Introduction to Computational Neuroscience
Professor Carl Cowen
TuTh 2:30-3:30 or by appointment
Some slides from Pulfrich Effect talk
Introduction to Computational Neuroscience is being offered Spring 2004
as both MATH 490N and
This course is intended for advanced undergraduate students or
graduate students in the biological or mathematical sciences.
MATH 490N meets TuTh 12:30 - 1:20 and Tu 1:30 - 2:20 in REC 122.
BIOL 595N meets TuTh 12:30 - 1:20 and Th 1:30 - 2:20 in REC 122.
That is, the courses meet together two hours per week and separately
one hour per week; this schedule will enable the course both to serve
different audiences and to enable interaction between them.
The prerequisite for MATH 490N is Math 366, Differential Equations.
The prerequisite for BIOL 595N is two semesters of Calculus, such as Math
223 and 224 or Math 161 and 162.
Leaders in the National Institutes of Health and the National Science
Foundation believe that computational and mathematical methods increasingly
will provide the foundation for advances in the biological sciences.
This course is intended to provide an introduction to mathematical modeling
of the biological processes involved in neuroscience. The course will
begin with a brief introduction to differential equations and the basic
biology underlying the electrical processes in neurons. Classical systems
of differential equations, such as those of Hodgkin-Huxley, FitzHugh-Nagumo,
and Morris-Lecar, used to describe firing of action potentials in neurons
and their propagation through networks will be developed and analyzed.
These ideas and these models describe a diverse set of biological systems
and organisms, from action potentials in the giant axon of the squid, to
control of insulin production in pancreatic beta cells, to understanding
the effect of dopamine in the thalamus of Parkinson's patients. The course
will introduce ideas from dynamical systems to understand the behavior of
these models, especially the ways in which the behavior changes as the
inputs and biological parameters change. Since systems of differential
equations of biological importance do not (usually) have closed form
solutions, software packages Neuron and XPPAUT will be used to do modeling
and computations with the resulting models. The course will emphasize
setting up the models of neural systems and interpreting the computed
solutions in the context in which the models arose and the dependence
of the predicted behavior on the inputs. An important goal of the course
is to help prepare students to work in an interdisciplinary environment
that includes both biological and mathematical scientists.
As would be expected, the BIOL 595N section will emphasize more of the
biological issues involved and the MATH 490N section will emphasize more
of the mathematical issues involved. For example, BIOL 595N will consider
the interpretation of mathematical models and their relationship with
the results of biological experiments and implications for future
modeling or experimentation. On the other hand, MATH 490N will include
more work on phase plane analysis and bifurcations and this work will be
supported computationally by XPPAUT.
Acknowledgement: The development of this course was supported by the Purdue
Mathematics Department, the Purdue Biology Department, and by an IGMS
grant from the National Science Foundation, DMS-0308897.
For more information on the NSF's Interdisciplinary Grants in the Mathematical Sciences, check the IGMS webpage.
BIOL 595N and MATH 490N will both use Computational Cell Biology, by C. P. Fall,
E. S. Marland, J. M. Wagner, and J. J. Tyson, editors, Springer, 2002.
In addition, MATH 490N will use Non-linear Dynamics and Chaos, by S. H.
Strogatz, Westview Press, 1994.
An Introduction to the Mathematics of Biology, by E. K. Yeargers,
R. W. Shonkwiler, and J. V. Herod, Birkhauser, 1996.
An Introduction to Dynamical Systems, Continuous and Discrete, by R. Clark
Robinson, Prentice-Hall, 2004.
The course grade will be based on a midterm exam, a final exam, homework
assignments that include computation using the packages Neuron and XPPAUT,
and a group report on a published model (chosen by the group members)
that was not covered in the lectures.
This course does NOT assume that students bring both mathematical and
biological sophistication to the course, but it is assumed that students
are at the Junior level or above in one of these areas. It is expected
that students will gain an appreciation
for the kinds of information that mathematical and computational approaches
can add to understanding the functioning of a neural system, for example,
to realize that some systems are inherently more sensitive to changes
in the input parameters than others. It is hoped that students who
have completed the course will be more willing and more able to incorporate
mathematical or computational approaches into their own biological work
or see ways in which their own mathematical work can be used in the
MATH 490N has been approved for graduation credit counting toward the
Math major in the Core, CS, and Applied Math options and will count
toward a Math minor. See your counselor for further clarification.
Software for the Course
There are several pieces of software that we will use in the course.
ITAP has made XPPAUT and NEURON available on both the PC's and the Mac's
in the ITAP labs. SNNAP is a Java program so it can be downloaded as
a "jar" file and accompanying documents into your home directory on
ITAP machines and then run by Java on those machines. In addition,
because all three programs are free, you can download them
onto your own computers for the "price" of doing the installation.
There are other programs that can solve systems of differential equations
and do other kinds of related computations and graphics that are not
free but are available on the ITAP machines. These programs include
MATLAB, MAPLE, and MATHEMATICA.
XPPAUT is a
program developed by mathematical biologist Bard Ermentrout at the University
of Pittsburgh that solves systems of ODE's, plots the phase diagrams, and
(more unusual) plots the bifurcation diagrams. It is idiosyncratic,
but useful! XPP Crib Sheet
Textbook's website has
.ode files for many of the illustrations of the book.
NEURON is a simulation
program developed at Yale University that can be used to model neurons
and networks of neurons. It
works from a biological description of the network and the differential
equations are hidden from the user. It is also available at a second
site at Duke University. Interestingly,
while the software is the same at both sites, the reference materials
available at the two sites are different.
