Synchrony suppression in ensembles of coupled oscillators via adaptive vanishing feedback

Feedback control of collective dynamics in an oscillator population with time-dependent connectivity

We present a numerical study of pulsatile feedback-based control of synchrony level in a highly-interconnected oscillatory network. We focus on a nontrivial case when the system is close to the synchronization transition point and exhibits collective rhythm with strong amplitude modulation. We pay special attention to technical but essential steps like causal real-time extraction of the signal of interest from a noisy measurement and estimation of instantaneous phase and amplitude. The feedback loop's parameters are tuned automatically to suppress synchrony. Though the study is motivated by neuroscience, the results are relevant to controlling oscillatory activity in ensembles of various natures and, thus, to the rapidly developing field of network physiology.

Desynchronizing oscillators coupled in multi-cluster networks through adaptively controlling partial networks

This article introduces an adaptive control scheme with a feedback delay, specifically designed for controlling partial networks, to achieve desynchronization in a coupled network with two or multiple clusters. The proposed scheme's effectiveness is validated through several representative examples of coupled neuronal networks with two interconnected clusters. The efficacy of this scheme is attributed to the rigorous and numerical analyses on the corresponding transcendental characteristic equation, which includes time delay and other network parameters. In addition to investigating the impact of time delay and inter-connectivity on the stability of an incoherent state, we also rigorously find that controlling only one cluster cannot realize the desynchronization in the coupled oscillators within three or more clusters. All these, we believe, can deepen the understanding of the deep brain stimulation techniques presently used in the clinical treatment of neurodegenerative diseases and suggest future avenues for enhancing these clinical techniques through adaptive feedback settings.

Optimizing charge-balanced pulse stimulation for desynchronization

Collective synchronization in a large population of self-sustained units appears both in natural and engineered systems. Sometimes this effect is in demand, while in some cases, it is undesirable, which calls for control techniques. In this paper, we focus on pulsatile control, with the goal to either increase or decrease the level of synchrony. We quantify this level by the entropy of the phase distribution. Motivated by possible applications in neuroscience, we consider pulses of a realistic shape. Exploiting the noisy Kuramoto-Winfree model, we search for the optimal pulse profile and the optimal stimulation phase. For this purpose, we derive an expression for the change of the phase distribution entropy due to the stimulus. We relate this change to the properties of individual units characterized by generally different natural frequencies and phase response curves and the population's state. We verify the general result by analyzing a two-frequency population model and demonstrating a good agreement of the theory and numerical simulations.

Real-time estimation of phase and amplitude with application to neural data

Computation of the instantaneous phase and amplitude via the Hilbert Transform is a powerful tool of data analysis. This approach finds many applications in various science and engineering branches but is not proper for causal estimation because it requires knowledge of the signal's past and future. However, several problems require real-time estimation of phase and amplitude; an illustrative example is phase-locked or amplitude-dependent stimulation in neuroscience. In this paper, we discuss and compare three causal algorithms that do not rely on the Hilbert Transform but exploit well-known physical phenomena, the synchronization and the resonance. After testing the algorithms on a synthetic data set, we illustrate their performance computing phase and amplitude for the accelerometer tremor measurements and a Parkinsonian patient's beta-band brain activity.

Eliminating synchronization of coupled neurons adaptively by using feedback coupling with heterogeneous delays

In this paper, we present an adaptive scheme involving heterogeneous delay interactions to suppress synchronization in a large population of oscillators. We analytically investigate the incoherent state stability regions for several specific kinds of distributions for delays. Interestingly, we find that, among the distributions that we discuss, the exponential distribution may offer great convenience to the performance of our adaptive scheme because this distribution renders an unbounded stability region. Moreover, we demonstrate our scheme in the realization of synchronization elimination in some representative, realistic neuronal networks, which makes it possible to deepen the understanding and even refine the existing techniques of deep brain stimulation in the treatment of some synchronization-induced mental disorders.

