Efficient estimation of phase-response curves via compressive sensing

Abstract

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.

A practical method for estimating coupling functions in complex dynamical systems

“…We begin by considering a network composed of interacting oscillator elements. In biology, such systems include a network of circadian cells in the suprachiasmatic nucleus [56], brain network composed of many spiking neurons [43,46], cardiac muscle cells in the heart [57], etc. In terms of nonlinear dynamics, such systems are described as a system of N coupled limit cycle oscillators:…”

“…The phase sensitivity function Z(θ) plays a vital role in the studies of coupled oscillators, since it describes one of the most fundamental properties of the oscillator element [58][59][60]. Numerous approaches have been proposed to estimate the phase sensitivity function from experimental data [43][44][45][46][47][48][49][50][51][52]. As an extension of our technique, the phase sensitivity function can be recovered from the coupling function [79].…”

“…Below, we compare the performance of the multiple-shooting method with that of least squares as the standard method of estimating the phase sensitivity function [43,46]. Here, the phase model (4.1) is integrated as…”

“…In fact, as the impulse strength is increased, the estimation error increases much more rapidly in the leastsquares method (dashed blue line) than the multiple shooting method (solid red line) (figure 3d). The least-squares method [43,46] assumes that phase of the oscillator evolves linearly in time according to the natural frequency. This approximation is effective as far as the external force is weak.…”

“…For a system of phase equations, a standard way to construct the coupling function is to measure the phase sensitivity function of an individual oscillator element and obtain the coupling function by averaging method that computes the amount of phase shift induced through interaction with another oscillator element [42]. However, a precisely measured phase sensitivity function is not always accessible, since it requires application of external perturbations to an individual oscillator, which cannot always be isolated from the rest of the system [43][44][45][46][47][48][49][50][51][52].…”