Controlling synchronization in an ensemble of globally coupled oscillators

External low energy electromagnetic fields affect heart dynamics: surrogate for system synchronization, chaos control and cancer patient's health

All cells in the human body, including cancer cells, possess specific electrical properties crucial for their functions. These properties are notably different between normal and cancerous cells. Cancer cells are characterized by autonomous oscillations and damped electromagnetic field (EMF) activation. Cancer reduces physiological variability, implying a systemic disconnection that desynchronizes bodily systems and their inherent random processes. The dynamics of heart rate, in this context, could reflect global physiological network instability in the sense of entrainment. Using a medical device that employs an active closed-loop system, such as administering specifically modulated EMF frequencies at targeted intervals and at low energies, we can evaluate the periodic oscillations of the heart. This procedure serves as a closed-loop control mechanism leading to a temporary alteration in plasma membrane ionic flow and the heart's periodic oscillation dynamics. The understanding of this phenomenon is supported by computer simulations of a mathematical model, which are validated by experimental data. Heart dynamics can be quantified using difference logistic equations, and it correlates with improved overall survival rates in cancer patients.

Simple model for the prediction of seizure durations

A simple model is used to simulate seizures in a population of spiking excitatory neurons experiencing a uniform effect from inhibitory neurons. A key feature is introduced into the model, i.e., a mechanism that weakens the firing thresholds. This weakening mechanism adds memory to the dynamics. We find a seizure-prone state in a "mode-switching" phase. In this phase, the system can suddenly switch from a "healthy" state with small scale-free avalanches to a "seizure" state with almost periodic large avalanches ("seizures"). Simulations of the model predict statistics for the average time spent in the seizure state (the seizure "duration") that agree with experiments and theoretical examples of similar behavior in neuronal systems. Our study points to. different connections between seizures and fracture and also offers an alternative view on the type of critical point controlling neuronal avalanches.

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.

Controlling neural activity: LPV modelling of optogenetically actuated Wilson-Cowan model. J Neural Eng 2024;21:036002. [PMID: 38653250 DOI: 10.1088/1741-2552/ad4212] [Citation(s) in RCA: 0] [Impact Index Per Article: 0]

Objective.This paper aims to bridge the gap between neurophysiology and automatic control methodologies by redefining the Wilson-Cowan (WC) model as a control-oriented linear parameter-varying (LPV) system. A novel approach is presented that allows for the application of a control strategy to modulate and track neural activity.Approach.The WC model is redefined as a control-oriented LPV system in this study. The LPV modelling framework is leveraged to design an LPV controller, which is used to regulate and manipulate neural dynamics.Main results.Promising outcomes, in understanding and controlling neural processes through the synergistic combination of control-oriented modelling and estimation, are obtained in this study. An LPV controller demonstrates to be effective in regulating neural activity.Significance.The presented methodology effectively induces neural patterns, taking into account optogenetic actuation. The combination of control strategies with neurophysiology provides valuable insights into neural dynamics. The proposed approach opens up new possibilities for using control techniques to study and influence brain functions, which can have key implications in neuroscience and medicine. By means of a model-based controller which accounts for non-linearities, noise and uncertainty, neural signals can be induced on brain structures.

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.

Coalescence of limit cycles in the presence of noise

Complex dynamical systems may exhibit multiple steady states, including time-periodic limit cycles, where the final trajectory depends on initial conditions. With tuning of parameters, limit cycles can proliferate or merge at an exceptional point. Here we ask how dynamics in the vicinity of such a bifurcation are influenced by noise. A pitchfork bifurcation can be used to induce bifurcation behavior. We model a limit cycle with the normal form of the Hopf oscillator, couple it to the pitchfork, and investigate the resulting dynamical system in the presence of noise. We show that the generating functional for the averages of the dynamical variables factorizes between the pitchfork and the oscillator. The statistical properties of the pitchfork in the presence of noise in its various regimes are investigated and a scaling theory is developed for the correlation and response functions, including a possible symmetry-breaking field. The analysis is done by perturbative calculations as well as numerical means. Finally, observables illustrating the coupling of a system with a limit cycle to a pitchfork are discussed and the phase-phase correlations are shown to exhibit nondiffusive behavior with universal scaling.

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.

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.

