Osmosis and Gap Junctions in Spreading Depression: A Mathematical Model
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Transcript of Osmosis and Gap Junctions in Spreading Depression: A Mathematical Model
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Osmosis and Gap Junctions in Spreading Depression:A Mathematical Model
Bruce E ShapiroDepartment of BiomathematicsUCLA School of Medicine
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Organization
Summary
Results
Methods
Background
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Background
What is Spreading
Depression?
How is SDInduced?
ClinicalSignificance
of SD
PreviousModels of SD
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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Depressed Activityup to 2 minutes
Electroencephalogram
What is Spreading Depression?
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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Depressed Activityup to 2 minutes
Electroencephalogram
Surface potentialPropagating Wave≈2 - ≈12 mm/minute
MembraneVoltage
What is Spreading Depression?
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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Depressed Activityup to 2 minutes
Electroencephalogram
Surface potentialPropagating Wave≈2 - ≈12 mm/minute
MembraneVoltage
Prodromal Spikes40 - 90 Hz
What is Spreading Depression?
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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Depressed Activityup to 2 minutes
Electroencephalogram
Surface potentialPropagating Wave≈2 - ≈12 mm/minute
MembraneVoltage
Prodromal Spikes40 - 90 Hz
[K]out3 mM35 mM
What is Spreading Depression?
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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Depressed Activityup to 2 minutes
Electroencephalogram
Surface potentialPropagating Wave≈2 - ≈12 mm/minute
MembraneVoltage
Prodromal Spikes40 - 90 Hz
[K]out3 mM35 mM
[Na]out, [Cl]out40-60 mM130-160 mM
What is Spreading Depression?
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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Depressed Activityup to 2 minutes
Electroencephalogram
Surface potentialPropagating Wave≈2 - ≈12 mm/minute
MembraneVoltage
Prodromal Spikes40 - 90 Hz
[K]out3 mM35 mM
[Na]out, [Cl]out40-60 mM130-160 mM
[Ca]out2 mM0.02 mM
What is Spreading Depression?
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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Other Features ofSpreading DepressionExtracellular space compressed ≈25% - ≈50%
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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Other Features of Spreading DepressionExtracellular space compressed ≈25% - ≈50%Followed by a vasodilatory period
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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Other Features of Spreading DepressionExtracellular space compressed ≈25% - ≈50%Followed by a vasodilatory periodPropagates only through grey matter
Usually stops at large sulci
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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Other Features of Spreading DepressionExtracellular space compressed ≈25% - ≈50%Followed by a vasodilatory periodPropagates 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
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Other Features of Spreading DepressionExtracellular space compressed ≈25% - ≈50%Followed by a vasodilatory periodPropagates only through grey matter
Usually stops at large sulci
Usually there is no residual injuryObserved 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
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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
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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High K+
Spreading Depression
“Droplet”PerfusionDialysisWet Tissue Paper
Induction Mechanisms
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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High K+
Mechanical
Spreading Depression
Inserting electrodes“Pricking” with a needleDropping a weightFocused ultrasonic irradiation
Induction Mechanisms
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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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
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High K+
ChemicalsNeurotransmitters
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
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High K+
ChemicalsNeurotransmitters
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
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Intense neuronal activity
High K+
ChemicalsNeurotransmitters
Hypoxia
Mechanical
Electrical Spontaneous
Spreading Depression
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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Intense neuronal activity
High K+
ChemicalsNeurotransmitters
Hypoxia
Mechanical
Electrical Spontaneous
Spreading Depression
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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Clinical Significance
Migraine• speed - comparable to SD
SD
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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Clinical Significance
Migraine• speed• blood flow changes
SD
Migraine: reduced blood flow?
SD: increased blood flow?
Woods, Iacoboni, and Mazziotta. New Eng J Med. 331:1689-1692 (1994)
Spontaneous migraine during PET
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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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).
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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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
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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
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Clinical Significance
TGA• wave of hippocampal SD?
