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

Speed modulated social influence in evacuating pedestrian crowds

Evacuation is a complex social phenomenon with individuals tending to exit a confined space as soon as possible. Social factors that influence an individual include collision avoidance and conformity with others with respect to the tendency to exit. While collision avoidance has been heavily focused on by the agent-based models used frequently to simulate evacuation scenarios, these models typically assume that all agents have an equal desire to exit the scene in a given situation. It is more likely that, out of those who are exiting, some are patient while others seek to exit as soon as possible. Here, we experimentally investigate the effect of different proportions of patient (no-rush) versus impatient (rush) individuals in an evacuating crowd of up to 24 people. Our results show that a) average speed changes significantly for individuals who otherwise tended to rush (or not rush) with both type of individuals speeding up in the presence of the other; and b) deviation rate, defined as the amount of turning, changes significantly for the rush individuals in the presence of no-rush individuals. We then seek to replicate this effect with Helbing's social force model with the twin purposes of analyzing how well the model fits experimental data, and explaining the differences in speed in terms of model parameters. We find that we must change the interaction parameters for both rush and no-rush agents depending on the condition that we are modeling in order to fit the model to the experimental data

An internal splash: Levitation of falling spheres in stratified fluids

We experimentally explore the motion of falling spheres in strongly stratified fluids in which the fluid transitions from low density at the top to high density at the bottom and document an internal splash in which the falling sphere may reverse its direction of motion (from falling, to rising, to falling again) as it penetrates a region of strong density transition. We present measurements of the sphere’s velocity and exhibit nonmonotonic sphere velocity profiles connecting the maximum and minimum terminal velocities, matching earlier measurements [J. Fluid Mech. 381, 175 (1999)], but further exhibit the new levitation phenomenon. We give a physical explanation of this motion which necessarily couples the sphere motion with the stratified fluid, and vice versa, and supplement this with a simplified, reduced mathematical model involving a nonlinear system of ordinary differential equations which captures the nonmonotonic transition and agrees with the measuredvelocity profiles at all depths except those in the vicinity of the sharp transition for which the model deviates from the measured speeds. We repeat the experiments adjusting the distance between the camera and falling sphere thereby reducing the optical blur associated with the change in optical refractive index associated with the strong density transition. By directly measuring the residual optical distortion with a center plane, vertical ruler, we exhibit that the measuredvelocity profile within the transition layer is strongly sensitive to the details of the measured optical distortion, and show subsequent improved agreement between the measurement and the model. Through direct measurement of the nonlinear mapping between physical and imaged coordinates we document measuredvelocity error trends which may occur from inaccurately accounting for this optical distortion. We suggest strategies for correcting this localized measurement detail generally

Model Pembelajaran Instruction, Doing, Dan Evaluating (Mpide) Dengan Video Kejadian Fisika Dalam Pembelajaran Fisika Di SMA

This research examines the “Model Pembelajaran Instruction, Doing, dan Evaluating (MPIDE)” with Physics Phenomenon Video in Physics Instruction at SMA. The research\u27s purpose are to determine the students activities, the effectiveness of model, and the students learning achievement retention. This research is a research action that implicated by one group pretest and posttest design for testing. This research conducted on SMA of Class XI with the techniques of data collection is the observation, interviews, and tests. Data analysis techniques are percentage, continuesly its described. The results of the students activities , the effectiveness of model, and the students learning achievement retention can improve each cycle. The average of the students activities from cycle one to cycle two is 69,17% to 73,33% with active activity category. The average of the effectiveness of model from cycle one to cycle two is 0.68 with anough effective category to 0.75 with effective category. The average of students learning achievement retention from cycle one to cycle two is 92.56% to 93.19% with high category. The research can be concluded that the model can improve the students activities, the effectiveness of model, and the students learning achievement retention when the model completed by dubbing on video

The effect of geography and citizen behavior on motor vehicle deaths in the United States.

Death due to motor vehicle collisions (MVCs) remains a leading cause of death in the US and alcohol plays a prominent role in a large proportion of these fatalities nationwide. Rates for these incidents vary widely among states and over time. Here, we explore the extent to which driving volume, alcohol consumption, legislation, political ideology, and geographical factors influence MVC deaths across states and time. We specify structural equation models for extracting associations between the factors and outcomes for MVC deaths and compute correlation functions of states' relative geographic and political positions to elucidate the relative contribution of these factors. We find evidence that state-level variation in MVC deaths is associated with time-varying driving volume, alcohol consumption, and legislation. These relationships are modulated by state spatial proximity, whereby neighboring states are found to share similar MVC death rates over the thirty-year observation period. These results support the hypothesis that neighboring states exhibit similar risk and protective characteristics, despite differences in political ideology

The effect of geography and citizen behavior on motor vehicle deaths in the United States.

Death due to motor vehicle collisions (MVCs) remains a leading cause of death in the US and alcohol plays a prominent role in a large proportion of these fatalities nationwide. Rates for these incidents vary widely among states and over time. Here, we explore the extent to which driving volume, alcohol consumption, legislation, political ideology, and geographical factors influence MVC deaths across states and time. We specify structural equation models for extracting associations between the factors and outcomes for MVC deaths and compute correlation functions of states' relative geographic and political positions to elucidate the relative contribution of these factors. We find evidence that state-level variation in MVC deaths is associated with time-varying driving volume, alcohol consumption, and legislation. These relationships are modulated by state spatial proximity, whereby neighboring states are found to share similar MVC death rates over the thirty-year observation period. These results support the hypothesis that neighboring states exhibit similar risk and protective characteristics, despite differences in political ideology