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The field of mathematical neuroscience is concerned with the modeling and interpretation of neuronal dynamics and associated phenomena. Neurons can be modeled individually, in small groups, or collectively as a large network. Mathematical models of single neurons typically involve either differential equations, discrete maps, or some combination of both. A number of two-dimensional spiking neuron models that combine continuous dynamics with an instantaneous reset have been introduced in the literature. The models are capable of reproducing a variety of experimentally observed spiking patterns, and also have the advantage of being mathematically tractable. Here an analysis of the transverse stability of orbits in the phase plane leads to sufficient conditions on the model parameters for regular spiking to occur. The application of this method is illustrated by three examples, taken from existing models in the neuroscience literature. In the first two examples the model has no equilibrium states, and regular spiking follows directly. In the third example there are equilibrium points, and some additional quantitative arguments are given to prove that regular spiking occurs. / Graduate

In this thesis mathematical techniques and models are applied to electroencephalographic (EEG) recordings to study mechanisms of idiopathic generalised epilepsy (IGE). First, we compare network structures derived from resting-state EEG from people with IGE, their unaffected relatives, and healthy controls. Next, these static networks are combined with a dynamical model describing the ac- tivity of a cortical region as a population of phase-oscillators. We then examine the potential of the differences found in the static networks and the emergent properties of the dynamic network as individual biomarkers of IGE. The emphasis of this approach is on discerning the potential of these markers at the level of an indi- vidual subject rather than their ability to identify differences at a group level. Finally, we extend a dynamic model of seizure onset to investigate how epileptiform discharges vary over the course of the day in ambulatory EEG recordings from people with IGE. By per- turbing the dynamics describing the excitability of the system, we demonstrate the model can reproduce discharge distributions on an individual level which are shown to express a circadian tone. The emphasis of the model approach is on understanding how changes in excitability within brain regions, modulated by sleep, metabolism, endocrine axes, or anti-epileptic drugs (AEDs), can drive the emer- gence of epileptiform activity in large-scale brain networks. Our results demonstrate that studying EEG recordings from peo- ple with IGE can lead to new mechanistic insight on the idiopathic nature of IGE, and may eventually lead to clinical applications. We show that biomarkers derived from dynamic network models perform significantly better as classifiers than biomarkers based on static network properties. Hence, our results provide additional ev- idence that the interplay between the dynamics of specific brain re- gions, and the network topology governing the interactions between these regions, is crucial in the generation of emergent epileptiform activity. Pathological activity may emerge due to abnormalities in either of those factors, or a combination of both, and hence it is essential to develop new techniques to characterise this interplay theoretically and to validate predictions experimentally.

In this thesis we model two very different movement-related neuronal circuits, both of which produce oscillatory patterns of activity. In one case we study oscillatory activity in the basal ganglia under both normal and Parkinsonian conditions. First, we used a detailed Hodgkin-Huxley type spiking model to investigate the activity patterns that arise when oscillatory cortical input is transmitted to the globus pallidus via the subthalamic nucleus. Our model reproduced a result from rodent studies which shows that two anti-phase oscillatory groups of pallidal neurons appear under Parkinsonian conditions. Secondly, we used a population model of the basal ganglia to study whether oscillations could be locally generated. The basal ganglia are thought to be organised into multiple parallel channels. In our model, isolated channels could not generate oscillations, but if the lateral inhibition between channels is sufficiently strong then the network can act as a rhythm-generating ``pacemaker'' circuit. This was particularly true when we used a set of connection strength parameters that represent the basal ganglia under Parkinsonian conditions. Since many things are not known about the anatomy and electrophysiology of the basal ganglia, we also studied oscillatory activity in another, much simpler, movement-related neuronal system: the spinal cord of the Xenopus tadpole. We built a computational model of the spinal cord containing approximately 1,500 biologically realistic Hodgkin-Huxley neurons, with synaptic connectivity derived from a computational model of axon growth. The model produced physiological swimming behaviour and was used to investigate which aspects of axon growth and neuron dynamics are behaviourally important. We found that the oscillatory attractor associated with swimming was remarkably stable, which suggests that, surprisingly, many features of axonal growth and synapse formation are not necessary for swimming to emerge. We also studied how the same spinal cord network can generate a different oscillatory pattern in which neurons on both sides of the body fire synchronously. Our results here suggest that under normal conditions the synchronous state is unstable or weakly stable, but that even small increases in spike transmission delays act to stabilise it. Finally, we found that although the basal ganglia and the tadpole spinal cord are very different systems, the underlying mechanism by which they can produce oscillations may be remarkably similar. Insights from the tadpole model allow us to predict how the basal ganglia model may be capable of producing multiple patterns of oscillatory activity.

