Modeling Perception and Cognition

MATH 126 - Current Problems in Applied Mathematics:

Mathematical Methods in Visual Neuroscience

Instructor: Daniel J. Graham, Ph.D.

email: daniel.j.graham@dartmouth.edu

campus phone: 6-9020

office: 310 Kemeny

http://math.dartmouth.edu/~dgraham/

Course Homepage: http://math.dartmouth.edu/~dgraham/math_126/Math126.htm

Period: Fall 2008 02A (2:00-3:50 pm)

Some make-up classes may be held during x-period (Weds. 4:15-5:05 pm)

Room: Kemeny 006

Overview: This course is designed to introduce graduate students and advanced undergraduates from a variety of disciplines to mathematical methods used to study and model neural mechanisms underlying the function of the human visual system. The course will focus primarily on neural coding strategies in the early visual system (retina, thalamus, primary visual cortex), though higher-level cortical representations and object recognition strategies will also be discussed. General topics will include neuronal and large-scale models, information-theoretic approaches, efficient coding theory, and nonlinearities. Mathematical tools including basic descriptive statistics, Fourier analysis, correlation analysis, and wavelets will be discussed. Some class periods will be conducted as seminars, where we will examine current papers in the field with discussions led by students.

Prerequisites: No background in neurobiology is expected. Experience with differential equations, linear algebra, computer programming (especially Matlab) and basic biology will be very useful.

Evaluation: Students will be evaluated based on problem sets (generally Matlab-based) and a final project.

Goals: One way of thinking about the class is as a survey of a vital area of neuroscience research, an area that is in special need of young researchers with strong applied math skills. A full understanding of this area also requires substantial knowledge of neurophysiology, which will be presented.

On a practical level, the course is designed in part to allow students to understand the following paper by Kay et al. about “mind reading” in the visual brain by the end of the term: http://www.nature.com/nature/journal/v452/n7185/abs/nature06713.html

This paper is particularly well-suited for this purpose since it bridges a variety of concepts, models, and techniques from neuroscience and mathematics. Students can expect to learn why the authors of this paper did what they did, why it “worked” and what the result means in the context of previous work. The first author of this paper, Kendrick Kay, will give a guest lecture in class Nov. 20.

Structure of the course: The current plan for the class is to begin with basic neurobiology, working our way from the retina to cortex. The emphasis will be on examining methods of study and models of visual coding in the early visual system. We will then look at advanced models, including ones that involve object recognition, scene categorization, and learning. The final unit will look at a variety of recent papers that deal with visual coding beyond primary visual cortex; with evolutionary and developmental constraints on vision coding; and with assorted nonlinear and emergent phenomena.

Reading

Readings will be drawn from a wide variety of sources. A number of books that are on reserve at Baker Library offer a great deal of useful background and related findings. All are worth owning if you intend to continue in this field, though no texts are required.

Mathematical modeling of the brain:

Theoretical Neuroscience by Dayan and Abbott (e-book avail. from library)

Introduction to the Theory of Neural Computation by Hertz et al.

Spikes by Rieke et al.

Biophysics of Computation by Koch

Information Theory, Inference and Learning Algorithms by Mackay (e-book avail.)

Digital Image Processing Using Matlab (DIPUM) by Gonzalez et al.

23 Problems in Systems Neuroscience by Leo van Hemmen & Sejnowski

Brain/Visual system background:

Brain Architecture by Swanson

Principles of Neural Science by Kandel et al.

Vision Science by Palmer (e-book avail. from library)

Neuroscience by Purves et al. (earlier ed. available as e-book)

Human Perception of Objects by Regan

Basic Vision, by Snowden et al.

Mathematics references:

The Fourier Transform and its Applications by Bracewell

Pattern Classification by Duda et al.

TENTATIVE SCHEDULE

Week 1

Introduction – Th. 9/25

· Structure and philosophy of the course

· Brain intro

· Basic properties of neurons

READING: Purves: Neuroscience, Chapter 1; Gonzalez: Mathematics Review

Week 2

Visual system overview – Tu. 9/30

· Introduction to the visual system

· Sensation and perception

READING: Palmer: pp. 15-43. On reserve at Baker and 1 copy available as an ebook through library website. Also: Wassle and Boycott, 1991; Meister and Berry, 1999;

ASSIGNMENT: MATLAB Intro tutorial—This should be a review. If this is unfamiliar, please go through and try out each command.

Retinal representations – Th. 10/2

· A brief word about light

· What do we mean by “representation”?

· Retinal representations

Week 3

Early Vision: Primary Visual Cortex (V1) – Tu. 10/7

· V1 anatomy

· V1 maps

· V1 receptive Fields

· Beyond V1

READING: Lennie: Receptive Field intro (very short), Lennie: Visual Cortical Organization (very long—skip sect. 5 for now). See also this nice demo of simple and complex cell receptive fields.

ASSIGNMENT: Problem Set I — Due 10/31. You will need this zip file, which contains DIPUM Ch. 3, image1.png, the function rotavg, and a paper by Atick and Redlich.