Ischemia
Migraine
SD
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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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
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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
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Clinical Significance
Concussion
Seizure
TGA
Ischemia
Migraine
SD
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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Clinical Significance
Concussion
Seizure
TGA
Ischemia
Migraine
SD
?
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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Published MathematicalModels
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
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Models of Spreading DepressionSingle 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: ∂c∂t
=D∂2c∂x2 +f(c)
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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Models of Spreading DepressionSingle Reaction/Diffusion Equation for K+
bistable equation: ∂c∂t
=D∂2c∂x2 +f(c)
f(c)=1
TK22 c−K0( ) K1−c( ) c−K2( )
K0K1K2f(c)cThresholdExcited StateQuiescent State
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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Models of Spreading DepressionSingle Reaction/Diffusion Equation for K+
bistable equation with cubic forcing termcdc/dtK2K1K3
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
Phase plane for traveling wave solutions
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Models of Spreading DepressionSingle Reaction/Diffusion Equation for K+
bistable equation with cubic forcing term
has an analytic solution:
c=K22
1+tanhx+VtVTC
−C⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
⎡
⎣ ⎢ ⎢
+K0K2
e−(x+Vt)VTCsechx+VtVTC
−C⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟ ⎤
⎦ ⎥ ⎥
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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Models of Spreading DepressionSingle 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
x[K+]out[K]=K0[K]=K2
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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Models of Spreading DepressionBistable Equation with Recovery Variable
(Reggia 1996-1999)Model:
Single R/D equation for PotassiumAdd Fitzhugh-Nagumo style recovery variable
∂K∂t
=D∂2K
∂x2 +f(K,R)
f(K,R)=A(K −K0)(K −K1)(K −K2)−RK
dRdt
=B(K −K0 −CR)
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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Models of Spreading DepressionBistable Equation with Recovery Variable
(Reggia 1996-1999)Model:
Single R/D equation for PotassiumAdd Fitzhugh-Nagumo style recovery variable
Results:Used to describe migraine aura and ischemic SDDesigned to describe effect of SD on surrounding tissueDoes 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
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Models of Spreading DepressionSystem 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, ClSingle membrane current for each ionic speciesSingle generic pump for each ionic species
∂cj,out∂t
=Dj∂2cj,out∂x2 + kijTi(V−Vj )
i∑ +pj 1−e
−rj (cj −C j )[ ]
∂cj,in∂t
=− kijTi(V−Vj )i∑ −pj 1−e
−rj (cj −Cj )[ ]
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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Models of Spreading DepressionSystem 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, ClSingle membrane current for each ionic speciesSingle generic pump for each ionic species
Results:Travelling Gaussian wave pulseFastest wave speed ≈0.6 mm/minReduced model - Na, Cl fixed ≈2 mm/min
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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What’s missing from these models?GliaOsmosisRange of Wave Speeds ObservedQualitative Shape
