Locally optimal extracellular stimulation for chaotic desynchronization of neural populations

Response of neuronal populations to phase-locked stimulation: model-based predictions and validation

Modulation of neuronal oscillations holds promise for the treatment of neurological disorders. Nonetheless, conventional stimulation in a continuous open-loop manner can lead to side effects and suboptimal efficiency. Closed-loop strategies such as phase-locked stimulation aim to address these shortcomings by offering a more targeted modulation. While theories have been developed to understand the neural response to stimulation, their predictions have not been thoroughly tested using experimental data. Using a mechanistic coupled oscillator model, we elaborate on two key predictions describing the response to stimulation as a function of the phase and amplitude of ongoing neural activity. To investigate these predictions, we analyze electrocorticogram recordings from a previously conducted study in Parkinsonian rats, and extract the corresponding phase and response curves. We demonstrate that the amplitude response to stimulation is strongly correlated to the derivative of the phase response ([Formula: see text] > 0.8) in all animals except one, thereby validating a key model prediction. The second prediction postulates that the stimulation becomes ineffective when the network synchrony is high, a trend that appeared missing in the data. Our analysis explains this discrepancy by showing that the neural populations in Parkinsonian rats did not reach the level of synchrony for which the theory would predict ineffective stimulation. Our results highlight the potential of fine-tuning stimulation paradigms informed by mathematical models that consider both the ongoing phase and amplitude of the targeted neural oscillation.Significance Statement This study validates a mathematical model of coupled oscillators in predicting the response of neural activity to stimulation for the first time. Our findings also offer further insights beyond this validation. For instance, the demonstrated correlation between phase response and amplitude response is indeed a key theoretical concept within a subset of mathematical models. This prediction can bring about clinical implications in terms of predictive power for manipulation of neural activity. Additionally, while phase dependence in modulation has been previously studied, we propose a general framework for studying amplitude dependence as well. Lastly, our study reconciles the seemingly contradictory views of pathologic hypersynchrony and theoretical low synchrony in Parkinson's disease.

Optimal phase-selective entrainment of electrochemical oscillators with different phase response curves

We investigate the entrainment of electrochemical oscillators with different phase response curves (PRCs) using a global signal: the goal is to achieve the desired phase configuration using a minimum-power waveform. Establishing the desired phase relationships in a highly nonlinear networked system exhibiting significant heterogeneities, such as different conditions or parameters for the oscillators, presents a considerable challenge because different units respond differently to the common global entraining signal. In this work, we apply an optimal phase-selective entrainment technique in both a kinetic model and experiments involving electrochemical oscillators in achieving phase synchronized states. We estimate the PRCs of the oscillators at different circuit potentials and external resistance, and entrain pairs and small sets of four oscillators in various phase configurations. We show that for small PRC variations, phase assignment can be achieved using an averaged PRC in the control design. However, when the PRCs are sufficiently different, individual PRCs are needed to entrain the system with the expected phase relationships. The results show that oscillator assemblies with heterogeneous PRCs can be effectively entrained to desired phase configurations in practical settings. These findings open new avenues to applications in biological and engineered oscillator systems where synchronization patterns are essential for system performance.

Data-driven control of oscillator networks with population-level measurement

Controlling complex networks of nonlinear limit-cycle oscillators is an important problem pertinent to various applications in engineering and natural sciences. While in recent years the control of oscillator populations with comprehensive biophysical models or simplified models, e.g., phase models, has seen notable advances, learning appropriate controls directly from data without prior model assumptions or pre-existing data remains a challenging and less developed area of research. In this paper, we address this problem by leveraging the network's current dynamics to iteratively learn an appropriate control online without constructing a global model of the system. We illustrate through a range of numerical simulations that the proposed technique can effectively regulate synchrony in various oscillator networks after a small number of trials using only one input and one noisy population-level output measurement. We provide a theoretical analysis of our approach, illustrate its robustness to system variations, and compare its performance with existing model-based and data-driven approaches.

