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