# Math Neuroscience

## Primary Reference

1. Homework 75%
2. Project 25%

Note that this course requires that you know some linear algebra and nonlinear dynamics. I will give you a good intro to the latter but you may want to look at some other books for more help. I recommend:

• Steven Strogatz Nonlinear Dynamics and Chaos
• Eugene Izhikevich Dynamical Systems in Neuroscience

I will also be doing some perturbation theory. The book by Jim Keener Principles of Applied Math is excellent for this.

You will also be required at some points to do some numerical stuff such as calculating phaseplanes, bifurcation diagrams, and possibly solving boundary value problems. For this, I recommend XPPAUT which runs on most platforms. If you prefer, there are other packages that are similar such as the Python interface to AUTO, PyDstool, and Matcont which works with MATLAB. As needed, I will provide some guidance here.

## Syllabus (rough and subject to change as the term goes on

• Week 1: Chapter 1.1-1.5; Chapter 2.1-2.4;
• Lectures 2 and 1
• Homework 1: page 28: 4,6,7; page 45:1,2,3,4,5,9 (Due Friday January 22) And this cool problem!
• Here is a nice long XPP tutorial on the dynamics of the Lecar model XPP Tutorial I don't want the HW, it is more for your practice. You need not go beyond two-dimensions for now. You can get the download info Here Look for windows/m,ac specific instructions. Please note that the setup in the tutorial is old and you should ignore it. The best thing to do is to download xpp via the website and then type in the ODEs.
• This Morris-Lecar tutorial has the odes in the correct syntax and is specific to the stuff we did in class.
• Week 2: Dynamics intro - Chapter 3. (see also the above lecture notes)
• Jan 29 - finish up with dynamics
• Feb 1-5: Channels and Bursting, chapters 4,5. Here is some homework due Feb 12
• File with many models
• Above is a plain text file that contains many different models in XPP format. There is a brief description of each of them.
• To simulate them, use the cut an paste option to cut out whichever you want and save it as a PLAIN text file and then run it with XPP. I think several have suggested values of parameters.
• You can also cut directly from this web link
• USE THIS FOR PROBLEM 2 PAGE 95 Just run it and type IG ILILIL then pull up AUTO. Then Run Steady state. Grab the second HB (point 3) and Run periodic!
• We will be discussing these current and channels for a week or so. If you go towards the end, there is a big file that has almost all the channels. You might want to follow along with this fun tutorial!
• Feb 15-19. Bursting bursting bursting
• HOMEWORK 3. Due Friday Mar 4

There are typos in the HW 3. Here they are:

• Problem 1. W' = - epsilon W
• Poblem 3. In the integral, in case it is not clear the denominator is r0 -[1/2+sqrt(1/4+p)]
• Problem 4. The frequency should be divided by pi.
• Problem 6 - use the f(V) that is in problem 5
• NOTE this HW doent have many sims!
• Notes on bursting and action potentials
• Weeks 5-6: Action ptential propagation, waves, and synapses Homework due Friday before spring break starts
• Week of Feb 29 - chapter 7 and start chapter 8 - finally some networks!!
• Feb 29-Mar 4 - we will start the sudy of oscillators & weak coupling (Chapter 8 Erm & Term)
• There will be class next 2/29-3/4
• March 21-25 Chapter 9 (relaxation oscillators) and maybe some noise!!
• Homework number 4 and FN ode and coupled FN ode in XPP format This is due on April 1
• Week of March 28 - Noise and firing rate models
• Week of Apr 4 Chapter 11
• Apr 11-22 Chapter 12
• Hopfield networks
• Spatial networks
• Waves
• Ring models & working memory & line attractors
• Hallucinations
• Probably some sort of chaotic states - random networks
• Maybe some learning - Kohonen algorithm??
• LAST HOMEWORK!!! Due the last day of class (4/29)
• Last week of class (Apr 25-29)- present projects. What is a project?
• You can pair up (preferred) or if you absolutely have do, go solo
• You could redo the simulations and analysis is a paper that you pick and run by me
• You could do part of one of the projects in the book (they are suggested after the exercises)
• You could make something up
• The project should not just be computational and should include some analytic or "semi-analytic" component. (The latter includes computing a bifurcation diagram or doing some sort of averaging or solving a Fokker-Planck equation, or something along those lines)
• You and your partner will give a 15 minute talk during the last week of class. April 25-29 . I do not need a write up