R-3 Repository :: Browsing by Author "Cox, Steven J."

Browsing by Author "Cox, Steven J."

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A Fast, Fully Implicit Backward Euler Solver for Dendritic Neurons
(2000-09) Cox, Steven J.; Griffith, Boyce E.
We develop and test a C++ implementation of a discretization of the Hodgkin-Huxley equations for dendritic neurons which employs backward Euler in time and finite differences in space. We make use of the sparse analytical Jacobian matrix to perform the nonlinear solve required at each time step via Newton's method.
A quantitative study of neuronal calcium signaling
(2004) Hartsfield, Jane Wall; Cox, Steven J.
Neurons have both a fast and slow mode of signaling. Fast signals are communicated by transmembrane voltage changes, while calcium levels within the cell communicate information on a much slower time scale. Calcium acts as a second messenger responsible for modulating neuronal excitability in many ways including the mediation of gene transcription in the cell and the sensitivity of the cell to further stimulus. I develop a voltage model of the neuron's electrical signal with ion diffusion and drift which includes voltage-gated calcium currents and calcium-dependent potassium currents. The influx of calcium resulting from the voltage model will prime the endoplasmic reticulum with calcium. A model of the dynamics of calcium induced calcium release from the endoplasmic reticulum via IP3 receptors which includes diffusion of calcium and IP3 as well as calcium buffering by the mitochondria results in a calcium wave similar to what has been observed experimentally.
A quantitative study of neuronal calcium signaling
(2006) Hartsfield, Jane Wall; Cox, Steven J.
Neurons have both a fast and slow mode of signaling. Fast signals are communicated by transmembrane voltage changes, while calcium levels within the cell communicate information on a much slower time scale. Calcium acts as a second messenger responsible for modulating neuronal excitability in many ways including the mediation of gene transcription in the cell and the sensitivity of the cell to further stimulus. I propose a means of determining calcium conductance density of the cell membrane from intracellular calcium concentration measurement data using a two step process. The first step is the inference of calcium current density from calcium concentration measurements using a least squares fit to the data. Once an estimate of the calcium current density is determined, the minimum value over time is used to determine the calcium conductance density. I develop a voltage model of the neuron's electrical signal with ion diffusion and drift which includes voltage-gated calcium currents and calcium-dependent potassium currents. The influx of calcium resulting from the voltage model will prime the endoplasmic reticulum with calcium. A model of the dynamics of calcium induced calcium release from the endoplasmic reticulum via IP3 receptors which includes diffusion of calcium and IP3 as well as calcium buffering by the mitochondria results in a calcium wave similar to what has been observed experimentally. Finally, I use a branch structure together with IP3 generation, calcium buffers in the cytosol and ER, cell membrane calcium transports (voltage-gated calcium channels, pumps, exchangers, and store-operated channels), and ER calcium transports (IP3 receptors, ryanodine receptors, pumps, leak channels) to show that calcium waves initiate in the apical trunk at the point where the stimulated oblique branches off.
A quantitative study of the lac operon
(2005) Turner, Jesse Hosea, III; Cox, Steven J.
The lac operon has been key in the study of genetic regulation. Consisting of three structural genes and a regulatory domain, the lac operon controls the manufacture of lactose-digesting enzymes in E. coli bacteria. The mechanisms through which it is repressed and activated are standard and apply to many operons in other genetic settings. As a result, scientists believe understanding the lac operon will help to decipher how more complicated genetic regulatory systems behave. A large amount of quantitative data has been generated from the lac operon, a consequence of both its small size and tractability. A variety of mathematical models have been employed to examine this data. Here, we will investigate how bifurcation theory, reverse engineering, and Gillespie's stochastic simulation method have all been used to uncover some aspect of the lac operon. Conclusions drawn from the results of these models promise to reveal new information about this operon.
Adaptive Reduction of Large Spiking Neurons
(2013-11-21) Du, Bosen; Sorensen, Danny C.; Cox, Steven J.; Embree, Mark; Antoulas, Athanasios C.
