Amplitude Response Curves Research Articles

How to design optimal brain stimulation to modulate phase-amplitude coupling?

Objective. Phase-amplitude coupling (PAC), the coupling of the amplitude of a faster brain rhythm to the phase of a slower brain rhythm, plays a significant role in brain activity and has been implicated in various neurological disorders. For example, in Parkinson’s disease, PAC between the beta (13–30 Hz) and gamma (30–100 Hz) rhythms in the motor cortex is exaggerated, while in Alzheimer’s disease, PAC between the theta (4–8 Hz) and gamma rhythms is diminished. Modulating PAC (i.e. reducing or enhancing PAC) using brain stimulation could therefore open new therapeutic avenues. However, while it has been previously reported that phase-locked stimulation can increase PAC, it is unclear what the optimal stimulation strategy to modulate PAC might be. Here, we provide a theoretical framework to narrow down the experimental optimisation of stimulation aimed at modulating PAC, which would otherwise rely on trial and error. Approach. We make analytical predictions using a Stuart–Landau model, and confirm these predictions in a more realistic model of coupled neural populations. Main results. Our framework specifies the critical Fourier coefficients of the stimulation waveform which should be tuned to optimally modulate PAC. Depending on the characteristics of the amplitude response curve of the fast population, these components may include the slow frequency, the fast frequency, combinations of these, as well as their harmonics. We also show that the optimal balance of energy between these Fourier components depends on the relative strength of the endogenous slow and fast rhythms, and that the alignment of fast components with the fast rhythm should change throughout the slow cycle. Furthermore, we identify the conditions requiring to phase-lock stimulation to the fast and/or slow rhythms. Significance. Together, our theoretical framework lays the foundation for guiding the development of innovative and more effective brain stimulation aimed at modulating PAC for therapeutic benefit.

Research on the intrinsic phase difference of Rayleigh scattering in distributed acoustic sensing

The differential cross-phase multiply (DCM) demodulation algorithm based on a 3 × 3 coupler is commonly employed in distributed acoustic systems (DAS) to recover phase information along the sensing fiber. This study investigates the amplitude-spatial resolution response characteristics of the DCM algorithm using 3 × 3 coupler demodulation. Firstly, we theoretically establish the variation of demodulation amplitude in DAS systems due to the intrinsic phase of the signal. Subsequently, we analyze the amplitude-pulse width response curves of the DCM algorithm based on the distribution of demodulated phase error obtained from both experimental and simulation data. The findings reveal a double peaks phenomenon in the amplitude response curves of the DCM demodulation algorithm based on 3 × 3 couplers. We also explore the amplitude response curves for multiple pulse widths and observe that, under the same arm length difference conditions, the double peaks persist. However, this double peak phenomenon is eliminated when the pulse width is less than 21 ns. Therefore, when the width determined by the pulse is greater than 40% of the difference between the light paths, the demodulation will appear double peaks phenomenon.

The dynamic regimes and the energy harvesting of the energy harvester based on the bistable piezoelectric composite laminate

The dynamic regimes and the energy harvesting of the bistable energy harvester based on the bistable piezoelectric composite laminate are elaborated here. To identify the all-around dynamic regimes and assess the bistable energy harvesting performance, the forward frequency sweep, the reverse frequency sweep and the resistance sweep are conducted. The peak-to-peak frequency–amplitude response curves on the displacement and the voltage output are graphically presented. The single-well vibration, the limit cycle oscillations, the intermittency between the limit cycle oscillations and the chaotic snapthrough, the intermittency between the multiple periods snapthrough and the chaotic snapthrough, the multiple periods snapthrough and the chaotic snapthrough are elaborated by the Time history diagram, the Phase diagram and the Poincare map. The dynamic regimes are graphically presented delivering the stability zone for the dynamic responses and the optimal region for the energy harvesting. The RMS power output and the Max voltage output are checked to be the value judgements. Due to the softening nonlinear effect and the hysteresis, the conspicuous biases between the forward and reverse frequency sweeps are found to exist illustrating that the limit cycle oscillations expand the bandwidth of the double-well regimes during the reverse frequency sweep. From the perspective of the energy harvesting capacity, the limit cycle oscillations are demonstrated to be optimal, the double-well responses generally outperform the single-well responses concerning the off-resonance and the single-well responses concerning the primary resonance around the reduced modal frequency may outperform the double-well responses except the limit cycle oscillations. The load resistance is crucial to the power generation performance and needs optimizing with the change of the external excitation. The load resistance, the external excitation amplitude and frequency are interrelated and can be optimized simultaneously for the maximum power generation efficiency.

