Osmosis and Gap Junctions in Spreading Depression: A Mathematical Model

1
Osmosis and Gap Junctions in Spreading
DepressionA Mathematical Model

• Bruce E Shapiro
• Department of Biomathematics
• UCLA School of Medicine

2
Organization
Background
Methods
Results
Summary
3
Background

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

How is SD Induced?
What is Spreading Depression?
Clinical Significance of SD
Previous Models 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
Other Features ofSpreading Depression

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

• Extracellular space compressed 25 - 50

11
Other Features of Spreading Depression

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

• Extracellular space compressed 25 - 50
• Followed by a vasodilatory period

12
Other Features of Spreading Depression

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

• Extracellular space compressed 25 - 50
• Followed by a vasodilatory period
• Propagates only through grey matter
• Usually stops at large sulci

13
Other Features of Spreading Depression

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

• 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
Other Features of Spreading Depression

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

• 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
Clinical Significance

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

Migraine speed - comparable to SD
SD
24
Clinical Significance

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

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.
3311689-1692 (1994)
25
Clinical Significance

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

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
Clinical Significance

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

Migraine speed blood flow changes aura -
occipital cortex
SD
... and tracked their progress
27
Clinical Significance

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

Migraine
Ischemia spontaneous ID in ischemic zone
SD in ischemic zone increases necrosis SD
may induce ischemic tolerance
SD
28
Clinical Significance

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

Migraine
Ischemia
SD
TGA wave of hippocampal SD?
29
Clinical Significance

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

Migraine
Ischemia
SD
Concussion mechanical simulation threshold
for concussion gt threshold for SD hence SD
probably occurs during concussion
TGA
30
Clinical Significance

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

Migraine
Seizure spikes resemble epiletiform
activity SD will not propagate into
seizure zone
Ischemia
SD
Concussion
TGA
31
Clinical Significance

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

Migraine
Seizure
Ischemia
SD
Concussion
TGA
32
Clinical Significance

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

Migraine
Seizure
?
Ischemia
Concussion
SD
TGA
33
Published MathematicalModels

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

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
Models of Spreading Depression

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

• Single Reaction/Diffusion Equation for K
• Attributed to Grafstein, Published in Bures,
  Buresová and Krívánèk(1974) The Mechanism and
  Applications of Leaõs Spreading Depression
• bistable equation

35
Models of Spreading Depression

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

• Single Reaction/Diffusion Equation for K
• bistable equation

36
Models of Spreading Depression

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

• Single Reaction/Diffusion Equation for K
• bistable equation with cubic forcing term

Phase plane for traveling wave solutions
37
Models of Spreading Depression

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

• Single Reaction/Diffusion Equation for K
• bistable equation with cubic forcing term
• has an analytic solution

38
Models of Spreading Depression

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

• 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
Models of Spreading Depression

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

• Bistable Equation with Recovery Variable (Reggia
  1996-1999)
• Model
• Single R/D equation for Potassium
• Add Fitzhugh-Nagumo style recovery variable

40
Models of Spreading Depression

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

• 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
Models of Spreading Depression

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

• 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
Models of Spreading Depression

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

• 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
Whats missing from these models?

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

44
Goals of the Present Study

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

45
Goals of the Present Study

• Background
• Methods
• Results
• Discussion

• What is SD?
• Induction
• Clinical significance
• Previous models
• Goals

46
Methods

• Conceptual model
• Electrophysiological
• Electrodiffusion equation
• Membrane currents
• Gap junctions
• Osmosis
• Implementation

• Background
• Methods
• Results
• Discussion

Conceptual Model
Electrophysiological Model
Mathematical Model
47
Methods

• Conceptual model
• Electrophysiological
• Electrodiffusion equation
• Membrane currents
• Gap junctions
• Osmosis
• Implementation

• Background
• Methods
• Results
• Discussion

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
51

• Conceptual model
• Electrophysiological
• Electrodiffusion equation
• Membrane currents
• Gap junctions
• Osmosis
• Implementation

