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Lecture 11: Multiscale Bio-Modeling and Visualization Organ Models I: Synapses and Transport Mechanisms. Chandrajit Bajaj http://www.cs.utexas.edu/~bajaj. The Brain Organ System I. Axonal transport of membranous organelles. Action Potentials. Neuronal Synapses. - PowerPoint PPT Presentation

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CCV presentationOrgan Models I: Synapses and Transport Mechanisms

Chandrajit Bajaj

http://www.cs.utexas.edu/~bajaj

November 2005

November 2005

November 2005

Action Potentials

November 2005

Neuronal Synapses

Neurons must be triggered by a stimulus to produce nerve impulses, which are waves of electrical charge moving along the nerve fibers. When the neuron receives a stimulus, the electrical charge on the inside of the cell membrane changes from negative to positive. A nerve impulse travels down the fiber to a synaptic knob at its end, triggering the release of chemicals (neurotransmitters) that cross the gap between the neuron and the target cell, stimulating a response in the target.

Synapse

The communication point between neurons (the synapse, enlarged at right) comprises the synaptic knob, the synaptic cleft, and the target site.

November 2005

Neuro-Muscular_coupling (synapses)

November 2005

Transport in Myocytes

Diffusion based Transport Mechanisms

Diffusion: the random walk of an ensemble of particles from regions of high concentration to regions of lower concentration

Conduction: heat migrates from regions of high heat to regions of low heat

November 2005

November 2005

Generalized Geometric Surface Diffusion Models

A curvature driven geometric evolution consists of finding a family M = {M(t): t >= 0} of smooth closed immersed orientable surface in IR3 which evolve according to the flow equation

Where x(t) – a surface point on M(t)

Vn(k1, k2, x) – the evolution speed of M(t)

N(x) – the unit normal of the surface at x(t)

November 2005

Curvature Computations for Surfaces/Images/Volumes

If surface M in 3D is the level set F(x,y,z) = 0 of the 3D Map Principal curvatures/directions are the Eigen-values/vectors of

If Surface M in 3D is the graph of an Image F(x,y) in 2D.

Principal Curvatures are Eigenvalues of H

Principal Curvature directions are Eigenvectors of C with

Similar for 3D Images or Maps F(x,y,z) (Volumes).

G = Structure Tensor. Rank of G is 1 and its Eigenvector (with nonzero Eigen-value ) is in the Normal direction

G’ is of Rank 2 and its Eigenvectors are in the tangent space of M with equal Eigenvalues

1.unknown

2.unknown

3.unknown

The mean curvature flow is area shrinking.

November 2005

2. Average Mean Curvature Surface Diffusion

The average mean curvature flow is volume preserving and area shrinking. The area shrinking stops if H = h.

November 2005

3. Isotropic L-B Surface Diffusion

The surface flow is area shrinking, but volume preserving. The area stops shrinking when the gradient of H is zero. That is, H is a surface with constant mean curvature.

November 2005

4. Higher Order Diffusional Models

The flow is volume preserving if K >= 2. The area/volume preserving/shrinking properties for the flows mentioned above are for closed surfaces.

November 2005

or

Preuer and Rumpf’s level set method in 3D

A triad of vectors on the level set:

two principal directions of curvature

and the normal

Let be the principal curvature directions of

at point

where

v

Diffusion tensor

No diffusion along normal direction

November 2005

Anisotropic Volumetric Diffusion

Three principal directions of curvature for volumes are used to construct the Diffusion tensor

The principal directions of curvature are the unit eigenvectors of a matrix

Principal curvatures are the corresponding eigenvalues

November 2005

a(x) is a symmetric, positive definite matrix (diffusion tensor)

Variational (weak) form

The linear system is solved by a conjugate gradient method.

Anisotropic Diffusion Filtering (contd)

The linearized Poisson-Boltzmann equation for the total average electrostatic potential in the presence of a membrane potential

where is the position-dependent dielectric constant at point r, is the total average electrostatic potential at point r, with potential charges scaled by , and imposed membrane potential , which governs the movement of charged species across the cell membrane.

is a Heaviside step-function equal to 0 on side I and 1 on side II, and

is the coupling parameter varying between 0 and 1 to scale the protein charges.

is the charge density of the solute.

November 2005

Poisson-Boltzmann voltage equation of the ion channel membrane system with asymmetrical solutions on sides I and II:

The step function is

The “pore” region is the region from which all ions other than the permeating species are excluded and

the “bulk” region contains the electrolytic solutions.

November 2005

Poisson-Nernst-Plank equations:

where is the diffusion coefficient, is the density, is an effective potential acting on the ions, is the charge density of the channel, is the position-dependent dielectric constant at point r, is the

average electrostatic potential arising from all the interactions in the system, is the charge of the ions.

November 2005

Additional Reading

C.Bajaj, G. Xu “Anisotropic Diffusion of Surfaces and Functions on Surfaces”, ACM Trans. On Graphics, 22, 4 – 32, 2003

G. Xu, Y. Pan, C. Bajaj “Discrete Surface Modelling Using PDE’s”, CAGD, 2005, in press

M. Meyer, M. Desbrun, P. Schroder, A. Barr, “Discrete Differential Geometry Operators for Triangulated 2-manifolds”, Proc. of Visual Math ’02, Germany

T. Weiss, “Cellular BioPhysics I: Transport ”, MIT Press, 1998

November 2005

November 2005

Time Direction Discretization – a semi-implicit Euler scheme.

