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

Optimal input reverberation and homeostatic self-organization toward the edge of synchronization

Transient or partial synchronization can be used to do computations, although a fully synchronized network is sometimes related to the onset of epileptic seizures. Here, we propose a homeostatic mechanism that is capable of maintaining a neuronal network at the edge of a synchronization transition, thereby avoiding the harmful consequences of a fully synchronized network. We model neurons by maps since they are dynamically richer than integrate-and-fire models and more computationally efficient than conductance-based approaches. We first describe the synchronization phase transition of a dense network of neurons with different tonic spiking frequencies coupled by gap junctions. We show that at the transition critical point, inputs optimally reverberate through the network activity through transient synchronization. Then, we introduce a local homeostatic dynamic in the synaptic coupling and show that it produces a robust self-organization toward the edge of this phase transition. We discuss the potential biological consequences of this self-organization process, such as its relation to the Brain Criticality hypothesis, its input processing capacity, and how its malfunction could lead to pathological synchronization and the onset of seizure-like activity.

Disrupting abnormal neuronal oscillations with adaptive delayed feedback control

Closed-loop neuronal stimulation has a strong therapeutic potential for neurological disorders such as Parkinson's disease. However, at the moment, standard stimulation protocols rely on continuous open-loop stimulation and the design of adaptive controllers is an active field of research. Delayed feedback control (DFC), a popular method used to control chaotic systems, has been proposed as a closed-loop technique for desynchronisation of neuronal populations but, so far, was only tested in computational studies. We implement DFC for the first time in neuronal populations and access its efficacy in disrupting unwanted neuronal oscillations. To analyse in detail the performance of this activity control algorithm, we used specialised in vitro platforms with high spatiotemporal monitoring/stimulating capabilities. We show that the conventional DFC in fact worsens the neuronal population oscillatory behaviour, which was never reported before. Conversely, we present an improved control algorithm, adaptive DFC (aDFC), which monitors the ongoing oscillation periodicity and self-tunes accordingly. aDFC effectively disrupts collective neuronal oscillations restoring a more physiological state. Overall, these results support aDFC as a better candidate for therapeutic closed-loop brain stimulation.

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.

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.

Suppression of beta oscillations by delayed feedback in a cortex-basal ganglia-thalamus-pedunculopontine nucleus neural loop model

Excessive neural synchronization of neural populations in the beta (β) frequency range (12-35 Hz) is intimately related to the symptoms of hypokinesia in Parkinson's disease (PD). Studies have shown that delayed feedback stimulation strategies can interrupt excessive neural synchronization and effectively alleviate symptoms associated with PD dyskinesia. Work on optimizing delayed feedback algorithms continues to progress, yet it remains challenging to further improve the inhibitory effect with reduced energy expenditure. Therefore, we first established a neural mass model of the cortex-basal ganglia-thalamus-pedunculopontine nucleus (CBGTh-PPN) closed-loop system, which can reflect the internal properties of cortical and basal ganglia neurons and their intrinsic connections with thalamic and pedunculopontine nucleus neurons. Second, the inhibitory effects of three delayed feedback schemes based on the external globus pallidum (GPe) on β oscillations were investigated separately and compared with those based on the subthalamic nucleus (STN) only. Our results show that all four delayed feedback schemes achieve effective suppression of pathological β oscillations when using the linear delayed feedback algorithm. The comparison revealed that the three GPe-based delayed feedback stimulation strategies were able to have a greater range of oscillation suppression with reduced energy consumption, thus improving control performance effectively, suggesting that they may be more effective for the relief of Parkinson's motor symptoms in practical applications.

Improved control effect of pathological oscillations by using delayed feedback stimulation in neural mass model with pedunculopontine nucleus

BACKGROUND

The role of delayed feedback stimulation in the discussion of Parkinson's disease (PD) has recently received increasing attention. Stimulation of pedunculopontine nucleus (PPN) is an emerging treatment for PD. However, the effect of PPN in regulating PD is ignored, and the delayed feedback stimulation algorithm is facing some problems in parameter selection.

METHODS

On the basis of a neural mass model, we established a new network for PPN. Four types of delayed feedback stimulation schemes were designed, such as stimulating subthalamic nucleus (STN) with the local field potentials (LFPs) of STN nucleus, globus pallidus (GPe) with the LFPs of Gpe nucleus, PPN with the LFPs of Gpe nucleus, and STN with the LFPs of PPN nucleus.

RESULTS

In this study, we found that all four kinds of delayed feedback schemes are effective, suggesting that the algorithm is simple and more effective in experiments. More specifically, the other three control schemes improved the control performance and reduced the stimulation energy expenditure compared with traditional stimulating STN itself only.

CONCLUSION

PPN stimulation can affect the new network and help to suppress pathological oscillations for each neuron. We hope that our results can gain an insight into the future clinical treatment.

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.

Decoupling of interacting neuronal populations by time-shifted stimulation through spike-timing-dependent plasticity

The synaptic organization of the brain is constantly modified by activity-dependent synaptic plasticity. In several neurological disorders, abnormal neuronal activity and pathological synaptic connectivity may significantly impair normal brain function. Reorganization of neuronal circuits by therapeutic stimulation has the potential to restore normal brain dynamics. Increasing evidence suggests that the temporal stimulation pattern crucially determines the long-lasting therapeutic effects of stimulation. Here, we tested whether a specific pattern of brain stimulation can enable the suppression of pathologically strong inter-population synaptic connectivity through spike-timing-dependent plasticity (STDP). More specifically, we tested how introducing a time shift between stimuli delivered to two interacting populations of neurons can effectively decouple them. To that end, we first used a tractable model, i.e., two bidirectionally coupled leaky integrate-and-fire (LIF) neurons, to theoretically analyze the optimal range of stimulation frequency and time shift for decoupling. We then extended our results to two reciprocally connected neuronal populations (modules) where inter-population delayed connections were modified by STDP. As predicted by the theoretical results, appropriately time-shifted stimulation causes a decoupling of the two-module system through STDP, i.e., by unlearning pathologically strong synaptic interactions between the two populations. Based on the overall topology of the connections, the decoupling of the two modules, in turn, causes a desynchronization of the populations that outlasts the cessation of stimulation. Decoupling effects of the time-shifted stimulation can be realized by time-shifted burst stimulation as well as time-shifted continuous simulation. Our results provide insight into the further optimization of a variety of multichannel stimulation protocols aiming at a therapeutic reshaping of diseased brain networks.

