Center for Applied Mathematics and Statistics

REPORT 0405-1:   Effectiveness of numerical techniques for calculating the quantity of calcium ion species during calcium sparksin heart muscle

D. A. Lott, M. Li and J. R. Berlin

An efficient numerical algorithm based on the convolution of functions and on finite difference approximations for the diffusion equation is utilized to determine the quantity of calcium ionsparticipating in unitary calcium ions release events, termed "calcium ions sparks," in heart muscle. Output images of localized increases in cytosolic calcium ion concentration, due predominantly to calcium ions release from intracellular storage sites, are obtained using fluorescent calcium indicators and confocal microscopy. To obtain the quantity of calcium ions underlying these localized increases of cytosolic calcium ion concentration, one-dimensional output images are deconvolved with a point spread function that describes the optical properties of the microscope. The resulting input image is then reconstructed assuming symmetry, in a three-dimensional image of calcium ion concentration and all calcium ion-bound species. Temporal information about
free and bound calcium ions species can be obtained by performing convolutions on a series of output images recorded in time and then accounting for the kinetics of calcium ions interactions with the fluorescent calcium indicator and other calcium ion binding species. The effect of microscope imaging properties on
measurements of local calcium ion concentration and the ability to reconstruct the underlying changes in calcium ions species during a calcium ions spark are presented.

REPORT 0405-2:   Time delay and amplitude estimation in underwater acoustics: a Gibbs Sampling approach

Zoi-Heleni Michalopoulou and Michele Picarelli

Multipath arrivals of a signal at a receiving sensor are frequently encountered in many areas of signal processing, including sonar, radar, and communication problems. In underwater acoustics, numerous approaches to source localization, geoacoustic inversion, and tomography rely on accurate multipath arrival extraction.  In this work, a novel method for estimation of time delays and amplitudes of arrivals with Maximum A Posteriori (MAP) estimation is presented. MAP estimation is optimal if appropriate statistical models are selected for the received data; implementation, requiring maximization of a multi-dimensional function, is computationally demanding. Gibbs Sampling is proposed as an efficient means for estimating the necessary posterior probability distributions, bypassing analytical calculations. The Gibbs Sampler estimates posterior distributions through an iterative process and includes as unknowns noise variance and number of arrivals as well as time delays and amplitudes of multipaths. Through Monte Carlo simulations, the method is shown to have a performance very close to that of analytical MAP estimation. The method is shown to be superior to Expectation-Maximization, which is often applied to time delay estimation. The Gibbs Sampling - MAP approach is demonstrated to be more informative than other time delay and amplitude estimation methods, providing complete posterior probability distributions compared to just point estimates; the probability distributions illustrate the uncertainty in the problem, presenting likely values of the unknowns that are different from simple point estimates.

REPORT 0405-3:  Existence of traveling wave solutions for Ginzburg-Landau-type problems in infinite cylinder

M. Lucia, C. B. Muratov and M. Novaga

We study a class of systems of reaction-diffusion equations in infinite cylinders. These systems of equations arise within the context of Ginzburg-Landau theories and describe the kinetics of phase transformation in second-order or weakly first-order phase transitions with non-conserved order parameter. We use a novel variational characterization to study existence of traveling wave solutions under very general assumptions on the nonlinearities. These solutions are a special class of the traveling wave solutions which are characterized by a fast exponential decay in the direction of propagation. Our main result is a simple verifiable criterion for existence of these traveling waves. We also prove boundedness, regularity, and some other properties of the obtained solutions, as well as several sufficient conditions for existence or non-existence of such traveling waves, and give rigorous upper and lower bounds for their speed. In addition, we prove that the speed of the obtained solutions gives a sharp upper bound for the propagation speed of a class of disturbances which are initially sufficiently localized. We give a sample application of our results using a computer-assisted approach.

