ion dynamics is for a large range of pump rates bistable without contact to an ion bath. This is interpreted as a threshold reduction of a new fundamental mechanism of ionic excitability that causes a long-lasting but transient FES as observed in pathological states. We can in particular conclude that a coupling of extracellular ion concentrations to a large glial-vascular bath can take a role as an inhibitory mechanism crucial in ion homeostasis, while the Na⁺/K⁺ pumps alone are insufficient to recover from FES. Our results provide the missing link between the HH formalism and activator-inhibitor models that have been successfully used for modeling migraine phenotypes, and therefore will allow us to validate the hypothesis that migraine symptoms are explained by disturbed function in ion channel subunits, Na⁺/K⁺ pumps, and other proteins that regulate ion homeostasis.

Dynamics of reaction-diffusion patterns controlled by asymmetric nonlocal coupling as a limiting case of differential advection

A one-component bistable reaction-diffusion system with asymmetric nonlocal coupling is derived as a limiting case of a two-component activator-inhibitor reaction-diffusion model with differential advection. The effects of asymmetric nonlocal couplings in such a bistable reaction-diffusion system are then compared to the previously studied case of a system with symmetric nonlocal coupling. We carry out a linear stability analysis of the spatially homogeneous steady states of the model and numerical simulations of the model to show how the asymmetric nonlocal coupling controls and alters the steady states and the front dynamics in the system. In a second step, a third fast reaction-diffusion equation is included which induces the formation of more complex patterns. A linear stability analysis predicts traveling waves for asymmetric nonlocal coupling, in contrast to a stationary Turing patterns for a system with symmetric nonlocal coupling. These findings are verified by direct numerical integration of the full equations with nonlocal coupling.

Migraine generator network and spreading depression dynamics as neuromodulation targets in episodic migraine

Migraine is a common disabling headache disorder characterized by recurrent episodes sometimes preceded or accompanied by focal neurological symptoms called aura. The relation between two subtypes, migraine without aura (MWoA) and migraine with aura (MWA), is explored with the aim to identify targets for neuromodulation techniques. To this end, a dynamically regulated control system is schematically reduced to a network of the trigeminal nerve, which innervates the cranial circulation, an associated descending modulatory network of brainstem nuclei, and parasympathetic vasomotor efferents. This extends the idea of a migraine generator region in the brainstem to a larger network and is still simple and explicit enough to open up possibilities for mathematical modeling in the future. In this study, it is suggested that the migraine generator network (MGN) is driven and may therefore respond differently to different spatio-temporal noxious input in the migraine subtypes MWA and MWoA. The noxious input is caused by a cortical perturbation of homeostasis, known as spreading depression (SD). The MGN might even trigger SD in the first place by a failure in vasomotor control. As a consequence, migraine is considered as an inherently dynamical disease to which a linear course from upstream to downstream events would not do justice. Minimally invasive and noninvasive neuromodulation techniques are briefly reviewed and their rational is discussed in the context of the proposed mechanism.

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.

Sleep, neuroengineering and dynamics

Modeling of consciousness-related phenomena and neuroengineering are fields that are rapidly growing together. We review recent approaches and developments and point out some promising directions of future research: Understanding the dynamics of consciousness states and associated oscillations, pathological oscillations as well as their treatment by stimulation, neuroprosthetics and brain-computer-interface approaches, and stimulation approaches that probe, influence and strengthen memory consolidation. In all these fields, computational models connect theory, neurophysiology and neuroengineering research and pave a way towards medical applications.

Experiments on autonomous Boolean networks

We realize autonomous Boolean networks by using logic gates in their autonomous mode of operation on a field-programmable gate array. This allows us to implement time-continuous systems with complex dynamical behaviors that can be conveniently interconnected into large-scale networks with flexible topologies that consist of time-delay links and a large number of nodes. We demonstrate how we realize networks with periodic, chaotic, and excitable dynamics and study their properties. Field-programmable gate arrays define a new experimental paradigm that holds great potential to test a large body of theoretical results on the dynamics of complex networks, which has been beyond reach of traditional experimental approaches.

Dynamics of localized structures in reaction-diffusion systems induced by delayed feedback

We are interested in stability properties of a single localized structure in a three-component reaction-diffusion system subjected to the time-delayed feedback. We shall show that variation in the product of the delay time and the feedback strength leads to complex dynamical behavior of the system, including formation of target patterns, spontaneous motion, and spontaneous breathing as well as various complex structures, arising from combination of different oscillatory instabilities. In the case of spontaneous motion, we provide a bifurcation analysis of the delayed system and derive an order parameter equation for the position of the localized structure, explicitly describing its temporal evolution in the vicinity of the bifurcation point. This equation is a subject to a nonlinear delay differential equation, which can be transformed to the normal form of the pitchfork drift bifurcation.