Efficient suppression of parkinsonian beta oscillations in a closed-loop model of deep brain stimulation with amplitude modulation

Introduction

Parkinson's disease (PD) is a movement disorder characterized by the pathological beta band (15-30 Hz) neural oscillations within the basal ganglia (BG). It is shown that the suppression of abnormal beta oscillations is correlated with the improvement of PD motor symptoms, which is a goal of standard therapies including deep brain stimulation (DBS). To overcome the stimulation-induced side effects and inefficiencies of conventional DBS (cDBS) and to reduce the administered stimulation current, closed-loop adaptive DBS (aDBS) techniques were developed. In this method, the frequency and/or amplitude of stimulation are modulated based on various disease biomarkers.

Methods

Here, by computational modeling of a cortico-BG-thalamic network in normal and PD conditions, we show that closed-loop aDBS of the subthalamic nucleus (STN) with amplitude modulation leads to a more effective suppression of pathological beta oscillations within the parkinsonian BG.

Results

Our results show that beta band neural oscillations are restored to their normal range and the reliability of the response of the thalamic neurons to motor cortex commands is retained due to aDBS with amplitude modulation. Furthermore, notably less stimulation current is administered during aDBS compared with cDBS due to a closed-loop control of stimulation amplitude based on the STN local field potential (LFP) beta activity.

Discussion

Efficient models of closed-loop stimulation may contribute to the clinical development of optimized aDBS techniques designed to reduce potential stimulation-induced side effects of cDBS in PD patients while leading to a better therapeutic outcome.

Long-Lasting Desynchronization of Plastic Neuronal Networks by Double-Random Coordinated Reset Stimulation

Hypersynchrony of neuronal activity is associated with several neurological disorders, including essential tremor and Parkinson's disease (PD). Chronic high-frequency deep brain stimulation (HF DBS) is the standard of care for medically refractory PD. Symptoms may effectively be suppressed by HF DBS, but return shortly after cessation of stimulation. Coordinated reset (CR) stimulation is a theory-based stimulation technique that was designed to specifically counteract neuronal synchrony by desynchronization. During CR, phase-shifted stimuli are delivered to multiple neuronal subpopulations. Computational studies on CR stimulation of plastic neuronal networks revealed long-lasting desynchronization effects obtained by down-regulating abnormal synaptic connectivity. This way, networks are moved into attractors of stable desynchronized states such that stimulation-induced desynchronization persists after cessation of stimulation. Preclinical and clinical studies confirmed corresponding long-lasting therapeutic and desynchronizing effects in PD. As PD symptoms are associated with different pathological synchronous rhythms, stimulation-induced long-lasting desynchronization effects should favorably be robust to variations of the stimulation frequency. Recent computational studies suggested that this robustness can be improved by randomizing the timings of stimulus deliveries. We study the long-lasting effects of CR stimulation with randomized stimulus amplitudes and/or randomized stimulus timing in networks of leaky integrate-and-fire (LIF) neurons with spike-timing-dependent plasticity. Performing computer simulations and analytical calculations, we study long-lasting desynchronization effects of CR with and without randomization of stimulus amplitudes alone, randomization of stimulus times alone as well as the combination of both. Varying the CR stimulation frequency (with respect to the frequency of abnormal target rhythm) and the number of separately stimulated neuronal subpopulations, we reveal parameter regions and related mechanisms where the two qualitatively different randomization mechanisms improve the robustness of long-lasting desynchronization effects of CR. In particular, for clinically relevant parameter ranges double-random CR stimulation, i.e., CR stimulation with the specific combination of stimulus amplitude randomization and stimulus time randomization, may outperform regular CR stimulation with respect to long-lasting desynchronization. In addition, our results provide the first evidence that an effective reduction of the overall stimulation current by stimulus amplitude randomization may improve the frequency robustness of long-lasting therapeutic effects of brain stimulation.

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.

Delayed Feedback-Based Suppression of Pathological Oscillations in a Neural Mass Model

Suppression of excessively synchronous beta frequency (12-35 Hz) oscillatory activity in the basal ganglia is believed to correlate with the alleviation of hypokinetic motor symptoms of the Parkinson's disease. Delayed feedback is an effective strategy to interrupt the synchronization and has been used in the design of closed-loop neuromodulation methods computationally. Although tremendous efforts in this are being made by optimizing delayed feedback algorithm and stimulation waveforms, there are still remaining problems in the selection of effective parameters in the delayed feedback control schemes. In most delayed feedback neuromodulation strategies, the stimulation signal is obtained from the local field potential (LFP) of the excitatory subthalamic nucleus (STN) neurons and then is administered back to STN itself only. The inhibitory external globus pallidus (GPe) nucleus in the excitatory-inhibitory STN-GPe reciprocal network has not been involved in the design of the delayed feedback control strategies. Thus, considering the role of GPe, this paper proposes three schemes involving GPe in the design of the delayed feedback strategies and compared their effectiveness to the traditional paradigm using STN only. Based on a neural mass model of STN-GPe network having capability of simulating the LFP directly, the proposed stimulation strategies are tested and compared. Our simulation results show that the four types of delayed feedback control schemes are all effective, even if with a simple linear delayed feedback algorithm. But the three new control strategies we propose here further improve the control performance by enlarging the oscillatory suppression space and reducing the energy expenditure, suggesting that they may be more effective in applications. This paper may guide a new approach to optimize the closed-loop deep brain stimulation treatment to alleviate the Parkinsonian state by retargeting the measurement and stimulation nucleus.

Long-Lasting Desynchronization Effects of Coordinated Reset Stimulation Improved by Random Jitters

Abnormally strong synchronized activity is related to several neurological disorders, including essential tremor, epilepsy, and Parkinson's disease. Chronic high-frequency deep brain stimulation (HF DBS) is an established treatment for advanced Parkinson's disease. To reduce the delivered integral electrical current, novel theory-based stimulation techniques such as coordinated reset (CR) stimulation directly counteract the abnormal synchronous firing by delivering phase-shifted stimuli through multiple stimulation sites. In computational studies in neuronal networks with spike-timing-dependent plasticity (STDP), it was shown that CR stimulation down-regulates synaptic weights and drives the network into an attractor of a stable desynchronized state. This led to desynchronization effects that outlasted the stimulation. Corresponding long-lasting therapeutic effects were observed in preclinical and clinical studies. Computational studies suggest that long-lasting effects of CR stimulation depend on the adjustment of the stimulation frequency to the dominant synchronous rhythm. This may limit clinical applicability as different pathological rhythms may coexist. To increase the robustness of the long-lasting effects, we study randomized versions of CR stimulation in networks of leaky integrate-and-fire neurons with STDP. Randomization is obtained by adding random jitters to the stimulation times and by shuffling the sequence of stimulation site activations. We study the corresponding long-lasting effects using analytical calculations and computer simulations. We show that random jitters increase the robustness of long-lasting effects with respect to changes of the number of stimulation sites and the stimulation frequency. In contrast, shuffling does not increase parameter robustness of long-lasting effects. Studying the relation between acute, acute after-, and long-lasting effects of stimulation, we find that both acute after- and long-lasting effects are strongly determined by the stimulation-induced synaptic reshaping, whereas acute effects solely depend on the statistics of administered stimuli. We find that the stimulation duration is another important parameter, as effective stimulation only entails long-lasting effects after a sufficient stimulation duration. Our results show that long-lasting therapeutic effects of CR stimulation with random jitters are more robust than those of regular CR stimulation. This might reduce the parameter adjustment time in future clinical trials and make CR with random jitters more suitable for treating brain disorders with abnormal synchronization in multiple frequency bands.