REPORT 0405-4:  On undercompressive shocks and flooding in countercurrent two-layer flows

T. M. Segin, B. S. Tilley and L. Kondic

We consider the countercurrent flow of two incompressible immiscible viscous fluids in an inclined channel. This configuration may lead to the phenomena of "flooding", i.e. the transition from a countercurrent regime to a cocurrent regime. This transition is marked by a variety of transient behavior, such as the development of large-amplitude waves that impede the flow of one of the fluids to the reversal of the flow of the denser fluid. From a lubrication approximation based on the ratio of the channel height to the downstream disturbance wavelength, we derive a nonlinear system of evolution equations that govern the interfacial shape separating the two fluids and the leading-order pressure. This system, which assumes fluids with disparate density and dynamic viscosity ratios, includes the effects of viscosity stratification, inertia, shear, and capillarity. Since the experimental constraints for this effective system are unclear, we consider two ways to drive the flow: either by fixing the volumetric flow rate of the gas phase or by fixing the total pressure drop over a downstream length of the channel. The latter forcing results in a single evolution equation whose dynamics depends nonlocally on the interfacial shape. From both of these driven systems, admissible criteria for Lax shocks, undercompressive shocks and rarefaction waves are investigated. Interestingly, these criteria, through a numerical verification, do not depend significantly on the inertial effects within the more dense layer. The choice of the local/nonlocal constraints appears to play a role in the transient growth of undercompressive shocks, and may relate to the phenomena observed near the onset of flooding.


REPORT 0405-5:  Bursting in 2-compartment neurons: A case study of the Pinsky-Rinzel model

Amitabha Bose and Victoria Booth

The two-compartment Pinsky-Rinzel model of a hippocampal CA3pyramidal neuron consists of electrically coupled soma and dendritic compartments, each with active ionic conductances. This model has been widely used in a variety of contexts, but little analysis has been performed on its bursting solutions. In this chapter, we provide a geometric framework to study the Pinsky-Rinzel model.
Using a combination of analysis and simulations, we identify neuronal mechanisms responsible for certain salient features of the model such as burst initiation and somatic-dendritic ping-pong.
Some of these mechanisms are then demonstrated in a conceptually simpler, two-compartment Morris-Lecar model.


REPORT 0405-6:  On the Blaschke Conjecture for 3-Webs

Vladislav V. Goldberg and Valentin V. Lychagin

We find relative differential invariants of orders eight and nine for a planar nonparallelizable 3-web such that their vanishing is necessary and sufficient for a 3-web to be linearizable. This solves the Blaschke conjecture for 3-webs. As a side result, we show that the number of linearizations in the Gronwall conjecture does not exceed fifteen and give criteria for rigidity of 3-webs.


REPORT 0405-7:  A computational model of electrically coupled, intrinsically distinct pacemaker neurons

Cristina Soto-Treviņo, Pascale Rabbah, Eve Marder and Farzan Nadim

Electrical coupling between neurons with similar properties is often studied. Nonetheless, the role of electrical coupling between neurons with widely different intrinsic properties also occurs, but is less well-understood. Inspired by the pacemaker group of the crustacean pyloric network, we developed a multi-compartment, conductance-based model of a small network of intrinsically distinct, electrically coupled neurons.  In the pyloric network, a small intrinsically bursting neuron, through gap-junctions, drives two larger, tonically spiking neurons to reliably burst in-phase with it. Each model neuron has two compartments, one responsible for spike-generation and the other for producing a slow, large amplitude oscillation.  We illustrate how these compartments interact, and determine the dynamics of the model neurons. Our model captures the dynamic oscillation range measured from the isolated and coupled biological neurons.  At the network level, we explore the range of coupling strengths for which synchronous bursting oscillations are possible.  The spatial segregation of ionic currents significantly enhances the ability of the two neurons to burst synchronously, and the oscillation range of the model pacemaker network depends not only on the strength of the electrical synapse but also on the identity of the neuron receiving inputs. We also compare the activity of the electrically coupled, distinct neurons with that of a network of coupled identical bursting neurons.  For small to moderate coupling strengths, the network of identical elements, when receiving asymmetrical inputs, can have a smaller dynamic range of oscillation than that of its constituent neurons in isolation.

REPORT 0405-8:  Effect of electrical coupling on ionic current and synaptic potential measurements

Pascale Rabbah, Jorge Golowasch and Farzan Nadim

Recent studies have found electrical coupling to be more ubiquitous than previously thought, and coupling through gap junctions is known to play a crucial role in neuronal function and network output. In particular, current spread through gap junctions may affect the activation of voltage-dependent conductances as well as chemical synaptic release. Using voltage-clamp recordings of two strongly electrically coupled neurons of the lobster stomatogastric ganglion and conductance-based models of these neurons, we identified effects of electrical coupling on the measurement of leak and voltage-gated outward currents, as well as synaptic potentials. Experimental measurements showed that both leak and voltage-gated outward currents are recruited via gap junctions from neurons coupled to the clamped cell. Nevertheless, in spite of the strong coupling between these neurons (~23%), the errors made in estimating voltage-gated conductance parameters were relatively minor (<10%). Thus, in many cases isolation of coupled neurons may not be required if a small degree of measurement error of the voltage-gated currents or the synaptic potentials is acceptable.
Modeling results show, however, that such errors may be as high as 20% if the gap junction position is near the recording site or as high as 90% when measuring smaller voltage-gated ionic currents. Paradoxically, improved space clamp increases the errors due to electrical coupling because voltage control across gap junctions is poor for even the highest realistic coupling conductances.
Furthermore, the common procedure of leak subtraction can add an extra error to the conductance measurement, the sign of which depends on the maximal conductance.

