Weighted spike-triggered average of a fluctuating stimulus yielding the phase response curve

Phase autoencoder for limit-cycle oscillators

We present a phase autoencoder that encodes the asymptotic phase of a limit-cycle oscillator, a fundamental quantity characterizing its synchronization dynamics. This autoencoder is trained in such a way that its latent variables directly represent the asymptotic phase of the oscillator. The trained autoencoder can perform two functions without relying on the mathematical model of the oscillator: first, it can evaluate the asymptotic phase and the phase sensitivity function of the oscillator; second, it can reconstruct the oscillator state on the limit cycle in the original space from the phase value as an input. Using several examples of limit-cycle oscillators, we demonstrate that the asymptotic phase and the phase sensitivity function can be estimated only from time-series data by the trained autoencoder. We also present a simple method for globally synchronizing two oscillators as an application of the trained autoencoder.

An extended Hilbert transform method for reconstructing the phase from an oscillatory signal

Rhythmic activity is ubiquitous in biological systems from the cellular to organism level. Reconstructing the instantaneous phase is the first step in analyzing the essential mechanism leading to a synchronization state from the observed signals. A popular method of phase reconstruction is based on the Hilbert transform, which can only reconstruct the interpretable phase from a limited class of signals, e.g., narrow band signals. To address this issue, we propose an extended Hilbert transform method that accurately reconstructs the phase from various oscillatory signals. The proposed method is developed by analyzing the reconstruction error of the Hilbert transform method with the aid of Bedrosian's theorem. We validate the proposed method using synthetic data and show its systematically improved performance compared with the conventional Hilbert transform method with respect to accurately reconstructing the phase. Finally, we demonstrate that the proposed method is potentially useful for detecting the phase shift in an observed signal. The proposed method is expected to facilitate the study of synchronization phenomena from experimental data.

Estimating asymptotic phase and amplitude functions of limit-cycle oscillators from time series data

We propose a method for estimating the asymptotic phase and amplitude functions of limit-cycle oscillators using observed time series data without prior knowledge of their dynamical equations. The estimation is performed by polynomial regression and can be solved as a convex optimization problem. The validity of the proposed method is numerically illustrated by using two-dimensional limit-cycle oscillators as examples. As an application, we demonstrate data-driven fast entrainment with amplitude suppression using the optimal periodic input derived from the estimated phase and amplitude functions.

Noninvasive inference methods for interaction and noise intensities of coupled oscillators using only spike time data

Identifying interactions is essential for understanding self-organized systems because they are a source of order, function, and complexity. However, distinguishing interaction and noise effect is generally difficult because they both critically affect the stability of an ordered state. We propose methods that enable us to simultaneously infer both interaction and noise intensities. Our methods use only the time series of periodic events such as spike time data and do not require any external stimuli. Moreover, it is not necessary to assume a function form to fit. We numerically demonstrate that our methods yield reasonable inference even for a relatively short time series. These features are particularly beneficial for application in biological and chemical complex systems.

Measurements of interaction intensity are generally achieved by observing responses to perturbations. In biological and chemical systems, external stimuli tend to deteriorate their inherent nature, and thus, it is necessary to develop noninvasive inference methods. In this paper, we propose theoretical methods to infer coupling strength and noise intensity simultaneously in two well-synchronized noisy oscillators through observations of spontaneously fluctuating events such as neural spikes. A phase oscillator model is applied to derive formulae relating each of the parameters to spike time statistics. Using these formulae, each parameter is inferred from a specific set of statistics. We verify these methods using the FitzHugh–Nagumo model as well as the phase model. Our methods do not require external perturbations and thus can be applied to various experimental systems.

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.

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.