of WaveformGap
JunctionsBiophysical Mechanisms
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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Spreading Depression
NeuronalGap Junctions
Osmosis
Goals of the Present Study
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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Goals of the Present Study
Spreading Depression
NeuronalGap Junctions
Osmosis
•Background•Methods•Results•Discussion
•What is SD?•Induction•Clinical significance•Previous models•Goals
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Methods
ConceptualModel
ElectrophysiologicalModel
MathematicalModel
•Background•Methods•Results•Discussion
•Conceptual model•Electrophysiological•Electrodiffusion equation•Membrane currents•Gap junctions•Osmosis•Implementation
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Methods
ConceptualModel
ElectrophysiologicalModel
MathematicalModel
•Background•Methods•Results•Discussion
•Conceptual model•Electrophysiological•Electrodiffusion equation•Membrane currents•Gap junctions•Osmosis•Implementation
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K Efflux↑ [K+]out
DepolarizationVGKCOpenSTIMULATION
A Conceptual Model
•Background•Methods•Results•Discussion
•Conceptual model•Electrophysiological•Electrodiffusion equation•Membrane currents•Gap junctions•Osmosis•Implementation
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K Efflux↑ [K+]out
DepolarizationVGKCOpenSTIMULATION
Glutamate ReleaseNMDA-R
activationRemove Mg++
Block
A Conceptual Model
•Background•Methods•Results•Discussion
•Conceptual model•Electrophysiological•Electrodiffusion equation•Membrane currents•Gap junctions•Osmosis•Implementation
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K Efflux↑ [K+]out
DepolarizationVGKCOpenSTIMULATION
Glutamate ReleaseNMDA-R
activationRemove Mg++
Block
VGCC Open
↑[Ca++]in ( ) K Ca Ca++
entryCICR
A Conceptual Model
•Background•Methods•Results•Discussion
•Conceptual model•Electrophysiological•Electrodiffusion equation•Membrane currents•Gap junctions•Osmosis•Implementation
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K Efflux↑ [K+]out
DepolarizationVGKCOpenSTIMULATION
Glutamate ReleaseNMDA-R
activationRemove Mg++
Block
VGCC Open
↑[Ca++]in ( ) K Ca Ca++
entryCICRGap
JunctionsOpen
Electro-Diffusionvia Gap
Junctions
↓ Interstitial VolumeCell
StretchOsmoticWaterEntry
↑[ 3] IPNa+,Cl-Entry
A Conceptual Model
•Background•Methods•Results•Discussion
•Conceptual model•Electrophysiological•Electrodiffusion equation•Membrane currents•Gap junctions•Osmosis•Implementation
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NeuronGap Junctions
Electrophysiological Model
•Background•Methods•Results•Discussion
•Conceptual model•Electrophysiological•Electrodiffusion equation•Membrane currents•Gap junctions•Osmosis•Implementation
Gray matter = dendrites + somata (excludes axons)
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NeuronGap Junctions
ION CHANNELSDRMAVGKC BKIKSKK(Ca) HVALVAVGCCFastPersistentVGNaCCaK, NaNMDACl
•Background•Methods•Results•Discussion
•Conceptual model•Electrophysiological•Electrodiffusion equation•Membrane currents•Gap junctions•Osmosis•Implementation
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NeuronGap Junctions
ION CHANNELSDRMAVGKC BKIKSKK(Ca) HVALVAVGCCFastPersistentVGNaCCaK, NaNMDACl
NaKCaNaCaATPHCO3ClPumps and Transporters
•Background•Methods•Results•Discussion
•Conceptual model•Electrophysiological•Electrodiffusion equation•Membrane currents•Gap junctions•Osmosis•Implementation
![Page 55: Osmosis and Gap Junctions in Spreading Depression: A Mathematical Model](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815a82550346895dc7ee06/html5/thumbnails/55.jpg)
NeuronGap Junctions
