Get your hands on mathematical neuroscience

Yesterday, I had the pleasure to introduce the speaker at a seminar: it was Bard Ermentrout. Already last year, Springer send me a copy of his book to review, so when, if not now, will be a better time to do this and reuse and expand my freshly crafted introduction?

In quite a few posts, I argued that math matters in neuroscience (e.g. see "Math Matters, Apply It To Neurology" or see here and here). Bard is one of the modern pioneers that began to use and develop mathematical methods to attack problems in neuroscience. I write modern pioneers because this field dates back to 19th century with Emil du Bois Reymond^* and Hermann von Helmholtz, known as "organic physics" then. So the term "modern" refers to methods of dynamical systems theory, a field within applied mathematics that describes complex behavior of dynamical systems by employing differential equations.

Trained as a mathematician, Bard got his PhD at University of Chicago in Biophysics, where he worked with Jack Cowan on oscillations in neuronal networks and a mathematical theory of visual hallucinations. This was in the late 1970's. While oscillations in neural networks is still a hot topic today, our common interest in visual  hallucinations is not shared by too many researchers nowadays. Anyway, to date, Bard Ermentrout is a Professor at the Department of Mathematics, University of Pittsburgh and has published about 170 scientific papers mostly in mathematical neuroscience but also other fields of biology.

Bard Ermentrout together with David Terman, who is a Professor of Mathematics at the Ohio State University, wrote the book "Mathematical Foundation of Neuroscience". It is one of the best textbooks around in this field. It gives you a solid understanding of the mathematics behind the Hodgkin-Huxley nerve model, dynamics in dendrites, synaptic communication, neuronal oscillators and networks of them, firing rate models, neural fields, and more.

And of course, these models should be tried out with XPP. XPP is a software tool Bard together with co-workers created to numerically solve nonlinear differential equations. If you want to get into this field, to my mind, this is the tool to go for (not Matlab or anything like this, although the book stands on his own, and if you have to use Matlab, well ...). I use both, his book and XPP in my lecture. 

XPP is free, runs fast, has many build-in features you need to visualize and understand dynamical systems (direction fields, nullclines etc. see below), and it has an interface to AUTO, another great free software package for continuation and bifurcation analysis. If these terms do not mean anything to you, you are probably a young student. You may want to read their book first, but the ultimate argument for you is probably that since March 19 XPP runs on the iPad. 

Screenshot of XPP on the iPad

Another screenshot of XPP's Steelers edition.

and soon there will be the Steelers edition, which can also be used by German BVB fans (much appreciated, as I lived for many years in Dortmund).

So one of the strength of the book "Mathematical Foundation of Neuroscience" is that the topics are illustrated throughout with hundreds of figures, 38 of them in color and hundreds of exercises that will help you to get your hands on mathematical neurscience.

Phase space representation of a neural oscillator.

 

Footnote

^* We had dinner on that Friday at the place where Emil du Bois Reymond used to live in Berlin, only that the buildnig is long gone.