Controlling collective synchrony in oscillatory ensembles by precisely timed pulses

We present an efficient technique for control of synchrony in a globally coupled ensemble by pulsatile action. We assume that we can observe the collective oscillation and can stimulate all elements of the ensemble simultaneously. We pay special attention to the minimization of intervention into the system. The key idea is to stimulate only at the most sensitive phase. To find this phase, we implement an adaptive feedback control. Estimating the instantaneous phase of the collective mode on the fly, we achieve efficient suppression using a few pulses per oscillatory cycle. We discuss the possible relevance of the results for neuroscience, namely, for the development of advanced algorithms for deep brain stimulation, a medical technique used to treat Parkinson's disease.

Reinforcement learning for suppression of collective activity in oscillatory ensembles

We present the use of modern machine learning approaches to suppress self-sustained collective oscillations typically signaled by ensembles of degenerative neurons in the brain. The proposed hybrid model relies on two major components: an environment of oscillators and a policy-based reinforcement learning block. We report a model-agnostic synchrony control based on proximal policy optimization and two artificial neural networks in an Actor-Critic configuration. A class of physically meaningful reward functions enabling the suppression of collective oscillatory mode is proposed. The synchrony suppression is demonstrated for two models of neuronal populations-for the ensembles of globally coupled limit-cycle Bonhoeffer-van der Pol oscillators and for the bursting Hindmarsh-Rose neurons using rectangular and charge-balanced stimuli.

Multisite Delayed Feedback for Electrical Brain Stimulation

Demand-controlled deep brain stimulation (DBS) appears to be a promising approach for the treatment of Parkinson's disease (PD) as revealed by computational, pre-clinical and clinical studies. Stimulation delivery is adapted to brain activity, for example, to the amount of neuronal activity considered to be abnormal. Such a closed-loop stimulation setup might help to reduce the amount of stimulation current, thereby maintaining therapeutic efficacy. In the context of the development of stimulation techniques that aim to restore desynchronized neuronal activity on a long-term basis, specific closed-loop stimulation protocols were designed computationally. These feedback techniques, e.g., pulsatile linear delayed feedback (LDF) or pulsatile nonlinear delayed feedback (NDF), were computationally developed to counteract abnormal neuronal synchronization characteristic for PD and other neurological disorders. By design, these techniques are intrinsically demand-controlled methods, where the amplitude of the stimulation signal is reduced when the desired desynchronized regime is reached. We here introduce a novel demand-controlled stimulation method, pulsatile multisite linear delayed feedback (MLDF), by employing MLDF to modulate the pulse amplitude of high-frequency (HF) DBS, in this way aiming at a specific, MLDF-related desynchronizing impact, while maintaining safety requirements with the charge-balanced HF DBS. Previously, MLDF was computationally developed for the control of spatio-temporal synchronized patterns and cluster states in neuronal populations. Here, in a physiologically motivated model network comprising neurons from subthalamic nucleus (STN) and external globus pallidus (GPe), we compare pulsatile MLDF to pulsatile LDF for the case where the smooth feedback signals are used to modulate the amplitude of charge-balanced HF DBS and suggest a modification of pulsatile MLDF which enables a pronounced desynchronizing impact. Our results may contribute to further clinical development of closed-loop DBS techniques.

Spontaneous symmetry breaking due to the trade-off between attractive and repulsive couplings

Spontaneous symmetry breaking is an important phenomenon observed in various fields including physics and biology. In this connection, we here show that the trade-off between attractive and repulsive couplings can induce spontaneous symmetry breaking in a homogeneous system of coupled oscillators. With a simple model of a system of two coupled Stuart-Landau oscillators, we demonstrate how the tendency of attractive coupling in inducing in-phase synchronized (IPS) oscillations and the tendency of repulsive coupling in inducing out-of-phase synchronized oscillations compete with each other and give rise to symmetry breaking oscillatory states and interesting multistabilities. Further, we provide explicit expressions for synchronized and antisynchronized oscillatory states as well as the so called oscillation death (OD) state and study their stability. If the Hopf bifurcation parameter (λ) is greater than the natural frequency (ω) of the system, the attractive coupling favors the emergence of an antisymmetric OD state via a Hopf bifurcation whereas the repulsive coupling favors the emergence of a similar state through a saddle-node bifurcation. We show that an increase in the repulsive coupling not only destabilizes the IPS state but also facilitates the reentrance of the IPS state.

