Jan Verwer

CWI and

Univ. of Amsterdam

A Scientific Computing Framework for Studying Axon Guidance

Computational Neuroscience Meeting, NWO, December 9, 2005

Centrum voor Wiskunde en Informatica

Scientific Computing

Scientific Computing

Computer based applied mathematics

Scientific Computing

Computer based applied mathematics, involving

• Modelling

• Analysis

• Simulation

Scientific Computing

Computer based applied mathematics, involving

• Modelling Prescription of a given problem in formulas, relations, equations. Approximating reality.

Here the application is prominent. • Analysis

• Simulation

Scientific Computing

Computer based applied mathematics, involving

• Modelling Prescription of a given problem in formulas, relations, equations. Approximating reality.

Here the application is prominent. • Analysis Study of mathematical and numerical issues (stability, conservation rules, etc).

Here the mathematics is prominent.

• Simulation

Scientific Computing

Computer based applied mathematics, involving

• Modelling Prescription of a given problem in formulas, relations, equations. Approximating reality.

Here the application is prominent. • Analysis Study of mathematical and numerical issues (stability, conservation rules, etc).

Here the mathematics is prominent.

• Simulation Programming, benchmark selection, testing, visualization, interpretation.

Here the computer is prominent.

Scientific Computing

Computer based applied mathematics, involving

• Modelling Prescription of a given problem in formulas, relations, equations. Approximating reality.

Here the application is prominent. • Analysis Study of mathematical and numerical issues (stability, conservation rules, etc).

Here the mathematics is prominent.

• Simulation Programming, benchmark selection, testing, visualization, interpretation.

Here the computer is prominent.

Scientific Computing

Computer based applied mathematics, involving

• Modelling This is critical.

• Analysis This is fun.

• Simulation This is hard work.

Axon Guidance

Results from the PhD thesis of J. Krottje (CWI):On the numerical solution of diffusion systems with localized, gradient-driven moving sources, UvA, November 17, 2005

Axon Guidance

Joint project between CWI (Verwer), NIBR (van Pelt) and VU (van Ooyen), carried out at CWI and funded by

Results from the PhD thesis of J. Krottje (CWI):On the numerical solution of diffusion systems with localized, gradient-driven moving sources, UvA, November 17, 2005

Axon Guidance

Axon Guidance

Axon Guidance

Axon Guidance Modelling

Axon Guidance Modelling

A first PDE model was built by Hentschel & van Ooyen ‘99

The model moves particles (axon heads) in attractant-repellent gradient fields

Axon Guidance Modelling

A first PDE model was built by Hentschel & van Ooyen ‘99

The model moves particles (axon heads) in attractant-repellent gradient fields

Axon Guidance Modelling

A first PDE model was built by Hentschel & van Ooyen ‘99

The model moves particles (axon heads) in attractant-repellent gradient fields

Axon Guidance Modelling

A first PDE model was built by Hentschel & van Ooyen ‘99

The model moves particles (axon heads) in attractant-repellent gradient fields

Axon Guidance Modelling

Krottje generalized their model and has developed the Matlab package: AG-tools

Axon Guidance Modelling

Mathematical Framework

Mathematical Framework

Three basic ingredients

• Domain

• States

• Fields

Mathematical Framework

Three basic ingredients

• Domain Physical environment of axons, neurons, chemical fields. Domain in 2D with smooth complicated boundary, possibly with holes. • States

• Fields

Mathematical Framework

Three basic ingredients

• Domain Physical environment of axons, neurons, chemical fields. Domain in 2D with smooth complicated boundary, possibly with holes. • States Growth cones, target cells, axon properties,

locations. Particle dynamics modelled by ordinary differential equations.

• Fields

Mathematical Framework

Three basic ingredients

• Domain Physical environment of axons, neurons, chemical fields. Domain in 2D with smooth complicated boundary, possibly with holes. • States Growth cones, target cells, axon properties,

locations. Particle dynamics modelled by ordinary differential equations.

• Fields Changing concentrations of guidance molecules due to diffusion, absorption, moving sources. Modelled by partial differential equations.

Three basic ingredients

• Domain

• States

• Fields

Mathematical Framework

Three basic ingredients

• Domain

• States

• Fields

Mathematical Framework

Three basic ingredients

• Domain

• States

• Fields

Mathematical Framework

Three basic ingredients

• Domain

• States

• Fields

Mathematical Framework

- Local function approximations- Arbitrary node sets- Unstructured Voronoi grids- Local refinement- Implicit-explicit Runge-Kutta integration

AGTools Example

AGTools Example

Ilustration of topographic mapping with 5 guidance fields(3 diffusive and 2 membrane bound) and 200 growth cones

Topographic Mapping Equations

Topographic Mapping Equations

No hard laws.Phenomenal setup.

Neuro Scientific Computing Challenges

• Modelling

• Analysis

• Simulation

Neuro Scientific Computing Challenges

• Modelling Here major steps are needed:

• Analysis

• Simulation

Neuro Scientific Computing Challenges

• Modelling Here major steps are needed: - e.g., dimensioned wires instead of point particles,

- in general, a less phenomenal setup, - realistic data (coefficients, parameters).

• Analysis

• Simulation

Neuro Scientific Computing Challenges

• Modelling Here major steps are needed: - e.g., dimensioned wires instead of point particles,

- in general, a less phenomenal setup, - realistic data (coefficients, parameters).

• Analysis Higher modelling level will require participation of PDE analysts.

• Simulation

Neuro Scientific Computing Challenges

• Modelling Here major steps are needed: - e.g., dimensioned wires instead of point particles,

- in general, a less phenomenal setup, - realistic data (coefficients, parameters).

• Analysis Higher modelling level will require participation of PDE analysts.

• Simulation 3D-model with many species and axons. Will require huge computer resources,

and presumably a different grid approach.