Mathematical modeling of the relationship between the functional activity and the structural wiring of the brain has largely been undertaken using non-linear and biophysically detailed mathematical models with regionally varying parameters. While this approach provides us a rich repertoire of multistable dynamics that can be displayed by the brain, it is computationally demanding. Moreover, although neuronal dynamics at the microscopic level are nonlinear and chaotic, it is unclear if such detailed nonlinear models are required to capture the emergent meso- (regional population ensemble) and macro-scale (whole brain) behavior, which is largely deterministic and reproducible across individuals. Indeed, recent modeling effort based on spectral graph theory has shown that an analytical model without regionally varying parameters can capture the empirical magnetoencephalography frequency spectra and the spatial patterns of the alpha and beta frequency bands accurately.
In this work, we demonstrate an improved hierarchical, linearized, and analytic spectral graph theory-based model that can capture the frequency spectra obtained from magnetoencephalography recordings of resting healthy subjects. We reformulated the spectral graph theory model in line with classical neural mass models, therefore providing more biologically interpretable parameters, especially at the local scale. We demonstrated that this model performs better than the original model when comparing the spectral correlation of modeled frequency spectra and that obtained from the magnetoencephalography recordings. This model also performs equally well in predicting the spatial patterns of the empirical alpha and beta frequency bands.
Almost everywhere convergence and divergence of Cesàro means with varying parameters of Walsh–Fourier series
