Instructor: Daniel J. Graham, Ph.D.
email: daniel.j.graham@dartmouth.edu
campus phone: 6-9020
office: 310 Kemeny
office hours: Weds. 1-3.
homepage: http://math.dartmouth.edu/~dgraham/
Course Homepage: http://math.dartmouth.edu/~dgraham/math_126/Math126.htm
Period: Fall 2009 02A (Tue & Thu 2:00-3:50 pm)
Some make-up classes/help sessions may be held during x-period (Weds. 4:15-5:05 pm)
Room: Kemeny 244
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), and higher-level cortical representations 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. We will also have a few guest lecturers.
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 (partly Matlab-based). Also, students will choose to either lead a discussion about a paper relevant to the course (the choices will be provided by the instructor) or to present an original project (e.g., a replication of a computational result or model, application of an existing model to a new context, etc.). A number of possibilities for projects will be suggested, and students are free to pursue their own interests outside of these choices. If students choose to perform an original project, they have the option to write a report on their work (~10 pgs., due during exam period) which will count as extra credit (amount of extra credit: up to the point value of one problem set). All students will present their chosen paper or project during the final class periods. More details about presentations will be given in coming weeks.
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. The goal of this course is to give students a strong background in current methods and models in vision science/visual neuroscience, and to inspire students to pursue new directions of research at the confluence of mathematics and brain science.
Structure of the course: We will 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 follow the Hyvarinen textbook (see below) for Weeks 2-6. Class meetings during this time will be primarily lecture-based. We will then look at advanced models, including ones that involve object recognition and categorization, as well as models of binocular rivalry. We 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. Guest lectures by Prof. Ming Meng (Psychological and Brain Sciences) and others will add breadth to the course. Classes will be more informal and discussion-based during the second portion of the course, culminating in student presentations/discussions.
Readings will be drawn from a wide variety of sources. For much of the course, we will follow Natural Image Statistics by Hyvarinen et al. The text is available as a free download here:
http://www.naturalimagestatistics.net/
A hardcopy is on reserve at Baker. It can also be purchased at Amazon for $80. A number of other books that are on reserve at Baker offer a great deal of useful background and related findings. We will read selected sections of some of these texts. All are worth owning if you intend to continue in this field, though none of these texts is required.
Matlab reference:
Matlab for Neuroscientists by Wallisch et al.
Brain/Visual system background:
Vision Science by Palmer (e-book available from library)
Neuroscience by Purves et al.
Brain Architecture by Swanson
Principles of Neural Science by Kandel et al.
Human Perception of Objects by Regan
Basic Vision, by Snowden et al.
Mathematical modeling of the brain:
Theoretical Neuroscience by Dayan and Abbott (e-book available from MIT Cognet)
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 available here)
Digital Image Processing Using Matlab (DIPUM) by Gonzalez et al.
23 Problems in Systems Neuroscience by Leo van Hemmen & Sejnowski
Mathematics references:
The Fourier Transform and its Applications by Bracewell
Pattern Classification by Duda et al.
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TENTATIVE SCHEDULE
Week 1
· Structure and philosophy of the course
· Brain intro
· Basic properties of neurons
READING: Purves: Neuroscience, Chapter 1; Hyvarinen also has a Crash Course in Linear Algebra (Chap. 19) that you should look at if you need review.
SUPPLEMENTARY READING: If you would like additional mathematics review, see Gonzalez: Linear Algebra Review
ASSIGNMENT: MATLAB
Intro tutorial pp. 12-30—If this is unfamiliar, please go through and
try out each command.
Visual system overview – Tu. 9/29
· Introduction to the visual system
· Sensation and perception
READING: Hyvarinen: Chapter 1 and Palmer: pp. 15-43. Palmer is on reserve at Baker and as an ebook through the library website.
ASSIGNMENT: MATLAB
Intro tutorial pp. 33-47—If this is unfamiliar, please go through and
try out each command.
· A brief word about light
· What do we mean by “representation”?
· Retinal representations – See Rodieck for ganglion cell model derivation, and Meister for comparison of model and experiment.
Week 3
· Bars and gratings
· Fourier transform
· Application to receptive fields – refers to Croner and Kaplan, 1995
READING: Hyvarinen Chapter 2
Supplementary Readings: Lennie: Receptive Field intro (very short). See also this nice demo of simple and complex cell receptive fields. For more background on image processing in Matlab, see this chapter from Gonzalez.
ASSIGNMENT: Problem Set I (All exercises from Hyvarinen Chapter 2)— Due 10/15.
· Autocorrelation
· Convolution
· Decorrelation in the Retina – refers to Atick and Redlich, 1992
Week 4
· Phase
· V1 anatomy
· V1 maps
· V1 receptive Fields
· Beyond V1
READING:; Hyvarinen Chapter 3
Supplementary Reading: Olshausen and Field, 2000
ASSIGNMENT: Problem Set II (All exercises from Hyvarinen Chapter 3) – Due 10/22
· Wavelets
· Review of Probability
· Spike triggered averages – see Chichilnisky, 2001 for derivation
Week 5
· Spike triggered averages, cont. – see Chichilnisky, 2001 for derivation
· An introduction to information theory
READING: Hyvarinen Chapter 4 and Chapter 8.1-8.3
ASSIGNMENT: Problem Set
III (problems listed below) – Due 10/29
Chapter
4: Mathematical Exercises #2 & #3, Computer Assignments #1 & #2
and
Chapter
8: Computer Assignment #1
Information theory and efficient coding – Th. 10/22
· Entropy of a Gaussian
· Estimating spike entropy
· Maximum entropy
· Maximum entropy in neural responses
· The efficient coding hypothesis
Week 6
· What do we mean by efficiency?
· Metabolic efficiency
· Representational efficiency
· Sparse coding
· Learning efficiency
READING: Hyvarinen
Chapter 5.1-5.7. Also: Field,
1994
· Sparse coding, cont.
· ICA: Minimizing mutual information
· Overcompleteness
· Face perception in cortex
·
Grandmother cell codes: extreme selectivity
and overcompleteness
· Invariance – Related to this paper by Wiskott
· Multidimensional Scaling of Similarities
· Representation as Similarity Space
READING: Edelman, Representation and Recognition in Vision, Chapter 1 and Chapter 2 (whole book is available on MIT Cognet); also Olshausen and Field, 2005
· The neural “dark matter” problem
Week 8
· Multi-voxel pattern analysis in cortex
READING: Kay paper + Kay Mathematics Handout (supplemental). Also: Heeger and Kemp
ASSIGNMENT: Email one question about EACH of the readings for Thursday to the class by midnight on Wednesday.
· Peter – Heeger/Attention Models
· Geethmala – Kemp/Finding Structure in Psychological Data
· Mathematical models of binocular rivalry
Guest Lecture: Cyrus McCandless – Th. 11/19
· Mathematical models of the vestibular system – Angelaki and Cullen.
Student Projects/Presentations – Tu. 11/24
· Ethan – Geisler/Contour Integration
· Nimit – Oliva/Statistics of Natural and Unnatural Images
ASSIGNMENT: Email one question about EACH of the readings for Tuesday to the class by midnight on Monday.
Student Projects/Presentations – Tu. 12/2
·
Sergey – Lewicki_1,
Lewicki_2/Efficient Auditory Coding
·
Zach – Bullmore & Sporns/Graph
Theoretic Measures of the Brain
ASSIGNMENT: Email one question about EACH of the readings for Tuesday to the class by midnight on Monday.