The Role of Spectral Complexity in Connectivity Estimation

Abstract

The study of functional connectivity from magnetoecenphalographic (MEG) data consists of quantifying the statistical dependencies among time series describing the activity of different neural sources from the magnetic field recorded outside the scalp. This problem can be addressed by utilizing connectivity measures whose computation in the frequency domain often relies on the evaluation of the cross-power spectrum of the neural time series estimated by solving the MEG inverse problem. Recent studies have focused on the optimal determination of the cross-power spectrum in the framework of regularization theory for ill-posed inverse problems, providing indications that, rather surprisingly, the regularization process that leads to the optimal estimate of the neural activity does not lead to the optimal estimate of the corresponding functional connectivity. Along these lines, the present paper utilizes synthetic time series simulating the neural activity recorded by an MEG device to show that the regularization of the cross-power spectrum is significantly correlated with the signal-to-noise ratio of the measurements and that, as a consequence, this regularization correspondingly depends on the spectral complexity of the neural activity.

Tuning Minimum-Norm regularization parameters for optimal MEG connectivity estimation

“…(2) The non-zero elements a i,j (k) of the coefficient matrices were drawn from a normal distribution of zero mean and standard deviation Γ, whose value was randomly drawn in the interval [0.1, 1] so to obtain time series with different spectral properties (Vallarino et al, 2021). We retained only coefficient matrices providing (i) a stable MVAR model (Lütkepohl, 2005) and (ii) pairs of signals (z 1 (t), z 2 (t)) ⊤ such that the ℓ 2 -norm of the strongest one was less than 3 times the ℓ 2 -norm of the weakest one.…”

“…When the two-step approach described in the previous section is used to infer functional connectivity, the regularization parameter λ in (5) needs to be set to solve the MEG inverse problem in step (i). Previous studies (Hincapié et al, 2016;Vallarino et al, 2020Vallarino et al, , 2021 have shown that the value of λ needs to be set differently depending on whether one wants to obtain the best possible estimate of local neural activity or the best possible connectivity estimate. In other words, the value of λ that yields the best source reconstruction does not by extension lead to the best results when it comes to estimating connectivity from the reconstructed time series.…”

“…In the more recent mathematical treatment of the problem Vallarino et al (2020) formally proved that the optimal regularization parameter for reconstruction of the cross-power spectrum, a largely used intermediate step in the connectivity estimation, is smaller than the optimal regularization parameter for source reconstruction; however, formal treatment required the strong assumption that neural signals and noise were white-noise Gaussian processes. In a numerical follow-up, the same authors showed that the inequality holds true when the data are generated by Multi-Variate Auto-Regressive models, and that the optimal regularization parameter for cross-spectrum estimation is related to a notion of spectrum complexity, or cross-spectrum signal-to-noise ratio (SNR) (Vallarino et al, 2021). However, these last two studies limited their analysis to the estimation of the cross-power spectrum, without considering the actual connectivity metric.…”

“…In this work, we considered four different connectivity metrics, namely the cross-power spectrum (CPS), the imaginary part of Coherence (imCOH) (Nolte et al, 2004), the corrected imaginary part of Phase Locking Value (ciPLV) (Bruña et al, 2018), and the weighted Phase Lag Index (wPLI) (Vinck et al, 2011). CPS was chosen in order to benchmark our results with those from previous works (Vallarino et al, 2020(Vallarino et al, , 2021, while imCOH, ciPLV and wPLI were chosen as they are among the most widely used connectivity metrics and have been specifically designed to mitigate the effects of volume conduction or field spread by ignoring instantaneous zero-phase interactions. For all these connectivity metrics we used the implementations available in the MNE-python package (Gramfort et al, 2013).…”

Differential Models, Numerical Simulations and Applications

“…The authors of [9] study the problem of functional connectivity by quantifying the statistical dependencies among time series describing the activity of different neural sources from the magnetic field recorded with magnetoencephalographic (MEG) exam. This problem can be addressed by utilizing connectivity measures whose computation in the frequency domain often relies on the evaluation of the cross-power spectrum of the neural time series, estimated by solving the MEG inverse problem.…”