One-to-one internal resonance of a cable-beam structure subjected to a concentrated load

Cable-beam composite structure, such as a cable-stayed bridge in cantilever construction and a tower crane hoisting load, will sustain dynamic excitation. In order to investigate the dynamic behaviors at this stage, a cable-supported cantilever beam model subjected to a concentrated harmonic excitation at the cantilever end of the beam is established. The interactions including the second-order effect of the axial compressive load on the beam and the dynamic interaction between the beam and the cable due to cable's dynamic elongation are considered. A reduced two-degree-of-freedom system is obtained by using Galerkin discretization. Multiple time scales method is applied to solve the ordinary differential equations. Based on the derived modulation equations and the ordinary differential equations, frequency response, amplitude response, phase plane and time history curves are provided to explore the dynamic behaviors of the system. Three different lengths of the cable-stayed cantilever beam system are considered to study the changes in dynamic behaviors. In order to verify the validity in each case, Runge-Kutta method is used to directly solve the original ordinary differential equations and two results have a satisfying consistence. The analysis results show that frequency responses of both the beam and the cable exhibit hardening spring properties; the cable has more rich and complicated dynamic behaviors than the beam, for example the cable has more significant jump phenomenon.

Autoparametric Internal Resonance in Coupled Oscillator: An Excitation Amplitude Insensitive Mass Sensing Scheme With a Roof Tilting

To overcome the effect of slowly changing excitation amplitude caused by temperature drift etc. on sensing performance, a slope-based working mechanism is exploited in autoparametric internal resonance systems. Through increasing modal damping, energy transfer between coupled modes is observed to be decreased, and the depressed cave on the amplitude response curve transits to a flat roof that is independent of both excitation amplitude and excitation frequency. A theoretical expression is established to characterize the flat roof, which tends to tilt when applying mass perturbations. The slope of the flat roof is found to be independent of the excitation amplitude but a function of the applied mass perturbations, while the frequency range of which broadens or narrows with the increase or decrease of the excitation amplitude respectively. Further concerning on the slopes under different excitation amplitudes, results from numerical study is also basically consistent with those from the theoretical prediction, apart a maximum relative deviation about 8%. An excitation insensitive mass sensing scheme via flat roof tilting is thus proposed. With applying different mass perturbations, the proposed scheme is mutually confirmed by both theoretical and numerical results, featured with a high linearity in a certain mass range. Compared to conventional frequency shift based or amplitude change based one, the proposed sensing scheme is insensitive to slow change of excitation amplitude, and is believed to have great potential in engineering scenarios with varying environmental temperature.

Amplitude resonance response and feedback control of cantilever beams with tip-mass under aerodynamic load

The dynamic of a cantilever beam with tip mass is studied under an aerodynamic loading. The effects of coupling is investigated by tacking into account the fluid flow. Using the multiple time scale method, the approximative solutions are found and the study of their stability is made by the Routh-Hurwitz stability criterion. The influence of parameters on the system is studied at the harmonic and subharmonic resonances. The results show that, the effects of tip mass can be neglected in harmonic resonance case, while they are more important in subharmonic resonance cases. The results equally demonstrate that an increase of the stable state fluid velocity reduces the amplitude of vibrations. In addition, the hysteresis phenomenon studies show that it is principally induced by nonlinearity coefficients. Finally, time-delay feedback control is applied and the effects of controlling are observed on amplitude response curve at the harmonic resonance, from where we note that optimized choice of control parameters can be useful in controlling vibrations.

