## Center for Applied Mathematics and Statistics

REPORT 0304-1:   A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type.

C. B. Muratov

We identified a variational structure associated with traveling waves for systems of reaction-diffusion equations of gradient type with equal diffusion coefficients defined inside an infinite cylinder, with either Neumann or Dirichlet boundary conditions. We show that the traveling wave solutions that decay sufficiently rapidly exponentially at one end of the cylinder are critical points of certain functionals. We obtain a global upper bound on the speed of these solutions. We also show that for a wide class of solutions of the initial value problem an appropriately defined instantaneous propagation speed approaches a limit at long times. Furthermore, under certain assumptions on the shape of the solution, there exists a reference frame in which the solution of the initial value problem converges to the traveling wave solution with this speed at least on a sequence of times. In addition, for a class of solutions we establish bounds on the shape of the solution in the reference frame associated with its leading edge and determine accessible limiting traveling wave solutions. For this class of solutions we find the upper and lower bounds for the speed of the leading edge.

REPORT 0304-2:   A shallow water ocean acoustics inverse problem.

David Stickler

A shallow water ocean acoustics experiment is described from which it is possible to determine in principle the scattering data necessary to recover the sound speed in the ocean bottom. This data is a reflection coefficient which is defined below. The essential assumptions are that the speed in the ocean is known but unknown in the bottom and that the measurements are made in the ocean layer.

REPORT 0304-3:   Instabilities in the flow of thin films on heterogeneous surfaces.

Lou Kondic and Javier A. Diez

We present computational and experimental results about instability development in the gravity driven flow of thin fluid films on heterogeneous surfaces. In particular, we concentrate on the dynamics of the fluid fronts, i.e., on the contact line. We show that heterogeneity of the solid surface can have significant effect on the flow dynamics. Since the effect of heterogeneity often competes with the basic instability mechanism that would occur even on homogeneous surfaces, the result is an elaborate interplay of various instability mechanisms. The computational results presented here outline both the flow on surfaces perturbed by regular patterns, and on surfaces perturbed by irregular, noise-like perturbations. We relate these computational results to the pattern formation process in our experiments of gravity driven flow down an incline. Good qualitative agreement is found between the simulations and the experiments.

REPORT 0304-4:   Elastic Energy, Fluctuations and Temperature for Granular Materials.

Lou Kondic and Robert P. Behringer

We probe, using a model system, elastic and kinetic energies for sheared granular materials. For large enough $P/E_y$ (pressure/Young's modulus) and $P/\rho v^2$ ($P/$kinetic energy density) elastic dominates kinetic energy, and energy fluctuations become primarily elastic in nature. This regime has likely been reached in recent experiments. We consider a generalization of the granular temperature, $T_g$, with both kinetic and elastic terms and that changes smoothly from one regime to the other. This $T_g$ is roughly consistent with a temperature adapted from equilibrium statistical mechanics.

REPORT 0304-5:   Spatially Discrete FitzHugh-Nagumo Equations.

Christopher E Elmer and Erik S Van Vleck

In this work we consider pulse and front solutions to a spatially discrete FitzHugh-Nagumo equation (with recovery) with a piecewise linear bistable nonlinearity, which until now is an unsolved problem. We demonstrate a technique for deriving candidate solutions, and present and apply solvability conditions necessary for existence.

REPORT 0304-6:   Dynamics of Monotone Traveling Fronts for Discretizations of Nagumo PDE's.

Christopher E Elmer and Erik S Van Vleck

When PDEs are discretized, the dynamics of the resulting equations differ from those of the original PDE. In this work, we study the dynamics of traveling wave solutions to the discretized Nagumo PDE ($A(\alpha)$ stable in time and/or uniform in space) with smooth bistable nonlinearities. In general, time discretization significantly speeds up traveling waves fronts and spatial discretization slows (or even halts) such fronts.

REPORT 0304-7:   Varieties with degenerate Gauss Maps with Multiple Foci and Twisted Cones

Maks A. Akivis and Vladislav V. Goldberg

The authors study in detail new types of varieties with degenerate Gauss maps: varieties with multiple foci and their particular case, the so-called twisted cones. They prove an existence theorem for twisted cones and describe their structure.

REPORT 0304-8:   Induced Connections on Submanifolds

Maks A. Akivis, Vladislav V. Goldberg, and Arto V. Chakmazyan

The authors establish a relation of the theory of varieties with degenerate Gauss maps in projective spaces with the theory of congruences and pseudocongruences of subspaces and show how these two theories can be applied to the construction of induced connections on submanifolds of projective spaces and other spaces endowed with a projective structure.