ION CHANNELSDRMAVGKC BKIKSKK(Ca) HVALVAVGCCFastPersistentVGNaCCaK, NaNMDACl
NaKCaNaCaATPHCO3ClPumps and Transporters
CaIP3RRyRCaEndoplasmic ReticulumIP3Stretch Receptors
OsmosisH2OCa,NaKLeak
•Background•Methods•Results•Discussion
•Conceptual model•Electrophysiological•Electrodiffusion equation•Membrane currents•Gap junctions•Osmosis•Implementation
![Page 56: Osmosis and Gap Junctions in Spreading Depression: A Mathematical Model](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815a82550346895dc7ee06/html5/thumbnails/56.jpg)
NeuronGap Junctions
ION CHANNELSDRMAVGKC BKIKSKK(Ca) HVALVAVGCCFastPersistentVGNaCCaK, NaNMDACl
NaKCaNaCaATPHCO3ClPumps and Transporters
CaIP3RRyRCaEndoplasmic ReticulumIP3Stretch Receptors
OsmosisH2OCa,NaKLeak
NaKKNaClGliaHCO3Cl
•Background•Methods•Results•Discussion
•Conceptual model•Electrophysiological•Electrodiffusion equation•Membrane currents•Gap junctions•Osmosis•Implementation
![Page 57: Osmosis and Gap Junctions in Spreading Depression: A Mathematical Model](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815a82550346895dc7ee06/html5/thumbnails/57.jpg)
Model Design
System of Reaction-Diffusion Equationselectrodiffusion term included in cytosolic equations
Interstitial reaction-diffusion equation:
One of each for K, Ca, Cl, Na (Eight equations)
∂cin∂t
=∂∂x
Dc,in∂cin∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟ +
zFRT
∂∂x
cinDc,in∂E∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟ −
AVJ c,m+sc
∂cout∂t
=Dc,out∂2cout∂x2 +
AV
f1− f
J c,m−J c,glia
•Background•Methods•Results•Discussion
•Conceptual model•Electrophysiological•Electrodiffusion eq•Membrane currents•Gap junctions•Osmosis•Implementation
![Page 58: Osmosis and Gap Junctions in Spreading Depression: A Mathematical Model](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815a82550346895dc7ee06/html5/thumbnails/58.jpg)
Reaction/Diffusion versus ElectrodiffusionParticle Conservation
Continuity Equation:
•Background•Methods•Results•Discussion
•Conceptual model•Electrophysiological•Electrodiffusion eq•Membrane currents•Gap junctions•Osmosis•Implementation
∂c∂t
= f −∇ •J
Change in concentration in some volume
Production inside volume element
Flux out of volume element
= –
![Page 59: Osmosis and Gap Junctions in Spreading Depression: A Mathematical Model](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815a82550346895dc7ee06/html5/thumbnails/59.jpg)
Reaction/Diffusion versus ElectrodiffusionParticle Conservation
Continuity Equation:
Brownian MotionFicks Law of Diffusion
Reaction/Diffusion Eq.
J =−D∇c
∂c∂t
=∇ ⋅ D∇c( )+f
•Background•Methods•Results•Discussion
•Conceptual model•Electrophysiological•Electrodiffusion eq•Membrane currents•Gap junctions•Osmosis•Implementation
∂c∂t
= f −∇ •J
On the average molecules tend to move from an area of high concentration to an area of low concentration
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Reaction/Diffusion versus ElectrodiffusionParticle Conservation
Continuity Equation:
Brownian MotionFicks Law of Diffusion
Reaction/Diffusion Eq.
∂c∂t
= f −∇ •J
J =−D ∇c+cFZRT
∇V⎛ ⎝
⎞ ⎠ J =−D∇c
Nernst-Planck Equation
Electrodiffusion Equation
∂c∂t
=∇ ⋅ D∇c( )+FzRT
∇ ⋅ Dc∇V( )+f∂c∂t
=∇ ⋅ D∇c( )+f
•Background•Methods•Results•Discussion
•Conceptual model•Electrophysiological•Electrodiffusion eq•Membrane currents•Gap junctions•Osmosis•Implementation
![Page 61: Osmosis and Gap Junctions in Spreading Depression: A Mathematical Model](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815a82550346895dc7ee06/html5/thumbnails/61.jpg)
Model Design
System of Reaction-Diffusion EquationsCurrents are due to individual membrane channels
and pumpsEquations for potassium:
∂[K+]in∂t
=∂∂x
DK,in∂[K+]in∂x
⎛
⎝ ⎜
⎞
⎠ ⎟ +
FRT
∂∂x
[K+]inDK,in∂E∂x
⎛ ⎝ ⎜
⎞ ⎠ ⎟ −
AVJ K
∂[K+]out∂t
=DK,out∂2[K+]out
∂x2 −J K,glia+AV
f1− f
J K
J K = jA +jM +jDR+jBK +jIK +jSK +jK,NMDA+jK,leak+jNa/K
