**
REPORT 0708-1: **
**Dynamics of a closed rod with twist and bend in fluid**

**Sookkyung Lim**, Mathematical Biosciences
Institute, Ohio State University

**Anca Ferent**, Courant Institute of Mathematical
Sciences, New York University

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

**Charles S.Peskin**, Courant Institute of Mathematical
Sciences, New York University

Abstract:

We investigate the instability and subsequent dynamics of a closed rod with twist and bend in a viscous, incompressible fluid. A new version of the immersed boundary (IB) method is used in which the immersed boundary applies torque as well as force to the surrounding fluid and in which the equations of motion of the immersed boundary involve the local angular velocity as well as the local velocity of the fluid. An important feature of the IB method in this context is that self-crossing of the rod is automatically avoided because the rod moves in a continuous (interpolated) velocity field. A rod with a uniformly distributed twist that has been slightly perturbed away from its circular equilibrium configuration is used as an initial condition, with the fluid initially at rest. If the twist in the rod is sufficiently small, the rod simply returns to its circular equilibrium configuration, but for larger twists that equilibrium configuration becomes unstable, and the rod undergoes large excursions before relaxing to a stable coiled configuration.

**
REPORT 0708-2: **
**Maintaining phase of the crustacean tri-phasic pyloric rhythm**

**Christina Mouser, **Department of
Mathematical Sciences, Medgar Evers College of CUNY

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

**Amitabha Bose, **
Department of Mathematical Sciences, New Jersey Institute of Technology

Abstract:

We construct and analyze a model network of the pyloric rhythm of the crustacean stomatogastric ganglion consisting of an oscillator neuron that inhibits two reciprocally inhibitory follower neurons. We derive analytic expressions that determine the phase of firing of the follower neurons with respect to the oscillator. An important aspect of the model is the inclusion of synapses that exhibit short-term synaptic depression. We show that these type of synapses allow there to be a complicated relationship between the intrinsic properties of the neurons and the synapses between them in determining phase relationships. Our analysis reveals the circumstances and ranges of cycle periods under which these properties work in concert with or independently from one another. In particular, we show that phase maintenance over a range of oscillator periods can be enhanced through the interplay of the two follower neurons if the synapses between these neurons are depressing. Since our model represents the core of the oscillatory pyloric network, the results of our analysis can be compared to experimental data and used to make predications about the biological network.

**REPORT 0708-3: ** **
A Newly Identified Extrinsic Input Triggers a Distinct Gastric Mill Rhythm via Activation of Modulatory Projection Neurons**

**Dawn M. Blitz, Rachel S. White, Shari R. Saideman, Aaron
Cook, **and **Michael P. Nusbaum, **Department of
Neuroscience, University of Pennsylvania School of Medicine

**Andrew E. Christie**, Department of Biology, University
of Washington, Seattle

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

Abstract:

Neuronal network flexibility enables animals to respond appropriately to changes in their internal and external states. We are using the isolated crab stomatogastric nervous system to determine how extrinsic inputs contribute to network flexibility. The stomatogastric system includes the well-characterized gastric mill (chewing) and pyloric (filtering of chewed food) motor circuits in the stomatogastric ganglion. Projection neurons with somata in the commissural ganglia (CoGs) regulate these rhythms. Previous work characterized a unique gastric mill rhythm that occurred spontaneously in some preparations, but whose origin remained undetermined. This rhythm includes a distinct protractor phase activity pattern, during which all active gastric mill circuit and projection neurons fire in a pyloric rhythm-timed activity pattern instead of the tonic firing pattern exhibited by these neurons during previously studied gastric mill rhythms. Here we identify a new extrinsic input, the post-oesophageal commissure (POC) neurons, whose relatively brief stimulation (30 sec) triggers a long-lasting (tens of minutes) activation of this novel gastric mill rhythm via its lasting activation of CoG projection neurons, including the previously identified MCN1 and CPN2. Immunocytochemical and electrophysiological data suggest that the POC neurons excite MCN1 and CPN2 by release of the neuropeptide Cancer borealis tachykinin-related peptide Ia (CabTRP Ia). These data further suggest that the CoG arborization of the POC neurons comprises the previously identified anterior commissural organ (ACO), a CabTPR Ia-containing neurohemal organ. This endocrine pathway thus appears to also have paracrine actions that include activation of a novel and lasting gastric mill rhythm.