A paramount question in the study of Calcium (Ca2+) signaling is how this ion regulates a wide spectrum of cellular processes, which include: fertilization, proliferation, learning, memory, and cell death. All of these processes are the result of synaptic strengthening and weakening. Part of the answer lies in the spatial-temporal interactions of Ca2+ at the extracellular and intracellular levels of a neuron. Within these levels of a neuron there is a complex concert of Ca2+ ion exchange and transport mechanisms that are activated (or inactivated) by external stimuli and it remains to be studied the role of these interactions at the ultrastructural scale. One mode of external stimulation is by Transcranial Magnetic Stimulation (TMS) and repetitive TMS (rTMS). TMS is a noninvasive brain stimulation method to modulate humanbrain activity by generating a strong magnetic field near the cranium. The magnetic field induces an electric field which depolarizes neurons; therefore, TMS is used in clinical applications to treat neuropsychiatric and neurological disorders. However, it is not well known the effect of TMS on intracellular Ca2+ interactions; therefore, I endeavor to determine the types of calcium interactions that occur when a neuron experiences TMS. I also determine how intracellular calcium mechanisms are affected by TMS stimuli. In particular, the cellular regulators of calcium are given by: the internal Ca2+ store (“calcium bank”) of a neuron called the endoplasmic reticulum (ER) with spine apparatus (SA), the voltage dependent calcium channels (VDCCs), and calcium influx at synaptic spines. Ultimately, the ER is responsible for synaptic plasticity and from here I determined under what conditions does TMS cause intracellular calcium to induce synaptic plasticity. For the first part of this dissertation I describe the neurobiology, model equations, and methods that are employed in understanding the role of intracellular calcium. Simulating calcium dynamics at the ultrastructural level is computationally expensive when including the effects of TMS in concert with intracellular calcium transport mechanism. Therefore, I also identify the numerical methodologies that provide the best results in terms of numerical accuracy to the physiology of the intracellullar dynamics and the parameters such as error and time step size that yield sufficient results. I will also describe the framework used in this study (i.e., UG4) and the pipeline for performing my studies, this includes: the process from microscopy to computational domains, generating and preserving mesh features, the choice of numerical methods, and the process of parallelizing the simulations. In the second part, I dive into the electro-dynamic mechanisms that cause voltage propagation through a neuron. This is of particular importance, because many ion membrane transport mechanisms depend on plasma membrane voltage. The simulations coded and executed in MatLab are used to drive calcium dynamics which is discussed in the third part of the dissertation. I will also take the opportunity to explain a case study involving virtual reality with the Hodgkin-Huxley electrical model for voltage propagation. Additionally, I incorporate synaptic communication which is driven by TMS protocols or simulated by voltage clamps, and both provide a mechanism by which intracellular calcium transients occurs. For the third chapter I discuss the calcium dynamic mechanisms that are inside of neurons and I discuss the methodology I take to setup a simulation and perform simulations. This includes the steps taken to process microscopy images to generate computational domains, implementing the model equations, and utilizing appropriate numerical schemes. I also discuss several preliminary examples as proof of concept to my simulation pipeline and I give results involving the regulation of calcium with respect to intracellular mechanisms. The fourth part of this dissertation describes the steps for running TMS simulations using voltage data from electrical simulations to drive calcium signaling events. In particular, I discuss the tool NeMo-TMS which uses voltage and calcium simulations together to draw conclusions with respect to intracellular calcium propagation. I describe the multi-scale paradigm that is used, model equations, and computational domains that are used and provide several examples of results from this modeling pipeline. Of particular importance, I provide discussion on the coupling of data from electrical simulations and biochemical simulations, i.e. I use TMS induced voltage data to drive voltage dependent calcium release and I examine the effects of TMS induced back propagating action potentials. / Mathematics

