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No description available.

No description available.

Calcium (Ca2+) is a universal second messenger that regulates the most important activities of the cell. Growing evidence shows Ca2+ is of importance in the pathogenesis of various brain diseases, such as Alzheimer’s Disease(AD). Modern science suggests that synaptic Ca2+ overload plays an important role in synaptic loss, which consequently increases the incidence of AD. This merits building and solving a calcium signaling model to predict calcium concentration in neurons. Accurately solving the spatial-temporal calcium signaling model via numerical method is fundamentally challenging because of (1) the detailed resolution of geometry, and (2) nonlinearity due to the complex membrane exchange mechanisms. Despite the modern computing power, the computational cost will be overwhelming when a neural network of actual size is considered.iv v The first part of the dissertation focuses on solving a partial differential equation system on cell level geometries by reducing three-dimensional problem to two-dimensional or one-dimensional problem. Firstly, we find the dominating physiologic and geometric parameters to activate stably propagating waves, and investigate how the wave velocity, duration, and amplitude depend on the dominating parameters. We implement the simulations on perfectly rotational symmetric dendrites for which the problem can be reduced to two dimensions. We show that the reduction is valid by a direct comparison of numeric solutions using 2D and 3D geometries. Then we study the threshold of parameters for stable waves. Secondly, we find empirical laws which express the wave velocity and wave amplitude as functions of dominating parameters. The second part of this dissertation is focusing on waves on branching dendrites. First of all, we apply a single stimulation to one of the child branches to determine if the calcium signal can propagate through the junction area and trigger a signal in the other branches. Various parameters of a branch point are considered, including branching angle, radius ratio of child dendrite to parent dendrite and ER radius. Numerical experiments are carried out and the corresponding physiological interpretations are given. The empirical laws enable predicting activating signal stability and strength at the branch point and consequently predicting wave properties in the other dendrites. / Mathematics

In many scientific fields, we are faced with extremely large, noisy datasets. Features of interest in these datasets may be difficult to explicitly define, obscured by noise, or simply lost in the magnitude of the dataset. Uncovering these features often necessitates the development of novel mathematical and statistical modeling approaches, and the utilization of powerful analysis tools. In this work, we present three distinct projects, all of which develop specific mathematical and statistical analysis to find features of interest amid large, noisy data. The first project measures cross-frequency coupling (CFC), i.e., the extent to which signals in different frequency bands interact, amid large, noisy neural voltage recordings. We use generalized linear models (GLMs) to define an accurate measure with confidence intervals and significance values. We show in simulation how this measure improves upon existing approaches, and apply this measure to analyze CFC during a human seizure. The second project develops a fully-automated detector of spike ripples, a powerful biomarker of epilepsy, which occur sparingly in long duration neural voltage recordings. The method applies convolutional neural networks (CNNs) to spectrogram data, and performs comparably to gold-standard expert classifications. We apply this measure to a population of patients with childhood epilepsy, and effectively separate them into high and low seizure risk groups. The final project studies the COVID-19 epidemic, modeling infections and deaths over time from large quantities of noisy, incomplete state-level observations. We use a statistical, data-driven analysis to estimate the basic reproduction number (R0), and use this estimate in multiple compartmental models, fitting unknown parameters for death and recovery rates using an ensemble Markov chain Monte Carlo (MCMC) method. We show consistent estimates of dynamics and parameters across multiple compartmental models, in alignment with our current epidemiological understanding of the disease. In all projects, we are able to uncover key features of interest amid the large, noisy data, providing key insights backed by mathematical and statistical rigor.

