Mathematical modeling at the molecular and cellular level

The goal of the class is to present methods in applied mathematics, modeling and simulations to study computational problems in neuroscience at the molecular and cellular level. We will study neuronal structures at different scales, form synapses to neural networks. From specific examples, we will develop mathematical models and their analysis to determine how the properties of neurons at the molecular level shape their activity and propagate to the network level. This change of scale will be formulated and analyzed using several tools such as partial differential equations, stochastic processes and numerical simulations.
We will first consider the classical electrical circuit analogy for modeling signal propagation in neurons, and its limitations. We will mainly develop two examples: a model of neuronal network (the pre-Botzinger Complex), and a model for the myelinated axon. In a second part, we will develop a model for short-term plasticity at the pre-synaptic terminal. Using the mean first passage time theory, we will coarsely grain the problem, initially formulated using a reaction-diffusion system in a complex geometry, into fast stochastic simulations.


 

Syllabus

Presentation of the course: lectures 1 to 5: the electrical circuit analogy, lecture 6: PNP, lecture 7 to 10: modeling tools at the molecular level. Pre-requisites and reminders. Basic neuronal physiology.

Lecture 1: Introduction (defining a model and its limits).

Lecture 2: Nernst potential, modeling ionic flows through the neuronal membrane (single compartment level). Biochemistry approach. Hodgkin Huxley and its simplifications (single compartment level).

Lecture 3: Modeling ionic channels, from HH to Markovian models.

Lecture 3: Modeling example with the pre-Botz (network of neurons, using a single compartment level).

Lecture 4: Signal propagation, Cable theory with HH. Application: myelin model (several compartments).

Lecture 5:  How to implement the Cable theory?  (coding tools).

Lecture 6: Limitation of the cable theory, PNP.

Lecture 7: Brownian motion and mean first passage time. Link with Poisson process, and Gillespie simulations.

Lecture 8: Chemical reactions.

Lecture 9: Multi-scale modeling: application to the modeling of the pre-synaptic terminal.