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In this thesis we discuss the synchronization of stochastic population oscillators in ecology and neuroscience. Traditionally, the synchronization of oscillators has been studied in deterministic systems with various modes of synchrony induced by coupling between the oscillators. However, recent developments have shown that an ensemble of uncoupled oscillators can be synchronized by a common noise source alone. By considering the effects of noise-induced synchronization on biological oscillators, we are able to explain various biological phenomena in ecological and neurobiological contexts - most importantly, the long-observed Moran effect. Our formulation of the systems as limit cycle oscillators arising from populations of individuals, each with a random element to its behaviour, also allows us to examine the interaction between an external noise source and this intrinsic stochasticity. This provides possible explanations as to why in ecological systems large-amplitude cycles may not be observed in the wild. In neural population oscillators, we were able to observe not just synchronization, but also clustering in some pa- rameter regimes. Finally, we are also able to extend our methods to include coupling in our models. In particular, we examine the competing effects of dispersal and extrinsic noise on the synchronization of a pair of Rosenzweig-Macarthur predator-prey systems. We discover that common environmental noise will ultimately synchronize the oscillators, but that the approach to synchrony depends on whether or not dispersal in the absence of noise supports any stable asynchronous states. We also show how the combination of correlated (shared) and uncorrelated (unshared) noise with dispersal can lead to a multistable steady-state probability density. Similar analysis on a coupled system of neural oscillators would be an interesting project for future work, which, among other future directions of research, is discussed in the concluding section of this thesis.

<p>    </p> <p>Over the last decade, measurement technology that records neural activity such as ECoG and Utah array has dramatically improved. These advancements have given researchers access to recordings from multiple neurons simultaneously. Efficient computational and statistical methods are required to analyze this data type successfully. The time-series model is one of the most common approaches for analyzing this data type. Unfortunately, even with all the advances made with time-series models, it is not always enough since these models often need massive amounts of data to achieve good results. This is especially true in the field of neuroscience, where the datasets are often limited, therefore imposing constraints on the type and complexity of the models we can use. Not only that, but the Signal-to- noise ratio tends to be lower than in other machine learning datasets. This paper will introduce different architectures and techniques to overcome constraints imposed by these small datasets. There are two major experiments that we will discuss. (1) We will strive to develop models for participants who lost the ability to speak by building upon the previous state-of-the-art model for decoding neural activity (ECoG data) into English text. (2) We will introduce two new models, RNNF and Neural RoBERTa. These new models impute missing neural data from neural recordings (Utah arrays) of monkeys performing kinematic tasks. These new models with the help of novel data augmentation techniques (dynamic masking) outperformed state-of-the-art models such as Neural Data Transformer (NDT) in the Neural Latents Benchmark competition. </p>