An Approach for Stabilizing Abnormal Neural Activity in ADHD Using Chaotic Resonance

Reduced integrity of neural pathways from frontal to sensory cortices has been suggested as a potential neurobiological basis of attention-deficit hyperactivity disorder. Neurofeedback has been widely applied to enhance reduced neural pathways in attention-deficit hyperactivity disorder by repeated training on a daily temporal scale. Clinical and model-based studies have demonstrated that fluctuations in neural activity underpin sustained attention deficits in attention-deficit hyperactivity disorder. These aberrant neural fluctuations may be caused by the chaos–chaos intermittency state in frontal-sensory neural systems. Therefore, shifting the neural state from an aberrant chaos–chaos intermittency state to a normal stable state with an optimal external sensory stimulus, termed chaotic resonance, may be applied in neurofeedback for attention-deficit hyperactivity disorder. In this study, we applied a neurofeedback method based on chaotic resonance induced by “reduced region of orbit” feedback signals in the Baghdadi model for attention-deficit hyperactivity disorder. We evaluated the stabilizing effect of reduced region of orbit feedback and its robustness against noise from errors in estimation of neural activity. The effect of chaotic resonance successfully shifted the abnormal chaos-chaos intermittency of neural activity to the intended stable activity. Additionally, evaluation of the influence of noise due to measurement errors revealed that the efficiency of chaotic resonance induced by reduced region of orbit feedback signals was maintained over a range of certain noise strengths. In conclusion, applying chaotic resonance induced by reduced region of orbit feedback signals to neurofeedback methods may provide a promising treatment option for attention-deficit hyperactivity disorder.

Suppression of chaotic bursting synchronization in clustered scale-free networks by an external feedback signal

Oscillatory activities in the brain, detected by electroencephalograms, have identified synchronization patterns. These synchronized activities in neurons are related to cognitive processes. Additionally, experimental research studies on neuronal rhythms have shown synchronous oscillations in brain disorders. Mathematical modeling of networks has been used to mimic these neuronal synchronizations. Actually, networks with scale-free properties were identified in some regions of the cortex. In this work, to investigate these brain synchronizations, we focus on neuronal synchronization in a network with coupled scale-free networks. The networks are connected according to a topological organization in the structural cortical regions of the human brain. The neuronal dynamic is given by the Rulkov model, which is a two-dimensional iterated map. The Rulkov neuron can generate quiescence, tonic spiking, and bursting. Depending on the parameters, we identify synchronous behavior among the neurons in the clustered networks. In this work, we aim to suppress the neuronal burst synchronization by the application of an external perturbation as a function of the mean-field of membrane potential. We found that the method we used to suppress synchronization presents better results when compared to the time-delayed feedback method when applied to the same model of the neuronal network.

Feedback-induced desynchronization and oscillation quenching in a population of globally coupled oscillators

Motivated from a wide range of applications, various methods to control synchronization in coupled oscillators have been proposed. Previous studies have demonstrated that global feedback typically induces three macroscopic behaviors: synchronization, desynchronization, and oscillation quenching. However, analyzing all of these transitions within a single theoretical framework is difficult, and thus the feedback effect is only partially understood in each framework. Herein, we analyze a model of globally coupled phase oscillators exposed to global feedback, which shows all of the typical macroscopic dynamical states. Analytical tractability of the model enables us to obtain detailed phase diagrams where transitions and bistabilities between different macroscopic states are identified. Additionally, we propose strategies to steer the oscillators into targeted states with minimal feedback strength. Our study provides a useful overview of the effect of global feedback and is expected to serve as a benchmark when more sophisticated feedback needs to be designed.

An Embedded Multi-Core Real-Time Simulation Platform of Basal Ganglia for Deep Brain Stimulation

Closed-loop deep brain stimulation (DBS) paradigm is gaining tremendous favor due to its potential capability of further and more efficient improvements in neurological diseases. Preclinical validation of closed-loop controller is quite necessary in order to minimize injury risks of clinical trials to patients, which can greatly benefit from real-time computational models and thus potentially reduce research and development costs and time. Here we developed an embedded multi-core real-time simulation platform (EMC-RTP) for a biological-faithful computational network model of basal ganglia (BG). The single neuron model is implemented in a highly real-time manner using a reasonable simplification. A modular mapping architecture with hierarchical routing organization was constructed to mimic the pathological neural activities of BG observed in parkinsonian conditions. A closed-loop simulation testbed for DBS validation was then set up using a host computer as the DBS controller. The availability of EMC-RTP and the testbed system was validated by comparing the performance of open-loop and proportional-integral (PI) controllers. Our experimental results showed that the proposed EMC-RTP reproduces abnormal beta bursts of BG in parkinsonian conditions while meets requirements of both real-time and computational accuracy as well. Closed-loop DBS experiments using the EMC-RTP suggested that the platform could perform reasonable output under different kinds of DBS strategies, indicating the usability of the platform.

Methods for Lowering the Power Consumption of OS-Based Adaptive Deep Brain Stimulation Controllers

The identification of a new generation of adaptive strategies for deep brain stimulation (DBS) will require the development of mixed hardware-software systems for testing and implementing such controllers clinically. Towards this aim, introducing an operating system (OS) that provides high-level features (multitasking, hardware abstraction, and dynamic operation) as the core element of adaptive deep brain stimulation (aDBS) controllers could expand the capabilities and development speed of new control strategies. However, such software frameworks also introduce substantial power consumption overhead that could render this solution unfeasible for implantable devices. To address this, in this work four techniques to reduce this overhead are proposed and evaluated: a tick-less idle operation mode, reduced and dynamic sampling, buffered read mode, and duty cycling. A dual threshold adaptive deep brain stimulation algorithm for suppressing pathological oscillatory neural activity was implemented along with the proposed energy saving techniques on an energy-efficient OS, YetiOS, running on a STM32L476RE microcontroller. The system was then tested using an emulation environment coupled to a mean field model of the parkinsonian basal ganglia to simulate local field potential (LFPs) which acted as a biomarker for the controller. The OS-based controller alone introduced a power consumption overhead of 10.03 mW for a sampling rate of 1 kHz. This was reduced to 12 μW by applying the proposed tick-less idle mode, dynamic sampling, buffered read and duty cycling techniques. The OS-based controller using the proposed methods can facilitate rapid and flexible testing and implementation of new control methods. Furthermore, the approach has the potential to become a central element in future implantable devices to enable energy-efficient implementation of a wide range of control algorithms across different neurological conditions and hardware platforms.