Pulsatile desynchronizing delayed feedback for closed-loop deep brain stimulation

High-frequency (HF) deep brain stimulation (DBS) is the gold standard for the treatment of medically refractory movement disorders like Parkinson’s disease, essential tremor, and dystonia, with a significant potential for application to other neurological diseases. The standard setup of HF DBS utilizes an open-loop stimulation protocol, where a permanent HF electrical pulse train is administered to the brain target areas irrespectively of the ongoing neuronal dynamics. Recent experimental and clinical studies demonstrate that a closed-loop, adaptive DBS might be superior to the open-loop setup. We here combine the notion of the adaptive high-frequency stimulation approach, that aims at delivering stimulation adapted to the extent of appropriately detected biomarkers, with specifically desynchronizing stimulation protocols. To this end, we extend the delayed feedback stimulation methods, which are intrinsically closed-loop techniques and specifically designed to desynchronize abnormal neuronal synchronization, to pulsatile electrical brain stimulation. We show that permanent pulsatile high-frequency stimulation subjected to an amplitude modulation by linear or nonlinear delayed feedback methods can effectively and robustly desynchronize a STN-GPe network of model neurons and suggest this approach for desynchronizing closed-loop DBS.

Dynamics of globally coupled oscillators: Progress and perspectives

In this paper, we discuss recent progress in research of ensembles of mean field coupled oscillators. Without an ambition to present a comprehensive review, we outline most interesting from our viewpoint results and surprises, as well as interrelations between different approaches.

Future of seizure prediction and intervention: closing the loop

The ultimate goal of epilepsy therapies is to provide seizure control for all patients while eliminating side effects. Improved specificity of intervention through on-demand approaches may overcome many of the limitations of current intervention strategies. This article reviews the progress in seizure prediction and detection, potential new therapies to provide improved specificity, and devices to achieve these ends. Specifically, we discuss (1) potential signal modalities and algorithms for seizure detection and prediction, (2) closed-loop intervention approaches, and (3) hardware for implementing these algorithms and interventions. Seizure prediction and therapies maximize efficacy, whereas minimizing side effects through improved specificity may represent the future of epilepsy treatments.

Designing a deep brain stimulator to suppress pathological neuronal synchrony

Some of neuropathologies are believed to be related to abnormal synchronization of neurons. In the line of therapy, designing effective deep brain stimulators to suppress the pathological synchrony among neuronal ensembles is a challenge of high clinical relevance. The stimulation should be able to disrupt the synchrony in the presence of latencies due to imperfect knowledge about parameters of a neuronal ensemble and stimulation impacts on the ensemble. We propose an adaptive desynchronizing deep brain stimulator capable of dealing with these uncertainties. We analyze the collective behavior of the stimulated neuronal ensemble and show that, using the designed stimulator, the resulting asynchronous state is stable. Simulation results reveal the efficiency of the proposed technique.

Controlling synchrony in oscillatory networks via an act-and-wait algorithm

The act-and-wait control algorithm is proposed to suppress synchrony in globally coupled oscillatory networks in the situation when the simultaneous registration and stimulation of the system is not possible. The algorithm involves the periodic repetition of the registration (wait) and stimulation (act) stages, such that in the first stage the mean field of the free system is recorded in a memory and in the second stage the system is stimulated with the recorded signal. A modified version of the algorithm that takes into account the charge-balanced requirement is considered as well. The efficiency of our algorithm is demonstrated analytically and numerically for globally coupled Landau-Stuart oscillators and synaptically all-to-all coupled FitzHugh-Nagumo as well as Hodgkin-Huxley neurons.