Dissertations / Theses on the topic 'Action potentials (Electrophysiology) – Mathematics'

This thesis is concerned with the development of a new approach to the modelling of cardiac action potentials. Electrophysiological models of the heart have become very accurate in recent years giving rise to extremely complicated systems of differential equations. Although describing the behaviour of cardiac cells well, the models are computationally demanding for numerical simulations and are very difficult to analyse from a mathematical (dynamical-systems) viewpoint. Simplified mathematical models that capture the underlying dynamics to a certain extent are therefore frequently used. However, from a physiological viewpoint these equations are unrealistic and often fail to reproduce important quantitative properties of the tissue. In this thesis we introduce a different approach to the mathematical modelling of cardiac action potentials with the aim of gaining a clearer insight into the origin of the dynamics of electrophysiological models. Chapter 1 contains an introduction to the research and outlines the main aims of the work. In Chapter 2 various background material is introduced. This includes some basic electrophysiology, ideas currently used in mathematical modelling of excitable media, and details of models previously developed for the study of cardiac tissue. In Chapter 3, following a detailed analysis of an early physiological model, we develop a mathematical model based on the currents involved. This model reproduces, to good accuracy, action potentials of heart tissue and we discuss the essential ideas behind the dynamics. In Chapter 4 the mathematical model developed in the previous chapter is analysed in more detail and simpler equations using similar ideas are introduced. Various types of action potentials of varying behaviours are studied. In Chapter 5 we investigate some spatial simulations of the new mathematical models. We principally concentrate on one-dimensional studies but towards the end of the chapter we look at some two-dimensional simulations. Finally, in Chapter 6, we discuss our conclusions and some possible ideas for further related work. Details of our methods of numerical simulation are included in Appendix A.

The work developed in this thesis focusses on the electrical activity of the heart, from the modeling of the action potential originating from cardiac cells and propagating through the heart, as well as its electrical manifestation at the body surface. The study is divided in two main parts: modeling the action potential, and numerical simulations. For modeling the action potential a dimensional and asymptotic analysis is done. The key advance in this part of the work is that this analysis gives the steps to reliably control the action potential. It allows predicting the time/space scales and speed of any action potential that is to say the shape of the action potential and its propagation. This can be done as the explicit relations on all the physiological constants are defined precisely. This method facilitates the integrative modeling of a complete human heart with tissue-specific ionic models. It even proves that using a single model for the cardiac action potential is enough in many situations. For efficient numerical simulations, a numerical method for solving the heart-torso coupling problem is explored according to a level set description of the domains. This is done in the perspective of using directly medical images for building computational domains. A finite element method is then developed to manage meshes not adapted to internal interfaces. Finally, an anisotropic adaptive remeshing methods for unstructured finite element meshes is used to efficiently capture propagating action potentials within complex, realistic two dimensional geometries.

Thesis (Ph.D.)--Biomedical Engineering, Georgia Institute of Technology, 2007.<br>Committee Chair: Potter, Steve; Committee Member: Butera, Robert; Committee Member: DeWeerth, Stephen; Committee Member: Schumacher, Eric; Committee Member: Wenner, Pete.

Auditory physiology is nearly unique in the human body because of its small-diameter neurons. When considering a single node on one neuron, the number of channels is very small, so ion fluxes exhibit randomness. Hodgkin and Huxley, in 1952, set forth a system of Ordinary Differential Equations (ODEs) to track the flow of ions in a squid motor neuron, based on a circuit analogy for electric current. This formalism for modeling is still in use today and is useful because coefficients can be directly measured. To measure auditory properties of Firing Efficiency (FE) and Post Stimulus Time (PST), we can simply measure the depolarization, or "upstroke," of a node. Hence, we reduce the four-dimensional squid neuron model to a two-dimensional system of ODEs. The stochastic variable m for sodium activation is allowed a random walk in addition to its normal evolution, and the results are drastic. The diffusion coefficient, for spreading, is inversely proportional to the number of channels; for 130 ion channels, D is closer to 1/3 than 0 and cannot be called negligible. A system of Partial Differential Equations (PDEs) is derived in these pages to model the distribution of states of the node with respect to the (nondimensionalized) voltage v and the sodium activation gate m. Initial conditions describe a distribution of (v,m) states; in most experiments, this would be a curve with mode at the resting state. Boundary conditions are Robin (Natural) boundary conditions, which gives conservation of the population. Evolution of the PDE has a drift term for the mean change of state and a diffusion term, the random change of state. The phase plane is broken into fired and resting regions, which form basins of attraction for fired and resting-state fixed points. If a stimulus causes ions to flow from the resting region into the fired region, this rate of flux is approximately the firing rate, analogous to clinically measuring when the voltage crosses a threshold. This gives a PST histogram. The FE is an integral of the population over the fired region at a measured stop time after the stimulus (since, in the reduced model, when neurons fire they do not repolarize). This dissertation also includes useful generalizations and methodology for turning other ODEs into PDEs. Within the HH modeling, parameters can be switched for other systems of the body, and may present a similar firing and non-firing separatrix (as in Chapter 3). For any system of ODEs, an advection model can show a distribution of initial conditions or the evolution of a given initial probability density over a state space (Chapter 4); a system of Stochastic Differential Equations can be modeled with an advection-diffusion equation (Chapter 5). As computers increase in speed and as the ability of software to create adaptive meshes and step sizes improves, modeling with a PDE becomes more and more efficient over its ODE counterpart.

