Phase Response Curve Research Articles

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.

Neural Networks In A Generalized Model Of The N-Pacemaker Phase Response Curve

In this publication, we generalize the proposed model of two interacting oscillators in the case of a strong difference in their periods (when the pacemaker pulses do not alternate) and propose a General model describing a network of oscillators coupled globally. Our goal is to make the model as simple as possible and enter the minimum number of parameters. Therefore, we will fully characterize the pacemaker of their internal lengths of the cycle and re-present them as pulse oscillators. Interaction of pacemakers is described by PRC.

Excitatory autapse induces different cases of reduced neuronal firing activities near Hopf bifurcation

Excitatory autapse with short time delay can lower the firing frequency of spiking behaviors near subcritical Hopf bifurcation, which is interpreted with type II phase response curve wherein an excitatory impulse current applied at the phase following a spike can induce the next spike delayed. Such a nonlinear phenomenon is different from the common viewpoint that excitatory effect typically induces enhancement of neuronal firing activities. In this paper, more cases of reduction of firing activities induced by excitatory autapse than the one mentioned above are simulated in the Hodgkin-Huxley model with relatively short time delay, and the dynamical mechanism is provided with phase trajectories modulated by excitatory autaptic current and bifurcations. For the monostable spiking near the subcritical Hopf bifurcation, one novel case is induced. The delayed autaptic current pulse evoked by a spike induces the trajectory moved to the neighborhood of the unstable focus, and after the pulse, the subthreshold oscillations with ascending amplitude appear until they evolve to the next spike, which results in mixed subthreshold oscillations and a single spike. For the spiking coexisting with the resting state, which locates between the subcritical Hopf bifurcation and saddle-node bifurcation of limit cycles, two novel cases are induced. One is the mixed-mode oscillations, and the trajectory of subthreshold oscillations begins from the neighborhood outside of the unstable limit cycle, which is induced by the delayed autaptic current pulse. The other is induced by the autaptic current pulse that can cause the trajectory to run into the unstable limit cycle, which leads to the appearance of subthreshold oscillations with descending amplitude after the pulse, and finally evolves to the resting state corresponding to the coexisting stable focus. The results present novel cases that excitatory effects can suppress rather than enhance firing activities and the corresponding dynamical mechanism related to bifurcations, which enrich the study of nonlinear dynamics and neurodynamics, and provide the potential functions of excitatory autapse.

Emergence of global synchronization in directed excitatory networks of type I neurons

The collective behaviour of neural networks depends on the cellular and synaptic properties of the neurons. The phase-response curve (PRC) is an experimentally obtainable measure of cellular properties that quantifies the shift in the next spike time of a neuron as a function of the phase at which stimulus is delivered to that neuron. The neuronal PRCs can be classified as having either purely positive values (type I) or distinct positive and negative regions (type II). Networks of type 1 PRCs tend not to synchronize via mutual excitatory synaptic connections. We study the synchronization properties of identical type I and type II neurons, assuming unidirectional synapses. Performing the linear stability analysis and the numerical simulation of the extended Kuramoto model, we show that feedforward loop motifs favour synchronization of type I excitatory and inhibitory neurons, while feedback loop motifs destroy their synchronization tendency. Moreover, large directed networks, either without feedback motifs or with many of them, have been constructed from the same undirected backbones, and a high synchronization level is observed for directed acyclic graphs with type I neurons. It has been shown that, the synchronizability of type I neurons depends on both the directionality of the network connectivity and the topology of its undirected backbone. The abundance of feedforward motifs enhances the synchronizability of the directed acyclic graphs.

Contribution of Phase Resetting to Adaptive Rhythm Control in Human Walking Based on the Phase Response Curves of a Neuromusculoskeletal Model.