REPORT 0405-9:  Target-specific short-term dynamics are important for the function of synapses in an oscillatory neural network.

Akira Mamiya and Farzan Nadim

Short-term dynamics such as facilitation and depression are present in most synapses and are often target-specific, even for synapses from the same type of neuron. We examine the dynamics and possible functions of two synapses from the same presynaptic neuron in the rhythmically active pyloric network of the spiny lobster. Using simultaneous recordings, we show that the synapses from the lateral pyloric (LP) neuron to the pyloric dilator (PD) and the pyloric constrictor (PY) neurons both show short-term depression. However, the postsynaptic potentials produced by the LP to PD synapse are larger in amplitude, depress less and recover faster than those produced by the LP to PY synapse. We show that the synapse from the LP neuron to the PD neuron, the latter a member of the pyloric pacemaker ensemble, functions to slow down the pyloric rhythm when it is fast and to speed it up when it is slow. In contrast, the synapse from the LP neuron to the PY neuron functions to delay the activity phase of the PY neuron at all cycle periods. Using a computational model of the pyloric network, we show that the short-term dynamics of synaptic depression observed for each of these synapses are tailored to their individual functions and that replacing the dynamics of either synapse with the other would disrupt these functions.
Together, the experimental and modeling results suggest that the target-specific features of short-term synaptic depression are functionally important for synapses efferent from the same presynaptic neuron.

REPORT 0405-10:  Postural Stability Index is a More Valid Measure of Stability Than Equilibrium Score

Hans Chaudhry, Thomas Findley, Karen S. Quigley, Zhiming Ji, Miriam Maney, Tiffany Sims, Bruce Bukiet, and Richard Foulds

Researchers, therapists and physicians often use Equilibrium Score (ES) from the Sensory Organization Test (SOT), a key test in the NeuroCom dynamic posturography system, to assess stability. ES reflects the overall coordination of the visual, proprioceptive, and vestibular systems for maintaining standing posture. In our earlier paper, we proposed a new measure of postural stability, the Postural Stability Index (PSI), which takes into account more biomechanical aspects than ES. In that paper, it was shown that PSI provides a clinically important adjunct to ES. In this paper, we show that PSI can provide an acceptable index even if a person falls during the trial, whereas ES assigns a zero score for any fall.
We also show that PSI is more closely related to ankle stiffness than ES, which is generally recognized as in indicator of postural stability. These results suggest that PSI is a more valid measure of stability than ES.

REPORT 0405-11:  Relationship Among Postural Stability, Weight, Height and Moment of Inertia of Normal Adults

Hans Chaudhry, Thomas Findley, Zhiming Ji, Bruce Bukiet, Richard Foulds, and Miriam Maney

Using the definition of postural stability index (PSI) developed by the authors in their earlier work, and the experimental results on 22 normal adults employing the Neurocom Equitest device, we demonstrate in this paper, that there is little correlation among postural stability, weight, height, the product of weight and height, and moment of inertia of a subject about the ankle joint. However, we observe, that there is a slight tendency for PSI and therefore stability to decrease as the height increases.


REPORT 0405-12:  The effects of varying the timing of inputs on a conditional oscillator

Christina Ambrosio, Farzan Nadim, Amitabha Bose

A conditional oscillator is one that requires input to oscillate. An example of such is the gastric mill network of the stomatogastric ganglion of the crab Cancer borealis which requires a modulatory input from outside the stomatogastric ganglion and input from the pyloric network of the animal in order to oscillate.
Here we study how the frequency of the gastric mill network is determined when it is receives rhythmic input from two different sources but where the timing of these inputs may differ. We find that over a certain range of the time difference one of the two rhythmic inputs plays no role what so ever in determining the network frequency, while in another range, both inputs work together to determine the frequency. The existence and stability of periodic solutions to model sets of equations are obtained analytically using geometric singular perturbation theory. The results are validated through numerical simulations. Comparisons to experiments are also presented.


