Ken Miller's math notes: Linear Algebra for Theoretical Neuroscience
These are some notes I've written to try to teach linear algebra and
related aspects of linear differential equations to students of
theoretical neuroscience. I've also included a nice set of notes written
by Philip (Flip) Sabes of UCSF
when we were co-teaching a course; these are best read after Part 3.
Most neuroscience students seem
to find they never make it through Part 4, which is on the Fourier
transform -- too much detail and too little motivation -- but the
rest seems to work reasonably well in getting the conceptual ideas
across to motivated biologists who don't have much background. Part 4 stands alone, and
can be omitted, but read it if you'd like to better understand the
Fourier transform and in particular understand that it is just another
coordinate transformation (a particular one that diagonalizes a
particular set of matrices or linear operators and hence is used
particularly often).
Although these notes work reasonably well -- particularly if they're used
in or followed by a course in which the ideas are used in the
context of real biological problems -- they also leave a lot to be
desired. They need many more
figures, many more neuroscience examples, and more and better problems.
I'd also like to include an
introductory chapter reminding people of basics of 1-dimensional
linear differential equations and the exponential function, before heading
into multiple dimensions. I'd like to include some chapters on
probability, and in particular on Poisson and Gaussian
distributions (and the linear algebra leads naturally to
understanding multi-dimensional Gaussian distributions). At that
point, it would probably become "Mathematics for Theoretical
Neuroscience" rather than "Linear Algebra for Theoretical Neuroscience".
Part 3, which deals with non-normal matrices -- matrices that do
not have a complete orthonormal basis of eigenvectors -- needs to
be completely rewritten: since it was written, I've learned that
non-normal matrices have many features not predicted by the
eigenvalues that are of great relevance in neurobiology and in
biology more generally, and the notes don't deal with this
(in the meantime, for the mathematical
aspects, see the book by L.N. Trefethen and M. Embree, Spectra and
Pseudospectra: The Behavior of Nonnormal Matrices and
Operators. Princeton University Press, 2005). And Part
4 runs out of steam where I start talking about the connections
between vectors and functions, matrices and linear operators,
Kronecker deltas and Dirac deltas, and even more where it talks about
multi-dimensional Fourier transforms. This all needs more work.
Just haven't had the time. If you are interested in taking on any
of these projects, particularly (but not limited to) figures,
examples, or problems, or in adding other useful pieces of mathematics,
let me know. Perhaps we can collaboratively build a useful resource.
All feedback on making these notes better will be appreciated.
I'm not certain when I'll have time to implement them, but hope
to.
I'd be very happy if you linked to this page. I'd prefer you link
rather than posting the material yourself, both because (1) that
way the creative commons license stays with the material and (2)
that way people are always pointed to the latest versions, in case
I should find the time to update.
Here are the notes:
And here's Flip Sabes' notes on linear algebraic
equations, SVD, and the pseudo-inverse:
Linear Algebra for Theoretical Neuroscience by Kenneth D. Miller is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License.
Linear Algebraic Equations, SVD, and the
Pseudo-Inversee by Phillip
N. Sabes is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License.
Ken Miller
Last modified: Thu Aug 28 11:45:45 EDT 2008