Humans walk adaptively in varying environments by manipulating their complicated and redundant musculoskeletal system. Although the central pattern generators in the spinal cord are largely responsible for adaptive walking through sensory-motor coordination, it remains unclear what neural mechanisms determine walking adaptability. It has been reported that locomotor rhythm and phase are regulated by the production of phase shift and rhythm resetting (phase resetting) for periodic motor commands in response to sensory feedback and perturbation. While the phase resetting has been suggested to make a large contribution to adaptive walking, it has only been investigated based on fictive locomotion in decerebrate cats, and thus it remains unclear if human motor control has such a rhythm regulation mechanism during walking. In our previous work, we incorporated a phase resetting mechanism into a motor control model and demonstrated that it improves the stability and robustness of walking through forward dynamic simulations of a human musculoskeletal model. However, this did not necessarily verify that phase resetting plays a role in human motor control. In our other previous work, we used kinematic measurements of human walking to identify the phase response curve (PRC), which explains phase-dependent responses of a limit cycle oscillator to a perturbation. This revealed how human walking rhythm is regulated by perturbations. In this study, we integrated these two approaches using a physical model and identification of the PRC to examine the hypothesis that phase resetting plays a role in the control of walking rhythm in humans. More specifically, we calculated the PRC using our neuromusculoskeletal model in the same way as our previous human experiment. In particular, we compared the PRCs calculated from two different models with and without phase resetting while referring to the PRC for humans. As a result, although the PRC for the model without phase resetting did not show any characteristic shape, the PRC for the model with phase resetting showed a characteristic phase-dependent shape with trends similar to those of the PRC for humans. These results support our hypothesis and will improve our understanding of adaptive rhythm control in human walking.

Amplitude Effects Allow Short Jet Lags and Large Seasonal Phase Shifts in Minimal Clock Models

Mathematical models of varying complexity have helped shed light on different aspects of circadian clock function. In this work, we question whether minimal clock models (Goodwin models) are sufficient to reproduce essential phenotypes of the clock: a small phase response curve (PRC), fast jet lag, and seasonal phase shifts. Instead of building a single best model, we take an approach where we study the properties of a set of models satisfying certain constraints; here, a 1h-pulse PRC with a range of 3h and clock periods between 22h and 26h is designed. Surprisingly, almost all these randomly parameterized models showed a 4h change in phase of entrainment between long and short days and jet lag durations of three to seven days in advance and delay. Moreover, intrinsic clock period influenced jet lag duration and entrainment amplitude and phase. Fast jet lag was realized in this model by means of an interesting amplitude effect: the association between clock amplitude and clock period termed “twist.” This twist allows amplitude changes to speed up and slow down clocks enabling faster shifts. These findings were robust to the addition of positive feedback to the model. In summary, the known design principles of rhythm generation – negative feedback, long delay, and switch-like inhibition (we review these in detail) – are sufficient to reproduce the essential clock phenotypes. Furthermore, amplitudes play a role in determining clock properties and must be always considered, although they are difficult to measure.

Phase-amplitude dynamics in terms of extended response functions: Invariant curves and arnold tongues

Phase response curves (PRCs) have been extensively used to control the phase of oscillators under perturbations. Their main advantage is the reduction of the whole model dynamics to a single variable (phase) dynamics. However, in some adverse situations (strong inputs, high-frequency stimuli, weak convergence,...), the phase reduction does not provide enough information and, therefore, PRC lose predictive power. To overcome this shortcoming, in the last decade, new contributions have appeared that allow to reduce the system dynamics to the phase plus some transversal variable that controls the deviations from the asymptotic behaviour. We call this setting extended response functions. In particular, we single out the phase response function (PRF, a generalization of the PRC) and the amplitude response function (ARF) that account for the above-mentioned deviations from the oscillating attractor. It has been shown that in adverse situations, the PRC misestimate the phase dynamics whereas the PRF-ARF system provides accurate enough predictions. In this paper, we address the problem of studying the dynamics of the PRF-ARF systems under periodic pulsatile stimuli. This paradigm leads to a two-dimensional discrete dynamical system that we call 2D entrainment map. By using advanced methods to study invariant manifolds and the dynamics inside them, we construct an analytico-numerical method to track the invariant curves induced by the stimulus as two crucial parameters of the system increase (the strength of the input and its frequency). Our methodology also incorporates the computation of Arnold tongues associated to the 2D entrainment map. We apply the method developed to study inner dynamics of the invariant curves of a canonical type II oscillator model. We further compare the Arnold tongues of the 2D map with those obtained with the map induced only by the PRC, which give already noticeable differences. We also observe (via simulations) how high-frequency or strong enough stimuli break up the oscillatory dynamics and lead to phase-locking, which is well captured by the 2D entrainment map.

Entrainment of the Circadian Clock of the Enteric Bacterium Klebsiella aerogenes by Temperature Cycles.