REPORT 0405-13:  Predictive Inference for Future Responses Given a Doubly Censored Sample from a Two Parameter Exponential Distribution

Hafiz M. R. Khan, M. Safiul Haq, and Serge B. Provost

In this paper, we derive the predictive distributions of one and several future responses including their average, on the basis of a type II doubly censored sample from a two parameter exponential life testing model. We also determine the highest predictive density interval for a future response. A numerical example is provided to illustrate the results.


REPORT 0405-14:  Predictive Inference for Future Responses from Two Component Systems

Hafiz M. R. Khan, M. Safiul Haq, and Serge B. Provost

Predictive distributions for bivariate future responses are derived for systems whose components are connected in parallel or in series under the assumption that the lifetimes of the components are exponentially distributed. The predictive reliability, moment generating and hazard rate functions are derived for the former case. Illustrative examples are provided for each type of system.


REPORT 0405-15:  Matching reflectances for the estimation of inherent optical properties

Zoi-Heleni Michalopoulou, Sima Bagheri, Lisa Axe

A novel approach based on an analytical bio-optical model is developed for the retrieval of Inherent Optical Properties, from which the water quality constituent concentrations can be obtained. The proposed method generates synthetic (sub)surface irradiance reflectances (R(0)) for different values of the unknown parameters and matches them to the measured reflectances; the values of the parameters that generate the best match are taken to be the parameter estimates. Through Monte Carlo simulations the method is shown to be superior to linear matrix inversion, consistently producing estimates very close to the true values of absorption and backscattering.


REPORT 0405-16:  Vector-soliton collision dynamics in nonlinear optical fibers

Roy H. Goodman and Richard Haberman

We consider the interactions of two identical, orthogonally polarized vector solitons in a nonlinear optical fiber with two polarization directions, described by a coupled pair of nonlinear Schrodinger equations. We study a low-dimensional model system of Hamiltonian ordinary differential equations (ODEs) derived by Ueda and Kath and also studied by Tan and Yang. We derive a further simplified model which has similar dynamics but is more amenable to analysis. Sufficiently fast solitons move by each other without much interaction, but below a critical velocity the solitons may be captured. In certain bands of initial velocities the solitons are initially captured, but separate after passing each other twice, a phenomenon known as the two-bounce or two-pass resonance. We derive an analytic formula for the critical velocity. Using matched asymptotic expansions for separatrix crossing, we determine the location of these resonance windows. Numerical simulations of the ODE models show they compare quite well with the asymptotic theory.


REPORT 0405-17:  Bayesian prediction for the log-normal model under Type II censoring

Hafiz M. R. Khan, M. Safiul Haq, and Serge B. Provost

Given a Type II censored sample and a Type II `median censored' sample from the two parameter log-normal distribution, we have derived the predictive distribution of future responses assuming a non-informative prior as well as an informative prior distribution for the parameters. We have also obtained various types of estimates of reliability functions. A numerical example illustrates the results.


REPORT 0405-18:  Strong Versions of the DFR Property

M.C. Bhattacharjee

This paper investigates two distinct subclasses of DFR distributions that exhibit strong forms of anti-aging behavior. Our results characterize such distributions through several representation theorems and consider associated consequences such as closure properties and reliability bounds for distributions which are DFR in the strong sense considered. The usefulness and prevalence of strong DFR properties are illustrated by examples and several applications.


REPORT 0405-19:  A simulation study of a Bayesian Hierarchical Changepoint Model with Covariates

Wonsuk Yoo and Elizabeth H. Slate

This paper presents a simulation study to investigate the behavior of estimates of covariate effects in a Bayesian hierarchical changepoint model. The model is a segmented linear regression model with one changepoint within a fully Bayesian framework. We introduce covariate effects into the model in three ways: as an affect on the interaction term, an affect on the slope after the changepoint, or as an affect on the timing of the changepoint. We estimate all parameters using Markov chain Monte Carlo and compute bias, relative bias and average mean square error of the posterior means of the parameters capturing the covariate effects. We also investigate the effects of model misspecification on the characteristics of the effect of using an inappropriate fitted model.


