MBI Publications

MBI Publications for 2012 (102)

  • J. Cushing, R. Costantino and S. Robertson
    Life stages: interactions and spatial patterns
    Bulletin of Mathematical BiologyVol. 74 (2012) pp. 491-508

    Abstract

  • K. Zhao
    Global dynamics of classical solutions to a model of mixing flow
    Journal of Dynamics and Differential EquationsVol. 12-F Issue 2 (2012)

    Abstract

    We study the long-time dynamics of classical solutions to an initial-boundary


    value problem for modeling equations of a two-component mixture. Time asymptotically,


    it is shown that classical solutions converge exponentially to constant equilibrium states as time goes to infinity for large initial data, due to diffusion and boundary effects
  • T. Li, K. Zhao, R. Pan and R. Pan
    Global dynamics of a chemotaxis model on bounded
    SIAM Journal on Applied Mathematics (2012)

    Abstract

    We prove global existence and qualitative behavior of classical solutions for
    a hyperbolic-parabolic system describing chemotaxis on bounded domains. It is shown
    that classical solutions to the initial-boundary value problem of the one-dimensional
    model exist globally in time for large initial data and the solutions converge to constant
    equilibrium states exponentially in time, which rigorously demonstrates the collapsing of
    cell populations in chemotaxis. Moreover, similar results are established for the multi-
    dimensional model when the initial data are small.
  • K. Zhao
    Long-time dynamics of a coupled Cahn-Hilliard-Boussinesq system
    Communications in Mathematical SciencesVol. 10 (2012)

    Abstract

    We study large-time asymptotic behavior of classical solutions to an initial-boundary

    value problem (IBVP) for a coupled Cahn-Hilliard-Boussinesq system on a bounded domain. Sufficient

    conditions are established under which classical solutions converge exponentially to constant

    states as time goes to infinity due to diffusion and boundary effects.
  • K. Zhao
    Large time behavior of density-dependent incompressible Navier-Stokes equations on bounded domains
    Journal of Mathematical Fluid Mechanics (2012)

    Abstract

    The large-time asymptotic behavior of classical solutions to the density-dependent incompressible NavierĆ¢ā?¬ā??Stokes

    equations driven by an external force on bounded domains in 2-D is studied. It is shown that the velocity field and its

    first-order derivatives converge to zero as time goes to infinity for large initial data and external forces.
  • M. Lai, R. Pan, K. Zhao and .
    Initial boundary value problem for 2D viscous Boussi- nesq equations
    Rational Mechanics and AnalysisVol. 199 (2012) pp. 739-760

    Abstract

    We study the initial boundary value problem of 2D viscous Boussinesq equations

    over a bounded domain with smooth boundary. We show that the equations have

    a unique classical solution for H3 initial data and no-slip boundary condition. In

    addition, we show that the kinetic energy is uniformly bounded in time.
  • T. Li and K. Zhao
    On a quasilinear hyperbolic system in blood flow modeling
    Discrete and Continuous Dynamical SystemsVol. 16 No. Series B (2012) pp. 333-344

    Abstract

    This paper aims at an initial-boundary value problem on bounded

    domains for a one-dimensional quasilinear hyperbolic model of blood

    ow with

    viscous damping. It is shown that, for given smooth initial data close to a

    constant equilibrium state, there exists a unique global smooth solution to the

    model. Time asymptotically, it is shown that the solution converges to the

    constant equilibrium state exponentially fast as time goes to infinity due to

    viscous damping and boundary eff ects.
  • K. Zhao
    Global regularity for a coupled Cahn-Hilliard-Boussinesq system on bounded domains
    Quarterly of Applied MathematicsVol. 69 (2012) pp. 331-356

    Abstract

    We study large-time asymptotic behavior of classical solutions to an initial-boundary

    value problem (IBVP) for a coupled Cahn-Hilliard-Boussinesq system on a bounded domain. Sufficient

    conditions are established under which classical solutions converge exponentially to constant

    states as time goes to infinity due to diffusion and boundary effects.
  • E. Green, A. Bassom and A. Friedman
    A mathematical model for cell-induced gel compaction in vitro
    Mathematical Models and Methods in Applied Sciences (2012) (Accepted)

    Abstract

  • R. Giedt, D. Pfeiffer, A. Matzavinos, C. Kao and B. Alevriadou
    Mitochondrial dynamics and motility inside living vascular endothelial cells: Role of bioenergetics
    Annals of Biomedical EngineeringVol. 40 No. 9 (2012) pp. 1903-1916