Engineering spatiotemporal patterns: information encoding, processing, and controllability in oscillator ensembles

The ability to finely manipulate spatiotemporal patterns displayed in neuronal populations is critical for understanding and influencing brain functions, sleep cycles, and neurological pathologies. However, such control tasks are challenged not only by the immense scale but also by the lack of real-time state measurements of neurons in the population, which deteriorates the control performance. In this paper, we formulate the control of dynamic structures in an ensemble of neuron oscillators as a tracking problem and propose a principled control technique for designing optimal stimuli that produce desired spatiotemporal patterns in a network of interacting neurons without requiring feedback information. We further reveal an interesting presentation of information encoding and processing in a neuron ensemble in terms of its controllability property. The performance of the presented technique in creating complex spatiotemporal spiking patterns is demonstrated on neural populations described by mathematically ideal and biophysical models, including the Kuramoto and Hodgkin-Huxley models, as well as real-time experiments on Wein bridge oscillators.

Data-Driven Control of Neuronal Networks with Population-Level Measurement

Controlling complex networks of nonlinear neurons is an important problem pertinent to various applications in engineering and natural sciences. While in recent years the control of neural populations with comprehensive biophysical models or simplified models, e.g., phase models, has seen notable advances, learning appropriate controls directly from data without any model assumptions remains a challenging and less developed area of research. In this paper, we address this problem by leveraging the network's local dynamics to iteratively learn an appropriate control without constructing a global model of the system. The proposed technique can effectively regulate synchrony in a neuronal network using only one input and one noisy population-level output measurement. We provide a theoretical analysis of our approach and illustrate its robustness to system variations and its generalizability to accommodate various physical constraints, such as charge-balanced inputs.

Leveraging deep learning to control neural oscillators

Modulation of the firing times of neural oscillators has long been an important control objective, with applications including Parkinson's disease, Tourette's syndrome, epilepsy, and learning. One common goal for such modulation is desynchronization, wherein two or more oscillators are stimulated to transition from firing in phase with each other to firing out of phase. The optimization of such stimuli has been well studied, but this typically relies on either a reduction of the dimensionality of the system or complete knowledge of the parameters and state of the system. This limits the applicability of results to real problems in neural control. Here, we present a trained artificial neural network capable of accurately estimating the effects of square-wave stimuli on neurons using minimal output information from the neuron. We then apply the results of this network to solve several related control problems in desynchronization, including desynchronizing pairs of neurons and achieving clustered subpopulations of neurons in the presence of coupling and noise.

Optimizing deep brain stimulation based on isostable amplitude in essential tremor patient models

OBJECTIVE

Deep brain stimulation is a treatment for medically refractory essential tremor. To improve the therapy, closed-loop approaches are designed to deliver stimulation according to the system's state, which is constantly monitored by recording a pathological signal associated with symptoms (e.g. brain signal or limb tremor). Since the space of possible closed-loop stimulation strategies is vast and cannot be fully explored experimentally, how to stimulate according to the state should be informed by modeling. A typical modeling goal is to design a stimulation strategy that aims to maximally reduce the Hilbert amplitude of the pathological signal in order to minimize symptoms. Isostables provide a notion of amplitude related to convergence time to the attractor, which can be beneficial in model-based control problems. However, how isostable and Hilbert amplitudes compare when optimizing the amplitude response to stimulation in models constrained by data is unknown.

APPROACH

We formulate a simple closed-loop stimulation strategy based on models previously fitted to phase-locked deep brain stimulation data from essential tremor patients. We compare the performance of this strategy in suppressing oscillatory power when based on Hilbert amplitude and when based on isostable amplitude. We also compare performance to phase-locked stimulation and open-loop high-frequency stimulation.

MAIN RESULTS

For our closed-loop phase space stimulation strategy, stimulation based on isostable amplitude is significantly more effective than stimulation based on Hilbert amplitude when amplitude field computation time is limited to minutes. Performance is similar when there are no constraints, however constraints on computation time are expected in clinical applications. Even when computation time is limited to minutes, closed-loop phase space stimulation based on isostable amplitude is advantageous compared to phase-locked stimulation, and is more efficient than high-frequency stimulation.

