REPORT 1314-1: A
Masking Index for Quantifying Hidden Glitches
Laure Berti-Equille - IRD ESPACE
DEV, Montepellier, France
Ji Meng Loh - New Jersey Institute of Technology
Tamraparni Dasu - AT&T Labs Research
Abstract: Data
glitches are errors in a data set; they are complex entities that often span
multiple attributes and records. When they co-occur in data, the presence of
one type of glitch can hinder the detection of another type of glitch. This
phenomenon is called masking. In this paper, we define two important types of
masking, and we propose a novel, statistically rigorous indicator called
masking index for quantifying the hidden glitches in four cases of masking:
outliers masked by missing values, outliers masked by duplicates, duplicates
masked by missing values, and duplicates masked by outliers. The masking index
is critical for data quality profiling and data exploration; it enables a user
to measure the extent of masking and hence the confidence in the data. In this
sense, it is a valuable data quality index for measuring the true cleanliness
of the data. It is also an objective and quantitative basis for choosing an
anomaly detection method that is best suited for the glitches that are present
in any given data set. We demonstrate the utility and effectiveness of the
masking index by intensive experiments on synthetic and real-world datasets.
REPORT 1314-2: New Strange Attractors for Discrete Dynamical Systems
Yogesh
Joshi - Department of Mathematics and Computer Science, Kingsborough Community
College, Brooklyn, NY 11235-2398 (email: yogesh.joshi@kbcc.cuny.edu)
Denis Blackmore - Department of
Mathematical Sciences and Center for Applied Mathematics and Statistics, New
Jersey Institute of Technology, Newark, NJ 07102-1982 (email:
deblac@m.njit.edu)
Abstract: A discrete dynamical system in
Euclidean m-space generated by the iterates of an asymptotically zero map f,
which goes to 0 as x goes to infinity, must have a compact global attracting
set A. The question of what additional hypotheses are sufficient to guarantee
that A has a minimal (invariant) subset A* that is a chaotic strange attractor
is answered in detail for a few types of asymptotically zero maps. These special cases happen to have many
applications (especially as mathematical models for a variety of processes in
ecological and population dynamics), some of which are presented as examples
and analyzed in considerable detail.
REPORT 1314-3: Frequency Preference in Two-dimensional Neural
Models: A Linear Analysis of the
Interaction between Resonant and Amplifying Currents
Horacio G. Rotstein
Department of Mathematical Sciences and
Center for Applied Mathematics and Statistics, New Jersey Institute of
Technology, Newark, NJ 07102-1982
Farzan Nadim
Department of Biological Sciences, NJIT and Department of
Mathematical Sciences, NJIT
Abstract: Many neuron types exhibit preferred frequency
responses in their voltage amplitude (resonance) or phase shift to subthreshold oscillatory currents, but the effect of
biophysical parameters on these properties is not well understood. We propose a
general framework to analyze the role of different ionic currents and their
interactions in shaping the properties of impedance amplitude and phase in
linearized biophysical models and demonstrate this approach in a
two-dimensional linear model with two effective conductances
gL and g1. We
compute the key attributes of impedance and phase (resonance frequency and
amplitude, zero-phase frequency, selectivity, etc.) in the gL-g1
parameter space. Using these attribute diagrams we identify two basic
mechanisms for the generation of resonance: an increase in the resonance
amplitude as g1 increases while the overall impedance is
decreased, and an increase in the maximal impedance, without any change in the
input resistance, as the ionic current time constant increases. We use the
attribute diagrams to analyze resonance and phase of the linearization of two biophysical
models that include resonant (Ih or
slow potassium) and amplifying currents (persistent sodium). In the absence of
amplifying currents, the two models behave similarly as the conductances
of the resonant currents is increased whereas, with the amplifying current present,
the two models have qualitatively opposite responses. This work provides a
general method for decoding the effect of biophysical parameters on linear
membrane resonance and phase by tracking trajectories, parametrized
by the relevant biophysical parameter, in pre-constructed attribute diagrams.