Model-Based Closed-Loop Suppression of Parkinsonian Beta Band Oscillations Through Origin Analysis

Excessive beta band (13-30 Hz) oscillations have been observed in the basal ganglia (BG) of patients with Parkinson's disease (PD). Understanding the origin and transmission of beta band oscillations are important to improve treatments of PD, such as closed-loop deep brain stimulation (DBS). This paper proposed a model-based closed-loop GPi stimulation system to suppress pathological beta band oscillations of BG. The feedback nucleus was selected through the analysis of GPi oscillations variation when different synaptic currents were blocked, mainly projections from globus pallidus external (GPe), the subthalamic nucleus (STN) and striatum. Since simulation results proved the important role of synaptic current from GPe in shaping the excessive GPi beta band oscillations, the local field potential (LFP) of GPe was chosen as the feedback signal. That is to say, the feedback nucleus was selected based on the origin analysis of the pathological GPi beta band oscillation. The closed-loop algorithm was the multiplication of linear delayed feedback of the filtered GPe-LFP and modeled synaptic dynamics from GPe to GPi. Thus, the formed stimulation waveform was synaptic current like shape, which was proved to be more energy efficient than open-loop continuous DBS in suppressing GPi beta band oscillation. With the development of DBS devices, the efficiency of this closed-loop stimulation could be testified in animal model and clinical.

Long-Lasting Desynchronization of Plastic Neural Networks by Random Reset Stimulation

Excessive neuronal synchrony is a hallmark of neurological disorders such as epilepsy and Parkinson's disease. An established treatment for medically refractory Parkinson's disease is high-frequency (HF) deep brain stimulation (DBS). However, symptoms return shortly after cessation of HF-DBS. Recently developed decoupling stimulation approaches, such as Random Reset (RR) stimulation, specifically target pathological connections to achieve long-lasting desynchronization. During RR stimulation, a temporally and spatially randomized stimulus pattern is administered. However, spatial randomization, as presented so far, may be difficult to realize in a DBS-like setup due to insufficient spatial resolution. Motivated by recently developed segmented DBS electrodes with multiple stimulation sites, we present a RR stimulation protocol that copes with the limited spatial resolution of currently available depth electrodes for DBS. Specifically, spatial randomization is realized by delivering stimuli simultaneously to L randomly selected stimulation sites out of a total of M stimulation sites, which will be called L/M-RR stimulation. We study decoupling by L/M-RR stimulation in networks of excitatory integrate-and-fire neurons with spike-timing dependent plasticity by means of theoretical and computational analysis. We find that L/M-RR stimulation yields parameter-robust decoupling and long-lasting desynchronization. Furthermore, our theory reveals that strong high-frequency stimulation is not suitable for inducing long-lasting desynchronization effects. As a consequence, low and high frequency L/M-RR stimulation affect synaptic weights in qualitatively different ways. Our simulations confirm these predictions and show that qualitative differences between low and high frequency L/M-RR stimulation are present across a wide range of stimulation parameters, rendering stimulation with intermediate frequencies most efficient. Remarkably, we find that L/M-RR stimulation does not rely on a high spatial resolution, characterized by the density of stimulation sites in a target area, corresponding to a large M. In fact, L/M-RR stimulation with low resolution performs even better at low stimulation amplitudes. Our results provide computational evidence that L/M-RR stimulation may present a way to exploit modern segmented lead electrodes for long-lasting therapeutic effects.

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.

Multifunctionality in a reservoir computer

Multifunctionality is a well observed phenomenological feature of biological neural networks and considered to be of fundamental importance to the survival of certain species over time. These multifunctional neural networks are capable of performing more than one task without changing any network connections. In this paper, we investigate how this neurological idiosyncrasy can be achieved in an artificial setting with a modern machine learning paradigm known as "reservoir computing." A training technique is designed to enable a reservoir computer to perform tasks of a multifunctional nature. We explore the critical effects that changes in certain parameters can have on the reservoir computers' ability to express multifunctionality. We also expose the existence of several "untrained attractors"; attractors that dwell within the prediction state space of the reservoir computer were not part of the training. We conduct a bifurcation analysis of these untrained attractors and discuss the implications of our results.

Dynamics of multiple interacting excitatory and inhibitory populations with delays

A network consisting of excitatory and inhibitory (EI) neurons is a canonical model for understanding local cortical network activity. In this study, we extended the local circuit model and investigated how its dynamical landscape can be enriched when it interacts with another excitatory (E) population with long transmission delays. Through analysis of a rate model and numerical simulations of a corresponding network of spiking neurons, we studied the transition from stationary to oscillatory states by analyzing the Hopf bifurcation structure in terms of two network parameters: (1) transmission delay between the EI subnetwork and the E population and (2) inhibitory couplings that induced oscillatory activity in the EI subnetwork. We found that the critical coupling strength can strongly modulate as a function of transmission delay, and consequently the stationary state can be interwoven intricately with the oscillatory state. Such a dynamical landscape gave rise to an isolated stationary state surrounded by multiple oscillatory states that generated different frequency modes, and cross-frequency coupling developed naturally at the bifurcation points. We identified the network motifs with short- and long-range inhibitory connections that underlie the emergence of oscillatory states with multiple frequencies. Thus, we provided a mechanistic explanation of how the transmission delay to and from the additional E population altered the dynamical landscape. In summary, our results demonstrated the potential role of long-range connections in shaping the network activity of local cortical circuits.

Coupled Möbius maps as a tool to model Kuramoto phase synchronization

We propose Möbius maps as a tool to model synchronization phenomena in coupled phase oscillators. Not only does the map provide fast computation of phase synchronization, it also reflects the underlying group structure of the sinusoidally coupled continuous phase dynamics. We study map versions of various known continuous-time collective dynamics, such as the synchronization transition in the Kuramoto-Sakaguchi model of nonidentical oscillators, chimeras in two coupled populations of identical phase oscillators, and Kuramoto-Battogtokh chimeras on a ring, and demonstrate similarities and differences between the iterated map models and their known continuous-time counterparts.

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.