SummaryThe gastrointestinal bacterium Klebsiella (née Enterobacter) aerogenes expresses an endogenously generated, temperature-compensated circadian rhythm in swarming motility. We hypothesized that this rhythm may be synchronized/entrained in vivo by body temperature (TB). To determine entrainment, cultures expressing bioluminescence were exposed to temperature cycles of 1°C (35°C–36°C) or 3°C (34°C–37°C) in amplitude at periods (T-cycles) of T = 22, T = 24, or T = 28 h. Bacteria entrained to all T-cycles at both amplitudes and with stable phase relationships. A high-amplitude phase response curve (PRC) in response to 1-h pulses of 3°C temperature spike (34°C–37°C) at different circadian phases was constructed, revealing a Type-0 phase resetting paradigm. Furthermore, real-time bioluminescence imaging revealed a spatiotemporal pattern to the circadian rhythm. These data are consistent with the hypothesis that the K. aerogenes circadian clock entrains to its host via detection of and phase shifting to the daily pattern of TB.

Statistical Estimation of Phase Response Curves Using Data Transformation

This study is aimed at developing a statistical method to estimate phase response curves (PRCs) while considering a strong correlation between the PRC explanatory and response variables. The correl...

Hilbert transform-based time-series analysis of the circadian gene regulatory network.

In this work, the authors propose the Hilbert transform (HT)‐based numerical method to analyse the time series of the circadian rhythms. They demonstrate the application of HT by taking both deterministic and stochastic time series that they get from the simulation of the fruit fly model Drosophila melanogaster and show how to extract the period, construct phase response curves, determine period sensitivity of the parameters to perturbations and build Arnold tongues to identify the regions of entrainment. They also derive a phase model that they numerically simulate to capture whether the circadian time series entrains to the forcing period completely (phase locking) or only partially (phase slips) or neither. They validate the phase model, and numerics with the experimental time series forced under different temperature cycles. Application of HT to the circadian time series appears to be a promising tool to extract the characteristic information about circadian rhythms.

A Geometric Approach to Phase Response Curves and Its Numerical Computation Through the Parameterization Method

The Phase Response Curve (PRC) is a tool used in neuroscience that measures the phase shift experienced by an oscillator due to a perturbation applied at different phases of the limit cycle. In this paper we present a new approach to PRCs based on the parameterization method. The underlying idea relies on the construction of a periodic system whose corresponding stroboscopic map has an invariant curve. We demonstrate the relationship between the internal dynamics of this invariant curve and the PRC, which yields a method to numerically compute the PRCs. Moreover, we link the existence properties of this invariant curve as the amplitude of the perturbation is increased with changes in the PRC waveform and with the geometry of isochrons. The invariant curve and its dynamics will be computed by means of the parameterization method consisting of solving an invariance equation. We show that the method to compute the PRC can be extended beyond the breakdown of the curve by means of introducing a modified invariance equation. The method also computes the amplitude response functions (ARCs) which provide information on the displacement away from the oscillator due to the effects of the perturbation. Finally, we apply the method to several classical models in neuroscience to illustrate how the results herein extend the framework of computation and interpretation of the PRC and ARC for perturbations of large amplitude and not necessarily pulsatile.

Autaptic Effects on Synchronization and Phase Response Curves of Neurons with a Chemical Synapse

Previously, we reported the effects of an autapse, a time-delayed self-feedback synapse, on synchronization of Morris-Lecar (ML) neurons with an electrical synapse [J. Korean Phys. Soc. 71, 63 (2017)]. Because communication between neurons via a chemical synapse is more common than that via a electrical synapse, the desire to study the autaptic effect on synchronization of neurons with a chemical synapse is natural. In this paper, we present the results for the autaptic effects on synchronization of neurons with a chemical synapse. We show that the phase response curves of a single neuron and a pair of neurons are useful for understanding the autaptic effects on synchronization. We introduce a model for a theoretical analysis of the autaptic effect on synchronization of parameter-mismatched neurons and find that the model qualitatively explains that effect well.

How to correctly quantify neuronal phase-response curves from noisy recordings.