Control of coupled neural oscillations using near-periodic inputs

Deep brain stimulation (DBS) is a commonly used treatment for medication resistant Parkinson's disease and is an emerging treatment for other neurological disorders. More recently, phase-specific adaptive DBS (aDBS), whereby the application of stimulation is locked to a particular phase of tremor, has been proposed as a strategy to improve therapeutic efficacy and decrease side effects. In this work, in the context of these phase-specific aDBS strategies, we investigate the dynamical behavior of large populations of coupled neurons in response to near-periodic stimulation, namely, stimulation that is periodic except for a slowly changing amplitude and phase offset that can be used to coordinate the timing of applied input with a specified phase of model oscillations. Using an adaptive phase-amplitude reduction strategy, we illustrate that for a large population of oscillatory neurons, the temporal evolution of the associated phase distribution in response to near-periodic forcing can be captured using a reduced order model with four state variables. Subsequently, we devise and validate a closed-loop control strategy to disrupt synchronization caused by coupling. Additionally, we identify strategies for implementing the proposed control strategy in situations where underlying model equations are unavailable by estimating the necessary terms of the reduced order equations in real-time from observables.

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.

Time-delayed Kuramoto model in the Watts-Strogatz small-world networks

We study the synchronization of small-world networks of identical coupled phase oscillators through the Kuramoto interaction and uniform time delay. For a given intrinsic frequency and coupling constant, we observe synchronization enhancement in a range of time delays and discontinuous transition from the partially synchronized state with defect patterns to a glassy phase, characterized by a distribution of randomly frozen phase-locked oscillators. By further increasing the time delay, this phase undergoes a discontinuous transition to another partially synchronized state. We found the bimodal frequency distributions and hysteresis loops as indicators of the discontinuous nature of these transitions. Moreover, we found the existence of Chimera states at the onset of transitions.

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.

Suppression of synchronous spiking in two interacting populations of excitatory and inhibitory quadratic integrate-and-fire neurons

Collective oscillations and their suppression by external stimulation are analyzed in a large-scale neural network consisting of two interacting populations of excitatory and inhibitory quadratic integrate-and-fire neurons. In the limit of an infinite number of neurons, the microscopic model of this network can be reduced to an exact low-dimensional system of mean-field equations. Bifurcation analysis of these equations reveals three different dynamic modes in a free network: a stable resting state, a stable limit cycle, and bistability with a coexisting resting state and a limit cycle. We show that in the limit cycle mode, high-frequency stimulation of an inhibitory population can stabilize an unstable resting state and effectively suppress collective oscillations. We also show that in the bistable mode, the dynamics of the network can be switched from a stable limit cycle to a stable resting state by applying an inhibitory pulse to the excitatory population. The results obtained from the mean-field equations are confirmed by numerical simulation of the microscopic model.

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.

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.

Deep brain stimulation: a review of the open neural engineering challenges

OBJECTIVE

Deep brain stimulation (DBS) is an established and valid therapy for a variety of pathological conditions ranging from motor to cognitive disorders. Still, much of the DBS-related mechanism of action is far from being understood, and there are several side effects of DBS whose origin is unclear. In the last years DBS limitations have been tackled by a variety of approaches, including adaptive deep brain stimulation (aDBS), a technique that relies on using chronically implanted electrodes on 'sensing mode' to detect the neural markers of specific motor symptoms and to deliver on-demand or modulate the stimulation parameters accordingly. Here we will review the state of the art of the several approaches to improve DBS and summarize the main challenges toward the development of an effective aDBS therapy.

APPROACH

We discuss models of basal ganglia disorders pathogenesis, hardware and software improvements for conventional DBS, and candidate neural and non-neural features and related control strategies for aDBS.

MAIN RESULTS

We identify then the main operative challenges toward optimal DBS such as (i) accurate target localization, (ii) increased spatial resolution of stimulation, (iii) development of in silico tests for DBS, (iv) identification of specific motor symptoms biomarkers, in particular (v) assessing how LFP oscillations relate to behavioral disfunctions, and (vi) clarify how stimulation affects the cortico-basal-ganglia-thalamic network to (vii) design optimal stimulation patterns.

SIGNIFICANCE

This roadmap will lead neural engineers novel to the field toward the most relevant open issues of DBS, while the in-depth readers might find a careful comparison of advantages and drawbacks of the most recent attempts to improve DBS-related neuromodulatory strategies.

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.

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.

