Center for Applied Mathematics and Statistics

REPORT 0001-1: The role of synaptic delay in organizing the behavior of networks of self-inhibiting neurons

S. Kunec & A. Bose

We consider a pair of mutually coupled inhibitory neurons in which each neuron is also self-inhibitory. We show that the size of the synaptic delay determines the existence and stability of solutions. For small delays, there is no synchronous solution, but a stable anti-phase and a stable on-state solution.  For long delays, only the synchronous solution is stable. For intermediate delays, either the anti-phase or synchronous solutions are stable. In contrast to prior work, for stability of synchrony, we only require the existence of a single slow process.

REPORT 0001-2: The small sample robust minimum distance estimators of shift in the two-sample location models

S. Dhar & D. Chen

A class of minimum distance estimator of the shift parameter is given. Some of these estimators have sample-wise breakdown point of 0.5. For symmetric distributions, these estimators are shown to be asymptotically normally distributed. Comparative computational study in terms of estimated mean squared errors and biases reveals that some of these minimum distance estimators exhibit small sample size robustness properties. They are good competitors of Hodges and Lehmann estimators, the Fines estimator and other efficient and estimators of the shift parameter in the asymmetric case.

REPORT 0001-3: A classification and examples of four-dimensional nonisoclinic three-webs

V. V. Goldberg

A classification and examples of 4-dimensional nonisoclinic 3-webs of codimension 2 are presented. The examples considered prove the existence for many classes of webs for which the general existence theorems are not proved yet.

REPORT 0001-4: Local equivalence of Sacksteder and Bourgain hypersurfaces

M. A. Akivis & V. V. Goldberg

Finding examples of submanifolds with degenerate Gauss maps in an Euclidean space R^4 that are  noncylindrical and without singularities is an important problem of differential geometry. The first example of such a hypersurface was constructed by Sacksteder in 1960. In 1995 Wu published an example of a noncylindrical tangentially degenerate algebraic hypersurface in R^4 whose Gauss map is of rank 2 and which is also without singularities. This example was constructed by Bourgain. In the current paper, the authors analyze Bourgain's example, prove that singular points of the Bourgain hypersurface are located in the hyperplane at infinity of the space R^4, and that the hypersurfaces of Sacksteder and Bourgain are locally equivalent.

REPORT 0001-5: On the structure of submanifolds with degenerate Gauss maps

M. A. Akivis & V. V. Goldberg

An n-dimensional submanifold X of a projective space P^N (C) is called  tangentially degenerate if the rank of its Gauss map \gamma: X ---> G (n, N) satisfies 0 < rank \gamma < n. The authors systematically study the geometry of tangentially degenerate submanifolds of a projective space P^N (C). By means of the focal images, three  basic types of submanifolds are discovered: cones, tangentially degenerate  hypersurfaces, and torsal submanifolds. Moreover, for  tangentially degenerate submanifolds, a structural theorem is proven. By this theorem, tangentially degenerate  submanifolds that do not belong to one of the basic types are foliated into submanifolds of basic types. In the proof the authors introduce irreducible, reducible, and completely reducible  tangentially degenerate  submanifolds. It is found  that cones and tangentially degenerate  hypersurfaces are irreducible, and torsal  submanifolds  are completely reducible while all other tangentially degenerate  submanifolds not belonging to basic types are reducible.

REPORT 0001-6: Goursat's (n+1)-webs

V. V. Goldberg

We consider the Goursat's (n+1)-webs of codimension one of two kinds on an n-dimensional manifold. They are characterized by the specific  closed form equations or by two special relations between components of the torsion tensor of the web. These relations allow us to establish a connection with solutions of two systems of nonlinear second-order PDEs investigated by Goursat in 1899. The integrability conditions of some distributions invariantly associated  with both kinds of Goursat's (n+1)-webs are also investigated.