This thesis develops adaptive reduction approaches for various models of large spiking neurons. Most neurons are like dendritic trees with many branches, and they communicate by nonlinear spiking behaviors. However, with the exception of Kellems' Strong-Weak model, most existing reduction approaches compromise the active ionic mechanisms that cause the spiking dynamics. The Strong-Weak model can predict the spikes caused by suprathreshold input traveling from the dendritic branches to the spike initiation zone (SIZ), but it is not able to reproduce the back propagation from SIZ to the dendritic branches after spikes. This thesis develops a new model called QAact, the mechanisms to incorporate QAact into the hybrid model to capture the back propagation behavior, and different model reduction techniques for each part of the new hybrid model where they are most advantageous. Computational tests of QAact and the new hybrid model as well as corresponding model reduction techniques on FitzHugh-Nagumo system, active nonuniform cable, and branched cell LGMD, demonstrate a significant reduction of dimension, computational complexity and running time.
An investigation of the optimal design of the tallest unloaded column
(1997) McCarthy, C. Maeve; Cox, Steven J.
The problem of the optimal design of the tallest unloaded column is revisited with a view towards clarifying the optimality of the design proposed by Keller & Niordson in 1966 (19). The first eigenvalue of a singular Sturm-Liouville problem must be maximized in order to yield the tallest column. The difficulty with the proposed design comes from the nature of the spectrum associated with it. If this spectrum were discrete, Keller & Niordson's techniques would have been appropriate. The proposed design is shown here to admit continuous spectra and hence warrants further investigation. Over a class of admissible designs admitting purely discrete spectra, it is shown that the decreasing rearrangement of the shape leads to a column at least as tall as the original. Existence of an optimal design follows from this fact. Similar methods are applied to design problems arising from other classes of Sturm-Liouville problems. The necessary conditions for optimality established by Keller & Niordson are re-established using generalized gradient methods. Using a finite element approximation to the boundary value problem, the first eigenvalue is maximized using MATLAB's Sequential Quadratic Programming implementation constr (15) to optimize and Radke's implementation of the Implicitly Restarted Arnoldi Method speig (28) to compute eigenvalues. Analysis of the convergence behavior of the higher eigenvalues leads to the conclusion that the spectrum of the Keller & Niordson design is made up of an essential spectrum and one isolated eigenvalue. This justifies the methods used by Keller & Niordson and confirms the optimality of their design. Finally, inverse spectral methods are investigated as a means by which to increase the height of a given column by adding or subtracting material appropriately.
Applying the Short-Time Direct Directed Transfer Function to Human Electrocorticographic Recordings from a Language Task
(2013-06-28) Whaley, Meagan; Cox, Steven J.; Embree, Mark; Tandon, Nitin; Dabaghian, Yuri
This thesis applied the short-time direct directed transfer function (SdDTF) to time series data recordings from intracranial electrodes that measure the brain's electrical activity to determine the causal influences that occurred between brain regions during a speech production task. The combination of high temporal and spatial resolution of the electrocorticography (ECoG) recordings directly from the cortex render these measurements of brain activity desirable, particularly when analyzing the fine cognitive dynamics involved in word generation. This research applied a new method to characterize the SdDTF results by compressing across time and high gamma frequencies, generating adjacency matrices, and graphing them to visualize the influences between anatomical regions over the duration of the entire task. This consolidated SdDTF analysis technique allowed for data from a total of seven patients to be combined, generating results which were consistent with current speech production models. The results from this thesis contribute to the expansion of language research by identifying areas relevant to word generation, providing information that will help surgeons avoid irreparable damage to crucial cortex during brain surgery.
Computation of eigenvalues for starlike domains
(1994) Book, Robert A.; Cox, Steven J.
In this paper, we present a software tool for the computation of eigenvalues of starlike domains defined by polar boundary functions. We also offer and numerically test a conjecture on the monotonicity of the fundamental eigenvalues of the members of a family of starlike domains.
Design against resonance
(1993) Spencer, Cynthia Genae; Cox, Steven J.
A method for maximizing the distance from the spectrum of an analytic, symmetric matrix with distinct eigenvalues from a given frequency is proposed. The method models the classical approach from optimization of finding the first derivative of the function to be maximized and setting it equal to zero. The function to be maximized is the norm of a resolvent in terms of the given perturbed matrix and the applied frequency. The problem of the function being nonsmooth is considered, and a solution is proposed. This solution entails using the concept of the generalized gradient. This method provides an efficient means for targeting the potential troublespots for resonance and avoiding them with relative ease.
Efficient and accurate simulation of integrate-and-fire neuronal networks in the hippocampus
(2007) Kellems, Anthony Richard; Cox, Steven J.
