The purpose of this class is to teach how to apply methods of differential equations to answer questions in neuroscience. If response B in a neuron occurs after stimulus A after time delay T, how do you describe it with an equation? If there is only one such neuron, you probably would not need mathematics. But a neuronal system behaves through interactions among many neurons and as such, it is impossible to understand the whole system without using mathematics. This course does not attempt to do formal training in differential equations per se, but rather to challenge students to think about underlying concepts behind the equations so they can feel comfortable with using the language of differential equations.

Week 1
Introduction: How to model a biological system? whiteboard-1 (pdf)

Week 2
How to model: A case of dripping water tank whiteboard-2 (pdf)

Week 3
A case of dripping water tank revisited whiteboard-3 (pdf)
and linear equations, plus some digressions
Supplementary textbook: Keith Hirst (2006) Calculus of One Variable. Springer.

Week 4
Gating of water drips and neural net consequences whiteboard-4 (pdf)
with preliminary ideas of Hodgkin-Huxley formalism

Week 5
A model of estrous cycles (or "Sanduhr" mechanism) whiteboard-5 (pdf)
and re-interpretation of Newton's first law

Week 6
A model of brain's metabilite clearance whiteboard-6 (pdf)
with a short note on linear systems in physics and Fourier transform
*Homework: basic metablite clearance modeling

Week 7
Solution to the metabolite clearance problem whiteboard-7 modified (pdf)
with a very short note on numerical methods

Week 8
Hodgkin-Huxley model of an "active gating valve" on the leaky water tank whiteboard-8 (pdf)

Week 9
Constructing your own active gating model & bifurcations in type I and II neurons whiteboard-9 (pdf)
with a recap on H-H model and Nernst potential
*Homework: possible dynamics of x given x_th in our simple active gating model

Week 10
Homework solution, Q&A, and numerical integration whiteboard-10 (pdf)

Week 11
Ideas and tips for a term project - self-organized criticality (SOC) whiteboard-11 (pdf)

Week 12
Synapse - calcium signaling, exocytosis and receptor pathways whiteboard-12 (pdf)
Also, further ideas and tips for a term project - water homeostasis modeling

Week 13
Cell signaling, molecular switch and amplification whiteboard-13 (pdf)

Week 14
Gene expression modeling (1) - historical background whiteboard-14 (pdf)

Week 15
Gene expression modeling (2) - more formalism whiteboard-15 (pdf)

Week 16
Thesis project (no class) no material.

Week 17
A water tank oscillator (wattankillator) whiteboard-17 (pdf)

Week 18 (last week)
Circadian clock model and corresponding modifications to wattankillator (now bucketillators) whiteboard-18 (pdf)
Mathematica notebook in pdf.

NB: All notes presented here are intended for enrolled students.