**
REPORT 0607-1: **
**Stretching of Heated Threads With Temperature-Dependent Viscosity: Asymptotic
Analysis **

**P.D. Howell**, Centre for Industrial and Applied Mathematics, Mathematical
Institute, University of Oxford, Oxford OX1 3LB UK

**J.J. Wylie**, Department of Mathematics, City University of Hong Kong, Kowloon,
Hong Kong

**H. Huang**, Department of Mathematics and Statistics, York University, Toronto,
Ontario M3J 1P3 Canada

**R.M. Miura**, Department of Mathematical Sciences, New Jersey Institute of
Technology, Newark, N.J. 07102 USA

Abstract:

We consider the stretching of a thin cylindrical thread with viscosity that depends on temperature. The thread is pulled with a prescribed force while receiving continuous heating from an external axially nonuniform heater. We use the canonical equations derived by Huang et al. (2006) and consider the limit of large dimensionless heating rate. We show that the asymptotic solution depends only on the local properties of the heating near its maximal heating value. We derive a uniformly valid asymptotic solution for the shape and the temperature profiles during the stretching process. We use a criterion to determine when breaking will occur and derive simple analytical expressions for the shape at breaking that clearly show the influence of heating strength and the degree of localization of the heating. The asymptotic shape profiles give good agreement with numerical simulations. These results are applied to the formation of glass microelectrodes.

**
REPORT 0607-2: **
**Neuromodulation Unmasks Short-Term Synaptic Facilitation in a Depressing
Synapse**

**Pascale Rabbah**, Department of Mathematical Sciences, New Jersey Institute of Technology, Department of
Biological Sciences, Rutgers University

**Seher Atamturktur**, Department of Biological Sciences, Rutgers University

**Yair Manor**, Life Sciences Department and Zlotowski Center for Neurosciences,
Ben-Gurion University of the Negev, Beer-Sheva, Israel

**Farzan Nadim**, Department of Mathematical Sciences, New Jersey Institute of
Technology, Department of
Biological Sciences, Rutgers University

Abstract:

Network output depends crucially on the strength of synapses among the network neurons. Synaptic strength is a dynamic variable and is subject to modifications due to short- and long-term plasticity and the actions of neuromodulatory substances. We show that a neuromodulator can act to reverse the direction of short-term synaptic dynamics by changing depression to facilitation. Moreover, depression or facilitation in the presence of the neuromodulator depends on the amplitude of the slow voltage waveform of the presynaptic neuron, which is an important component of synaptic release in bursting neurons. We examined the effects of the neuropeptide proctolin on the short-term dynamics of the inhibitory synapse from the lateral pyloric (LP) to the pyloric dilator (PD) neuron in the crab pyloric network. This synapse is the only chemical feedback to the pacemaker group of neurons in this central pattern generating network. In response to periodic injection of high amplitude (?40 mV) waveforms in the voltage-clamped LP neuron, the LP to PD synapse showed depression in both control and in the presence of proctolin. In contrast, low-amplitude (?30 mV) waveforms resulted in depression in control and facilitation in proctolin. These results show that a synapse may change from depressing to facilitating depending only on the amplitude and not the frequency of the presynaptic signal. Such a neuromodulator-induced switch in short-term dynamics may allow the synapse to maintain a steady level of release despite variations in the presynaptic bursting amplitude, thus contributing to network stability.

**
REPORT 0607-3: **
**Regression Level Set Estimation -- The Excess Mass Approach in the Fixed
Design Case
**

Abstract:

Considering the d-dimensional nonparametric regression d greater than or equal to 1 with fixed design, the estimation of regression level sets is studied based on the maximization of empirical excess masses. By applying modern marked empirical process theory to excess masses, consistency and rates of convergence for the resulting estimators are studied under the conditions on bracketing numbers and modulus of continuity, and a functional central limit theorem for standardized excess mass processes is derived. The consistency and rates of convergence for test statistic of multi-modality are also presented.

