Mathematical Biology at the University of Houston

Department of Mathematics

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Giles Auchmuty is a Professor of Mathematics at the University of Houston.maintains a continuing interest in the analysis of nonlinear equations and electromagnetic effects arising in biological problems. Recently he has worked with Mandri Obeysekere and Edwin Tecarro on ODE models of the cell-cycle. Also John Alford has recently graduated with a Ph.D. thesis on the existence and computation of rotating wave solutions of the Fitzhugh Nagumo equations. This study was motivated by models investigated by Leon Glass for tachycardiac arrhythmias. 1) with M.N. Obeyesekere and E.S. Tecarro, "Analysis of a model of the Mammalian Cell cycle's G1 phase", Nonlinear Analysis and Applications, (to appear). 2) with M.N. Obeyesekere, E.S. Tecarro and S.O. Zimmerman, "A model of cell cycle behavior dominated by kinetics of a pathway stimulated by growth factors", Bull. Math. Biology, 61, (1999), 917-934 3) with J. He, "An Integral Inequality in Population Modeling", Solution of problem 95-13, SIAM Review 40, (1998), 710-713.

Suncica Canic is an Associate Professor of Mathematics. Her interests are in partial differential equations; theory and numerics and biomathematics. She is interested in modeling blood flow with application in the design of stents. Recently, she has also supervised an REU program in the department. She collaborates with aerospace engineer Dr. Ravi-Chandar, UT Austin, mollecular biologist Dr. Doreen Rosenstrauch, M.D. with cardiologist Dr. Zvonko Krajcer, the Texas Heart Institute and with Dr. Andro Mikelic, University of Lyon 1, France. 1. S. Canic. Blood flow through compliant vessels after endovascular repair: wall deformations induced by the discontinuous wall properties. Computing and Visualization in Science. Springer-Verlag. 4(3) (2002) 147-155. 2. S. Canic and E-H. Kim. Mathematical Analysis of Quasilinear Effects in a Hyperbolic Model of Blood Flow through Compliant Axi-Symmetric Vessels, Mathematical Methods in Applied Sciences, accepted (2002). 3. S. Canic and A. Mikelic, Effective Equations Describing the Flow of a Viscous Incompressible Fluid Through a Long Elastic Tube, Comptes Rendus Mechanique Acad. Sci. Paris 330 (2002) pp. 661-666. 4. Canic , S. and A. Mikelic, Effective equations modeling the flow of a viscous incompressible fluid through a long elastic tube arising in the study of blood flow through small arteries. Submitted to SIAM J. Appl. Dyn. Sys. (2002).

William E. Fitzgibbon is Professor and Chairman of the Department of Mathematics. He is interested application of reaction-diffusion equations and other coupled distributed parameter systems to problems in mathematical biology especially in population biology, epidemiology and the environment. Recent publications include:: 1. "Weakly coupled hyperbolic systems modeling the circulation of FeLV in structured feline populations" (with M. Langlais), Mathematical Biosciences, 165 (2000), 79-95. 2. "Modeling the Spread of Feline Leukemia in Heterogeneous Habitats", Fields Institute Communications (with M. Langlais and J. Morgan), 29 (2001), 133-146. 3. "A mathematical model of the spread of Feline Leukemia Virus Through a Highly Heterogeneous Domain" (with M. Langlais and J. Morgan), SIAM J. Mathematical Analysis, 33 (2001), 570-588. 4. "An application of homogenization techniques to population dynamics models" (with B.E. Ainseba, M. Langlais, J.J Morgan), Communications in Pure and Applied Mathematics, 1 (2002), 19-33.

Roland Glowinski is a Cullen Distinguished Professor of Mathematics and Mechanical Engineering at the University of Houston. He is interested in numerical methods in partial differential equations. Methods he has developed have been fundamental in modeling various phenomena in biological systems.

Martin Golubitsky is a Cullen Distinguished Professor of Mathematics at the University of Houston. His work in mathematical neuroscience has emphasized the use of symmetry methods to study locomotor central pattern generators for quadrupedal gaits and to study geometric visual hallucinations via pattern formation on the primary visual cortex. More generally, he is investigating the generic dynamics of coupled cell networks based on system architecture. He is a coorganizer of the Nonlinear Dynamics and Neuroscience seminar series. 1) M. Golubitsky, I. Stewart, P.L. Buono and J.J. Collins. The Role of Symmetry in Locomotor Central Pattern Generators and Animal Gaits. Nature. 401 (1999) 693-695. 2) P.L. Buono and M. Golubitsky. Models of central pattern generators for quadruped locomotion: I. primary gaits. J. Math. Biol. 42 No. 4 (2001) 291-326. 3) P.C. Bressloff, J.D. Cowan, M. Golubitsky, P.J. Thomas, and M.C. Wiener. Geometric visual hallucinations, Euclidean symmetry, and the functional architecture of striate cortex. Phil. Trans. Royal Soc. London B 356 (2001) 299-330. 4) M. Golubitsky, L-J. Shiau, and A. Torok. Bifurcation on the visual cortex with weakly anisotropic lateral coupling. SIAM J. Appl. Dynam. Sys. To appear. 5) I. Stewart, M. Golubitsky, and M. Pivato. Symmetry Groupoids and Patterns of Synchrony in Coupled Cell Networks. SIAM J. Appl. Dynam. Sys. Submitted.

