I am interested in a number of topics in applied mathematics. Most of these concern applications of the theory of regular and stochastic dynamical systems to problems in neuroscience and genetics.

• Democratization in a passive dendritic tree: an analytical investigation Submitted (2007).
• Learning by structural remodeling in a classof single cell models Submitted (2007).
• Analysis of a Renormalization Group Method for Solving Perturbed Ordinary Differential Equations Submitted (2007).
• Unstable Solutions of Non-autonomous Linear Differential Equations To appear in SIAM Review (2007).
• Correlation between neural spike trains increases with firing rate To appear in Nature (2007).

• Universal properties of correlation transfer in integrate-and-fire neurons. preprint (2007).
• Branching Dendrites with Resonant Membrane: A sum-over-trips approach. To appear in Biological Cybernetics (2007).
• Non-uniform decay of predictability and return of skill in stochastic oscillatory models To appear in Physica D (2006).
• Is there a nucleotide composition constraint on genome sequences? To appear in Computational Biology and Chemistry (2006).
• Network structure and spatiotemporally symmetric dynamics Physica D 224(1-2), 52-68 (2006)

• Neuronal firing in the presence of stochastic trains of strong synaptic inputs Neural Computation 19, 1251-1294 (2007).
• Toward blueprints for network architecture, biophysical dynamics, and signal transduction Proceedings of the Royal Society A 364, 3301-3318 (2006).
• A mathematical model of cancer radiovirotherapy Mathematical Biosciences, vol. 199, issue 1, pp. 55-78 (2006) (Epublished Dec 22, 2005).
• Rotation Sets for Networks of Circle Maps Chaos, vol. 16, issue 1, p. 015115 (2006).
• Winding Numbers and Average Frequencies in Phase Oscillator Networks Journal of Nonlinear Science, vol. 16, issue 4, pp. 201-231, (2006).

• Deriving information about architecture from activity patterns in coupled cell systems SIAM Journal of Applied Dynamical Systems, vol. 4, p. 53-77, (2005).
• Applications of Asymptotic Normal Form Theory and its Connections with the Renormalization Group Method preprint (2004).
• Synchronization in Networks of General Weakly Non-Linear Oscillators Journal of Physics A, vol. 39, p. 11801-11817 (2004).
• Irrational Phase Synchronization Physical Review E, vol. 69, number 056228 (2004).
• The Structure of Synchronization Sets for Noninvertible Systems Chaos, vol. 14:2, pp. 249-262 (2004).

• Singular perturbations of quadratic maps International Journal of Bifurcations and Chaos, vol. 14:1, pp. 161-171 (2004).
• Playing catchup with iterated exponentials American Mathematical Monthly, 111:8, pp. 704-709, October, 2004.
• The geometry of chaos synchronization Chaos, vol 13:1 pp. 151-164 (2003).
• A geometric theory of chaotic phase synchronization Chaos, vol 13:1 pp. 247-258 (2003).
• Limits to the Experimental Detection of Nonlinear Synchrony Physical Review E, 65, article 046225 (2002).

• Phase Synchronization of Chaotic Systems with Small Phase Diffusion Physical Review E, 64, article 056234 (2001).
• Accessible Points of Julia Sets of Stable Exponentials Discrete and Continuous Dynamical Systems B, 1:3, p. 299-318 (2001).
• Phase Synchronization of Chaotic Systems unpublished (2001).
• The Mathematics of Musical Instruments American Mathematical Monthly, 108:4, p. 347-357 (2001).
• Planetary Motion and the Duality of Force Laws SIAM Review, 42:1, p. 114-125 (2000).

• Dynamics of a Diffusively Coupled Ring of Lorenz Oscillators Journal of Statistical Physics, 98:1, p. 1-30 (2000).
• Synchronization of Chaotic Systems and Invariant Manifolds Nonlinearity, 13:4, pp. 1321-1336 (2000).
• Invariant Manifolds and Synchronization of Coupled Dynamical Systems Physical Review Letters, 80:14, pp. 3053-3056 (1998).
• Local Bifurcations in the Symmetric Model of Selection with Fertility Differences Journal of Theoretical Biology, 189, pp. 291-295 (1997).
• My Complete Thesis

• Optimization of tumor virotherapy with recombinant measles viruses To appear in Optimization in Medicine and Biology edited by Gino J. Lim and Eva K. Lee (2007).
• Scholarpedia entry on isochrons (*)
• Scholarpedia entry on periodic orbits (*)
• Bursting in Coupled Systems in Bursting: The genesis of rhythm in the nervous system, edited by Stephen Coombes and Paul C. Bressloff (2005).
• An Unfolding Theory Approach to Bursting in Fast-Slow Systems chapter in Global Analysis of Dynamical Systems, dedicated to Floris Takens (2001).

The following is a program that Yakov Shapiro has developed while working with me in the summer of 2001. We were interested in studying the bifurcations of the reversible, area preserving complex Henon Map. The program is in principle able to parse all quadratic polynomial maps in C^2. You can download the .txt files for information on how to install and use the program.

The program StCurve.exe will draw the intersection of the stable manifold of the saddle point of the Henon Map with a plane that can be specified by the user.

This program will unfortunately only run on a PC.

• Readme.txt
• Install.txt
• The main program Henon.exe
• Stable manifolds.