Facilitating the propagation of spiking activity in feedforward networks by including feedback

Transient oscillations in network activity upon sensory stimulation have been reported in different sensory areas of the brain. These evoked oscillations are the generic response of networks of excitatory and inhibitory neurons (EI-networks) to a transient external input. Recently, it has been shown that this resonance property of EI-networks can be exploited for communication in modular neuronal networks by enabling the transmission of sequences of synchronous spike volleys (’pulse packets’), despite the sparse and weak connectivity between the modules. The condition for successful transmission is that the pulse packet (PP) intervals match the period of the modules’ resonance frequency. Hence, the mechanism was termed communication through resonance (CTR). This mechanism has three severe constraints, though. First, it needs periodic trains of PPs, whereas single PPs fail to propagate. Second, the inter-PP interval needs to match the network resonance. Third, transmission is very slow, because in each module, the network resonance needs to build up over multiple oscillation cycles. Here, we show that, by adding appropriate feedback connections to the network, the CTR mechanism can be improved and the aforementioned constraints relaxed. Specifically, we show that adding feedback connections between two upstream modules, called the resonance pair, in an otherwise feedforward modular network can support successful propagation of a single PP throughout the entire network. The key condition for successful transmission is that the sum of the forward and backward delays in the resonance pair matches the resonance frequency of the network modules. The transmission is much faster, by more than a factor of two, than in the original CTR mechanism. Moreover, it distinctly lowers the threshold for successful communication by synchronous spiking in modular networks of weakly coupled networks. Thus, our results suggest a new functional role of bidirectional connectivity for the communication in cortical area networks.

The cortex is organized as a modular system, with the modules (cortical areas) communicating via weak long-range connections. It has been suggested that the intrinsic resonance properties of population activities in these areas might contribute to enabling successful communication. A module’s intrinsic resonance appears in the damped oscillatory response to an incoming spike volley, enabling successful communication during the peaks of the oscillation. Such communication can be exploited in feedforward networks, provided the participating networks have similar resonance frequencies. This, however, is not necessarily true for cortical networks. Moreover, the communication is slow, as it takes several oscillation cycles to build up the response in the downstream network. Also, only periodic trains of spikes volleys (and not single volleys) with matching intervals can propagate. Here, we present a novel mechanism that alleviates these shortcomings and enables propagation of synchronous spiking across weakly connected networks with not necessarily identical resonance frequencies. In this framework, an individual spike volley can propagate by local amplification through reverberation in a loop between two successive networks, connected by feedforward and feedback connections: the resonance pair. This overcomes the need for activity build-up in downstream networks, causing the volley to propagate distinctly faster and more reliably.

Impact of number of stimulation sites on long-lasting desynchronization effects of coordinated reset stimulation

Excessive neuronal synchrony is a hallmark of several neurological disorders, e.g., Parkinson's disease. An established treatment for medically refractory Parkinson's disease is high-frequency deep brain stimulation. However, it provides only acute relief, and symptoms return shortly after cessation of stimulation. A theory-based approach called coordinated reset (CR) has shown great promise in achieving long-lasting effects. During CR stimulation, phase-shifted stimuli are delivered to multiple stimulation sites to counteract neuronal synchrony. Computational studies in plastic neuronal networks reported that synaptic weights reduce during stimulation, which may cause sustained structural changes leading to stabilized desynchronized activity even after stimulation ceases. Corresponding long-lasting effects were found in recent preclinical and clinical studies. We study long-lasting desynchronization by CR stimulation in excitatory recurrent neuronal networks of integrate-and-fire neurons with spike-timing-dependent plasticity (STDP). We focus on the impact of the stimulation frequency and the number of stimulation sites on long-lasting effects. We compare theoretical predictions to simulations of plastic neuronal networks. Our results are important regarding CR calibration for two reasons. We reveal that long-lasting effects become most pronounced when stimulation parameters are adjusted to the characteristics of STDP-rather than to neuronal frequency characteristics. This is in contrast to previous studies where the CR frequency was adjusted to the dominant neuronal rhythm. In addition, we reveal a nonlinear dependence of long-lasting effects on the number of stimulation sites and the CR frequency. Intriguingly, optimal long-lasting desynchronization does not require larger numbers of stimulation sites.

Global phase-amplitude description of oscillatory dynamics via the parameterization method

In this paper, we use the parameterization method to provide a complete description of the dynamics of an n-dimensional oscillator beyond the classical phase reduction. The parameterization method allows us, via efficient algorithms, to obtain a parameterization of the attracting invariant manifold of the limit cycle in terms of the phase-amplitude variables. The method has several advantages. It provides analytically a Fourier-Taylor expansion of the parameterization up to any order, as well as a simplification of the dynamics that allows for a numerical globalization of the manifolds. Thus, one can obtain the local and global isochrons and isostables, including the slow attracting manifold, up to high accuracy, which offer a geometrical portrait of the oscillatory dynamics. Furthermore, it provides straightforwardly the infinitesimal phase and amplitude response functions, that is, the extended infinitesimal phase and amplitude response curves, which monitor the phase and amplitude shifts beyond the asymptotic state. Thus, the methodology presented yields an accurate description of the phase dynamics for perturbations not restricted to the limit cycle but to its attracting invariant manifold. Finally, we explore some strategies to reduce the dimension of the dynamics, including the reduction of the dynamics to the slow stable submanifold. We illustrate our methods by applying them to different three-dimensional single neuron and neural population models in neuroscience.

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.

Adaptive delivery of continuous and delayed feedback deep brain stimulation - a computational study

Adaptive deep brain stimulation (aDBS) is a closed-loop method, where high-frequency DBS is turned on and off according to a feedback signal, whereas conventional high-frequency DBS (cDBS) is delivered permanently. Using a computational model of subthalamic nucleus and external globus pallidus, we extend the concept of adaptive stimulation by adaptively controlling not only continuous, but also demand-controlled stimulation. Apart from aDBS and cDBS, we consider continuous pulsatile linear delayed feedback stimulation (cpLDF), specifically designed to induce desynchronization. Additionally, we combine adaptive on-off delivery with continuous delayed feedback modulation by introducing adaptive pulsatile linear delayed feedback stimulation (apLDF), where cpLDF is turned on and off using pre-defined amplitude thresholds. By varying the stimulation parameters of cDBS, aDBS, cpLDF, and apLDF we obtain optimal parameter ranges. We reveal a simple relation between the thresholds of the local field potential (LFP) for aDBS and apLDF, the extent of the stimulation-induced desynchronization, and the integral stimulation time required. We find that aDBS and apLDF can be more efficient in suppressing abnormal synchronization than continuous simulation. However, apLDF still remains more efficient and also causes a stronger reduction of the LFP beta burst length. Hence, adaptive on-off delivery may further improve the intrinsically demand-controlled pLDF.