The study of cardiac action potentials has many medical applications. Dr. Dennis Noble first used mathematical models to study cardiac action potentials in the 1960s. We begin our study of cardiac action potentials with one form of the Fitzhugh-Nagumo partial differential equation. We use the non-classical method to produce a closed form solution for the decoupled Fitzhugh Nagumo equation. Using voltage recording data of action potentials in a cardiac myocyte and in purkinje fibers, we estimate parameter values for the closed form solution with standard linear and non-linear regression methods. Results are limited, thus leading us to perturb the solution to obtain a better fit. We turn to singular perturbation theory to justify our pole-based approach. Finally, we test our model on independent action potential data sets to evaluate our model and to draw conclusions on how our model can be applied.

Thesis (Ph. D.)--Mechanical Engineering, Georgia Institute of Technology, 2008.<br>Committee Chair: Steve M. Potter; Committee Member: Eric Schumacher; Committee Member: Robert J. Butera; Committee Member: Stephan P. DeWeerth; Committee Member: Thomas D. DeMarse.

Event related potentials (ERP's) carry very important information that relates to the performance of the brain functions of the human being. Further studies have identified that one component, in particular, P 300, is affected by the memory process. Matched filter is used to improved the SNR of signal ERP' s. We use the output of the matched filter to distinguish the difference of the waveforms between normal subjects and memory-impaired subjects. In our study, we found that the peak values of the matched filtering output were different between normal subjects and memoryimpaired subjects. Also, as an application, wavelet transform is introduced to the ERP analysis. Local maximum of wavelet transform was used as a local feature to find the relationship between the sharp variation points and the memory process. A comparison between matched filtering and wavelet transform was made and also the correlation coefficients of the peaks and sharp variation points are calculated to find the relationship between the important moments in a memory process.

Wavelet transform provides an alternative to the classical Short-Time Fourier Transform (STFT). In contrast to the STFT, which uses a single analysis window, the Wavelet Transform uses shorter windows at higher frequencies and longer windows at lower frequencies. For some particular wavelet functions, the local maxima of the wavelet transform correspond to the sharp variation points of the signal. As an application, wavelet transform is introduced to the character recognition. Local maximum of wavelet transform is used as a local feature to describe character boundary. The wavelet method performs well in the presence of noise. The maximum of wavelet transform is also an important feature for analyzing the properties of brain wave. In our study, we found the maximum of wavelet transform was related to the P300 latency. It provides an easy and efficient way to measure P300 latency.

