Bayesian estimation of phase response curves

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.

A practical method for estimating coupling functions in complex dynamical systems

A foremost challenge in modern network science is the inverse problem of reconstruction (inference) of coupling equations and network topology from the measurements of the network dynamics. Of particular interest are the methods that can operate on real (empirical) data without interfering with the system. One such earlier attempt (Tokuda et al. 2007 Phys. Rev. Lett. 99, 064101. (doi:10.1103/PhysRevLett.99.064101)) was a method suited for general limit-cycle oscillators, yielding both oscillators' natural frequencies and coupling functions between them (phase equations) from empirically measured time series. The present paper reviews the above method in a way comprehensive to domain-scientists other than physics. It also presents applications of the method to (i) detection of the network connectivity, (ii) inference of the phase sensitivity function, (iii) approximation of the interaction among phase-coherent chaotic oscillators, and (iv) experimental data from a forced Van der Pol electric circuit. This reaffirms the range of applicability of the method for reconstructing coupling functions and makes it accessible to a much wider scientific community. This article is part of the theme issue 'Coupling functions: dynamical interaction mechanisms in the physical, biological and social sciences'.

Higher-order frequency locking of an organ pipe: A measurement study based on synchronization theory

Higher-order frequency locking of an organ pipe was investigated in terms of relationships between the locking phenomena and the harmonics of the pipe sound and an external force acting onto the pipe. The authors first assumed the pipe as a phase oscillator that is used in synchronization theory and predicted frequency ratios that can cause frequency locking. The authors then forced an actual pipe using a pure tone with frequency ratios of 1 : 1, 1 : 2, 2 : 1, 1 : 3, and 2 : 3. In addition, experiments were conducted using complex tones to investigate effects of higher harmonics of the external force on frequency locking. As a result, frequency locking occurred only at frequency ratios of 1 : 1, 1 : 2, and 1 : 3 in the case of the pure tone in agreement with the prediction of synchronization theory. For the complex tone, the authors succeeded in inducing 2 : 1 locking. The results show that the frequency of a harmonic component of the external force was close to that of the pipe sound when frequency locking occurred. Frequency locking of an organ pipe was therefore enhanced through the proximity of the harmonics of the pipe sound and the external force.

Neurons as oscillators

Regularly spiking neurons can be described as oscillators. In this article we review some of the insights gained from this conceptualization and their relevance for systems neuroscience. First, we explain how a regularly spiking neuron can be viewed as an oscillator and how the phase-response curve (PRC) describes the response of the neuron's spike times to small perturbations. We then discuss the meaning of the PRC for a single neuron's spiking behavior and review the PRCs measured from a variety of neurons in a range of spiking regimes. Next, we show how the PRC can be related to a number of common measures used to quantify neuronal firing, such as the spike-triggered average and the peristimulus histogram. We further show that the response of a neuron to correlated inputs depends on the shape of the PRC. We then explain how the PRC of single neurons can be used to predict neural network behavior. Given the PRC, conduction delays, and the waveform and time course of the synaptic potentials, it is possible to predict neural population behavior such as synchronization. The PRC also allows us to quantify the robustness of the synchronization to heterogeneity and noise. We finally ask how to combine the measured PRCs and the predictions based on PRC to further the understanding of systems neuroscience. As an example, we discuss how the change of the PRC by the neuromodulator acetylcholine could lead to a destabilization of cortical network dynamics. Although all of these studies are grounded in mathematical abstractions that do not strictly hold in biology, they provide good estimates for the emergence of the brain's network activity from the properties of individual neurons. The study of neurons as oscillators can provide testable hypotheses and mechanistic explanations for systems neuroscience.

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.