Nonstationary coherence-incoherence patterns in nonlocally coupled heterogeneous phase oscillators

We consider a large ring of nonlocally coupled phase oscillators and show that apart from stationary chimera states, this system also supports nonstationary coherence-incoherence patterns (CIPs). For identical oscillators, these CIPs behave as breathing chimera states and are found in a relatively small parameter region only. It turns out that the stability region of these states enlarges dramatically if a certain amount of spatially uniform heterogeneity (e.g., Lorentzian distribution of natural frequencies) is introduced in the system. In this case, nonstationary CIPs can be studied as stable quasiperiodic solutions of a corresponding mean-field equation, formally describing the infinite system limit. Carrying out direct numerical simulations of the mean-field equation, we find different types of nonstationary CIPs with pulsing and/or alternating chimera-like behavior. Moreover, we reveal a complex bifurcation scenario underlying the transformation of these CIPs into each other. These theoretical predictions are confirmed by numerical simulations of the original coupled oscillator system.

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.

Two scenarios for the onset and suppression of collective oscillations in heterogeneous populations of active rotators

We consider the macroscopic regimes and the scenarios for the onset and the suppression of collective oscillations in a heterogeneous population of active rotators composed of excitable or oscillatory elements. We analyze the system in the continuum limit within the framework of Ott-Antonsen reduction method, determining the states with a constant mean field and their stability boundaries in terms of the characteristics of the rotators' frequency distribution. The system is established to display three macroscopic regimes, namely the homogeneous stationary state, where all the units lie at the resting state, the global oscillatory state, characterized by the partially synchronized local oscillations, and the heterogeneous stationary state, which includes a mixture of resting and asynchronously oscillating units. The transitions between the characteristic domains are found to involve a complex bifurcation structure, organized around three codimension-two bifurcation points: a Bogdanov-Takens point, a cusp point, and a fold-homoclinic point. Apart from the monostable domains, our study also reveals two domains admitting bistable behavior, manifested as coexistence between the two stationary solutions or between a stationary and a periodic solution. It is shown that the collective mode may emerge via two generic scenarios, guided by a saddle-node of infinite period or the Hopf bifurcation, such that the transition from the homogeneous to the heterogeneous stationary state under increasing diversity may follow the classical paradigm, but may also be hysteretic. We demonstrate that the basic bifurcation structure holds qualitatively in the presence of small noise or small coupling delay, with the boundaries of the characteristic domains shifted compared to the noiseless and the delay-free case.

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.

Stabilization of steady states in an array of all-to-all coupled oscillators

An array of globally all-to-all coupled FitzHugh-Nagumo-type oscillators is considered. We suggest an adaptive first-order stable filter control feedback technique to stabilize the steady states of the oscillators. The overall system includes separate networks of coupling and control. Therefore, the controller does not depend on the intrinsic parameters of coupling between the oscillators. We have investigated stabilization of the steady states in an array of nonidentical oscillators analytically, numerically, and experimentally.

Stochastic bursting in unidirectionally delay-coupled noisy excitable systems

We show that "stochastic bursting" is observed in a ring of unidirectional delay-coupled noisy excitable systems, thanks to the combinational action of time-delayed coupling and noise. Under the approximation of timescale separation, i.e., when the time delays in each connection are much larger than the characteristic duration of the spikes, the observed rather coherent spike pattern can be described by an idealized coupled point process with a leader-follower relationship. We derive analytically the statistics of the spikes in each unit, the pairwise correlations between any two units, and the spectrum of the total output from the network. Theory is in good agreement with the simulations with a network of theta-neurons.

Dynamical complexity as a proxy for the network degree distribution

We explore the relation between the topological relevance of a node in a complex network and the individual dynamics it exhibits. When the system is weakly coupled, the effect of the coupling strength against the dynamical complexity of the nodes is found to be a function of their topological roles, with nodes of higher degree displaying lower levels of complexity. We provide several examples of theoretical models of chaotic oscillators, pulse-coupled neurons, and experimental networks of nonlinear electronic circuits evidencing such a hierarchical behavior. Importantly, our results imply that it is possible to infer the degree distribution of a network only from individual dynamical measurements.

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.

