Hyperbolic Brain Modelling and Neurocognitive Decline Analysis for Disease Detection

Mapping hierarchical brain networks within traditional Euclidean space causes significant structural distortion, undermining neuroimaging diagnostic frameworks. While hyperbolic models like the Poincaré ball preserve these nested topologies, they demand heavy computational overhead due to intricate Möbius operations and curved geodesics. This paper introduces a highly efficient non-Euclidean framework for analyzing neurocognitive decline utilizing the Beltrami-Klein ball model. By projecting hyperbolic geodesics as Euclidean straight lines, this approach converts complex distance calculations into simple dot products, radically reducing processing demands. We validated our methodology against state-of-the-art Poincaré and Lorentz baselines using datasets for Schizophrenia, Parkinson's Disease, and Alzheimer's Disease. The Klein-based framework demonstrates superior performance, delivering both higher diagnostic precision and accelerated processing velocities across all three neurocognitive disorders.