This dissertation consists of two parts: Neurodegenerative damage reduces firing coherence in a continuous attractor model of grid cells, and damage impact on the gird-cell population codes for animal's locations.

The work in Part I is motivated by the discovery of grid cells and their specific grid-like firing pattern: Grid cells, firstly found in the dorsolateral band of the medial entorhinal cortex(dMEC) in 2005, display strikingly regular periodic firing patterns on a lattice of positions in 2-D space. This helps animals to encode relative spatial location without reference to external cues. The dMEC is damaged in the early stages of Alzheimer's Disease, which affects navigation ability of a disease victim, reducing the synaptic density of neurons in the network. Within an established 2-dimensional continuous attractor neural network model of grid cell activity, we introduce neural sheet damage parameterized by radius and by the strength of the synaptic output for neurons in the damaged region. The mean proportionality of the grid field flow rate in the dMEC to the velocity of the model animal is maintained, but there is a broadened distribution of flow rates in the damaged case. This flow rate-to-velocity proportionality is essential to establish coherent grid firing fields for individual grid cells for a roaming animal. When we examine the coherence of the grid cell firing field by studying Bragg Peaks of the Fourier transformed lattice firing field intensity in both damaged and undamaged regions, we find that for a wide range of damage radius and reduced synaptic strength that for undamaged model grid cells there is an incoherent firing field structure with only a single central peak. In the radius-damage plane this is adjacent to narrow bands of striped lattices (two additional Bragg peaks), which about an orthorhombic pattern (four additional Bragg peaks), that abut the undamaged hexagonal region (six additional Bragg peaks). Within the damaged region, grid cells show no Bragg peaks outside the central one which shows reduced intensity with increasing damage, and outside the damaged region the central Bragg peak strength is largely unaffected. There is a re-entrant region of normal grid firing fields for very large damage area. We anticipate that the modified grid cell behavior can be observed in non-invasive fMRI imaging of the dMEC.

The work in Part II is motivated by a broad goal to explain navigation system: The brain is a remarkable information engine and it's efficiency may come from a hierarchy organization of neurons. At the same time, A unique topographical representation of space is found in the concerted activity of grid cells in the medial entorhinal cortex. Many in this region exhibit a hexagonal firing pattern with grid spacing. And grid spacing has been found to increase along the dorsoventral axis of dMEC but in discrete steps. Such a modular structure provides a new place-coding theory that explains why grid cells has hierarchy organization identified by different spacing. Compared with classical population code (CPC) theory, the hierarchy in grid population code (GPC) improves the coding efficiency and the noise robustness. We developed Sammeet Sreenivasan and Ila Fiete's network model (readout-grid cell network) to construct the GPC process from input signal, through grid cells modules, to the place cells sensory. The largely stable consistency between input location and inferred location by place cells proves the practicality of readout-grid cell network. Within the completed multiple-layers neural network model of grid coding, we introduce grid layers damage parameterized by radius and by the strength of the synaptic output for neurons in the damaged region. The self-consistence between location signal and inferred location is distributed within reduced coding range. For M layers of the N grid cells, damage within a single layer doesn't destroy the accurate place coding considering the maximum possible coding range ($R \sim N^M$) overloads the reduced coding range ($R_l < 500cm$). We construct the landscape of heat-map showing influence of damage in all situations, and noticed that the layers with bigger spacing (top layers) show more severe disruption given the same condition. This proves the hierarchy theory of GPC that the top panels dominate place coding and fluctuations on big-spacing modules bring more errors.

Working memory refers to information actively held in the brain and is essential for advanced functions like thinking, decision-making, or learning. In this thesis, I focus on two key problems in working memory: 1) how to maintain a graded amplitude of a local memory in a classical ring architecture, and 2) how to maintain a novel pattern of graded neural activity in an unstructured network. To address these problems, I start with reexamining previous methodologies to limit modeling possibilities. Then, I propose two neural circuit models with dendritic bistability, each of which is treated analytically and which are robust against various perturbations. This work links physiological properties to functionality, contributing to a deeper understanding of the mechanisms underlying working memory.