SIGNIFICANCE

Our results suggest a potential benefit to using isostable amplitude more broadly for model-based optimization of stimulation in neurological disorders.

Stabilization of Weakly Unstable Fixed Points as a Common Dynamical Mechanism of High-Frequency Electrical Stimulation

While high-frequency electrical stimulation often used to treat various biological diseases, it is generally difficult to understand its dynamical mechanisms of action. In this work, high-frequency electrical stimulation is considered in the context of neurological and cardiological systems. Despite inherent differences between these systems, results from both theory and computational modeling suggest identical dynamical mechanisms responsible for desirable qualitative changes in behavior in response to high-frequency stimuli. Specifically, desynchronization observed in a population of periodically firing neurons and reversible conduction block that occurs in cardiomyocytes both result from bifurcations engendered by stimulation that modifies the stability of unstable fixed points. Using a reduced order phase-amplitude modeling framework, this phenomenon is described in detail from a theoretical perspective. Results are consistent with and provide additional insight for previously published experimental observations. Also, it is found that sinusoidal input is energy-optimal for modifying the stability of weakly unstable fixed points using periodic stimulation.

Emerging technologies for improved deep brain stimulation

Deep brain stimulation (DBS) is an effective treatment for common movement disorders and has been used to modulate neural activity through delivery of electrical stimulation to key brain structures. The long-term efficacy of stimulation in treating disorders, such as Parkinson's disease and essential tremor, has encouraged its application to a wide range of neurological and psychiatric conditions. Nevertheless, adoption of DBS remains limited, even in Parkinson's disease. Recent failed clinical trials of DBS in major depression, and modest treatment outcomes in dementia and epilepsy, are spurring further development. These improvements focus on interaction with disease circuits through complementary, spatially and temporally specific approaches. Spatial specificity is promoted by the use of segmented electrodes and field steering, and temporal specificity involves the delivery of patterned stimulation, mostly controlled through disease-related feedback. Underpinning these developments are new insights into brain structure-function relationships and aberrant circuit dynamics, including new methods with which to assess and refine the clinical effects of stimulation.

Phase reduction and phase-based optimal control for biological systems: a tutorial

A powerful technique for the analysis of nonlinear oscillators is the rigorous reduction to phase models, with a single variable describing the phase of the oscillation with respect to some reference state. An analog to phase reduction has recently been proposed for systems with a stable fixed point, and phase reduction for periodic orbits has recently been extended to take into account transverse directions and higher-order terms. This tutorial gives a unified treatment of such phase reduction techniques and illustrates their use through mathematical and biological examples. It also covers the use of phase reduction for designing control algorithms which optimally change properties of the system, such as the phase of the oscillation. The control techniques are illustrated for example neural and cardiac systems.

Phase-Dependent Suppression of Beta Oscillations in Parkinson's Disease Patients

Synchronized oscillations within and between brain areas facilitate normal processing, but are often amplified in disease. A prominent example is the abnormally sustained beta-frequency (∼20 Hz) oscillations recorded from the cortex and subthalamic nucleus of Parkinson's disease patients. Computational modeling suggests that the amplitude of such oscillations could be modulated by applying stimulation at a specific phase. Such a strategy would allow selective targeting of the oscillation, with relatively little effect on other activity parameters. Here, activity was recorded from 10 awake, parkinsonian patients (6 male, 4 female human subjects) undergoing functional neurosurgery. We demonstrate that stimulation arriving on a particular patient-specific phase of the beta oscillation over consecutive cycles could suppress the amplitude of this pathophysiological activity by up to 40%, while amplification effects were relatively weak. Suppressive effects were accompanied by a reduction in the rhythmic output of subthalamic nucleus (STN) neurons and synchronization with the mesial cortex. While stimulation could alter the spiking pattern of STN neurons, there was no net effect on firing rate, suggesting that reduced beta synchrony was a result of alterations to the relative timing of spiking activity, rather than an overall change in excitability. Together, these results identify a novel intrinsic property of cortico-basal ganglia synchrony that suggests the phase of ongoing neural oscillations could be a viable and effective control signal for the treatment of Parkinson's disease. This work has potential implications for other brain diseases with exaggerated neuronal synchronization and for probing the function of rhythmic activity in the healthy brain.SIGNIFICANCE STATEMENT In Parkinson's disease (PD), movement impairment is correlated with exaggerated beta frequency oscillations in the cerebral cortex and subthalamic nucleus (STN). Using a novel method of stimulation in PD patients undergoing neurosurgery, we demonstrate that STN beta oscillations can be suppressed when consecutive electrical pulses arrive at a specific phase of the oscillation. This effect is likely because of interrupting the timing of neuronal activity rather than excitability, as stimulation altered the firing pattern of STN spiking without changing overall rate. These findings show the potential of oscillation phase as an input for "closed-loop" stimulation, which could provide a valuable neuromodulation strategy for the treatment of brain disorders and for elucidating the role of neuronal oscillations in the healthy brain.

