Humans and other animals constantly accumulate sensory evidence, engage short term memory, and navigate their dynamic environment. These neural computations rely on the brain’s ability to robustly integrate information. Our group is extending existing theory and developing new mathematical techniques to understand how the brain��s complex networks and associated cognitive computations robustly integrate spatially structured input. Our work leverages stochastic and spatially-extended models, breaking new ground in the fields of mathematical neuroscience, nonlinear waves, statistical inference, and stochastic processes. A core goal is to develop a theory of the algorithms that underlie organismal behavior using a combination of computational modeling and field and laboratory data analysis. Key to this work is the close interaction with experimental collaborators whose labs: (a) record and image propagating activity in cortical tissue; (b) administer decision-making tasks to humans and non-human primates while simultaneously imaging neural activity as well as taking pupillometric readings; and (c) identify mechanisms underlying animal movement during foraging in social groups.
keywords
probabilistic inference/Bayesian models of decision making, stochastic processes in spatiotemporal systems, large-scale neuronal network dynamics, analyzing recurrent neural networks, dynamics of collective decisions and animal movement, nonlinear analysis of waves and patterns in partial differential equations, data-driven modeling of neuronal networks underlying sensory processing, working memory, stochastic models of foraging
APPM 2360 - Introduction to Differential Equations with Linear Algebra
Primary Instructor
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Spring 2019
Introduces ordinary differential equations, systems of linear equations, matrices, determinants, vector spaces, linear transformations, and systems of linear differential equations. Credit not granted for this course and both MATH 2130 and MATH 3430.
APPM 3010 - Chaos in Dynamical Systems
Primary Instructor
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Fall 2021
Introduces undergraduate students to chaotic dynamical systems. Topics include smooth and discrete dynamical systems, bifurcation theory, chaotic attractors, fractals, Lyapunov exponents, synchronization and networks of dynamical systems. Applications to engineering, biology and physics will be discussed.
APPM 3570 - Applied Probability
Primary Instructor
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Spring 2018
Studies axioms, counting formulas, conditional probability, independence, random variables, continuous and discrete distribution, expectation, joint distributions, moment generating functions, law of large numbers and the central limit theorem. Degree credit not granted for this course and ECEN 3810 or MATH 4510. Same as STAT 3100.
APPM 4370 - Computational Neuroscience
Primary Instructor
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Spring 2020 / Fall 2021 / Fall 2023 / Fall 2024
Applies mathematical and computational methods to neuroscience. Techniques from linear algebra, differential equations, introductory dynamical systems, probability, stochastic processes, model validation, and machine learning will be learned and used. Neuroscience topics include neural spiking, network dynamics, probabilistic inference, learning, and plasticity. Will learn how the brain uses computational principles to enact decision making, vision, and memory. Recommended background includes linear algebra, differential equations, probability, and programming. Students will hone programming skills in MATLAB/Python and TensorFlow. Recommended prerequisite: APPM 3570/STAT 3100, STAT 2600 or CSCI 3022. Same as APPM 5370.
APPM 5370 - Computational Neuroscience
Primary Instructor
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Spring 2020 / Fall 2021 / Fall 2023 / Fall 2024
Applies mathematical and computational methods to neuroscience. Techniques from linear algebra, differential equations, introductory dynamical systems, probability, stochastic processes, model validation, and machine learning will be learned and used. Neuroscience topics include neural spiking, network dynamics, probabilistic inference, learning, and plasticity. Will learn how the brain uses computational principles to enact decision making, vision, and memory. Recommended background includes linear algebra, differential equations, probability, and programming. Students will hone programming skills in MATLAB/Python and TensorFlow. Recommended prerequisites: APPM 2360 and APPM 3310 and STAT 4000 or equivalent courses. Same as APPM 4370.
APPM 5470 - Methods of Applied Mathematics: Partial Differential and Integral Equations
Primary Instructor
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Fall 2018 / Fall 2019 / Fall 2020 / Fall 2024
Studies properties and solutions of partial differential equations. Covers methods of characteristics, well-posedness, wave, heat and Laplace equations, Green's functions and related integral equations. Department enforced prerequisites: APPM 4350 or MATH 4470 and APPM 4360 or MATH 3450.
APPM 5480 - Methods of Applied Mathematics: Approximation Methods
Primary Instructor
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Spring 2021
Covers asymptotic evaluation of integrals (stationary phase and steepest descent), perturbation methods (regular and singular methods, and inner and outer expansions), multiple scale methods and applications to differential and integral equations. Department enforced prerequisite: APPM 5470.
APPM 6920 - Professional Internship
Primary Instructor
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Summer 2024
This class provides a structure for Applied Mathematics graduate students to receive academic credit for internships with industry partners that have an academic component to them suitable for graduate-level work. Participation in the program will consist of an internship agreement between a student and an industry partner who will employ the student in a role that supports the academic goals of the internship. Instructor participation will include review of internship agreement, facilitation of mid-term and final assessments of student performance, and support for any academic-related issues that may arise during the internship period. May be repeated up to 6 total credit hours.
APPM 6950 - Master's Thesis
Primary Instructor
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Fall 2020 / Spring 2021
May be repeated up to 6 total credit hours.
APPM 7400 - Topics in Applied Mathematics
Primary Instructor
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Spring 2024
Provides a vehicle for the development and presentation of new topics with the potential of being incorporated into the core courses in applied mathematics. May be repeated up to 6 total credit hours.
APPM 8400 - Mathematical Biology Seminar
Primary Instructor
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Spring 2018 / Spring 2019 / Fall 2019 / Spring 2020
Introduces advanced topics and research in mathematical and computational biology. Instructor consent required.
