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Abstract

The complexity of our brains can be described as a multi-layer network: neurons, neural agglomerates, and lobes. Neurological diseases are often related to malfunctions in this network. We propose a conceptual model of the brain, describing the disease as the result of an operator affecting and disrupting the network organization. We adopt the formalism of operators, matrices, and tensor products adapted from theoretical physics. This novel approach can be tested and instantiated for different diseases, balancing mathematical formalism and data-driven findings, including pathologies where aging is included as a risk factor. We quantitatively model the K-operator from real data of Parkinson’s Disease, from the Parkinson’s Progression Markers Initiative (PPMI) upon concession by the University of Southern California. The networks are reconstructed from fMRI analysis, resulting in a matrix acting on the healthy brain and giving as output the diseased brain. We finally decompose the K-operator into the tensor product of its submatrices and we are able to assess its action on each region of interest (ROI) characterizing the brain for the specific considered samples. We also approximate the time-dependent K-operator from the fMRI of the same patient at the baseline and at the first follow-up. Our results confirm the findings of the literature on the topic. Also, these applications confirm the feasibility of the proposed analytic technique. Further research developments can compare operators for different patients and for different diseases, looking for commonalities and aiming to develop a comprehensive theoretical approach.