    Abstract

  • D. Chen and G. Wei
    Quantum dynamics in continuum models for proton transport-Generalized correlatio
    Journal of Chemical PhysicsVol. 136 No. 134109 (2012)

    Abstract

  • J. Chifman
    The core control system of intracellular iron homeostasis: A mathematical model
    J Theor BiolVol. 300 (2012) pp. 91-99

    Abstract

  • M. Golubitsky and C. Postlethwaite
    Feed-forward networks, center manifolds, and forcing
    Discrete and Continuous Dynamical Systems - Series AVol. 32 (2012) pp. 2913-2935

    Abstract

  • M. Golubitsky, D. Romano and Y. Wang
    Network periodic solutions: patterns of phase-shift synchrony
    NonlinearityVol. 25 (2012) pp. 1045-1074

    Abstract

  • J. Feder, R. Gejji, S. Yeaman and P. Nosil
    Establishment of new mutations under divergence and genome hitchhiking
    Philosophical Transactions of the Royal Society B: Biological SciencesVol. 367 No. 1587 (2012) pp. 461-474

    Abstract

  • D. Janies, J. Aaronson, S. Handelman, J. Hardman, L. Kawalec, T. Bitterman and W. Wheeler
    Analysis and visualization of H7 influenza using genomic, evolutionary and geographic information in a modular web service
    Cladistics (2012) (In Press)

    Abstract

  • Y. Kim and H. Othmer
    A hybrid model of tumor-stromal interactions in breast cancer
    Bull. Math. Biol. (2012) (Submitted)

    Abstract

  • Y. Kim, S. Lee, Y. Kim, Y. Kim, Y. Gho and H. hwang
    Regulation of Th1/Th2 cells in asthma development A mathematical model
    J. Math. Biol. (2012) (Submitted)

    Abstract

  • W. Lo, L. Chen, M. Wang and Q. Nie
    A Efficient and Robust Method for Steady State Patterns in Reaction-Diffusion Systems
    Journal of Computational Physics (2012) (In Press)

    Abstract

  • W. Lo, S. Zhou, A. Lander and Q. Nie
    Robust and Precise Morphogen-mediated Patterning: Tradeoffs, Constraints and Mechanisms.
    (2012) (Submitted)

    Abstract

  • B. Shtylla and J. Keener
    A mathematical model of ParA filament-mediated chromosome movement in Caulobacter crescentus.
    Journal of Theoretical BiologyVol. 307 (2012) pp. 82-95

    Abstract

  • R. Tien and S. Ellner
    Variable cost of prey defense and coevolution in predator-prey systems
    Ecological Monographs (2012) (In Press)

    Abstract

  • A. Friedman, B. Hu and C. Xue
    A three dimensional model of chronic wound healing: analysis and computation
    DCDS-B (2012) (In Preparation)

    Abstract

  • F. Hinkelmann and A. Jarrah
    Inferring Biologically Relevant Models: Nested Canalyzing Functions
    ISRN Biomathematics (2012) (Accepted)

    Abstract

  • S. Zhou, W. Lo, J. Suhalim, M. Digman, E. Gratton, Q. Nie and A. Lander
    Free Extracellular Diffusion Creates the Dpp Morphogen Gradient of the Drosophila Wing Disc
    Current Biology (2012) (In Press)

    Abstract

  • V. Gay, P. Hemond, D. Schmidt, M. O Boyle, Z. Hemond, J. Best, L. O Farrell and K. Suter
    Hormone secretion in transgenic rats and their electrophysiological activity in gonadotrophin releasing-hormone (GnRH) neurons
    (2012) (Submitted)

    Abstract

  • R. Grima, D. Schmidt and T. Newman
    Exact solution of the master equation of a gene regulatory network with a transcriptional feedback loop
    (2012) (Submitted)

    Abstract

  • C. Xue, C. Chou, C. Kao, C. Sen and A. Friedman
    Propagation of Cutaneous Thermal Injury: A Mathematical Model
    Wound Repair and RegenVol. 20 No. 1 (2012) pp. 114-122

    Abstract

  • H. Othmer and C. Xue
    The mathematical analysis of biological aggregation and dispersal: progress, problems and perspectives
    Dispersal, individual movement and spatial ecology: A mathematical perspective (2012) (In Preparation)

    Abstract

  • H. Jain, N. Moldovan and H. Byrne
    Modeling stem/progenitor cell-induced neovascularization and oxygenation around solid implants
    Tissue Eng. Part C Methods (2012) (In Press)