Wing-Keung Sim.<br>Thesis (M.Phil.)--Chinese University of Hong Kong, 1999.<br>Includes bibliographical references (leaves 109-114).<br>Abstracts in English and Chinese.<br>Abstract --- p.2<br>摘要 --- p.3<br>Acknowledgements --- p.4<br>Contents --- p.5<br>List of Figures --- p.8<br>List of Tables --- p.9<br>Introductions --- p.10<br>Chapter 1.1 --- Motivation & Aims --- p.10<br>Chapter 1.2 --- Contributions --- p.11<br>Chapter 1.3 --- Structure of Thesis --- p.11<br>Electrophysiological Spike Discrimination --- p.13<br>Chapter 2.1 --- Introduction --- p.13<br>Chapter 2.2 --- Cellular Physiology --- p.13<br>Chapter 2.2.1 --- Action Potential --- p.13<br>Chapter 2.2.2 --- Recording of Spikes Activities --- p.15<br>Chapter 2.2.3 --- Demultiplexing of Multi-Neuron Recordings --- p.17<br>Chapter 2.3 --- Application of Clustering for Mixed Spikes Train Separation --- p.17<br>Chapter 2.3.1 --- Design Principles for Spike Discrimination Procedures --- p.17<br>Chapter 2.3.2 --- Clustering Analysis --- p.18<br>Chapter 2.3.3 --- Comparison of Clustering Techniques --- p.19<br>Chapter 2.4 --- Literature Review --- p.19<br>Chapter 2.4.1 --- Template Spike Matching --- p.19<br>Chapter 2.4.2 --- Reduced Feature Matching --- p.20<br>Chapter 2.4.3 --- Artificial Neural Networks --- p.21<br>Chapter 2.4.4 --- Hardware Implementation --- p.21<br>Chapter 2.5 --- Summary --- p.22<br>Correlation of Perceived Headphone Sound Quality with Physical Parameters --- p.23<br>Chapter 3.1 --- Introduction --- p.23<br>Chapter 3.2 --- Sound Quality Evaluation --- p.23<br>Chapter 3.3 --- Headphone Characterization --- p.26<br>Chapter 3.3.1 --- Frequency Response --- p.26<br>Chapter 3.3.2 --- Harmonic Distortion --- p.26<br>Chapter 3.3.3 --- Voice-Coil Driver Parameters --- p.27<br>Chapter 3.4 --- Statistical Correlation Measurement --- p.29<br>Chapter 3.4.1 --- Correlation Coefficient --- p.29<br>Chapter 3.4.2 --- t Test for Correlation Coefficients --- p.30<br>Chapter 3.5 --- Summary --- p.31<br>Algorithms --- p.32<br>Chapter 4.1 --- Introduction --- p.32<br>Chapter 4.2 --- Principal Component Analysis --- p.32<br>Chapter 4.2.1 --- Dimensionality Reduction --- p.32<br>Chapter 4.2.2 --- PCA Transformation --- p.33<br>Chapter 4.2.3 --- PCA Implementation --- p.36<br>Chapter 4.3 --- Traditional Clustering Methods --- p.37<br>Chapter 4.3.1 --- Online Template Matching (TM) --- p.37<br>Chapter 4.3.2 --- Online Template Matching Implementation --- p.40<br>Chapter 4.3.3 --- K-Means Clustering --- p.41<br>Chapter 4.3.4 --- K-Means Clustering Implementation --- p.44<br>Chapter 4.4 --- Unsupervised Neural Learning --- p.45<br>Chapter 4.4.1 --- Neural Network Basics --- p.45<br>Chapter 4.4.2 --- Artificial Neural Network Model --- p.46<br>Chapter 4.4.3 --- Simple Competitive Learning (SCL) --- p.47<br>Chapter 4.4.4 --- SCL Implementation --- p.49<br>Chapter 4.4.5 --- Adaptive Resonance Theory Network (ART). --- p.50<br>Chapter 4.4.6 --- ART2 Implementation --- p.53<br>Chapter 4.6 --- Summary --- p.55<br>Experimental Design --- p.57<br>Chapter 5.1 --- Introduction --- p.57<br>Chapter 5.2 --- Electrophysiological Spike Discrimination --- p.57<br>Chapter 5.2.1 --- Experimental Design --- p.57<br>Chapter 5.2.2 --- Extracellular Recordings --- p.58<br>Chapter 5.2.3 --- PCA Feature Extraction --- p.59<br>Chapter 5.2.4 --- Clustering Analysis --- p.59<br>Chapter 5.3 --- Correlation of Headphone Sound Quality with physical Parameters --- p.61<br>Chapter 5.3.1 --- Experimental Design --- p.61<br>Chapter 5.3.2 --- Frequency Response Clustering --- p.62<br>Chapter 5.3.3 --- Additional Parameters Measurement --- p.68<br>Chapter 5.3.4 --- Listening Tests --- p.68<br>Chapter 5.3.5 --- Confirmation Test --- p.69<br>Chapter 5.4 --- Summary --- p.70<br>Results --- p.71<br>Chapter 6.1 --- Introduction --- p.71<br>Chapter 6.2 --- Electrophysiological Spike Discrimination: A Comparison of Methods --- p.71<br>Chapter 6.2.1 --- Clustering Labeled Spike Data --- p.72<br>Chapter 6.2.2 --- Clustering of Unlabeled Data --- p.78<br>Chapter 6.2.3 --- Remarks --- p.84<br>Chapter 6.3 --- Headphone Sound Quality Control --- p.89<br>Chapter 6.3.1 --- Headphones Frequency Response Clustering --- p.89<br>Chapter 6.3.2 --- Listening Tests --- p.90<br>Chapter 6.3.3 --- Correlation with Measured Parameters --- p.90<br>Chapter 6.3.4 --- Confirmation Listening Test --- p.92<br>Chapter 6.4 --- Summary --- p.93<br>Conclusions --- p.97<br>Chapter 7.1 --- Future Work --- p.98<br>Chapter 7.1.1 --- Clustering Analysis --- p.98<br>Chapter 7.1.2 --- Potential Applications of Clustering Analysis --- p.99<br>Chapter 7.2 --- Closing Remarks --- p.100<br>Appendix --- p.101<br>Chapter A.1 --- Tables of Experimental Results: (Spike Discrimination) --- p.101<br>Chapter A.2 --- Tables of Experimental Results: (Headphones Measurement) --- p.104<br>Bibliography --- p.109<br>Publications --- p.114