**
REPORT 0607-4: **
**Distinct Synaptic Dynamics of Heterogeneous Pacemaker Neurons in an
Oscillatory Network**

**Pascale Rabbah** and **Farzan Nadim**

Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102 Department of Biological Sciences, Rutgers University, Newark, NJ 07102

Abstract:

Many rhythmically active networks involve heterogeneous populations of pacemaker neurons with potentially distinct synaptic outputs that can be differentially targeted by extrinsic inputs or neuromodulators, thereby increasing possible network output patterns. In order to understand the roles of heterogeneous pacemaker neurons, we characterized differences in synaptic output from the anterior burster (AB) and pyloric dilator (PD) neurons in the lobster pyloric network. The intrinsically distinct AB and PD neurons are strongly electrically coupled and constitute the pyloric pacemaker ensemble. During pyloric oscillations, these co-active neurons produce compound inhibitory synapses to the follower pyloric neurons including the lateral pyloric (LP) and pyloric constrictor (PY) neurons.

Using pharmacological blockers, we separated the synapses efferent from the AB and PD neurons and investigated their temporal dynamics using square pulses and realistic pre-recorded waveforms. These synapses exhibited distinct short-term dynamics, depending on the presynaptic neuron type, and had different relative contributions to the total synaptic output depending on waveform shape and cycle frequency. However, paired comparisons revealed that the amplitude or dynamics of synapses from either AB or PD neuron did not depend on the postsynaptic neuron type, LP or PY. To address the functional implications of these findings, we examined the correlation between synaptic inputs from the pacemakers and the burst onset phase of the LP and PY neurons in the ongoing pyloric rhythm. These comparisons showed that the activity of the LP and PY neurons weakly depend on the peak phase and amplitude of the IPSPs they receive from the pacemaker neurons.

**
REPORT 0607-5: **
**Influence of Surfactant on the Deformation and Breakup of a Bubble in a
Viscous Surrounding**

**M. Hameed**, Department of Mathematics and Computer Science, University of
South Carolina Upstate, Spartanburg, SC 29303, USA

**M. Siegel, Y.-N. Young, M. R. Booty, D. T. Papageorgiou**, Department of
Mathematical Sciences, NJIT, Newark, NJ, 07102, USA

**J. Li**, Department of Engineering, University of Cambridge, Cambridge, CB21PZ,
UK

Abstract:

The influence of surfactant on the breakup of a bubble in a quiescent viscous surrounding is studied by a combination of direct numerical simulation and the solution of a longwave asymptotic model. The direct numerical simulations describe the evolution toward breakup of an inviscid bubble, while the effects of small but nonzero interior viscosity are readily included in the longwave model in the Stokes flow limit.

The direct numerical simulations use a specific but realizable and representative initial bubble shape to compare the evolution toward breakup of a clean or surfactant-free bubble and a bubble that is coated with insoluble surfactant. A distinguishing feature of the evolution in the presence of surfactant is the interruption of bubble breakup by formation of a slender quasi-steady thread of the interior fluid.

The longwave asymptotic model, for a thread with periodic boundary conditions, explains the principal mechanism of slender thread formation and confirms, for example, the relatively minor role played by the Marangoni stress. The large-time evolution of the slender thread and the precise location of its breakup are, however, influenced by effects such as the Marangoni stress and surface diffusion of surfactant.

**
REPORT 0607-6:
Capturing the Bursting Dynamics of a Two-Cell Inhibitory Network Using a
One-Dimensional Map**

**Victor Matveev**^{1}, **Amitabha Bose**^{1}, and
**Farzan Nadim** ^{1,2}

^{1} Department of Mathematical Sciences, New Jersey Institute of Technology,
NJ 07102, USA

^{2} Department of Biological Sciences, Rutgers University, Newark, NJ 07102,
USA

Abstract:

Out-of-phase bursting is a functionally important behavior displayed by central pattern generators and other neural circuits. Understanding this complex activity requires the knowledge of the interplay between the intrinsic cell properties and the properties of synaptic coupling between the cells. Here we describe a simple method that allows us to investigate the existence and stability of anti-phase bursting solutions in a network of two spiking neurons, each possessing a T-type calcium current and coupled by reciprocal inhibition. We derive a one-dimensional map which fully characterizes the genesis and regulation of anti-phase bursting arising from the interaction of the T-current properties with the properties of synaptic inhibition. This map is the burst length return map formed as the composition of two distinct one-dimensional maps that are each regulated by a different, but mutually overlapping, set of model parameters. Although each map is constructed using the properties of a single isolated model neuron, the composition of the two maps accurately captures the behavior of the full network. We analyze the parameter sensitivity of these maps to determine the influence of both the intrinsic cell properties and the synaptic properties on the burst length, and to find the conditions under which multistability of several bursting solutions is achieved. Although the derivation of the map relies on a number of simplifying assumptions, we discuss how the principle features of this dimensional reduction method could be extended to more realistic model networks.

