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

Organization Summary Results Methods Background

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

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

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

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

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

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

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

Other Features of Spreading Depression Extracellular space compressed ≈25% - ≈50% 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 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 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 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 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) Background Methods Results Discussion What is SD? Induction Clinical significance Previous models Goals

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

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

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

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

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

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

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

Clinical Significance Migraine speed blood flow changes SD Migraine: reduced blood flow? SD: increased blood flow? Woods, Iacoboni, and Mazziotta. New Eng J Med. 331: (1994) Spontaneous migraine during PET 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: (1941). 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 Background Methods Results Discussion What is SD? Induction Clinical significance Previous models Goals

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

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

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

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

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

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

Published Mathematical Models R/D + Recovery Term (Fitzhugh-Nagumo Method) (Reggia & Montgomery) R/D equation for each extracellular ionic species (Tuckwell) Single Reaction/Diffusion Equation for K + (Grafstein) 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: 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: 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 Background Methods Results Discussion What is SD? Induction Clinical significance Previous models Goals Phase plane for traveling wave solutions

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

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

Models of Spreading Depression Bistable Equation with Recovery Variable (Reggia )  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) Background Methods Results Discussion What is SD? Induction Clinical significance Previous models Goals

Models of Spreading Depression System of Reaction-Diffusion Equations (Tuckwell )  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 Background Methods Results Discussion What is SD? Induction Clinical significance Previous models Goals

Models of Spreading Depression System of Reaction-Diffusion Equations (Tuckwell )  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 Background Methods Results Discussion What is SD? Induction Clinical significance Previous models Goals

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

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

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

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

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

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

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

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

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

Electrophysiological Model Background Methods Results Discussion Conceptual model Electrophysiological Electrodiffusion equation Membrane currents Gap junctions Osmosis Implementation Gray matter = dendrites + somata (excludes axons)

Background Methods Results Discussion Conceptual model Electrophysiological Electrodiffusion equation Membrane currents Gap junctions Osmosis Implementation

Background Methods Results Discussion Conceptual model Electrophysiological Electrodiffusion equation Membrane currents Gap junctions Osmosis Implementation

Background Methods Results Discussion Conceptual model Electrophysiological Electrodiffusion equation Membrane currents Gap junctions Osmosis Implementation

Background Methods Results Discussion Conceptual model Electrophysiological Electrodiffusion equation Membrane currents Gap junctions Osmosis Implementation

Model Design System of Reaction-Diffusion Equations  electrodiffusion term included in cytosolic equations  Interstitial reaction-diffusion equation:  One of each for K, Ca, Cl, Na (Eight equations) Background Methods Results Discussion Conceptual model Electrophysiological Electrodiffusion eq Membrane currents Gap junctions Osmosis Implementation

Reaction/Diffusion versus Electrodiffusion Particle Conservation  Continuity Equation: Background Methods Results Discussion Conceptual model Electrophysiological Electrodiffusion eq Membrane currents Gap junctions Osmosis Implementation Change in concentration in some volume Production inside volume element Flux out of volume element =–

Reaction/Diffusion versus Electrodiffusion Particle Conservation  Continuity Equation: Brownian Motion  Ficks Law of Diffusion  Reaction/Diffusion Eq. Background Methods Results Discussion Conceptual model Electrophysiological Electrodiffusion eq Membrane currents Gap junctions Osmosis Implementation On the average molecules tend to move from an area of high concentration to an area of low concentration

Reaction/Diffusion versus Electrodiffusion Particle Conservation  Continuity Equation: Brownian Motion  Ficks Law of Diffusion  Reaction/Diffusion Eq.  Nernst-Planck Equation  Electrodiffusion Equation Background Methods Results Discussion Conceptual model Electrophysiological Electrodiffusion eq Membrane currents Gap junctions Osmosis Implementation

Model Design System of Reaction-Diffusion Equations  Currents are due to individual membrane channels and pumps  Equations for potassium: Background Methods Results Discussion Conceptual model Electrophysiological Electrodiffusion eq Membrane currents Gap junctions Osmosis Implementation

Model Design System of Reaction-Diffusion Equations Hodgkin/Huxley Formalism  29 state variables  14 membrane currents and ion pumps Typical current: potassium delayed rectifier: Background Methods Results Discussion Conceptual model Electrophysiological Electrodiffusion equation Membrane currents Gap junctions Osmosis Implementation

Model Design System of Reaction-Diffusion Equations Hodgkin/Huxley Formalism Inter-neuronal gap junctions  modeled by cytosolic diffusion Background Methods Results Discussion Conceptual model Electrophysiological Electrodiffusion equation Membrane currents Gap junctions Osmosis Implementation

Model Design System of Reaction-Diffusion Equations Hodgkin/Huxley Formalism Inter-neuronal gap junctions Osmosis and volume changes  time dependent model Background Methods Results Discussion Conceptual Model Electrophysiological Electrodiffusion Equation Membrane Currents Gap junctions Osmosis Implementation

Model Design System of Reaction-Diffusion Equations Hodgkin/Huxley Formalism Inter-neuronal gap junctions Osmosis and volume changes  time dependent model  steady state model: after each integration step, f jumps instantaneously to steady state Background Methods Results Discussion Conceptual model Electrophysiological Electrodiffusion equation Membrane currents Gap junctions Osmosis Implementation