Inferring the dynamics of oscillatory systems using recurrent neural networks

We investigate the predictive power of recurrent neural networks for oscillatory systems not only on the attractor but in its vicinity as well. For this, we consider systems perturbed by an external force. This allows us to not merely predict the time evolution of the system but also study its dynamical properties, such as bifurcations, dynamical response curves, characteristic exponents, etc. It is shown that they can be effectively estimated even in some regions of the state space where no input data were given. We consider several different oscillatory examples, including self-sustained, excitatory, time-delay, and chaotic systems. Furthermore, with a statistical analysis, we assess the amount of training data required for effective inference for two common recurrent neural network cells, the long short-term memory and the gated recurrent unit.

Inferring the phase response curve from observation of a continuously perturbed oscillator

Phase response curves are important for analysis and modeling of oscillatory dynamics in various applications, particularly in neuroscience. Standard experimental technique for determining them requires isolation of the system and application of a specifically designed input. However, isolation is not always feasible and we are compelled to observe the system in its natural environment under free-running conditions. To that end we propose an approach relying only on passive observations of the system and its input. We illustrate it with simulation results of an oscillator driven by a stochastic force.

Multicellularity enriches the entrainment of Arabidopsis circadian clock

The phase response curve (PRC) of the circadian clock provides one of the most significant indices for anticipating entrainment of outer cycles, despite the difficulty of making precise PRC determinations in experiments. We characterized the PRC of the Arabidopsisthaliana circadian clock on the basis of its phase-locking property to variable periodic pulse perturbations. Experiments revealed that the PRC changed remarkably from continuous to discontinuous fashion, depending on the oscillation amplitude. Our hypothesis of amplitude-dependent adaptability to outer cycles was successfully clarified by elucidation of this transition of PRC as a change in the collective response of the circadian oscillator network. These findings provide an essential criterion against which to evaluate the precision of PRC measurement and an advanced understanding of the adaptability of plant circadian systems to environmental conditions.

Measurement of phase resetting curves using optogenetic barrage stimuli

BACKGROUND

The phase resetting curve (PRC) is a primary measure of a rhythmically firing neuron's responses to synaptic input, quantifying the change in phase of the firing oscillation as a function of the input phase. PRCs provide information about whether neurons will synchronize due to synaptic coupling or shared input. However, PRC estimation has been limited to in vitro preparations where stable intracellular recordings can be obtained and background activity is minimal, and new methods are required for in vivo applications.

NEW METHOD

We estimated PRCs using dense optogenetic stimuli and extracellular spike recording. Autonomously firing neurons in substantia nigra pars reticulata (SNr) of Thy1-channelrhodopsin 2 (ChR2) transgenic mice were stimulated with random barrages of light pulses, and PRCs were determined using multiple linear regression.

RESULTS

The PRCs obtained were type-I, showing only phase advances in response to depolarizing input, and generally sloped upward from early to late phases. Secondary PRCs, indicating the effect on the subsequent ISI, showed phase delays primarily for stimuli arriving at late phases. Phase models constructed from the optogenetic PRCs accounted for a large fraction of the variance in ISI length and provided a good approximation of the spike-triggered average stimulus.

COMPARISON WITH EXISTING METHODS

Compared to methods based on intracellular current injection, the new method sacrifices some temporal resolution. However, it should be much more widely applicable in vivo, because only extracellular recording and optogenetic stimulation are required.

CONCLUSIONS

These results demonstrate PRC estimation using methods suitable for in vivo applications.

Evaluation of the Phase-Dependent Rhythm Control of Human Walking Using Phase Response Curves