Suppression of Phase Synchronization in Scale-Free Neural Networks Using External Pulsed Current Protocols

The synchronization of neurons is fundamental for the functioning of the brain since its lack or excess may be related to neurological disorders, such as autism, Parkinson’s and neuropathies such as epilepsy. In this way, the study of synchronization, as well as its suppression in coupled neurons systems, consists of an important multidisciplinary research field where there are still questions to be answered. Here, through mathematical modeling and numerical approach, we simulated a neural network composed of 5000 bursting neurons in a scale-free connection scheme where non-trivial synchronization phenomenon is observed. We proposed two different protocols to the suppression of phase synchronization, which is related to deep brain stimulation and delayed feedback control. Through an optimization process, it is possible to suppression the abnormal synchronization in the neural network.

What Can Computational Models Contribute to Neuroimaging Data Analytics?

Over the past years, nonlinear dynamical models have significantly contributed to the general understanding of brain activity as well as brain disorders. Appropriately validated and optimized mathematical models can be used to mechanistically explain properties of brain structure and neuronal dynamics observed from neuroimaging data. A thorough exploration of the model parameter space and hypothesis testing with the methods of nonlinear dynamical systems and statistical physics can assist in classification and prediction of brain states. On the one hand, such a detailed investigation and systematic parameter variation are hardly feasible in experiments and data analysis. On the other hand, the model-based approach can establish a link between empirically discovered phenomena and more abstract concepts of attractors, multistability, bifurcations, synchronization, noise-induced dynamics, etc. Such a mathematical description allows to compare and differentiate brain structure and dynamics in health and disease, such that model parameters and dynamical regimes may serve as additional biomarkers of brain states and behavioral modes. In this perspective paper we first provide very brief overview of the recent progress and some open problems in neuroimaging data analytics with emphasis on the resting state brain activity. We then focus on a few recent contributions of mathematical modeling to our understanding of the brain dynamics and model-based approaches in medicine. Finally, we discuss the question stated in the title. We conclude that incorporating computational models in neuroimaging data analytics as well as in translational medicine could significantly contribute to the progress in these fields.

Bayesian adaptive dual control of deep brain stimulation in a computational model of Parkinson's disease

In this paper, we present a novel Bayesian adaptive dual controller (ADC) for autonomously programming deep brain stimulation devices. We evaluated the Bayesian ADC's performance in the context of reducing beta power in a computational model of Parkinson's disease, in which it was tasked with finding the set of stimulation parameters which optimally reduced beta power as fast as possible. Here, the Bayesian ADC has dual goals: (a) to minimize beta power by exploiting the best parameters found so far, and (b) to explore the space to find better parameters, thus allowing for better control in the future. The Bayesian ADC is composed of two parts: an inner parameterized feedback stimulator and an outer parameter adjustment loop. The inner loop operates on a short time scale, delivering stimulus based upon the phase and power of the beta oscillation. The outer loop operates on a long time scale, observing the effects of the stimulation parameters and using Bayesian optimization to intelligently select new parameters to minimize the beta power. We show that the Bayesian ADC can efficiently optimize stimulation parameters, and is superior to other optimization algorithms. The Bayesian ADC provides a robust and general framework for tuning stimulation parameters, can be adapted to use any feedback signal, and is applicable across diseases and stimulator designs.

Activity patterns with silent states in a heterogeneous network of gap-junction coupled Huber-Braun model neurons

A mathematical model of a network of nearest neighbor gap-junction coupled neurons has been used to examine the impact of neuronal heterogeneity on the networks' activity during increasing coupling strength. Heterogeneity has been introduced by Huber-Braun model neurons with randomization of the temperature as a scaling factor. This leads to neurons of an enormous diversity of impulse pattern, including burst discharges, chaotic activity, and two different types of tonic firing-all of them experimentally observed in the peripheral as well as central nervous system. When the network is composed of all these types of neurons, randomly selected, a particular phenomenon can be observed. At a certain coupling strength, the network goes into a completely silent state. Examination of voltage traces and inter-spike intervals of individual neurons suggests that all neurons, irrespective of their original pattern, go through a well-known bifurcation scenario, resembling those of single neurons especially on external current injection. All the originally spontaneously firing neurons can achieve constant membrane potentials at which all intrinsic and gap-junction currents are balanced. With limited diversity, i.e., taking out neurons of specific patterns from the lower and upper temperature range, spontaneous firing can be reinstalled with further increasing coupling strength, especially when the tonic firing regimes are missing. Reinstalled firing develops from slowly increasing subthreshold oscillations leading to tonic firing activity with already fairly well synchronized action potentials, while the subthreshold potentials can still be significantly different. Full in phase synchronization is not achieved. Additional studies are needed elucidating the underlying mechanisms and the conditions under which such particular transitions can appear.

Suppressing bursting synchronization in a modular neuronal network with synaptic plasticity

Excessive synchronization of neurons in cerebral cortex is believed to play a crucial role in the emergence of neuropsychological disorders such as Parkinson's disease, epilepsy and essential tremor. This study, by constructing a modular neuronal network with modified Oja's learning rule, explores how to eliminate the pathological synchronized rhythm of interacted busting neurons numerically. When all neurons in the modular neuronal network are strongly synchronous within a specific range of coupling strength, the result reveals that synaptic plasticity with large learning rate can suppress bursting synchronization effectively. For the relative small learning rate not capable of suppressing synchronization, the technique of nonlinear delayed feedback control including differential feedback control and direct feedback control is further proposed to reduce the synchronized bursting state of coupled neurons. It is demonstrated that the two kinds of nonlinear feedback control can eliminate bursting synchronization significantly when the control parameters of feedback strength and feedback delay are appropriately tuned. For the former control technique, the control domain of effective synchronization suppression is similar to a semi-elliptical domain in the simulated parameter space of feedback strength and feedback delay, while for the latter one, the effective control domain is similar to a fan-shaped domain in the simulated parameter space.

The emergence of multistability and chaos in a two-mode van der Pol generator versus different connection types of linear oscillators

In this work, we study the multistability and chaos phenomena in a classical two-mode van der Pol generator which consists of a nonlinear element and two linear oscillators. We show that the configuration of the connections of the linear oscillators in the two-mode self-oscillating system significantly affects its oscillation regimes and bifurcational transitions. In the case of the feedback loop including one oscillator, the two-mode system demonstrates the well-known effect of frequency entrainment, including bistability and hysteresis phenomena. If the feedback loop involves both linear oscillators, the entrainment effect disappears; however, two new complex regimes of quasi-periodicity and chaotic self-oscillations emerge. We present here the results of the bifurcation analysis of the multistability formation and transition to chaos.