Delayed feedback induces motion of localized spots in reaction-diffusion systems

We study the formation of localized structures, often called localized spots, in reaction-diffusion systems subject to time delayed feedback control. We focus on the regime close to a second-order critical point marking the onset of a hysteresis loop. We show that the space-time dynamics of the FitzHugh-Nagumo model in the vicinity of that critical point could be described by the delayed Swift-Hohenberg equation. We show that the delayed feedback induces a spontaneous motion of localized spots. We characterize this motion by computing analytically the velocity and the threshold above which localized structures start to move in an arbitrary direction. Numerical solutions of the governing equation are in close agreement with those obtained from the delayed Swift-Hohenberg equation.

Instabilities of localized structures in dissipative systems with delayed feedback

We report on a novel behavior of solitary localized structures in a real Swift-Hohenberg equation subjected to a delayed feedback. We shall show that variation in the product of the delay time and the feedback strength leads to nontrivial instabilities resulting in the formation of oscillons, soliton rings, labyrinth patterns, or moving structures. We provide a bifurcation analysis of the delayed system and derive a system of order parameter equations explicitly describing the temporal behavior of the localized structure in the vicinity of the bifurcation point. We demonstrate that a normal form of the bifurcation, responsible for the emergence of moving solitary structures, can be obtained and show that spontaneous motion to the lowest order occurs without change of the shape.

Reduced dynamics for delayed systems with harmonic or stochastic forcing

The analysis of nonlinear delay-differential equations (DDEs) subjected to external forcing is difficult due to the infinite dimensionality of the space in which they evolve. To simplify the analysis of such systems, the present work develops a non-homogeneous center manifold (CM) reduction scheme, which allows the derivation of a time-dependent order parameter equation in finite dimension. This differential equation captures the major dynamical features of the delayed system. The forcing is assumed to be small compared to the amplitude of the autonomous system, in order to cause only small variations of the fixed points and of the autonomous CM. The time-dependent CM is shown to satisfy a non-homogeneous partial differential equation. We first briefly review CM theory for DDEs. Then we show, for the general scalar case, how an ansatz that separates the CM into one for the autonomous problem plus an additional time-dependent order-two correction leads to satisfying results. The paper then details the application to a transcritical bifurcation subjected to single or multiple periodic forcings. The validity limits of the reduction scheme are also highlighted. Finally, we characterize the specific case of additive stochastic driving of the transcritical bifurcation, where additive white noise shifts the mode of the probability density function of the state variable to larger amplitudes.

Synchronization transitions in coupled time-delay electronic circuits with a threshold nonlinearity

Experimental observations of typical kinds of synchronization transitions are reported in unidirectionally coupled time-delay electronic circuits with a threshold nonlinearity and two time delays, namely feedback delay τ(1) and coupling delay τ(2). We have observed transitions from anticipatory to lag via complete synchronization and their inverse counterparts with excitatory and inhibitory couplings, respectively, as a function of the coupling delay τ(2). The anticipating and lag times depend on the difference between the feedback and the coupling delays. A single stability condition for all the different types of synchronization is found to be valid as the stability condition is independent of both the delays. Further, the existence of different kinds of synchronizations observed experimentally is corroborated by numerical simulations and from the changes in the Lyapunov exponents of the coupled time-delay systems.

Inferring local dynamics and connectivity of spatially extended systems with long-range links based on steady-state stabilization

A method is presented for system identification of spatially extended systems with structural inhomogeneities of local dynamics and additional long-range links. The proposed identification procedure is based on steady-state stabilization and is illustrated with an inhomogeneous two-dimensional grid of coupled FitzHugh-Nagumo models.

Diverse self-sustained oscillatory patterns and their mechanisms in excitable small-world networks

Diverse self-sustained oscillatory patterns and their mechanisms in small-world networks (SWNs) of excitable nodes are studied. Spatiotemporal patterns of SWNs are sensitive to long-range connection probability P and coupling intensity D . By varying P in wide range with fixed D , we observe totally six types of asymptotic states: pure spiral waves, pure self-sustained target waves, patterns of mixtured spirals and target waves, pseudospiral turbulence, synchronizing oscillations, and rest state. The parameter conditions for all these states are specified, and the mechanisms of these states are heuristically explained. In particular, the mechanism of emergence and annihilation of synchronizing oscillations is explained by using the shortest path length analysis.