SNNAP, the "Simulator for Neural
Networks and Action Potentials," is a more modest program than NEURON
developed at the University of Texas at Houston Medical College. SNNAP
appears to be simpler to use, and apparently requires less programming,
We will discuss these programs in class.
Approximate Course Outline
This outline will be dynamic, updated as the topics and activities are
decided. This page will be kept as up to date as possible.
For each date, the first entry is the common class meeting. Entries beginning
with (Math) are meetings of Math 490N and entries beginning with (Biol)
are meetings of Biol 595N. You are welcome to come to class meetings
of both courses if you wish.
1/13 Course Organization, Introduction, Mathematical Models (slides)
1/13 (Math) Cells, Neurons, the Nervous System (slides)
1/15 Differential Equations
1/15 (Biol) Math Refresher, Some Simple Differential Equations
Homework 1, Due Tuesday, 1/27
1/20 More Differential Equations, Phase Portraits
1/20 (Math) Linear Systems and Linearization of Non-linear Systems
Strogatz, Chapter 2
1/22 Bifurcations, Bifurcation Diagrams
1/22 (Biol) Graphing for Differential Equations
Strogatz, Chapter 3
1/27 Intro to Computational Modeling in Biology
Voltage Gated Ion Channels
1/27 (Math) Description of Activation and Inactivation of Channels
Fall, et al., Chapters 1, 2
1/29 Voltage Gated Ion Channels, Morris-Lecar Models
1/29 (Biol) Morris-Lecar on the Barnacle Giant Muscle (slides)
Fall, et al., Chapter 2
Homework 2, Due Tuesday, 2/10
2/3 More on the Morris-Lecar Model for the Barnacle Giant Muscle
2/3 (Math) Using Fast and Slow Equations
2/5 Synapses and Ion Channels (Professor Christie Sahley)
2/5 (Biol) Synapses and Ion Channels (Professor Christie Sahley)
2/10 Pictures from XPP-Auto
2/10 (Math) More on Using XPP
2/12 Synapses and Ion Channels (Professor Christie Sahley)
2/12 (Biol) Synapses and Ion Channels (Professor Christie Sahley)
2/17 Calcium control in Cell and ER (Text, Section 5.1)
2/17 (Math) Phase Planes and more on using XPP
Homework 3 Due Tuesday, 2/24: From Strogatz
Use XPP to get bifurcation diagram for problems
3.1.2, 3.2.1, 3.2.2, 3.4.1, 3.4.7
2/19 Calcium control in Cell and ER, II (slides)
2/19 (Biol) Papers on oscillations in the bullfrog ganglion
Special! Brian A. Wandell, Stanford University,
"Computational Neuroimaging: Cortical Color
Responses in Human and Macaque"
3:30p Stewart, Room 202
2/24 Calcium control in Cell and ER, III BFSG closed system ode file
2/24 (Math) Some Differential Equations Theory
2/26 Calcium control in Cell and ER, IV BFSG open system ode file
2/26 (Biol) Biological interpretions from the XPP data
3/2 Calcium control in Cell and ER, V BFSG open system, reduced, ode file
Calcium in the Pituitary Gonadotrophs, I (slides)
(Math) More on Differential Equations Theory a sample problem
3/4 Calcium in the Pituitary Gonadotrophs, II Pituitary Gonadotroph, closed system, ode file
(Biol) Calcium oscillations and exocytosis in the
Pituitary Gonadotrophs (slides)
Special! David Crews,
"Evolution of Neuroendocrine Mechanisms Controlling
Male-and Female-Typical Sexual Behaviors"
3:30p Stewart, Room 214D
(Math) More on Differential Equations Theory
Solution of System Example
3/11 Midterm Test 12:30 - 1:20 Topic List
3/11 Extra time to complete Midterm Test 1:20 - 2:20
Special! Alan Slater, University of Exeter,
"Vision in the Young Infant:
From Sensation to Perception to Representation"
3:30p Stewart, Room 202
3/16 SPRING BREAK! No Classes
3/18 SPRING BREAK! No Classes
3/22 Registrar: Last day to drop a course, 5:00pm.
3/23 Discussion of Midterm Test
(Math) Existence and uniqueness, stability (from Robinson)
3/25 Report topics due for approval
3/30 No Class
4/1 Gap junction connected Morris-Lecar Neurons A (ode file), B odefile.
4/6 Synaptic, mutually excitatory Morris-Lecar Neurons excitatory (ode file), inhibitory (ode file).
(Math) Relating stability of non-linear system to linearization
4/8 No Class
4/13 Discussion on BOLD, Cortical Activity in Movie Watching
SEE Reading Assignment
(Math) no meeting
4/15 Integrate-and-Fire Models; the Central Pattern Generator
for swimming in Tritonia diomedea (slides)
(Biol) Getting the parameters to support computational models
4/20 Group Presentations, Group 5, then Group 2,
(Math) Periodic orbits (Robinson, Chapter 6)
4/22 Group Presentations, Group 1, then Group 4,
(Biol) Parkinson's disease and modeling of activity in
the globus pallidus and subthalamic nucleus (slides) Terman files
4/27 Group Presentations, Group 3
(Math) More on periodic orbits (Robinson, Chapter 6) Poincare Bendixson example ode file
Written Reports Due
Group 1: Paths to Diabetes (Refs omitted)
Group 2: Dopaminergic Modulation of Na+ Currents
Group 3: Leech Heart CPG
Group 4: Retinal Ganglion
Group 5: Binocular Rivalry
(Biol) No official class meeting
Optional presentation on Pulfrich effect and axonal transport
5/7 Final Exam: Friday, May 7, 1:00p - 3:00p, REC 122 Topic List
Last Update: May 2, 2004
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