Osmosis and Gap Junctions in Spreading Depression: A Mathematical Model

1. Osmosis and Gap Junctions in Spreading Depression:A Mathematical Model Bruce E Shapiro Department of Biomathematics UCLA School of Medicine

2. Organization Background Methods Results Summary

3. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Background How is SD Induced? What is Spreading Depression? Previous Models of SD Clinical Significance of SD

4. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals What is Spreading Depression?

5. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals What is Spreading Depression?

6. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals What is Spreading Depression?

7. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals What is Spreading Depression?

8. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals What is Spreading Depression?

9. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals What is Spreading Depression?

10. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Other Features ofSpreading Depression • Extracellular space compressed ≈25% - ≈50%

11. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Other Features of Spreading Depression • Extracellular space compressed ≈25% - ≈50% • Followed by a vasodilatory period

12. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Other Features of Spreading Depression • Extracellular space compressed ≈25% - ≈50% • Followed by a vasodilatory period • Propagates only through grey matter • Usually stops at large sulci

13. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Other Features of Spreading Depression • Extracellular space compressed ≈25% - ≈50% • Followed by a vasodilatory period • Propagates only through grey matter • Usually stops at large sulci • Usually there is no residual injury

14. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Other Features of Spreading Depression • Extracellular space compressed ≈25% - ≈50% • Followed by a vasodilatory period • Propagates only through grey matter • Usually stops at large sulci • Usually there is no residual injury • Observed in-vitro and in-vivo • Primates, mammals, fish, amphibians, reptiles, insects • cortex, cerebellum, retina, hippocampus, striatum, spinal ganglia, amygdala, hypothalamus

15. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals James MF, et. al. (2000) Cortical spreading depression in the gyrencephalic feline brain studied by magnetic resonance imaging, J Cereb Bl Fl Metab (in press) http://www-user.uni-bremen.de/~bockhors/Literatur/J_Physiol_full_21th.html

16. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Induction Mechanisms High K+ “Droplet” Perfusion Dialysis Wet Tissue Paper Spreading Depression

17. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Induction Mechanisms High K+ Spreading Depression Mechanical Inserting electrodes “Pricking” with a needle Dropping a weight Focused ultrasonic irradiation

18. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Induction Mechanisms High K+ Spreading Depression • Hinder/block SD • naloxine • 4AP • octanol • heptanol • conotoxins • Facilitate/Stimulate SD • opiods (meta, leu-enk) • oubain • veratrine • theophylline • ethanol Mechanical Chemicals

19. Facilitate or Stimulate SD • glutamatergic agonists • proline • at high concentrations • cholonergic modulators • e.g., ach, protigmine, • nicotine, cytisine • D1 agonists • Hinder or block SD • proline • at low concentrations • chol modulators • e.g., curare, atropine, • mecamlyamine, carbachol • D2 agonists • 5HT modulators • e.g., d-fen, sumatriptan • Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals High K+ Spreading Depression Mechanical Chemicals Neurotransmitters

20. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals • hypoxia: reduced oxygen level • ischemia: reduction in blood flow • infarct: area of ischemic damage • MCAO: middle cerebral artery occlusion High K+ Spreading Depression Mechanical Hypoxia Chemicals Neurotransmitters

21. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Intense neuronal activity High K+ Electrical Spontaneous Spreading Depression Mechanical Hypoxia Chemicals Neurotransmitters

22. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Intense neuronal activity High K+ Electrical Spontaneous Spreading Depression Mechanical Hypoxia Chemicals Neurotransmitters

23. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Clinical Significance Migraine • speed - comparable to SD SD

24. Background • Methods • Results • Discussion • What is SD? • Induction • Clinicalsignificance • Previous models • Goals Clinical Significance Migraine • speed • blood flow changes Migraine: reduced blood flow? SD: increased blood flow? SD Spontaneous migraine during PET Woods, Iacoboni, and Mazziotta. New Eng J Med. 331:1689-1692 (1994)

25. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Clinical Significance Migraine • speed • blood flow changes • aura - occipital cortex SD Lashley diagrammed his own auras ... Lashley, K. S. ,Arch. Neurol Psyc. 46: 331-339 (1941).

26. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Clinical Significance Migraine • speed • blood flow changes • aura - occipital cortex SD ... and tracked their progress

27. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Clinical Significance Migraine SD Ischemia • spontaneous ID in ischemic zone • SD in ischemic zone increases necrosis • SD may induce ischemic tolerance

28. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Clinical Significance Migraine SD Ischemia TGA • wave of hippocampal SD?

29. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Clinical Significance Migraine SD Ischemia Concussion • mechanical simulation threshold for concussion > threshold for SD • hence SD probably occurs during concussion TGA

30. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Clinical Significance Migraine Seizure • spikes resemble epiletiform activity • SD will not propagate into seizure zone SD Ischemia Concussion TGA

31. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Clinical Significance Migraine Seizure SD Ischemia Concussion TGA

32. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Clinical Significance Migraine Seizure ? Ischemia Concussion SD TGA

33. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Published MathematicalModels R/D + Recovery Term (Fitzhugh-Nagumo Method) (Reggia & Montgomery) Single Reaction/Diffusion Equation for K+ (Grafstein) R/D equation for each extracellular ionic species (Tuckwell)

34. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Models of Spreading Depression • Single Reaction/Diffusion Equation for K+ Attributed to Grafstein, Published in Bures, Buresová and Krívánèk(1974) The Mechanism and Applications of Leaõ’s Spreading Depression • bistable equation:

35. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Models of Spreading Depression • Single Reaction/Diffusion Equation for K+ • bistable equation:

36. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Models of Spreading Depression • Single Reaction/Diffusion Equation for K+ • bistable equation with cubic forcing term Phase plane for traveling wave solutions

37. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Models of Spreading Depression • Single Reaction/Diffusion Equation for K+ • bistable equation with cubic forcing term • has an analytic solution:

38. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Models of Spreading Depression • Single Reaction/Diffusion Equation for K+ • bistable equation with cubic forcing term • has an analytic solution • traveling wave front • not a wave pulse • does not model recovery • no biophysical model

39. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Models of Spreading Depression • Bistable Equation with Recovery Variable (Reggia 1996-1999) • Model: • Single R/D equation for Potassium • Add Fitzhugh-Nagumo style recovery variable

40. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Models of Spreading Depression • Bistable Equation with Recovery Variable (Reggia 1996-1999) • Model: • Single R/D equation for Potassium • Add Fitzhugh-Nagumo style recovery variable • Results: • Used to describe migraine aura and ischemic SD • Designed to describe effect of SD on surrounding tissue • Does not provide any biophysical mechanism for shape of the forcing term (such was not the goal of the model)

41. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Models of Spreading Depression • System of Reaction-Diffusion Equations (Tuckwell 1978-81) • Model: • One R/D equation each for: interstitial K, Ca, Na, Cl • One PDE each for: cytoplasmic K, Ca, Na, Cl • Single membrane current for each ionic species • Single generic pump for each ionic species

42. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Models of Spreading Depression • System of Reaction-Diffusion Equations (Tuckwell 1978-81) • Model: • One R/D equation each for: interstitial K, Ca, Na, Cl • One PDE each for: cytoplasmic K, Ca, Na, Cl • Single membrane current for each ionic species • Single generic pump for each ionic species • Results: • Travelling Gaussian wave pulse • Fastest wave speed ≈0.6 mm/min • Reduced model - Na, Cl fixed ≈2 mm/min

43. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals What’s missing from these models?

44. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Goals of the Present Study

45. Background • Methods • Results • Discussion • What is SD? • Induction • Clinical significance • Previous models • Goals Goals of the Present Study

46. Conceptual model • Electrophysiological • Electrodiffusion equation • Membrane currents • Gap junctions • Osmosis • Implementation • Background • Methods • Results • Discussion Methods Conceptual Model Electrophysiological Model Mathematical Model

47. Conceptual model • Electrophysiological • Electrodiffusion equation • Membrane currents • Gap junctions • Osmosis • Implementation • Background • Methods • Results • Discussion Methods Conceptual Model Electrophysiological Model Mathematical Model

48. Conceptual model • Electrophysiological • Electrodiffusion equation • Membrane currents • Gap junctions • Osmosis • Implementation • Background • Methods • Results • Discussion A Conceptual Model

49. Conceptual model • Electrophysiological • Electrodiffusion equation • Membrane currents • Gap junctions • Osmosis • Implementation • Background • Methods • Results • Discussion A Conceptual Model

50. Conceptual model • Electrophysiological • Electrodiffusion equation • Membrane currents • Gap junctions • Osmosis • Implementation • Background • Methods • Results • Discussion A Conceptual Model