REPORT 0405-20:  Hyperpolarizabilities for the one-dimensional infinite single-electron periodic systems: I. Analytical solutions under dipole-dipole correlations

Shidong Jiang and Minzhong Xu

The analytical solutions for the general-four-wave-mixing hyperpolarizabilities on infinite chains under both Su-Shrieffer-Heeger and Takayama-Lin-Liu-Maki models of trans-polyacetylene are obtained through the scheme of dipole-dipole correlation. Analytical expressions of DC Kerr effect, DC-induced second harmonic generation, optical Kerr effect and DC-electric-field-induced optical rectification are derived. By including or excluding the gradient terms in the calculations, comparisons show that the intraband contributions dominate the hyperpolarizabilities if they are included. The intraband transition leads to the break of the overall permutation symmetry in hyperpolarizabilities even for the low frequency and non-resonant regions. Hence it breaks the Kleinman symmetry that is directly based on the overall permutation symmetry. Our calculations provide a clear understanding of the Kleinman symmetry breaks that are widely observed in many experiments. Finally, we also suggest a feasible experiment on hyperpolarizabilities to test the validity of overall permutation symmetry and our theoretical prediction.


REPORT 0405-21:  Hyperpolarizabilities for the one-dimensional infinite single-electron periodic systems: II. Dipole-dipole versus current-current correlations

Minzhong Xu and Shidong Jiang

Based on Takayama-Lin-Liu-Maki model, analytical expressions for the third-harmonic generation, DC Kerr effect, DC-induced second harmonic optical Kerr effect, optical Kerr effect or intensity-dependent index of refraction and DC-electric-field-induced optical rectification are derived under the static current-current correlation for one-dimensional infinite chains. The results of hyperpolarizabilities under the current-current correlation are then compared with those obtained using the dipole-dipole correlation. The comparison shows that the conventional current-current correlation, albeit quite successful for the linear case, is incorrect for studying the nonlinear optical properties of periodic systems.


REPORT 0405-22:  Breaking of the overall permutation symmetry in nonlinear optical susceptibilities of periodic systems

Minzhong Xu and Shidong Jiang

Based on infinite one-dimensional periodic chain models (Su-Shrieffer-Heeger and Takayama-Lin-Liu-Maki) of trans-polyacetylene, we show analytically that the overall permutation symmetry of nonlinear optical susceptibilities is, albeit preserved in the molecular systems with only bound states, no longer generally held for the periodic systems. Hence it breaks the Kleinman symmetry in the off-resonant regions. Our theory qualitatively explain the widely observed deviations of Kleinman symmetry in experiments. New nonlinear optical experiments are proposed to verify the overall symmetry break.


REPORT 0405-23:  Preclinical Cardiac Safety Assessment of Pharmaceutical Compounds Using an Integrated Systems-Based Computer Model of the Heart

Dean Bottino, Chrstian Penland, Andrew Stamps, Martin Traebert, Berengere Dumotier, Anna Georgieva, Gabriel Helmlinger and Scott Lett

Blockade of the delayed rectifier potassium channel current,IKr, has been associated with drug-induced QT prolongation in the ECG and life-threatening cardiac arrythmias. However, it is increasingly clear that compound-induced interactions with multiple cardiac ion channels may significantly affect QT prolongation that would result from inhibition of only IKr (Redfern, Carlsson et al, 2003). Such an assessment may not be feasible in vitro, due to multi-factorial processes that are also time-dependent and highly non-linear. Limited preclinical data, IKr hERG assay and canine Purkinje fiber action potentials (Gintant, Limberis et al. 2001), were used for two test compounds in a systems-based modeling platform of cardiac electrophysiology (Muzikant and Penland 2002) to (i) convert a canine myocyte model to a Purkinje fiber model by training functional current parameters to the action potential data; (ii) reverse-engineer the compounds’ effects on five channel currents other than IKr, predicting significant IC50 values for INa+,sustained and ICa2+, L-type , which were subsequently experimentally validated; (iii) ICa2+use the predicted (INa+,sustained and ICa2+, L-type) and measured (IKr) IC50 values to simulate dose-dependent effects of the compounds on action potentials in Endo-, M-, and Epi-cardiac ventricular cells; and (iv) integrate the three types of cellular responses into a tissue-level spatial model, which quantifiably predicted no potential for the test compounds to induce either QT prolongation or increased transmural dispersion of repolarization in a dose-dependent and reverse rate-dependent fashion, despite their inhibition of IKr in vitro.


REPORT 0405-24:  Some generalizations of the First Fredholm Theorem to Hammerstein Equations and the Number of Solutions

P.S. Milojevic

We prove some generalizations of the First Fredholm theorem for Hammerstein operator equations in Banach spaces and study the number of their solutions using a projection like method. The linear part is assumed to be either selfadjoint or nonselfadjoint while the nonlinearities are such that the corresponding map is ( pseudo) approximation proper. In particular, the nonlinearities can be either of monotone type, or of type (S+), or condensing, or the sum of such maps.


