A time-series model of phase amplitude cross frequency coupling and comparison of spectral characteristics with neural data.
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We introduce the full stochastic process time series model as a summation of three component weak-sense stationary (WSS) processes, namely, θ, γ, and η, with η a 1/f (α) noise process.The γ process is constructed as a product of a latent and unobserved high-frequency process x with a function of the lagged, low-frequency oscillatory component (θ).After demonstrating that the model process is WSS, an appropriate method of simulation is introduced based upon the WSS property.
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Affiliation: The Department of Mathematics & Statistics, Boston University, 111 Cummington Mall, Boston, MA 02215, USA.
ABSTRACT
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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 (θ) 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, θ, γ, and η, with η a 1/f (α) noise process. The γ process is constructed as a product of a latent and unobserved high-frequency process x with a function of the lagged, low-frequency oscillatory component (θ). 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. Related in: MedlinePlus |
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Mentions: Let t be the integer-valued time-index of the length n WSS zero-mean random process θ with autocovariance sequence r(θ):(1)rτ(θ)=Eθtθt+τ−EθtEθt+τ,=Eθtθt+τ.Here E{X} denotes the expected value of the random variable X. Similarly, specify WSS zero-mean processes for the γ rhythm and noise, η, components of the model. That is, (2)rτ(γ)=Eγtγt+τ,rτ(η)=Eηtηt+τ.Further, specify the θ and η components as uncorrelated. Because both θ and η are also zero-mean, it follows that E{θtηt′} is equal to zero. These components are jointly Gaussian, uncorrelated, and hence they are independent. The components θ and γ are linked to model CFC. This linking and its consequence are discussed in Section 2.1.2. It remains to specify the autocovariance sequences r(θ), r(γ), and r(η). As described, this is accomplished by specificying their respective spectra, S(θ), S(γ), and S(η), and using the example relation obtained by applying the discrete-time analog to the Wiener-Khintchine theorem [13]: (3)rτ(θ)=Δ∫−fNfNSθ(f)ei2πfτΔdf.Here fN is the Nyquist frequency, equal to (2Δ)−1 (in Hz), specified in terms of the sample period Δ (in s). Figures 1 and 2 depict the specified model autocovariance sequences and spectra for the θ and η components (resp.). The γ component is further detailed in Section 2.1.2. |
View Article: PubMed Central - PubMed
Affiliation: The Department of Mathematics & Statistics, Boston University, 111 Cummington Mall, Boston, MA 02215, USA.