REPORT 0001-7: Multidimensional (n+1)-webs with reducible subwebs

V. V. Goldberg

We find three characterizations for a multidimensional (n+1)-web W possessing a reduct reducible subweb: its closed form equations, the integrability of an invariant distribution associated with W, and the relations between the components of its torsion tensor. In the case of codimension one, the latter criterion establishes a relation with solutions of a system of nonlinear second-order PDEs. Some particular cases of this system were considered by Goursat in 1899.

REPORT 0001-8: Bistable oscillations arising from synaptic depression

A. Bose, Y. Manor & Farzan Nadim

Synaptic depression is a common form of short-term plasticity in the central and peripheral nervous systems. We show that in a network of two reciprocally connected neurons a single depressing synapse can produce two distinct oscillatory regimes. These distinct periodic behaviors can be studied by varying the maximal conductance, $\gbarinh$, of the depressing synapse. For small $\gbarinh$, the network has a short period cell-dominated solution. For large $\gbarinh$, the solution has much longer period and is synapse-dominated. We show that in an intermediate range of $\gbarinh$ values both stable periodic solutions exist simultaneously. Thus, the network can switch oscillatory modes either by changing $\gbarinh$, or, for fixed $\gbarinh$, by changing initial conditions.

REPORT 0001-9: Neural mechanisms for generating rate and temporal codes in model CA3 pyramidal cells

V. Booth & A. Bose

The effect of synaptic inhibition on burst firing of a two-compartment model of a  CA3 pyramidal cell is considered. We show that depending on its timing, a short dose of fast decaying synaptic inhibition can either delay or advance the timing of firing of subsequent bursts.  Moreover, increasing the  strength of the inhibitory input is shown to modulate the burst profile from a full complex burst, to a burst with multiple spikes,  to single spikes. We additionally show how slowly decaying inhibitory input can be used to synchronize a network of pyramidal cells. Implications for the phase precession phenomenon of hippocampal place cells  and for the generation of temporal and rate codes are discussed.

REPORT 0001-10: A variational method for eigenvalue problems nonlinearly dependent on the spectral parameter

R. I. Andrushkiw & V. V. Slastikov

Eigenvalue problems nonlinear in the spectral parameter,  involving operator pencils with unbounded symmetrizable operators, are investigated in a suitable Hilbert space.  A variational method for approximating the eigenvalues  of the above problems is developed, which generalizes and extends some of the results previously obtained for  eigenvalue problems with quadratic or  polynomial operator pencils.  The theory is illustrated with  a numerical example.

REPORT 0001-11: Dynamical properties of discrete Lotka-Voltera equations

D. Blackmore, J. Chen, J. Perez & M. Savescu

Discrete versions of the Lotka-Volterra equations have been found to provide good predictive models for competing
populations. The dynamics  of a general discrete, two-dimensional Lotka-Volterra dynamical system is  analyzed in detail. It is found that for certain  parameter ranges the system exhibits period-doubling bifurcation and also  chaos in the form of "twisted horseshoe" dynamics.

REPORT 0001-12: Regulating firing rate of networks of pyramidal cells

V. Booth & A. Bose

In a minimal model of pyramidal cells and interneurons, we show how excitation and inhibition cooperate to produce firing rate changes in pyramidal cells that are consistent with observed place cell firing in region CA3 of the hippocampus. We show that inhibition from a common interneuron can synchronize networks of pyramidal cells with no direct connections. Moreover, recurrent excitation together with common inhibition can modulate burst profiles of synchronously firing cells from complex bursts to bursts with multiple spikes to single  spikes.

