Slow population activities (SPAs) are population activities in the brain with frequencies of less than 5 Hz. SPAs are prominent in many brain structures including the neocortex and the hippocampus. Examples of SPAs include the neocortical EEG δ waves and the hippocampal large amplitude irregular activities during NREM sleep. These in vivo SPAs are believed to play a fundamental role in brain plasticity. However, despite many experimental attempts to understand SPAs, their mechanisms are still not well understood. It is unclear how the individual neurons can sustain low frequency activities on the network as a whole. In this thesis, we demonstrate that a mathematical and computational perspective is indispensable in understanding slow population phenomena and generating testable hypotheses for future experiments. Our focus is on a hippocampal slice preparation exhibiting spontaneous, inhibitory-based SPAs (hippocampal SPAs). We develop a multi-pronged approach consisting of parameter extraction, simulation, and mathematical analysis to elucidate the mechanisms responsible for hippocampal SPAs. Our results suggest that hippocampal SPAs are an emergent phenomenon. In other words, the network “slowness” is not directly represented by any particular individual element within the network. Instead, the low frequency activities on the network are the result of interactions between synaptic and intrinsic characteristics of individual inhibitory interneurons. Our simulations quantify these characteristics which underlie hippocampal SPAs. Specifically, our simulations predict that individual interneurons should 1) be moderately fast-spiking above threshold before the increase in spike frequency slows down with increasing drive, and 2) be well connected with one another for SPAs to occur. We also predict that excitatory noise levels have a larger influence on hippocampal SPAs than mean excitatory drive. Subsequent mathematical analyses show that the synaptic and intrinsic conditions of individual interneurons as predicted by simulations promote network multi-stability. Hippocampal SPAs occur when the network switches from one network firing state to another. Since many of the parameters we use for simulations are extracted from experiments, our simulation model is likely a reasonable representation of actual biological mechanisms in hippocampal networks.

Slow population activities (SPAs) are population activities in the brain with frequencies of less than 5 Hz. SPAs are prominent in many brain structures including the neocortex and the hippocampus. Examples of SPAs include the neocortical EEG δ waves and the hippocampal large amplitude irregular activities during NREM sleep. These in vivo SPAs are believed to play a fundamental role in brain plasticity. However, despite many experimental attempts to understand SPAs, their mechanisms are still not well understood. It is unclear how the individual neurons can sustain low frequency activities on the network as a whole. In this thesis, we demonstrate that a mathematical and computational perspective is indispensable in understanding slow population phenomena and generating testable hypotheses for future experiments. Our focus is on a hippocampal slice preparation exhibiting spontaneous, inhibitory-based SPAs (hippocampal SPAs). We develop a multi-pronged approach consisting of parameter extraction, simulation, and mathematical analysis to elucidate the mechanisms responsible for hippocampal SPAs. Our results suggest that hippocampal SPAs are an emergent phenomenon. In other words, the network “slowness” is not directly represented by any particular individual element within the network. Instead, the low frequency activities on the network are the result of interactions between synaptic and intrinsic characteristics of individual inhibitory interneurons. Our simulations quantify these characteristics which underlie hippocampal SPAs. Specifically, our simulations predict that individual interneurons should 1) be moderately fast-spiking above threshold before the increase in spike frequency slows down with increasing drive, and 2) be well connected with one another for SPAs to occur. We also predict that excitatory noise levels have a larger influence on hippocampal SPAs than mean excitatory drive. Subsequent mathematical analyses show that the synaptic and intrinsic conditions of individual interneurons as predicted by simulations promote network multi-stability. Hippocampal SPAs occur when the network switches from one network firing state to another. Since many of the parameters we use for simulations are extracted from experiments, our simulation model is likely a reasonable representation of actual biological mechanisms in hippocampal networks.

Sea-ice ecosystem modelling is a novel field of research. In this thesis, the main organism studied is sea-ice algae. A basic introduction to algae and its importance in the aquatic food web is given first. An introduction to modeling and its purposes is presented, and this is followed by a brief description of ice algae models in practice with some physical conditions which influence ecosystem modelling. In the following Chapter, a simple mathematical model to represent the algae population is derived, and analyzed using pseudo spectral numerical methods implemented with MATLAB. The behaviour of the algae population and the boundary layers are discussed by examining the numerical results. Perturbation and asymptotic analysis is used for further analysis of the system using Maple. In the following Chapter a Nutrient Phytoplankton Zooplankton Detritus (or NPZD) model, which is a commonly used type of model in marine ecosystem modelling, is developed based on the framework of Soetaert and Herman. The model is examined under five different experimental setups (herein we mean numerical experiments) and the results are discussed. The NPZD model implemented is compared with a well-studied model in the literature. Our model can be considered somewhat simpler than other models in the literature (though it still has a much larger parameter space than the idealized model discussed in the previous Chapter). Finally we discuss future directions for research.