Investigation of the Scruton Number Effects of Wind-Induced Unsteady Galloping Responses of Bridge Suspenders

When the critical wind speed of vortex-induced resonance is close to that of quasi-steady galloping, a type of coupled wind-induced vibration that is different from divergent galloping can easily occur in a rectangular bar. It is a type of “unsteady galloping” phenomenon wherein the response amplitude increases linearly with the increase in the wind speed, while a limit cycle oscillation is observed at each wind speed, whose mechanism is still in research. Mass and damping are the key parameters that affect the coupling degree and amplitude response estimation. For a set of rectangular section member models with a width-to-height ratio of 1.2, by adjusting the equivalent stiffness, equivalent mass, and damping ratio of the model system and performing comparative tests on the wind-induced vibration response of the same mass with different damping ratios, it is possible to achieve the same damping ratio with different masses and the same Scruton number with different masses and damping combinations under the same Reynolds number. The results show that the influence of the mass and damping parameters on the “unsteady galloping” amplitude response is independent, and the weight is the same in the coupling state. The Scruton number “locked interval” (12.4–30.6) can be found in the “unsteady galloping” amplitude response, and the linear slope of the dimensionless wind speed amplitude response curve does not change with the Scruton number in the “locked interval.” In addition, a “transition interval” (26.8–30.6) coexists with the “locked interval” wherein the coupling state of the wind-induced vibration is converted into the uncoupled state. The empirical formula for estimating the “unsteady galloping” response amplitude is modified and can be used to predict the amplitude within the design wind speed range of similar engineering members.

Thermal Effects on the Nonlinear Forced Responses of Hinged-Clamped Beam with Multimodal interaction

Nonlinear analysis of a forced geometrically nonlinear Hinged-Clamped beam involving three modes interactions with internal resonance and submitted to thermal and mechanical loadings is investigated. Based on the extended Hamilton’s principle, the PDEs governing the thermoelastic vibration of planar motion is derived. Galerkin’s orthogonalization method is used to reduce the governing PDEs of motion into a set of nonlinear non-autonomous ordinary differential equations. The system is solved for the three modes involved by the use of the multiple scales method for amplitudes and phases. For steady states responses, the Newton-Raphson shooting technique is used to solve the three systems of six parametric nonlinear algebraic equations obtained. Results are presented in terms of influences of temperature variations on the response amplitudes of different substructures when each of the modes is excited. It is observed for all substructures and independent of the mode excited a shift within the frequency axis of the temperature influenced amplitude response curves on either side of the temperature free-response curve. Moreover, it is found that thermal loads diversely influence the interacting substructures. Depending on the directly excited mode, higher oscillation amplitudes are found in some substructures under negative temperature difference, while it is observed in others under positive temperature change and in some others for temperature free-response curves.

AN EFFICIENT ANCIENT CHINESE ALGORITHM TO INVESTIGATE THE DYNAMICS RESPONSE OF A FRACTAL MICROGRAVITY FORCED OSCILLATOR

In this paper, the ancient Chinese algorithm is applied to derive fractal microgravity forced oscillator’s frequency–amplitude response curves using the two-scale fractal transform derivative. Depending on the fractal parameter values, it is shown that the system restoring forces exhibit hardening or softening behavior. Furthermore, the proposed solution approach is simple, efficient, and accurate for obtaining the approximate steady-state solution of a fractal microgravity nonlinear oscillator with a purely nonlinear power-form restoring force.

Exact steady-state solution of fractals damped, and forced systems

In this article, the methodology for deriving the exact steady-state solutions of forced oscillatory systems has been extended to obtain the solution of fractal damped and forced differential equations using the two-scale fractal dimension transform. Finally, a numerical example is given to illustrate the applicability of the proposed approach in obtaining the frequency–amplitude response curves for a forced and damped fractal oscillator. Numerical results show that the fractal parameter values shift the frequency–amplitude curves to the left and right from the resonance curve. The present method sheds new light on solving damped and forced fractal oscillators.

Nonlinear modeling and study of rigid-flexible piezoelectric energy harvester with additional frame

Based on a rigid-flexible piezoelectric energy harvester (PEH) composed of antisymmetric double plates with a rigid frame, which has three operating bandwidths in a low-frequency range, nonlinearity is used to further broaden the operating bandwidth. Firstly, finite-element simulation is used to study the influence of structural parameters, such as the width of the frame and the distance between two plates, on the natural frequencies. Bending deformation of cantilever plates and rigid motion of frame are considered comprehensively to obtain the theoretical mode functions of the structure. The nonlinear electromechanical coupling model is established via von Karman nonlinear geometric assumptions and the Hamilton principle. In order to study the broadening of output bandwidth caused by nonlinearity, the average equations with 1:1.5:2 internal resonance ratio are derived by the multi-scale method. On the basis of the calculated optimal resistance, the voltage response and dimensionless amplitude response curves induced by the first three order modes are numerically calculated. Theoretical analysis shows that the proposed PEH not only obtains three peaks of voltage output in a low-frequency range, but also utilizes hard spring effect and internal resonance to widen the output bandwidth of each resonance interval.