Crank-Nicholson Integration Algorithms tested in Mathematica v.4.0  allows fast prototype design  includes Livermore mathematical libraries Final implementation in FORTRAN  Absoft Pro-FORTRAN/F77 v.6.0  Apple iMac/233 MHz  Approximately 8000 lines of code Results plotted in Excel Background Methods Results Discussion Conceptual model Electrophysiological Electrodiffusion equation Membrane currents Gap junctions Osmosis Implementation

Results Initial Conditions (Stimulation Protocol) Typical Waveform Gap Junctions Volume Changes Simulation of Channel Block Calcium Waves Glial Contribution Background Methods Results Discussion Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia

Stimulation Protocol (initial conditions) Increase [K + ] out at t = 0 Typical values used: c stim =50 mM,  =150  m Results relatively insensitive to changes in these parameters Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion

Start of a Typical Wave Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion

Typical DC-Voltage Shift Waveform Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion

Typical Ionic Shifts observed at a fixed point Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion

Gap Junctions To Simulate Gap Junction Block, reduce Diffusion Constant Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion

Gap Junctions To Simulate Gap Junction Block, reduce Diffusion Constant Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion

Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion Volume Changes During Wave Passage observed at a fixed point

Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion

Effect of osmotic time constant Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion

Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion Effect of osmotic time constant

Extracellular Packing Wave propagation may not be possible in tightly packed tissue Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion

NMDA Channels To Simulate Channel Block, reduce conductance Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion

NMDA Channels To Simulate Channel Block, reduce conductance Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion

NMDA Channels NMDA antagonists usually impede or block SD To Simulate Channel Block, reduce conductance Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion

K(Ca) Currents: BK To Simulate Channel Block, reduce conductance Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion

K(Ca) Currents: BK To Simulate Channel Block, reduce conductance Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion

K(Ca) Currents: BK To Simulate Channel Block, reduce conductance Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion

K(Ca) Channels Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion

K(Ca) Channels Facilitates SD? Inhibits SD? Observation: Apamin can induce seizure Observation:TEA sometimes inhibits SD Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion

Voltage Gated K+ Channels Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion

Facilitates SD? Observation: TEA sometimes inhibits SD Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion Voltage Gated K+ Channels

Inhibits SD? Facilitates SD? Observation: 4AP may induce SD Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion Voltage Gated K+ Channels

Sodium Channels Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion

Sodium Channels Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion

Sodium Channels Inhibitory? Facilitatory? Mixed effect Waves still propagate even under 100% block Observation: TTX does not block SD but it does prevent spikes Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion

Calcium and Calcium Channels Simulation of Channel Block Simulation of removal from bath This prediction is similar to observations of removal of Ca ++ from the bath Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion

Calcium Waves Ca wave propagates at same speed as SD... Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion

Calcium Waves Ca wave propagates at same speed as SD and roughly coincides with DC voltage shift Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion

Neuroglia Normal working glia act to prevent SD and maintain homeostasis Observation: Glial poisons do not prevent SD Stimulation & waveform Gap junctions Osmosis & volume Currents: NMDA, K(Ca), DR, A, Na, Ca Ca waves Glia Background Methods Results Discussion

Summary Goal: to model and predict the importance of  volume changes  inter-neuronal gap junctions in the propagation of spreading depression Basic Assumptions  osmotic forces cause water entry/efflux  cytoplasmic voltage gradients may be significant  ions propagate between neurons via gap junctions Background Methods Results Discussion Summary Major predictions Contributions Critique Conclusions

Predictions SD will not propagate unless cells can expand  predicted volume changes consistent with results of Kraig and Nicholson (1978) and Jing, Aitken and Somjen (1994)  SD is easier to induce is species with less tightly packed neuropil Blocking gap junctions prevents SD  consistent with results of Martins-Ferreira and Ribeiro (1995), Nedergaard, Cooper and Goldman (1995), and Largo (1996) Glial poisons should not prevent SD  consistent with results of Largo (1996, 1997) Background Methods Results Discussion Summary Major predictions Contributions Critique Conclusions

Predictions Calcium waves accompany SD  observed via optical imaging during SD NMDA, BK, DR, Na +, and HVA-Ca ++ facilitate SD  NMDA blockers long known to prevent SD  Observations in Ca-free media suggest SD more difficult to induce and has a reduced onset-slope  Predicted slope change is qualitatively similar to observed SK, A, and glial currents impede SD  Spontaneous SD observed after A-blocker 4-AP applied  Spontaneous seizures observed in after SK-blocker apamin applied Background Methods Results Discussion Summary Major predictions Contributions Critique Conclusions

Additional Contributions First use of Hodgkin-Huxley formalism in SD First use of standard biophysical models of membrane ion currents First model of gap junctions in spreading depression First mathematical formulation of osmotic volume changes during spreading depression First application of electrodiffusion equation to study spreading depression Background Methods Results Discussion Summary Major predictions Contributions Critique Conclusions

Critique Future Directions Extracellular geometry  Connectivity  Glial, vascular, axonal compartments  same model with different parameters should work for glia  two/three dimensions  anatomical Intracellular geometry  Calcium compartments, multiple calcium waves  Sodium channels, spiking  Channel distribution Gap junctions  distribution  activation Background Methods Results Discussion Summary Major predictions Contributions Critique Conclusions

Conclusion Predictions are consonant with findings that  gap junction poisons block SD  glial poisons do not block SD The predictions are qualitatively consistent with all published observations of SD Predictions support the theories that  cytoplasmic diffusion via gap junctions  osmosis and volume changes are important mechanisms underlying spreading depression Background Methods Results Discussion Summary Major predictions Contributions Critique Conclusions