This thesis evaluates a method of computing highly accurate solutions for network simulations of integrate-and-fire (IAF) neurons. Simulations are typically evolved using time-stepping, but since the IAF model is composed of linear first-order ODEs with hard thresholds, explicit solutions in terms of integrals of exponentials exist and can be approximated using quadrature. The technique presented here utilizes Clenshaw-Curtis quadrature to approximate these integrals to high accuracy. It uses the secant method to more precisely identify spike times, thus yielding more accurate solutions than do time-stepping methods. Additionally, modeling synaptic input with delta functions permits the quadrature method to be practical for simulating largescale networks. I determine general conditions under which the quadrature method is faster and more accurate than time-stepping methods. In order to make these methods accessible to other researchers, I introduce and develop software designed for simulating networks of IAF hippocampal cells.
Engineering Deep Brain Stimulation as a Treatment for Parkinson's Disease: from Models to Materials
(2014-04-25) Summerson, Samantha Rose; Aazhang, Behnaam; Kemere, Caleb T.; Baraniuk, Richard G.; Cox, Steven J.; Robinson, Jacob T.
This thesis analyzes deep brain stimulation (DBS) as a treatment for the motor symptoms of Parkinson's disease (PD) at multiple levels. Although this treatment is currently used on human patients, little is understood about the mechanism of action which allows patients to experience therapeutic benefits. The work here investigates efficacy of DBS in computational and experimental manners in order to enhance the understanding of the effects on neural activity and behavior. First, I examine computational models of the nuclei within the motor circuit of the brain and used these models to test novel electrical stimulation signal designs. I show that irregular spacing of stimulation pulses allows for increased variability in neuronal firing rate responses within the basal ganglia. Also, I develop a model of the stimulation-frequency-dependent nature of antidromic spiking induced in the motor cortex as a result of DBS. Second, I use the hemi-Parkinsonian rat model to demonstrate motor and cognitive behavioral effects of DBS in the globus pallidus internus (GPi). The work validates this animal model for translational research on DBS of the GPi and demonstrates results consistent with reports for DBS of the subthalamic nucleus (STN) in the same model. Additionally I study recorded neural activity in the motor cortex while stimulating the STN in order to characterize the corresponding changes in neural activity. I found that regular and irregular stimulation patterns both decrease Parkinsonian entropic noise in the output layer of the motor cortex, with irregular stimulation having the greatest benefit towards reducing this noise. Third, I consider a new material for its biocompatibility and applicability as a material for stimulating electrodes. In the rat model that I previously validated, I verify that behavioral results using a stimulating electrode made from carbon nanotube fibers (CNTf) match results from previous experiments using standard platinum iridium (PtIr) electrodes. Additionally, it is shown that CNTf electrodes produce lower inflammation, gliosis and damage to the blood brain barrier. Together, all three aspects of the work demonstrate significant contributions to the functionality and engineering of DBS as a neuromodulation therapy for PD.
Exploiting balanced trees in the computation of elementary flux modes via breadth-first search
(2006) Acosta, Fernando; Cox, Steven J.
In order to rationally design bacteria to produce large quantities of their precious metabolic by-products, we must first understand the metabolic pathways that lead from a starting material (e.g., glucose) to a metabolic product (e.g., hydrogen). Given a set of reactions, elementary flux modes can be used to mathematically define all metabolic pathways that are stoichiometrically and thermodynamically feasible. The elementary flux modes are the intersections of positive halfspaces, or the vertices of a convex polyhedron. Although linear programing has long been used to examine polyhedra, biologists have yet to use linear programming's full functionality to rationally design bacteria. The simplex method is one relevant linear programming function that pivots from one vertex of a polyhedron to another. The breadth-first search uses a simplex-like pivot to list every possible vertex of a polyhedron. When listing a polyhedron's vertices, it is important not to repeat vertices and thus a robust search algorithm is needed. I have employed balanced trees to speed up the breadth-first search. Specific examples, including the maximization of succinate production, will be investigated and the vertices of the mixed acid fermentation process in E. coli will be enumerated. The open source Matlab code along with a GUI will be provided, as well as links to m-files, which are available on the Internet.
Fast Electron Spectroscopy of Enhanced Plasmonic Nanoantenna Resonances
(2014-07-31) Day, Jared K.; Halas, Naomi J.; Natelson, Douglas; Nordlander, Peter; Cox, Steven J.