    Abstract

  • Y. Kim and S. Roh
    A hybrid model of cell proliferation and migration in glioblastoma
    Discrete and Continuous Dynamical Systems -B (2012) (Submitted)

    Abstract

  • B. Szomolay, T. Eubank, R. Roberts, C. Marsh and A. Friedman
    Modeling inhibition of breast cancer growth by GM-CSF
    J. of Theor. Biol. (2012) (Accepted)

    Abstract

  • R. Gejji, Y. Lou, D. Munther and J. Peyton
    Evolutionary Convergence to Ideal Free Dispersal Strategies and Coexistence
    Bulletin of Mathematical BiologyVol. 74 No. 2 (2012) pp. 257-299

    Abstract

  • A. Sharma and D. Chowdhury
    Error correction during DNA replication
    Physical Review EVol. 86 No. 011913 (2012)

    Abstract

    DNA polymerase (DNAP) is a dual-purpose enzyme that plays two opposite roles in two different situations
    during DNA replication. It plays its a normal role as a polymerase catalyzing the elongation of a new DNA
    molecule by adding a monomer. However, it can switch to the role of an exonuclease and shorten the same
    DNA by cleavage of the last incorporated monomer from the nascent DNA. Just as misincorporated nucleotides
    can escape exonuclease causing a replication error, the correct nucleotide may get sacrificed unnecessarily by
    erroneous cleavage. The interplay of polymerase and exonuclease activities of a DNAP is explored here by
    developing a minimal stochastic kinetic model of DNA replication. Exact analytical expressions are derived for
    a few key statistical distributions; these characterize the temporal patterns in the mechanical stepping and the
    chemical (cleavage) reaction. The Michaelis-Menten-like analytical expression derived for the average rates of
    these two processes not only demonstrate the effects of their coupling, but are also utilized to measure the extent
    of replication error and erroneous cleavage.
  • Y. Wang, P. Paszek, C. Horton, H. Yue, M. White, D. Kell, M. Muldoon, M. Muldoon and D. Broomhead
    A systematic study of the response of a NF-kB signalling pathway to TNFalpha stimulation.
    Journal of Theoretical BiololgyVol. 297 (2012) pp. 137-147

    Abstract

  • J. Tian, Z. Jin and T. Xie
    Mathematical model for two germline stem cells competing for niche occupancy
    Bulletin of Mathematical BiologyVol. 74 No. 5 (2012) pp. 1207-1225

    Abstract

  • D. Yang, J. Tian and J. Wang
    A solvable hyperbolic free boundary problem modeling tumor regrowth
    Applicable Analysis (2012)

    Abstract

  • J. Tian and J. Wang
    Global stability for cholera epidemic models
    Mathematical SciencesVol. 232 No.1 (2012) pp. 31-41

    Abstract

  • J. Tian
    The replicability of oncolytic virus: defining conditions on tumor virotherapy
    Mathematical Sciences and EngineeringVol. 8 No. 3 (2012) pp. 841-860

    Abstract

  • T. Li, R. Pan and K. Zhao
    Global dynamics of a chemotaxis model on bounded domains with large data
    SIAM Journal on Applied MathematicsVol. 72 No. 1 (2012)

    Abstract

  • J. Tian, A. Friedman, J. Wang and E. Chiocca
    Modeling the Effects of Resection, Radiation and Chemotherapy in Glioblastoma
    Journal of Neuro-OncologyVol. 91 No. 3 (2012) pp. 287-293

    Abstract

  • J. Tian
    Finite-time perturbations of dynamical systems and applications to tumor therapy
    Discrete and Continuous Dynamical Systems - BVol. 12 No. 2 (2012) pp. 469-479

    Abstract

  • J. Tian and S. Lin
    The Mutation Process in Colored Coalescent Theory
    Bulletin of Mathematical BiologyVol. 71 No. 8 (2012) pp. 1873-1889

    Abstract

  • J. Tian
    Evolution Algebras and Their Applications (research monograph)
    Lecture Notes in MathematicsVol. 1921 (2012)

    Abstract

  • F. Rassoul-Agha, M. Balazs and T. Seppalainen
    Existence of the zero range process and a deposition model with superlinear growth rates
    Ann. Probab.Vol. 35 (2012) pp. 1209-1249

    Abstract

  • F. Rassoul-Agha, M. Balazs and T. Seppalainen
    Random average process and random walk in a space-time random environment in one dimension.
    Commun. Math. Phys.Vol. 266 (2012) pp. 499-545

    Abstract

  • W. Just and G. Enciso
    Analogues of the Hirsch and Smale theorems for cooperative Boolean and discrete systems
    Journal of Difference Equations and ApplicationsVol. 18 No. 2 (2012) pp. 233-238