This dissertation is split into two parts. In part 1, we discuss plant cytokinesis, a fundamental process of plant life which involves de novo formation of a “cell plate” partitioningthe cytoplasm of dividing cells. Cell plate formation is directed by orchestrated delivery, fusion of cytokinetic vesicles, and membrane maturation to form a nascent cell wall by timely deposition of polysaccharides. During cell plate maturation, the fragile membrane network transitions to a fenestrated sheet and finally a young cell wall. Here, we approximated cell plate sub-structures with testable shapes and adopted the Helfrich-free energy model for membranes, including a stabilizing and spreading force, to understand the transition from a vesicular network to a fenestrated sheet and mature cell plate. Regular cell plate development in the model was possible, with suitable bending modulus, for a two-dimensional late stage spreading force of 2–6pN/nm, an osmotic pressure difference of 2–10kP a, and spontaneous curvature between 0 and 0.04nm−1 . With these conditions, stable membrane conformation sizes and morphologies emerged in concordance with stages of cell plate development. To reach a mature cell plate, our model required the late-stage onset of a spreading/stabilizing force coupled with a concurrent loss of spontaneous curvature. Absence of a spreading/stabilizing force predicts failure of maturation. The proposed model provides a framework to interrogate different players in late cytokinesis and potentially other membrane networks that undergo such transitions. Callose, is a polysaccharide that accumulates transiently during cell plate maturation. Callose-related observations were consistent with the proposed model’s concept, suggesting that it is one of the factors involved in establishing the spreading force. In part 2, we discuss three different in-silico studies of SARS-CoV-2. These studies involve the use of molecular dynamics simulations, endpoint free-energy estimates, as well as predictive neural networks (AlphaFold). In Chapter 4, we present binding strength estimates of three critical fitness parameters (RBD/ACE2 binding, furin enzyme binding, antibody escape) of the SARS-CoV-2 omicron variant. We show that our results align with the preliminary observations noted with the variant, i.e. weakened RBD/ACE2 binding, but increased antibody escape. In Chapter 5, we present an in-depth analysis of the most commonly observed sequences in the Furin Cleavage Domain (FCD) and their interaction with the furin enzyme. We show that the Delta variant exhibits the strongest possible binding with the furin enzyme, and we identify key observed and unobserved sequences that could exhibit the same binding strength. In Chapter 6, we present a computational design of a humanized ACE2 decoy to be used as a possible therapeutic or diagnostic agent based on the principles of competitive binding. We show that our design binds favorably well to multiple SARS-CoV-2 target RBDs, including the delta and omicron variants.

Oscillations in olfactory bulb activity are modulated by neurodegenerative disease pathology. Particularly, Alzheimer's disease pathology has been found to strengthen the gamma-band oscillatory activity that arises from the interface between mitral and granule cells. Using two different computational models of the olfactory bulb network, this work investigates the effect of synaptic damage on olfactory bulb gamma oscillations and explores various mechanisms involved in gamma oscillation generation. The first model treats the olfactory bulb as a set of coupled nonlinear oscillators and propagates synaptic weight degradation locally and globally. Gamma oscillations are enhanced at moderate levels of damage, followed by a loss of oscillatory activity predictable by linearized analysis. The second set of simulations uses the large-scale mechanistic model presented by Kersen, Tavoni, and Balasubramanian, implementing Izhikevich dynamical equations. This work adds synapse-specific graded inhibition to the mitral cells, resulting in activity-dependent inhibition critically involving granule cell NMDA current, similar to recent experimental findings. The modified model also implements tonic inhibition to the granule cell population, which reduces granule cell firing and increases mitral cell spike participation and network synchrony. The present iterations of this model do not yield robust gamma oscillations. However, the same model framework could feasibly achieve gamma activity via an entrainment mechanism by using resonant mitral cells with subthreshold oscillations. Finally, the last chapter proposes an extension of the model that implements inhibition-dependent backpropagation of mitral cell action potentials. Previous work theorized that such a mechanism could underlie the increased gamma power and decreased gamma frequency which is observed after modest application of GABA antagonist, an experimental result not yet captured by computational modeling. Thus this work aims to provide critical elements and potentially fruitful directions in the continued investigations of the generation and perturbation of olfactory bulb gamma oscillations.