REPORT 0001-13: Combining incomplete information from independent assessment surveys for estimating masonry deterioration

W. B. Fairley, A. Izemann & S. M. Crunk

Construction materials used in building structures, such as masonry, wood and reinforced concrete, deteriorate over time because of many factors including poor design, defective materials or manufacture, and poor workmanship. This article is concerned with estimates of masonry deterioration and the effects of covariates on the damage to bricks on the walls of five multiple-story buildings of a residential complex located in the Bronx, New York. In this case study, the damage of primary interest was a "spall", a physical separation of a portion of the brick face from the body of the brick. Eventually, the face becomes so damaged that portions fall off. The result is an unattractive appearance and a hazard to passersby. In this study, spall damage was assessed by means of three different and independent condition assessment surveys; an expensive, precise and and hence very limited scaffold drop survey and two additional inexpensive, but more detailed photographic and visual surveys. The photographic survey was obtained by photographing the walls of the entire residential complex, and the visual survey was done by individuals walking around the periphery of each building and making a visual assessment of the damage to each wall. In the photographic survey, a large amount of incomplete data was unavoidable because of either poor photo angles of various physical obstructions. A binomial regression model using four categorical explanatory variables or factors was fitted to the observed photographic spall data. Sparseness of the data, the presence of outliers, and overdispersion were major problems encountered in selecting and fitting a suitable model. A small pilot survey, in which the relevant portions of the photographic and visual surveys were matched to 11 drop locations of the scaffold survey, recorded spall counts using each survey method. From this pilot survey, photographic and visual spall data were calibrated to the scaffold drop survey data. It was determined that of the two surveys, only the photographic spall survey was needed to predict scaffold spalls. The estimate of total damage from the photographic survey was then adjusted using the calibration results. Finally, a multiple imputation procedure was used to impute values for the missing data and obtain an estimate (and its standard error) of the true spall rate over the entire residential complex. Source of uncertainty to factfinders in legal trials are discussed and illustrated through the present case.

REPORT 0001-14: Synchrony and frequency regulation by synaptic delay in networks of self-inhibiting neurons

A. Bose & S. Kunec

We show that a pair of mutually coupled self-inhibitory neurons can display stable synchronous oscillations provided only that the delay to the onset of inhibition is sufficiently long.  The frequency of these oscillations is determined either entirely by the length of the synaptic delay, or by the synaptic delay and intrinsic time constants. We also show how cells can exhibit transient synchronous oscillations where the length of the transients is determined by the synaptic delay, but where the frequency is largely independent of the delay.

REPORT 0001-15: Control of network output by synaptic depression

F. Nadim, Y. Manor & A. Bose

We show how synaptic depression in small networks can be used to switch between high and low frequency oscillations. The period of solutions is controlled by intrinsic cell parameters (high frequency case) or synaptic parameters (low frequency case). Using simulations, we compute the bifurcation diagram which shows how the distinct solutions arise as a function of the maximal conductance of the depressing synapse. We also show that there is a large region of bistability where both solutions exist.

REPORT 0001-16: Neurons and neural networks: computational models

F. Nadim & Y. Manor

This is a review of this subject for the Encyclopedia of Life Sciences.

REPORT 0001-17: Frequency regulation demonstrated by coupling a model and a biological neuron

Y. Manor & F. Nadim

Using a 2-cell reciprocally inhibitory network consisting of a biological pacemaker neuron and a model neuron integrated in real time, we show that if both synapses are depressing there is a wide range of bistability in the system. The two stable outputs are both oscillatory, but one is controlled by the intrinsic properties of the biological pacemaker neuron, whereas the other is controlled by the strength of the depressing synapses.

REPORT 0001-18: Electromagnetic propagation in periodic translationally invariant media