REPORT 1314-4: Abrupt and Gradual Transitions between Low and Hyperexcited Firing Frequencies in Neuronal Models with
Fast Synaptic Excitation: A Comparative Study
Horacio G. Rotstein
Department of Mathematical Sciences and
Center for Applied Mathematics and Statistics, New Jersey Institute of
Technology, Newark, NJ 07102-1982
Abstract:
Hyperexcitability of neuronal networks is one
of the hallmarks of epileptic brain seizure generation, and results from a net
imbalance between excitation and inhibition that promotes excessive abnormal
firing frequencies. The transition
between low and high firing frequencies as the levels of recurrent AMPA excitation change can occur either gradually or
abruptly. We used modeling, numerical
simulations and dynamical systems tools to investigate the biophysical and
dynamic
mechanisms that underlie these two identified modes of transition in
recurrently connected neurons via AMPA excitation. We compare our results and demonstrate that
these two modes of transition are qualitatively different and can be linked to
different intrinsic properties of the participating neurons.
REPORT 1314-5: Preferred
Frequency Responses to Oscillatory Inputs in an Electrochemical Cell Model:
Linear Amplitude and Phase Resonance
Horacio G. Rotstein
Department of Mathematical Sciences and
Center for Applied Mathematics and Statistics, New Jersey Institute of
Technology, Newark, NJ 07102-1982
Abstract:
We investigate the dynamic mechanisms of generation of amplitude and
phase resonance in a phenomenological electrochemical cell model in response to
sinusoidal inputs. We describe how the attributes of the impedance and
phase profiles change as the participating physicochemical parameters vary
within a range corresponding to the existence of stable nodes and foci in the
corresponding autonomous system, thus extending previous work that considered
systems close to limit cycle regimes. The method we use permits to understand
how changes in these parameters generate amplifications of the cell's response
at the resonant frequency band and captures some important nonlinear effects.
REPORT 1314-6: Neurons and
Neural Networks: Computational Models
Horacio G. Rotstein
Department of Mathematical Sciences and
Center for Applied Mathematics and Statistics, New Jersey Institute of
Technology, Newark, NJ 07102-1982
Abstract:
Neural networks produce electrical activity that is generated by the
biophysical properties of the constituent neurons and synapses. Individual
neurons produce electrical signals through processes that are highly nonlinear
and communicate these signals to one another through synaptic interactions,
resulting in emergent network outputs. The output of neural networks underlies
behaviors in all higher animals. 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. In this chapter, we briefly review the
current mathematical and computational techniques involved in modeling neurons
and neural networks.
REPORT 1314-7: Subthreshold Amplitude and Phase Resonance in Single Cells
Horacio
G. Rotstein
Department of Mathematical Sciences and
Center for Applied Mathematics and Statistics, New Jersey Institute of
Technology, Newark, NJ 07102-1982
Abstract: In this paper we review the linear properties
of the voltage response of neuronal models to oscillatory current inputs.
REPORT 1314-8:
Mixed-mode Oscillations in Single Neurons
Horacio
G. Rotstein
Department of Mathematical Sciences and
Center for Applied Mathematics and Statistics, New Jersey Institute of
Technology, Newark, NJ 07102-1982
Abstract: In this paper we review the dynamic
mechanisms of generation of mixed-mode oscillations (patterns consisting of subthreshold oscillations interspersed with spikes) in
biophysical (conductance-based) neuronal models.