**
REPORT 0607-7:
Calculation of Complex Singular Solutions to the 3D Incompressible Euler
Equations**

**Michael Siegel**, Department of Mathematical Sciences, New Jersey Institute
of Technology, NJ 07102, USA

Abstract:

We describe an approach for the construction of singular solutions to the 3D Euler equations for complex initial data. The approach is based on a numerical simulation of complex traveling wave solutions with imaginary wave speed, originally developed by Caflisch for axisymmetric flow with swirl. Here, we simplify and generalize this construction to calculate traveling wave solutions in a fully 3D (nonaxisymmetric) geometry. We also discuss a semi-analytic approach to the problem of Euler singularities based on numerical computation of the complex traveling wave solutions, followed by perturbation construction of a real solution. The perturbation analysis depends on a small amplitude of the singularity in the traveling wave solution; techniques for producing such a small amplitude are described. This is joint work with Russ Caflisch.

**
REPORT 0607-8:
Dominant Ionic Mechanisms Explored in Spiking and Bursting Using Local
Low-Dimensional Reductions of a Biophysically Realistic Model Neuron**

**Robert Clewley**, Department of Mathematics, Cornell University

**Cristina Soto-Treviño**, Department of Mathematical Sciences, New Jersey Institute of Technology

**Farzan Nadim**, Department of Mathematical Sciences, New Jersey Institute of Technology and Department of Biological Sciences,Rutgers University

Abstract:

The large number of variables involved in most biophysical models often makes analysis of the dynamics of solutions and the transitions between different modes of activity prohibitively difficult. To address this issue, we use a novel model reduction method, based on “scales of dominance”, to systematically reduce the dimension of a large conductance-based model in order to analyze its different modes of activity. We demonstrate this technique using a two-compartment conductance-based model of a crustacean pyloric dilator (PD) neuron that exhibits distinct modes of oscillation—tonic spiking, intermediate bursting and strong bursting. From the full sixteen-variable model we obtain a globally-reduced, nine-variable model, which is then divided along its solution trajectory into regions dominated by a smaller number of variables. The dominant variables in each region are identified by their influence level (dominance scale) on the membrane potential according to a pre-determined threshold. This technique results in a reduced hybrid model that can have dimension as low as two in some temporal regimes. The reduced model exhibits the same modes of oscillation as the full 16-dimensional model over a comparable parameter range. Using these reduced models, we investigate the low-dimensional organizing structure of the model dynamics and the dependence of the model oscillations on parameters such as the maximal conductances of calcium currents. The techniques demonstrated in this study could be used to build hybrid low-dimensional models from any large conductance-based model in order to analyze transitions between different modes of activity.

**
REPORT 0607-9:
On Breakup of Fluid Films of Finite and Infinite Extent**

**Javier A. Diez**, Instituto de Fisica Arroyo Seco, Universidad Nacional del
Centro de la Provincia de Buenos Aires, Pinto 399, 7000, Tandil, Argentina

**Lou Kondic**, Department of Mathematical Sciences, New Jersey Institute of
Technology, Newark, NJ 07102

Abstract:

We study the dewetting process of thin fluid films that
partially wet a solid surface. Using long wave (lubrication) approximation, we
formulate a nonlinear partial differential equation governing the evolution of
the film thickness, h. This equation includes the effects of capillarity,
gravity, and additional conjoining/disjoining pressure term to account for
intermolecular forces. We perform standard linear stability analysis of an
infinite flat film, and identify the corresponding stable, unstable and
metastable regions. Within this framework, we analyze the evolution of a
semi-infinite film of length L in one direction. The numerical simulations show
that for long and thin films, the dewetting fronts of the film generate a
pearling process involving successive formation of ridges at the film ends and
consecutive pinch-off behind these ridges. On the other hand, for shorter and
thicker films, the evolution ends up by forming a single drop. The time
evolution as well as the final drops pattern shows a competition between the
dewetting mechanisms caused by nucleation and by free surface instability. We
find that precise computations, requiring quadruple precision of computer
arithmetic, are often needed to avoid spurious results.