Shear-stress-controlled dynamics of nematic complex fluids

Based on a mesoscopic theory we investigate the nonequilibrium dynamics of a sheared nematic liquid, with the control parameter being the shear stress σ xy (rather than the usual shear rate, γ). To this end we supplement the equations of motion for the orientational order parameters by an equation for γ, which then becomes time dependent. Shearing the system from an isotropic state, the stress-controlled flow properties turn out to be essentially identical to those at fixed γ. Pronounced differences occur when the equilibrium state is nematic. Here, shearing at controlled γ yields several nonequilibrium transitions between different dynamic states, including chaotic regimes. The corresponding stress-controlled system has only one transition from a regular periodic into a stationary (shear-aligned) state. The position of this transition in the σ xy-γ plane turns out to be tunable by the delay time entering our control scheme for σ xy. Moreover, a sudden change in the control method can stabilize the chaotic states appearing at fixed γ.

Delay stabilization of periodic orbits in coupled oscillator systems

We study diffusively coupled oscillators in Hopf normal form. By introducing a non-invasive delay coupling, we are able to stabilize the inherently unstable anti-phase orbits. For the super- and subcritical cases, we state a condition on the oscillator's nonlinearity that is necessary and sufficient to find coupling parameters for successful stabilization. We prove these conditions and review previous results on the stabilization of odd-number orbits by time-delayed feedback. Finally, we illustrate the results with numerical simulations.

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.

Feedback control of subcritical Turing instability with zero mode

A global feedback control of a system that exhibits a subcritical monotonic instability at a nonzero wave number (short-wave or Turing instability) in the presence of a zero mode is investigated using a Ginzburg-Landau equation coupled to an equation for the zero mode. This system is studied analytically and numerically. It is shown that feedback control, based on measuring the maximum of the pattern amplitude over the domain, can stabilize the system and lead to the formation of localized unipulse stationary states or traveling solitary waves. It is found that the unipulse traveling structures result from an instability of the stationary unipulse structures when one of the parameters characterizing the coupling between the periodic pattern and the zero mode exceeds a critical value that is determined by the zero mode damping coefficient.

Time-delayed feedback in neurosystems

The influence of time delay in systems of two coupled excitable neurons is studied in the framework of the FitzHugh-Nagumo model. A time delay can occur in the coupling between neurons or in a self-feedback loop. The stochastic synchronization of instantaneously coupled neurons under the influence of white noise can be deliberately controlled by local time-delayed feedback. By appropriate choice of the delay time, synchronization can be either enhanced or suppressed. In delay-coupled neurons, antiphase oscillations can be induced for sufficiently large delay and coupling strength. The additional application of time-delayed self-feedback leads to complex scenarios of synchronized in-phase or antiphase oscillations, bursting patterns or amplitude death.

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.

Multistability, local pattern formation, and global collective firing in a small-world network of nonleaky integrate-and-fire neurons

We investigate numerically the collective dynamical behavior of pulse-coupled nonleaky integrate-and-fire neurons that are arranged on a two-dimensional small-world network. To ensure ongoing activity, we impose a probability for spontaneous firing for each neuron. We study network dynamics evolving from different sets of initial conditions in dependence on coupling strength and rewiring probability. Besides a homogeneous equilibrium state for low coupling strength, we observe different local patterns including cyclic waves, spiral waves, and turbulentlike patterns, which-depending on network parameters-interfere with the global collective firing of the neurons. We attribute the various network dynamics to distinct regimes in the parameter space. For the same network parameters different network dynamics can be observed depending on the set of initial conditions only. Such a multistable behavior and the interplay between local pattern formation and global collective firing may be attributable to the spatiotemporal dynamics of biological networks.

Control of coherence resonance in semiconductor superlattices

We study the effect of time-delayed feedback control and Gaussian white noise on the spatiotemporal charge dynamics in a semiconductor superlattice. The system is prepared in a regime where the deterministic dynamics is close to a global bifurcation, namely, a saddle-node bifurcation on a limit cycle. In the absence of control, noise can induce electron charge front motion through the entire device, and coherence resonance is observed. We show that with appropriate selection of the time-delayed feedback parameters the effect of coherence resonance can be either enhanced or destroyed, and the coherence of stochastic domain motion at low noise intensity is dramatically increased. Additionally, the purely delay-induced dynamics in the system is investigated, and a homoclinic bifurcation of a limit cycle is found.

Introduction to focus issue: design and control of self-organization in distributed active systems

Spatiotemporal self-organization is found in a wide range of distributed dynamical systems. The coupling of the active elements in these systems may be local or global or within a network, and the interactions may be diffusive or nondiffusive in nature. The articles in this focus issue describe biological and chemical systems designed to exhibit spatiotemporal dynamics and the control of such dynamics through feedback methods.