Phase model-based neuron stabilization into arbitrary clusters

Deep brain stimulation (DBS) is a common method of combating pathological conditions associated with Parkinson's disease, Tourette syndrome, essential tremor, and other disorders, but whose mechanisms are not fully understood. One hypothesis, supported experimentally, is that some symptoms of these disorders are associated with pathological synchronization of neurons in the basal ganglia and thalamus. For this reason, there has been interest in recent years in finding efficient ways to desynchronize neurons that are both fast-acting and low-power. Recent results on coordinated reset and periodically forced oscillators suggest that forming distinct clusters of neurons may prove to be more effective than achieving complete desynchronization, in particular by promoting plasticity effects that might persist after stimulation is turned off. Current proposed methods for achieving clustering frequently require either multiple input sources or precomputing the control signal. We propose here a control strategy for clustering, based on an analysis of the reduced phase model for a set of identical neurons, that allows for real-time, single-input control of a population of neurons with low-amplitude, low total energy signals. After demonstrating its effectiveness on phase models, we apply it to full state models to demonstrate its validity. We also discuss the effects of coupling on the efficacy of the strategy proposed and demonstrate that the clustering can still be accomplished in the presence of weak to moderate electrotonic coupling.

Toward adaptive deep brain stimulation in Parkinson's disease: a review

Clinical deep brain stimulation (DBS) is now regarded as the therapeutic intervention of choice at the advanced stages of Parkinson's disease. However, some major challenges of DBS are stimulation induced side effects and limited pacemaker battery life. Side effects and shortening of pacemaker battery life are mainly as a result of continuous stimulation and poor stimulation focus. These drawbacks can be mitigated using adaptive DBS (aDBS) schemes. Side effects resulting from continuous stimulation can be reduced through adaptive control using closed-loop feedback, while those due to poor stimulation focus can be mitigated through spatial adaptation. Other advantages of aDBS include automatic, rather than manual, initial adjustment and programming, and long-term adjustments to maintain stimulation parameters with changes in patient's condition. Both result in improved efficacy. This review focuses on the major areas that are essential in driving technological advances for the various aDBS schemes. Their challenges, prospects and progress so far are analyzed. In addition, important advances and milestones in state-of-the-art aDBS schemes are highlighted – both for closed-loop adaption and spatial adaption. With perspectives and future potentials of DBS provided at the end.

Predicting seizure by modeling synaptic plasticity based on EEG signals - a case study of inherited epilepsy

This paper explores the internal dynamical mechanisms of epileptic seizures through quantitative modeling based on full brain electroencephalogram (EEG) signals. Our goal is to provide seizure prediction and facilitate treatment for epileptic patients. Motivated by an earlier mathematical model with incorporated synaptic plasticity, we studied the nonlinear dynamics of inherited seizures through a differential equation model. First, driven by a set of clinical inherited electroencephalogram data recorded from a patient with diagnosed Glucose Transporter Deficiency, we developed a dynamic seizure model on a system of ordinary differential equations. The model was reduced in complexity after considering and removing redundancy of each EEG channel. Then we verified that the proposed model produces qualitatively relevant behavior which matches the basic experimental observations of inherited seizure, including synchronization index and frequency. Meanwhile, the rationality of the connectivity structure hypothesis in the modeling process was verified. Further, through varying the threshold condition and excitation strength of synaptic plasticity, we elucidated the effect of synaptic plasticity to our seizure model. Results suggest that synaptic plasticity has great effect on the duration of seizure activities, which support the plausibility of therapeutic interventions for seizure control.

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.

The use of intermuscular coherence analysis as a novel approach to detect age-related changes on postural muscle synergy

The overall goal of this study was to investigate potential adaptations brought about by the natural processes of aging on the coordination of postural muscles. Considering the progressive and non-homogeneous deterioration of sensorimotor and neuromuscular systems as the individual grows older, it was hypothesized that aging is associated with a reorganization of synergistic mechanisms controlling postural muscles. Therefore, the presence, distribution, and strength of correlated neural inputs to three posterior postural muscles were measured by intermuscular coherence estimations at a low frequency band (0-55Hz). Nine healthy young adults and thirteen healthy older adults performed ten trials of a perturbed task: bipedal stance while holding a five kg load for fifteen seconds. Estimates of intermuscular coherence for each pair of electromyographic signals (soleus and biceps femoris, soleus and erector spinae, and biceps femoris and erector spinae) were computed. Results revealed significantly stronger levels of synchronization of posterior muscles within 0-10Hz in seniors compared to young adults. In addition, seniors presented similar spectra of intermuscular coherence within 0-55Hz for all three muscle pairs analyzed. These findings provide valuable information regarding compensatory mechanisms adopted by older adults to control balance. The age-related reorganization of neural drive controlling posterior postural muscles revealing a stronger synchronization within 0-10Hz might be related to the faster body sway and muscle co-activation patterns usually observed in this population. Finally, this study supports the use of Intermuscular Coherence Analysis as a sensitive method to detect age-related changes in multi-muscle control.

Advances in closed-loop deep brain stimulation devices

BACKGROUND

Millions of patients around the world are affected by neurological and psychiatric disorders. Deep brain stimulation (DBS) is a device-based therapy that could have fewer side-effects and higher efficiencies in drug-resistant patients compared to other therapeutic options such as pharmacological approaches. Thus far, several efforts have been made to incorporate a feedback loop into DBS devices to make them operate in a closed-loop manner.

METHODS

This paper presents a comprehensive investigation into the existing research-based and commercial closed-loop DBS devices. It describes a brief history of closed-loop DBS techniques, biomarkers and algorithms used for closing the feedback loop, components of the current research-based and commercial closed-loop DBS devices, and advancements and challenges in this field of research. This review also includes a comparison of the closed-loop DBS devices and provides the future directions of this area of research.

RESULTS

Although we are in the early stages of the closed-loop DBS approach, there have been fruitful efforts in design and development of closed-loop DBS devices. To date, only one commercial closed-loop DBS device has been manufactured. However, this system does not have an intelligent and patient dependent control algorithm. A closed-loop DBS device requires a control algorithm to learn and optimize the stimulation parameters according to the brain clinical state.

CONCLUSIONS

The promising clinical effects of open-loop DBS have been demonstrated, indicating DBS as a pioneer technology and treatment option to serve neurological patients. However, like other commercial devices, DBS needs to be automated and modernized.