**
REPORT 0607-10:
Dynamic Structure Formation at the Fronts of Volatile Liquid Drops**

**Y. Gotkis**, KLA-Tencor Corporation, San-Jose, CA

**I. Ivanov**, Blue29, Sunnyvale, CA

**N. Murisic**, Department of Mathematical Sciences, New Jersey Institute of
Technology, Newark, NJ 07102

**L. Kondic**, Department of Mathematical Sciences, New Jersey Institute of
Technology, Newark, NJ 07102

Abstract:

We report on instabilities during spreading of volatile
liquids, with emphasis on the novel instability observed when isopropyl alcohol
(IPA)
is deposited on a monocrystaline Si wafer. This instability is characterized by
emission of drops ahead of the expanding front, with each drop followed by
smaller, satellite droplets, forming the structures which we nickname `octopi'
due to their appearance. A less volatile liquid, or a substrate of larger heat
conductivity, suppress this instability. We have formulated a theoretical model
that reproduces the main features of the experiment.

**
REPORT 0607-11:
On Velocity Profiles and Stresses in Sheared and Vibrated Granular Systems
Under Variable Gravity**

**Oleh Baran**, ExxonMobil Research and Engineering,1545 Route 22 East,
Annandale, NJ, USA, 08801

**Lou Kondic**, Department of Mathematical Sciences, New Jersey Institute of
Technology, Newark, NJ 07102, USA

Abstract:

We employ discrete element 3D simulations (DES) that include realistic modeling of physical system boundaries to determine the influence of gravity on velocity profiles and stresses for frictional inelastic particles that are confined in an angular Couette cell, and sheared by a rotated upper wall. In addition to Earth gravity, we consider other gravitational fields, in particular those of the Moon and Mars. The computational techniques are based on hard sphere simulations of polydisperse particles at relatively high volume fraction (50-55%). We find that the presence of gravity induces significant changes of the velocity profiles and stresses. We also consider systems which are vibrated in addition to being sheared, since vibrations are one of several important methods for agitating (e.g., fluidizing and/or unjamming) granular systems.

REPORT 0607-12:Mesoscopic Simulation of Ostwald Ripening

David J. Horntrop, Department of Mathematical Sciences, New Jersey Institute of Technology

Abstract:

The self-organization of particles in a two phase system in the coexistence region through a diffusive mechanism is known as Ostwald ripening. This phenomenon is an example of a multiscale problem in that the microscopic level interaction of the particles can greatly impact the macroscale or observable morphology of the system. Ostwald ripening is studied here through the use of a mesoscopic model which is a stochastic partial integrodifferential equation that is derived from a spin exchange Ising model.

This model is studied through the use of recently developed and benchmarked spectral schemes for the simulation of solutions to stochastic partial differential equations.

The typical cluster size is observed to grow like t^(1/3) over range of times with faster growth at later times.

The results included here also demonstrate the effect of adjusting the interparticle interaction on the morphological evolution of the system at the macroscopic level.

**
REPORT 0607-13: **
** ****Interface
Dynamics for Quasi-Stationary Stefan Problem**

**R. Andrushkiw**, Department of Mathematical Sciences, New Jersey Institute of
Technology, Newark, N.J. 07102 USA

**V. Gafiychuk**, Institute of Computer Modeling, Krakow University of
Technolology, 31155 Krakow, Poland

**B. Datsko**, Institute of Applied Problems of Mechanics and Mathematics,
National Academy of Sciences of Ukraine, Lviv, 79053 Ukraine

Abstract:

We investigate the interface dynamics in a Laplacian growth model, using conformal mapping technique. Starting from the governing equation obtained by B.Shraiman and D. Bensimon, we derive integro-differential evolution equation of interface dynamics. It is shown that such representation of the conformal mapping technique is convenient for computer simulation of the quasi-stationary Stefan problem.

**
REPORT 0607-14:****
Mathematical Analysis of Applied Loads on Skeletal
Muscles in Osteopathic Manual Treatment **

**Hans Chaudhry**, Ph.D.^{1,3}, **Bruce Bukiet**,
Ph.D.^{2 }**Thomas Findley**, M.D., Ph.D.^{1,3}

^{1} Department of Biomedical Engineering, New Jersey
Institute of Technology, Newark, New Jersey 07102, USA.

^{2} Department of Mathematical Sciences, Center for
Applied Mathematics and Statistics, New Jersey Institute of Technology,
Newark, New Jersey 07102, USA.