REPORT 0304-9:   Dually Degenerate Varieties and the Generalized Griffiths-Harris Theorem

Maks A. Akivis and Vladislav V. Goldberg

The dual variety X* for a smooth variety X of the projective space P^N is a set of the tangent hyperplanes to X. If dim X = n, then in the general case, the variety X is a hypersurface in the dual space P^N*. If dim X* < N - 1, then the variety X is called dually degenerate. The authors refine these definitions for a variety X of the projective space P^N with a degenerate Gauss map of rank r. For such a variety, in the general case, the dimension of its dual variety X* is N - l - 1, where l = n - r, and X is dually degenerate if X* < N - l - 1. In 1979 Griffiths and Harris proved that a smooth variety X of the projective space P^N is dually degenerate if and only if all its second fundamental forms are degenerate. The authors generalize this theorem for a variety X of the projective space P^N with a degenerate Gauss map of rank r.

REPORT 0304-10:   On the Structure of Varieties with Degenerate Gauss Mappings

M. A. Akivis, V. V. Goldberg and J. M. Landsberg

We exhibit examples of projective varieties with degenerate Gauss mappings and determine numerical invariants of such varieties. Our examples provide counter-examples to an asserted structure theorem of Griffiths and Harris (Ann. Sci. ENS 1979).

REPORT 0304-11:   Linearizability of d-Webs, d > 3, on Two-Dimensional Manifolds

Maks A. Akivis, Vladislav V. Goldberg and Valentin V. Lychagin

We find d - 2 relative differential invariants for a d-web, d > 3, on a two-dimensional manifold and prove that their vanishing is necessary and sufficient for a d-web to be linearizable. If one writes the above invariants in terms of web functions f (x, y) and g_4 (x, y), ... , g_d (x, y), then necessary and sufficient conditions of linearizabilty of a d-web are two PDEs of the fourth order with respect to f and g_4, and d - 4 PDEs of the second order with respect to f and g_4,...,g_d. For d = 4, this result confirms Blaschke's conjecture on the nature of conditions of linearizabilty of a 4-web. We also give Mathematica codes for testing 4- and 5-webs for linearizability and examples of their usage.

REPORT 0304-12:   Analysis of Stress and Pressure in the Human Alveolar Wall Before Bursting

H. R. Chaudhry, B. Bukiet, S. Kirshblum

A strain energy function for the human alveolar wall in vivo is developed based on its length-tension properties. Using large deformation theory, this function is employed to determine the relationship between circumferential stress and stretch ratio as well as between alveolar pressure and stretch ratio. We find that both the circumferential stress and the pressure in the alveolar wall rise very rapidly if the wall is stretched even slightly more than that which occurs at the pressures attained during mechanical in-exsufflation (MI-E). MI-E involves producing a high inspiratory air pressure and then quickly switching to a large negative air pressure in order to remove mucus from the airways.

REPORT 0304-13:   Measures of Postural Stability

H. R. Chaudhry, T. Findley, K. S. Quigley, B. Buckiet, Z. Ji, T. Sims, M. Maney

Dynamic posturography has become an important tool for understanding standing balance in clinical settings. A key test in the NeuroCom dynamic posturography system, the Sensory Organization Test (SOT) provides information about the integration of multiple components of balance. The SOT test leads to an outcome measure called the equilibrium score (ES), which reflects the overall coordination of the visual, proprioceptive, and vestibular systems for maintaining standing posture. Researchers and physicians often use the ES from the SOT as a clinically relevant measure of standing balance. We discuss here the formula used for evaluating the ES and propose an additional measure of postural stability, called the Postural Stability Index (PSI) that takes into account additional biomechanical aspects of postural stability during quiet standing. We propose that this new measure provides a clinically important adjunct to the current SOT.

REPORT 0304-14:   Computational Method to Evaluate Ankle Postural Stiffness with Ground Reaction Forces

Zhiming Ji, Thomas Findley, H. Chaudhry, and B. Bukiet

We examine an existing method for evaluating postural sway based on force plate technology. Through an improved mathematical model of postural dynamics, we propose a new method which better evaluates postural sway, and in addition, computes ankle moment and ankle postural stiffness directly from the measured ground reaction force. An example is detailed that demonstrates the utility of this approach. The proposed method does not involve filtering or numerical integration, and takes into account the platform inclination. Results from normal subjects show a linear relation between the ankle moment and the sway angle during quiet standing.