Humans and animals control their walking rhythms to maintain motion in a variable environment. The neural mechanism for controlling rhythm has been investigated in many studies using mechanical and electrical stimulation. However, quantitative evaluation of rhythm variation in response to perturbation at various timings has rarely been investigated. Such a characteristic of rhythm is described by the phase response curve (PRC). Dynamical simulations of human skeletal models with changing walking rhythms (phase reset) described a relation between the effective phase reset on stability and PRC, and phase reset around touch-down was shown to improve stability. A PRC of human walking was estimated by pulling the swing leg, but such perturbations hardly influenced the stance leg, so the relation between the PRC and walking events was difficult to discuss. This research thus examines human response to variations in floor velocity. Such perturbation yields another problem, in that the swing leg is indirectly (and weakly) perturbed, so the precision of PRC decreases. To solve this problem, this research adopts the weighted spike-triggered average (WSTA) method. In the WSTA method, a sequential pulsed perturbation is used for stimulation. This is in contrast with the conventional impulse method, which applies an intermittent impulsive perturbation. The WSTA method can be used to analyze responses to a large number of perturbations for each sequence. In the experiment, perturbations are applied to walking subjects by rapidly accelerating and decelerating a treadmill belt, and measured data are analyzed by the WSTA and impulse methods. The PRC obtained by the WSTA method had clear and stable waveforms with a higher temporal resolution than those obtained by the impulse method. By investigation of the rhythm transition for each phase of walking using the obtained PRC, a rhythm change that extends the touch-down and mid-single support phases is found to occur.

Humans and animals tune their walking rhythms when motion is disturbed, such that they hesitate before making the transition from stance to swing phase. The effectiveness of rhythm control for stability has also been shown, and thus the elucidation of rhythm responses is important to understanding human strategies for walking control. In this research, how and when humans change their walking rhythm in response to disturbance is analyzed over the complete walking cycle. Phase response of human walking has previously been estimated by pulling the swing leg. The problem with this perturbation is that it hardly disturbs the stance leg, so here we apply the perturbation by changing floor velocity. However, perturbation from the floor yields another problem in that it weakly influences the swing leg, decreasing the precision of the PRC. The present research tackles this problem by introducing a new method for identifying rhythm characteristics by use of high-frequency perturbation, which allows us to obtain results with clear temporal resolution. We found that the human walking rhythm changes by lengthening the touch-down and mid-single support phases. These phase responses are compared with neural mechanisms for rhythm control, and relevance to the cutaneous and proprioceptive originated responses is shown.

On the firing rate dependency of the phase response curve of rat Purkinje neurons in vitro

Synchronous spiking during cerebellar tasks has been observed across Purkinje cells: however, little is known about the intrinsic cellular mechanisms responsible for its initiation, cessation and stability. The Phase Response Curve (PRC), a simple input-output characterization of single cells, can provide insights into individual and collective properties of neurons and networks, by quantifying the impact of an infinitesimal depolarizing current pulse on the time of occurrence of subsequent action potentials, while a neuron is firing tonically. Recently, the PRC theory applied to cerebellar Purkinje cells revealed that these behave as phase-independent integrators at low firing rates, and switch to a phase-dependent mode at high rates. Given the implications for computation and information processing in the cerebellum and the possible role of synchrony in the communication with its post-synaptic targets, we further explored the firing rate dependency of the PRC in Purkinje cells. We isolated key factors for the experimental estimation of the PRC and developed a closed-loop approach to reliably compute the PRC across diverse firing rates in the same cell. Our results show unambiguously that the PRC of individual Purkinje cells is firing rate dependent and that it smoothly transitions from phase independent integrator to a phase dependent mode. Using computational models we show that neither channel noise nor a realistic cell morphology are responsible for the rate dependent shift in the phase response curve.

Spectral reconstruction of phase response curves reveals the synchronization properties of mouse globus pallidus neurons