^{3} War-Related Illness and Injury Study Center, VA
Medical Center, East Orange, NJ 07018, USA.

Abstract:

Context: To determine the loads to produce compression, shear, extension and twist on biceps muscle in osteopathic manual treatment.

Methods: Mathematical Analysis valid for the in vivo state of transversely isotropic biceps muscle is performed to determine the loads under elastic and viscoelastic biceps models.

Results: 7% lower loads are needed to produce 10% deformation using the viscoelastic model compared to the elastic model. Using the viscoelastic model, it is found that stress relaxes by 18% of its maximum value for the case in which muscle is deformed by 10% over a period of 60 seconds and held in that deformed state up to 200 seconds. It is observed that with quick maneuvers, the viscoelasticity effect is decreased, i.e. greater loads need to be applied for a given deformation.

Conclusions: The biceps muscle is 15 times stiffer in the directions parallel to the muscle fibers compared to the perpendicular direction. The results in this paper can be useful to the manual therapists to adjust their technique to tissue properties. Since the biceps muscle is viscoelastic, the results obtained in this paper for the viscoelastic model are more realistic for determination of viscoelastic stresses compared to those from using the elastic model.

**
REPORT 0607-15:****
**
Dynamics of Central Pattern Generating Networks: Locus of Control

Farzan Nadim and Amitabha Bose

Department of Mathematical Sciences, New Jersey Institute of Technology

Abstract:

None

**
REPORT 0607-16:****
**
Neurons and Neural Networks: Computational Models

**Horacio Rotstein **and** Farzan
Nadim**

Department of Mathematical Sciences, New Jersey Institute of Technology

Abstract:

Neural networks produce electrical activity that is generated by the biophysical properties of the constituent neurons and synapses. Mathematical equations can be used to describe the electrical activity of neurons and neural networks and the underlying biophysical properties. These equations give rise to computational models of neurons and networks that can be analyzed using mathematical techniques or numerically simulated with computers.

**
REPORT 0607-17:****
****
Cortical Spreading Depression: An Enigma**

Robert M. Miura,Department of Mathematics, New Jersey Institute of Technology

Huaxiong Huang,Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada M3J 1P3

Jonathan J. Wylie,Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

Abstract:

The brain is a complex organ with active components composed largely
of neurons, glial cells, and blood vessels. There exists an enormous
experimental and theoretical

literature on the mechanisms involved in the functioning of the brain,
but we still do not have a good understanding of how it works on a gross
mechanistic level. In general, the brain maintains a homeostaticstate
with relatively small ion concentration changes, the major ions being
sodium, potassium, and chloride, as well as a very important ion,
calcium. Cortical spreading depression (CSD for short) was discovered
over 60 years ago by A.A.P. Leao, a Brazilian physiologist doing his
doctoral research on epilepsy at Harvard University, ``Spreading
depression of activity in the cerebral cortex," J. Neurophysiol., 7
(1944), pp. 359-390. Cortical spreading depression is characterized by
massive changes in ionic concentrations and slow nonlinear chemical
waves, with speeds on the order of mm/min, in the cortex of different
brain structures in various experimental animals. In humans, CSD is
associated with migraine with aura, where a light scintillation in the
visual field propagates, then disappears, and is followed by a sustained
headache.

To date, CSD remains an enigma, and further detailed experimental and
theoretical investigations are needed to develop a comprehensive picture
of the diverse

mechanisms involved in producing CSD. A number of mechanisms have been
hypothesized to be important for CSD wave propagation. In this paper, we
briefly

describe several characteristics of CSD wave propagation, and examine
some of the mechanisms that are believed to be important, including ion
diffusion, membrane

ionic currents, osmotic effects, spatial buffering, neurotransmitter
substances, gap junctions, metabolic pumps, and synaptic connections.
Continuum models of CSD,

consisting of coupled nonlinear diffusion equations for the ion
concentrations, and a discrete lattice-Boltzmann method approach will be
described. Also, we will

describe some open problems and remaining challenges.

