Applied Mathematics

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This is the collection for the University of Waterloo's Department of Applied Mathematics.

Research outputs are organized by type (eg. Master Thesis, Article, Conference Paper).

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Now showing 1 - 6 of 6

Bifurcation Analysis of Large Networks of Neurons
(University of Waterloo, 2015-12-23) Nicola, Wilten; Campbell, Sue Ann
The human brain contains on the order of a hundred billion neurons, each with several thousand synaptic connections. Computational neuroscience has successfully modeled both the individual neurons as various types of oscillators, in addition to the synaptic coupling between the neurons. However, employing the individual neuronal models as a large coupled network on the scale of the human brain would require massive computational and financial resources, and yet is the current undertaking of several research groups. Even if one were to successfully model such a complicated system of coupled differential equations, aside from brute force numerical simulations, little insight may be gained into how the human brain solves problems or performs tasks. Here, we introduce a tool that reduces large networks of coupled neurons to a much smaller set of differential equations that governs key statistics for the network as a whole, as opposed to tracking the individual dynamics of neurons and their connections. This approach is typically referred to as a mean-field system. As the mean-field system is derived from the original network of neurons, it is predictive for the behavior of the network as a whole and the parameters or distributions of parameters that appear in the mean-field system are identical to those of the original network. As such, bifurcation analysis is predictive for the behavior of the original network and predicts where in the parameter space the network transitions from one behavior to another. Additionally, here we show how networks of neurons can be constructed with a mean-field or macroscopic behavior that is prescribed. This occurs through an analytic extension of the Neural Engineering Framework (NEF). This can be thought of as an inverse mean-field approach, where the networks are constructed to obey prescribed dynamics as opposed to deriving the macroscopic dynamics from an underlying network. Thus, the work done here analyzes neuronal networks through both top-down and bottom-up approaches.
Clustering behaviour in networks with time delayed all-to-all coupling
(University of Waterloo, 2017-08-30) Wang, Zhen; Campbell, Sue Ann; Campbell, Sue Ann
Networks of coupled oscillators arise in a variety of areas. Clustering is a type of oscillatory network behavior where elements of a network segregate into groups. Elements within a group oscillate synchronously, while elements in different groups oscillate with a fixed phase difference. In this thesis, we study networks of N identical oscillators with time delayed, global circulant coupling with two approaches. We first use the theory of weakly coupled oscillators to reduce the system of delay differential equations to a phase model where the time delay enters as a phase shift. We use the phase model to determine model independent existence and stability results for symmetric cluster solutions. We show that the presence of the time delay can lead to the coexistence of multiple stable clustering solutions. We then perform stability and bifurcation analysis to the original system of delay differential equations with symmetry. We first study the existence of Hopf bifurcations induced by coupling time delay, and then use symmetric Hopf bifurcation theory to determine how these bifurcations lead to different patterns of symmetric cluster oscillations. We apply our results to two specfi c examples: a network of FitzHugh-Nagumo neurons with diffusive coupling and a network of Morris-Lecar neurons with synaptic coupling. In the case studies, we show how time delays in the coupling between neurons can give rise to switching between different stable cluster solutions, coexistence of multiple stable cluster solutions and solutions with multiple frequencies.
Collective Dynamics of Large-Scale Spiking Neural Networks by Mean-Field Theory
(University of Waterloo, 2024-05-31) Chen, Liang; Campbell, Sue Ann
The brain contains a large number of neurons, each of which typically has thousands of synaptic connections. Its functionality, whether function or dysfunction, depends on the emergent collective dynamics arising from the coordination of these neurons. Rather than focusing on large-scale realistic simulations of individual neurons and their synaptic coupling to understand these macroscopic behaviors, we emphasize the development of mathematically manageable models in terms of macroscopic observable variables. This approach allows us to gain insight into the underlying mechanisms of collective dynamics from a dynamical systems perspective. It is the central idea of this thesis. We analytically reduce large-scale neural networks to low-dimensional mean-field mod- els that account for spike frequency adaptation, time delay between neuron communication, and short-term synaptic plasticity. These mean-field descriptions offer a precise correspondence between the microscopic dynamics of individual neurons and the macroscopic dynamics of the neural network, valid in the limit of infinitely many neurons in the network. Bifurcation analysis of the mean-field systems is capable of predicting net- work transitions between asynchronous and synchronous states, or different patterns of synchronization, such as slow-fast nested collective oscillations. We discuss how these dynamics are closely related to normal brain functions and neurological disorders. We also investigate the influence on these dynamic transitions induced by current heterogeneity, adaptation intensity, and delayed coupling. By integrating a kinetic model of synapses into the neural network, we describe calcium-dependent short-term synaptic plasticity in a relatively simple mathematical form. Through our mean-field modeling approach, we explore the impact of synaptic dynamics on collective behaviors, particularly the effect of muscarinic activation at inhibitory hippocampal synapses. Together, this thesis provides a tractable and reliable tool for model-based inference of neurological mechanisms from the perspective of theoretical neuroscience.
