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 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 BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007 International Conference on FUTURE OF APPLIED ANALYSIS - 20
Outline Outline • Cortical Spreading Depression • Ion Movements in the ECS and ICS • Spatial Buffering and Cell Swelling • Applications BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007 International Conference on FUTURE OF APPLIED ANALYSIS - 20
Cortical Spreading Depression Cortical Spreading Depression • A.A. Leao - 1944 - Ph.D. Harvard, Epilepsy in rabbit • Depression of the EEG - ~1-3 min Bures, Buresova, and Krivanek (1974) BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007 International Conference on FUTURE OF APPLIED ANALYSIS - 20
Why Study Cortical Spreading Cortical Spreading Depression? Depression? Why Study • 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 BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007 International Conference on FUTURE OF APPLIED ANALYSIS - 20
Cortical Spreading Depression (SD) Spreading Depression (SD) Cortical • Stimuli - chemical, electrical, mechanical • Animals - rabbit, cat, rat, human, others • Structures - cerebral cortex, retina, hippocampus, etc. Initial • Functional significance: Stimulus • Physiologists - nuisance Waves of • Psychologists - learning and behavior Spreading Depression • Physicians - migraine with aura A Cortex Recording Electrode B BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007 International Conference on FUTURE OF APPLIED ANALYSIS - 20
Cortical Spreading Depression Cortical Spreading Depression Time and space scales • Slow wave phenomena - 1-15 mm/min • Long wavelengths relative to cell size 32 • Large changes in ionic concentrations of Na + , K + , Cl - , Ca 2+ 16 [K + ] 0 Ion Concentrations 8 4 time .8 1 min .4 .2 [Ca ++ ] 0 .1 [mM/liter] BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007 International Conference on FUTURE OF APPLIED ANALYSIS - 20
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.) BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007 International Conference on FUTURE OF APPLIED ANALYSIS - 20
Simplified Model Equations Simplified Model Equations Continuum model, considering only potassium and calcium: K o D K o ρ (I P ), = + + t K xx 1 K K α K ρ (I P ), i = − + t 1 K K 1 − α C D C ρ (I P ), o o = + + t C xx 2 Ca Ca α C ρ (I P ), i = − + t 2 Ca Ca 1 − α x , t 0. − ∞ < < ∞ > Tortuosity and volume fraction. H.C. Tuckwell and R.M. Miura, “A mathematical model for spreading cortical depression,” Biophysical J. 23 (1978), 257-276. BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007 International Conference on FUTURE OF APPLIED ANALYSIS - 20
Solution of the SD Equations Solution of the SD Equations in One Space Dimension (K + ,Ca 2+ ) in One Space Dimension (K + ,Ca 2+ ) BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007 International Conference on FUTURE OF APPLIED ANALYSIS - 20
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) BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007 International Conference on FUTURE OF APPLIED ANALYSIS - 20
Modelling Geometric Structure Modelling Geometric Structure of the Brain- -Cell Microenvironment Cell Microenvironment of the Brain Retrieving geometry from electron micrograph Retrieved components of the system: 1. ECS and ICS structure 2. Cell shape and membrane BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007 International Conference on FUTURE OF APPLIED ANALYSIS - 20
Solving the Diffusion Processes using LBE Solving the Diffusion Processes using LBE Ions move along the lattice nodes. The densities at each node, the a 2 a 1 LBE rule, and the corresponding a 3 diffusion coefficients are given a 4 by: 4 � � ∑ y C (r,t) = N (r,t), i,o i,o j j=0 x � � � N (r,t) N (r + v ,t + ) i,o i,o → τ j j j � 4 � � � ∑ N i,o (r,t) = p (r,t)N i,o (r -v ,t), l j j,l l l=0 1-p λ 2 D = (1-p ), p = p 0,K,Na,Cl = 0, j K,Na,Cl K,Na,Cl j,l 4 τ 4 BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007 International Conference on FUTURE OF APPLIED ANALYSIS - 20
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. BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007 International Conference on FUTURE OF APPLIED ANALYSIS - 20
• Potassium in the ICS Potassium diffusion in the ICS at t=0.125ms and t=5ms. The injection stops at t=2.5 ms. BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007 International Conference on FUTURE OF APPLIED ANALYSIS - 20
Spatial Buffering Spatial Buffering Single cell microenvironment with injection of potassium 30 ECS node ECS recording site ICS recording site ECS injection site 25 20 Membrane potential Y (node) 15 10 5 0 0 10 20 30 40 50 60 70 80 90 100 X (node) Chen and Nicholson, Biophys. J. 78 (2000), 2776-2797. Steinberg, Wang, Huang, and Miura, Math. Biosci. Engin. 2 (2005), 675-702. BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007 International Conference on FUTURE OF APPLIED ANALYSIS - 20
Osmosis & Cell Volume Change Osmosis & Cell Volume Change • During SD, the extracellular space is compressed to about 25-50% of 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 BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007 International Conference on FUTURE OF APPLIED ANALYSIS - 20
Applications Applications • SD is a cause of migraine with aura • Diffusion tensor imaging • Blood vessels (video) BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007 International Conference on FUTURE OF APPLIED ANALYSIS - 20
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 BIRS Canada-China Workshop on Industrial Mathematics August 5-10, 2007 International Conference on FUTURE OF APPLIED ANALYSIS - 20
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