REPORT 0304-15:   Mathematical Modeling of Air Flow in the Branches of the Lungs

B. Bukiet, H. Chaudhry, and S. Kirshblum

Understanding the air flow in the lungs is important for evaluating treatments for patients with weak respiratory systems. These patients experience build-up of mucus in the lungs which can lead to infection and hospitalization. We develop a simple method for modeling the air flow in the branches of the human lung. We take ideas from incompressible, viscous, Poiseuille flow in cylindrically symmetric tubes and implement them in the context of a quasi-steady model. Numerical computations have been performed using physiological data and varying the number of levels of branching in the model. These computations show that when a high input pressure is given and held constant, increasing the number of levels of branching causes the pressure to approach its limiting value in the alveoli more slowly. This work is one step in a program to evaluate the effectiveness and safety of treatments to clear secretions from the lungs of patients with weak respiratory systems.

REPORT 0304-16:   Two-oscillator model of ventilatory rhythmogenesis in the frog

Amitabha Bose, Timothy Lewis, and Richard Wilson

Frogs produce two distinct yet highly coordinated ventilatory behaviors, buccal and lung. Lung ventilation occurs in short episodes, interspersed with periods of buccal ventilation. Recent data suggests that two brainstem oscillators are involved in generating these behaviours, one primarily responsible for buccal ventilation, the other for lung. Here we use a modeling approach to demonstrate that the episodic pattern of lung ventilation might be an emergent property of the coupling between the oscillators, and may not require a perturbing input from another, as yet unidentified but previously postulated, neuronal oscillator.

REPORT 0304-17:   The Effect of Modulatory Neuronal Input on Gastric Mill Frequency

Christina Ambrosio, Amitabha Bose, and Farzan Nadim

We show how fast inhibition from the pyloric network interacts with a slow modulatory input to control the frequency of the gastric mill rhythm. We deduce that the timing of the pyloric input is crucial in determining what affect it will have on the frequency of the gastric network. Over one set of timings, the modulatory input and the pyloric input work together to determine the frequency and over another set of timings, the affect of the pyloric input is mitigated by the modulatory input.

REPORT 0304-18:   Strong NLS Soliton-Defect Interactions

Roy Goodman, Philip Holmes and Michael Weinstein

We consider the interaction of a nonlinear Schroedinger soliton with a spatially localized (point) defect in the medium through which it travels. Using numerical simulations, we find parameter regimes under which the soliton may be reflected, transmitted, or captured by the defect. We propose a mechanism of resonant energy transfer to a nonlinear standing wave mode supported by the defect. Extending Forinash et. al., we then derive a finite-dimensional model for the interaction of the soliton with the defect via a collective coordinates method. The resulting system is a three degree-of-freedom Hamiltonian with an additional conserved quantity. We study this system both numerically and using the tools of dynamical systems theory, and find that it exhibits a variety of interesting behaviors, largely determined by the structures of stable and unstable manifolds of special classes of periodic orbits. We use this geometrical understanding to interpret the simulations of the finite-dimensional model, compare them with the nonlinear Schroedinger simulations, and

REPORT 0304-19:   Interaction of sine-Gordon kinks with defects: The two-bounce resonance

Roy H. Goodman and Richard Haberman

A model of soliton-defect interactions in the sine-Gordon equations is studied using singular perturbation theory. Melnikov theory is used to derive a critical velocity for strong interactions, which is shown to be exponentially small for weak defects. Matched asymptotic expansions for nearly heteroclinic orbits are constructed for the initial value problem, which are then used to derive analytical formulas for the locations of the well known two- and three-bounce resonance windows, as well as several other phenomena seen in numerical simulations.

REPORT 0304-20:   Localized activity patterns in excitatory neuronal networks

Jonathan Rubin and Amitabha Bose

The existence of localized activity patterns, or bumps, has been investigated in a variety of spatially distributed neuronal network models that contain both excitatory and inhibitory coupling between cells. Here we show that a neuronal network with purely excitatory synaptic coupling can exhibit localized activity. Bump formation ensues from an initial transient synchrony of a localized group of cells, followed by the emergence of desynchronized activity within the group. Transient synchrony is shown to promote recruitment of cells into the bump, while desynchrony is shown to be good for curtailing recruitment and sustaining oscillations of those cells already within the bump. These arguments are based on the geometric structure of the phase space in which solutions of the model equations evolve. We explain why bump formation and bump size are very sensitive to initial conditions and changes in parameters in this type of purely excitatory network, and we examine how short-term synaptic depression influences the characteristics of bump formation.