<p dir="ltr">Frontloading is an alcohol drinking pattern where intake is skewed toward the onset of access. The goal of the current study was to identify brain regions involved in frontloading using whole brain imaging. 63 C57Bl/6J (32 female and 31 male) mice underwent 8 days of binge drinking using drinking-in-the-dark (DID). Three hours into the dark cycle, mice received 20% (v/v) alcohol or water for two hours on days 1-7. Intake was measured in 1-minute bins using volumetric sippers, which facilitated analyses of drinking patterns. Mice were perfused 80 minutes into the day 8 DID session and brains were extracted and processed for iDISCO clearing and c-fos immunohistochemistry. For brain network analyses, day 8 drinking patterns were used to characterize mice as frontloaders or non-frontloaders using a change-point analysis described in our recent ACER publication (Ardinger et al., 2022). Groups were female frontloaders (n = 20), female non-frontloaders (n = 2), male frontloaders (n = 13) and male non-frontloaders (n = 8). There were no differences in total alcohol intake as a function of frontloading status. Water drinkers had an n of 10 for each sex. As only two female mice were characterized as non-frontloaders, it was not possible to construct a functional correlation network for this group. Following light sheet imaging, ClearMap2.1 was used to register brains to the Allen Brain Atlas and detect fos+ cells. Functional correlation matrices were calculated for each group from log₁₀ c-fos values. Euclidean distances were calculated from these R values and hierarchical clustering was used to determine modules (highly connected groups of brain regions) at a tree-cut height of 50%. In males, alcohol access decreased modularity (3 modules in both frontloaders and non-frontloaders) as compared to water drinkers (7 modules). In females, an opposite effect was observed. Alcohol access (9 modules) increased modularity as compared to water drinkers (5 modules). These results suggest sex differences in how alcohol consumption reorganizes the functional architecture of networks. Next, key brain regions in each network were identified. Connector hubs, which primarily facilitate communication between modules, and provincial hubs, which facilitate communication within modules, were of specific interest for their important and differing roles. In males, 4 connector hubs and 17 provincial hubs were uniquely identified in frontloaders (i.e., were brain regions that did not have this status in male non-frontloaders or water drinkers). These represented a group of hindbrain regions (e.g., locus coeruleus and the pontine gray) connected to striatal/cortical regions (e.g., cortical amygdalar area) by the paraventricular nucleus of the thalamus. In females, 16 connector and 17 provincial hubs were uniquely identified which were distributed across 8 of the 9 modules in the female alcohol drinker network. Only one brain region (the nucleus raphe pontis) was a connector hub in both sexes, suggesting that frontloading in males and females may be driven by different brain regions. In conclusion, alcohol consumption led to fewer, but more densely connected, groups of brain regions in males but not females, and recruited different hub brain regions between the sexes. These results suggest target brain regions for future studies to try to manipulate frontloading behavior and more broadly contribute to the literature on alcohol’s effect on neural networks.</p>

<p dir="ltr">Multi-omic studies ranging from genomics, transcriptomics (e.g., gene expression) to proteomics data exploration have been widely applied to interpret findings from genome wide association studies (GWAS) of Alzheimer's disease (AD). However, previous studies examine each -omics data type individually and the functional interactions between genetic variations, genes and proteins are only used after discovery to interpret the findings, but not beforehand. In this case, multi-omic findings are likely not functionally related and therefore it is challenging for result interpretation. To handle this challenge, we present new modularity constrained least absolute shrinkage and selection operator (M-LASSO), new modularity constrained logistic regression (M-Logistic), new interpretable multi-omic graph fusion neural network model (MoFNet) and new transfer learning framework integrated graph fusion neural network model (TransFuse) to integrate prior biological knowledge to model the functional interactions of multi-omic data. These approaches aim to identify functional connected sub-networks predictive of AD. In this thesis, the intrepretable model MoFNet and TransFuse incorporate prior biological connected multi-omics network, and for the first time model the dynamic information flow from deoxyribonucleic acid (DNA) to ribonucleic acid (RNA) and proteins. While applying the proposed models on multi-omic data from the religious orders study/memory and aging project (ROS/MAP) cohort, MoFNet and TransFuse outperformed all other state-of-art classifiers. Instead of targeting individual markers, the proposed methods identified multi-omic sub-networks associated with AD. MoFNet and TransFuse, produced sub-network and pathway findings that were robustly validated in another independent cohort. These identified gene/protein networks highlight potential pathways involved in AD pathogenesis and could offer systematic overview for understanding the molecular mechanisms of the disease. Investigating these identified pathways in more detail could help uncover the mechanisms causing synaptic dysfunction in AD and guide future research into potential therapeutic targets.</p>