REPORT 1314-9: Multistability Arising from Synaptic Dynamics
Amitabha Bose: Department
of Mathematical Sciences, NJIT
Farzan Nadim: Department of Biological Sciences, NJIT Department of
Mathematical Sciences, NJIT
Abstract: The strength of a synapse imparted by a presynaptic neuron onto its postsynaptic target can change as a function of the activity of the presynaptic neuron. This change is referred to as short-term synaptic plasticity. Networks of neurons connected with plastic synapses have the potential ability to display multiple stable solutions either at different parameter values or for the same set of parameters. This latter property is known as multistability. Self-consistency between the network frequency and the level of plasticity is needed to ensure multistability. In this paper, we show different ways in which a network uses short-term synaptic plasticity to create multiple stable solutions.
REPORT 1314-10: Effects of Synaptic Plasticity on Phase and Period Locking of a Network of Two Oscillatory Neurons
Zeynep Ackay: Department of Mathematical Sciences, NJIT
Amitabha Bose: Department of Mathematical Sciences,
NJIT
Farzan Nadim: Department of Biological Sciences, NJIT Department of
Mathematical Sciences, NJIT
Abstract: We study
the effects of synaptic plasticity on the determination of firing period and
relative phases in a network of two oscillatory neurons coupled with reciprocal
inhibition. We combine the phase response curves of the neurons with the
short-term synaptic plasticity properties of the synapses to define Poincaré maps for the activity of an oscillatory network.
Fixed points of these maps correspond to the phase locked modes of the network.
These maps allow us to analyze the
dependence of the resulting network activity on the properties of network
components. Using a combination of analysis and simulations, we show how
various parameters of the model affect the existence and stability of
phase-locked solutions. We find conditions on the synaptic plasticity profiles
and the phase response curves of the neurons for the network to be able to
maintain a constant firing period, while varying the phase of locking between
the neurons or vice versa. A generalization to cobwebbing for two dimensional
maps is also discussed.
REPORT 1314-11: Reduced Dynamical Models for 1D Tapping of Particle Columns
(1) Denis Blackmore, Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102-1982 (email: deblac@m.njit.edu)
(2) Anthony Rosato, Department of Mechanical & Industrial Engineering, New Jersey Institute of Technology, Newark, NJ 07102-1982 (email: anthony.rosato@njit.edu)
(3) Xavier Tricoche, Computer Science Department, Purdue University, West Lafayette, IN 47907-2107 (email: xmt@purdue.edu)
(4) Kevin Urban, Department of Physics, New Jersey Institute of Technology, Newark, NJ 07102-1982 (email: kdu2@njit.edu)
(5) Luo Zou, Department of Mechanical & Industrial Engineering, New Jersey Institute of Technology, Newark, NJ 07102-1982 (email: lz39@njit.edu )
Abstract: A lower-dimensional center-of-mass dynamical model is devised as a simplified means of approximately predicting some important aspects of the motion of a vertical column comprised of a large number of particles subjected to gravity and periodic vertical tapping. This model is investigated first as a continuous dynamical system using analytical, simulation and visualization techniques. Then, by employing an approach analogous to that used for a bouncing ball on an oscillating flat plate dynamics, it is modeled as a discrete dynamical system and analyzed to determine transitions to chaotic motion and other properties. An alternative procedure for obtaining a similar discrete dynamical systems model is also briefly described. The predictions of the analysis are then compared with the visualization and simulation results of the reduced continuous model, and ultimately with simulations of the complete system dynamics.
REPORT 1314-12: Numerical Simulation of Drop and Bubble Dynamics
with Soluble Surfactant
Qiming Wang1, Michael Siegel2, and
Michael R. Booty2
1 Department of Mathematics
and Statistics, York University, Toronto, Ontario M3J 1P3, Canada and
2 Department
of Mathematical Sciences and Center for Applied Mathematics and Statistics, New
Jersey Institute of Technology, University Heights, Newark, NJ 07102, USA
Abstract: Numerical
computations are presented to study the effect of soluble surfactant on the
deformation and breakup of an axisymmetric drop or bubble stretched by an
imposed linear strain flow in a viscous fluid.