**REPORT 0708-4: ** **
Op****tical Fiber Drawing and Dopant Transport**

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

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

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

Abstract:

Optical fibers are made of glass with different
refractive indices in the (inner) core and the (outer) cladding regions. The
difference in refractive index arises due to a rapid transition in the
concentration of a dopant across the boundary between these two regions. Fibers
are normally drawn from a heated glass preform, and the different dopant
concentrations in the two regions will change due to dopant diffusion and
convective transport induced by the flow. In this paper, we analyze a
mathematical model for the dynamics of dopant concentration changes during the
fiber drawing process. Using a long-wave approximation, we show that the
governing equations can be reduced to a simple diffusion equation. As a result,
we are able to identify key dimensionless parameters that contribute to the
diffusion process.

We
also derive asymptotic solutions for the temperature, cross-sectional area, and
effective diffusion coefficient when there are strong temperature dependencies
in the viscosity and the diffusion coefficient. Our simplified model and
asymptotic solutions reduce the need for extensive numerical simulations and can
be used to devise control strategies to limit excess dopant diffusion.

**REPORT 0708-5: ** **On P****roperties of the Ising Model for Complex Energy/Temperature and Magnetic Field**

**Victor Matveev**, Department of Mathematical
Sciences, New Jersey Institute of Technology, Newark, NJ 17102, USA

**Robert Shrock, **C. N. Yang Institute for
Theoretical Physics, State University of New York, Stony Brook, NY 11794, USA

Abstract:

We study some properties of the Ising model in the plane of the complex (energy/temperature)-dependent variable u = e4K, where K = J=(kBT), for nonzero external magnetic eld, H. Exact results are given for the phase diagram in the u plane for the model in one dimension and on in nite-length quasi-one-dimensional strips. In the case of real h = H=(kBT), these results provide new insights into features of our earlier study of this case. We also consider complex h = H=(kBT) and = e2h. Calculations of complex-u zeros of the partition function on sections of the square lattice are presented. For the case of imaginary h, i.e., = ei, we use exact results for the quasi-1D strips together with these partition function zeros for the model in 2D to infer some properties of the resultant phase diagram in the u plane. We nd that in this case, the phase boundary Bu contains a real line segment extending through part of the physical ferromagnetic interval 0 u 1, with a right-hand endpoint urhe at the temperature for which the Yang-Lee edge singularity occurs at = ei. Conformal field theory arguments are used to relate the singularities at urhe and the Yang-Lee edge.

**REPORT 0708-6: ** **Loss of Phase-Locking in Non-Weakly Coupled Inhibitory Networks with Finite Synaptic Decay Time**

**Myongkeun Oh** and **Victor Matveev**, Department of
Mathematical Sciences and the Center for Applied Mathematics and Statistics, New
Jersey Institute of Technology, Newark, NJ 07102

Abstract:

The study of synchronization of coupled oscillators is important for the understanding of rhythmic activity in networks of excitable cells. Much of existing work on synchronization is centered on the weak coupling theory, which can only predict the existence of phase-locked activity. However, it is known that strong coupling can destabilize phase-locked firing in some networks. Here we show that such loss of phase-locking is a generic property of inhibitory networks of type I cells that are close to their excitation thresholds. We analyze the dynamics of two identical Morris-Lecar model neurons coupled by reciprocal inhibition with non-negligible synaptic decay time, and find that an increase in coupling strength destabilizes phase-locking through a period-doubling cascade, leading to the 2:2 frequency-locked alternating-order (”leap-frog”) spiking, as well as more complex n:n periodic modes and chaotic dynamics. Similar behavior was recently reported by Maran and Canavier (2007) in heterogeneous networks of strongly coupled higher-dimensional type-I excitable cells. We find that leap-frog spiking can be maintained by a completely homogeneous two-cell network, and that it arises when the synaptic input is sufficiently strong to transiently bring the postsynaptic cell past the excitation threshold to a fixed equilibrium, pushing the trajectory off the limit cycle. We give an intuitive geometric description of the observed dynamics, and analyze quantitatively these activity states using the first-order spike-time response curve of each cell. Finally, we show that an inter-spike interval return map based on a simple quadratic spike-time response curve can reproduce the entire coupling-strength bifurcation diagram characterizing the dynamics of two coupled type-I oscillators, which illustrates the universality of the described network behavior.