    Abstract

  • B. Dasgupta, G. Enciso, E. Sontag and Y. Zhang
    Algorithmic & complexity results for decompositions of biological networks into monotone subsystems
    Springer Lecture Notes in Computer Science (2012) pp. 253-264

    Abstract

  • W. Just and A. Nevai
    A Kolmogorov-type competition model with finitely supported allocation profiles and its applications to plant competition for sunlight.
    J. Biol. DynVol. 3 (2012) pp. 599-619

    Abstract

  • B. Stigler and A. Veliz-Cuba
    Boolean Models Can Explain Bistability in the lac Operon
    Journal of Computational BiologyVol. 16 No. 3 (2012) pp. 783-794

    Abstract

  • J. Day, A. Friedman and L. Schlesinger
    Modeling the immune rheostat of macrophages in the lung in response to infection.
    Proc. Natl Acad Sci USAVol. 106 No. 27 (2012) pp. 11246-11251

    Abstract

  • J. Day, J. Rubin and C. Chow
    Competition between transients in the rate of approach to a fixed point.
    SIAM J. Appl. Dyn. Syst.Vol. 8 No. 4 (2012) pp. 1523-1563

    Abstract

  • R. Leander, S. Lenhart and V. Protopopescu
    Using optimal control theory to identify network structures that foster synchrony
    Physica D.Vol. 241 (2012) pp. 574-582 (Submitted)

    Abstract

  • M. Popesco, S. Lin, Z. Wang, Z. Ma, L. Friedman, A. Frostholm and A. Rotter
    Serial Analysis of Gene Expression profiles of adult and aged mouse cerebellum
    Neurobiology of AgingVol. 29 (2012) pp. 774-788

    Abstract

  • P. Budu-Grajdeanu, R. Schugart, A. Friedman, D. Birmingham and B. Rovin
    Predicting renal interstitial inflammation levels using urine biomarkers and artificial neural networks
    (2012) (In Preparation)

    Abstract

  • S. Sun, P. Yan, T. Huang and S. Lin
    Identifying differentially methylated genes using mixed effect and generalized least square models
    BMC BioinformaticsVol. 10 (2012) pp. 404

    Abstract

  • C. Greenwood, S. Sun, J. Veenstra, N. Hamel, B. Niell, S. Gruber and W. Foulkes
    How old is this mutation? - a study of three Ashkenazi Jewish founder mutations
    BMC GenetVol. 11 No. Series 39 (2012)

    Abstract

  • L. Han, S. Zheng, S. Sun, T. Huang and Z. Zhao
    Genome-wide DNA methylation profiling in 40 breast cancer cell lines
    Advanced Intelligent Computing Theories and ApplicationsVol. 6215 (2012) pp. 277-284

    Abstract

  • S. Sun, Z. Chen, P. Yan, Y. HUANG, T. Huang and S. Lin
    Identifying hypermethylated CpG islands using a quantile regression model
    BMC bioinformaticsVol. 12 No. Series 54 (2012)

    Abstract

  • S. Sun, Y. HUANG, P. Yan, T. Huang and S. Lin
    Preprocessing differentially methylation hybridization microarray data
    BioData MiningVol. 4 No. 14 (2012)

    Abstract

  • Y. Kim and K. Boushaba
    A mathematical model of tumor dormancy and secondary metastasis
    Systems of Tumor Dormancy (2012) (Under Revision)

    Abstract

  • C. Xue, H. hwang, K. Painter and R. Erban
    Traveling waves in hyperbolic chemotaxis equations
    Bul. Math. BiolVol. 73 No. 8 (2012) pp. 1695-1733

    Abstract

  • C. Xue, C. Chou, C. Kao, C. Sen and A. Friedman
    Propagation of Cutaneous Thermal Injury: A Mathematical Model
    Wound Repair and RegenVol. 20 No. 1 (2012) pp. 114-122

    Abstract

  • C. Xue, A. Friedman and B. Hu
    A three dimensional model of chronic wound healing: analysis and computation
    DCDS-B (2012) (Submitted)

    Abstract

  • H. Jain and H. Byrne
    Qualitative analysis of an integro-differential equation model of periodic chemotherapy
    Appl Math Lett. (2012) (Under Revision)

    Abstract

  • H. Jain and A. Friedman
    Modeling prostate cancer response to continuous versus intermittent androgen ablation therapy
    Discrete Cont Dyn-B (2012) (Under Revision)