The question how aquatic populations persist in rivers when individuals are constantly lost due to downstream drift has been termed the ``drift paradox." Reaction-diffusion-advection models have been used to describe the spatial-temporal dynamics of stream population and they provide some qualitative explanations to the paradox. Here random undirected movement of individuals in the environment is described by passive diffusion, and an advective term is used to describe the directed movement in a river caused by the flow. In this work, the effect of spatially varying Allee effect growth rate on the dynamics of reaction-diffusion-advection models for the stream population is studied. In the first part, a reaction-diffusion-advection equation with strong Allee effect growth rate is proposed to model a single species stream population in a unidirectional flow. Under biologically reasonable boundary conditions, the existence of multiple positive steady states is shown when both the diffusion coefficient and the advection rate are small, which lead to different asymptotic behavior for different initial conditions. On the other hand, when the advection rate is large, the population becomes extinct regardless of initial condition under most boundary conditions. It is shown that the population persistence or extinction depends on Allee threshold, advection rate, diffusion coefficient and initial conditions, and there is also rich transient dynamical behavior before the eventual population persistence or extinction. The dynamical behavior of a reaction-diffusion-advection model of a stream population with weak Allee effect type growth is studied in the second part. Under the open environment, it is shown that the persistence or extinction of population depends on the diffusion coefficient, advection rate and type of boundary condition, and the existence of multiple positive steady states is proved for intermediate advection rate using bifurcation theory. On the other hand, for closed environment, the stream population always persists for all diffusion coefficients and advection rates. In the last part, the dynamics of a reaction-diffusion-advection benthic-drift population model that links changes in the flow regime and habitat availability with population dynamics is studied. In the model, the stream is divided into drift zone and benthic zone, and the population is divided into two interacting compartments, individuals residing in the benthic zone and individuals dispersing in the drift zone. The benthic population growth is assumed to be of strong Allee effect type. The influence of flow speed and individual transfer rates between zones on the population persistence and extinction is considered, and the criteria of population persistence or extinction are formulated and proved. All results are proved rigorously using the theory of partial differential equation, dynamical systems. Various mathematical tools such as bifurcation methods, variational methods, and monotone methods are applied to show the existence of multiple steady state solutions of models.

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Steps in wave-ray construction are as follows: 1) Select wave periods and approach angles for each series of rays to be constructed. 2) Prepare a grid of depth values for the area of investigation. 3) Use a computer program to obtain a table of water depths and related wave velocities for each wave pe:r.'iod selected in (1). 4) Make a grid of wave-velocity values for each wave period selected in (1), for the area of investigation, using a second computer program which takes as input the depth grid of ( 2) and tho appropriate depth-velocity table of (3). 5) Derive, using a third computer program, matrices for use in the linear interpolation scheme of the computer program of (6). 6) Calculate for each wave period specified in (1), the points along a wave ray using a computer program which takes for input: (a) the appropriate velocity grid of (4), (b) the matrices of (5), (c) the origin points and approach angles of (1) for given wave rays. A linear-interpolation scheme (using the least-squares method) is used in determination of wave velocity at a given point along a ray. Ray curvature is then calculated at this point and an iteration procedure is solved to obtain the position of the next point. The ray terminates at the shore or gird border. The procedure outlined is in the developmental stage, and suggestions fo1" improvements are given that should offer a quick, accurate, and objective method of constructing wave rays.

I consider the problem of Reinforcement Learning (RL) in a biologically feasible neural network model, as a proxy for investigating RL in the brain itself. Recent research has demonstrated that synaptic plasticity in the higher regions of the brain (such as the cortex and striatum) depends on neuromodulatory signals which encode, amongst other things, a response to reward from the environment. I consider which forms of synaptic plasticity rules might arise under the guidance of an Evolutionary Algorithm (EA), when an agent is tasked with making decisions in response to noisy stimuli (perceptual decision making). By proposing a general framework which captures many proposed biologically feasible phenomenological synaptic plasticity rules, including classical SpikeTime-Dependent Plasticity (STDP) rules and the triplet rules, and rate-based rules such as Oja's Rule and BCM rules, as well as their reward-modulated extensions (such as Reward-Modulated Spike-Time-Dependent Plasticity (R-STDP)), I allow a general biologically feasible neural network the ability to evolve the rules best suited for learning to solve perceptual decision-making tasks.

In the retina, the activity of ganglion cells, which feed information through the optic nerve to the rest of the brain, is all that our brain will ever know about the visual world. The interactions between many neurons are essential to processing visual information and a growing body of evidence suggests that the activity of populations of retinal ganglion cells cannot be understood from knowledge of the individual cells alone. Modelling the probability of which cells in a population will fire or remain silent at any moment in time is a difficult problem because of the exponentially many possible states that can arise, many of which we will never even observe in finite recordings of retinal activity. To model this activity, maximum entropy models have been proposed which provide probabilistic descriptions over all possible states but can be fitted using relatively few well-sampled statistics. Maximum entropy models have the appealing property of being the least biased explanation of the available information, in the sense that they maximise the information theoretic entropy. We investigate this use of maximum entropy models and examine the population sizes and constraints that they require in order to learn nontrivial insights from finite data. Going beyond maximum entropy models, we investigate autoencoders, which provide computationally efficient means of simplifying the activity of retinal ganglion cells.