•Background•Methods•Results•Discussion
•Conceptual model•Electrophysiological•Electrodiffusion eq•Membrane currents•Gap junctions•Osmosis•Implementation
![Page 62: Osmosis and Gap Junctions in Spreading Depression: A Mathematical Model](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815a82550346895dc7ee06/html5/thumbnails/62.jpg)
Model DesignSystem of Reaction-Diffusion EquationsHodgkin/Huxley Formalism
29 state variables14 membrane currents and ion pumps
Typical current: potassium delayed rectifier:
jDR =gDRm
2hF
V−RTF
ln(K[ ]out/ K[ ]in( )
dmdt
= α V( )+β V( )( )α V−20( )
α V−20( )+β V−20( )−m
⎛ ⎝ ⎜
⎞ ⎠ ⎟
dhdt
=1τh
1+e(V+25)/ 4( )
−1−h
⎛ ⎝ ⎜
⎞ ⎠ ⎟
α V( ) =0.0047(V+12) 1+e(V+25)/4( )
−1
β V( ) =e−(V+147)/30
-20-1001020304050-75-50-25025Steady State jDR μ /amps cm2-90 mV-60 mV-30 mV0E, mV
•Background•Methods•Results•Discussion
•Conceptual model•Electrophysiological•Electrodiffusion equation•Membrane currents•Gap junctions•Osmosis•Implementation
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Model Design
System of Reaction-Diffusion EquationsHodgkin/Huxley FormalismInter-neuronal gap junctions
modeled by cytosolic diffusion
Deffective=D
1+0.00161−Δ
Δ
10-610-510-40.0010.010.111.00.80.60.40.20clumped aggregratesuniformly distributedDeffDinΔ
•Background•Methods•Results•Discussion
•Conceptual model•Electrophysiological•Electrodiffusion equation•Membrane currents•Gap junctions•Osmosis•Implementation
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Model Design
System of Reaction-Diffusion EquationsHodgkin/Huxley FormalismInter-neuronal gap junctions Osmosis and volume changes
time dependent model
d( fV)dt
=PfVWS OSMin −OSMout( )
dfdt
=1τ
f∞(1− f∞)f (1− f)
f∞ − f( )
•Background•Methods•Results•Discussion
•Conceptual Model•Electrophysiological•Electrodiffusion Equation•Membrane Currents•Gap junctions•Osmosis•Implementation
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Model Design
System of Reaction-Diffusion EquationsHodgkin/Huxley FormalismInter-neuronal gap junctions Osmosis and volume changes
time dependent modelsteady 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
![Page 66: Osmosis and Gap Junctions in Spreading Depression: A Mathematical Model](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815a82550346895dc7ee06/html5/thumbnails/66.jpg)
Implementation
Crank-Nicholson IntegrationAlgorithms tested in Mathematica v.4.0
allows fast prototype design includes Livermore mathematical libraries
Final implementation in FORTRANAbsoft Pro-FORTRAN/F77 v.6.0Apple iMac/233 MHzApproximately 8000 lines of code
Results plotted in Excel
•Background•Methods•Results•Discussion
•Conceptual model•Electrophysiological•Electrodiffusion equation•Membrane currents•Gap junctions•Osmosis•Implementation
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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
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Stimulation Protocol(initial conditions)
Increase [K+]out at t = 0
Typical values used: cstim=50 mM, =150 μm
Results relatively insensitive to changes in these parameters
c x,0( )=crest+ cstim−crest( )e−x2 2σ2
10 mM100 μmx = 0
•Stimulation & waveform•Gap junctions•Osmosis & volume•Currents: NMDA, K(Ca), DR, A, Na, Ca•Ca waves•Glia
•Background•Methods•Results•Discussion
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Start of a Typical Wave
118120122124126128130132134136138-1.0-0.500.51.0x, mm05101520253035404550[K+]out mMt=0t=5t=2t=4t=3t=1[K+]in mM
•Stimulation & waveform•Gap junctions•Osmosis & volume•Currents: NMDA, K(Ca), DR, A, Na, Ca•Ca waves•Glia
•Background•Methods•Results•Discussion
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Typical DC-VoltageShift Waveform
-100-90-80-70-60-50-40-3000.511.522.5 x, mmt=30 sect=25 secV, mVVrest
•Stimulation & waveform•Gap junctions•Osmosis & volume•Currents: NMDA, K(Ca), DR, A, Na, Ca•Ca waves•Glia