    Abstract

  • M. Rempe, J. Best and D. Terman
    A Mathematical Model of the sleep/wake cycle
    J Math BiolVol. 60 (2012) pp. 615-644

    Abstract

  • S. Ahn, B. Smith, A. Borisyuk and D. Terman
    Analyzing Neuronal Networks Using Discrete-Time Dynamic
    Physica D: Nonlinear phenomenaVol. 239 No. 9 (2012) pp. 515-528

    Abstract

  • D. Janies, P. Goloboff and D. Pol
    Large-scale phylogenetic analysis for the study of zoonosis and assessment of influenza surveillance
    Scalable Computing: Practice and ExperienceVol. 8 (2012) pp. 143-146

    Abstract

  • P. Goel, A. Sherman and A. Friedman
    Multiscale modeling of electrical and intracellular activity in the pancreas: The islet Tridomain equations
    SIAM Multiscale Modeling and SimulationVol. 7 (2012) pp. 1609-1642

    Abstract

  • S. Wei, C. balch, H. Paik, Y. Kim, R. Baldwin, S. Liyanarachchi, L. Li, Z. Wang, J. Wan, R. Davuluri, B. Karlan, G. Gifford, R. Brown, S. Kim, T. Huang and K. Nephew
    Prognostic DNA Methylation Biomarkers in Ovarian Cancer
    Clin Cancer Res.Vol. 12 No. 9 (2012) pp. 2788-2794

    Abstract

  • B. Szomolay, M. Dindos and N. Cogan
    Effect of Periodic Disinfection on Persisters in a One-Dimensional Biofilm Model
    (2012) (In Preparation)

    Abstract

  • D. Schmidt, J. Best and M. Blumberg
    Linking network structure and stochastic dynamics to neural activity patterns involved in sleep-wake regulation
    (2012) (In Preparation)

    Abstract

  • F. Hinkelmann, B. Delidow and R. Laubenbacher
    Downregulation of LRP6 inhibits growth of melanoma cells
    (2012) (Under Review)

    Abstract

  • D. Siegal-Gaskins, J. Ash and S. Crosson
    Model-based Deconvolution of Cell Cycle Time-series Data Reveals Gene Expression Details at High Resolution
    BMC Syst BiolVol. 3 No. 96 (2012)

    Abstract

  • R. Leander, S. Dai, L. Schlesinger and A. Friedman
    A Mathematical Model of CR3/TLR2 Crosstalk in the context of Francisella Tularensis infection
    PLoS Comput Biol (2012) (Under Revision)

    Abstract

  • R. Leander, S. Lenhart and V. Protopopescu
    Using optimal control theory to identify network structures that foster synchrony
    Physica D. No. 241 (2012) pp. 574-582

    Abstract

  • D. Chen, J. Roda, C. Marsh, T. Eubank and A. Friedman
    Hypoxia Inducible Factors-mediated inhibition of cancer by GM-CSF: A mathematical model
    (2012) (Under Review)

    Abstract

  • D. Chen and A. Friedman
    Analysis of a two-phase free boundary problem for a parabolic-hyperbolic system: an application to tumor growth.
    (2012) (Submitted)

    Abstract

  • R. Leander and A. Friedman
    Modulation of the cAMP response by G alpha i and G beta gamma: a computational study of G protein signaling in immune cells
    PLoS One (2012) (Under Review)

    Abstract

  • S. Riechert, R. Leander and S. Lenhart
    A role-playing exercise that demonstrates the process of evolution by natural selection: Caching squirrels in a world of pilferers
    Am. Biol. TeachVol. 73 (2012) pp. 208-216

    Abstract

  • M. Golubitsky and A. Comanici
    Patterns on growing square domains via mode interactions
    Dynamical SystemsVol. 23 No. 2 (2012) pp. 167-206

    Abstract

  • J. Keener and S. Dai
    Using noise to determine cardiac restitution with memory
    Phys. Rev. E.Vol. 85 No. 061902 (2012)

    Abstract

  • Y. Guo, C. Park, M. Rong, R. Worth and L. Rubchinsky
    Thalamocortical relay modulation by basal ganglia in Parkinsonā??s disease and dystonia
    (2012) (Submitted)

    Abstract

  • Y. Guo
    Existence and Stability of Traveling Fronts in a Lateral Inhibition Neural Network
    SIAM Journal on Applied Dynamical Systems (2012) (Accepted)

    Abstract

  • S. Dai, D. Li and K. Zhao
    Finite-time quenching of competing species with constrained border evaporation
    DCDS-B (2012) (Submitted)

