A Mathematical Model for the Variation of Cerebral Electrical Conductivity and the Amount of β-Amyloid Protein Values Due to Alzheimer’s Disease

Abstract

Background/Objectives: This study presents a time-dependent mathematical model that describes how progressive amyloid-β (Aβ) accumulation drives the gradual decline of cerebral electrical conductivity during Alzheimer’s disease (AD). Methods: The formulation captures the coupled evolution of molecular burden and electrophysiological function through a pair of interconnected dynamical processes, enabling a mechanistic link between early biochemical alterations and large-scale neural degradation. Results: Simulations reveal a characteristic pattern in which Aβ levels rise steadily toward a pathological plateau, while conductivity follows a delayed but persistent downward trajectory that stabilizes at an impaired state consistent with advanced neurodegeneration. The model reproduces key phenomena reported in experimental and clinical studies, including the slow, irreversible reduction in cortical conductivity and the strong inverse relationship between amyloid burden and electrophysiological integrity. Conclusions: Although intentionally minimal, the framework offers a tractable basis for interpreting disease progression and can be extended to incorporate additional pathological pathways such as tau aggregation, inflammatory responses, or spatial heterogeneity. By providing a compact yet biologically meaningful representation of the interplay between molecular pathology and electrical dysfunction, the model supports the development of computational biomarkers and contributes to a more integrated understanding of AD progression.