• Background
• Methods
• Results
• Discussion

A Conceptual Model
52
Electrophysiological Model

• Conceptual model
• Electrophysiological
• Electrodiffusion equation
• Membrane currents
• Gap junctions
• Osmosis
• Implementation

• Background
• Methods
• Results
• Discussion

Gray matter dendrites somata (excludes axons)
53

• Conceptual model
• Electrophysiological
• Electrodiffusion equation
• Membrane currents
• Gap junctions
• Osmosis
• Implementation

• Background
• Methods
• Results
• Discussion

54

• Conceptual model
• Electrophysiological
• Electrodiffusion equation
• Membrane currents
• Gap junctions
• Osmosis
• Implementation

• Background
• Methods
• Results
• Discussion

55

• Conceptual model
• Electrophysiological
• Electrodiffusion equation
• Membrane currents
• Gap junctions
• Osmosis
• Implementation

• Background
• Methods
• Results
• Discussion

56

• Conceptual model
• Electrophysiological
• Electrodiffusion equation
• Membrane currents
• Gap junctions
• Osmosis
• Implementation

• Background
• Methods
• Results
• Discussion

57
Model Design

• Conceptual model
• Electrophysiological
• Electrodiffusion eq
• Membrane currents
• Gap junctions
• Osmosis
• Implementation

• Background
• Methods
• Results
• Discussion

• 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)

58
Reaction/Diffusion versus Electrodiffusion

• Conceptual model
• Electrophysiological
• Electrodiffusion eq
• Membrane currents
• Gap junctions
• Osmosis
• Implementation

• Background
• Methods
• Results
• Discussion

• Particle Conservation
• Continuity Equation

Change in concentration in some volume
Production inside volume element
Flux out of volume element


59
Reaction/Diffusion versus Electrodiffusion

• Conceptual model
• Electrophysiological
• Electrodiffusion eq
• Membrane currents
• Gap junctions
• Osmosis
• Implementation

• Background
• Methods
• Results
• Discussion

• Particle Conservation
• Continuity Equation
• Brownian Motion
• Ficks Law of Diffusion
• Reaction/Diffusion Eq.

On the average molecules tend to move from an
area of high concentration to an area of low
concentration
60
Reaction/Diffusion versus Electrodiffusion

• Conceptual model
• Electrophysiological
• Electrodiffusion eq
• Membrane currents
• Gap junctions
• Osmosis
• Implementation

• Background
• Methods
• Results
• Discussion

• Particle Conservation
• Continuity Equation
• Brownian Motion
• Ficks Law of Diffusion
• Reaction/Diffusion Eq.

• Nernst-Planck Equation
• Electrodiffusion Equation

61
Model Design

• Conceptual model
• Electrophysiological
• Electrodiffusion eq
• Membrane currents
• Gap junctions
• Osmosis
• Implementation

• Background
• Methods
• Results
• Discussion

• System of Reaction-Diffusion Equations
• Currents are due to individual membrane channels
  and pumps
• Equations for potassium

62
Model Design

• Conceptual model
• Electrophysiological
• Electrodiffusion equation
• Membrane currents
• Gap junctions
• Osmosis
• Implementation

• Background
• Methods
• Results
• Discussion

• System of Reaction-Diffusion Equations
• Hodgkin/Huxley Formalism
• 29 state variables
• 14 membrane currents and ion pumps
• Typical current potassium delayed rectifier

63
Model Design

• Conceptual model
• Electrophysiological
• Electrodiffusion equation
• Membrane currents
• Gap junctions
• Osmosis
• Implementation

• Background
• Methods
• Results
• Discussion

• System of Reaction-Diffusion Equations
• Hodgkin/Huxley Formalism
• Inter-neuronal gap junctions
• modeled by cytosolic diffusion

64
Model Design

• Conceptual Model
• Electrophysiological
• Electrodiffusion Equation
• Membrane Currents
• Gap junctions
• Osmosis
• Implementation