REPORT 0607-18:Nonparametric and Semiparametric Bayesian Reliability Analysis

Kaushik Ghosh,Department of Mathematics, New Jersey Institute of Technology

Abstract:

In this article, we first provide an overview of some nonparametric Bayesian methods of inference using life-history data. These methods include those that use Dirichlet process, gamma process and beta process as the prior. We then present a semiparametric Bayesian method for estimating the reliability of a component in a multi-component system using lifetime data from several systems. This method assumes that the component lifetimes have a parametric distribution with a Dirichlet process prior on the distribution of the parameters. The semiparametric method is illustrated with a simulation study using a data augmentation procedure.

REPORT 0607-19:Using Dynamic Programming Methods in Repair and Replacement Problems

Manish Bhattacharjee, Department of Mathematics, New Jersey Institute of Technology

Abstract:

Applications of dynamic programming to problems in reliability, with a specific emphasis on repair and replacement are surveyed. A brief summary of early classic work on such uses of dynamic programming is followed by a review of later approaches adopted by different researchers towards the problem of modeling plausible and pragmatic notions of imperfect repairs. We then provide a synthesis of how such different approaches for the study of repairable systems can be subsumed within a unified setting of a stochastic dynamic programming framework with some illustrative applications of such a formulation.

REPORT 0607-20:Ca2+-Dependent Inactivation of CaV1.2 Channels Prevents Gd3+ Block: Does Ca2+Block the Pore

of Inactivated Channels?

**Victor Matveev, **Department of Mathematical Sciences, New Jersey Institute of Technology,
Newark, NJ 07102

**Olga Babich, Andrew L. Harris, **and** Roman Shirokov**, Department of Pharmacology and Physiology, UMDNJ - New Jersey Medical School,
Newark, NJ 07103

Abstract:

Lanthanide gadolinium (Gd3+) blocks CaV1.2 channels at the selectivity
filter. Here we investigated whether Gd3+ block interferes with Ca2+-dependent
inactivation, which requires Ca2+ entry through the same site. Using brief
pulses to 200 mV that relieve Gd3+ block but not inactivation, we monitored how
the proportions of open and openblocked channels change during inactivation. We
found that blocked channels inactivate much less. This is expected for Gd3+
block of the Ca2+ influx that enhances inactivation. However, we also found that
the extent of Gd3+ block did not change when inactivation was reduced by
abolition of Ca2+/calmodulin interaction, showing that Gd3+ does not block the
inactivated channel. Thus, Gd3+ block and inactivation are mutually exclusive,
suggesting action at a common site. These observations suggest that inactivation
causes a change at the selectivity filter that either hides the Gd3+ site, or
reduces its affinity, or that Ca2+ occupies the binding site at the selectivity
filter in inactivated channels. The latter possibility is supported by previous
findings that the EEQE mutation of the selectivity EEEE locus is void of

Ca2+-dependent inactivation (Zong Z.Q., J.Y. Zhou, and T. Tanabe. 1994.
Biochem Biophys Res. Commun. 201(3):1117-11123), and that Ca2+-inactivated
channels conduct Na+ when Ca2+ is removed from the extracellular medium (Babich
O., D. Isaev, and R. Shirokov. 2005. J. Physiol. 565.3;709-717). Based on these
results, we propose that inactivation increases affinity of the selectivity
filter for Ca2+ so that Ca2+ ion blocks the pore. A minimal model, in which the
inactivation “gate” is an increase in affinity of the selectivity filter for
permeating ions, successfully simulates the characteristic U-shaped voltage
dependence of inactivation in Ca2+.

REPORT 0607-21:An Iterative Matrix-Free Method in Implicit Immersed Boundary/Continuum Methods

**X. Sheldon Wang, **Department of Mathematical Sciences, New Jersey Institute of Technology,
Newark, NJ 07102

Abstract:

The objective of this paper is to present an iterative solution strategy for implicit immersed boundary/continuum methods. An overview of the newly proposed immersed continuum method in conjunction with the traditional immersed boundary method will also be presented. As a key ingredient of the fully implicit time integration, a matrix-free combination of Newton-Raphson iteration and GMRES iterative linear solver is proposed.

REPORT 0607-22:A Reduced Model for Flame-Flow Interaction

**Peter Gordon**, Department of Mathematical Sciences, New Jersey Institute of Technology,
University Heights, Newark, NJ 07102

**Michael L. Frankel**, Department of Mathematical Sciences, Indiana University Purdue University,
Indianapolis, IN 46202

**Gregory I. Sivashinsky**, Department of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978,
Israel

Abstract:

The paper is concerned with an extension of the classical relation between the flame speed and the curvature-flow stretch, valid only for high Lewis numbers (diffusively stable flames). At low Lewis numbers the corresponding flame-flow system suffers short-wavelength instability, making the associated initial value problem ill-posed. In this study the difficulty is resolved by incorporation of higher-order effects. As a result one ends up with a reduced model based on a coupled system of second-order dynamic equations for the flame interface and its temperature. As an illustration the new model is applied for description of diffusively unstable stagnation-point flow flames.

