BioMed Research International

Volume 2015, Article ID 140837, 8 pages

http://dx.doi.org/10.1155/2015/140837

## A Time-Series Model of Phase Amplitude Cross Frequency Coupling and Comparison of Spectral Characteristics with Neural Data

The Department of Mathematics & Statistics, Boston University, 111 Cummington Mall, Boston, MA 02215, USA

Received 25 January 2015; Accepted 5 March 2015

Academic Editor: Tjeerd Boonstra

Copyright © 2015 Kyle Q. Lepage and Sujith Vijayan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Stochastic processes that exhibit cross-frequency coupling (CFC) are introduced. The ability of these processes to model observed CFC in neural recordings is investigated by comparison with published spectra. One of the proposed models, based on multiplying a pulsatile function of a low-frequency oscillation () with an unobserved and high-frequency component, yields a process with a spectrum that is consistent with observation. Other models, such as those employing a biphasic pulsatile function of a low-frequency oscillation, are demonstrated to be less suitable. We introduce the full stochastic process time series model as a summation of three component weak-sense stationary (WSS) processes, namely, , , and , with a noise process. The process is constructed as a product of a latent and unobserved high-frequency process with a function of the lagged, low-frequency oscillatory component (). After demonstrating that the model process is WSS, an appropriate method of simulation is introduced based upon the WSS property. This work may be of interest to researchers seeking to connect inhibitory and excitatory dynamics directly to observation in a model that accounts for known temporal dependence or to researchers seeking to examine what can occur in a multiplicative time-domain CFC mechanism.

#### 1. Introduction

Cross frequency coupling (CFC) is a statistical relation between the phase or amplitude of a low frequency and the phase or amplitude of a high frequency. In this work, focus is placed upon phase-amplitude CFC, which can be thought of as the correlation of the amplitude of a relatively high-frequency oscillation () with the phase of a lower frequency oscillation ().

The rationale behind this proposal is based on the experimental observations that (i) relatively slow frequency oscillations tend to be coordinated over large regions of neural tissue, unlike higher frequency oscillations [1, 2], and (ii) oscillatory activity reflects changes in the excitability of neural tissue [3]. Hence, a lower frequency oscillation may provide time intervals in which high-frequency activity may occur and may consequently coordinate high-frequency oscillations that are spatially disparate.

Recently, the theoretical interest in phase-amplitude CFC has been reinforced. Recent observations of the phenomenon have occurred in varied species [4–6], brain regions [7–9], and states of vigilance [10]. Experimental studies have shown that the nature of phase-amplitude CFC can be altered by predictive cues and attentional demands; the phase of the low-frequency oscillation can be reset such that a stimulus of attentional interest arrives at the phase of maximal excitability [11]. Furthermore, phase-amplitude CFC has been implicated in learning and memory [6, 12], and the dynamics of phase-amplitude CFC have been shown to change over the course of a cognitive task [8].

In this work, in a fashion akin to that classically employed for parametrically modeling the spectra of time-series, stochastic time-series models are constructed which exhibit cross frequency coupling (see, e.g., [13–15]). Through simulation it is shown that these models can produce time-series that exhibit CFC similar to that exhibited in neural recordings. Some mathematical properties of these models are given, and some consequences are discussed.

#### 2. Methodology

Stochastic processes exhibiting cross frequency coupling are introduced. The ability for these processes to model observed cross frequency coupling in neural recordings is investigated and mathematical properties of the new models are given. Investigations are conducted through the use of mathematical analysis and simulation.

##### 2.1. Model Specification

The stochastic, or random, processes introduced in this work model CFC phenomena in the following way. For each recorded measurement a random variable is introduced; the measurement is modeled as a realization of this random variable. The collection of random variables comprises a discrete-time random process. The observed time-series is modeled as the corresponding collection of realizations of each of the random variables in the random, or stochastic, process. This is the standard setup in classical time-series analysis [13] and it includes the independent and identically distributed (IID) random sample as a special case.

Each of the random processes is characterized by all of the possible moments between the random variables; in this work consideration will be restricted to random processes whose joint distributions are Gaussian. In this situation, specification of the first two joint moments completely specifies the model.

Without restriction upon the time dependence of pair-wise correlation, the number of unique pair-wise correlations increases quadratically with every new observation (in a single trial). This is a more challenging regime to perform inference than is typically considered, as in this case the number of unknowns is growing rapidly with increasing observation length. Contemporary work deals with this issue by recording many trials, or by using models with other restrictions.

Here, as is often customary, the introduced models are weak-sense stationary (WSS) random processes [13–15]. The weak-sense stationary property implies that the correlation between any pair of random variables in the process does not depend upon absolute time, but, rather, only upon the difference in the times associated with the pair. It also implies that the mean of each of the random variables in the process is equal; thus the process mean is also independent of time. Gaussian weak-sense stationary random processes are amongst the simplest of time-series models and the number of pair-wise correlations for these processes is of the same order as the number of measurements.

In this work, the random processes exhibiting CFC are constructed from component WSS processes modeling -rhythm, -rhythm, background activity, and sensor noise. The mean of these processes is taken to be zero, consistent with randomly observed neural phenomena (nonevoked). Based upon observed spectra, the autocorrelation of the , , and noise components is specified in the Fourier domain. Based upon the discrete-time analog of the Wiener-Khintchine theorem, the autocorrelation of each of these component processes is obtained by inverse discrete-time Fourier transforming the specified spectra [13].

###### 2.1.1. The Component Processes

Let be the integer-valued time-index of the length WSS zero-mean random process with autocovariance sequence :Here denotes the expected value of the random variable . Similarly, specify WSS zero-mean processes for the rhythm and noise, , components of the model. That is, Further, specify the and components as uncorrelated. Because both and are also zero-mean, it follows that is equal to zero. These components are jointly Gaussian, uncorrelated, and hence they are independent. The components and are linked to model CFC. This linking and its consequence are discussed in Section 2.1.2. It remains to specify the autocovariance sequences , , and . As described, this is accomplished by specificying their respective spectra, , , and , and using the example relation obtained by applying the discrete-time analog to the Wiener-Khintchine theorem [13]: Here is the Nyquist frequency, equal to (in Hz), specified in terms of the sample period (in ). Figures 1 and 2 depict the specified model autocovariance sequences and spectra for the and components (resp.). The component is further detailed in Section 2.1.2.