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

1. (sept 2) Overview and some “toy models” (integrate & fire) [1]

2. (sept 9) Membrane biophysics [2]

   1. Electrodiffusion theory for ion fluxes: resting potential; flux through open channels (optional)

   2. Channel gating; deterministic and stochastic treatments

3. (sept 16 & 23, a, b, c) Excitability and action potentials (APs) [3-4]

   1. The Hodgkin-Huxley model

      1. development of the model, I-V rel’ns

      2. phase plane treatment of V-m, and V-n/h reductions

      3. repetitive firing, Hopf bifurc’n

   2. The Morris-Lecar model

      1. phase plane analysis of AP, single and multiple steady states

      2. repetitive firing: Type I (saddle-node), II (Hopf)

      3. bistability of various types

   3. Functional diversity of currents

   4. (sept 30 & oct 7) Other firing dynamics [5-6]

      1. Optional: XPPAUT (sept 30?)

      2. onset firing: Type III, input slope detector

      3. adaptation; post-inhibitory rebound

      4. bursting w/ fast/slow geometric analysis

      5. Optional: stochastic firing; phase-locking

4. (oct 14 & 21) Cable equation, axonal propagation [7-8]

   1. Traveling AP and AP trains; sing perturb’n treatment; dispersion relation; kinematics

   2. Stimulus-response properties, including Hopf bifurcation in cable.

   3. Effects of inhomogeneities; propagation in myelinated axon [recent work of Longtin?]

5. (oct 28 nov 4 & 11) Dendritic signaling [9-12] possible Guest Lecturers: complex neuron model (Lytton) and local

   branch computations (Jayetta Basu)

   1. Rall’s passive model: equivalent cylinder approximation; estimating neuronal parameters; compartmental treatment a&b for [9]

   2. Signal attenuation in passive branching trees; dendrodendritic synapses

   3. (nov 4) Active dendrites; reduced models with few compartments; segregated currents c & d for [10]

   4. (nov 11) Coincidence detection with dendrites; dendritic spines

   5. (nov 18) Multi-compt models, local & selective dendritic computation (Basu)

6. (nov 25) Thanksgiving break.

7. (dec 2 & 9) Synaptic transmission and dynamics [13 & 14]

   1. Models for postsynaptic conductance dynamics

   2. Presynaptic considerations: calcium domains, pool depletion

   3. Facilitation, depression, plasticity (LTP, STDP, NMDA)

   4. Optional: Effects on cell interactions - synchronization or not.

8. (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%)