Theoretical Principles of Multiscale Spatiotemporal Control of Neuronal Networks: A Complex Systems Perspective

Success in the fine control of the nervous system depends on a deeper understanding of how neural circuits control behavior. There is, however, a wide gap between the components of neural circuits and behavior. We advance the idea that a suitable approach for narrowing this gap has to be based on a multiscale information-theoretic description of the system. We evaluate the possibility that brain-wide complex neural computations can be dissected into a hierarchy of computational motifs that rely on smaller circuit modules interacting at multiple scales. In doing so, we draw attention to the importance of formalizing the goals of stimulation in terms of neural computations so that the possible implementations are matched in scale to the underlying circuit modules.

Numerical optimization of coordinated reset stimulation for desynchronizing neuronal network dynamics

Excessive synchronization in neural activity is a hallmark of Parkinson's disease (PD). A promising technique for treating PD is coordinated reset (CR) neuromodulation in which a neural population is desynchronized by the delivery of spatially-distributed current stimuli using multiple electrodes. In this study, we perform numerical optimization to find the energy-optimal current waveform for desynchronizing neuronal network with CR stimulation, by proposing and applying a new optimization method based on the direct search algorithm. In the proposed optimization method, the stimulating current is described as a Fourier series, and each Fourier coefficient as well as the stimulation period are directly optimized by evaluating the order parameter, which quantifies the synchrony level, from network simulation. This direct optimization scheme has an advantage that arbitrary changes in the dynamical properties of the network can be taken into account in the search process. By harnessing this advantage, we demonstrate the significant influence of externally applied oscillatory inputs and non-random network topology on the efficacy of CR modulation. Our results suggest that the effectiveness of brain stimulation for desynchronization may depend on various factors modulating the dynamics of the target network. We also discuss the possible relevance of the results to the efficacy of the stimulation in PD treatment.

Evolution of extrema features reveals optimal stimuli for biological state transitions

The ability to define the unique features of an input stimulus needed to control switch-like behavior in biological systems is an important problem in computational biology and medicine. We show in this study how highly complex and intractable optimization problems can be simplified by restricting the search to the signal's extrema as key feature points, and evolving the extrema features towards optimal solutions that closely match solutions derived from gradient-based methods. Our results suggest a model-independent approach for solving a class of optimization problems related to controlling switch-like state transitions.

From Maps to Multi-dimensional Network Mechanisms of Mental Disorders

The development of advanced neuroimaging techniques and their deployment in large cohorts has enabled an assessment of functional and structural brain network architecture at an unprecedented level of detail. Across many temporal and spatial scales, network neuroscience has emerged as a central focus of intellectual efforts, seeking meaningful descriptions of brain networks and explanatory sets of network features that underlie circuit function in health and dysfunction in disease. However, the tools of network science commonly deployed provide insight into brain function at a fundamentally descriptive level, often failing to identify (patho-)physiological mechanisms that link system-level phenomena to the multiple hierarchies of brain function. Here we describe recently developed techniques stemming from advances in complex systems and network science that have the potential to overcome this limitation, thereby contributing mechanistic insights into neuroanatomy, functional dynamics, and pathology. Finally, we build on the Research Domain Criteria framework, highlighting the notion that mental illnesses can be conceptualized as dysfunctions of neural circuitry present across conventional diagnostic boundaries, to sketch how network-based methods can be combined with pharmacological, intermediate phenotype, genetic, and magnetic stimulation studies to probe mechanisms of psychopathology.