G. A. Kriegsmann

A variational technique is employed to compute approximate propagation constants for electromagnetic waves in a periodic, translationally invariant, dielectric media.  The fundamental cell, in the periodic structure, is composed of a pore and the surrounding host media.  The pore is a circle of radius $R_0$ filled with a dielectric $\epsilon_1$ and the host dielectric characterized by $\epsilon_2$.  The size of the cell is characterized by the length $A$ which is $\sim R_0$. Two limiting cases are considered.  In the first the pore size is assumed to be much smaller than the wavelength; this limit is motivated by microwave heating of porous material.  The approximate propagation constants are explicitly computed for this case and are shown to depend upon the two dielectric constants, the relative areas of the two regions in the cell, and on a modal number.  They are not given by a simple mixture formula. In the second limit the pore size is taken to be of the same order as the wavelength; this limit is motivated by the propagation of light in a holy fiber.  In this case our variational argument directly yields the dispersion relationship recently derived by Ferrando, et. al. [5], using intuitive and physical reasoning.  Thus, our method puts theirs into a mathematical framework from which other approximations might be deduced.

REPORT 0001-19: Dynamics and rupture of planar electrified liquid sheets

B. S. Tilley, P. G. Petropoulos & D. T. Papageorgiou

We investigate the stability of a thin two-dimensional liquid film when a uniform electric field is applied in a direction parallel to the initially flat bounding fluid interfaces. We consider the distinct physical effects of surface tension and electrically induced forces for an inviscid, incompressible and electrically nonconducting fluid. The film is assumed to be thin enough and the surface forces large enough that gravity can be ignored to leading order. Our aim is to analyze the nonlinear stability of the flow. We achieve this by deriving a set of nonlinear evolution equations for the local film thickness and local horizontal velocity. The equations are valid for waves which are long compared to the average film thickness and for symmetrical interfacial perturbations. The electric field effects enter non-locally and the resulting system contains terms which are reminiscent of the Kortweg de-Vries as well as the Benjamin-Ono equations. Traveling wave solutions are calculated and their stability is studied using time dependent simulations. Of particular interest is the phenomenon of film rupture. We present extensive simulations that show that the presence of the electric field causes a nonlinear stabilization of the flow in that it delays singularity (rupture) formation. Our results also indicate that terminal singularity structures are surface tension dominated irrespective of the electric field.

REPORT 0001-20: Pattern formation in a gravity driven flow of thin films: Constant flux flow

L. Kondic & J. Diez

We present fully nonlinear time-dependent simulations of a thin liquid film flowing down an inclined plane.  Within the lubrication approximation, and assuming complete wetting, we find that varying the inclination angle considerably modifies the shape of the emerging patterns: finger-shaped patterns result for the flow down a vertical plane, while saw-tooth patterns develop for the flows down an inclined plane.  However, in all of our simulations, the roots always move, indicating that the shape of the patterns is not necessarily related to the surface coverage, a technologically important feature of the flow.  Furthermore, we find that triangular steady state patterns may be produced for the flow down an incline, while the fingers typically grow in length for all explored times.  We find quantitative agreement with reported experiments, and suggest new ones.

REPORT 0001-21: Computer-aided  pattern recognition method for cytogenetic diagnosis of breast cancer and fibroadenomatosis

Yu.I.Petunin, D.A.Klyushin, R.I.Andrushkiw, K.P.Ganina & N.V.Boroday

A computer-aided pattern recognition method for the diagnosis of breast cancer (CMG) and fibroadenometosis (FAM) is developed, based on the analysis of the  DNA content of the nuclei of cells obtained from  patients buccal scrapes. Based on a single analysis,  the probability of error in the diagnosis of CMG and the probability of non-acceptance of decision do not exceed 6%.  For FAM the probability of error in the diagnosis  is practically zero, and the probability of non-acceptance of decision is 43%.  If the decision is not accepted, repeated analysis is performed taking more trials (buccal scrapes). If the results of the analysis are similar after n independent trials, then the probability of non-acceptance of decision is approximately (equation). The method may be applied in screening  women for early detection of breast cancer and identification of  those  who are in a high risk category.

REPORT 0001-22: Membrane Equilibrium Equations

D. Stickler

Description not available.