• Background
• Methods
• Results
• Discussion

• System of Reaction-Diffusion Equations
• Hodgkin/Huxley Formalism
• Inter-neuronal gap junctions
• Osmosis and volume changes
• time dependent model

65
Model Design

• Conceptual model
• Electrophysiological
• Electrodiffusion equation
• Membrane currents
• Gap junctions
• Osmosis
• Implementation

• Background
• Methods
• Results
• Discussion

• 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

66
Implementation

• Conceptual model
• Electrophysiological
• Electrodiffusion equation
• Membrane currents
• Gap junctions
• Osmosis
• Implementation

• Background
• Methods
• Results
• Discussion

• 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

67
Results

• Stimulation waveform
• Gap junctions
• Osmosis volume
• Currents NMDA, K(Ca), DR, A, Na, Ca
• Ca waves
• Glia

• Background
• Methods
• Results
• Discussion

Initial Conditions (Stimulation Protocol)
Typical Waveform
Gap Junctions
Volume Changes
Simulation of Channel Block
Calcium Waves
Glial Contribution
68
Stimulation Protocol(initial conditions)

• Stimulation waveform
• Gap junctions
• Osmosis volume
• Currents NMDA, K(Ca), DR, A, Na, Ca
• Ca waves
• Glia

• Background
• Methods
• Results
• Discussion

• Increase Kout at t 0
• Typical values used cstim50 mM, s150 mm
• Results relatively insensitive to changes in
  these parameters

69
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

70
Typical DC-VoltageShift Waveform

• Stimulation waveform
• Gap junctions
• Osmosis volume
• Currents NMDA, K(Ca), DR, A, Na, Ca
• Ca waves
• Glia

• Background
• Methods
• Results
• Discussion

71
Typical Ionic Shiftsobserved 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

72
Gap Junctions

• Stimulation waveform
• Gap junctions
• Osmosis volume
• Currents NMDA, K(Ca), DR, A, Na, Ca
• Ca waves
• Glia

• Background
• Methods
• Results
• Discussion

To Simulate Gap Junction Block , reduce Diffusion
Constant
73
Gap Junctions

• Stimulation waveform
• Gap junctions
• Osmosis volume
• Currents NMDA, K(Ca), DR, A, Na, Ca
• Ca waves
• Glia

• Background
• Methods
• Results
• Discussion

To Simulate Gap Junction Block , reduce Diffusion
Constant
74
Volume Changes During Wave Passageobserved 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

75
Volume Changes During Wave Passageobserved 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

76
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

77
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

78
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

79
NMDA Channels

• Stimulation waveform
• Gap junctions
• Osmosis volume
• Currents NMDA, K(Ca), DR, A, Na, Ca
• Ca waves
• Glia

• Background
• Methods
• Results
• Discussion

To Simulate Channel Block , reduce conductance
80
NMDA Channels

• Stimulation waveform
• Gap junctions
• Osmosis volume
• Currents NMDA, K(Ca), DR, A, Na, Ca
• Ca waves
• Glia

• Background
• Methods
• Results
• Discussion

To Simulate Channel Block , reduce conductance
81
NMDA Channels

• Stimulation waveform
• Gap junctions
• Osmosis volume
• Currents NMDA, K(Ca), DR, A, Na, Ca
• Ca waves
• Glia

• Background
• Methods
• Results
• Discussion

NMDA antagonists usually impede or block SD
To Simulate Channel Block , reduce conductance
82
K(Ca) Currents BK

• Stimulation waveform
• Gap junctions
• Osmosis volume
• Currents NMDA, K(Ca), DR, A, Na, Ca
• Ca waves
• Glia

• Background
• Methods
• Results
• Discussion

To Simulate Channel Block , reduce conductance
83
K(Ca) Currents BK

• Stimulation waveform
• Gap junctions
• Osmosis volume
• Currents NMDA, K(Ca), DR, A, Na, Ca
• Ca waves
• Glia

• Background
• Methods
• Results
• Discussion

To Simulate Channel Block , reduce conductance
84
K(Ca) Currents BK

• Stimulation waveform
• Gap junctions
• Osmosis volume
• Currents NMDA, K(Ca), DR, A, Na, Ca
• Ca waves
• Glia