REPORT 0304-21:   Dynamic interaction of oscillatory neurons coupled with reciprocally inhibitory depressing synapses acts to stabilize the rhythm period

In the rhythmically active pyloric circuit of the spiny lobster, the pyloric dilator (PD) neurons are members of the pacemaker group of neurons that make inhibitory synapses on to the follower lateral pyloric (LP) neuron. The LP neuron, in turn, makes a depressing inhibitory synapse to the PD neurons, providing the sole inhibitory feedback from the pyloric network to its pacemakers. This study investigates in biologically realistic conditions, the dynamic interaction between the pyloric cycle period, the two types of neurons and the feedback synapse in the reciprocally inhibitory loop. When the rhythm period was changed, the membrane potential waveform of the LP neuron was affected with a consistent pattern. These changes in the LP neuron waveform directly affected the dynamics of the LP to PD synapse and caused the postsynaptic potential (PSP) in the PD neurons to both peak earlier in phase and become larger in amplitude. Using an artificial synapse implemented in dynamic clamp, we show that when the LP to PD PSP occurs early in phase, it acts to speed up the pyloric rhythm and larger PSPs further strengthen this trend. Together, these results indicate that interactions between these two types of neurons can dynamically change in response to increases in the rhythm period, and this dynamic change provides a negative feedback to the pacemaker group that could work to stabilize the rhythm period.

REPORT 0304-22:   On undercompressive shocks in constrained 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. The lower fluid is denser than the upper fluid which is relevant to air-water systems. Flow is driven by an imposed pressure gradient and gravity. 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 includes the physical effects of advection, capillarity, inertia and hydrostatic pressure. Our interest is to compare the dynamics of the solutions under different flow constraints. In particular, we compare the dynamics when the liquid volumetric flow rate and the downstream pressure drop are held fixed to the case when the gas volumetric flow rate and the interfacial height at ends of the channel are held fixed.

REPORT 0304-23:   On nontrivial traveling waves in thin film flows including contact lines

Lou Kondic and Javier Diez

We discuss dynamics of thin liquid films spreading down an inclined plane. The fronts of these films are known to be unstable with respect to formation of finger-like and triangular patterns. In this work, we concentrate on one particular aspect of these flows, and that is the existence of nonlinear traveling waves. We find evidence for presence of these waves for all inclination angles less than $90^\circ$. To understand better the relevant pattern formation mechanism, we explore via numerical simulations the bifurcation structure of the stability diagram close to the critical wavenumber. The recovered structure is consistent with the existence of a subcritical bifurcation. We discuss the connection between the bifurcation diagram and the existence of nontrivial traveling waves.

REPORT 0304-24:   Unstable spreading of a fluid filament on a vertical plane: Experiments and simulations

J. Diez, A. G. Gonzalez, J. Gomba, R. Gratton, and L. Kondic

We present results of experiments and numerical simulations on the spreading of a constant volume of fluid flowing down a vertical plate in the form of a micrometric thin film. In the experiments, the initial condition is generated from a horizontal fluid filament (PDMS) of typical diameter $\approx 0.4~mm$, and the flow is probed with two optical techniques: one based on an anamorphic system, and the other on the schlieren method. The first one yields the thickness profile and the second one captures the bidimensional pattern of the transversal film instability. The numerical simulations are performed under the lubrication approximation and using a precursor film to overcome the contact line divergence. The comparison between numerical and experimental profiles shows a very good agreement, and therefore allows to estimate the thickness of the precursor film needed to model the flow. We find that this thin precursor (of thickness $43~nm$) is very demanding for the numerical description of the instability, since it requires the use of a very fine grid. We show that the use of thicker precursor allows to obtain numerical results which describe qualitatively well the experimental data. In order to study the early times of the instability, we develop a linear model to account for the evolution of the modal amplitudes of the spatial Fourier spectrum of the contact line. The model is in good agreement with both experiments and simulations for the appropriate precursor film thickness in each case. We find that the precursor film thickness mainly influences the growth rates of the unstable modes, but it does not modify the main features of instability development.