Analysis of Harmonically Forced Duffing Oscillator with Time Delay State Feedback by Incremental Harmonic Balance Method

Periodic solutions of a harmonically forced Duffing oscillator with time-delay state feedback are investigated using the incremental harmonic balance method. In the process of solving, the explicit effect matrix of the time delay term was derived. The stability of the periodic solutions was determined by a method which combines the continuous time approximation and multivariable Floquet theory. On this basis, the frequency–amplitude response curve and stability characteristics of the primary resonance and the 1/3 subharmonic resonance were obtained. The stable results were compared with results obtained by the numerical method, which demonstrated the effectiveness and accuracy of the incremental harmonic balance method for the analysis of strongly nonlinear equations with time delays. The influence of the time delay and feedback control parameters on the primary and 1/3 subharmonic resonance is investigated. The periodicity of the effect of the time delay is also discussed.

Inhibition and facilitation of nociceptively evoked muscular activity by fentanyl or dexmedetomidine in isoflurane-anaesthetized pigs

ObjectiveTo investigate motor and cardiovascular responses to dexmedetomidine or fentanyl in isoflurane-anaesthetized pigs. Study designExperimental, balanced, block randomized, two-group design. AnimalsA group of 16 crossbred pigs, 55 ± 8 days (mean ± standard deviation) old. MethodsDeltoid electromyography (EMG) was recorded during isoflurane anaesthesia. Electrical stimulation using 5, 10, 20 and 40 mA of the distal right thoracic limb elicited a nociceptive withdrawal reflex (NWR), quantified by the area under the curve (AUC) for the simulation intensity versus EMG amplitude response curve. Latency to movement evoked by clamping a claw for maximum 60 seconds was noted. Arterial blood pressure and pulse rate were recorded. Data were sampled at baseline and during dexmedetomidine 0.25, 0.5, 1.0, 2.0, 4.0 and 8.0 μg kg–1 hour–1 or fentanyl 5, 10, 20, 40, 80 and 160 μg kg–1 hour–1 infusions. The influence of infusion rate on NWR AUC and spontaneous EMG was analysed using a mixed model, with p < 5%. ResultsNWR AUC increased at fentanyl 5 μg kg–1 hour–1 but decreased at fentanyl 40, 80 and 160 μg kg–1 hour–1 and dexmedetomidine 4.0 and 8.0 μg kg–1 hour–1. All pigs at fentanyl 80 μg kg–1 hour–1, and three pigs at dexmedetomidine 8.0 μg kg–1 hour–1 had mechanical latencies greater than 60 seconds. Spontaneous EMG activity increased accompanied by visually evident ‘shivering’ at fentanyl 5, 10 and 20 μg kg–1 hour–1 but decreased at dexmedetomidine 2, 4 and 8 μg kg–1 hour–1. Clinically relevant effects of increasing infusion rates on blood pressure or pulse rate were not observed. Conclusion and clinical relevanceIf anaesthetic plane or antinociception is evaluated in pigs, response to claw clamping and NWR will not necessarily give uniform results when comparing drugs. If only one method is used, results should be interpreted cautiously.

Duffing-type Equations: Singular Points of Amplitude Profiles and Bifurcations

We study the Duffing equation and its generalizations with polynomial nonlinearities. Recently, we have demonstrated that metamorphoses of the amplitude response curves, computed by asymptotic methods in implicit form as $F\left( \Omega ,\ A\right) =0$, permit prediction of qualitative changes of dynamics occurring at singular points of the implicit curve $F\left(\Omega ,\ A\right) =0$. In the present work we determine a global structure of singular points of the amplitude profiles computing bifurcation sets, i.e. sets containing all points in the parameter space for which the amplitude profile has a singular point. We connect our work with independent research on tangential points on amplitude profiles, associated with jump phenomena, characteristic for the Duffing equation. We also show that our techniques can be applied to solutions of form $\Omega _{\pm }=f_{\pm }\left( A\right) $, obtained within other asymptotic approaches.

