Delayed feedback control of collective synchrony: an approach to suppression of pathological brain rhythms

Dynamics of globally delay-coupled neurons displaying subthreshold oscillations

We study an ensemble of neurons that are coupled through their time-delayed collective mean field. The individual neuron is modelled using a Hodgkin-Huxley-type conductance model with parameters chosen such that the uncoupled neuron displays autonomous subthreshold oscillations of the membrane potential. We find that the ensemble generates a rich variety of oscillatory activities that are mainly controlled by two time scales: the natural period of oscillation at the single neuron level and the delay time of the global coupling. When the neuronal oscillations are synchronized, they can be either in-phase or out-of-phase. The phase-shifted activity is interpreted as the result of a phase-flip bifurcation, also occurring in a set of globally delay-coupled limit cycle oscillators. At the bifurcation point, there is a transition from in-phase to out-of-phase (or vice versa) synchronized oscillations, which is accompanied by an abrupt change in the common oscillation frequency. This phase-flip bifurcation was recently investigated in two mutually delay-coupled oscillators and can play a role in the mechanisms by which the neurons switch among different firing patterns.

Analysis of the mechanism of action of deep brain stimulation using the concepts of dither injection and the equivalent nonlinearity

Deep brain stimulation (DBS) is a widely applied clinical procedure for the alleviation of pathological neural activity, and is particularly effective in suppressing the symptoms of Parkinson's disease. The mechanisms of action of DBS remain to be fully elucidated. In this paper, we present an application to DBS of the concepts of dither injection and equivalent nonlinearity from the theory of nonlinear feedback control systems. We propose that this model provides a framework for understanding the mechanism by which an injected high-frequency signal can quench undesired oscillations in closed-loop systems of interacting neurons in the brain.

Method to control the coupling function using multilinear feedback

Methods to control the dynamics of coupled oscillators have been developed owing to various medical and technological demands. In this study, we develop a method to control coupled oscillators in which the coupling function expressed in a phase model is regulated by the multilinear feedback. The present method has wide applicability because we do not need to measure an individual output from each oscillator, but only measure the sum of the outputs from all the oscillators. Moreover, it allows us to easily control the coupling function up to higher harmonics. The validity of the present method is confirmed through a simulation.

Finding quasi-optimal network topologies for information transmission in active networks

This work clarifies the relation between network circuit (topology) and behaviour (information transmission and synchronization) in active networks, e.g. neural networks. As an application, we show how one can find network topologies that are able to transmit a large amount of information, possess a large number of communication channels, and are robust under large variations of the network coupling configuration. This theoretical approach is general and does not depend on the particular dynamic of the elements forming the network, since the network topology can be determined by finding a Laplacian matrix (the matrix that describes the connections and the coupling strengths among the elements) whose eigenvalues satisfy some special conditions. To illustrate our ideas and theoretical approaches, we use neural networks of electrically connected chaotic Hindmarsh-Rose neurons.

Feedback suppression of neural synchrony in two interacting populations by vanishing stimulation

We discuss the suppression of collective synchrony in a system of two interacting oscillatory networks. It is assumed that the first network can be affected by the stimulation, whereas the activity of the second one can be monitored. The study is motivated by ongoing attempts to develop efficient techniques for the manipulation of pathological brain rhythms. The suppression mechanism we consider is related to the classical problem of interaction of active and passive systems. The main idea is to connect a specially designed linear oscillator to the active system to be controlled. We demonstrate that the feedback loop, organized in this way, provides an efficient suppression. We support the discussion of our approach by a theoretical treatment of model equations for the collective modes of both networks, as well as by the numerical simulation of two coupled populations of neurons. The main advantage of our approach is that it provides a vanishing-stimulation control, i.e., the stimulation reduces to the noise level as soon as the goal is achieved.

Controlling limit-cycle behaviors of brain activity

The limit cycles of brain activity are studied using a compact continuum model that reproduces the main features of electroencephalographic signals, including bifurcations of fixed points and limit cycles in seizures. Frequencies and amplitudes are predicted analytically and related to physiology. Gaussian stimuli yield two distinct evoked responses in the linearly stable zone, consistent with experiment. Limit cycles can be initiated or suppressed by control signals or stimuli.