REPORT 0304-25:    Facilitation through Buffer Saturation: Constraints on Endogenous Buffering Properties

Victor Matveev, Robert S. Zucker and Arthur Sherman

Synaptic facilitation (SF) is a ubiquitous form of short-term plasticity, regulating synaptic dynamics on fast time scales. Although SF is known to depend on the presynaptic accumulation of Ca2+, its precise mechanism is still under debate. Recently it has been shown that at certain central synapses SF results at least in part from the progressive saturation of an endogenous Ca2+ buffer (Blatow et al., 2003), as proposed by Klingauf and Neher (1997). Using computational modeling, we study the magnitude of SF that can be achieved by a buffer saturation mechanism (BSM), and explore its dependence on the endogenous buffering properties. We find that a high SF magnitude can be obtained either by a global saturation of a highly
mobile buffer in the entire presynaptic terminal, or a local saturation of a completely immobilized buffer. A characteristic feature of BSM in both cases is that SF magnitude depends non-monotonically on the buffer concentration. In agreement with results of Blatow et al. (2003), we find that SF grows with increasing distance from the Ca2+ channel cluster, and increases with increasing external Ca2+, [Ca2+]ext, for small levels of [Ca2+]ext. We compare our modeling results with the experimental
properties of SF at the crayfish neuromuscular junction (NMJ), and find that the saturation of an endogenous mobile buffer can explain the observed SF magnitude and its supralinear accumulation time course. However, we show that the BSM predicts slowing of the SF decay rate in the presence of exogenous Ca2+ buffers, contrary to experimental observations at the crayfish NMJ. Further modeling and data are required to resolve this aspect of the BSM.

REPORT 0304-26:   The activity phase of postsynaptic neurons in a simplified rhythmic network

A. Bose, Y. Manor and F. Nadim

Many inhibitory rhythmic networks produce activity in a range of frequencies. The relative phase of activity between neurons in these networks is often a determinant of the network output. This relative phase is determined by the interaction between synaptic inputs to the neurons and their intrinsic properties. We show, in a simplified network consisting of an oscillator inhibiting a follower neuron, how the interaction between synaptic depression and a transient potassium current in the follower neuron determines the activity phase of this neuron.  We derive a mathematical expression to determine at what phase of the oscillation the follower neuron becomes active . This expression can be used to understand which parameters determine the phase of activity of the follower as the frequency of the oscillator is changed. We show that in the presence of synaptic depression, there can be three distinct frequency intervals, in which the phase of the follower neuron is determined by different sets of parameters. Alternatively, when the synapse is not depressing, only one set of parameters determines the phase of activity at all frequencies.

REPORT 0304-27:   An egalitarian network model for the emergence of simple and complex cells in visual cortex

Louis Tao, Michael Shelley, David McLaughlin and Robert Shapley

We explain how Simple and Complex cells arise in a large-scale neuronal network model of the primary visual cortex of the macaque. Our model consists of over 16,000 integrate-and-fire, conductance-based point neurons, representing the cells in a small, 1 mm squared patch of an input layer of the primary visual cortex.  In the model the local connections are isotropic and nonspecific, and convergent input from the lateral geniculate nucleus confers cortical cells with orientation and spatial phase preference.  The balance between lateral connections and LGN drive determines whether individual neurons in this recurrent circuit are Simple or Complex.  The model reproduces qualitatively the experimentally observed distributions of both extracellular and intracellular measures of Simple and Complex response.

REPORT 0304-28:   A New Kinetic Representation of Fluctuation-Driven Neuronal Networks with Application to Simple & Complex Cells in Primary Visual Cortex

David Cai, Louis Tao, Michael Shelley & David McLaughlin

A new coarse-grained representation of the dynamics of neuronal networks is introduced and developed in terms of kinetic equations, which, via a novel moments \emph{closure}, are derived mathematically, directly from the original large-scale integrate-and-fire (I&F) network. This powerful kinetic theory can capture the full dynamic range of neuronal networks --- from the mean-driven limit (a limit as the number of neurons N goes to infinity, in which the fluctuations vanish) to the fluctuation-dominated limit (such as in small N networks). Comparison with full numerical simulations of the original I&F network establishes that the reduced dynamics is very accurate and numerically efficient over all dynamic ranges. Both analytical insights and numerical scale-up of neuronal computation can be achieved via this kinetic approach. Here the theory is illustrated by a study of the dynamical properties of networks of various architectures, including excitatory and inhibitory neurons of both simple and complex type, which exhibit rich phenomena, e.g., transition to bistability and hysteresis, even in the presence of large fluctuations. The implication for possible connections between the structure of the bifurcations and the behavior of complex cells is discussed. Finally, I&F networks and kinetic theory are used to discuss orientation selectivity of complex cells for "ring model" architectures which characterize changes in the response of neurons located from near to far from orientation pinwheel centers.