**
REPORT 0708-7:
**
**Electric Discharge Sintering: A
Mathematical Model**

**G. A. Kriegsmann**, Department of Mathematical Sciences, Center for Applied
Mathematics and Statistics, New Jersey Institute of Technology, University
Heights, Newark, NJ 07102

Abstract:

In this paper we mathematically model the densification of metallic powders and the sintering of ceramic powders by electric discharge. The ordinary and partial differential equations governing these processes are the same with the exception of the effective electrical conductivity. This function is a monotonically decreasing (increasing) function of temperature for the metallic (ceramic) powders. We employ asymptotic methods to approximate the solution to these equations in the limit as ǫ → 0, where ǫ is the ratio of the discharge to diffusion time scales. We find on the shortest time scale that the temperature, voltage, and density satisfy a system of nonlinear, coupled ordinary equation. We solve these and find the relationship between the temperature and density, as functions of the input energy. The results on the short or discharge time scale do not take into account diffusion and heat loss into the surrounding medium. These occur on a much longer time scale which we identify and exploit to deduce a new approximation. On this time scale the capacitor has no more energy to deposit into the powder. The temperature relaxes to that of its surroundings and the density increases to its final value. Our results show the functional relationship between the final density and the initial energy stored in the capacitor, as well as the initial density of the powder.

**REPORT 0708-8: ** **Electromagnetic Wave Propagation in Periodic Porous Structures**

**I. David Abrahams**, School of Mathematics,
University of Manchester, Oxford Road, Manchester M13 9PL, UK

**G. A.
Kriegsmann**, Department of Mathematical Sciences, Center for Applied
Mathematics and Statistics, New Jersey Institute of Technology, University
Heights, Newark, NJ 07102

Abstract:

We employ a homogenization procedure to describe the propagation of electromagnetic waves in a dielectric structure which is doubly-periodic in the X-Y plane and of arbitrary variation in the direction of propagation, Z. The fundamental cell is composed of an arbitrarily shaped pore filled with a dielectric and the host by another. Our analysis yields the structure of the electromagnetic fields at the micro level and gives an effective medium equation at the macro level. The latter contains a simple arithmetic average of the dielectric constants and a correction term which involves a line integral around the pore. The integrand of this integral depends upon the polarization of the wave and the solution to a canonical potential problem. We approximately solve this problem for small pore volumes and for large contrasts, the pore dielectric constant being much larger than the host. We also provide an equivalent variational formulation for the potential problem and use a simple Raleigh-Ritz procedure to determine an approximate solution. For all these approximations, we provide a simple macroscopic description of electromagnetic wave propagation in our structure.

**REPORT 0708-9: **
**Complete transmission
through a periodically perforated rigid slab**

**Lin Zhou**,
Department of Mathematical Sciences**, **
University of Delaware, Newark, Delaware 19711

**G.
A. Kriegsmann**, Department of Mathematical Sciences, Center for Applied
Mathematics and Statistics, New Jersey Institute of Technology, University
Heights, Newark, NJ 07102

Abstract:

The propagation of a normally incident plane acoustic
wave through a three-dimensional rigid slab with periodically placed
holes is modeled and analyzed. The spacing of the holes *A *and *B*,
the wavelength ,
and the thickness of the slab
*L *are order one
parameters compared to the characteristic size *D *of the holes, which is a small
quantity. Scattering matrix techniques are used to derive expressions
for the transmission and reflection coefficients of the lowest mode.
These expressions depend only on the transmission coefficient, 0,
of an infinitely long slab with the same configuration. The
determination of 0 requires the solution of an infinite
set of algebraic equations. These equations are approximately solved by
exploiting the small parameter *D*/*AB*.
Remarkably, this structure is transparent at certain frequencies and
opaque for all others. Such a structure may be useful in constructing
narrow-band filters and resonators.

**REPORT 0708-10: **
**Hazard estimation from right censored
data with missing censoring indicators**

**Sundarraman Subramanian**, Department of Mathematical Sciences, New
Jersey Institute of Technology, Newark, New Jersey

**Derek Bean**, Department of Statistics, University
of California, Berkeley, California

Abstract:

The kernel smoothed Nelson-Aalen estimator has been well investigated, but is unsuitable when some of the censoring indicators are missing. A representation introduced by Dikta, however, facilitates hazard estimation when there are missing censoring indicators. In this article, we investigate (i) a kernel smoothed semiparametric hazard estimator and (ii) a kernel smoothed \pre-smoothed" Nelson Aalen estimator. We derive the asymptotic normality of the proposed estimators and compare their asymptotic variances.

