Bayesian inference of a spectral graph model for brain oscillations

The relationship between brain functional connectivity and structural connectivity has caught extensive attention of the neuroscience community, commonly inferred using mathematical modeling. Among many modeling approaches, spectral graph model (SGM) is distinctive as it has a closed-form solution of the wideband frequency spectra of brain oscillations, requiring only global biophysically interpretable parameters. While SGM is parsimonious in parameters, the determination of SGM parameters is non-trivial. Prior works on SGM determine the parameters through a computational intensive annealing algorithm, which only provides a point estimate with no confidence intervals for parameter estimates. To fill this gap, we incorporate the simulation-based inference (SBI) algorithm and develop a Bayesian procedure for inferring the posterior distribution of the SGM parameters. Furthermore, using SBI dramatically reduces the computational burden for inferring the SGM parameters. We evaluate the proposed SBI-SGM framework on the resting-state magnetoencephalography recordings from healthy subjects and show that the proposed procedure has similar performance to the annealing algorithm in recovering power spectra and the spatial distribution of the alpha frequency band. In addition, we also analyze the correlations among the parameters and their uncertainty with the posterior distribution which cannot be done with annealing inference. These analyses provide a richer understanding of the interactions among biophysical parameters of the SGM. In general, the use of simulation-based Bayesian inference enables robust and efficient computations of generative model parameter uncertainties and may pave the way for the use of generative models in clinical translation applications.


A.1. Spectral graph model
Notation.All the vectors and matrices are written in boldface and the scalars are written in normal font.The frequency  of a signal is specified in Hertz (Hz), and the corresponding angular frequency  = 2 is used to obtain the Fourier transforms.The connectivity matrix is defined as  =   , where   is the connectivity strength between regions  and , normalized by the row degree.

Mesoscopic model
Given region  out of  regions, we denote the local excitatory signal as   (), local inhibitory signal as   (), and the long-range macroscopic signals as   (𝑡) ) .

Macroscopic model
Accounting for long-range connections between brain regions, the macroscopic signal   is assumed to conform to the following evolution model: where,   is the graph characteristic time constant,  is the global coupling constant,   are elements of the connectivity matrix,    is the delay in signals reaching from the th to the th region,  is the corticocortical fiber conduction speed with which the signals are transmitted.The delay    is calculated as   ∕, where   is the distance between regions  and  and   () +   () is the input signal determined from Eqs. ( 3) and ( 4).The Gamma-shaped   () is written as ) .
The neural gain   is kept as 1 to ensure parameter identifiability, therefore, SGM only includes 7 identifiable parameters as listed in Table 1.

Closed-form model solution in the fourier domain
A salient feature of SGM is that it provides a closed-form solution of brain oscillations under the frequency domain.Let  be the Fourier transform at angular frequency  = 2 .Note that the mesoscopic models for different regions share the same parameters, therefore, without loss of generality, we can drop the subscript .
The solutions for   () and   () under the frequency domain are =   () (), In the 10 repetitions with t(3) noise, the PSD Pearson's correlation between the reconstructed and empirical PSDs is changed in [0.9049, 0.9066] which is very similar to the results with the Gaussian noise ([0.905, 0.907] in Section 3.4).We also present the results from the representative experiment that yields a correlation closest to the mean level in the 10 repetitions for both noise types in Fig. S.1, including the reconstructed PSD, the PSD and spatial (in alpha band) correlations and the posterior densities of the 7 SGM parameters.All the results are very similar under both noise types.
The comparison between the two error types indicates that the SBI-SGM is robust to the selection of the noise distribution.