    Abstract

  • D. Yang and Y. Guo
    Localized states in 1-D homogeneous neural field models with general coupling and firing rate functions.
    (2012) (Submitted)

    Abstract

  • L. Zheng, H. Othmer and H. Kang
    The effect of the signaling scheme on the robustness of pattern formation in development
    Interface FocusVol. 2 No. 4 (2012) pp. 465-486 (To Appear)

    Abstract

  • P. Hurtado
    Within-Host Dynamics of Mycoplasma Infections: Conjunctivitis in Wild Passerine Birds
    Journal of Theoretical BiologyVol. 306 (2012) pp. 73-92

    Abstract

  • A. Beri, R. Azencott and I. Timofeyev
    Calibration of Stochastic Volatility Model under Indirect Observability of the Volatility Process
    (2012) (In Preparation)

    Abstract

  • R. Azencott , A. Beri, Y. Gadhyan, N. Joseph, C. Lehalle and M. Rowley
    Realtime Market Microstructure Analysis: Online Transaction Cost Analysis
    (2012) (In Preparation)

    Abstract

  • A. Beri, D. Chowdhury and H. Jain
    Agent-based and Macroscopic PDE models for foraging dynamics of competing ant colonies, and associated boundary interactions
    (2012) (In Preparation)

    Abstract

  • J. Wenstrup, M. Deacon, I. Comeras, V. Schumemann, D. Fix, S. Liyanarachchi, S. Sun, M. Baze, M. Clendenning, S. Thibodeau, H. Lynch, H. Hampel and A. De La Chapelle
    Origins and prevalence of the American Founder Mutation of MSH2
    Cancer ResVol. 68 (2012) pp. 2145-2153

    Abstract

  • D. Chen, J. Roda, C. Marsh, T. Eubank and A. Friedman
    Hypoxia inducible factors mediated-inhibition of cancer by GM-CSF: A mathematical model
    Bulletin of Mathematical Biology (2012)

    Abstract

    Under hypoxia, tumor cells, and tumor-associated macrophages produce VEGF (vascular endothelial growth factor), a signaling molecule that induces angiogenesis. The same macrophages, when treated with GM-CSF (granulocyte/macrophage colony-stimulating factor), produce sVEGFR-1 (soluble VEGF receptor-1), a soluble protein that binds with VEGF and inactivates its function. The production of VEGF by macrophages is regulated by HIF-1Ī± (hypoxia inducible factor-1Ī±), and the production of sVEGFR-1 is mediated by HIF-2Ī±. Recent experiments measured the effect of inhibiting tumor growth by GM-CSF treatment in mice with HIF-1Ī±-deļ¬?cient or HIF-2Ī±-deļ¬?cient macrophages. In the present paper, we represent these experiments by a mathematical model based on a system of partial differential equations. We show that the model simulations agree with the above experiments. The model can then be used to suggest strategies for inhibiting tumor growth. For example, the model qualitatively predicts the extent to which GM-CSF treatment in combination with a small molecule inhibitor that stabilizes HIF-2Ī± will reduce tumor volume and angiogenesis.
  • M. Schwemmer and T. Lewis
    The robustness of phase-locking in neurons with dendro-dendritic electrical coupling
    Journal of Mathematical Biology (2012)

    Abstract

    We examine the effects of dendritic filtering on the existence, stability, and robustness of phase-locked states to heterogeneity and noise in a pair of electrically coupled ball-and-stick neurons with passive dendrites. We use the theory of weakly coupled oscillators and analytically derived filtering properties of the dendritic coupling to systematically explore how the electrotonic length and diameter of dendrites can alter phase-locking. In the case of a fixed value of the coupling conductance ( gc ) taken from the literature, we find that repeated exchanges in stability between the synchronous and anti-phase states can occur as the electrical coupling becomes more distally located on the dendrites. However, the robustness of the phase-locked states in this case decreases rapidly towards zero as the distance between the electrical coupling and the somata increases. Published estimates of gc are calculated from the experimentally measured coupling coefficient ( CC ) based on a single-compartment description of a neuron, and therefore may be severe underestimates of gc . With this in mind, we re-examine the stability and robustness of phase-locking using a fixed value of CC , which imposes a limit on the maximum distance the electrical coupling can be located away from the somata. In this case, although the phase-locked states remain robust over the entire range of possible coupling locations, no exchanges in stability with changing coupling position are observed except for a single exchange that occurs in the case of a high somatic firing frequency and a large dendritic radius. Thus, our analysis suggests that multiple exchanges in stability with changing coupling location are unlikely to be observed in real neural systems.
  • D. Chen and A. Friedman
    A two-phase free boundary problem with discontinuous velocity: Application to tumor model
    Journal of Mathematical Analysis and Applications (2012)