Control of spatially patterned synchrony with multisite delayed feedback

We present an analytical study describing a method for the control of spatiotemporal patterns of synchrony in networks of coupled oscillators. Delayed feedback applied through a small number of electrodes effectively induces spatiotemporal dynamics at minimal stimulation intensities. Different arrangements of the delays cause different spatial patterns of synchrony, comparable to central pattern generators (CPGs), i.e., interacting clusters of oscillatory neurons producing patterned output, e.g., for motor control. Multisite delayed feedback stimulation might be used to restore CPG activity in patients with incomplete spinal cord injury or gait ignition disorders.

Control of unstable steady states by extended time-delayed feedback

Time-delayed feedback methods can be used to control unstable periodic orbits as well as unstable steady states. We present an application of extended time delay autosynchronization introduced by Socolar [Phys. Rev. E 50, 3245 (1994)] to an unstable focus. This system represents a generic model of an unstable steady state which can be found, for instance, in Hopf bifurcation. In addition to the original controller design, we investigate effects of control loop latency and a bandpass filter on the domain of control. Furthermore, we consider coupling of the control force to the system via a rotational coupling matrix parametrized by a variable phase. We present an analysis of the domain of control and support our results by numerical calculations.

Zero delay synchronization of chaos in coupled map lattices

We show that two coupled map lattices that are mutually coupled to one another with a delay can display zero delay synchronization if they are driven by a third coupled map lattice. We analytically estimate the parametric regimes that lead to synchronization and show that the presence of mutual delays enhances synchronization to some extent. The zero delay or isochronal synchronization is reasonably robust against mismatches in the internal parameters of the coupled map lattices, and we analytically estimate the synchronization error bounds.

Chaotic phase synchronization in scale-free networks of bursting neurons

There is experimental evidence that the neuronal network in some areas of the brain cortex presents the scale-free property, i.e., the neuron connectivity is distributed according to a power law, such that neurons are more likely to couple with other already well-connected ones. From the information processing point of view, it is relevant that neuron bursting activity be synchronized in some weak sense. A coherent output of coupled neurons in a network can be described through the chaotic phase synchronization of their bursting activity. We investigated this phenomenon using a two-dimensional map to describe neurons with spiking-bursting activity in a scale-free network, in particular the dependence of the chaotic phase synchronization on the coupling properties of the network as well as its synchronization with an externally applied time-periodic signal.

Optimal deep brain stimulation of the subthalamic nucleus--a computational study

Deep brain stimulation (DBS) of the subthalamic nucleus, typically with periodic, high frequency pulse trains, has proven to be an effective treatment for the motor symptoms of Parkinson's disease (PD). Here, we use a biophysically-based model of spiking cells in the basal ganglia (Terman et al., Journal of Neuroscience, 22, 2963-2976, 2002; Rubin and Terman, Journal of Computational Neuroscience, 16, 211-235, 2004) to provide computational evidence that alternative temporal patterns of DBS inputs might be equally effective as the standard high-frequency waveforms, but require lower amplitudes. Within this model, DBS performance is assessed in two ways. First, we determine the extent to which DBS causes Gpi (globus pallidus pars interna) synaptic outputs, which are burstlike and synchronized in the unstimulated Parkinsonian state, to cease their pathological modulation of simulated thalamocortical cells. Second, we evaluate how DBS affects the GPi cells' auto- and cross-correlograms. In both cases, a nonlinear closed-loop learning algorithm identifies effective DBS inputs that are optimized to have minimal strength. The network dynamics that result differ from the regular, entrained firing which some previous studies have associated with conventional high-frequency DBS. This type of optimized solution is also found with heterogeneity in both the intrinsic network dynamics and the strength of DBS inputs received at various cells. Such alternative DBS inputs could potentially be identified, guided by the model-free learning algorithm, in experimental or eventual clinical settings.