•Background•Methods•Results•Discussion
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Typical Ionic Shiftsobserved at a fixed point
030K0600102030t, secΔV0140Cl0160Na02Ca124138Kin
•Stimulation & waveform•Gap junctions•Osmosis & volume•Currents: NMDA, K(Ca), DR, A, Na, Ca•Ca waves•Glia
•Background•Methods•Results•Discussion
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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
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Gap Junctions
196049010 mV5 sec980
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
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0.840.870.900.930204060f t, sec
ΔV
•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 Passageobserved at a fixed point
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Volume Changes During Wave Passageobserved at a fixed point
0.840.870.900.930204060f t, sec
85%2 %f[K+]in[Na+]out[Cl-]out[K+]outt, sec46810
•Stimulation & waveform•Gap junctions•Osmosis & volume•Currents: NMDA, K(Ca), DR, A, Na, Ca•Ca waves•Glia
•Background•Methods•Results•Discussion
ΔV
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Effect of osmotic time constantVolumetime, secondsτ2τ3τ4τ5τ6τ7τ
•Stimulation & waveform•Gap junctions•Osmosis & volume•Currents: NMDA, K(Ca), DR, A, Na, Ca•Ca waves•Glia
•Background•Methods•Results•Discussion
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Volumetime, secondsτ2τ3τ4τ5τ6τ7τ
•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
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Extracellular PackingWave 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
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NMDA Channels10002025303540451010010000gNMDA, pS/μm2Wave Magnitude [K+]out, mM 100gDR=500150200250
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
![Page 80: Osmosis and Gap Junctions in Spreading Depression: A Mathematical Model](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815a82550346895dc7ee06/html5/thumbnails/80.jpg)
NMDA Channels10002025303540451010010000gNMDA, pS/μm2Wave Magnitude [K+]out, mM 100gDR=500150200250 0234567110010000Wave Speed, mm/min250200150100500gNMDA, pS/μm2
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
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NMDA Channels10002025303540451010010000gNMDA, pS/μm2Wave Magnitude [K+]out, mM 100gDR=500150200250 0234567110010000Wave Speed, mm/min250200150100500gNMDA, pS/μm2
1000350gDRThreshold gK,NMDA 100 10100
NMDA antagonistsusually 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
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K(Ca) Currents: BK
2530354045503010010003002902802501000300500gDR=1000gBK, pS/μm2Maximum [K+]out, mM
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
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K(Ca) Currents: BK
2530354045503010010003002902802501000300500gDR=1000gBK, pS/μm2Maximum [K+]out, mM 01234567891001000gBK, pS/μm230030gDR=10005003002902802501000Wave Speed, mm/min
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
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K(Ca) Currents: BK
2530354045503010010003002902802501000300500gDR=1000gBK, pS/μm2Maximum [K+]out, mM 01234567891001000gBK, pS/μm230030gDR=10005003002902802501000Wave Speed, mm/minThreshold gBK 0 2500300gDR
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
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K(Ca) Channels10 sec20 mVControl90% BlockBK-CHANNEL
5 sec10 mV3 ✕ ControlControlSK-CHANNEL
•Stimulation & waveform•Gap junctions•Osmosis & volume•Currents: NMDA, K(Ca), DR, A, Na, Ca•Ca waves•Glia
•Background•Methods•Results•Discussion
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K(Ca) Channels10 sec20 mVControl90% BlockBK-CHANNEL
5 sec10 mV3 ✕ ControlControlSK-CHANNEL
Facilitates SD?