Analogue closed-loop optogenetic modulation of hippocampal pyramidal cells dissociates gamma frequency and amplitude

Gamma-band oscillations are implicated in modulation of attention, integration of sensory information and flexible communication among anatomically connected brain areas. How networks become entrained is incompletely understood. Specifically, it is unclear how the spectral and temporal characteristics of network oscillations can be altered on rapid timescales needed for efficient communication. We use closed-loop optogenetic modulation of principal cell excitability in mouse hippocampal slices to interrogate the dynamical properties of hippocampal oscillations. Gamma frequency and amplitude can be modulated bi-directionally, and dissociated, by phase-advancing or delaying optogenetic feedback to pyramidal cells. Closed-loop modulation alters the synchrony rather than average frequency of action potentials, in principle avoiding disruption of population rate-coding of information. Modulation of phasic excitatory currents in principal neurons is sufficient to manipulate oscillations, suggesting that feed-forward excitation of pyramidal cells has an important role in determining oscillatory dynamics and the ability of networks to couple with one another.

Optimizing stimulus waveforms for suppressing epileptic activity reveals a counterbalancing mechanism

Electrical stimulation is used to treat drug- resistant epilepsy, and excessive stimulation can lead to adverse effects for patients. In this article, we use an extrema featured stochastic search algorithm to find energy-efficient stimulus waveforms that suppress seizure activity in two different computational models of epilepsy. We infer general principles that may provide insight into future design of energy efficient stimulus for epilepsy treatments.

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.

How stimulation frequency and intensity impact on the long-lasting effects of coordinated reset stimulation

Several brain diseases are characterized by abnormally strong neuronal synchrony. Coordinated Reset (CR) stimulation was computationally designed to specifically counteract abnormal neuronal synchronization processes by desynchronization. In the presence of spike-timing-dependent plasticity (STDP) this may lead to a decrease of synaptic excitatory weights and ultimately to an anti-kindling, i.e. unlearning of abnormal synaptic connectivity and abnormal neuronal synchrony. The long-lasting desynchronizing impact of CR stimulation has been verified in pre-clinical and clinical proof of concept studies. However, as yet it is unclear how to optimally choose the CR stimulation frequency, i.e. the repetition rate at which the CR stimuli are delivered. This work presents the first computational study on the dependence of the acute and long-term outcome on the CR stimulation frequency in neuronal networks with STDP. For this purpose, CR stimulation was applied with Rapidly Varying Sequences (RVS) as well as with Slowly Varying Sequences (SVS) in a wide range of stimulation frequencies and intensities. Our findings demonstrate that acute desynchronization, achieved during stimulation, does not necessarily lead to long-term desynchronization after cessation of stimulation. By comparing the long-term effects of the two different CR protocols, the RVS CR stimulation turned out to be more robust against variations of the stimulation frequency. However, SVS CR stimulation can obtain stronger anti-kindling effects. We revealed specific parameter ranges that are favorable for long-term desynchronization. For instance, RVS CR stimulation at weak intensities and with stimulation frequencies in the range of the neuronal firing rates turned out to be effective and robust, in particular, if no closed loop adaptation of stimulation parameters is (technically) available. From a clinical standpoint, this may be relevant in the context of both invasive as well as non-invasive CR stimulation.

Resumption of dynamism in damaged networks of coupled oscillators

Deterioration in dynamical activities may come up naturally or due to environmental influences in a massive portion of biological and physical systems. Such dynamical degradation may have outright effect on the substantive network performance. This requires us to provide some proper prescriptions to overcome undesired circumstances. In this paper, we present a scheme based on external feedback that can efficiently revive dynamism in damaged networks of active and inactive oscillators and thus enhance the network survivability. Both numerical and analytical investigations are performed in order to verify our claim. We also provide a comparative study on the effectiveness of this mechanism for feedbacks to the inactive group or to the active group only. Most importantly, resurrection of dynamical activity is realized even in time-delayed damaged networks, which are considered to be less persistent against deterioration in the form of inactivity in the oscillators. Furthermore, prominence in our approach is substantiated by providing evidence of enhanced network persistence in complex network topologies taking small-world and scale-free architectures, which makes the proposed remedy quite general. Besides the study in the network of Stuart-Landau oscillators, affirmative influence of external feedback has been justified in the network of chaotic Rössler systems as well.

Effects of stochastic time-delayed feedback on a dynamical system modeling a chemical oscillator

We examine how stochastic time-delayed negative feedback affects the dynamical behavior of a model oscillatory reaction. We apply constant and stochastic time-delayed negative feedbacks to a point Field-Körös-Noyes photosensitive oscillator and compare their effects. Negative feedback is applied in the form of simulated inhibitory electromagnetic radiation with an intensity proportional to the concentration of oxidized light-sensitive catalyst in the oscillator. We first characterize the system under nondelayed inhibitory feedback; then we explore and compare the effects of constant (deterministic) versus stochastic time-delayed feedback. We find that the oscillatory amplitude, frequency, and waveform are essentially preserved when low-dispersion stochastic delayed feedback is used, whereas small but measurable changes appear when a large dispersion is applied.