Flow-induced vibrations of a hinged cavity at the rear of a blunt-based body subject to laminar flow

We perform numerical simulations to characterize the flow-induced vibrations (FIV) of a rear cavity with elastically hinged rigid plates, placed as a passive device at the base of a blunt body that is subject to a laminar flow of Reynolds number Re=400. The dynamic response and forcing of plates, wake features and force coefficients are investigated for the range of reduced velocity U*=[0,30]. Three different regimes of the rotational oscillations are identified. An initial branch of low oscillation amplitude is defined for U*<2.5, where the plates oscillate in counter-phase (varicose mode) with a frequency fp that corresponds to the harmonic of the wake vortex shedding frequency fp≃2fw, and is similar to the natural frequency of the plates, fp≃fn. For intermediate values of U*, the plates oscillate in phase (sinuous mode) at their natural frequency, with respect to a closer averaged location of plates. Such synchronization regime amplifies the vibration magnitude and defines the upper branch in the amplitude response curve, whose maximum is attained at U*=4.7. Due to such enhanced vibration, the vortex shedding frequency is now locked-in at the natural frequency of plates, so that fp=fn=fw. Finally, for larger values of U*, a lower branch of moderate amplitude response is defined, which is characterized by the in-phase oscillation of plates, with respect to an more open average position, governed again by the shedding frequency, fp=fw>fn. Additionally, a multibody model has been developed to retrieve, from the plates rotational motion, the resultant forces and moments that produce the plates vibration. Such inverse dynamics model is formulated to allow its generalization for configurations of higher dynamical order, and validated against the results obtained from the numerical simulations. The analysis shows that the synchronization regime is mainly promoted by a reduced fluid damping and a forcing moment that acts in phase with the plates motion. The switch in such phase from 0∘ to 180∘ occurs after the lock-in, what attenuates the plates response at large U*. In general, the FIV of plates alters the vortex shedding and near wake pressure, especially during the synchronization regime, inducing an overall increase of the global force coefficients with respect to the static cavity. Thus, the performance of hinged plates enhances generally the mean drag, although a 25% reduction is reported for the lift amplitude.

Numerical analysis of the flow-induced vibrations in the laminar wake behind a blunt body with rear flexible cavities

We present a numerical study on the fluid–structure interaction of an incompressible laminar flow around a slender blunt-based body implementing a rear cavity of flexible plates. The study focuses on the use of this type of device to control the wake dynamics and the aerodynamic forces acting on the body, as well as to harvest energy from the flow. To that aim, the effects on the plates flow-induced vibrations (FIV) and on the dynamics of the flow of three control parameters, namely the reduced velocity, 0≤U∗<12, the mass ratio, m∗=[500,1000] and the cavity height hc, are evaluated at Reynolds number Re=500. Four different branches are identified in terms of dynamic response, as U∗ increases, regardless of the values of hc and m∗. At low values of U∗, an initial branch with a local peak of moderate amplitude response is observed, where the plates oscillate in counter-phase, in a varicose mode, with a frequency fp that is synchronized with the second harmonic of the vortex shedding frequency, 2fvs, and similar to the natural frequency of the solid, fn≃f1,n. For intermediate values of U∗, a transition towards a sinuous mode of oscillation occurs, where the plate frequency is synchronized with the vortex shedding frequency, fp=fvs. Initially, after such transition, the oscillation frequency is lower than the natural frequency of the plates fp<f1,n, so that a weak response defines a lower branch in the amplitude response curve. When U∗ increases, a lock-in regime develops with fp=fvs=f1,n, that is characterized by a response amplification, giving rise to the emergence of the upper branch, where the FIV amplitude of plates is the highest. Finally, a fourth desynchronization branch of moderate amplitude appears for larger values of U∗, where the plates present a complex, multi-mode regime, characterized by the vibration at multiple frequencies within the range 5f1,n<fp<6.5f1,n. The analysis of the plates deformation through POD analysis reveals that they mainly oscillate following the first Euler–Bernoulli mode for the first three branches, and a combination of the second and first Euler–Bernoulli modes for the desynchronization branch. The effect of increasing the mass ratio is to mitigate the response at higher values of U∗, while reducing the cavity height hc leads to the fostering of oscillations within the upper and desynchronization branches. In general, the FIV response of plates alters the wake dynamics and the force coefficient, especially within the lock-in regime, where the drag is strongly amplified, although the fluctuations of the lift are attenuated (nearly a 40% reduction). Besides, reductions of the drag of approximately 2% are obtained within the initial and lower branches. Finally, a simple quantification of the energy transfer from the plates, through a linearized Euler–Bernoulli approach, shows that the lock-in regime represents the best regime to extract energy from the flow.