Surface plasmons are elementary excitations of the collective and coherent oscillations of conductive band electrons coupled with photons at the surface of metals. Surface plasmons of metallic nanostructures can efficiently couple to light making them a new class of optical antennas that can confine and control light at nanometer scale dimensions. Nanoscale optical antennas can be used to enhance the energy transfer between nanoscale systems and freely-propagating radiation. Plasmonic nanoantennas have already been used to enhance single molecule detection, diagnosis and treat cancer, harvest solar energy, to create metamaterials with new optical properties and to enhance photo-chemical reactions. The applications for plasmonic nanoantennas are only limited by the fundamental understanding of their unique optical properties and the rational design of new coupled antenna systems. It is therefore necessary to interrogate and image the local electromagnetic response of nanoantenna systems to establish intuition between near-field coupling dynamics and far-field optical properties. This thesis focuses on the characterization and enhancement of the longitudinal multipolar plasmonic resonances of Au nanorod nanoantennas. To better understand these resonances fast electron spectroscopy is used to both visualize and probe the near- and far-field properties of multipolar resonances of individual nanorods and more complex nanorod systems through cathodoluminescence (CL). CL intensity maps show that coupled nanorod systems enhance and alter nanorod resonances away from ideal resonant behavior creating hybridized longitudinal modes that expand and relax at controllable locations along the nanorod. These measurements show that complex geometries can strengthen and alter the local density of optical states for nanoantenna designs with more functionality and better control of localized electromagnetic fields. Finally, the electron excitations are compared to plane wave optical stimulation both experimentally and through Finite Difference Time Domain simulations to begin to develop a qualitative picture of how the local density of optical states affects the far-field optical scattering properties of plasmonic nanoantennas.
Force and Heat Generation in a Conducting Sphere in an Alternating Magnetic Field
(1995) Sathuvalli, Udaya Bhaskar R.; Bayazitoglu, Yildiz; Chapman, Alan J.; Akin, John Edward.; Cox, Steven J.
The interaction of an electrically conducting sphere with a time varying magnetic ﬁeld is useful in the study of "containerless" processing methods such as electromagnetic levitation melting. The fundamental quantities of interest in this interaction are the rate of heat generation in the sphere, the Lorentz force and magnetic pressure on it. These quantities depend upon the nature of the current sources that create the magnetic field, and the material properties of the sphere. In this work, the Maxwell equations for the interaction of a sphere with an arbitrary external alternating magnetic ﬁeld are ﬁrst formulated. Then, the density of the induced currents in the sphere is found as a function of the external current sources and the material properties of the sphere. The current density is now used to calculate the heat generated in the sphere. Next, a method to calculate the Lorentz force on an electrically conducting sphere placed in an arbitrary sinusoidally varying magnetic ﬁeld is developed and a formula for the force on the sphere is given. This formula is used to derive the special case of a sphere in an axisymmetric system of circular current loops. Numerical results for the force on a sphere on the axis of a stack of loops are presented as a function of the stack geometry. The results for the heat generation and the Lorentz force obtained in this study are compared with the results obtained by a previously used model (known as the "homogeneous model") which assumes that the external magnetic ﬁeld is uniform and unidirectional. It is shown that the homogeneous model is a special case of the present model and that it underestimates heat generation signiﬁcantly, and overestimates the Lorentz force. In addition, as the size of the sphere decreases, the homogeneous model gives erroneous results, approaching an order of magnitude for heat generation in a very small sphere. Subsequently, a procedure to determine the magnetic pressure distribution on the surface of a levitated liquid metal droplet is developed. The pressure distribution is calculated in terms of the geometry of the coil that creates the ﬁeld. Finally, the magnetic ﬁelds of helical windings that are commonly used in the laboratory for levitation melting are calculated.
Functional inference of conductances in the LGMD neuron
(2013-08-27) Ackermann, Etienne; Cox, Steven J.; Embree, Mark; Sorensen, Danny C.; Dabaghian, Yuri
This thesis develops an approach to determine spatially-varying ionic channel conductances throughout the dendrites of the LGMD neuron from distal transmembrane potential recordings in response to distributed subthreshold current injections. In particular this approach is demonstrated on a straight cable approximation to the LGMD neuron with leak and hyperpolarization-activated h-currents. Knowledge of the underlying channel conductances can help neuroscientists to characterize, better understand, and predict neuronal behavior---and topographic integration in the LGMD neuron in particular---but it is extremely difficult to measure these conductances directly. As a consequence, these conductances are commonly estimated by searching for several parameters that lead to simulated responses that are consistent with recorded behavior. In contrast, the approach presented here uses the method of moments to directly recover the underlying conductances, eliminating the need to simulate responses, making this approach both faster and more robust than typical optimization approaches since the solution cannot get trapped in local minima.