Conversion of stability in systems close to a Hopf bifurcation by time-delayed coupling

We propose a control method with time delayed coupling which makes it possible to convert the stability features of systems close to a Hopf bifurcation. We consider two delay-coupled normal forms for Hopf bifurcation and demonstrate the conversion of stability, i.e., an interchange between the sub- and supercritical Hopf bifurcation. The control system provides us with an unified method for stabilizing both the unstable periodic orbit and the unstable steady state and reveals typical effects like amplitude death and phase locking. The main method and the results are applicable to a wide class of systems showing Hopf bifurcations, for example, the Van der Pol oscillator. The analytical theory is supported by numerical simulations of two delay-coupled Van der Pol oscillators, which show good agreement with the theory.

Refuting the odd-number limitation of time-delayed feedback control

We refute an often invoked theorem which claims that a periodic orbit with an odd number of real Floquet multipliers greater than unity can never be stabilized by time-delayed feedback control in the form proposed by Pyragas. Using a generic normal form, we demonstrate that the unstable periodic orbit generated by a subcritical Hopf bifurcation, which has a single real unstable Floquet multiplier, can in fact be stabilized. We derive explicit analytical conditions for the control matrix in terms of the amplitude and the phase of the feedback control gain, and present a numerical example. Our results are of relevance for a wide range of systems in physics, chemistry, technology, and life sciences, where subcritical Hopf bifurcations occur.

Feedback suppression of neural synchrony by vanishing stimulation

We suggest a method for suppression of collective synchrony in an ensemble of all-to-all interacting units. The suppression is achieved by organizing an interaction between the ensemble and a passive oscillator. Technically, this can be easily implemented by a simple feedback scheme. The important feature of our approach is that the feedback signal vanishes as soon as the control is successful. The technique is illustrated by the simulation of a model of an isolated population of neurons. We discuss the possible application of the technique in neuroscience.

All-optical noninvasive control of unstable steady states in a semiconductor laser

All-optical noninvasive control of a multisection semiconductor laser by means of time-delayed feedback from an external Fabry-Perot cavity is realized experimentally. A theoretical analysis, in both a generic model as well as a device-specific simulation, points out the role of the optical phase. Using phase-dependent feedback we demonstrate stabilization of the continuous-wave laser output and noninvasive suppression of intensity pulsations.

Noise-induced cooperative dynamics and its control in coupled neuron models

We investigate feedback control of the cooperative dynamics of two coupled neural oscillators that is induced merely by external noise. The interacting neurons are modeled as FitzHugh-Nagumo systems with parameter values at which no autonomous oscillations occur, and each unit is forced by its own source of random fluctuations. Application of delayed feedback to only one of two subsystems is shown to be able to change coherence and time scales of noise-induced oscillations either in the given subsystem, or globally. It is also able to induce or to suppress stochastic synchronization under certain conditions.

Delayed feedback control of chaos

Time-delayed feedback control is well known as a practical method for stabilizing unstable periodic orbits embedded in chaotic attractors. The method is based on applying feedback perturbation proportional to the deviation of the current state of the system from its state one period in the past, so that the control signal vanishes when the stabilization of the target orbit is attained. A brief review on experimental implementations, applications for theoretical models and most important modifications of the method is presented. Recent advancements in the theory, as well as an idea of using an unstable degree of freedom in a feedback loop to avoid a well-known topological limitation of the method, are described in detail.

Energy aspects of the synchronization of model neurons

We have deduced an energy function for a Hindmarsh-Rose model neuron and we have used it to evaluate the energy consumption of the neuron during its signaling activity. We investigate the balance of energy in the synchronization of two bidirectional linearly coupled neurons at different values of the coupling strength. We show that when two neurons are coupled there is a specific cost associated to the cooperative behavior. We find that the energy consumption of the neurons is incoherent until very near the threshold of identical synchronization, which suggests that cooperative behaviors without complete synchrony could be energetically more advantageous than those with complete synchrony.

