Propriedades estatísticas do espectro de redes cerebrais funcionais

Abstract

The brain is one of the most complex networks known, composed of an enormous number of neurons and connections capable of performing countless synapses, anatomically organized across different spatial scales and interactive across various temporal scales, forming a structure of complex architecture. Understanding the functioning of this system is essential for guiding neuroscience studies, offering new perspectives for the treatment of brain disorders that still lack effective therapies. Due to theoretical advances in mathematics and statistical physics, with increasingly precise models for the description of real data, along with technological advances in computational data analysis techniques and brain mapping exams, it has become possible to approach the connectome, defined as the matrix of functional connectivities of the anatomical regions of the brain, thus allowing the brain to be studied as a complex network. This approach enables, for example, the application of Random Matrix Theory (RMT) to study the spectral statistics of this type of system. In this work, we analyzed a dataset of functional brain connections obtained by functional magnetic resonance imaging (fMRI) of individuals in a resting state. We studied the distribution of consecutive spacings (DCS), as well as the distribution of ratios between consecutive spacings (DRCS) of the adjacency matrix spectra of the data. We used the Brody distribution to fit the DCS and proposed a semi-empirical expression for the DRCS to assess the degree of correlation of the eigenvalues in the ensemble of adjacency matrices. These semi-empirical distributions depend on a parameter β that varies continuously between zero, when the spectral statistics are described by the Poisson distribution, and one, when described by the Gaussian Orthogonal Ensemble (GOE) of RMT. The fits performed provided estimates of β = 0.89 for the DCS and β = 0.87 for the DRCS, suggesting an intermediate spectral statistic between the Poisson distribution and the GOE.