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http://www.utdallas.edu/~kilgard/brain.jpg BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007 International Conference on FUTURE OF APPLIED ANALYSIS - 20 Modeling Cortical Spreading Depression Modeling Cortical Spreading

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International Conference on FUTURE OF APPLIED ANALYSIS - 20

BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007

http://www.utdallas.edu/~kilgard/brain.jpg

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International Conference on FUTURE OF APPLIED ANALYSIS - 20

BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007

Modeling Cortical Spreading Depression Modeling Cortical Spreading Depression

Robert M. Miura

Departments of Mathematical Sciences and of Biomedical Engineering New Jersey Institute of Technology, Newark, New Jersey, USA Huaxiong Huang – York University Anisha Banerjee – NJ Inst. of Technology Yuqing Wang – Royal Bank of Canada Ben Steinberg – University of Toronto Longxiang Dai - Maryland Hideo Ikeda – Toyama University Henry Tuckwell - France

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International Conference on FUTURE OF APPLIED ANALYSIS - 20

BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007

Outline Outline

  • Cortical Spreading Depression
  • Ion Movements in the ECS and ICS
  • Spatial Buffering and Cell Swelling
  • Applications
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International Conference on FUTURE OF APPLIED ANALYSIS - 20

BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007

  • A.A. Leao - 1944 - Ph.D. Harvard, Epilepsy in rabbit
  • Depression of the EEG - ~1-3 min

Cortical Spreading Depression Cortical Spreading Depression

Bures, Buresova, and Krivanek (1974)

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International Conference on FUTURE OF APPLIED ANALYSIS - 20

BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007

Why Study Why Study Cortical Spreading Cortical Spreading Depression? Depression?

  • Discovered in 1944, but still do not understand SD
  • Functional reasons

– Seizures – Connection with (classic) migraine with aura

  • Structure of the brain and diffusion paths of ions

– Volume fraction – Tortuosity

  • Ionic concentrations in the microenvironment of neurons

– Large ion concentration changes in the ECS – Maintain balance of ions during neural activity

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International Conference on FUTURE OF APPLIED ANALYSIS - 20

BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007

Cortical Cortical Spreading Depression (SD) Spreading Depression (SD)

B

Recording Electrode Cortex A Initial Stimulus Waves of Spreading Depression

  • Stimuli - chemical, electrical, mechanical
  • Animals - rabbit, cat, rat, human, others
  • Structures - cerebral cortex, retina, hippocampus, etc.
  • Functional significance:
  • Physiologists - nuisance
  • Psychologists - learning and behavior
  • Physicians - migraine with aura
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International Conference on FUTURE OF APPLIED ANALYSIS - 20

BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007

Cortical Spreading Depression Cortical Spreading Depression

[K+]0 [Ca++]0

1 min

32 16 8 4 .8 .4 .2 .1 [mM/liter]

Time and space scales

  • Slow wave phenomena - 1-15 mm/min
  • Long wavelengths relative to cell size
  • Large changes in ionic concentrations of

Na+, K+, Cl-, Ca2+

time Ion Concentrations

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International Conference on FUTURE OF APPLIED ANALYSIS - 20

BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007

Models of Spreading Depression Models of Spreading Depression

  • Analog to conduction of impulses in cardiac muscle

(Wiener and Rosenblueth, Shibata and Bures)

  • Computer simulation (Reshodko and Bures)
  • Potassium, action potentials (Grafstein)
  • Neurotransmitter mechanism (Tuckwell and M.)
  • Osmosis and neuronal gap junctions (Shapiro, Kager

et al.)

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International Conference on FUTURE OF APPLIED ANALYSIS - 20

BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007

Simplified Model Equations Simplified Model Equations

Continuum model, considering only potassium and calcium:

= + + = − + − = + + = − + −

  • t

K xx 1 K K i t 1 K K

  • t

C xx 2 Ca Ca i t 2 Ca Ca

K D K ρ (I P ), K ρ (I P ), 1 C D C ρ (I P ), C ρ (I P ), 1 α α α α

0. t , x > < < − ∞ ∞

Tortuosity and volume fraction.

