A major goal in mathematical neuroscience is to understand how the dynamics of individual neurons contribute to computational capabilities of the brain. Mathematical models of neurons help to foster new experiments, further our understanding of the brain, and create and answer interesting mathematical questions. Generally these models are divided into two distinct classes; biophysical and phenomenological. Biophysical models, such as Hodgkin-Huxley models, attempt to incorporate as many known biological facts and data as necessary to achieve quantitative results. These models are often very complicated and are therefore not amenable to mathematical analysis. On the other hand, phenomenological models attempt to grasp the qualitative features of a biological system and are thus more easily analyzed with mathematics. Since phenomenological models are qualitative in nature, these models are often lacking important biological features. In my research, I have used a phenomenological model to reproduce the responses of neurons.
More generally, I am interested in applied mathematical techniques in modeling of biological problems including dynamical systems and bifurcation theory, differential equations, perturbation methods and numerical simulations.
- Neural Models
- Mathematical Physiology
- Genetic Evolution Models
- Population Dynamics
- Epidemiological Models
- Stochastic Modeling
Selected Publications (*student co-authors):
- Daneshbod, Y. and J. Latulippe. A Look at Damped Harmonic Oscillators Through the Phase Plane. Teaching Mathematics and its Applications.
- Kim, P.*, J. Latulippe, S. Muehlbacher*, E. Shen*, and K. Shun*. Genetic Algorithm and the Pendulum Problem. The Mathematical Scientist.
- Latulippe, J. and J. Switkes. Some Interesting Numerical Differentiation Formulae.The Mathematics Magazine.
- Daneshbod, Y. and J. Latulippe. The Geometry of Undamped Harmonic Oscillators. The Mathematical Scientist (2010) 35:43-53.
- Latulippe, C. and J. Latulippe. Discovering the Beauty of Science. Loci (March 2010), DOI: 10.4169/loci003445.
- Latulippe, J. and M. Pernarowski. A Non-autonomous Phenomenological Model for On and Off Responses of Cells in Sensory Neurons. Bulletin of Mathematical Biology (2009) 71:162-188.
- Latulippe, J. A Non-autonomous Phenomenological Bursting Model for Neurons, Department of Mathematical Sciences, Montana State University. Ph.D. Dissertation supervised by Prof. M. Pernarowski, 2007.
Undergraduate Student Projects:
Listed below are some of the undergraduate students projects I have supervised at Cal Poly Pomona and at Norwich University.
- Stephen Walz (2011)- “A Mathematical Model for Falling Objects in Viscous Liquids”. Senior Capstone Project. Norwich University.
- Karen Wood (2010) – “Volcanic Magma flows”. Research supported by the Women’s Educational Equity Act (WEEA). Cal Poly Pomona.
- Christopher Conow (2010) – “A Stochastic Model for Synaptic Transmission”. Project for the Honors College at Cal Poly Pomona. Cal Poly Pomona.
- Vanessa Ynez (2010) – “A Mathematical Model for Synaptic Transmission in Neurons”. Research supported by the Women’s Educational Equity Act (WEEA). Cal Poly Pomona.
- Randy Sierra (2009) – “A Mathematical Approach to Scapholunate Dissociation Healing Process”. Research supported by California State Polytechnic University Pomona College Cost Reduction and Access Act (CCRAA) Research-Apprentice Program. Cal Poly Pomona.
- Eddie Acevedo (2009) – “Modeling the effects of varied structural components in concrete columns”. Research supported by CSU-LSAMP Senior Alliance NSF Grant (HRD-0802628). Cal Poly Pomona.
- Silviana Carstea (2009) – “A Mathematical Model for Avian Influenza”. Research supported by the Women’s Educational Equity Act (WEEA). Cal Poly Pomona.
- Melissa Bilboa (2008) – “Modeling the Genetics of Autism”. Research supported by the Women’s Educational Equity Act (WEEA). Melissa presented her research by giving a talk at the Pacific Coast Undergraduate Mathematics Conference at LMU in 2008. Cal Poly Pomona.
Graduate Student Projects:
- A Multiple Scales Approach to a Weakly Nonlinear Oscillator, Randy Sierra, M.S. thesis, To be completed in Spring 2012. California State Polytechnic University, Pomona.
- Solution Methods for the Black-Scholes Model, Yoshi Moto Yama, M.S. thesis, December 2009, California State Polytechnic University, Pomona.
- A Discrete Time Predator-Prey Model Involving Cannibalism, Michael Grisgby, M.S. thesis, to be completed Spring 2011, California State Polytechnic University, Pomona.