REPORT 0001-23: Theory of phase separation kinetics in polymer-liquid crystal systems

C. B. Muratov & W. E

We introduce a kinetic model describing the phase separation in the mixture of long rod-like molecules and long chain-like molecules. The model uses the angular distribution function for the orientations of the rods as a dynamical variable. The energetics is based on the non-local Onsager theory for the rods combined with a non-local extension of the Flory-Huggins theory. The kinetics explicitly takes into account the preferential diffusion along the rods. We computed the phase diagrams in this model and found a number of transitions leading to phase separation including a microphase separation transition. We performed numerical simulations of the kinetics of quenches of the system into the unstable states and the resulting morphologies.

REPORT 0001-24: Microwave joining of two long hollow ceramic tubes: a combined asymptotic and numerical analysis

G. A. Kriegsmann & J. H. C. Luke

Description not available.

REPORT 0001-25: Microwave joining of two long hollow tubes: an asymptotic theory and numerical simulations

G. A. Kriegsmann & J. H. C. Luke

A nonlinear heat equation which models the microwave assisted joining of two large $SiC$ tubes is analyzed. By exploiting the small fineness ratio of the structure and disparate time scales an asymptotic theory for this problem is systematically deduced. Specifically, a one-dimensional nonlinear heat equation is described which governs the temperature in the outer region. This is a numerically well posed problem and it is efficiently solved using standard methods. This solution is not valid in the inner region which includes the microwave source. An inner asymptotic approximation is derived to describe the temperature in this region. This approximation yields two unknown functions which are determined from matching to the outer solution. The results of the asymptotic theory are compared to calculations done on the full problem. Since the full problem is numerically ill conditioned, the asymptotic theory yields enormous savings in computational time and effort.

REPORT 0001-26: Surface water waves involving a vertical barrier in the presence of an ice-cover

A. Chakrabarti, M. S. Rao & D. S. Ahluwalia

A class of boundary value problems involving propagation of two-dimensional surface water waves, associated with deep water and a plane vertical rigid barrier is nvestigated under the assumption that the surface is covered by a thin sheet of ice. Assuming that the ice-cover behaves like a thin isotropic elastic plate, the problems under consideration lead to those of solving two-dimensional Laplace equation in a quarter-plane, under a Neumann boundary condition on the vertical boundary and a condition involving upto fifth order derivatives of the unknown function on the horizontal ice-covered boundary, along with two appropriate edge-conditions, ensuring the uniqueness of the solutions. Two different methods are employed to solve the mixed boundary value problems completely, by determining the unique solution of a special type of integral equation of the first kind and by exploiting the analyticity property of the Fourier cosine transform in the second method.

REPORT 0001-27: A mathematical analysis of heart reduction surgery

H. R. Chaudry, B. Bukiet, R. Arora, T. Regan & A. B. Ritter

Heart reduction surgery in patients with dilated cardiomyopathy has been reported to have beneficial effects on overall cardiac function. The usual explanation for this phenomenon is based on Laplace's law. This explanation is oversimplified. The purpose of this paper is to demonstrate how to analyze mathematically heart reduction surgery based on rigorous biological solid mechanics. We employ a large deformation theory for a transversely isotropic cylindrical shell model of the canine heart with fibers in the myocardium and take into account residual stresses. In particular, a model is developed that may assist in estimating the change in ejection fraction for a specified inner radius as a result of surgery. We observe that residual stresses are responsible for reducing the transmural circumferential stress and stress gradients, at the end diastolic state, which enable all the layers of the myocardium to function in a more collective and uniform pattern. We find that computed stress depends strongly on ejection fraction (if material properties do not vary) - the lower the ejection fraction, the lower the stress. We also observe that in the normal heart compressive stresses outweigh the active tension at the end systolic state but for the dilated and reduced (low ejection fraction) heart the active tension makes the overall stress tensile in nature.