REPORT 0607-23: Influence of Insoluble Surfactant on the Deformation and Breakup of a Bubble or Thread in a Viscous Fluid

M. Hameed^{1},M. Siegel^{2},Y.-N. Young^{2},J. Li^{3},M. R. Booty^{2}andD. T. Papageorgiou^{2}

^{1}Division of Mathematics and Computer Science, Unversity of South Carolina Upstate, Spartanburg, SC 29303, USA

^{2}Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, NJIT, Newark, NJ, 07102, USA

^{3}Department of Engineering, University of Cambridge, Cambridge, CB2 1PZ, UK

Abstract:

The influence of surfactant on the breakup of a prestretched bubble in a quiescent viscous surrounding is studied by a combination of direct numerical simulation and the solution of a longwave asymptotic model. The direct numerical simulations describe the evolution toward breakup of an inviscid bubble, while the effects of small but nonzero interior viscosity are readily included in the longwave model for a fluid thread in the Stokes flow limit. The direct numerical simulations use a specific but realizable and representative initial bubble shape to compare the evolution toward breakup of a clean or surfactant-free bubble and a bubble that is coated with insoluble surfactant. A distinguishing feature of the evolution in the presence of surfactant is the interruption of bubble breakup by formation of a slender quasi-steady thread of the interior fluid. This forms since the decrease in surface area causes a decrease in the surface tension and capillary pressure, until at a small but nonzero radius equilibrium occurs between the capillary pressure and interior fluid pressure. The longwave asymptotic model, for a thread with periodic boundary conditions, explains the principal mechanism of the slender thread's formation and con¯rms, for exam- ple, the relatively minor role played by the Marangoni stress. The large-time evolution of the slender thread and the precise location of its breakup are, however, influenced by effects such as the Marangoni stress and surface diffusion of surfactant.

REPORT 0607-24:A Stretch-Coil Transition and Transport of Fibers in Cellular Flows

Y.-N. Young^{1}andM. Shelley^{2}

^{1}Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, 07102, USA

^{2}Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

Abstract:

It is shown that a slender elastic fiber moving in a Stokesian fluid can be susceptible to a buckling instability--termed the "stretch-coil" instability--when moving in the neighborhood of a hyperbolic stagnation point of the flow. When the stagnation point is embedded in an extended cellular flow, it is found that immersed fibers can move as random walkers across time-independent closed-streamline flow. It is also found that the flow is segregated into transport regions around hyperbolic stagnation points and their manifolds, and closed entrapment regions around elliptic points.

REPORT 0607-25:Comparing Projection Neuron- and Neuromodulator-Elicited Oscillations in an Oscillatory Network

**Nickolas Kintos**, Department of Mathematical Sciences, New Jersey
Institute of Technology

**Michael P. Nusbaum**, Department of Neuroscience, University of
Pennsylvania School of Medicine

**Farzan Nadim**, Department of Mathematical Sciences, New Jersey
Institute of Technology

Abstract:

Many central pattern generating networks are influenced by synaptic input from modulatory projection neurons. The network response to a projection neuron is sometimes mimicked by bath applying the neuronally-released modulator, despite the absence of network interactions with the projection neuron. One interesting example occurs in the crab stomatogastric ganglion (STG), where bath applying the neuropeptide pyrokinin (PK) elicits a gastric mill rhythm which is similar to that elicited by the projection neuron MCN1, despite the absence of PK in MCN1 and the fact that MCN1 is not active during the PK-elicited rhythm. MCN1 terminals have fast and slow synaptic actions on the gastric mill network and are presynaptically inhibited by this network in the STG. These local connections are inactive in the PK-elicited rhythm, and the mechanism underlying this rhythm is unknown. We use mathematical and biophysically-realistic modeling to propose potential mechanisms by which PK can elicit a gastric mill rhythm that is similar to the MCN1-elicited rhythm. We analyze slow-wave network oscillations using simplified mathematical models and, in parallel, develop biophysically-realistic models that account for fast, action potential-driven oscillations and some spatial structure of the network neurons. Our results illustrate how the actions of bath-applied neuromodulators can mimic those of descending projection neurons through mathematically similar but physiologically distinct mechanisms.