Identification of Regional Variation in the Constitutive Response of Axisymmetric Membranes
(2000-09) Boriek, Aladin M.; Cox, Steven J.
We demonstrate that the equilibrium equations for an axisymmetric, nonlinear, anisotropic membrane under hydrostatic pressure allow explicit representation of the longitudinal and azimuthal stresses in terms of the associated longitudinal and azimuthal strains. We apply this result in a numerical simulation of the canine diaphragm. More precisely, we compute the deformation of the membrane under a quasi static increase in the intensity of the applied hydrostatic load. The associated strains are easily estimated via finite differences. As the membrane is inflated the set of achieved strains grows and as a result of our explicit representation formula, we recover larger and larger patches of the associated stress surfaces.
Magnetic damping of an elastic conductor
(2009) Hokanson, Jeffrey M.; Embree, Mark; Cox, Steven J.
Many applications call for a design that maximizes the rate of energy decay. Typical problems of this class include one dimensional damped wave operators, where energy dissipation is caused by a damping operator acting on the velocity. Two damping operators are well understood: a multiplication operator (known as viscous damping) and a scaled Laplacian (known as Kelvin---Voigt damping). Paralleling the analysis of viscous damping, this thesis investigates energy decay for a novel third operator known as magnetic damping, where the damping is expressed via a rank-one self-adjoint operator, dependent on a function a. This operator describes a conductive monochord embedded in a spatially varying magnetic field perpendicular to the monochord and proportional to a. Through an analysis of the spectrum, this thesis suggests that unless a has a singularity at one boundary for any finite time, there exist initial conditions that give arbitrarily small energy decay at any time.
Matrix Analysis
(Rice University, 2005-09-05) Cox, Steven J.
Equilibria and the solution of linear and linear least squares problems. Dynamical systems and the eigenvalue problem with the Jordan form and Laplace transform via complex integration.
Minimizers of the vector-valued coarea formula
(2012-09-05) Carroll, Colin; Hardt, Robert M.; Wolf, Michael; Cox, Steven J.
The vector-valued coarea formula provides a relationship between the integral of the Jacobian of a map from high dimensions down to low dimensions with the integral over the measure of the fibers of this map. We explore minimizers of this functional, proving existence using both a variational approach and an approach with currents. Additionally, we consider what properties these minimizers will have and provide examples. Finally, this problem is considered in metric spaces, where a third existence proof is given.
Model reduction of large spiking neurons
(2010) Kellems, Anthony Richard; Cox, Steven J.
This thesis introduces and applies model reduction techniques to problems associated with simulation of realistic single neurons. Neurons have complicated dendritic structures and spatially-distributed ionic kinetics that give rise to highly nonlinear dynamics. However, existing model reduction methods compromise the geometry, and thus sacrifice the original input-output relationship. I demonstrate that linear and nonlinear model reduction techniques yield systems that capture the salient dynamics of morphologically accurate neuronal models and preserve the input-output maps while using significantly fewer variables than the full systems. Two main dynamic regimes characterize the voltage response of a neuron, and I demonstrate that different model reduction techniques are well-suited to each regime. Small perturbations from the neuron's rest state fall into the subthreshold regime, which can be accurately described by a linear system. By applying Balanced Truncation (BT), a model reduction technique for general linear systems, I recover subthreshold voltage dynamics, and I provide an efficient Iterative Rational Krylov Algorithm (IRKA), which makes large problems of interest tractable. However, these approximations are not valid once the input to the neuron is sufficient to drive the voltage into the spiking regime, which is characterized by highly nonlinear behavior. To reproduce spiking dynamics, I use a proper orthogonal decomposition (POD) to reduce the number of state variables and a discrete empirical interpolation method (DEIM) to reduce the complexity of the nonlinear terms. The techniques described above are successful, but they inherently assume that the whole neuron is either passive (linear) or active (nonlinear). However, in realistic cells the voltage response at distal locations is nearly linear, while at proximal locations it is very nonlinear. With this observation, I fuse the aforementioned models together to create a reduced coupled model in which each reduction technique is used where it is most advantageous, thereby making it possible to more accurately simulate a larger class of cortical neurons.

Browsing by Author "Cox, Steven J."

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