H.C. Tuckwell and R.M. Miura, “A mathematical model for spreading cortical depression,” Biophysical J. 23 (1978), 257-276.

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International Conference on FUTURE OF APPLIED ANALYSIS - 20

BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007

Solution of the SD Equations Solution of the SD Equations in One Space Dimension (K in One Space Dimension (K+

+,Ca

,Ca2+

2+)

)

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International Conference on FUTURE OF APPLIED ANALYSIS - 20

BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007

Difficulties in Modelling and Computations Difficulties in Modelling and Computations

  • Complicated 3-D geometric structures of ICS and ECS
  • Different kinds of cells and processes, such as neurons, glial cells,

axons, synapses

  • Many different kinds of ions with distinct diffusion coefficients and

coupled dynamics

  • Connections between neurons (synapses, gap junctions) and between

glial cells (gap junctions)

  • Cell membranes have spatial distributions of ion channel densities
  • Cell swelling (moving cell membranes)
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International Conference on FUTURE OF APPLIED ANALYSIS - 20

BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007

Modelling Geometric Structure Modelling Geometric Structure

  • f the Brain
  • f the Brain-
  • Cell Microenvironment

Cell Microenvironment

Retrieving geometry from electron micrograph Retrieved components

  • f the system:

1. ECS and ICS structure 2. Cell shape and membrane

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International Conference on FUTURE OF APPLIED ANALYSIS - 20

BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007

Solving the Diffusion Processes using LBE Solving the Diffusion Processes using LBE

Ions move along the lattice nodes. The densities at each node, the LBE rule, and the corresponding diffusion coefficients are given by:

→ =

∑ ∑

  • 4

i,o i,o j j=0 i,o i,o j j j 4 i,o i,o l j j,l l l=0 2 0,K,Na,Cl 0, j K,Na,Cl K,Na,Cl j,l

C (r,t) = N (r,t), N (r,t) N (r + v ,t + ) N (r,t) = p (r,t)N (r -v ,t), 1-p λ D = (1-p ), p = p 4τ 4 τ

x y a2 a1 a3 a4

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International Conference on FUTURE OF APPLIED ANALYSIS - 20

BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007

Simulation of a Small System Simulation of a Small System with Permeable Membranes with Permeable Membranes

  • Potassium injected in the ECS

Potassium diffusion in the ECS at t=0.125ms and t=5ms. The injection stops at t=2.5 ms.

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International Conference on FUTURE OF APPLIED ANALYSIS - 20

BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007

  • Potassium in the ICS

Potassium diffusion in the ICS at t=0.125ms and t=5ms. The injection stops at t=2.5 ms.

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International Conference on FUTURE OF APPLIED ANALYSIS - 20

BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007

Spatial Buffering Spatial Buffering

Single cell microenvironment with injection of potassium

10 20 30 40 50 60 70 80 90 100 5 10 15 20 25 30 X (node) Y (node) ECS node ECS recording site ICS recording site ECS injection site

Membrane potential

Chen and Nicholson, Biophys. J. 78 (2000), 2776-2797. Steinberg, Wang, Huang, and Miura, Math. Biosci. Engin. 2 (2005), 675-702.

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International Conference on FUTURE OF APPLIED ANALYSIS - 20

BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007

Osmosis & Cell Volume Change Osmosis & Cell Volume Change

  • During SD, the extracellular space is compressed to about 25-50%
  • f its original fraction of 20%.
  • The swelling of cells caused by osmosis due to the movement of

water molecules across a semi-permeable membrane.

  • Na+, K+, Cl-, and water move through channels in the membrane.
  • X molecules are trapped

inside the cell.

  • Isotonicity
  • Electroneutrality
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International Conference on FUTURE OF APPLIED ANALYSIS - 20

BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007

Applications Applications

  • SD is a cause of migraine with aura
  • Diffusion tensor imaging
  • Blood vessels (video)
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International Conference on FUTURE OF APPLIED ANALYSIS - 20

BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007

Summary Summary

  • Spreading Cortical Depression
  • Brain-Cell Microenvironment
  • Ion Movements in the ECS and ICS
  • Spatial Buffering
  • Cell Swelling
  • Applications to Migraine with Aura and

Diffusion Tensor Imaging