This thesis provides the study of graphene’s electrostatic interaction with the substrate surrounding it. Mathematical models based on current experimental configurations of graphene field-effect transistors (FET) are developed and analyzed. The conductivity and mobility of charge carriers in graphene are examined in the presence of impurities trapped in the substrate near graphene. The impurities encompass a wide range of possible structures and parameters, including different types of impurities, their distance from graphene, and the spatial correlation between them. Furthermore, we extend our models to analyze the influence of impurities on the fluctuations of the electrostatic potential and the charge carrier density in the plane of graphene. The results of our mathematical models are compared with current experimental results in the literature.

The traditional view of cancer asserts that a malignant tumour is composed of a population of cells, all of which share the ability to divide without limit. Within the last decade, however, this notion has lost ground to the emerging cancer stem cell hypothesis, which counters that only a (typically small) sub-population of so-called `cancer stem cells' has the capacity to proliferate indefinitely, and hence to drive and maintain tumour growth. Cancer stem cells have been putatively identified in leukemias and, more recently, in a variety of solid tumours including those of the breast and brain. The cancer stem cell hypothesis helps to explain certain clinically-observed phenomena, including the apparent inability of conventional anti-cancer therapies to eradicate the disease despite (transient) reduction of overall tumour bulk -- presumably these treatments fail to kill the underlying cancer stem cells. Herein, we develop stochastic and deterministic temporal models of tumour growth based on the cancer stem cell hypothesis, and apply these models to discussions of the treatment of glioblastoma multiforme, a common type of brain cancer believed to be maintained by cancer stem cells, and to the phenomenon of the epithelial-mesenchymal transition, a process thought to be important in generating cancer cells capable of metastasis.

In this thesis, I present a computational neural model that reproduces the high-level behavioural results of well-known fear conditioning experiments: first-order conditioning, second-order conditioning, sensory preconditioning, context conditioning, blocking, first-order extinction and renewal (AAB, ABC, ABA), and extinction and renewal after second-order conditioning and sensory preconditioning. The simulated neural populations used to account for the behaviour observed in these experiments correspond to known anatomical regions of the mammalian brain. Parts of the amygdala, periaqueductal gray, cortex and thalamus, and hippocampus are included and are connected to each other in a biologically plausible manner. The model was built using the principles of the Neural Engineering Framework (NEF): a mathematical framework that allows information to be encoded and manipulated in populations of neurons. Each population represents information via the spiking activity of simulated neurons, and is connected to one or more other populations; these connections allow computations to be performed on the information being represented. By specifying which populations are connected to which, and what functions these connections perform, I developed an information processing system that behaves analogously to the fear conditioning circuit in the brain.

The traditional view of cancer asserts that a malignant tumour is composed of a population of cells, all of which share the ability to divide without limit. Within the last decade, however, this notion has lost ground to the emerging cancer stem cell hypothesis, which counters that only a (typically small) sub-population of so-called `cancer stem cells' has the capacity to proliferate indefinitely, and hence to drive and maintain tumour growth. Cancer stem cells have been putatively identified in leukemias and, more recently, in a variety of solid tumours including those of the breast and brain. The cancer stem cell hypothesis helps to explain certain clinically-observed phenomena, including the apparent inability of conventional anti-cancer therapies to eradicate the disease despite (transient) reduction of overall tumour bulk -- presumably these treatments fail to kill the underlying cancer stem cells. Herein, we develop stochastic and deterministic temporal models of tumour growth based on the cancer stem cell hypothesis, and apply these models to discussions of the treatment of glioblastoma multiforme, a common type of brain cancer believed to be maintained by cancer stem cells, and to the phenomenon of the epithelial-mesenchymal transition, a process thought to be important in generating cancer cells capable of metastasis.