REPORT 0304-29:   Bistability and Tunable Gating in a Recurrent Neuronal Network with Synaptic Failure and Hidden Neurons

Louis Tao, David Cai and Michael Shelley

We study networks of all-to-all coupled, integrate-and-fire, excitatory neurons, with a portion of the population receiving feedforward drive and the remaining, hidden'', portion receiving no feedforward input. We demonstrate that this system undergoes a subcritical bifurcation as either the input or recurrent coupling is varied.  Because of this transition, the network gates feedforward inputs and exhibits bistability and hysteresis.  The hidden population allows this network response to be continuously tunable in the forcing and coupling strengths. Furthermore, these features persist in the presence of synaptic failure and for small network sizes, suggesting that the computations discussed here could be implemented in biological networks. Finally, we demonstrate that a long network correlation, orders of magnitude longer than the synaptic time, emerges from the recurrent coupling. This correlation time scales with network size and with synaptic connection probability.

REPORT 0304-30:   Solvability and the number of solutions of Hammerstein equations

P.S. Milojevic

We study the solvability and the number of solutions to Hammerstein operator equations in Banach spaces using a projection like method and degree theory for corresponding vector fields.  The linear part is assumed to be either selfadjoint or non-selfadjoint. We  present also applications to Hammerstein integral equations.

REPORT 0304-31:    Inter-Circuit Control via Rhythmic Regulation of Projection Neuron Activity

Debra E. Wood, Yair Manor, Farzan Nadim and Michael P. Nusbaum1

Synaptic feedback from rhythmically active neuronal circuits commonly causes their descending inputs to exhibit the rhythmic activity pattern generated by that circuit.  In most cases, however, the function of this rhythmic feedback is unknown.  In fact, generally these inputs can still activate the target circuit when driven in a tonic activity pattern.  We are using the crab stomatogastric nervous system (STNS) to test the hypothesis that the neuronal circuit-mediated rhythmic activity pattern in projection neurons contributes to inter-circuit regulation.  The crab STNS contains an identified projection neuron, modulatory commissural neuron 1 (MCN1), whose tonic stimulation activates and modulates the gastric mill (chewing) and pyloric (filtering of chewed food) motor circuits in the stomatogastric ganglion (STG).  Moreover, during tonic stimulation of MCN1, the pyloric circuit regulates both gastric mill cycle frequency and gastro/pyloric coordination via a synapse onto a gastric mill neuron in the STG.  However, when MCN1 is spontaneously active it has a pyloric-timed activity pattern, due to a synaptic input from the pyloric circuit.  This pyloric-timed activity in MCN1 provides the pyloric circuit with a second pathway for regulating the gastric mill rhythm.  At these times, the STG synapse from the pyloric circuit to the gastric mill circuit is not necessary for pyloric regulation of the gastric mill rhythm.  However, in the intact system, these two pathways play complementary roles in this inter-circuit regulation. Thus, one role for rhythmicity in modulatory projection neurons is to enable them to mediate the interactions between distinct but related neuronal circuits.

REPORT 0304-32:    Diagnostic method for cancer of the digestive tract based on the analysis of blood mechanoemmission curves

V.E. Orel, D.A.Klyushin, A.V.Romanov, Yu.I.Petunin, and R.I.Andrushkiw

Mechanoemission (ME) is the emission of electrons, ions, electromagnetic and acoustics waves, caused by mechanical activation of a substance. Autocorrelation and phase map analysis of blood mechanoemission curves show an increase in chaotic blood ME in patients with gastric cancer and inflammatory gastric mucosa, as compared with healthy individuals. This work  is devoted to the development of a diagnostic method for cancer of the digestive tract based on the analysis of patient’s blood mechanoemission curves, using mathematical/statistical theory of pattern recognition.

REPORT 0304-33:   On the existence of trapped waves involving a two-layer fluid

A. Chakrabarti and D.S. Ahluwalia

Selection of the appropriate positive root of a quadratic equation, in conjunction with consistency requirements involving Fourier Transforms is shown to lead to the correct perturbation method to be employed and to establish the existence of only one set of trapped mode frequencies, instead of two, in the case of two layers of fluid and a submerged cylinder of arbitrary cross section, in the lower layer.