    Abstract

    We consider a two-phase free boundary problem consisting of a hyperbolic equation for w and a parabolic equation for u, where w and u represent, respectively, densities of cells and cytokines in a simpliļ¬?ed tumor growth model. The tumor region Ω(t) is enclosed by the free boundary Γ(t), and the exterior of the tumor, D(t), consists of a healthy normal tissue. Due to cancer cells proliferation, the convective velocity ~v of cells is discontinuous across the free boundary; the motion of the free boundary Γ(t) is determined by ~v. We prove the existence and uniqueness of a solution to this system in the radially symmetric case for a small time interval 0 t T, and apply the analysis to the full tumor growth model.
  • L. Hu, D. Chen and G. Wei
    High-order fractional partial differential equation for molecular surface construction
    Molecular based Mathematical Biology (2012)

    Abstract

    Fractional derivative or fractional calculus plays a signifcant role in theoretical modeling of scientiļ¬?c and engineering problems. However, only relatively low order fractional derivatives are used at present. In general, it is not obvious what role a high fractional derivative can play and how to make use of arbitrarily high-order fractional derivatives. This work introduces arbitrarily high-order fractional partial differential equations (PDEs) to describe fractional hyperdiffusions. The fractional PDEs are constructed via fractional variational principle. A fast fractional Fourier transform (FFFT) is proposed to numerically integrate the high-order fractional PDEs so as to avoid stringent stability constraints in solving high-order evolution PDEs. The proposed high-order fractional PDEs are applied to the surface generation of proteins. We first validate the proposed method with a variety of test examples in two and three-dimensional settings. The impact of high-order fractional derivatives to surface analysis is examined. We also construct fractional PDE transform based on arbitrarily high-order fractional PDEs. We demonstrate that the use of arbitrarily high-order derivatives gives rise to time-frequency localization, the control of the spectral distribution, and the regulation of the spatial resolution in the fractional PDE transform. Consequently, the fractional PDE transform enables the mode decomposition of images, signals, and surfaces. The effect of the propagation time on the quality of resulting molecular surfaces is also studied. Computational efficiency of the present surface generation method is compared with the MSMS approach in Cartesian representation. We further validate the present method by examining some benchmark indicators of macromolecular surfaces, i.e., surface area, surface enclosed volume, surface electrostatic potential and solvation free energy. Extensive numerical experiments and comparison with an established surface model indicate that the proposed high-order fractional PDEs are robust, stable and efficient for biomolecular surface generation.
  • P. Thomas, C. Wilson and C. Diekman
    Spontaneous autoresuscitation in a model of respiratory control
    Conf Proc IEEE Eng Med Biol Soc (2012) pp. 6669-6672

    Abstract

    We introduce a closed-loop model of respiratory control incorporating a conductance-based central pattern generator (CPG), low-pass filtering of CPG output by the respiratory musculature, gas exchange in the lung, metabolic oxygen demand, and chemosensation. The CPG incorporates Butera, Rinzel and Smith (BRS)ā??s (1999) conditional pacemaker model. BRS model cells can support quiescent, bursting, or beating activity depending on the level of excitatory drive; we identify these activity modes with apnea (cessation of breathing), eupnea (normal breathing), and tachypnea (excessively rapid breathing). We demonstrate the coexistence of two dynamically stable behaviors in the closed-loop model, corresponding respectively to eupnea and tachypnea. The latter state represents a novel failure mode within a respiratory control model. In addition, the closed-loop system exhibits a form of autoresuscitation: conductances intrinsic to the BRS model buffer the CPG against brief episodes of hypoxia, steering the system away from catastrophic collapse as can occur with tachypnea.
  • Y. Wang, T. McMillen, M. Golubitsky and C. Diekman
    Reduction and dynamics of a generalized rivalry network with two learned patterns
    SIAM Journal of Applied Dynamical SystemsVol. 11 (2012) pp. 1270-1309