Inhibits SD?
Observation: Apamincan 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
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Voltage Gated K+ Channels
mm/min2025303540451001000 gDR pS/μm2[K+]out mM0123456710 Wave SpeedWave Magnitude
D ELAYED RECTIFIER
•Stimulation & waveform•Gap junctions•Osmosis & volume•Currents: NMDA, K(Ca), DR, A, Na, Ca•Ca waves•Glia
•Background•Methods•Results•Discussion
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mm/min2025303540451001000 gDR pS/μm2[K+]out mM0123456710 Wave SpeedWave Magnitude
D ELAYED RECTIFIER
10 sec20 mV90% BlockControlFacilitates 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
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mm/min2025303540451001000 gDR pS/μm2[K+]out mM0123456710 Wave SpeedWave Magnitude
D ELAYED RECTIFIER
10 sec20 mV90% BlockControl
Inhibits SD?
Facilitates SD?
A-CHANNELControl5 sec10 mV90% Block
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
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Sodium Channels
•Stimulation & waveform•Gap junctions•Osmosis & volume•Currents: NMDA, K(Ca), DR, A, Na, Ca•Ca waves•Glia
•Background•Methods•Results•Discussion
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Sodium Channels
5 sec10 mVControl90% Block
•Stimulation & waveform•Gap junctions•Osmosis & volume•Currents: NMDA, K(Ca), DR, A, Na, Ca•Ca waves•Glia
•Background•Methods•Results•Discussion
![Page 92: Osmosis and Gap Junctions in Spreading Depression: A Mathematical Model](https://reader035.fdocuments.us/reader035/viewer/2022062500/56815a82550346895dc7ee06/html5/thumbnails/92.jpg)
Sodium Channels
5 sec10 mVControl90% BlockInhibitory?
Facilitatory?
• Mixed effect
• Waves still propagate even under 100% block
Observation: TTX does not block SDbut 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
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Calcium and Calcium Channels
5 sec10 mV10 mV5 sec0.2 mM90% Block2 mMHVA Ca-CHANNEL[Ca++]outControl
Simulationof ChannelBlock
Simulation of removal from bath
This prediction is similarto observations of removal ofCa++ from the bath
•Stimulation & waveform•Gap junctions•Osmosis & volume•Currents: NMDA, K(Ca), DR, A, Na, Ca•Ca waves•Glia
•Background•Methods•Results•Discussion
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Calcium Waves
0204060801001201400.70.80.91.01.11.2x, mm[Ca++]in μM10.5 sec11.5 sec
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
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Calcium Waves
0204060801001201400.70.80.91.01.11.2x, mm[Ca++]in μM10.5 sec11.5 sec
Ca wave propagates at same speed as SD ...
... and roughly coincides with DC voltage shift
05010015068101214t, sec602040ΔV mV[Ca]inΔV
•Stimulation & waveform•Gap junctions•Osmosis & volume•Currents: NMDA, K(Ca), DR, A, Na, Ca•Ca waves•Glia
•Background•Methods•Results•Discussion
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Neuroglia
293031320246810Maximum Glial Pump Rate mm/liter-secWave
Magnitude
0123456 Wave Speed[K+]out, mMmm/min
Normal working glia act to prevent SD and maintainhomeostasis
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
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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/effluxcytoplasmic voltage gradients may be significant ions propagate between neurons via gap junctions
•Background•Methods•Results•Discussion
•Summary•Major predictions•Contributions•Critique•Conclusions
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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
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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
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Additional ContributionsFirst use of Hodgkin-Huxley formalism in SDFirst use of standard biophysical models of
membrane ion currentsFirst model of gap junctions in spreading
depressionFirst mathematical formulation of osmotic volume
changes during spreading depressionFirst application of electrodiffusion equation to
study spreading depression
•Background•Methods•Results•Discussion
•Summary•Major predictions•Contributions•Critique•Conclusions
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CritiqueFuture 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
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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