This dissertation is split into 3 parts. In the first part (Chapter 2) we look at shapesor manifolds on which neural activity over time lies. In a neural state space, where each axis represents a neuron, neural activity over time forms a point cloud. This point cloud often occupies a small region in the space of all possible activity patterns thus revealing structure in data. We consider point clouds from neural activities from common population codes known as “tuning curve models”. In these models, the firing rate of each neuron is a function of a latent variable which might be a stimulus variable or a variable related to an internal state and a tuning curve parameter which labels each neuron. We address the question: How close are point clouds formed by such models to a linear subspace? To answer this, we define the linear dimension of the data to be the number of dimensions which captures a very high fraction of variance, for example 95% variance in this data. We show that the linear dimension grows exponentially with the number of latent variables encoded by the population. Thus the manifolds formed by the neural activities from these models are extremely non-linear. Linear dimension is not a good measure for the intrinsic dimension of the manifold on which this point cloud lies. In the second part (Chapter 3), we model connections between distant brain regions by sparse random connections. We start by observing that such a network has a special property known as the expander property. Using this property it can be shown that information can be transmitted efficiently from a source region to a target region even if the target region has fewer neurons than the source region. We also consider if the compressed patterns in the target region can be re-coded or expanded to perform some computation. We show that the compressed patterns can be re-expanded by algorithms known as Locally Competitive Algorithms (LCA) and the re-expanded patterns can be separated by a downstream neuron into arbitrarily defined classes. We next consider whether long range reciprocal connections between two regions can be used to maintain persistent activity in both the regions. Such activity is thought to be a substrate for working memory, the ability to hold things in mind. We show that the network can indeed maintain sparse patterns of activity through simple network dynamics. We conclude that sparse random connections can be used to transmit information effectively and improve the performance of certain computations compared to dense random connections. In the last part (Chapter 4), we built a computational rate model for the pre-cortex biological neural circuit responsible for the localisation of sound in the vertical plane. Interaction of incoming sound waves with the outer ear filters out energy from specific frequency bands in the spectrum of the incoming sound. The frequency bands with zero or reduced power in them are known as notches. The position of the notches is a function of the angle of elevation of the sound source. There is a dedicated set of neurons in the auditory pathway which are sensitive to the position of these notches and hence thought to be responsible for the localization of sound in the vertical plane. These neurons show different levels of excitation or inhibition above or below their spontaneous rates for different combinations of frequencies and intensities of sound. We built a computational model to probe how this complex set of responses arise from the interaction between the various populations of neurons in the auditory pathway.

Uridine 5'-diphosphate-N-acetylglucosamine (UDP-GlcNAc) acyltransferase (LpxA) catalyzes a reversible reaction for adding an O-acyl group to the GlcNAc in UDP-GlcNAc in the first step of lipid A biosynthesis. Lipid A constitutes a major component of lipopolysaccharides, also referred to as endotoxins, which form the outer monolayer of the outer membrane of Gram-negative bacteria. Ligand-free and UDP-GlcNAc-bound crystal structures of LpxA from Bacteroides fragilis NCTC 9343, the most common pathogenic bacteria found in abdominal abscesses, have been determined and are presented here. The enzyme crystallizes in a cubic space group, with the crystallographic threefold axis generating the biological functional homotrimer and with each monomer forming a nine-rung left-handed β-helical (LβH) fold in the N-terminus followed by an α-helical motif in the C-terminus. The structure is highly similar to LpxA from other organisms. Yet, despite sharing a similar LβH structure with LpxAs from Escherichia coli and others, previously unseen calcium ions are observed on the threefold axis in B. fragilis LpxA to help stabilize the trimeric assembly.