Modeling Neurobiological systems, a mathematical approach Weizmann Institute 2004, D. Holcman

Examples Where are the mathematical problems? Synaptic plasticity: Receptors movements Sensor cells: Photo-transduction Dynamics of transient process

Synaptic plasticity: Receptor trafficking

Synapse

Receptor trafficking

Mathematical Modeling How long it takes to escape from micro-domains How to compute a coarse-grained diffusion constant? Answers: Formulate a stochastic equation and solve the associated Partial Differential equations

Exit from a small opening

Photo-transduction

diffusion in a single cone

Geometry of the cone outer- segment

Response curves of photon detection

Dark noise in the outer-segment of photo receptor cells

Two dimensional random walk of a Rhodopsin molecules

Mathematical modeling How to model amplification: 1-Photon  change at the cellular level. 2-Single photon response-curve Amplification, how to model 1-chemical reactions, diffusion 2-Noise 3- explain cone rods difference.

Mathematical tools What is a chemical reaction at a molecular level. Computation of chemical constant: forward a backward binding rate Reaction-Diffusion equations Analyze the role of the cell-geometry Noise analysis: solve PDE and stochastic PDE

Dynamics in microstructures: dendritic spines

Dendritic spines

Calcium dynamics in a spine

Model transient dynamics Model effect of few ions: 1-Chemical reactions 2-effect of the geometry 3-find coarse-grained approach Produce a simulation, based at a molecular level

Simulation of Ca dynamics in a dendritic spine D.Holcman et.al, Biophysical J. 2004

Conclusion Purpose of the class Describe microbiological systems and predict the function. Organization of the class Stochastic, Brownian motion Stochastic equations, Ito calculus. PDE( elliptic and parabolic, linear and nonlinear) Asymptotic analysis examples: compute Chemical reaction constants Neurobiological examples