# Math Neuroscience

## Primary References

The grades are based on
1. Homework 75%
2. Project 25%

## Syllabus (rough)

(Ignore the dates, these will change)
• Single cells
• Nernst equation, CFE, etc
• Homework: page 10: 1,2,4,6,7
• Equivalent circuits
• Homework: page 13: 2,3,4.
• Dendrites/compartmental modeling
• Homework: page 18:1,2,3,4
• HW Due Jan 18
• Cable equations
• Linear theory
• Transients and steady states; BVPs
• Green's functions
• HW Due Jan 25 last section of chapter,1b,c; 2,4,7
• Week of January 23 - Feb 8 chapter 3 (active channels)
• Introduction to voltage-gated channels
• Hodgkin-Huxley equations 3.2 exercises 2-5 (p44)
• ODE files from the book
• Morris-Lecar model
• Bifurcations to limit cycles
• Type I/Type II
• Other channels
• Simple models page 55 exercise 4,6,8,9,11,13 (Due wed Feb 15)
• Integrate and fire
• Theta model
• Week of Feb 13th
• Chapter five - synaptic channels (one day only, no exercises)
• Analysis of Neural excitability and Figures - all about bursting in here!
• We start Noise - chapter 6 A short version of the material
• Here is an XPP file for computing first passage times for a simple scalar neuron model. The figure shows some histograms for (red I=0,sigma=0.2;green I=0,sigma=0.4; white I=-.1,sigma=.2)
• Here is an XPP file for computing the FI curve for a simple scalar neuron model. The plot shows the FI curve as sigma goes from 0 to 0.5.
• Here is a very complex XPP file to solve the first passage BVP using AUTO. The plot shows the results of this for sigma=1 and 0.1 Note AUTO has a bit of a problem when I is negative and large.
• The FI curve with noise can be approximated by the following function. Let F(I) be the deterministic curve and define

L(x,b) =x/(1-exp(-b*x))

As b gets large, L(x,b) approaches the curve max(x,0). So, 1/b is like the amount of noise and FI(I) = F(L(I,b)) is a decent fit. We will use something like this later when we create firing rate models.

• Noise related homework: page 153: 1,2 (computer exercise);4,6,8
• Neural oscillators
• Start reading the chapter on oscillators
• HOMEWORK: oscillator chapter: 7,23,26,27. I'd assign many more, but you wouldnt do 'em anyway.
• Coupled cells
• Slice models/wave propagation
• Single spike assumptions
• Firing rate models
• Asynchronous states
• Neural networks Neural net dynamics - Randy Beer
• Spatial models
• Bumps
• Waves
• Spatially periodic behavior

XPP tutorial