**REPORT 0708-11: **
**Homeomorphisms and Finite Solvability of
Their Perturbations for Fredholm Maps of Index Zero with Applications**

**P. S. Milojevic**, Department of
Mathematical Sciences and CAMS, New Jersey Institute of Technology,
Newark, NJ, USA

Abstract:

We prove a number of homeomorphism results for nonlinear Fredholm maps of index zero and their perturbations. Moreover, we show that k-ball and k-set perturbations of these homeomorphisms are again homeomorphisms or that the corresponding equations are finitely solvable. Various generalized first Fredholm theorems are given and finite solvability of general ( odd ) Fredholm maps is also studied. We apply these results to finite solvability of quasilinear elliptic equations on RNas well as on bounded domains. The basic tool used are the recent degree theories for nonlinear C1-Fredholm maps of index zero and their perturbations.

**REPORT 0708-12: **
**Proximity measure between samples
with repetition factor greater than one**

**
R.I. Andrushkiw, **Department
Mathematical Sciences and Center for Applied Mathematics and Statistics,
New Jersey Institute of Technology, Newark, N.J., USA

**D.A. Klyushin and Yu.I. Petunin, **
Department of
Cybernetics, Kyiv National Taras Shevchenko University, Kyiv, Ukraine

Abstract:

A new proximity measure between empirical samples, having values k x that may occur in a sample x more than once, is constructed. This proximity measure is based on confidence intervals containing the bulk of population constructed by means of order statistics.

**
REPORT 0708-13: ** **On
Explicit/Implicit and Incompressible/Compressible Issues of Immersed
Boundary/Continuum Methods**

**Xiaodong (Sheldon) Wang**, Department of
Mathematical Sciences, New Jersey Institute of Technology Newark, NJ 07102

Abstract:

In addition to an overview of the immersed boundary/continuum methods and their finite element formulations, explicit vs. implicit and incompressible vs. compressible issues are discussed. The recent finite element formulations retain the same strategies employed in the original immersed boundary method, namely, the independent Lagrangian solid mesh moves on top of a fixed or prescribed background Eulerian fluid mesh. The added features in recent finite element formulations are the generality of the immersed solid which can occupy a finite volume in the fluid and be impermeable, compressible, and highly deformable. Furthermore, a matrix-free Newton-Krylov iterative solution technique also resolves the time step limitation issues related to stiff spring supports from the boundary and the high elasticity moduli of the immersed solid. This implicit iterative approach enables the application of immersed methods to many engineering problems some of which are documented here for illustrative purposes.

**
REPORT 0708-14: **
**Computer-Aided Cytogenetic Method of Breast Cancer Diagnosis**

**
R.I. Andrushkiw, **Department
Mathematical Sciences and Center for Applied Mathematics and Statistics,
New Jersey Institute of Technology, Newark, N.J., USA

**Yu.I. Petunin **and** D.A. Klyushin**,
Department of Cybernetics, Kyiv National Taras ShevchenkoUniversity,
Kyiv, Ukraine

**N.V. Boroday, **R.E.Kavetsky Institute of
Oncology and Radiobiology, National Academy of Sciences of Ukraine, Kyiv,
Ukraine

**I.V. Dosenko**,
Institute of Oncology of the Academy of Medical Sciences, Kyiv, Ukraine

Abstract:

A Computer-aided cytogenetic method for the diagnosis of breast cancer is proposed. The method is based on mathematical/statistical analysis of the indexes of interphase nuclei of buccal epitheliocytes, calculated with respect to their RGBimage after Feulgen staining. Key words: breast cancer, fibroadenomatosis, buccal epithelium, discriminant analysis.

**
REPORT 0708-15: **
**Mathematical and
Computational Analysis of the effect of the A-current in a Follower
Neuron in an Inhibitory Network**

**Yu Zhang, Amitabha Bose, **and** Farzan Nadim**, Department of Mathematical
Sciences, New Jersey Institute of Technology

Abstract:

The transient potassium A-current is present in most neurons and plays an important role in determining the onset of activity. We examine the role of the A-current on the activity time of a follower neuron in a rhythmic feed-forward inhibitory network. Using geometric analysis of dynamical systems, we determine both the transient and steady state behavior of the follower neuron in a number of different parameter regimes. We find that the fate of the follower cell is determined by the time constant associated with the A-current inactivation in some parameter regimes whereas, in other regimes, the time constant of the recovery variable determines this fate and yet, in a third regime, they both do. Our analyses show that, even with a simple 3D model used here, the interaction of the A-current parameters and other intrinsic parameters can lead to a number of distinct periodic or possibly chaotic behaviors. The geometric and analytic tools we used for this simplified model can be generalized to understand the role of the A-current in more complex systems, including the neurons of the crustacean pyloric network which is the biological motivation for this study.