REPORT 0405-25:  The "exact" confidence limits for unknown probability in Bernoulli models.

R.I. Andrushkiw, D.A. Klyushin, Yu.I. Petunin and M.Yu. Savkina,

The application of mathematical-statistical models in medical diagnostics often requires the construction of an "exact" confidence interval for the unknown probability p of success in Bernoulli models (so called binomial proportion, or proportion of population). This problem was considered in a number of papers (for example, see [1-5] and references cited). The web-site BioMed Central gives more than 200 citations devoted to this theme. The purpose of our paper is to construct an "exact" confidence interval for unknown probability p of success in classical and generalized Bernoulli models.


REPORT 0405-26:  An Embedded Network Approach for Scale-Up of Fluctuation-driven Systems with Preservation of Spike Information

David Cai, Louis Tao and David McLaughlin

To address computational "scale-up" issues in modeling of large regions of the cortex, many coarse-graining procedures have been invoked to obtain effective descriptions of neuronal network dynamics. However, because of local averaging in space and time, these methods do not contain detailed spike information, and, thus, cannot be used to investigate, e.g., cortical mechanisms which are encoded through detailed spike-timing patterns. To retain spike information, we develop a hybrid theoretical framework which embeds a sub-network of point neurons within, and fully interacting with, a coarse-grained network of dynamical background. We employ our newly developed kinetic theory for the description of the coarse-grained background, in combination of a Poisson spike reconstruction procedure to ensure that our method works for the fluctuation-driven regime as well as the mean-driven regime. This embedded network approach is verified to be dynamically accurate and numerically efficient. As an example, we use this embedded representation to construct "reverse-time correlations" as spiked-triggered averages in a ring model of orientation tuning dynamics.


REPORT 0405-27:  Orientation Selectivity in Visual Cortex by Fluctuation-Controlled Criticality

Louis Tao, David Cai, David McLaughlin, Robert Shapley and Michael Shelley

We examine how synaptic fluctuations modify the effects of strong recurrent network amplification to produce orientation selectivity in a large-scale neuronal network model of the macaque primary visual cortex. Previously, we showed that the model reproduces many of the experimentally observed properties of simple and complex cells, through a balance between feedforward and recurrent excitation. However, strong cortical amplification leads to network instabilities, unrealistically high firing rates and complex cells that are not orientation selective, even in the presence of strong cortical inhibition. In this paper, we show that large fluctuations in the cortico-cortical conductances can stabilize the network, allowing strong cortical gain and the emergence of orientation selective complex cells. By increasing the strength of synaptic fluctuations, say, through sparsifying the network connectivity, we identify a transition between two types of dynamics, mean- and fluctuation-driven. In a network with strong recurrent excitation, this fluctuation-controlled transition is signified by a near hysteretic behavior and a rapid rise of network firing rates as the synaptic drive or stimulus input is increased. We discuss the connection between this transition and orientation selectivity in our model of primary visual cortex.


REPORT 0405-28:  Stratified Kolmogorov flow. Part 2.

N. J. Balmforth and Y.-N. Young

Forced stratified flows are shown to suffer two types of linear long-wave instability: a 'viscous'
instability which is related to the classical instability of Kolmogorov flow, and a 'conductive instability', with the form of a large-scale, negative thermal diffusion. The nonlinear dynamics of both instabilities is explored with weakly nonlinear theory and numerical computations. The introduction of stratification arises with stronger stratification and creates a prominent staircase in the buoyancy field; the steps of the staircase evolve over long timescales by approaching one another, colliding and merging (coarsening the staircase).


REPORT 0405-29:  Registration-based morphing of active contours for segmentation of CT scans

Y.-N. Young and D. Levy

We present a new algorithm for segmenting organs in CT scans for radiotherapy treatment planning. Given a contour of an organ that is segmented in one image, our algorithm proceeds to segment contours that identify the same organ in the consecutive images. Our technique combines partial differential equations-based morphing active contours with algorithms for joint segmentation and registration. The coupling between these different techniques is done in order to deal with the complexity of segmenting ``real" images, where boundaries are not always well defined, and the initial contour is not an isophote of the image.