Extracellular potentials due to firing of action potentials are computed around cortical neurons and populations of cortical neurons. These extracellular potentials are calculated as a sum of contributions from ionic currents passing through the cell membrane at various locations using Maxwell's equations in the quasi-static limit. These transmembrane currents are found from simulations of anatomically reconstructed cortical neurons implemented as multi-compartmental models in the simulation tool NEURON. Extracellular signatures of action potentials of single neurons are calculated both in the immediate vicinity of the neuron somas and along vertical axes. For the neuronal populations only vertical axis distributions are considered. The vertical-axis calculations were performed to investigate the contributions of action potential firing to laminar-electrode recordings. Results for high-pass (750 - 3000 Hz) filtered potentials are also given to mimic multi-unit activity (MUA) recordings. Extracellular traces from single neurons and populations (both synchronous and asynchronous) of neurons are shown for three different neuron types: layer 3 pyramid, layer 4 stellate and layer 5 pyramid cell. The layer 3 cell shows a 'closed-field' configuration, while the layer 5 pyramid demonstrates an 'open-field' appearance for singe neuron simulations which is less apparent in population simulations. The layer 4 stellate cell seems to fall somewhere in between the open- and closed-field scenarios. Comparing single neuron and synchronous populations, the amplitudes of the extracellular traces increase as population radii increase, though the shapes are generally similar. Asynchronous populations produce small amplitudes due to a time convolution of various neuron contributions.<br>Thesis (M.Sc.)-University of KwaZulu-Natal, Pietermaritzburg, 2005

abstract: It is increasingly common to see machine learning techniques applied in conjunction with computational modeling for data-driven research in neuroscience. Such applications include using machine learning for model development, particularly for optimization of parameters based on electrophysiological constraints. Alternatively, machine learning can be used to validate and enhance techniques for experimental data analysis or to analyze model simulation data in large-scale modeling studies, which is the approach I apply here. I use simulations of biophysically-realistic cortical neuron models to supplement a common feature-based technique for analysis of electrophysiological signals. I leverage these simulated electrophysiological signals to perform feature selection that provides an improved method for neuron-type classification. Additionally, I validate an unsupervised approach that extends this improved feature selection to discover signatures associated with neuron morphologies - performing in vivo histology in effect. The result is a simulation-based discovery of the underlying synaptic conditions responsible for patterns of extracellular signatures that can be applied to understand both simulation and experimental data. I also use unsupervised learning techniques to identify common channel mechanisms underlying electrophysiological behaviors of cortical neuron models. This work relies on an open-source database containing a large number of computational models for cortical neurons. I perform a quantitative data-driven analysis of these previously published ion channel and neuron models that uses information shared across models as opposed to information limited to individual models. The result is simulation-based discovery of model sub-types at two spatial scales which map functional relationships between activation/inactivation properties of channel family model sub-types to electrophysiological properties of cortical neuron model sub-types. Further, the combination of unsupervised learning techniques and parameter visualizations serve to integrate characterizations of model electrophysiological behavior across scales.<br>Dissertation/Thesis<br>Doctoral Dissertation Applied Mathematics 2020