Mucus clears from the trachea in a helix: a new twist to understanding airway diseases.

BACKGROUND: Mucociliary clearance (MCC) is critical to lung health and is impaired in many diseases. The path of MCC may have an important impact on clearance but has never been rigorously studied. The objective of this study is to assess the three-dimensional path of human tracheal MCC in disease and health. METHODS: Tracheal MCC was imaged in 12 ex-smokers, 3 non-smokers (1 opportunistically imaged during acute influenza and repeated after recovery) and 5 individuals with primary ciliary dyskinesia (PCD). Radiolabelled macroaggregated albumin droplets were injected into the trachea via the cricothyroid membrane. Droplet movement was tracked via scintigraphy, the path of movement mapped and helical and axial models of tracheal MCC were compared. MEASUREMENTS AND MAIN RESULTS: In 5/5 participants with PCD and 1 healthy participant with acute influenza, radiolabelled albumin coated the trachea and did not move. In all others (15/15), mucus coalesced into globules. Globule movement was negligible in 3 ex-smokers, but in all others (12/15) ascended the trachea in a helical path. Median cephalad tracheal MCC was 2.7 mm/min ex-smokers vs 8.4 mm/min non-smokers (p=0.02) and correlated strongly to helical angle (r=0.92 (p=0.00002); median 18o ex-smokers, 47o non-smokers (p=0.036)), but not to actual speed on helical path (r=0.26 (p=0.46); median 13.6 mm/min ex-smokers vs 13.9 mm/min non-smokers (p=1.0)). CONCLUSION: For the first time, we show that human tracheal MCC is helical, and impairment in ex-smokers is often caused by flattened helical transit, not slower movement. Our methodology provides a simple method to map tracheal MCC and speed in vivo.

Effects of quasi-steady-state reduction on biophysical models with oscillations.

Many biophysical models have the property that some variables in the model evolve much faster than others. A common step in the analysis of such systems is to simplify the model by assuming that the fastest variables equilibrate instantaneously, an approach that is known as quasi-steady state reduction (QSSR). QSSR is intuitively satisfying but is not always mathematically justified, with problems known to arise, for instance, in some cases in which the full model has oscillatory solutions; in this case, the simplified version of the model may have significantly different dynamics to the full model. This paper focusses on the effect of QSSR on models in which oscillatory solutions arise via one or more Hopf bifurcations. We first illustrate the problems that can arise by applying QSSR to a selection of well-known models. We then categorize Hopf bifurcations according to whether they involve fast variables, slow variables or a mixture of both, and show that Hopf bifurcations that involve only slow variables are not affected by QSSR, Hopf bifurcations that involve fast and slow variables (i.e., singular Hopf bifurcations) are generically preserved under QSSR so long as a fast variable is kept in the simplified system, and Hopf bifurcations that primarily involve fast variables may be eliminated by QSSR. Finally, we present some guidelines for the application of QSSR if one wishes to use the method while minimising the risk of inadvertently destroying essential features of the original model.

The Role of Cell Volume in the Dynamics of Seizure, Spreading Depression, and Anoxic Depolarization.

Cell volume changes are ubiquitous in normal and pathological activity of the brain. Nevertheless, we know little of how cell volume affects neuronal dynamics. We here performed the first detailed study of the effects of cell volume on neuronal dynamics. By incorporating cell swelling together with dynamic ion concentrations and oxygen supply into Hodgkin-Huxley type spiking dynamics, we demonstrate the spontaneous transition between epileptic seizure and spreading depression states as the cell swells and contracts in response to changes in osmotic pressure. Our use of volume as an order parameter further revealed a dynamical definition for the experimentally described physiological ceiling that separates seizure from spreading depression, as well as predicted a second ceiling that demarcates spreading depression from anoxic depolarization. Our model highlights the neuroprotective role of glial K buffering against seizures and spreading depression, and provides novel insights into anoxic depolarization and the relevant cell swelling during ischemia. We argue that the dynamics of seizures, spreading depression, and anoxic depolarization lie along a continuum of the repertoire of the neuron membrane that can be understood only when the dynamic ion concentrations, oxygen homeostasis,and cell swelling in response to osmotic pressure are taken into consideration. Our results demonstrate the feasibility of a unified framework for a wide range of neuronal behaviors that may be of substantial importance in the understanding of and potentially developing universal intervention strategies for these pathological states.

A geometric understanding of how fast activating potassium channels promote bursting in pituitary cells.