The propensity of a neuron to synchronize is captured by its infinitesimal phase response curve (iPRC). Determining whether an iPRC is biphasic, meaning that small depolarizing perturbations can actually delay the next spike, if delivered at appropriate phases, is a daunting experimental task because negative lobes in the iPRC (unlike positive ones) tend to be small and may be occluded by the normal discharge variability of a neuron. To circumvent this problem, iPRCs are commonly derived from numerical models of neurons. Here, we propose a novel and natural method to estimate the iPRC by direct estimation of its spectral modes. First, we show analytically that the spectral modes of the iPRC of an arbitrary oscillator are readily measured by applying weak harmonic perturbations. Next, applying this methodology to biophysical neuronal models, we show that a low-dimensional spectral reconstruction is sufficient to capture the structure of the iPRC. This structure was preserved reasonably well even with added physiological scale jitter in the neuronal models. To validate the methodology empirically, we applied it first to a low-noise electronic oscillator with a known design and then to cortical pyramidal neurons, recorded in whole cell configuration, that are known to possess a monophasic iPRC. Finally, using the methodology in conjunction with perforated-patch recordings from pallidal neurons, we show, in contrast to recent modeling studies, that these neurons have biphasic somatic iPRCs. Biphasic iPRCs would cause lateral somatically targeted pallidal inhibition to desynchronize pallidal neurons, providing a plausible explanation for their lack of synchrony in vivo.

Statistical method for detecting phase shifts in alpha rhythm from human electroencephalogram data

We developed a statistical method for detecting discontinuous phase changes (phase shifts) in fluctuating alpha rhythms in the human brain from electroencephalogram (EEG) data obtained in a single trial. This method uses the state space models and the line process technique, which is a Bayesian method for detecting discontinuity in an image. By applying this method to simulated data, we were able to detect the phase and amplitude shifts in a single simulated trial. Further, we demonstrated that this method can detect phase shifts caused by a visual stimulus in the alpha rhythm from experimental EEG data even in a single trial. The results for the experimental data showed that the timings of the phase shifts in the early latency period were similar between many of the trials, and that those in the late latency period were different between the trials. The conventional averaging method can only detect phase shifts that occur at similar timings between many of the trials, and therefore, the phase shifts that occur at differing timings cannot be detected using the conventional method. Consequently, our obtained results indicate the practicality of our method. Thus, we believe that our method will contribute to studies examining the phase dynamics of nonlinear alpha rhythm oscillators.

Parameterized phase response curves for characterizing neuronal behaviors under transient conditions

Phase response curves (PRCs) are a simple model of how a neuron's spike time is affected by synaptic inputs. PRCs are useful in predicting how networks of neurons behave when connected. One challenge in estimating a neuron's PRCs experimentally is that many neurons do not have stationary firing rates. In this article we introduce a new method to estimate PRCs as a function of firing rate of the neuron. We call the resulting model a parameterized PRC (pPRC). Experimentally, we perturb the neuron applying a current with two parts: 1) a current held constant between spikes but changed at the onset of a spike, used to make the neuron fire at different rates, and 2) a pulse to emulate a synaptic input. A model of the applied constant current and the history is made to predict the interspike interval (ISI). A second model is then made to fit the modulation of the spike time from the expected ISI by the pulsatile stimulus. A polynomial with two independent variables, the stimulus phase and the expected ISI, is used to model the pPRC. The pPRC is validated in a computational model and applied to pyramidal neurons from the CA1 region of the hippocampal slices from rat. The pPRC can be used to model the effect of changing firing rates on network synchrony. It can also be used to characterize the effects of neuromodulators and genetic mutations (among other manipulations) on network synchrony. It can also easily be extended to account for more variables.

Efficient estimation of phase-response curves via compressive sensing

The phase-response curve (PRC), relating the phase shift of an oscillator to external perturbation, is an important tool to study neurons and their population behavior. It can be experimentally estimated by measuring the phase changes caused by probe stimuli. These stimuli, usually short pulses or continuous noise, have a much wider frequency spectrum than that of neuronal dynamics. This makes the experimental data high dimensional while the number of data samples tends to be small. Current PRC estimation methods have not been optimized for efficiently discovering the relevant degrees of freedom from such data. We propose a systematic and efficient approach based on a recently developed signal processing theory called compressive sensing (CS). CS is a framework for recovering sparsely constructed signals from undersampled data and is suitable for extracting information about the PRC from finite but high-dimensional experimental measurements. We illustrate how the CS algorithm can be translated into an estimation scheme and demonstrate that our CS method can produce good estimates of the PRCs with simulated and experimental data, especially when the data size is so small that simple approaches such as naive averaging fail. The tradeoffs between degrees of freedom vs. goodness-of-fit were systematically analyzed, which help us to understand better what part of the data has the most predictive power. Our results illustrate that finite sizes of neuroscientific data in general compounded by large dimensionality can hamper studies of the neural code and suggest that CS is a good tool for overcoming this challenge.

