Non-linear auto-regressive models for cross-frequency coupling in neural time series

Abstract

We address the issue of reliably detecting and quantifying cross-frequency coupling (CFC) in neural time series. Based on non-linear auto-regressive models, the proposed method provides a generative and parametric model of the time-varying spectral content of the signals. As this method models the entire spectrum simultaneously, it avoids the pitfalls related to incorrect filtering or the use of the Hilbert transform on wide-band signals. As the model is probabilistic, it also provides a score of the model “goodness of fit” via the likelihood, enabling easy and legitimate model selection and parameter comparison; this data-driven feature is unique to our model-based approach. Using three datasets obtained with invasive neurophysiological recordings in humans and rodents, we demonstrate that these models are able to replicate previous results obtained with other metrics, but also reveal new insights such as the influence of the amplitude of the slow oscillation. Using simulations, we demonstrate that our parametric method can reveal neural couplings with shorter signals than non-parametric methods. We also show how the likelihood can be used to find optimal filtering parameters, suggesting new properties on the spectrum of the driving signal, but also to estimate the optimal delay between the coupled signals, enabling a directionality estimation in the coupling.

Measuring spectrally-resolved information transfer for sender- and receiver-specific frequencies

“…The network simulation was generated as follows: first, a source S 0 was coupled to a target with a CFIT. We adapted an example of [27] to simulate a more complex scenario with a non-sinusoidal driver, since in nonlinear systems such as the brain, perfect sinusoidal are often an exception [28]. The CFIT coupling was between f 1 = 6 Hz and f 2 = 80 Hz.…”

Driver Estimation in Non-Linear Autoregressive Models

“…However, since the model is based only on the driver's value, it does not disentangle the ascending phase from the descending phase of the driver. To fix this issue and obtain phase invariance, the parametrization can be improved using a complex-valued driver x = x re + jx im [13]. The parametrization is now:…”

“…Our work builds upon driven AR (DAR) models [12], which have been used in particular to estimate cross-frequency coupling (CFC) in neural time-series [13]. In a word, CFC is an inter-frequency coupling phenomenon observed in electrophysiology signals, that is believed to play a central role in functional interactions between neural ensembles [14].…”

“…A limitation of DAR models is the assumption that the driver is given. In practice, the driver is obtained by filtering an exogenous time-series, which requires to search for filter parameters over a grid of values [13].…”

“…For the former, we propose a fast optimization scheme based on quasi-Newton L-BFGS algorithm [15]. For the latter, we refer the reader to [13]. This paper is organized as follows.…”