REPORT 0304-34:   Large Eddy Simulation of Rotating Turbulent Convection Using the Estimation Subgrid Scale Model

S. Kimmel and F.P. Deek

Numerical simulations of turbulent convection under the influence of rotation will help to study mixing in oceanic flows. In this study, a large eddy simulation (LES) with the Smagorinsky subgrid scale model is used to compute the time evolution of a rotating convection flow generated by a buoyancy source of finite size at a relatively high Rayleigh number. The computed velocity and temperatures are in better agreement with a direct numerical simulation (DNS) than LES simulations with constant eddy viscosity. These results also demonstrate that the qualitative behavior of vorticies which form under the source depend on the aspect ratio of the flow. For source diameters that are small compared to the size of the domain, the vortices propagate away from the source. On the other hand, if the ratio of source diameter to domain size is relatively large, the vortices remain under the source. Evidence from other studies suggests that a rim current around the edge of the source develops for intermediate values of this aspect ratio. Though the results are qualitatively similar to a direct numerical simulation (DNS) and other LES, in this simulation the flow remains laminar much longer than the DNS predicts. This particular flow is complicated by the turbulence transition between the convective plume and the quiescent ambient fluid, and an eddy viscosity model is inadequate to accurately model this type of flow. In addition, the Smagorinsky model is not consistent in a noninertial reference frame. A more accurate simulation of rotating convection requires an alternate subgrid scale model. In particular, the estimation model has demonstrated better results for other types of rotating flows and is the recommended subgrid scale model for future work.

REPORT 0304-35:   Comparison of random Sums in some Integral Orderings and Applications

M.C. Bhattacharjee

Some new order preservation properties of stopped sums of independent nonnegative random variables, when the stopping variable is independent of the summands, is investigated. We show that such randomly stopped sums preserve the stochastic Laplace as well as the integral harmonic mean residual life orders. For the case of Laplace orders, there is a suitable converse for each of the order preservation results. Exponential distributions are characterized within the class of random sums with geometric stopping times, via simple moment conditions on the summand obeying a suitably weak aging hypothesis.

REPORT 0304-36:    Adaptive Economic Choices under Recurrent Disasters : a Bayesian Perspective

M.C. Bhattacharjee

We consider an economic choice problem in a stochastic regime of repeated natural disasters. In a Bayesian formulation of the problem, the role of inverted beta distributions, which appear as distributions of disaster times under a suitable family of conjugate priors, is illustrated. Return of the optimal Bayesian policy and its asymptotic behaviour is explored.

REPORT 0304-37:   Reflection and Transmission from a thin inhomogeneous cylinder in a rectangular TE10 Waveguide

M. Booty and G.A. Kriegsmann

We study the scattering problem for a thin cylindrical target that is placed with arbitrary orientation in a rectangular TE10 waveguide and subjected to an imposed electromagnetic field.  The scattered far-field is expressed in terms of the scattered field inside the target and the far-field expansion of the dyadic Green's function for the waveguide.  In order to capture features of interest in microwave heating applications, we allow the target material's electrical properties to be arbitrary functions of position along the thin cylindrical target's axis.   Reflection and transmission coefficients for such a target, and an expression for the rate of deposition of electromagnetic energy within it are derived.

REPORT 0304-38:   Spatial Buffering Mechanism: Mathematical Model and Computer Simulations

Ben Steinberg, Yuqing Wang, Huaxiong Huang, and Robert M. Miura

It is generally accepted that the spatial buffering mechanism is important to buffer extracellular-space potassium in the brain-cell microenvironment. In the past, this phenomenon, generally associated with glial cells, has been treated analytically and numerically using a simplified one-dimensional description. The present study extends the previous research by using a novel numerical scheme for the analysis of potassium buffering mechanisms in the extracellular brain-cell microenvironment. In particular, a lattice-cellular automaton was employed to simulate a detailed two-compartment model of a two-dimensional brain-cell system. With this numerical approach, the present study elaborates upon previous theoretical work on spatial buffering (SB) by incorporating a more realistic structure of the brain cell microenvironment, which was not feasible earlier. We use the experimental paradigm consisting of iontophoretic injection of KCl to study the SB mechanism. Our simulations confirmed the results reported in the literature obtained by an averaged model. The results also show that the additional effects captured by a simplified two-dimensional geometry do not alter significantly the conclusions obtained from the averaged model. The details of applying such a numerical method to the study of ion movements in cellular environments, as well as its potential for future study, are discussed.

REPORT 0304-39:   Nonlinear Toys

Morikazu Toda and Robert M. Miura

(no abstract)

REPORT 0304-40:   On velocity profiles, stresses and Bagnold scaling of sheared granular system in zero gravity

O. Baran and L. Kondic

We report the results of three-dimensional molecular dynamics simulations of sheared granular system in Couette geometry.\cite{movies} The simulations use realistic boundary conditions that may be expected in physical experiments. For a range of boundary properties we report velocity and density profiles, as well as forces on the boundaries. In particular, we find that the results for the velocity profiles throughout the shearing cell depend strongly on the interaction of the system particles with the physical boundaries.  Even frictional boundaries can allow for significant slippage of the particles, therefore, reducing the shear in the system. Next, we present shear rate dependence of stress, including mean force and force fluctuations, both for controlled volume, and for controlled stress configurations. We discuss the dependence of solid volume fraction on shear rate under the constant pressure condition and relate it to Bagnold scaling in volume controlled experiments. In addition, we study the influence of oscillatory driving on the system properties.