The electrical activity of endocrine pituitary cells is mediated by a plethora of ionic currents and establishing the role of a single channel type is difficult. Experimental observations have shown however that fast-activating voltage- and calcium-dependent potassium (BK) current tends to promote bursting in pituitary cells. This burst promoting effect requires fast activation of the BK current, otherwise it is inhibitory to bursting. In this work, we analyze a pituitary cell model in order to answer the question of why the BK activation must be fast to promote bursting. We also examine how the interplay between the activation rate and conductance of the BK current shapes the bursting activity. We use the multiple timescale structure of the model to our advantage and employ geometric singular perturbation theory to demonstrate the origin of the bursting behaviour. In particular, we show that the bursting can arise from either canard dynamics or slow passage through a dynamic Hopf bifurcation. We then compare our theoretical predictions with experimental data using the dynamic clamp technique and find that the data is consistent with a burst mechanism due to a slow passage through a Hopf.

Mixed mode oscillations as a mechanism for pseudo-plateau bursting.

We combine bifurcation analysis with the theory of canard-induced mixed mode oscillations to investigate the dynamics of a novel form of bursting. This bursting oscillation, which arises from a model of the electrical activity of a pituitary cell, is characterized by small impulses or spikes riding on top of an elevated voltage plateau. Oscillations with these characteristics have been called "pseudo-plateau bursting". Unlike standard bursting, the subsystem of fast variables does not possess a stable branch of periodic spiking solutions, and in the case studied here the standard fast/slow analysis provides little information about the underlying dynamics. We demonstrate that the bursting is actually a canard-induced mixed mode oscillation, and use canard theory to characterize the dynamics of the oscillation. We also use bifurcation analysis of the full system of equations to extend the results of the singular analysis to the physiological regime. This demonstrates that the combination of these two analysis techniques can be a powerful tool for understanding the pseudo-plateau bursting oscillations that arise in electrically excitable pituitary cells and isolated pancreatic beta-cells.

Understanding anomalous delays in a model of intracellular calcium dynamics.

In many cell types, oscillations in the concentration of free intracellular calcium ions are used to control a variety of cellular functions. It has been suggested [J. Sneyd et al., "A method for determining the dependence of calcium oscillations on inositol trisphosphate oscillations," Proc. Natl. Acad. Sci. U.S.A. 103, 1675-1680 (2006)] that the mechanisms underlying the generation and control of such oscillations can be determined by means of a simple experiment, whereby a single exogenous pulse of inositol trisphosphate (IP(3)) is applied to the cell. However, more detailed mathematical investigations [M. Domijan et al., "Dynamical probing of the mechanisms underlying calcium oscillations," J. Nonlinear Sci. 16, 483-506 (2006)] have shown that this is not necessarily always true, and that the experimental data are more difficult to interpret than first thought. Here, we use geometric singular perturbation techniques to study the dynamics of models that make different assumptions about the mechanisms underlying the calcium oscillations. In particular, we show how recently developed canard theory for singularly perturbed systems with three or more slow variables [M. Wechselberger, "A propos de canards (Apropos canards)," Preprint, 2010] applies to these calcium models and how the presence of a curve of folded singularities and corresponding canards can result in anomalous delays in the response of these models to a pulse of IP(3).

Neural Excitability and Singular Bifurcations.

We discuss the notion of excitability in 2D slow/fast neural models from a geometric singular perturbation theory point of view. We focus on the inherent singular nature of slow/fast neural models and define excitability via singular bifurcations. In particular, we show that type I excitability is associated with a novel singular Bogdanov-Takens/SNIC bifurcation while type II excitability is associated with a singular Andronov-Hopf bifurcation. In both cases, canards play an important role in the understanding of the unfolding of these singular bifurcation structures. We also explain the transition between the two excitability types and highlight all bifurcations involved, thus providing a complete analysis of excitability based on geometric singular perturbation theory.

Excitable neurons, firing threshold manifolds and canards.

We investigate firing threshold manifolds in a mathematical model of an excitable neuron. The model analyzed investigates the phenomenon of post-inhibitory rebound spiking due to propofol anesthesia and is adapted from McCarthy et al. (SIAM J. Appl. Dyn. Syst. 11(4):1674-1697, 2012). Propofol modulates the decay time-scale of an inhibitory GABAa synaptic current. Interestingly, this system gives rise to rebound spiking within a specific range of propofol doses. Using techniques from geometric singular perturbation theory, we identify geometric structures, known as canards of folded saddle-type, which form the firing threshold manifolds. We find that the position and orientation of the canard separatrix is propofol dependent. Thus, the speeds of relevant slow synaptic processes are encoded within this geometric structure. We show that this behavior cannot be understood using a static, inhibitory current step protocol, which can provide a single threshold for rebound spiking but cannot explain the observed cessation of spiking for higher propofol doses. We then compare the analyses of dynamic and static synaptic inhibition, showing how the firing threshold manifolds of each relate, and why a current step approach is unable to fully capture the behavior of this model.

Changes in the criticality of Hopf bifurcations due to certain model reduction techniques in systems with multiple timescales.