    Abstract

    We use the theory of coupled cell systems to analyze a neuronal network model for generalized rivalry posed by H. Wilson. We focus on the case of rivalry between two patterns and identify conditions under which large networks of n attributes and m intensity levels can reduce to a model consisting of two or three cells depending on whether or not the patterns have any attribute levels in common. (The two-cell reduction is equivalent to certain recent models of binocular rivalry.) Notably, these reductions can lead to large recurrent excitation in the reduced network even though the individual cells in the original network may have none. We also show that symmetry-breaking Takensā??Bogdanov (TB) bifurcations occur in the reduced networks, and this allows us to further reduce much of the dynamics to a planar system. We analyze the dynamics of the quotient systems near the TB singularity, discussing how variation of the input parameter I organizes the dynamics. This variation leads to a degenerate path through the unfolding of the TB point. We also discuss how the network structure affects recurrent excitation in the reduced networks, and the consequences for the dynamics.
  • H. Kang
    A multiscale approximation in a heat shock response model in E.coli
    BMC Systems BiologyVol. 6 No. 1 (2012) pp. 143

    Abstract

    Background: A heat shock response model of Escherichia coli developed by Srivastava, Peterson, and Bentley (2001) has multiscale nature due to its species numbers and reaction rate constants varying over wide ranges. Applying the method of separation of time-scales and model reduction for stochastic reaction networks extended by Kang and Kurtz (2012), we approximate the chemical network in the heat shock response model.
    Results: Scaling the species numbers and the rate constants by powers of the scaling parameter, we embed the model into a one-parameter family of models, each of which is a continuous-time Markov chain. Choosing an appropriate set of scaling exponents for the species numbers and for the rate constants satisfying balance conditions, the behavior of the full network in the time scales of interest is approximated by limiting models in three time scales. Due to the subset of species whose numbers are either approximated as constants or are averaged in terms of other species numbers, the limiting models are located on lower dimensional spaces than the full model and have a simpler structure than the full model does.
    Conclusions: The goal of this paper is to illustrate how to apply the multiscale approximation method to the biological model with significant complexity. We applied the method to the heat shock response model involving 9 species and 18 reactions and derived simplified models in three time scales which capture the dynamics of the full model. Convergence of the scaled species numbers to their limit is obtained and errors between the scaled species numbers and their limit are estimated using the central limit theorem.
  • H. Kang, L. Zheng and H. Othmer
    A new method for choosing the computational cell in stochastic reaction-diffusion systems
    Journal of Mathematical BiologyVol. 65 No. Series 6-7 (2012) pp. 1017-1099

    Abstract

    How to choose the computational compartment or cell size for the stochastic simulation of a reactionā€“diffusion system is still an open problem, and a number of criteria have been suggested. A generalized measure of the noise for finite-dimensional systems based on the largest eigenvalue of the covariance matrix of the number of molecules of all species has been suggested as a measure of the overall fluctuations in a multivariate system, and we apply it here to a discretized reactionā€“diffusion system. We show that for a broad class of first-order reaction networks this measure converges to the square root of the reciprocal of the smallest mean species number in a compartment at the steady state. We show that a suitably re-normalized measure stabilizes as the volume of a cell approaches zero, which leads to a criterion for the maximum volume of the compartments in a computational grid. We then derive a new criterion based on the sensitivity of the entire network, not just of the fastest step, that predicts a grid size that assures that the concentrations of all species converge to a spatially-uniform solution. This criterion applies for all orders of reactions and for reaction rate functions derived from singular perturbation or other reduction methods, and encompasses both diffusing and non-diffusing species. We show that this predicts the maximal allowable volume found in a linear problem, and we illustrate our results with an example motivated by anterior-posterior pattern formation in Drosophila, and with several other examples.
  • C. Diekman, C. Wilson and P. Thomas
    Spontaneous autoresuscitation in a model of respiratory control.
    Conference proceedings : ... Annual International Conference of the IEEE Engineering in Medicine and Biology Society. IEEE Engineering in Medicine and Biology Society. ConferenceVol. 2012 (2012) pp. 6669-72

    Abstract

    We introduce a closed-loop model of respiratory control incorporating a conductance-based central pattern generator (CPG), low-pass filtering of CPG output by the respiratory musculature, gas exchange in the lung, metabolic oxygen demand, and chemosensation. The CPG incorporates Butera, Rinzel and Smith (BRS)'s (1999) conditional pacemaker model. BRS model cells can support quiescent, bursting, or beating activity depending on the level of excitatory drive; we identify these activity modes with apnea (cessation of breathing), eupnea (normal breathing), and tachypnea (excessively rapid breathing). We demonstrate the coexistence of two dynamically stable behaviors in the closed-loop model, corresponding respectively to eupnea and tachypnea. The latter state represents a novel failure mode within a respiratory control model. In addition, the closed-loop system exhibits a form of autoresuscitation: conductances intrinsic to the BRS model buffer the CPG against brief episodes of hypoxia, steering the system away from catastrophic collapse as can occur with tachypnea.

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