Subharmonic resonance of a clamped-clamped buckled beam with 1:1 internal resonance under base harmonic excitations

The subharmonic resonance and bifurcations of a clamped-clamped buckled beam under base harmonic excitations are investigated. The nonlinear partial integro-differential equation of the motion of the buckled beam with both quadratic and cubic nonlinearities is given by using Hamilton’s principle. A set of second-order nonlinear ordinary differential equations are obtained by spatial discretization with Galerkin’s method. A high-dimensional model of the buckled beam is derived, concerning nonlinear coupling. The incremental harmonic balance (IHB) method is used to achieve the periodic solutions of the high-dimensional model of the buckled beam to observe the nonlinear frequency response curve and nonlinear amplitude response curve, and the Floquet theory is used to analyze the stability of the periodic solutions. Attention is focused on the subharmonic resonance caused by the internal resonance as the excitation frequency near twice of the first natural frequency of the buckled beam with/without the anti-symmetric modes being excited. Bifurcations including the saddle-node, Hopf, period-doubling, and symmetry-breaking bifurcations are observed. Furthermore, quasi-periodic motion is observed by using the fourth-order Runge-Kutta method, which results from the Hopf bifurcation of the response of the buckled beam with the anti-symmetric modes being excited.

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.

Nonlinear energy harvester with coupled Duffing oscillators

Structural vibrations are very common in aerospace and mechanical engineering systems, where dynamic analysis of modern aerospace structures and industrial machines has become an indispensable step in their design. Suppression of unwanted vibrations and their exploitation for energy harvesting at the same time would be the most desirable scenario. The dynamical system presented in this communication is based on a discrete model of energy harvesting device realized in such a manner as to achieve both vibration suppression and harvesting of vibration energy by introducing the nonlinear energy sink concept. The mechanical model is formed as a two-degree of freedom nonlinear oscillator with an oscillating magnet and harmonic base excitation. The corresponding mathematical model is based on the system of nonlinear nonhomogeneous Duffing type differential equations. To explore complex dynamical behaviour of the presented model, periodic solutions and their bifurcations are found by using the incremental harmonic balance (IHB) and continuation methods. For the detection of unstable periodic orbits, the Floquet theory is applied and an interesting harmonic response of the presented nonlinear dynamical model is detected. The main advantage of the presented approach is its ability to obtain approximated periodic responses in terms of Fourier series and estimate the voltage output of an energy harvester for a system with strong nonlinearity. The accuracy of the presented methodology is verified by comparing the results obtained in this work with those obtained by a standard numerical integration method and results from the literature. Numerical examples show the effects of different physical parameters on amplitude-frequency, response amplitude - base amplitude and time response curves, where a qualitative change is explored and studied in detail. Presented theoretical results demonstrate that the proposed system has advanced performance in both system requirements - vibration suppression, and energy harvesting.

Phase-dependence of response curves to deep brain stimulation and their relationship: from essential tremor patient data to a Wilson\u2013Cowan model

Essential tremor manifests predominantly as a tremor of the upper limbs. One therapy option is high-frequency deep brain stimulation, which continuously delivers electrical stimulation to the ventral intermediate nucleus of the thalamus at about 130 Hz. Constant stimulation can lead to side effects, it is therefore desirable to find ways to stimulate less while maintaining clinical efficacy. One strategy, phase-locked deep brain stimulation, consists of stimulating according to the phase of the tremor. To advance methods to optimise deep brain stimulation while providing insights into tremor circuits, we ask the question: can the effects of phase-locked stimulation be accounted for by a canonical Wilson–Cowan model? We first analyse patient data, and identify in half of the datasets significant dependence of the effects of stimulation on the phase at which stimulation is provided. The full nonlinear Wilson–Cowan model is fitted to datasets identified as statistically significant, and we show that in each case the model can fit to the dynamics of patient tremor as well as to the phase response curve. The vast majority of top fits are stable foci. The model provides satisfactory prediction of how patient tremor will react to phase-locked stimulation by predicting patient amplitude response curves although they were not explicitly fitted. We also approximate response curves of the significant datasets by providing analytical results for the linearisation of a stable focus model, a simplification of the Wilson–Cowan model in the stable focus regime. We report that the nonlinear Wilson–Cowan model is able to describe response to stimulation more precisely than the linearisation.