REPORT 0405-30:  Long-wave linear stability theory for two-fluid channel flow

Tetyana M. Segin, Lou Kondic and Burt S. Tilley

We present the linear stability of the laminar flow of an immiscible system of a compressible gas and incompressible liquid separated by an interface with large surface tension in a thin inclined channel. The flow is driven by an applied pressure drop and gravity. Following the air-water case, which is found in a variety of engineering systems, the ratio of the characteristic values of the gas and liquid densities and viscosities are assumed to be disparate. Under lubrication approximation, and assuming ideal gas behavior and isothermal conditions, this approach leads to a coupled nonlinear system of partial differential equations describing the evolution of the interface between the gas and the liquid and the streamwise density distribution of the gas. This system also includes the effects of viscosity stratification, inertia, shear, and capillarity. A linear stability analysis that allows for physically relevant nonzero pressure-drop base state is then performed. In contrast to zero-pressure drop case which is amenable to the classical normal-mode approach, this configuration requires solving numerically a nonautonomous boundary-value problem for the gas density and interfacial deviations from the base state in the streamwise coordinate. We find that the effect of gas compressibility on the interfacial stability in the limit of vanishingly small wavenumber is destabilizing, even for Stokes flow in the liquid. However, for finite wavenumber disturbances, compressibility may have stabilizing effects. In this regime, sufficient shear is required to destabilize the flow.


REPORT 0405-31:  Velocity Profiles in Repulsive Athermal Systems under Shear

Ning Xu, Corey S. O'Hern and Lou Kondic

We conduct molecular dynamics simulations of athermal systems undergoing boundary-driven planar shear flow in two and three spatial dimensions. We find that these systems possess nonlinear mean velocity profiles when the velocity $u$ of the shearing wall exceeds a critical value $u_c$. Above $u_c$, we also show that the packing fraction and mean-square velocity profiles become spatially-dependent with dilation and enhanced velocity fluctuations near the moving boundary. In systems with overdamped dynamics, $u_c$ is only weakly dependent on packing fraction $\phi$. However, in systems with underdamped dynamics, $u_c$ is set by the speed of shear waves in the material and tends to zero as $\phi$ approaches $\phi_c$. In the small damping limit, $\phi_c$ approaches values for random close-packing obtained in systems at zero temperature. For underdamped systems with $\phi<\phi_c$, $u_c$ is zero and thus they possess nonlinear velocity profiles at any nonzero boundary velocity.


REPORT 0405-32:  Dopamine Modulation of Phasing of Activity in a Rhythmic Motor Network: Contribution of Synaptic and Intrinsic Modulatory Actions

Bruce R. Johnson, Lauren Schneider, Farzan Nadim, Ronald M. Harris-Warrick

The phasing of neuronal activity in a rhythmic motor network is determined by a neuron's intrinsic firing properties and synaptic inputs; these could vary in their relative importance under different modulatory conditions. In the lobster pyloric network, the firing of eight follower PY neurons is shaped by their intrinsic rebound after pacemaker inhibition and by synaptic input from the LP neuron, which inhibits all PY neurons and is electrically coupled to a subset of them. Under control conditions, LP inhibition is weak and has little influence on PY firing. We examined modulation that could theoretically enhance the LP's synaptic contribution to PY firing. We measured the effects of dopamine (DA) on LPaPY synapses, driving the LP neuron with trains of realistic waveforms constructed from prerecorded control and DA LP oscillations, which differed in shape and duration. Under control conditions, chemical inhibition underwent severe depression and disappeared; in the mixed synapses, electrical coupling dominated. Switching between control and DA LP waveforms (with or without DA present) caused only subtle changes in synaptic transmission. DA markedly enhanced synaptic inhibition, reduced synaptic depression and weakened electrical coupling, reversing the sign of the mixed synapses. Despite this, removal of the LP from the intact network still had only weak effects on PY firing. DA also enhances PY intrinsic rebound properties, which still control the onset of PY firing. Thus, in a rhythmic network, the functional importance of synaptic modulation can only be understood in the context of parallel modulation of intrinsic properties.


REPORT 0405-33:  Integrated Data Analysis for Genotyping Microarrays

Kai Zhang, Marc Q. Ma, Hui-Yun Wang, Yu Wang, Frank Shih

We present TIMDA (a Toolkit for Integrated Microarray Data Analysis), a Matlab-based software framework designed for spotted single nucleotide polymorphism (SNP) genotyping microarray data analysis. TIMDA features seamlessly integration of numerical computation, analysis, visualization and algorithms as well as excellent extensibility and maintainability. The framework consists of modules designed for image processing, intermediate data conversion, genotype calling, and loss-of-heterozygosity (LOH) study with text or graphics output. Each of these modules can work smoothly with others or independently from each other. Meanwhile, data from other software also can be integrated into TIMDA.