REPORT 0304-41:   Global existence, singular solutions and ill-posedness for the Muskat problem

Michael Siegel, Russel E. Caflisch,  and Sam Howison

The Muskat, or Muskat--Leibenzon, problem describes the evolution of the interface between two immiscible fluids in a porous medium or Hele-Shaw cell under applied pressure gradients or fluid injection/extraction.  In contrast to the Hele-Shaw problem (the one-phase version of the Muskat problem), there are few nontrivial exact solutions or analytic results for the Muskat problem.  For the stable, forward Muskat problem, in which the higher viscosity fluid expands into the lower viscosity fluid, we show global in time existence for initial data that is a small perturbation of a flat interface. The initial data in this result may contain weak (e.g., curvature) singularities. For the unstable, backward problem, in which the higher viscosity fluid contracts, we construct singular solutions that start off with smooth initial data, but develop a point of infinite curvature at finite time.

REPORT 0304-42:   Exact solutions for the evolution of a bubble in Stokes flow: a Cauchy transform approach

Darren Crowdy and Michael Siegel

A Cauchy transform approach to the problem of determining the free surface evolution of a single bubble in Stokes flow is developed. A number of exact solutions to a class of problems have been derived in the literature using conformal mapping theory, and these solutions are retrieved and further generalized using the new formulation. Certain quantities, which are conserved by the dynamics are also identified, the existence of which had not previously been pointed out. A principal purpose of this paper is to use the new formulation to understand when it is possible to externally specify the evolution of the bubble area in such classes of exact solution. It is found to be possible only for certain types of far-field boundary conditions.

REPORT 0304-43:   The evolution of slender non-axisymmetric drop in an extensional flow

P. D. Howell and M. Siegel

An asymptotic method for analysing slender non-axisymmetric bubbles and jets in a general straining flow is developed. The method relies on the slenderness of the geometry to reduce the three-dimensional equations to a sequence of weakly coupled, quasi-two-dimensional Stokes flow problems for the cross-sectional evolution.  Exact solution techniques for the flow outside a bubble in two-dimensional Stokes flow are generalised to solve for the transverse flow field, allowing large non-axisymmetric deformations of the bubble to be described.  A generalisation to the case where the interior of the bubble contains a slightly viscous fluid is also presented.

REPORT 0304-44:   Persistence of memory in drop breakup: The breakdown of universality

Pankaj Doshi, Itai Cohen, Wendy W. Zhang, Michael Siegel, Peter Howell, Osman A. Basaran, and Sidney R. Nagel

A low-viscosity drop breaking apart inside a viscous fluid is encountered whenever air bubbles, entrained in thick syrup or honey, rise and break apart. Experiment, simulation and theory show that the breakup in the situation where the interior viscosity can be neglected produces an exceptional form of singularity. In contrast to previous studies of drop breakup, universality is violated so that the final shape at breakup retains an imprint of the initial and boundary conditions. A finite interior viscosity, no matter how small, cuts off this form of singularity and produces an unexpectedly long and slender thread. If exterior viscosity is large enough, however, the cut-off does not occur because the minimum drop radius reaches sub-atomic dimensions first.

REPORT 0304-45:   Evolution of material voids for highly anisotropic surface energy

M. Siegel, M.J. Miksis, and P.W. Voorhees

In this paper we consider the evolution by surface diffusion of material voids in a linearly elastic solid, focusing on the evolution of voids with large surface energy anisotropy. It is well known that models for the time evolution of similar material surfaces can become mathematically ill-posed when the surface energy is highly anisotropic. In some cases, this ill-posedness has been  associated with the formation of corners along the interface. Here the ill-posedness is removed through a regularization which incorporates higher order terms in the surface energy. Spectrally  accurate numerical simulations are performed to calculate the steady-state solution branches and time-dependent evolution of voids, with a particular emphasis on inferring trends in the zero  regularization limit. For steady voids with large anisotropy we find that apparent corners form as the regularization tends to zero. In the presence of elastic stresses  the limiting corner angles are most often found to differ from angles found on the  Wulff  shape. For large elastic stresses we find that steady solutions no longer exist; instead the void steadily lengthens via a filamenting instability referred to as tip streaming.