Hyperbolic Brain Modelling and Neurocognitive Decline Analysis for Disease Detection

Abstract

Mapping hierarchical brain networks within traditional Euclidean space causes significant structural distortion, undermining neuroimaging diagnostic frameworks. While hyperbolic models like the Poincare ball preserve these nested topologies, they demand heavy computational overhead due to intricate Mobius 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 Poincare and Lorentz baselines using datasets for Schizophrenia, Parkinsons Disease, and Alzheimers Disease. The Klein-based framework demonstrates superior performance, delivering both higher diagnostic precision and accelerated processing velocities across all three neurocognitive disorders.