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Kilpatrick, Zachary P Associate Professor

Positions

Research Areas research areas

Research

research overview

  • 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 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. 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 swarm intelligence in house-hunting honeybees.

keywords

  • stochastic processes in spatiotemporal systems, large-scale neuronal network dynamics, probabilistic inference models of decision making, analyzing recurrent neural networks, dynamics of swarm/collective decisions, nonlinear analysis of waves and patterns in partial differential equations, data-driven modeling of neuronal networks underlying sensory processing, motor output, working memory

Publications

selected publications

Teaching

courses taught

  • APPM 2360 - Introduction to Differential Equations with Linear Algebra
    Primary Instructor - 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 3570 - Applied Probability
    Primary Instructor - 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 - Spring 2020
    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 - Spring 2020
    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. Same as APPM 4370.
  • APPM 5470 - Methods of Applied Mathematics: Partial Differential and Integral Equations
    Primary Instructor - Fall 2018 / Fall 2019
    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 6950 - Master's Thesis
    Primary Instructor - Fall 2019 / Spring 2020
    May be repeated up to 6 total credit hours.
  • APPM 8000 - Colloquium in Applied Mathematics
    Primary Instructor - Spring 2018
    Introduces graduate students to the major research foci of the Department of Applied Mathematics.
  • APPM 8400 - Mathematical Biology Seminar
    Primary Instructor - Spring 2018 / Spring 2019 / Fall 2019 / Spring 2020
    Introduces advanced topics and research in mathematical and computational biology. Instructor consent required.

Background

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Github

  • zpkilpat