Steady-State Poisson-Nernst-Planck Systems: Asymptotic expansions and applications to ion channels

Important properties of ion channels can be described by a steady state Poisson-Nernst-Plank system for electrodiffusion. The solution to the PNP system gives a relation between the current and electric potential of the ions in the channel, called the I-V curve. In this thesis, we will discuss the matched asymptotic expansions method of solving a singularly perturbed system and apply this method to find an approximate solution to the steady-state Poisson-Nernst-Planck system. In general, for nonlinear systems, it is impossible to obtain any representations of solutions. Due to the presence of a singular parameter in the PNP system, we can treat the system as a singularly perturbed problem. This system with specific nonlinearity has special structures that are crucial for the explicit higher order asymptotic expansions of the solutions. Although the ion channel problem considers only one cell, the I-V relation obtained in this thesis is consistent with the cubic-like assumption of the I-V relation in the FitzHugh-Nagumo model for action potential involving a population of ion channels. However, applications of the results of this thesis to ion channels are limited, since we considered a simplified model with two species of ions and a zero permanent charge in the channel

Asymptotic Expansions of I-V Relations via a Poisson–Nernst–Planck System

This is the published version, also available here: http://dx.doi.org/10.1137/070691322.We investigate higher order matched asymptotic expansions of a steady-state Poisson–Nernst–Planck (PNP) system with particular attention to the I-V relations of ion channels. Assuming that the Debye length is small relative to the diameter of the narrow channel, the PNP system can be viewed as a singularly perturbed system. Special structures of the zeroth order inner and outer systems make it possible to provide an explicit derivation of higher order terms in the asymptotic expansions. For the case of zero permanent charge, our results concerning the I-V relation for two oppositely charged ion species are (i) the first order correction to the zeroth order linear I-V relation is generally quadratic in V; (ii) when the electro-neutrality condition is enforced at both ends of the channel, there is NO first order correction, but the second order correction is cubic in V. Furthermore (Theoremsigmo), up to the second order, the cubic I-V relation has (except for a very degenerate case) three distinct real roots that correspond to the bistable structure in the FitzHugh–Nagumo simplification of the Hodgkin–Huxley model

Gait Detection in Children with and without Hemiplegia Using Single-Axis Wearable Gyroscopes

In this work, we develop a novel gait phase detection algorithm based on a hidden Markov model, which uses data from foot-mounted single-axis gyroscopes as input. We explore whether the proposed gait detection algorithm can generate equivalent results as a reference signal provided by force sensitive resistors (FSRs) for typically developing children (TD) and children with hemiplegia (HC). We find that the algorithm faithfully reproduces reference results in terms of high values of sensitivity and specificity with respect to FSR signals. In addition, the algorithm distinguishes between TD and HC and is able to assess the level of gait ability in patients. Finally, we show that the algorithm can be adapted to enable real-time processing with high accuracy. Due to the small, inexpensive nature of gyroscopes utilized in this study and the ease of implementation of the developed algorithm, this work finds application in the on-going development of active orthoses designed for therapy and locomotion in children with gait pathologies. © 2013 Abaid et al