Kresimir Josic is an Assistant Professor in the Department of Mathematics. He is interested in applications of the theory of dynamical systems to neuroscience, particularly the understanding of synchronous behavior between complex systems. Synchronization has been shown to play a fundamental role in cognition, and the analysis of synchronized behavior between complex system and in networks with complex architecture provides many mathematical challenges. He is a coorganizer of the Nonlinear Dynamics and Neuroscience seminar series. 1) M. Beck and K. Josic. A geometric theory of chaotic phase synchronization. To appear in Chaos (2002). 2) P. So, E. Barreto, K. Josic, E. Sander, and S.J. Schiff. Limits to the Experimental Detection of Nonlinear Synchrony. Physical Review E, 65, article 046225 (2002). 3) M. Golubitsky, K. Josic and T.J. Kaper. An Unfolding Theory Approach to Bursting in Fast-Slow Systems. Chapter in Global Analysis of Dynamical Systems, dedicated to Floris Takens (2001).

Courses in Mathematical Biology and Related Topics

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This course is scheduled to be taught in the Fall of 2003 by Prof. Josic.

This undergraduate course introduces students to standard mathematical models of individual neurons (Hodgkin-Huxley, `integrate and fire', etc.) and the synaptic events by which neurons communicate. We will also spend some time discussing simple models of signal propagation along neurons. Next we will study small networks consisting of excitatory and inhibitory neurons -- giving some sense of the collective behavior required for sensory perception, information processing, short and long term memory, and learning.

The course is designed for advanced undergraduate and graduate students in mathematics, physics, engineering and the biological sciences and will be centered on differential equation models of neurons. The software package i XPP will be used to simulate the behavior of small networks of neurons. There are no biological pre-requisites. The course material will prepare interested students for continuing research projects in the area of theoretical and computational neurobiology.

We will loosely follow the book by Keener and Sneyd, Mathematical Physiolgy, Springer Verlag (1998). We will cover only the first 8 chapters, however we will use other references to explore much of the material in more detail.

The techniques of nonlinear dynamics have become an indespensable tool in the analysis of mathematical models of biological systems. This is a two semester course which serves as an introduction to the subject.

The first semester of the course is typically devoted to discrete dynamical systems. The topics covered include the qualitative analysis of discrete dynamical systems and ergodic theory.

The second semesters is devoted to a study of ordinary differential equations, and, in particular, their the qualitative properties of their solutions. We are currelnty using Strogat's book Nonlinear Dynamics and Chaos in this course whic contains a great number of applications. All the abstract are illustrated in pertinent examples taken from biology, chemistry, engineering. These applications have been the focus of a recent course.

Course: 6394 PDEs and Applications

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Prerequisites are courses in Multivariable Calculus, Real and Complex Analysis.

Partial differential equations are very frequently used to model various phenomena in biology. This course covers a review of basic linear PDEs, an introduction to fundamentals of fluid mechanics (basic equations of motion: continuity, momentum, energy, vorticity), incompressible/compressible flow examples, analysis of quasilinear PDEs with the focus on hyperbolic conservation laws, and basic numerical methods.

Once these tools are developed, they are applied to the analysis and numerical simulation arising in the study of blood flow through compliant blood vessels.

The texts used in the course include Strauss's PDEs, R. LeVeques's "Conservation Laws", Chorin and Marsden: "Fluid Mechanics", Y.C. Fung: "Circulation", and Research Papers

Differential equations are fundamental in any type of modeling, and modeling of biological systems is no exception. The department offers several graduate and undergraduate courses on this topic:

MATH 3331 --- Differential Equations
Prerequisites: MATH 2433 and MATH 2431. Systems of ordinary differential equations and topics in linear algebra. Existence, uniqueness, and stability of solutions; initial value problems; elementary bifurcation theory; Jordan normal form; higher order equations and Laplace transforms. Computer assignments will be given and limited computer facilities will be made available.

MATH 6324 --- Ordinary Differential Equations
This course will emphasize: phase portrait analysis for linear systems; general existence theorems for nonlinear systems; Linearization theorems including the stable and unstable manifold theorems; theory of discrete dynamical systems; standard well known examples of systems of ODEs.

Math 6325 --- Ordinary Differential Equations II
This course will stress the local bifurcation theory of dynamical systems through codimension two, including Liapunov--Schmidt and center manifold reductions, normal form theory, steady-state bifurcation, Hopf bifurcation, Takens--Bogdanov bifurcations and other codimension two mode interactions. Some aspects of chaotic dynamics, including Melnikov's method and Smale horseshoes, will be presented. Emphasis will be on the mathematical ideas (rather than on formal proofs) and how to apply these ideas.

Computer simulations have become indespensible in modeling biological systems, and the need to understand the ideas underlying the numerical methods used in such simulations has grown proportionately. The mathematics department has an exceptionally strong group of people working on computational methods, and thus a number of courses on the undergraduate and graduate level are offered in this area.

4364;4365 : Numerical Analysis
Cr. 3 per semester. (3-0). Prerequisites: MATH 2431, 3331; COSC 1301 or 2101 or equivalent; or consent of instructor. Topics selected from numerical linear algebra, approximation of functions, numerical integration and differentiation, interpolation, approximate solutions of ordinary and partial differential equations, Fourier methods, optimization.