Model for a cortical circuit associated with childhood absence epilepsy
(University of Waterloo, 2019-08-23) Ahmed, Maliha; Campbell, Sue Ann
Childhood absence epilepsy is suspected to result from mutations in genes which encode ion channels including sodium channels. Our purpose in this thesis is to explore some of the factors that alter the function of neurons in the cerebral cortex. In particular, we investigate the consequence of these alterations on neuronal network activity associated with this disorder. In this regard, we create a small network consisting of deep layer cortical pyramidal neurons and an interneuron, each described by a single-compartment Hodgkin-Huxley style model. We investigate factors that convert a normal network into a hyperexcitable one, including impairment of $GABA_A$ synapses and sodium channel defects resulting from mutations in Scn genes. Our model agrees with experimental results indicating the role of GABA impairment in generating a hyperexcitable network. In particular, our cortical network is capable of generating its own spike-and-wave oscillations analogous to those in a thalamocortical network. Our results also suggest that the co-existence of multiple $Na^{+}$- channel mutations alters individual neuronal function to increase or decrease the likelihood of the network exhibiting seizure-like behaviour.
Model for the RE-TC thalamic circuit with application to childhood absence epilepsy
(University of Waterloo, 2019-08-12) Newman, Jennie; Campbell, Sue Ann
Childhood absence epilepsy (CAE) is an idiopathic neurological disorder aﬀecting roughly 2-8 children per 100,000 children worldwide. It is characterized by absence seizures, or short lapses in consciousness, and the appearance of slow wave discharge (SWD) patterns on an electroencephalogram (EEG). With the cause of onset and recovery still unknown, much research has been conducted in order to determine the set of genes responsible for this disorder. Experimental animal models and mathematical models of neural networks have so far suggested the thalamocortical network as the site of seizure initiation, and the 𝐶𝐴𝐶𝑁𝐴1𝐻 gene as a promoter of SWD patterns in the brain. In this thesis, we develop a mathematical model of part of the thalamocortical network, comprised of thalamic reticular (RE) and thalamocortical (TC) neurons. We then use this model to study the eﬀects of previously suggested CAE factors, such as the 𝐶𝐴𝐶𝑁𝐴1𝐻 gene mutation, GABA_A synapse conductance and T-type Ca²⁺ channel conductance, on the formation of SWD patterns in the network. We ﬁnd a link between a decreased GABA_A conductance and increased SWD activity in our network, as well as the dependence of SWD activity on the interactions between the multiple factors of our study. Our results imply that CAE and SWD activity may have a multifactorial cause, and that the thalamus may not be solely responsible for the generation and propagation of SWDs in the thalamocortical network.
Persistent oscillations in the Aplysia bag cell network
(University of Waterloo, 2017-08-24) Keplinger, Keegan; Campbell, Sue Ann
Persistence is a phenomena by which a resting neuron enters a state of persistent behavior following a brief stimulus. Persistent neural systems can exhibit long-term responses that remain after the stimulus is removed, switching from excitable, steady-state dynamics to a period of tonic spiking or bursting. In Aplysia, such behavior, known as the afterdischarge, is exhibited by the bag cell neuron and regulated by second messenger calcium dynamics. In this thesis, we construct a model for the electrical activity of the Aplysia bag cell neuron is constructed based on experimental data. The model includes many features of the bag cell, including use-dependence, non-selective cation channels, a persistent calcium current, and the afterdischarge. Each of these features contributes to the onset of afterdischarge. Several methods are used to ﬁt experimental data and construct the model, including hand tuning, parameter forcing, genetic algorithms for optimization, and continuation analysis. These methods help to address common modeling issues such as degeneracy and sensitivity. Use-dependence in the calcium channels of Aplysia is thought to depend on calcium. The model developed in this thesis veriﬁes calcium as a viable driver for use-dependence. The literature often emphasizes two potassium channels in the context of bag cell afterdischarge, but we show that afterdischarge behavior is produced with only a single potassium channel in simulations. Experimental evidence suggests that nonselective cation channels are a primary driver of afterdischarge behavior. In the model developed here, the nonselective current is required for in silica afterdischarge to take place. Continuation analysis is used to determine and tune the location and stability of ﬁxed points in the model. For exploratory analysis, an electrically-coupled network model is constructed to simulate the observed dynamics in bag cell clusters in vivo. A simple two-neuron network reproduces some experimental results. Larger networks are considered. Little is known about the topology of Aplysia bag cell cluster. The ﬁnal chapter of this thesis explores diﬀerent topologies in a 100-neuron network, including a ring topology, a cluster ring topology, and a randomly-connected scatter network, exploring how the coupling constant, topology, and size of the network aﬀect the ability of the network to synchronize.

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Browsing Applied Mathematics by Author "Campbell, Sue Ann"