Sample Syllabus: Modeling of Neuronal Dynamics
J Rinzel, Fall 2020 Wednesday, 1:25 pm-3:15 pm, WWH 517.
Blended mode: in-person (12 attendees) and remote: video & notes (asynchronous) MATH-GA 2863 Advanced Topics in Math Physiology (Courant Inst)
NEURL-GA 3042 Special Topics in Neural Science (CNS) BIOL-GA 2855 Special Topics in Math Physiology (Biology)
Contact info: rinzeljm@gmail.com
Office: Rm 919 in Courant, Rm 753 in CNS; phone: x83308
(sept 2) Overview and some “toy models” (integrate & fire) [1]
(sept 9) Membrane biophysics [2]
Electrodiffusion theory for ion fluxes: resting potential; flux through open channels (optional)
Channel gating; deterministic and stochastic treatments
(sept 16 & 23, a, b, c) Excitability and action potentials (APs) [3-4]
The Hodgkin-Huxley model
development of the model, I-V rel’ns
phase plane treatment of V-m, and V-n/h reductions
repetitive firing, Hopf bifurc’n
The Morris-Lecar model
phase plane analysis of AP, single and multiple steady states
repetitive firing: Type I (saddle-node), II (Hopf)
bistability of various types
Functional diversity of currents
(sept 30 & oct 7) Other firing dynamics [5-6]
Optional: XPPAUT (sept 30?)
onset firing: Type III, input slope detector
adaptation; post-inhibitory rebound
bursting w/ fast/slow geometric analysis
Optional: stochastic firing; phase-locking
(oct 14 & 21) Cable equation, axonal propagation [7-8]
Traveling AP and AP trains; sing perturb’n treatment; dispersion relation; kinematics
Stimulus-response properties, including Hopf bifurcation in cable.
Effects of inhomogeneities; propagation in myelinated axon [recent work of Longtin?]
(oct 28 nov 4 & 11) Dendritic signaling [9-12] possible Guest Lecturers: complex neuron model (Lytton) and local
branch computations (Jayetta Basu)
Rall’s passive model: equivalent cylinder approximation; estimating neuronal parameters; compartmental treatment a&b for [9]
Signal attenuation in passive branching trees; dendrodendritic synapses
(nov 4) Active dendrites; reduced models with few compartments; segregated currents c & d for [10]
(nov 11) Coincidence detection with dendrites; dendritic spines
(nov 18) Multi-compt models, local & selective dendritic computation (Basu)
(nov 25) Thanksgiving break.
(dec 2 & 9) Synaptic transmission and dynamics [13 & 14]
Models for postsynaptic conductance dynamics
Presynaptic considerations: calcium domains, pool depletion
Facilitation, depression, plasticity (LTP, STDP, NMDA)
Optional: Effects on cell interactions - synchronization or not.
(dec 16, tentative date) Project presentations
Some references (on reserve in Courant Library)
Koch, C. Biophysics of Computation, Oxford Univ Press, 1998.
Ermentrout B & Terman D. Mathematical Foundations of Neuroscience. Springer, 2010.
Izhikevich, EM. Dynamical Systems in Neuroscience. The Geometry of Excitability and Bursting. MIT Press, 2007. Strogatz, S. Nonlinear Dynamics and Chaos. Addison-Wesley, 1994.
Available on course web site (NYU Classes)
Rinzel & Ermentrout. Analysis of neural excitability and oscillations. In Koch & Segev (see above). Also “Live” on www.pitt.edu/~phase/
Borisyuk A & Rinzel J. Understanding neuronal dynamics by geometrical dissection of minimal models. In, Chow et al, eds: Models and Methods
in Neurophysics (Les Houches Summer School 2003), Elsevier, 2005, 19-72.
Peskin lecture notes, 2000. https://www.math.nyu.edu/faculty/peskin/neuronotes/index.html
What’s expected:
Homework: 4 or so assignments (~ 40% of grade)
Modeling project: written and oral presentation (~ 40%)
Your model or from literature; your question.
Report: 5-7 pgs. Intro, Methods, Results, Conclusions; figs & captions.
Oral presentation at end of term
Abstract due 30nov
Other: participation etc (~20%)