[SS25] Unravelling Double Trouble: Using Mathematical Models to Tackle Ecological Challenges and Neurological Disorders

[SS25-06] A Geometrical aspect of calcium dysfunction hypothesis of Alzheimer’s disease in astrocytes and their networks

*Debasish Pradhan¹, Ranjit Kumar Upadhyay¹ (1. Indian Institute of Technology (Indian School of Mines), Dhanbad, India (India))

Keywords:

Calcium dysfunction,GSPT,Canard,Network,Synchronisation

Exposure of astrocytes to amyloid beta is believed to trigger the dysregulation of intracellular calcium oscillations, suggesting the calcium dysfunction hypothesis in Alzheimer’s disease (AD). In terms of the mathematical modelling of AD, mixed-mode oscillation and mixed-mode burst oscillations (MMOs and MMBOs) are the resulting patterns for the aberrant calcium signalling. Our study explores a mathematical model of AD based on the calcium dysfunction hypothesis in astrocytes, focusing on multi-pathway flux contributions. Our motivation lies in studying the geometrical aspects behind the emergence of the MMOs and MMBOs in our model, as well as the behaviour of the cells in the network type systems. For this, we analyse the model in two directions. First, we introduce the main model and derive an equivalent form to establish positivity, boundedness, and equilibrium conditions. The equivalent model captures diverse oscillatory behaviours, including quiescence, periodic spiking, MMOs, and MMBOs. Using geometric singular perturbation analysis (GSPT), we investigate three parameter regimes to understand the geometric mechanisms behind MMOs and MMBOs. These regimes involve one fast and two slow variables (canard dynamics) in the 1st regime, one fast, one intermediate, and one slow variable (three-timescale analysis) in the 2nd regime, and relaxation-type oscillations in the 3rd regime with two fast and one slow variable. We hypothesize convergence between the first and second regimes at some point and combine all three to provide a comprehensive geometric perspective. Moreover, the bifurcation analysis reveals that amyloid beta significantly induces calcium oscillations, explaining the appearance and disappearance of spontaneous astrocytic calcium oscillations. In the second direction, we extend our main model to a fractional-order system with an additional flux, enabling mixed-mode oscillations with fractional exponents. A coupled fractional-order model with linear and nonlinear gap junctions for IP3 diffusion is introduced and extended to a network model to study the synchronized firings. Results show that linear coupling achieves synchronization at higher coupling strengths, while nonlinear coupling fails under similar conditions. For a network model with linear coupling, the system displays evidence of synchronized behaviour for higher strength values, highlighting the intracellular oscillations explanation during astrocytic wave propagation. We believe that these findings will provide new insights into the calcium dysfunction hypothesis in AD.