At the level of individual neurons, various coding properties can be inferred from the input-output relationship of a cell. For small inputs, this relation is captured by the phase-response curve (PRC), which measures the effect of a small perturbation on the timing of the subsequent spike. Experimentally, however, an accurate experimental estimation of PRCs is challenging. Despite elaborate measurement efforts, experimental PRC estimates often cannot be related to those from modeling studies. In particular, experimental PRCs rarely resemble the characteristic theoretical PRC expected close to spike initiation, which is indicative of the underlying spike-onset bifurcation. Here, we show for conductance-based model neurons that the correspondence between theoretical and measured phase-response curve is lost when the stimuli used for the estimation are too large. In this case, the derived phase-response curve is distorted beyond recognition and takes on a generic shape that reflects the measurement protocol and masks the spike-onset bifurcation. We discuss how to identify appropriate stimulus strengths for perturbation and noise-stimulation methods, which permit to estimate PRCs that reliably reflect the spike-onset bifurcation - a task that is particularly difficult if a lower bound for the stimulus amplitude is dictated by prominent intrinsic neuronal noise.

Calcium conductance-dependent network synchronization is differentially modulated by firing frequency

Synchronous oscillations in certain frequencies of the sub-thalamic nucleus (STN) neurons are closely related to the physical symptoms of Parkinson’s disease (PD). Recent results have highlighted the importance of calcium channels in the synchronization properties and regulation of STN neurons. In this paper, a novel hybrid neuron model which can capture the electrophysiological signature of neurons with low or high density of calcium channels is used to explore the synchronization propensity and regulation by firing frequencies of neurons. Numerical simulations show that the synchronization propensity of networks consisting of the novel hybrid neurons is quite distinguishing in low and high calcium conductance modes, especially, the synchronization can be differentially modulated by network frequencies in the two modes. By analyzing the firing frequency and phase response curve of the individual neuron, we find that a single parameter of the hybrid neuron, which is a direct image of the calcium conductance, is crucial in determining the excitability and response properties of the neuron. Different phase response properties of single neurons in different calcium conductance modes lead to network synchronization discrepancies.

Chimera States on a Ring of Strongly Coupled Relaxation Oscillators

Weakly coupled oscillators can exhibit seemingly incongruous synchronization patterns comprised of coherent and incoherent spatial domains known as chimera states. However, the weak coupling approximation is invalid when the characteristic phase response curve of an oscillator does not scale linearly with the coupling strength and instead changes its shape. In chemical experiments with photo-coupled relaxation oscillators, we find that beyond weak coupling chimera patterns consist of different coexisting cluster states. Numerical modeling reveals that the observed cluster states result from a phase-dependent excitability that is also commonly observed in neural tissue and cardiac pacemaker cells.

Lifespan is unaffected by size and direction of daily phase shifts in Nasonia, a hymenopteran insect with strong circadian light resetting

Most organisms have an endogenous circadian clock with a period length of approximately 24 h that enables adaptation, synchronization and anticipation to environmental cycles. The circadian system (circa = about or around, diem = a day) may provide evolutionary benefits when entrained to the 24-h light-dark cycle. The more the internal circadian period (τ) deviates from the external light-dark cycle, the larger the daily phase shifts need to be to synchronize to the environment. In some species, large daily phase shifts reduce survival rate. Here we tested this ‘resonance fitness hypothesis’ on the diurnal wasp Nasonia vitripennis, which exhibits a large latitudinal cline in free-running period with longer circadian period lengths in the north than in the south. Longevity was measured in northern and southern wasps placed into light-dark cycles (T-cycles) with periods ranging from 20 h to 28 h. Further, locomotor activity was recorded to estimate range and phase angle of entrainment under these various T-cycles. A light pulse induced phase response curve (PRC) was measured in both lines to understand entrainment results. We expected a concave survival curve with highest longevity at T = τ and a reduction in longevity the further τ deviates from T (τ/T<>1). Our results do not support this resonance fitness hypothesis. We did not observe a reduction in longevity when τ deviates from T. Our results may be understood by the strong circadian light resetting mechanism (type 0 PRC) to single light pulses that we measured in Nasonia, resulting in: (1) the broad range of entrainment, (2) the wide natural variation in circadian free-running period, and (3) the lack of reduced survival when τ/T ratio’s deviates from 1. Together this indicates that circadian adaption to latitude may lead to changes in circadian period and light response, without negative influences on survival.

Compensating for Sensor Error in the Model Predictive Control of Circadian Clock Phase.