Control of neuronal synchrony by nonlinear delayed feedback

We present nonlinear delayed feedback stimulation as a technique for effective desynchronization. This method is intriguingly robust with respect to system and stimulation parameter variations. We demonstrate its broad applicability by applying it to different generic oscillator networks as well as to a population of bursting neurons. Nonlinear delayed feedback specifically counteracts abnormal interactions and, thus, restores the natural frequencies of the individual oscillatory units. Nevertheless, nonlinear delayed feedback enables to strongly detune the macroscopic frequency of the collective oscillation. We propose nonlinear delayed feedback stimulation for the therapy of neurological diseases characterized by abnormal synchrony.

Stimulus-locked responses of two phase oscillators coupled with delayed feedback

For a system of two phase oscillators coupled with delayed self-feedback we study the impact of pulsatile stimulation administered to both oscillators. This system models the dynamics of two coupled phase-locked loops (PLLs) with a finite internal delay within each loop. The delayed self-feedback leads to a rich variety of dynamical regimes, ranging from phase-locked and periodically modulated synchronized states to chaotic phase synchronization and desynchronization. Remarkably, for large coupling strength the two PLLs are completely desynchronized. We study stimulus-locked responses emerging in the different dynamical regimes. Simple phase resets may be followed by a response clustering, which is intimately connected with long poststimulus resynchronization. Intriguingly, a maximal perturbation (i.e., maximal response clustering and maximal resynchronization time) occurs, if the system gets trapped at a stable manifold of an unstable saddle fixed point due to appropriately calibrated stimulus. Also, single stimuli with suitable parameters can shift the system from a stable synchronized state to a stable desynchronized state or vice versa. Our result show that appropriately calibrated single pulse stimuli may cause pronounced transient and/or long-lasting changes of the oscillators' dynamics. Pulse stimulation may, hence, constitute an effective approach for the control of coupled oscillators, which might be relevant to both physical and medical applications.

Bifurcation control of a seizing human cortex

We consider as a mathematical model of human cortical electrical activity a system of fourteen ordinary differential equations. With appropriate parameters, the model produces activity characteristic of a seizure. To prevent such seizures, we incorporate feedback controllers into the model dynamics. We show that three controllers--a linear feedback controller, a differential controller, and a filter controller--can be used to eliminate seizing activity in the model system. We show how bifurcations induced by the linear controller alter those present in the original dynamics.

Delayed feedback control of the Lorenz system: an analytical treatment at a subcritical Hopf bifurcation

We develop an analytical approach for the delayed feedback control of the Lorenz system close to a subcritical Hopf bifurcation. The periodic orbits arising at this bifurcation have no torsion and cannot be stabilized by a conventional delayed feedback control technique. We utilize a modification based on an unstable delayed feedback controller. The analytical approach employs the center manifold theory and the near identity transformation. We derive the characteristic equation for the Floquet exponents of the controlled orbit in an analytical form and obtain simple expressions for the threshold of stability as well as for an optimal value of the control gain. The analytical results are supported by numerical analysis of the original system of nonlinear differential-difference equations.

Nonlinear multivariate analysis of neurophysiological signals

Multivariate time series analysis is extensively used in neurophysiology with the aim of studying the relationship between simultaneously recorded signals. Recently, advances on information theory and nonlinear dynamical systems theory have allowed the study of various types of synchronization from time series. In this work, we first describe the multivariate linear methods most commonly used in neurophysiology and show that they can be extended to assess the existence of nonlinear interdependence between signals. We then review the concepts of entropy and mutual information followed by a detailed description of nonlinear methods based on the concepts of phase synchronization, generalized synchronization and event synchronization. In all cases, we show how to apply these methods to study different kinds of neurophysiological data. Finally, we illustrate the use of multivariate surrogate data test for the assessment of the strength (strong or weak) and the type (linear or nonlinear) of interdependence between neurophysiological signals.

Random delays and the synchronization of chaotic maps

We investigate the dynamics of an array of chaotic logistic maps coupled with random delay times. We report that for adequate coupling strength the array is able to synchronize, in spite of the random delays. Specifically, we find that the synchronized state is a homogeneous steady state, where the chaotic dynamics of the individual maps is suppressed. This synchronization behavior is largely independent of the connection topology and depends mainly on the average number of links per node. We carry out a statistical linear stability analysis that confirms the numerical results and provides a better understanding of the nontrivial roles of random delayed interactions.