Measurement of infinitesimal phase response curves from noisy real neurons

We sought to measure infinitesimal phase response curves (iPRCs) from rat hippocampal CA1 pyramidal neurons. It is difficult to measure iPRCs from noisy neurons because of the dilemma that either the linearity or the signal-to-noise ratio of responses to external perturbations must be sacrificed. To overcome this difficulty, we used an iPRC measurement model formulated as the Langevin phase equation (LPE) to extract iPRCs in the Bayesian scheme. We then simultaneously verified the effectiveness of the measurement model and the reliability of the estimated iPRCs by demonstrating that LPEs with the estimated iPRCs could predict the stochastic behaviors of the same neurons, whose iPRCs had been measured, when they were perturbed by periodic stimulus currents. Our results suggest that the LPE is an effective model for real oscillating neurons and that many theoretical frameworks based on it may be applicable to real nerve systems.

Phase resetting of collective rhythm in ensembles of oscillators

Phase resetting curves characterize the way a system with a collective periodic behavior responds to perturbations. We consider globally coupled ensembles of Sakaguchi-Kuramoto oscillators, and use the Ott-Antonsen theory of ensemble evolution to derive the analytical phase resetting equations. We show the final phase reset value to be composed of two parts: an immediate phase reset directly caused by the perturbation and the dynamical phase reset resulting from the relaxation of the perturbed system back to its dynamical equilibrium. Analytical, semianalytical and numerical approximations of the final phase resetting curve are constructed. We support our findings with extensive numerical evidence involving identical and nonidentical oscillators. The validity of our theory is discussed in the context of large ensembles approximating the thermodynamic limit.

Bayesian estimation of phase response curves

Phase response curve (PRC) of an oscillatory neuron describes the response of the neuron to external perturbation. The PRC is useful to predict synchronized dynamics of neurons; hence, its measurement from experimental data attracts increasing interest in neural science. This paper introduces a Bayesian method for estimating PRCs from data, which allows for the correlation of errors in explanatory and response variables of the PRC. The method is implemented with a replica exchange Monte Carlo technique; this avoids local minima and enables efficient calculation of posterior averages. A test with artificial data generated by the noisy Morris-Lecar equation shows that the proposed method outperforms conventional regression that ignores errors in the explanatory variable. Experimental data from the pyramidal cells in the rat motor cortex is also analyzed with the method; a case is found where the result with the proposed method is considerably different from that obtained by conventional regression.

A comparison of methods to determine neuronal phase-response curves

The phase-response curve (PRC) is an important tool to determine the excitability type of single neurons which reveals consequences for their synchronizing properties. We review five methods to compute the PRC from both model data and experimental data and compare the numerically obtained results from each method. The main difference between the methods lies in the reliability which is influenced by the fluctuations in the spiking data and the number of spikes available for analysis. We discuss the significance of our results and provide guidelines to choose the best method based on the available data.

Coding of information in limit cycle oscillators

Starting from a general description of noisy limit cycle oscillators, we derive from the Fokker-Planck equations the linear response of the instantaneous oscillator frequency to a time-varying external force. We consider the time series of zero crossings of the oscillator's phase and compute the mutual information between it and the driving force. A direct link is established between the phase response curve summarizing the oscillator dynamics and the ability of a limit cycle oscillator, such as a heart cell or neuron, to encode information in the timing of peaks in the oscillation.