REPORT 0001-28: Testing a hypothesis in developmental biology: modeling and computational analysis of autocrine loops in Drosophila oogenesis

S. Y. Shvartsman, C. B. Muratov & D. A. Lauffenburger

Spatially distributed cell signaling networks establish chemical blueprints guiding morphogenesis in a developing organism. Autocrine signaling through the Epidermal Growth Factor Receptor (EGFR) is highly conserved across species and operates at various stages of development, specifying patterning events in tissue and organogenesis. A recent hypothesis suggested that a distributed network of positive and negative EGFR autocrine feedback loops in Drosophila oogenesis may be capable of spatially modulating a simple single-peaked input into a more complex two-peaked signaling pattern, specifying the formation of a pair organ. Here we integrate genetic and biochemical information about the EGFR/Rhomboid/Spitz/Argos network into a mechanistic model of patterning events in the formation of a pair of respiratory appendages in the Drosophila oogenesis. Using large-scale dynamical systems analysis to evaluate the model with spatially distributed control loops, we conclude that the proposed network is, indeed, a robust module that first amplifies a signaling pattern and then splits it in two. We argue that versatility in signaling mediated by autocrine loops can be systematically explored using experiment-based mechanistic models and their computational analysis.

REPORT 0001-29: A role for depressing synapses in frequency regulation

Y. Manor & F. Nadim

Synaptic depression is a common dynamical property observed in central synapses. We show that synaptic depression can play an important role in frequency regulation of rhythmic networks, such as central pattern generators. Experiments were performed by coupling biological pacemaker neuron to a model neuron using artificial inhibitory depressing synapses. This hybrid biological-model network exhibited two modes of oscillation, one where the oscillation frequency was determined solely by the intrinsic properties of the biological pacemaker and the other where the frequency was largely affected by the dynamics of the depressing synapses. These two modes of oscillation were often present in the same parameter range to produce bistability. The activity of the network could be switched from one mode to the other by engaging a regenerative process that increased, or decreased, the strength of both synapses. In the bistable regime, this process could be triggered, for example, by a brief current pulse injected in either cell. This mechanism suggests that, in rhythmically active systems, network reconfiguration can occur by allowing depressed synapses to recover from or decay to their depressed state.

REPORT 0001-30: Solution of Belousov's problem

M. A. Akivis & V. V. Goldberg

The authors prove that a local n-quasigroup defined by the equation

x_{n+1} = F (x_1, ... , x_n) = [f_1 (x_1) + ... + f_n (x_n)]/[x_1 + ... + x_n]

where f_i (x_i), i, j = 1, ..., n, are arbitrary functions, is irreducible if and only if any two functions f_i (x_i) and f_j (x_j), i \neq j, are not both linear homogeneous, or these functions are linear homogeneous but f_i (x_i)/x_i \neq f_j (x_j)/x_j.

REPORT 0001-31: On the exponentially self-regulating population model

J. Chen & D. Blackmore

Description not available.

REPORT 0001-32: Approximating the surface impedance of a homogeneous lossy half-space: an example of "dialable" accuracy

P. G. Petropoulos

We present an approximation by exponentials of the time-domain surface impedance of a lossy half-space. Gauss-Chebyshev quadrature of order N-1 is employed to approximate an integral representation of the modified Bessel functions comprising the time-domain impedance kernel. An explicit error estimate is obtained in terms of the physical parameters, the computation time, and the number of quadrature points N. We show our approximation is as accurate as other approaches which do not come with such an error estimate. The conditions under which the error estimate derived herein also applies to the approximation in [5] are investigated.

REPORT 0001-33: Inversion in an uncertain environment using linearization and ray path arrivals

X. Ma & Z-H. Michalopoulou

This work inversigates the potential of using disting ray path arrivals in order to extract information on source location, bathymetry, and sound speed. The method, not requiring full field matching, takes advantage of sharp peaks in received time series resulting from the transmission of relatively high-frequenc broadband pulses. Identifying the peaks (arrivals) and matching them to the unknown parameters through a linear system can lead to an efficient and accurate estimation scheme.

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