REPORT 0405-34:  Modeling and Simulation of Soluble Guanylyl Cyclase

Yu Wang, Kentaro Sugino, Marc Q. Ma, Annie V. Beuve

Soluble guanylyl cyclase (sGC) is a heterodimeric enzyme that catalyzes the formation of cGMP from GTP. The mechanisms regulating the catalytic activity of sGC still remain unclear despite extensive experimental studies. We detail the steps used in homology modeling for constructing the 3D structure of sGC's catalytic core region. Many homology-based models are obtained, and their quality is evaluated using various programs including PROCHECK. Based on our homology models, we perform classical molecular dynamics (MD) simulations to study the conformational changes of the protein complex during the allosteric regulation, and a tentative mechanism is established, which can be used to guide the screening and design of new vasodilation drugs.


REPORT 0405-35:  Variable Selection of a Bayesian hierarchical changepoint Model for longitudinal biomarkers of Prostate Cancer

Wonsuk Yoo, Elizabeth H. Slate

Prostate specific antigen (PSA) is a common biomarker used to aid detection of prostate cancer (PCa). This research aims to develop and implement models for longitudinal biomarkers for prostate cancer, and to use these models to develop a diagnostic rule for early detection. We generalize a fully Bayesian hierarchical changepoint model, similar to that proposed by Slate and Clark (1999), by incorporating risk factors for prostate cancer as covariates. The changepoint, which is specific to each individual, represents the age of PCa onset. Our model permits the covariates to affect an individual's PSA three ways: the overall level, the age at which cancer initiates (changepoint), and the growth rate following the changepoint. We use Markov chain Monte Carlo (MCMC) to estimate all model parameters, including, especially, the subject-specific changepoints. Using data obtained from the Nutritional Prevention of Cancer Trial (Clark et al, 1996), we investigate the effects of smoking status, alcohol consumption and body mass index (BMI) on PSA growth. Moreover, we consider whether PSA velocity varies with the stage of prostate cancer at diagnosis.

We select the most useful combination of covariates in the model by examining the Bayesian credible intervals for the associated parameters, by computing conditional predictive ordinate (CPO) values (Gelfand et al., 1992) and pseudo Bayes factors. A retrospective receiver operating characteristic (ROC) curve method is applied to verify a potential best model.


REPORT 0405-36:  Signal transmission between gap-junctionally coupled passive cables occurs at an optimal cable diameter

Farzan Nadim, Jorge Golowasch

Theoretical studies have shown that gap junctional coupling can lead to complex network activity. We examine the consequences of current flow between two passive cables coupled by gap junctions at one end and derive analytical solutions for the steady state case. This architecture could represent gap-junctionally coupled processes of two neurons or coupling between two muscle fibers. We find that the transfer of electrical signals across the gap junction between two coupled cables does not always increase with increased cable diameter as expected from the increase in length constants, but that an optimal diameter exists. The optimal cable diameter arises because the gap junction acts as a current limiter. As current flows from one cable to the second, voltage attenuation along the latter follows two opposing rules. At low diameters, voltage attenuation along cable 2 decreases with increasing diameter, while at high diameters, it increases. This is due to an increase in leak conductance in cable 2 and the limited current flow through the gap junction. This optimal diameter depends on the gap junction resistance as well as the membrane properties and lengths of the cables. Moreover, in branched cables dependence on diameter is local and thus may serve to functionally compartmentalize branches that are independently coupled to other cells. Such compartmentalization may be important when periodic signals or action potentials cause the current flow across gap junctions. We predict that, in biological networks, coupled processes may have optimal diameters that maximize signal transfer between cells.


REPORT 0405-37:  Analysis of Equivalent Distorted Ratchet Potentials

Yiming Cheng, Robert M. Miura

Different kinds of charged particles undergoing Brownian motion can be separated by subjecting them periodically to an asymmetrical spatially periodic electric field, sometimes referred to as a Brownian ratchet. It has been found numerically that equal fluxes of particles can be obtained from two different potentials under special distortions. Although a simple physical explanation for this equivalency can be given, there is no corresponding mathematical proof. Here, we pose the mathematical problem and discuss the difficulty in carrying out the analytical proof.