At the high values of bulk Peclet number Pe in typical
fluid-surfactant systems, there is a thin transition layer near the interface
in which the surfactant concentration varies rapidly. The large surfactant gradients are resolved
using a fast and accurate `hybrid' numerical method that incorporates a
separate, singular perturbation analysis of the dynamics in the transition
layer into a full numerical solution of the free boundary problem. The method is used to investigate the
dependence of drop deformation on parameters that characterize surfactant
solubility. We also compute resolved
examples of tipstreaming, and investigate its
dependence on parameters such as flow rate and bulk surfactant concentration.
REPORT 1314-13: Network Symmetry and Binocular Rivalry Experiments
Casey O. Diekman* and Martin Golubitsky**
* Department of Mathematical Sciences, New Jersey Institute
of Technology, Newark, NJ 07102
** Mathematical Biosciences Institute, The
Ohio State University, Columbus, OH 43210
Abstract: Hugh Wilson
has proposed treating higher-level decision making as a competition between
patterns, where patterns are coded in the brain as levels of a set of
attributes in an appropriately defined network. In this paper, we propose that
symmetry-breaking Hopf bifurcation from fusion states
in suitably modified Wilson networks, which we call rivalry networks, can be
used in an algorithmic way to explain the surprising percepts that have been
observed in a number of binocular rivalry experiments. These rivalry networks
modify and extend Wilson networks by permitting different kinds of attributes
and different types of coupling. We apply this algorithm to psychophysics
experiments discussed by Kovacs et al., Shevell and
Hong, and Suzuki and Grabowecky. We also analyze an
experiment with four colored dots (a simplified version of a 24-dot experiment
performed by Kovacs, and a three-dot analogue of the four-dot experiment. Our
algorithm predicts surprising differences between the three- and four-dot
experiments.
REPORT 1314-14: Generalized Linear Model under the Extended Negative Multinomial Model and Cancer Incidence
Sunil Kumar Dhar
Soumi Lahiri
Center for Applied Mathematics and Statistics, Department of
Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102
Abstract: The generalized linear model for a multi-way contingency table
for several independent populations that follow the extended negative
multinomial distributions is introduced. This model represents an ex-
tension of negative multinomial log-linear model. The parameters of
the new model are estimated by the quasi-likelihood method and the
corresponding score function, which gives a close form estimate of the
regression parameters. The goodness-of-fit test for the model is also
discussed. An application of the log-linear model under the generalized
inverse sampling scheme representing cancer incidence data is given as
an example to demonstrate the effectiveness of the model.
REPORT 1314-15: Self-trapping and Josephson tunneling solutions to the nonlinear Schrödinger / Gross-Pitaevskii Equation
R. H. Goodman, NJIT
J. L. Marzuola, University of North Carolina
M. I. Weinstein, Columbia University
Abstract: We study the long-time behavior of solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation (NLS/GP) with a symmetric double-well potential, continuing work of Marzuola and Weinstein 2010. NLS/GP governs nearly-monochromatic guided optical beams in weakly coupled waveguides with both linear and nonlinear (Kerr) refractive indices and zero absorption. The optical power (L^2 norm) is conserved with propagation distance. At low optical power, the beam energy executes beating oscillations between the two waveguides. There is an optical power threshold above which the set of guided mode solutions splits into two families of solutions. One type of solution corresponds to an optical beam which is concentrated in either waveguide, but not both. Solutions in the second family undergo tunneling oscillations between the two waveguides. NLS/GP can also model the behavior of Bose-Einstein condensates. A finite dimensional reduction (system of ODEs) well-approximates the PDE dynamics on long time scales. In particular, we derive this reduction, find a class of exact solutions and prove the very long-time shadowing of these solutions by applying the approach of Marzuola and Weinstein.