Dynamical responses to external stimuli for both cases of excitatory and inhibitory synchronization in a complex neuronal network

For studying how dynamical responses to external stimuli depend on the synaptic-coupling type, we consider two types of excitatory and inhibitory synchronization (i.e., synchronization via synaptic excitation and inhibition) in complex small-world networks of excitatory regular spiking (RS) pyramidal neurons and inhibitory fast spiking (FS) interneurons. For both cases of excitatory and inhibitory synchronization, effects of synaptic couplings on dynamical responses to external time-periodic stimuli S(t) (applied to a fraction of neurons) are investigated by varying the driving amplitude A of S(t). Stimulated neurons are phase-locked to external stimuli for both cases of excitatory and inhibitory couplings. On the other hand, the stimulation effect on non-stimulated neurons depends on the type of synaptic coupling. The external stimulus S(t) makes a constructive effect on excitatory non-stimulated RS neurons (i.e., it causes external phase lockings in the non-stimulated sub-population), while S(t) makes a destructive effect on inhibitory non-stimulated FS interneurons (i.e., it breaks up original inhibitory synchronization in the non-stimulated sub-population). As results of these different effects of S(t), the type and degree of dynamical response (e.g., synchronization enhancement or suppression), characterized by the dynamical response factor [Formula: see text] (given by the ratio of synchronization degree in the presence and absence of stimulus), are found to vary in a distinctly different way, depending on the synaptic-coupling type. Furthermore, we also measure the matching degree between the dynamics of the two sub-populations of stimulated and non-stimulated neurons in terms of a "cross-correlation" measure [Formula: see text]. With increasing A, based on [Formula: see text], we discuss the cross-correlations between the two sub-populations, affecting the dynamical responses to S(t).

Behavioral synchronization induced by epidemic spread in complex networks

During the spread of an epidemic, individuals in realistic networks may exhibit collective behaviors. In order to characterize this kind of phenomenon and explore the correlation between collective behaviors and epidemic spread, in this paper, we construct several mathematical models (including without delay, with a coupling delay, and with double delays) of epidemic synchronization by applying the adaptive feedback motivated by real observations. By using Lyapunov function methods, we obtain the conditions for local and global stability of these epidemic synchronization models. Then, we illustrate that quenched mean-field theory is more accurate than heterogeneous mean-field theory in the prediction of epidemic synchronization. Finally, some numerical simulations are performed to complement our theoretical results, which also reveal some unexpected phenomena, for example, the coupling delay and epidemic delay influence the speed of epidemic synchronization. This work makes further exploration on the relationship between epidemic dynamics and synchronization dynamics, in the hope of being helpful to the study of other dynamical phenomena in the process of epidemic spread.

Dynamics of oscillators globally coupled via two mean fields

Many studies of synchronization properties of coupled oscillators, based on the classical Kuramoto approach, focus on ensembles coupled via a mean field. Here we introduce a setup of Kuramoto-type phase oscillators coupled via two mean fields. We derive stability properties of the incoherent state and find traveling wave solutions with different locking patterns; stability properties of these waves are found numerically. Mostly nontrivial states appear when the two fields compete, i.e. one tends to synchronize oscillators while the other one desynchronizes them. Here we identify normal branches which bifurcate from the incoherent state in a usual way, and anomalous branches, appearance of which cannot be described as a bifurcation. Furthermore, hybrid branches combining properties of both are described. In the situations where no stable traveling wave exists, modulated quasiperiodic in time dynamics is observed. Our results indicate that a competition between two coupling channels can lead to a complex system behavior, providing a potential generalized framework for understanding of complex phenomena in natural oscillatory systems.

Emergence of ultrafast sparsely synchronized rhythms and their responses to external stimuli in an inhomogeneous small-world complex neuronal network

We consider an inhomogeneous small-world network (SWN) composed of inhibitory short-range (SR) and long-range (LR) interneurons, and investigate the effect of network architecture on emergence of synchronized brain rhythms by varying the fraction of LR interneurons p_long. The betweenness centralities of the LR and SR interneurons (characterizing the potentiality in controlling communication between other interneurons) are distinctly different. Hence, in view of the betweenness, SWNs we consider are inhomogeneous, unlike the "canonical" Watts-Strogatz SWN with nearly the same betweenness centralities. For small p_long, the load of communication traffic is much concentrated on a few LR interneurons. However, as p_long is increased, the number of LR connections (coming from LR interneurons) increases, and then the load of communication traffic is less concentrated on LR interneurons, which leads to better efficiency of global communication between interneurons. Sparsely synchronized rhythms are thus found to emerge when passing a small critical value p_long^(c)(≃0.16). The population frequency of the sparsely synchronized rhythm is ultrafast (higher than 100 Hz), while the mean firing rate of individual interneurons is much lower (∼30 Hz) due to stochastic and intermittent neural discharges. These dynamical behaviors in the inhomogeneous SWN are also compared with those in the homogeneous Watts-Strogatz SWN, in connection with their network topologies. Particularly, we note that the main difference between the two types of SWNs lies in the distribution of betweenness centralities. Unlike the case of the Watts-Strogatz SWN, dynamical responses to external stimuli vary depending on the type of stimulated interneurons in the inhomogeneous SWN. We consider two cases of external time-periodic stimuli applied to sub-populations of the LR and SR interneurons, respectively. Dynamical responses (such as synchronization suppression and enhancement) to these two cases of stimuli are studied and discussed in relation to the betweenness centralities of stimulated interneurons, representing the effectiveness for transfer of stimulation effect in the whole network.

Closed-loop deep brain stimulation by pulsatile delayed feedback with increased gap between pulse phases

Computationally it was shown that desynchronizing delayed feedback stimulation methods are effective closed-loop techniques for the control of synchronization in ensembles of interacting oscillators. We here computationally design stimulation signals for electrical stimulation of neuronal tissue that preserve the desynchronizing delayed feedback characteristics and comply with mandatory charge deposit-related safety requirements. For this, the amplitude of the high-frequency (HF) train of biphasic charge-balanced pulses used by the standard HF deep brain stimulation (DBS) is modulated by the smooth feedback signals. In this way we combine the desynchronizing delayed feedback approach with the HF DBS technique. We show that such a pulsatile delayed feedback stimulation can effectively and robustly desynchronize a network of model neurons comprising subthalamic nucleus and globus pallidus external and suggest this approach for desynchronizing closed-loop DBS. Intriguingly, an interphase gap introduced between the recharging phases of the charge-balanced biphasic pulses can significantly improve the stimulation-induced desynchronization and reduce the amount of the administered stimulation. In view of the recent experimental and clinical studies indicating a superiority of the closed-loop DBS to open-loop HF DBS, our results may contribute to a further development of effective stimulation methods for the treatment of neurological disorders characterized by abnormal neuronal synchronization.