The circadian oscillator regulates many critical biological functions; misalignment between the phase of this oscillator and the environment has been linked to adverse health outcomes. Thus, shifting the circadian phase of the oscillator to align with the environment using either light or small molecule pharmaceuticals as control inputs is desired. One challenge to controlling circadian phase is that the magnitude and direction of the phase shift caused by these inputs is dependent on the phase at which the input is delivered. Simulations show that model predictive control (MPC) can successfully shift the phase of the circadian clock using perfect knowledge of the current phase of the system. However, methods to assess circadian phase continuously in real time, as would be needed to implement MPC in vivo, are limited in their accuracy. Here, we explore the impact of imperfect sensing on our ability to control circadian phase. While some pathological patterns of sensor error can make control impossible, we show that by assuming errors in the phase sensor are bounded to be sufficiently small, we can bound the error of our MPC algorithm. We propose using the expected phase response curve to improve control when sensor error is present.

Effects of Sodium Lighting On Circadian Rhythms in Rats.

Rodent studies often must be conducted during an animal's active phase (that is, in darkness) yet also during a typical day shift for staff. Low-pressure sodium lighting (LPSL), to which human retinas are more sensitive than rodents' at low intensity, has been used to facilitate study conduct in dark phase. The assumption was that LPSL would be equivalent to total darkness due to low rodent retinal sensitivity but provide enough lighting for safe technical manipulations due to higher human retinal sensitivity. Unlike other light sources, LPSL has been tested for effects on circadian rhythm specific to locomotive activities in albino mice. Whether LPSL affects circadian rhythms in rats is unknown. In this study, circadian endpoints were derived from body temperature and locomotor activity via telemeters in 8 adult male Wistar rats. When moved from a 12:12-h white-light (that is, cold white fluorescent light):dark (LD) cycle to a 12:12-h white-light:sodium-light cycle, rats demonstrated free-running and disrupted circadian rhythms (that is, lengthened circadian period and reduced circadian robustness and amplitude). Body temperature and locomotor activity were significantly lower in the LPSL phase as compared with dark phase under the baseline condition. When exposed to a 12:12 h sodium-light:dark (SD) cycle, rats entrained with a circadian period similar to 12:12-h white-light:dark (LD), but significantly different from the period under constant darkness (DD). Circadian onset and acrophase were delayed under SD compared with LD. When illuminated with a LPSL pulse under DD, rats showed phase shifts similar to white-light pulse effects, consistent with the phase response curve. To determine whether the image-forming photoreceptors are involved in this process, we used electroretinography. Compared with white light, 589-nm light generated during electroretinography elicited rod photoreceptors responses with longer latency and cone photoreceptor responses with lower amplitude. These results indicate that LPSL is a weaker zeitgeber than white light and may alter the circadian system in rats. Furthermore, because LPSL appeared to be visible to rats, it may not be an appropriate substitute for actual darkness.

Asymptotic Synchronization of Phase Oscillators With a Single Input

We consider asymptotic synchronization of multiple phase oscillators represented by first-order reduced dynamics with a phase response curve (PRC) function. A single control input is employed to synchronize the oscillators. We design a feedback control that achieves asymptotic synchronization of the oscillators. We also show that the undesired equilibria of the closed-loop system are unstable, thereby guaranteeing almost global synchronization. We extend the design to address the scenarios where the control is symmetrically or asymmetrically bounded and where the coefficients in the PRC function are unknown constants. Simulation results are presented to validate the proposed control algorithms.

Ensemble Controllability of Cellular Oscillators

Many diseases including cancer, Parkinson’s disease and heart diseases are caused by loss or malfunction of regulatory mechanism of an oscillatory system. Successful treatment of these diseases might involve recovering the healthy behavior of the oscillators in the system, i.e., achieving synchrony or a desired distribution of the oscillators on their periodic orbit. In this letter, we consider the problem of controlling the distribution of a population of cellular oscillators described in terms of phase models. Different practical limitations on the observability and controllability of cellular states naturally lead to an ensemble control formulation in which a population-level feedback law for achieving a desired distribution is sought. A systems theoretic approach to this problem leads to Lyapunov- and LaSalle-like arguments, from which we develop our main contribution, novel necessary and sufficient conditions for the controllability of phase distributions in terms of the Fourier coefficients of the phase response curve. Since our treatment is based on a rather universal formulation of phase models, the results and methods proposed in this letter are readily applicable to the control of a wide range of other types of oscillating populations, such as circadian clocks, and spiking neurons.