• Background
• Methods
• Results
• Discussion

To Simulate Channel Block , reduce conductance
85
K(Ca) Channels

• Stimulation waveform
• Gap junctions
• Osmosis volume
• Currents NMDA, K(Ca), DR, A, Na, Ca
• Ca waves
• Glia

• Background
• Methods
• Results
• Discussion

86
K(Ca) Channels

• Stimulation waveform
• Gap junctions
• Osmosis volume
• Currents NMDA, K(Ca), DR, A, Na, Ca
• Ca waves
• Glia

• Background
• Methods
• Results
• Discussion

Facilitates SD?
ObservationTEA sometimes inhibits SD
Inhibits SD?
Observation Apamin can induce seizure
87
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

88
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
89
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 4AP may induce SD
Inhibits SD?
90
Sodium Channels

• Stimulation waveform
• Gap junctions
• Osmosis volume
• Currents NMDA, K(Ca), DR, A, Na, Ca
• Ca waves
• Glia

• Background
• Methods
• Results
• Discussion

91
Sodium Channels

• Stimulation waveform
• Gap junctions
• Osmosis volume
• Currents NMDA, K(Ca), DR, A, Na, Ca
• Ca waves
• Glia

• Background
• Methods
• Results
• Discussion

92
Sodium Channels

• Stimulation waveform
• Gap junctions
• Osmosis volume
• Currents NMDA, K(Ca), DR, A, Na, Ca
• Ca waves
• Glia

• Background
• Methods
• Results
• Discussion

Inhibitory?
Facilitatory?

• Mixed effect
• Waves still propagate even under 100 block

Observation TTX does not block SD but it does
prevent spikes
93
Calcium and Calcium Channels

• Stimulation waveform
• Gap junctions
• Osmosis volume
• Currents NMDA, K(Ca), DR, A, Na, Ca
• Ca waves
• Glia

• Background
• Methods
• Results
• Discussion

Simulation of Channel Block
Simulation of removal from bath
This prediction is similar to observations of
removal of Ca from the bath
94
Calcium Waves

• Stimulation waveform
• Gap junctions
• Osmosis volume
• Currents NMDA, K(Ca), DR, A, Na, Ca
• Ca waves
• Glia

• Background
• Methods
• Results
• Discussion

Ca wave propagates at same speed as SD ...
95
Calcium Waves

• Stimulation waveform
• Gap junctions
• Osmosis volume
• Currents NMDA, K(Ca), DR, A, Na, Ca
• Ca waves
• Glia

• Background
• Methods
• Results
• Discussion

Ca wave propagates at same speed as SD ...
... and roughly coincides with DC voltage shift
96
Neuroglia

• Stimulation waveform
• Gap junctions
• Osmosis volume
• Currents NMDA, K(Ca), DR, A, Na, Ca
• Ca waves
• Glia

• Background
• Methods
• Results
• Discussion

Normal working glia act to prevent SD and
maintain homeostasis
Observation Glial poisons do not prevent SD
97
Summary

• Background
• Methods
• Results
• Discussion

• Summary
• Major predictions
• Contributions
• Critique
• Conclusions

• 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

98
Predictions

• Background
• Methods
• Results
• Discussion

• Summary
• Major predictions
• Contributions
• Critique
• Conclusions

• 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)

99
Predictions

• Background
• Methods
• Results
• Discussion

• Summary
• Major predictions
• Contributions
• Critique
• Conclusions

• 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

100
Additional Contributions

• Background
• Methods
• Results
• Discussion

• Summary
• Major predictions
• Contributions
• Critique
• Conclusions

• 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

101
CritiqueFuture Directions

• Background
• Methods
• Results
• Discussion

• Summary
• Major predictions
• Contributions
• Critique
• Conclusions

• 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

102
Conclusion

• Background
• Methods
• Results
• Discussion

• Summary
• Major predictions
• Contributions
• Critique
• Conclusions

• 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