A major obstacle in the analysis of many physiological models is the issue of model simplification. Various methods have been used for simplifying such models, with one common technique being to eliminate certain 'fast' variables using a quasi-steady-state assumption. In this article, we show when such a physiological model reduction technique in a slow-fast system is mathematically justified. We provide counterexamples showing that this technique can give erroneous results near the onset of oscillatory behaviour which is, practically, the region of most importance in a model. In addition, we show that the singular limit of the first Lyapunov coefficient of a Hopf bifurcation in a slow-fast system is, in general, not equal to the first Lyapunov coefficient of the Hopf bifurcation in the corresponding layer problem, a seemingly counterintuitive result. Consequently, one cannot deduce, in general, the criticality of a Hopf bifurcation in a slow-fast system from the lower-dimensional layer problem.

The dynamics underlying pseudo-plateau bursting in a pituitary cell model.

Pituitary cells of the anterior pituitary gland secrete hormones in response to patterns of electrical activity. Several types of pituitary cells produce short bursts of electrical activity which are more effective than single spikes in evoking hormone release. These bursts, called pseudo-plateau bursts, are unlike bursts studied mathematically in neurons (plateau bursting) and the standard fast-slow analysis used for plateau bursting is of limited use. Using an alternative fast-slow analysis, with one fast and two slow variables, we show that pseudo-plateau bursting is a canard-induced mixed mode oscillation. Using this technique, it is possible to determine the region of parameter space where bursting occurs as well as salient properties of the burst such as the number of spikes in the burst. The information gained from this one-fast/two-slow decomposition complements the information obtained from a two-fast/one-slow decomposition.

The selection of mixed-mode oscillations in a Hodgkin-Huxley model with multiple timescales.

In recent work [J. Rubin and M. Wechselberger, Biol. Cybern. 97, 5 (2007)], we explained the appearance of remarkably slow oscillations in the classical Hodgkin-Huxley (HH) equations, modified by scaling a time constant, using recently developed theory about mixed-mode oscillations (MMOs). This theory is only rigorously valid, however, for epsilon sufficiently small, where epsilon is a parameter that arises from nondimensionalization of the HH system. Here, we illustrate how the parameter regime over which MMOs exist, and the features of the MMO patterns within this regime, vary with respect to several key parameters in the nondimensionalized HH equations, including epsilon. Moreover, we explain our findings in terms of the effects that these parameters are expected to have on certain organizing structures within the corresponding flow, generalized from analysis done previously in the singular limit.

Giant squid-hidden canard: the 3D geometry of the Hodgkin-Huxley model.

This work is motivated by the observation of remarkably slow firing in the uncoupled Hodgkin-Huxley model, depending on parameters tau( h ), tau( n ) that scale the rates of change of the gating variables. After reducing the model to an appropriate nondimensionalized form featuring one fast and two slow variables, we use geometric singular perturbation theory to analyze the model's dynamics under systematic variation of the parameters tau( h ), tau( n ), and applied current I. As expected, we find that for fixed (tau( h ), tau( n )), the model undergoes a transition from excitable, with a stable resting equilibrium state, to oscillatory, featuring classical relaxation oscillations, as I increases. Interestingly, mixed-mode oscillations (MMO's), featuring slow action potential generation, arise for an intermediate range of I values, if tau( h ) or tau( n ) is sufficiently large. Our analysis explains in detail the geometric mechanisms underlying these results, which depend crucially on the presence of two slow variables, and allows for the quantitative estimation of transitional parameter values, in the singular limit. In particular, we show that the subthreshold oscillations in the observed MMO patterns arise through a generalized canard phenomenon. Finally, we discuss the relation of results obtained in the singular limit to the behavior observed away from, but near, this limit.

Ionic channels and conductance-based models for hypothalamic neuronal thermosensitivity.

Thermoregulatory responses are partially controlled by the preoptic area and anterior hypothalamus (PO/AH), which contains a mixed population of temperature-sensitive and insensitive neurons. Immunohistochemical procedures identified the extent of various ionic channels in rat PO/AH neurons. These included pacemaker current channels [i.e., hyperpolarization-activated cyclic nucleotide-gated channels (HCN)], background potassium leak channels (TASK-1 and TRAAK), and transient receptor potential channel (TRP) TRPV4. PO/AH neurons showed dense TASK-1 and HCN-2 immunoreactivity and moderate TRAAK and HCN-4 immunoreactivity. In contrast, the neuronal cell bodies did not label for TRPV4, but instead, punctate labeling was observed in traversing axons or their terminal endings. On the basis of these results and previous electrophysiological studies, Hodgkin-Huxley-like models were constructed. These models suggest that most PO/AH neurons have the same types of ionic channels, but different levels of channel expression can explain the inherent properties of the various types of temperature-sensitive and insensitive neurons.