How to desynchronize quorum-sensing networks

In this paper we investigate how so-called quorum-sensing networks can be desynchronized. Such networks, which arise in many important application fields, such as systems biology, are characterized by the fact that direct communication between network nodes is superimposed to communication with a shared, environmental variable. In particular, we provide a new sufficient condition ensuring that the trajectories of these quorum-sensing networks diverge from their synchronous evolution. Then, we apply our result to study two applications.

Algorithmic design of a noise-resistant and efficient closed-loop deep brain stimulation system: A computational approach

Advances in the field of closed-loop neuromodulation call for analysis and modeling approaches capable of confronting challenges related to the complex neuronal response to stimulation and the presence of strong internal and measurement noise in neural recordings. Here we elaborate on the algorithmic aspects of a noise-resistant closed-loop subthalamic nucleus deep brain stimulation system for advanced Parkinson’s disease and treatment-refractory obsessive-compulsive disorder, ensuring remarkable performance in terms of both efficiency and selectivity of stimulation, as well as in terms of computational speed. First, we propose an efficient method drawn from dynamical systems theory, for the reliable assessment of significant nonlinear coupling between beta and high-frequency subthalamic neuronal activity, as a biomarker for feedback control. Further, we present a model-based strategy through which optimal parameters of stimulation for minimum energy desynchronizing control of neuronal activity are being identified. The strategy integrates stochastic modeling and derivative-free optimization of neural dynamics based on quadratic modeling. On the basis of numerical simulations, we demonstrate the potential of the presented modeling approach to identify, at a relatively low computational cost, stimulation settings potentially associated with a significantly higher degree of efficiency and selectivity compared with stimulation settings determined post-operatively. Our data reinforce the hypothesis that model-based control strategies are crucial for the design of novel stimulation protocols at the backstage of clinical applications.

Desynchronization of stochastically synchronized chemical oscillators

Experimental and theoretical studies are presented on the design of perturbations that enhance desynchronization in populations of oscillators that are synchronized by periodic entrainment. A phase reduction approach is used to determine optimal perturbation timing based upon experimentally measured phase response curves. The effectiveness of the perturbation waveforms is tested experimentally in populations of periodically and stochastically synchronized chemical oscillators. The relevance of the approach to therapeutic methods for disrupting phase coherence in groups of stochastically synchronized neuronal oscillators is discussed.

An energy-optimal approach for entrainment of uncertain circadian oscillators

We develop an approach to find an energy-optimal stimulus that entrains an ensemble of uncertain, uncoupled limit cycle oscillators. Furthermore, when entrainment occurs, the phase shift between oscillators is constrained to be less than a predetermined amount. This approach is illustrated for a model of Drosophila circadian activity, for which it performs better than a standard 24-h light-dark cycle. Because this method explicitly accounts for uncertainty in a given system and only requires information that is experimentally obtainable, it is well suited for experimental implementation and could ultimately represent what is believed to be a novel treatment for patients suffering from advanced/delayed sleep-phase syndrome.

Optimal entrainment of heterogeneous noisy neurons

We develop a methodology to design a stimulus optimized to entrain nonlinear, noisy limit cycle oscillators with uncertain properties. Conditions are derived which guarantee that the stimulus will entrain the oscillators despite these uncertainties. Using these conditions, we develop an energy optimal control strategy to design an efficient entraining stimulus and apply it to numerical models of noisy phase oscillators and to in vitro hippocampal neurons. In both instances, the optimal stimuli outperform other similar but suboptimal entraining stimuli. Because this control strategy explicitly accounts for both noise and inherent uncertainty of model parameters, it could have experimental relevance to neural circuits where robust spike timing plays an important role.