@

Chandrajit Bajaj

http://www.cs.utexas.edu/~bajaj

November 2005

November 2005

November 2005

Action Potentials

November 2005

Neuronal Synapses

Neurons must be triggered by a stimulus to produce nerve impulses, which are waves of electrical charge moving along the nerve fibers. When the neuron receives a stimulus, the electrical charge on the inside of the cell membrane changes from negative to positive. A nerve impulse travels down the fiber to a synaptic knob at its end, triggering the release of chemicals (neurotransmitters) that cross the gap between the neuron and the target cell, stimulating a response in the target.

Synapse

The communication point between neurons (the synapse, enlarged at right) comprises the synaptic knob, the synaptic cleft, and the target site.

November 2005

Neuro-Muscular_coupling (synapses)

November 2005

Transport in Myocytes

Diffusion based Transport Mechanisms

Diffusion: the random walk of an ensemble of particles from regions of high concentration to regions of lower concentration

Conduction: heat migrates from regions of high heat to regions of low heat

November 2005

November 2005

Generalized Geometric Surface Diffusion Models

A curvature driven geometric evolution consists of finding a family M = {M(t): t >= 0} of smooth closed immersed orientable surface in IR3 which evolve according to the flow equation

Where x(t) – a surface point on M(t)

Vn(k1, k2, x) – the evolution speed of M(t)

N(x) – the unit normal of the surface at x(t)

November 2005

Curvature Computations for Surfaces/Images/Volumes

If surface M in 3D is the level set F(x,y,z) = 0 of the 3D Map Principal curvatures/directions are the Eigen-values/vectors of

If Surface M in 3D is the graph of an Image F(x,y) in 2D.

Principal Curvatures are Eigenvalues of H

Principal Curvature directions are Eigenvectors of C with

Similar for 3D Images or Maps F(x,y,z) (Volumes).

G = Structure Tensor. Rank of G is 1 and its Eigenvector (with nonzero Eigen-value ) is in the Normal direction

G’ is of Rank 2 and its Eigenvectors are in the tangent space of M with equal Eigenvalues

1.unknown

2.unknown

3.unknown

The mean curvature flow is area shrinking.

November 2005

2. Average Mean Curvature Surface Diffusion

The average mean curvature flow is volume preserving and area shrinking. The area shrinking stops if H = h.

November 2005

3. Isotropic L-B Surface Diffusion

The surface flow is area shrinking, but volume preserving. The area stops shrinking when the gradient of H is zero. That is, H is a surface with constant mean curvature.

November 2005

4. Higher Order Diffusional Models

The flow is volume preserving if K >= 2. The area/volume preserving/shrinking properties for the flows mentioned above are for closed surfaces.

November 2005

or

Preuer and Rumpf’s level set method in 3D

A triad of vectors on the level set:

two principal directions of curvature

and the normal

Let be the principal curvature directions of

at point

where

v

Diffusion tensor

No diffusion along normal direction

November 2005

Anisotropic Volumetric Diffusion

Three principal directions of curvature for volumes are used to construct the Diffusion tensor

The principal directions of curvature are the unit eigenvectors of a matrix

Principal curvatures are the corresponding eigenvalues

November 2005

a(x) is a symmetric, positive definite matrix (diffusion tensor)

Variational (weak) form

The linear system is solved by a conjugate gradient method.

Anisotropic Diffusion Filtering (contd)

The linearized Poisson-Boltzmann equation for the total average electrostatic potential in the presence of a membrane potential

where is the position-dependent dielectric constant at point r, is the total average electrostatic potential at point r, with potential charges scaled by , and imposed membrane potential , which governs the movement of charged species across the cell membrane.

is a Heaviside step-function equal to 0 on side I and 1 on side II, and

is the coupling parameter varying between 0 and 1 to scale the protein charges.

is the charge density of the solute.

November 2005

Poisson-Boltzmann voltage equation of the ion channel membrane system with asymmetrical solutions on sides I and II:

The step function is

The “pore” region is the region from which all ions other than the permeating species are excluded and

the “bulk” region contains the electrolytic solutions.

November 2005

Poisson-Nernst-Plank equations:

where is the diffusion coefficient, is the density, is an effective potential acting on the ions, is the charge density of the channel, is the position-dependent dielectric constant at point r, is the

average electrostatic potential arising from all the interactions in the system, is the charge of the ions.

November 2005

Additional Reading

C.Bajaj, G. Xu “Anisotropic Diffusion of Surfaces and Functions on Surfaces”, ACM Trans. On Graphics, 22, 4 – 32, 2003

G. Xu, Y. Pan, C. Bajaj “Discrete Surface Modelling Using PDE’s”, CAGD, 2005, in press

M. Meyer, M. Desbrun, P. Schroder, A. Barr, “Discrete Differential Geometry Operators for Triangulated 2-manifolds”, Proc. of Visual Math ’02, Germany

T. Weiss, “Cellular BioPhysics I: Transport ”, MIT Press, 1998

November 2005

November 2005

Time Direction Discretization – a semi-implicit Euler scheme.

@

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