Math Neuroscience

Bard Ermentrout
Thackeray 502
x48324
bard@pitt.edu
www.pitt.edu/~phase

Primary References

• Neural Excitability-HTML or PDFs: TEXT and FIGURES
• Neural Networks as Spatio-temporal pattern forming systems
• Review of coupled oscillators

Grades

1. Homework X%
2. Exam Y%
3. Project (100-X-Y)%

Syllabus (rough)

• Single cells
  • Homework 1 and the associated XPP file
  • Channels and bifurcations
    • Type I; saddle-nodes
    • Type II; Hopf bifurcation
  • Phaseplanes and reduced models MORE FUN HOMEWORK 2
  • Bursting and slow passage through bifurcations
  • Numerical bifurcation analysis and phase-planes; Quick tour of qualitative analysis with XPP and bifurcation of HH eqns
  • Noise and first-passage Some notes and HW
• Action potential propagation Homework and explanations
  • Cable equations
    • Linear theory
    • Transients and steady states; BVPs
    • Green's functions
  • Nonlinear fronts
  • Singular perturbation and APs
  • Stability of pulses and Evans functions
• Coupled cells
  • Weak coupling of oscillators Notes and HW and an ODE file
  • Kuramoto theory Strogatz'review
  • Weak coupling near bifurcations Homework
    • Hopf
    • SNIC
  • Slow coupling and firing rate reductions
• 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

Simulation software