osmosis and gap junctions in spreading depression: a mathematical model

Osmosis and Gap Junctions in Spreading Depression:A Mathematical Model

Bruce E ShapiroDepartment of BiomathematicsUCLA School of Medicine

Organization

Summary

Results

Methods

Background

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

Depressed Activityup to 2 minutes

Electroencephalogram

What is Spreading Depression?





Surface potentialPropagating Wave≈2 - ≈12 mm/minute

MembraneVoltage







MembraneVoltage

Prodromal Spikes40 - 90 Hz







MembraneVoltage


[K]out3 mM35 mM







MembraneVoltage


[K]out3 mM35 mM

[Na]out, [Cl]out40-60 mM130-160 mM







MembraneVoltage


[K]out3 mM35 mM

[Na]out, [Cl]out40-60 mM130-160 mM

[Ca]out2 mM0.02 mM




Other Features ofSpreading DepressionExtracellular space compressed ≈25% - ≈50%



Other Features of Spreading DepressionExtracellular space compressed ≈25% - ≈50%Followed by a vasodilatory period



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





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



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



High K+

Spreading Depression

“Droplet”PerfusionDialysisWet Tissue Paper

Induction Mechanisms



High K+

Mechanical


Inserting electrodes“Pricking” with a needleDropping a weightFocused ultrasonic irradiation




High K+

Chemicals

Mechanical


Facilitate/Stimulate SD• opiods (meta, leu-enk)• oubain• veratrine• theophylline• ethanol

Hinder/block SD• naloxine• 4AP• octanol• heptanol• conotoxins




High K+

ChemicalsNeurotransmitters

Mechanical


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



High K+


Hypoxia

Mechanical


• hypoxia: reduced oxygen level• ischemia: reduction in blood flow• infarct: area of ischemic damage• MCAO: middle cerebral artery occlusion



Intense neuronal activity

High K+


Hypoxia

Mechanical

Electrical Spontaneous




Clinical Significance

Migraine• speed - comparable to SD

SD




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




Migraine• speed• blood flow changes• aura - occipital cortex

SD

Lashley diagrammed his own auras ...Lashley, K. S. ,Arch. Neurol Psyc. 46: 331-339 (1941).




Migraine• speed• blood flow changes• aura - occipital cortex

SD

... and tracked their progress




Ischemia• spontaneous ID in ischemic zone• SD in ischemic zone increases necrosis• SD may induce ischemic tolerance

Migraine

SD




TGA• wave of hippocampal SD?

Ischemia

Migraine

SD




Concussion• mechanical simulation threshold for concussion > threshold for SD• hence SD probably occurs during concussion

TGA

Ischemia

Migraine

SD




Concussion

Seizure• spikes resemble epiletiform activity• SD will not propagate into seizure zone

TGA

Ischemia

Migraine

SD




Concussion

Seizure

TGA

Ischemia

Migraine

SD




Concussion

Seizure

TGA

Ischemia

Migraine

SD

?



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)



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)




bistable equation: ∂c∂t

=D∂2c∂x2 +f(c)

f(c)=1

TK22 c−K0( ) K1−c( ) c−K2( )

K0K1K2f(c)cThresholdExcited StateQuiescent State




bistable equation with cubic forcing termcdc/dtK2K1K3



Phase plane for traveling wave solutions


bistable equation with cubic forcing term

has an analytic solution:

c=K22

1+tanhx+VtVTC

−C⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎡

⎣ ⎢ ⎢

+K0K2

e−(x+Vt)VTCsechx+VtVTC

−C⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ ⎤

⎦ ⎥ ⎥




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



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)



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)



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 )[ ]



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



What’s missing from these models?GliaOsmosisRange of Wave Speeds ObservedQualitative Shape

of WaveformGap

JunctionsBiophysical Mechanisms




NeuronalGap Junctions

Osmosis

Goals of the Present Study



Goals of the Present Study


NeuronalGap Junctions

Osmosis



Methods

ConceptualModel

ElectrophysiologicalModel

MathematicalModel


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

K Efflux↑ [K+]out

DepolarizationVGKCOpenSTIMULATION

A Conceptual Model



K Efflux↑ [K+]out


Glutamate ReleaseNMDA-R

activationRemove Mg++

Block

A Conceptual Model



K Efflux↑ [K+]out




Block

VGCC Open

↑[Ca++]in ( ) K Ca Ca++

entryCICR

A Conceptual Model



K Efflux↑ [K+]out




Block

VGCC Open

↑[Ca++]in ( ) K Ca Ca++

entryCICRGap

JunctionsOpen

Electro-Diffusionvia Gap

Junctions

↓ Interstitial VolumeCell

StretchOsmoticWaterEntry

↑[ 3] IPNa+,Cl-Entry

A Conceptual Model



NeuronGap Junctions

Electrophysiological Model



Gray matter = dendrites + somata (excludes axons)