**
REPORT 0708-16: **
**Internet Search Result Probabilities Heaps’ Law and Word
Associativity**

Jonathan C. LanseyandBruce Bukiet,Center for Applied Mathematics and Statistics, Department of Mathematical Sciences, New Jersey Institute of Technology

Abstract:

We study the number of internet search results returned from multi-word queries based on the number of results returned when each word is searched for individually. We derive a model to describe search result values for multi-word queries using the total number of pages indexed by Google and by applying the Zipf power law to the word per page distribution on the internet and Heaps’ law for unique word counts. Based on data from 351 word pairs each with exactly one hit when searched for together, and a Zipf law coefficient determined in other studies, we approximate the Heaps’ law coefficient for the indexed world wide web (about 8 billion pages) to be . Previous studies used under 20,000 pages.

We demonstrate through examples how the model could be used to analyze automatically the relatedness of word pairs assigning each a value we call “Strength of Associativity.” We demonstrate the validity of our method with word triplets and through two experiments conducted eight months apart. We then use our model to compare the index sizes of competing search giants Yahoo and Google.

**
REPORT 0708-17: **
**How do drops evaporate?**

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

Abstract:

We consider evaporation of pure liquid drops on a thermally conductive substrate. Two evaporative models are considered: one that concentrates on the liquid phase in determining evaporative flux, and the other one that centers on the gas/vapor phase. A single governing equation for the evolution of drop thickness, including both models, is developed.

Experiments are used to estimate relevant parameters. We show how the derived governing equation can be used to predict which evaporation model is appropriate under considered experimental conditions.

REPORT 0708-18:Signal propagation through dense granular systems

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

**O. M. Dybenko**, Department of
Mechanical Engineering, New Jersey Institute of Technology, Newark, NJ 07102

**R. P. Behringer**, Department of
Physics and Center for Nonlinear and Complex Systems, Duke University, Durham,
NC 27708

Abstract:

The manner in which signals propagate through dense granular systems in both space and time is not well understood. In order to learn more about this process, we carry out discrete element simulations of the system response to excitations where we control the driving frequency and wavelength independently.

Fourier analysis shows that properties of the signal depend strongly on the spatial and temporal scales introduced by the perturbation. The features of the response provide a test-bed for any continuum theory attempting to predict signal properties.

We illustrate this connection between micro-scale physics and macro-scale behavior by comparing the system response to a simple elastic model with damping.

REPORT 0708-19:Instabilities and Taylor dispersion in isothermal binary thin fluid films

**Z. Borden**, Franklin W. Olin College of Engineering,
Needham, MA

**H. Grandjean**, Ecole Polytechnique, Palaiseau Cedex,
France

**A.E. Hosoi**, Hatsopoulos Microfluids Laboratory,
Massachusetts Institute of Technology, Cambridge, MA

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

**B.S. Tilley**, Franklin W. Olin College of Engineering,
Needham, MA

Abstract:

Experiments with glycerol-water thin films flowing down an inclined plane reveal a localized instability that is primarily three dimensional. These transient structures, referred to as ``dimples'', appear initially as nearly isotropic depressions on the interface. A linear stability analysis of a binary mixture model in which barodiffusive effects dominate over the thermophoresis (i.e. the Soret effect) reveals unstable modes when the components of the mixture have different bulk densities and surface tensions. This instability occurs when Fickian diffusion and Taylor dispersion effects are small, and is driven by Marangoni stresses arising from gradients in concentration of one component, across the depth of the film. Qualitative comparision between the experiments and the linear stability results over a wide range of parameters is presented.