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.

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.

Robust Weak Chimeras in Oscillator Networks with Delayed Linear and Quadratic Interactions

We present an approach to generate chimera dynamics (localized frequency synchrony) in oscillator networks with two populations of (at least) two elements using a general method based on a delayed interaction with linear and quadratic terms. The coupling design yields robust chimeras through a phase-model-based design of the delay and the ratio of linear and quadratic components of the interactions. We demonstrate the method in the Brusselator model and experiments with electrochemical oscillators. The technique opens the way to directly bridge chimera dynamics in phase models and real-world oscillator networks.

Emergence of localized patterns in globally coupled networks of relaxation oscillators with heterogeneous connectivity

Oscillations in far-from-equilibrium systems (e.g., chemical, biochemical, biological) are generated by the nonlinear interplay of positive and negative feedback effects operating at different time scales. Relaxation oscillations emerge when the time scales between the activators and the inhibitors are well separated. In addition to the large-amplitude oscillations (LAOs) or relaxation type, these systems exhibit small-amplitude oscillations (SAOs) as well as abrupt transitions between them (canard phenomenon). Localized cluster patterns in networks of relaxation oscillators consist of one cluster oscillating in the LAO regime or exhibiting mixed-mode oscillations (LAOs interspersed with SAOs), while the other oscillates in the SAO regime. Because the individual oscillators are monostable, localized patterns are a network phenomenon that involves the interplay of the connectivity and the intrinsic dynamic properties of the individual nodes. Motivated by experimental and theoretical results on the Belousov-Zhabotinsky reaction, we investigate the mechanisms underlying the generation of localized patterns in globally coupled networks of piecewise-linear relaxation oscillators where the global feedback term affects the rate of change of the activator (fast variable) and depends on the weighted sum of the inhibitor (slow variable) at any given time. We also investigate whether these patterns are affected by the presence of a diffusive type of coupling whose synchronizing effects compete with the symmetry-breaking global feedback effects.

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.

Representing delayed force feedback as a combination of current and delayed states

To adapt to deterministic force perturbations that depend on the current state of the hand, internal representations are formed to capture the relationships between forces experienced and motion. However, information from multiple modalities travels at different rates, resulting in intermodal delays that require compensation for these internal representations to develop. To understand how these delays are represented by the brain, we presented participants with delayed velocity-dependent force fields, i.e., forces that depend on hand velocity either 70 or 100 ms beforehand. We probed the internal representation of these delayed forces by examining the forces the participants applied to cope with the perturbations. The findings showed that for both delayed forces, the best model of internal representation consisted of a delayed velocity and current position and velocity. We show that participants relied initially on the current state, but with adaptation, the contribution of the delayed representation to adaptation increased. After adaptation, when the participants were asked to make movements with a higher velocity for which they had not previously experienced with the delayed force field, they applied forces that were consistent with current position and velocity as well as delayed velocity representations. This suggests that the sensorimotor system represents delayed force feedback using current and delayed state information and that it uses this representation when generalizing to faster movements.NEW & NOTEWORTHY The brain compensates for forces in the body and the environment to control movements, but it is unclear how it does so given the inherent delays in information transmission and processing. We examined how participants cope with delayed forces that depend on their arm velocity 70 or 100 ms beforehand. After adaptation, participants applied opposing forces that revealed a partially correct representation of the perturbation using the current and the delayed information.

Local complexity predicts global synchronization of hierarchically networked oscillators

We study the global synchronization of hierarchically-organized Stuart-Landau oscillators, where each subsystem consists of three oscillators with activity-dependent couplings. We considered all possible coupling signs between the three oscillators, and found that they can generate different numbers of phase attractors depending on the network motif. Here, the subsystems are coupled through mean activities of total oscillators. Under weak inter-subsystem couplings, we demonstrate that the synchronization between subsystems is highly correlated with the number of attractors in uncoupled subsystems. Among the network motifs, perfect anti-symmetric ones are unique to generate both single and multiple attractors depending on the activities of oscillators. The flexible local complexity can make global synchronization controllable.

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).

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.