Radiofrequency ablation (RFA) is a clinic tool for the treatment of various target tissues. However, one of the major limitations with RFA is the ‘small’ size of target tissues that can be effectively ablated. By small it is meant the size of the target tissue is less than 3 cm in diameter of the tissue otherwise ‘large’ size of tissue in this thesis. A typical problem with RFA for large target tissue is the incompleteness of tumour ablation, which is an important reason for tumour recurring. It is widely agreed that two reasons are responsible for the tumour recurring: (1) the tissue charring and (2) the ‘heat-sink’ effect of large blood vessels (i.e. ≥3 mm in diameter). This thesis study was motivated to more quantitatively understand tissue charring during the RFA procedure and to develop solutions to increase the size of target tissues to be ablated. The thesis study mainly performed three tasks: (1) evaluation of the existing devices and protocols to give a clear understanding of the state of arts of RFA devices in clinic, (2) development of an accurate mathematical model for the RFA procedure to enable a more quantitative understanding of the small target tissue size problem, and (3) development of a new protocol based on the existing device to increase the size of target tissues to be ablated based on the knowledge acquired from (1) and (2). In (1), a design theory called axiomatic design theory (ADT) was applied in order to make the evaluation more objective. In (2), a two-compartment finite element model was developed and verified with in vitro experiments, where liver tissue was taken and a custom-made RFA system was employed; after that, three most commonly used internally cooled RFA systems (constant, pulsed, and temperature-controlled) were employed to demonstrate the maximum size of tumour that can be ablated. In (3) a novel feedback temperature-controlled RFA protocol was proposed to overcome the small target tissue size problem, which includes (a) the judicious selection of control areas and target control temperatures and (b) the use of the tissue temperature instead of electrode tip temperature as a feedback for control. The conclusions that can be drawn from this thesis are given as follows: (1) the decoupled design in the current RFA systems can be a critical reason for the incomplete target tissue necrosis (TTN), (2) using both the constant RFA and pulsed RFA, the largest TTN can be achieved at the maximum voltage applied (MVA) without the roll-off occurrence. Furthermore, the largest TTN sizes for both constant RFA and pulsed RFA are all less than 3 cm in diameter, (3) for target tissues of different sizes, the MVA without the roll-off occurrence is different and it decreases with increase of the target tissue size, (4) the largest TTN achieved by using temperature-controlled RFA under the current commercial protocol is still smaller 3 cm in diameter, and (5) the TTN with and over 3 cm in diameter can be obtained by using temperature-controlled RFA under a new protocol developed in this thesis study, in which the temperature of target tissue around the middle part of electrode is controlled at 90 ℃ for a standard ablation time (i.e. 720 s). There are a couple of contributions with this thesis. First, the underlying reason of the incomplete TTN of the current commercially available RFA systems was found, which is their inadequate design (i.e. decoupled design). This will help to give a guideline in RFA device design or improvement in the future. Second, the thesis has mathematically proved the empirical conclusion in clinic that the limit size of target tissue using the current RFA systems is 3 cm in diameter. This has advanced our understanding of the limit of the RFA technology in general. Third, the novel protocol proposed by the thesis is promising to increase the size of TTN with RFA technology by about 30%. The new protocol also reveals a very complex thermal control problem in the context of human tissues, and solving this problem effectively gives implication to similar problems in other thermal-based tumour ablation processes.

Sea-ice ecosystem modelling is a novel field of research. In this thesis, the main organism studied is sea-ice algae. A basic introduction to algae and its importance in the aquatic food web is given first. An introduction to modeling and its purposes is presented, and this is followed by a brief description of ice algae models in practice with some physical conditions which influence ecosystem modelling. In the following Chapter, a simple mathematical model to represent the algae population is derived, and analyzed using pseudo spectral numerical methods implemented with MATLAB. The behaviour of the algae population and the boundary layers are discussed by examining the numerical results. Perturbation and asymptotic analysis is used for further analysis of the system using Maple. In the following Chapter a Nutrient Phytoplankton Zooplankton Detritus (or NPZD) model, which is a commonly used type of model in marine ecosystem modelling, is developed based on the framework of Soetaert and Herman. The model is examined under five different experimental setups (herein we mean numerical experiments) and the results are discussed. The NPZD model implemented is compared with a well-studied model in the literature. Our model can be considered somewhat simpler than other models in the literature (though it still has a much larger parameter space than the idealized model discussed in the previous Chapter). Finally we discuss future directions for research.

This thesis provides the study of graphene’s electrostatic interaction with the substrate surrounding it. Mathematical models based on current experimental configurations of graphene field-effect transistors (FET) are developed and analyzed. The conductivity and mobility of charge carriers in graphene are examined in the presence of impurities trapped in the substrate near graphene. The impurities encompass a wide range of possible structures and parameters, including different types of impurities, their distance from graphene, and the spatial correlation between them. Furthermore, we extend our models to analyze the influence of impurities on the fluctuations of the electrostatic potential and the charge carrier density in the plane of graphene. The results of our mathematical models are compared with current experimental results in the literature.