**
REPORT 0708-20:
** **Systems of Coupled Diffusion
Equations with Degenerate Nonlinear Source Terms: Linear Stability and Traveling
Waves**

Jonathan J. WYLIE^{1}, Huaxiong HUANG^{2}, and
Robert M. MIURA^{3}

^{1} Department of Mathematics, City University of
Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong;
hhuang@yorku.ca

^{2} Department of Mathematics and Statistics, York
University, Toronto, Ontario, Canada M3J 1P3; hhuang@yorku.ca

^{3} Department of Mathematical Sciences and Center
for Applied Mathematics and Statistics, New Jersey Institute of Technology,
Newark, New Jersey 07102, USA; miura@njit.edu

Abstract:

Diffusion equations with degenerate nonlinear source terms arise in many different applications, e.g., in the theory of epidemics, in models of cortical spreading depression, and in models of evaporation and condensation in porous media. In this paper, we consider a generalization of these models to a system of n coupled diffusion equations with identical nonlinear source terms. We determine simple conditions that ensure the linear stability of uniform rest states and show that traveling wave trajectories connecting two stable rest states can exist generically only for discrete wave speeds. Furthermore, we show that families of traveling waves with a continuum of wave speeds cannot exist.

**
REPORT 0708-21:
** **Bootstrap Bandwidth for
Estimation in the Missing Censoring Indicator Model **

**Sundar Subramanian**, Department of Mathematical
Sciences New Jersey Institute of Technology Newark, New Jersey

**Derek Bean**, Department of Statistics University of
California Berkeley

Abstract:

In cancer survival studies, death certificate information can be missing, or incidental and fatal occurrences may be indistinguishable for some subjects, leading to missing censoring indicators (MCIs). For the framework of right censored data with MCIs, sub-density function kernel estimators play a significant role for estimating a survival function. Data-driven bandwidths for computing these kernel estimators are proposed. The bandwidths are obtained as minimizers of certain estimates of the mean integrated squared error (MISE). It is shown that the smoothed bootstrap oŽers a motivation for choosing the proposed MISE estimates for minimization. The eącacy of the proposed procedures is investigated through several simulation studies. Three illustrations are provided, using a mice data set, a data set extracted from the SEER database, and the well-known Primary Biliary Cirrhosis data. KEY WORDS: Integrated squared bias, Inverse probability weighted, Least squares cross validation, Missing at random, Pilot bandwidth, Plug-in bandwidth selectors 1This research was supported by a National Institute of Health grant CA 103845. 1

**
REPORT 0708-22:
**
**Semiparametric Models for Left Truncation and Right**
**Censoring with Missing Censoring Indicators**

**Sundar Subramanian**, Department of Mathematical
Sciences New Jersey Institute of Technology Newark, New Jersey

**Dipankar Bandyopadhyay**, Department of Biostatistics,
Bioinformatics and Epidemiology, Medical University of South
Carolina,Charleston, SC 29425

Abstract:

We derive the asymptotic distributions of a semiparametric Dikta-type estimator and an inverse probability weighted type estimator of a survival function for a missing censoring indicator left truncated model and provide a theoretical comparison study.

**
REPORT 0708-23:
** **Predicting the
activity phase of a follower neuron with A-current in an inhibitory network**

**Yu Zhang**, Department of Mathematical Sciences, New Jersey
Institute of Technology, Newark, NJ 07102, yu.zhang@njit.edu

**Amitabha Bose**, Department of Mathematical
Sciences, New Jersey Institute of Technology, Newark, NJ 07102,
bose@njit.edu

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

Department of Biological Sciences, Rutgers University, Newark, NJ 07102, farzan@njit.edu

Abstract:

The transient potassium A-current is present in most neurons and plays an important role in determining the timing of action potentials. We examine the role of the A-current on the activity phase of a follower neuron in a rhythmic feed-forward inhibitory network with a reduced three-variable model and conduct experiments to verify the usefulness of our model. Using geometric analysis of dynamical systems, we explore the factors that determine the onset of activity in a follower neuron following release from inhibition. We first analyze the behavior of the follower neuron in a single cycle and find that the phase plane structure of the model can be used to predict the potential behaviors of the follower neuron following release from inhibition. We show that, depending on the relative scales of the inactivation time constant of the A-current and the time constant of the recovery variable, the follower neuron may or may not reach its active state following inhibition. Our simple model is used to derive a recursive set of equations to predict the contribution of the A-current parameters in determining the activity phase of a follower neuron as a function of the duration and frequency of the inhibitory input it receives. These equations can be used to demonstrate the dependence of activity phase on the period and duty cycle of the periodic inhibition, as seen by comparing the predictions of the model with the activity of the pyloric constrictor PY neurons in the crustacean pyloric network.