Abstract

Motivated by the growing interest in modeling the nonlinear propagation dynamics of species, such as prion-like proteins, within complex networks like brain connectomes, this thesis introduces a comprehensive modeling framework comprising four key stages: (I) model development and analysis, (II) biologically informed parameterization, (III) a priori identifiability, and (IV) parameter estimation and practical identifiability, with a focus on modeling tauopathy progression in the brain. In Stage I, a generalization of the celebrated Fisher-Kolmogorov-Petrovsky-Piskunov (Fisher-KPP) reaction-diffusion (Re-Di) equation using fractional polynomial (FP) terms is proposed and rigorously analyzed, incorporating a nonlinear graph Laplacian to capture complex transport dynamics characterized by heterogeneity, directional bias, and subdiffusion over directed networks. In Stage II, we present an empirical framework for parameterizing the extended FP-Fisher-KPP Re-Di equations, grounded in a phenomenological description of tauopathy propagation, a leading hallmark of neurodegenerative diseases such as Alzheimer’s disease (AD), responsible for 60-80% of dementia cases worldwide. In Stage III, we propose a framework for a priori identifiability, formulated under idealized, noise-free experimental scenarios in a generic sense for the parameterized FP-Fisher-KPP Re-Di equations with multi-experimental designs, effortlessly adaptable to any analytic and meromorphic system of differential equations. This framework offers four main contributions: (i) highlighting the limitations and gaps in certain prior results by conducting a rigorous literature survey and presenting illustrative counterexamples, (ii) formulating the a priori identifiability of parameters and the observability of states simultaneously for systems with partially known parameterized initial conditions in a generic sense, (iii) formalizing the concept of generic local minimal dependence using one-parameter Lie groups of transformations, and (iv) devising a decomposition method to explore this new concept. The Allen Mouse Brain Connectivity Atlas (AMBCA) dataset is also used to model tauopathy progression in the mouse brain for implementing the proposed framework for a priori identifiability. In Stage IV, parameter estimation and practical identifiability are carried out for the proposed tauopathy model using experimental data under realistic conditions. Practical identifiability is evaluated via singular value decomposition of the Fisher information matrix (FIM), accounting for noise and experimental constraints such as limited measurement time points.