Spatial pattern analysis of A Aβ-monomer model with inflammation processes for Alzheimer’s disease

¹ Inria, and Université de Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, F-69603 Villeurbanne, France
² Universidad de Granada, Departamento de Matemática Aplicada, Granada, Andalusia, Spain
³ Université Claude Bernard Lyon 1, CNRS, Centrale Lyon, INSA Lyon, Université Jean Monnet, ICJ UMR5208, Inria, 69622 Villeurbanne, France
⁴ Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur 208016, India

^* Corresponding author: maxime.estavoyer@inria.fr

Received: 31 October 2024
Accepted: 17 July 2025

Abstract

We study the emergence of spatial patterns for a system of reaction-diffusion equations, modeling the progression of Alzheimer’s disease through the interaction of Aβ-monomers, oligomers, microglial cells, and interleukins with neurons. In our work, these spatial patterns stand for inert amyloid plaques, which are extracellular deposits of Aβ-proteins and a characteristic feature of this neurodegenerative disease. Using linear analysis and numerical simulations, we show the existence of spatially heterogeneous solutions and exhibit a wide variety of possible spatially-dependent solutions: time-oscillating, low-amplitude, and high-amplitude patterns. Moreover, we carry out an extensive analysis of high-amplitude patterns in the one- and two-dimensional domains. In particular, we study the stability of branches of heterogeneous steady states through bifurcation diagrams and their selection. From this numerical bifurcation analysis, we develop some conjectures concerning the influence of inflammation and microglial cells in the formation of amyloid plaques. These findings offer insights into potential anti-inflammatory treatments that might be used to mitigate the progression of Alzheimer’s disease and the emergence of inert amyloid plaques.