Document Type

Thesis

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Faculty/School

Faculty of Science

First Advisor

Prof. Roderick Melnik

Advisor Role

Supervision

Abstract

The complex nature of the human brain, with its intricate organic structure and multiscale spatio-temporal characteristics ranging from synapses to the entire brain, presents a major obstacle in brain modelling. Capturing this complexity poses a significant challenge for researchers. The complex interplay of coupled multiphysics and biochemical activities within this intricate system shapes the brain's capacity, functioning within a structure-function relationship that necessitates a specific mathematical framework. Advanced mathematical modelling approaches that incorporate the coupling of brain networks and the analysis of dynamic processes are essential for advancing therapeutic strategies aimed at treating neurodegenerative diseases (NDDs), which afflict millions of individuals worldwide.

This thesis aims to develop novel multiscale brain network models to investigate the properties of biological processes spanning multiple spatial and temporal scales, potentially offering new insights into the fundamental aspects of brain networks in healthy and diseased states. Among various factors, our particular attention is devoted to brain electrical activity, which serves as a significant characteristic of NDDs such as Alzheimer's disease (AD), cortical spreading depression (CSD), and Parkinson's disease (PD).

Our analysis starts with a focus on large-scale brain network models within the context of AD and PD. Importantly, AD is characterized by pathological accumulations of amyloid-beta plaques ($A\beta$) and neurofibrillary tangles ($\tau$). Specifically, the propagation of intracellular calcium ($Ca^{2+}$) signalling within non-neuronal cells, such as astrocytes, has an impact on the tripartite synapse propagation in neuronal cells, leading to synaptic failure and neuronal death. To gain insights into the pathogenesis of AD, particularly the role of astrocytes in the presence of misfolded proteins ($A\beta$ and $\tau$), we propose an improved large-scale brain network model. This model incorporates the concept of astrocytic clearance, which aids in the fragmentation and elimination of toxic $A\beta$. We obtained the data from the Human Connectome Project (HCP) and proposed a hybrid multiscale modelling strategy based on a large-scale brain network model. Furthermore, PD is characterized by involuntary or uncontrolled movements such as shaking, stiffness, and difficulties with balance and coordination. In our study, we have developed a small-scale brain network model that aims to shed light on the intricate dynamics of PD and its impact on the brain. Within the context of PD, we have explored the optimization of deep brain stimulation (DBS) procedures, a well-established neuromodulation technique for managing PD symptoms. By leveraging data from the HCP, we propose a closed-loop DBS approach based on a brain network model.

Additionally, CSD is characterized by a slowly propagating wave that disrupts the brain's homeostasis, leading to a temporary impairment in the normal functioning of neurons. In this thesis, our focus is to model the propagation of CSD in the brain using two different approaches found in the literature. Firstly, we employ a simplified model consisting of six coupled equations of the reaction-diffusion type in two spatial dimensions. Secondly, we utilize a more complex one-dimensional neuronal model that incorporates ionic currents and ionic pumps. Furthermore, we have developed a comprehensive coupled neuronal-glial model to investigate the impact of temperature on the activation and inactivation of ionic channels, specifically focusing on their influence on calcium-mediated exosomal dynamics in PD and AD. This model specifically examines controlled therapeutic exosomal release by evaluating the modulated release rate and the concentration of released exosomes. Importantly, it considers the temperature threshold dependence on $Ca^{2+}$ dynamics by incorporating cold-sensing neurons. The outcomes of our investigation highlight the significant role played by $TRPM8$ and voltage-gated $Ca^{2+}$ channels in determining temperature-dependent activation and inactivation at various threshold levels.

Notably, the primary objective of this thesis is to develop multiscale deterministic and stochastic models that can capture the overall trends and average behaviour of the system, providing insights into the mechanisms underlying these diseases. The etiology of AD can be described as a multi-state disease process utilizing the approximate Bayesian computation (ABC) method. In this context, we employ ADNI data from 2-year visits for AD patients and apply this method to investigate the interplay between $A\beta$ and $Ca^{2+}$ levels at different stages of disease development. Astrocytes play a crucial role in NDDs as they secrete neurotrophins, regulate synaptogenesis, facilitate the formation of neural networks, and influence synaptic plasticity underlying learning, memory, and disease progression. Thus, we employ strong and weak astrocyte effect models on $A\beta$ dynamics to explore the biological mechanisms within the central nervous system. In this regard, we analyze ADNI data for $A\beta$ concentration and fit it to the developed stochastic models using the ABC technique, allowing us to refine and validate the model based on clinical data.

It is expected that the coupled multiscale models developed in this thesis will provide novel insight into disease origin and progression and a better understanding of the key mechanisms underlying the dynamics of the pathological brain.

Convocation Year

2024

Convocation Season

Spring

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