Counterion density profiles at charged flexible membranes

Counterion distributions at charged soft membranes are studied using perturbative analytical and simulation methods in both weak coupling (mean-field or Poisson-Boltzmann) and strong coupling limits. The softer the membrane, the more smeared out the counterion density profile becomes and counterions pentrate through the mean-membrane surface location, in agreement with anomalous scattering results. Membrane-charge repulsion leads to a short-scale roughening of the membrane.Comment: 4 pages, 4 figure

Computation of saddle type slow manifolds using iterative methods

This paper presents an alternative approach for the computation of trajectory segments on slow manifolds of saddle type. This approach is based on iterative methods rather than collocation-type methods. Compared to collocation methods, that require mesh refinements to ensure uniform convergence with respect to $\epsilon$, appropriate estimates are directly attainable using the method of this paper. The method is applied to several examples including: A model for a pair of neurons coupled by reciprocal inhibition with two slow and two fast variables and to the computation of homoclinic connections in the FitzHugh-Nagumo system.Comment: To appear in SIAM Journal of Applied Dynamical System

The what and where of adding channel noise to the Hodgkin-Huxley equations

One of the most celebrated successes in computational biology is the Hodgkin-Huxley framework for modeling electrically active cells. This framework, expressed through a set of differential equations, synthesizes the impact of ionic currents on a cell's voltage -- and the highly nonlinear impact of that voltage back on the currents themselves -- into the rapid push and pull of the action potential. Latter studies confirmed that these cellular dynamics are orchestrated by individual ion channels, whose conformational changes regulate the conductance of each ionic current. Thus, kinetic equations familiar from physical chemistry are the natural setting for describing conductances; for small-to-moderate numbers of channels, these will predict fluctuations in conductances and stochasticity in the resulting action potentials. At first glance, the kinetic equations provide a far more complex (and higher-dimensional) description than the original Hodgkin-Huxley equations. This has prompted more than a decade of efforts to capture channel fluctuations with noise terms added to the Hodgkin-Huxley equations. Many of these approaches, while intuitively appealing, produce quantitative errors when compared to kinetic equations; others, as only very recently demonstrated, are both accurate and relatively simple. We review what works, what doesn't, and why, seeking to build a bridge to well-established results for the deterministic Hodgkin-Huxley equations. As such, we hope that this review will speed emerging studies of how channel noise modulates electrophysiological dynamics and function. We supply user-friendly Matlab simulation code of these stochastic versions of the Hodgkin-Huxley equations on the ModelDB website (accession number 138950) and http://www.amath.washington.edu/~etsb/tutorials.html.Comment: 14 pages, 3 figures, review articl

The stroke oxygen pilot study: a randomized control trial of the effects of routine oxygen supplementation early after acute stroke--effect on key outcomes at six months

Introduction: Post-stroke hypoxia is common, and may adversely affect outcome. We have recently shown that oxygen supplementation may improve early neurological recovery. Here, we report the six-month outcomes of this pilot study. Methods: Patients with a clinical diagnosis of acute stroke were randomized within 24 h of admission to oxygen supplementation at 2 or 3 L/min for 72 h or to control treatment (room air). Outcomes (see below) were assessed by postal questionnaire at 6 months. Analysis was by intention-to-treat, and statistical significance was set at p#0.05. Results: Out of 301 patients randomized two refused/withdrew consent and 289 (148 in the oxygen and 141 in the control group) were included in the analysis: males 44%, 51%; mean (SD) age 73 (12), 71 (12); median (IQR) National Institutes of Health Stroke Scale score 6 (3, 10), 5 (3, 10) for the two groups respectively. At six months 22 (15%) patients in the oxygen group and 20 (14%) in the control group had died; mean survival in both groups was 162 days (p= 0.99). Median (IQR) scores for the primary outcome, the modified Rankin Scale, were 3 (1, 5) and 3 (1, 4) for the oxygen and control groups respectively. The covariate-adjusted odds ratio was 1.04 (95% CI 0.67, 1.60), indicating that the odds of a lower (i.e. better) score were non-significantly higher in the oxygen group (p= 0.86). The mean differences in the ability to perform basic (Barthel Index) and extended activities of daily living (NEADL), and quality of life (EuroQol) were also non-significant. Conclusions: None of the key outcomes differed at 6 months between the groups. Although not statistically significant and generally of small magnitude, the effects were predominantly in favour of the oxygen group; a larger trial, powered to show differences in longer-term functional outcomes, is now on-going. Trial Registration: Controlled-Trials.com ISRCTN12362720; Eudract.ema.europa.eu 2004-001866-4

Computing Slow Manifolds of Saddle Type

Slow manifolds are important geometric structures in the state spaces of dynamical systems with multiple time scales. This paper introduces an algorithm for computing trajectories on slow manifolds that are normally hyperbolic with both stable and unstable fast manifolds. We present two examples of bifurcation problems where these manifolds play a key role and a third example in which saddle-type slow manifolds are part of a traveling wave profile of a partial differential equation. Initial value solvers are incapable of computing trajectories on saddle-type slow manifolds, so the slow manifold of saddle type (SMST) algorithm presented here is formulated as a boundary value method. We take an empirical approach here to assessing the accuracy and effectiveness of the algorithm.Comment: preprint version - for final version see journal referenc

Managerial delegation in a dynamic renewable resource oligopoly

I propose a differential oligopoly game of resource extraction under (quasi-static) open-loop and nonlinear feedback strategies, where firms are managerial and two alternative types of delegation contract are considered. Under open-loop information, delegation expands the residual steady state resource stock. Conversely, under nonlinear feedback information the outcome depends on the structure of managerial incentives. If sales are used, once again delegation favours resource preservation. On the contrary, if market shares are included in the delegation contract, this combines with an underlying voracity effect in shrinking the steady state volume of the resource

Adaptive and Phase Selective Spike Timing Dependent Plasticity in Synaptically Coupled Neuronal Oscillators

We consider and analyze the influence of spike-timing dependent plasticity (STDP) on homeostatic states in synaptically coupled neuronal oscillators. In contrast to conventional models of STDP in which spike-timing affects weights of synaptic connections, we consider a model of STDP in which the time lags between pre- and/or post-synaptic spikes change internal state of pre- and/or post-synaptic neurons respectively. The analysis reveals that STDP processes of this type, modeled by a single ordinary differential equation, may ensure efficient, yet coarse, phase-locking of spikes in the system to a given reference phase. Precision of the phase locking, i.e. the amplitude of relative phase deviations from the reference, depends on the values of natural frequencies of oscillators and, additionally, on parameters of the STDP law. These deviations can be optimized by appropriate tuning of gains (i.e. sensitivity to spike-timing mismatches) of the STDP mechanism. However, as we demonstrate, such deviations can not be made arbitrarily small neither by mere tuning of STDP gains nor by adjusting synaptic weights. Thus if accurate phase-locking in the system is required then an additional tuning mechanism is generally needed. We found that adding a very simple adaptation dynamics in the form of slow fluctuations of the base line in the STDP mechanism enables accurate phase tuning in the system with arbitrary high precision. Adaptation operating at a slow time scale may be associated with extracellular matter such as matrix and glia. Thus the findings may suggest a possible role of the latter in regulating synaptic transmission in neuronal circuits