NeuronGap Junctions

ION CHANNELSDRMAVGKC BKIKSKK(Ca) HVALVAVGCCFastPersistentVGNaCCaK, NaNMDACl



NeuronGap Junctions


NaKCaNaCaATPHCO3ClPumps and Transporters



NeuronGap Junctions



CaIP3RRyRCaEndoplasmic ReticulumIP3Stretch Receptors

OsmosisH2OCa,NaKLeak



NeuronGap Junctions



CaIP3RRyRCaEndoplasmic ReticulumIP3Stretch Receptors

OsmosisH2OCa,NaKLeak

NaKKNaClGliaHCO3Cl



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


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

Reaction/Diffusion versus ElectrodiffusionParticle Conservation

Continuity Equation:



∂c∂t

= f −∇ •J

Change in concentration in some volume

Production inside volume element

Flux out of volume element

= –



Brownian MotionFicks Law of Diffusion

Reaction/Diffusion Eq.

J =−D∇c

∂c∂t

=∇ ⋅ D∇c( )+f



∂c∂t

= f −∇ •J

On the average molecules tend to move from an area of high concentration to an area of low concentration



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



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



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



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



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( )


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

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



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



Results

Initial Conditions (Stimulation Protocol)

Typical Waveform

Gap Junctions

Volume Changes

Simulation of Channel Block

Calcium Waves

Glial Contribution


•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: 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



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



Typical DC-VoltageShift Waveform

-100-90-80-70-60-50-40-3000.511.522.5 x, mmt=30 sect=25 secV, mVVrest



Typical Ionic Shiftsobserved at a fixed point

030K0600102030t, secΔV0140Cl0160Na02Ca124138Kin



Gap Junctions

To Simulate Gap Junction Block , reduce Diffusion Constant



Gap Junctions

196049010 mV5 sec980

To Simulate Gap Junction Block , reduce Diffusion Constant



0.840.870.900.930204060f t, sec

ΔV



Volume Changes During Wave Passageobserved at a fixed point

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



ΔV

Effect of osmotic time constantVolumetime, secondsτ2τ3τ4τ5τ6τ7τ



Volumetime, secondsτ2τ3τ4τ5τ6τ7τ



Effect of osmotic time constant

Extracellular PackingWave propagation may not be possible in tightly packed tissue



NMDA Channels10002025303540451010010000gNMDA, pS/μm2Wave Magnitude [K+]out, mM 100gDR=500150200250

To Simulate Channel Block , reduce conductance



NMDA Channels10002025303540451010010000gNMDA, pS/μm2Wave Magnitude [K+]out, mM 100gDR=500150200250 0234567110010000Wave Speed, mm/min250200150100500gNMDA, pS/μm2




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




K(Ca) Currents: BK

2530354045503010010003002902802501000300500gDR=1000gBK, pS/μm2Maximum [K+]out, mM




K(Ca) Currents: BK

2530354045503010010003002902802501000300500gDR=1000gBK, pS/μm2Maximum [K+]out, mM 01234567891001000gBK, pS/μm230030gDR=10005003002902802501000Wave Speed, mm/min




K(Ca) Currents: BK

2530354045503010010003002902802501000300500gDR=1000gBK, pS/μm2Maximum [K+]out, mM 01234567891001000gBK, pS/μm230030gDR=10005003002902802501000Wave Speed, mm/minThreshold gBK 0 2500300gDR




K(Ca) Channels10 sec20 mVControl90% BlockBK-CHANNEL

5 sec10 mV3 ✕ ControlControlSK-CHANNEL



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



Voltage Gated K+ Channels

mm/min2025303540451001000 gDR pS/μm2[K+]out mM0123456710 Wave SpeedWave Magnitude

D ELAYED RECTIFIER




D ELAYED RECTIFIER

10 sec20 mV90% BlockControlFacilitates SD?

Observation:TEA sometimes inhibits SD





D ELAYED RECTIFIER

10 sec20 mV90% BlockControl

Inhibits SD?

Facilitates SD?

A-CHANNELControl5 sec10 mV90% Block

Observation:4AP may induce SD




Sodium Channels



Sodium Channels

5 sec10 mVControl90% Block



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



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



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 ...



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



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



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


•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)



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



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



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



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



osmosis and gap junctions in spreading depression: a mathematical model

Documents

cortical spreading depression

vasodilatory periodpropagates

grey matterusually

residual injuryobserved

magnetic resonance imaging

j cereb bl fl metab

gyrencephalic feline

spinal ganglia