REPORT 0607-26:Sustained Rhythmic Activity in Gapneurons Depends on the Diameter of Coupled Dendrites

Juliane Gansert, Department of Mathematical Sciences, NJIT, [Currently at: European Neuroscience Institute Göttingen, Germany]

Jorge Golowasch, Department of Mathematical Sciences, NJIT, Federated Department of Biological Sciences, Rutgers University and NJITFarzan Nadim, Department of Mathematical Sciences, NJIT, Federated Department of Biological Sciences, Rutgers University and NJIT

Abstract:

Gap junctions are known to be important for many network functions such as synchronization of activity and the generation of waves and oscillations. Gap junctions have also been proposed to be essential for the generation of early embryonic activity. We have previously shown that the amplitude of electrical signals propagating across gapjunctionally coupled passive cables is maximized at a unique diameter. This suggests that threshold-dependent signals may propagate through gap junctions for a finite range of diameters around this optimal value. Here we examine the diameter dependence of action potential propagation across model networks of dendro-dendritically coupled neurons. The neurons in these models have passive soma and dendrites and an action potential generating axon. We show that propagation of action potentials across gap junctions occurs only over a finite range of dendritic diameters and that propagation delay depends on this diameter. Additionally, in networks of gap-junctionally coupled neurons, rhythmic activity can emerge when closed loops (reentrant paths) occur but again only for a finite range of dendrite diameters. The frequency of such rhythmic activity depends on the length of the path and the dendrite diameter. For large networks of randomly coupled neurons, we find that the reentrant paths that underlie rhythmic activity also depend on dendrite diameter. These results underline the potential importance of dendrite diameter as a determinant of network activity in gap-junctionally coupled networks, such as network rhythms that are observed during early nervous system development.

REPORT 0607-27:Neuromodulators, Not Activity, Control Coordinated Expression of Ionic Currents

Olga Khorkova, Federated Department of Biological Sciences, NJITJorge Golowasch, Department of Mathematical Sciences, NJIT

Abstract:

Neurons express wide variability in the ionic currents that determine their output, but their electric activity is stable over long periods. However, neuronal mechanisms may reduce variability and thus enhance output stability by coordinately regulating the expression of multiple ionic currents. Studying identified neurons of the Cancer borealis pyloric network we discovered that the removal of neuromodulatory input to this network (decentralization) was accompanied by the loss of the coordinated regulation of ionic current levels. Additionally, decentralization induces large changes in the absolute levels of several ionic currents. The loss of co-regulation and the absolute level changes were prevented with exogenous application of the naturally occurring peptidergic neuromodulator proctolin. This peptide is known not to exert fast regulatory actions of any of the currents affected over the long-term. We conclude that neuromodulatory inputs to the pyloric network exert a novel long-term control of ionic current expression: they regulate the coordinated expression of multiple voltage-gated ionic currents that they do not acutely modulate. We discuss the possible functional significance of this type of regulation.

REPORT 0607-28:On Regularizing the Strongly Nonlinear Internal Wave Model

**Tae-Chang Jo**, Department of Mathematics, Inha University

**Wooyoung Choi**, Department of Mathematical Science, New Jersey
Institute of Technology

Abstract:

A strongly nonlinear asymptotic model describing the evolution of large amplitude internal waves in a two-layer system is studied numerically. While the steady model has been demonstrated to capture correctly the characteristics of large amplitude internal solitary waves, a local stability analysis shows that the time-dependent inviscid model suffers from the Kelvin-Helmholtz instability due to a tangential velocity discontinuity across the interface accompanied by the interfacial deformation. An attempt to represent the viscous effect that is missing in the model is made with eddy viscosity, but this simple ad-hoc model is shown to fail to suppress unstable short waves. Alternatively, when a smooth low-pass Fourier filter is applied, it is found that a large amplitude internal solitary wave propagates stably without change of form, and mass and energy are conserved well. The head-on collision of two counter-propagating solitary waves is studied using the filtered strongly nonlinear model and its numerical solution is compared with the weakly nonlinear asymptotic solution.