Rianka Bhattacharya and Sundarraman Subramanian
Center for
Applied Mathematics and Statistics
Department of Mathematical Sciences
New Jersey Institute of Technology
USA
Abstract:
a model
for the conditional probability that an observation is uncensored given the
observed minimum. The extension to the two-sample setting assumes the
availability of good fitting models for the groupspecific conditional
probabilities. When the models are correctly specified for each group, the new
location and scale estimators are shown to be asymptotically as or more
efficient than the estimators obtained using the Kaplan–Meier based quantiles.
Individual and joint confidence intervals for the parameters are developed.
Simulation studies show that the proposed method produces confidence intervals
that have correct empirical coverage and that are more informative. The proposed
method is illustrated using two
real data sets.
Shoubhik Mondal, Sundar Subramanian
Center for Applied Mathematics and Statistics
Department of Mathematical Sciences
New Jersey Institute of Technology
USA
Abstract:
In the first part, entitled ``Model assisted Cox regression" and
published in Journal of Multivariate Analysis (JMVA), it was shown
that standard Cox regression, combined with Dikta's semiparametric
random censorship models, provides an effective framework for
obtaining improved parameter estimates. Here, this methodology is
exploited to construct simultaneous confidence bands (SCBs) for
subject-specific survival curves. Simulation results are presented to
compare the performance of the proposed SCBs with the SCBs that are
based only on standard Cox. The new SCBs provide correct empirical
coverage and are more informative. The proposed SCBs are illustrated
with two real examples. An extension to handle missing censoring
indicators is also outlined.
REPORT 1314-18:
Glen Atlas
Rutgers New Jersey Medical School
Dept. of Anesthesiology
Newark, New Jersey, USA
atlasgm@njms.rutgers.edu
and
Stevens Institute of Technology
Dept. of Chemistry, Chemical Biology, and Biomedical
Engineering
Hoboken, New Jersey, USA
Sunil Dhar
New Jersey Institute of Technology
Dept. of Mathematical Sciences
Newark, New Jersey, USA
Abstract: The aortic pressure flow relationship is typically described using traditional integer calculus. This paper uses fractional calculus to relate the velocity of aortic blood flow to aortic pressure. The basis for this research is a Taylor series model of the velocity of aortic blood flow with subsequent term-by-term fractional integration as well as fractional differentiation. Fractional calculus may be a useful mathematical tool in hemodynamic modelling.
Keywords: fractional calculus, aortic blood flow, esophageal Doppler monitor, differintegral, differintegration
Center for Applied Mathematics and Statistics
Department of Mathematical Sciences
New Jersey Institute of Technology
USA
and
Michael A. Zazanis
Dept. of Statistics, Athens University of Economics and
Business
Athens 10434, Greece
Abstract
Casey O.
Diekman
Department of
Mathematical Sciences
New Jersey
Institute of Technology
Newark, NJ
07102
Kohinoor
Dasgupta and Vijay Nair
Department of
Statistics, University of Michigan, Ann Arbor, MI 48109
and
K.P.
Unnikrishnan
Center for
Biomedical Research Informatics
NorthShore
University Health System
Evanston, IL
60201
Abstract:
Department of
Mathematical Sciences
New Jersey
Institute of Technology
Newark, NJ
07102
Abstract: The research of Horacio G. Rotstein focuses on the dynamics of neuronal networks and other chemical and biological systems. His long-term goal is to understand how coherent patterns of activity emerge in oscillatory networks, how these networks process information, what are their computational properties, and how all this depends on the intrinsic properties of the nodes (or cells) and the network topology. He primarily focuses on rhythmic oscillations on networks of neurons in the nervous system over a wide spectrum of interacting levels of organization, ranging from the subcellular, through the cellular, to the network level. The specific goal is to understand the biophysical and dynamic mechanisms of generation of these oscillations and their functional role in cognition and motor behavior in both health and disease. Additional areas of interest are oscillatory chemical and biochemical reactions. We use mathematical modeling, numerical simulations and develop dynamical systems tools. The research group have ongoing collaborations with experimental labs both 'in vivo' and 'in vitro' and other theoretical scientistists.