The Role of Oxygen on the Dynamics of Seizure and ......Yina Wei Allen Ins?tute for Brain Science Feb 14, 2018 Mathema(cal Modeling of Cor(cal Spreading Depression (SD) and Related

TheRoleofOxygenontheDynamicsofSeizureandSpreadingDepression

YinaWeiAllenIns?tuteforBrainScience

Feb14,2018

Mathema(calModelingofCor(calSpreadingDepression(SD)andRelatedPhenomena,UniversityofMinnesota

2/14/18 2

Outline• Whatisseizure?• Whatisspreadingdepression?•  Commonality?• Whatismissingincurrentmodel?•  Energyconsump?oninthebrain•  PartI:oxygenandseizuredynamics•  PartII:oxygenandspreadingdepression

WhatisSeizure?Epilepsy•  Oneofthemostcommonbraindisorders•  1%worldpopula(on,3millionpeopleinUS•  characterizedastheoccurrenceofrepe((veSeizures

32/14/18

Ziburkusetal.2006JNeurophysiol.

ViViVo

Seizure•  Asuddenabnormal

excessiveneuronalac(vity•  Lastsfromafewsecondsto

afewminutes

Cause•  Headtrauma,stroke,braininfec(onetc.

WhatisSpreadingDepression?

ImagefromWikipedia

SpreadingDepression(SD)•  apathophysiologicalphenomenonoccurredduring

migraine,headtrauma,andstroke.•  nearlycompletedepolariza(on•  propagates2-5mm/min

42/14/18

SDcanbeinducedby Hypoxia Highpotassium

Czehetal.,1993BrainResearch BrissonandAndrew,2012,JNeurophysiol

2/14/18

Seizures HypoxiaSpreadingDepression

Commonali?es:•  Shi]sofextracellularpoten(al•  Redistribu(onofionsbetweenintracellularandextracellular

space•  Canbeinducedbylowoxygenandhighpotassium,etc.

Czehetal.,1993BrainResearch

TheCommonali?es

5

Ziburkusetal.2006JNeurophysiol.

ViViVo

HighK+MakesSeizuresandSD

20 s

10 min Timeline Spont. Seizures

Traynelis&Dingledine1988

8.5mM

Anderson&Andrew,2002JNeurophys Zhouetal.,2010CerebCtx

26mM 40mM

62/14/18

BrainsHypoxicduringSeizuresandSDInVitroSeizure

Baharetal.,2006NeuroReport Takanoetal2007

[O2]DeterminesDura(onSD

72/14/18

2/14/18

WhatisMissinginCurrentModels?Hodgkin–HuxleyModel:

OutIn

Na+ K+ Cl-2K+

3Na+

Na/Kpump

ATP

Na+ K+

SeizuresorSD��lossofionichomeostasis.

Kager2000,2002,2007:SDmodel

Cressmanetal.,2009,Ullahetal.,2009:Seizuremodel

assumesenergy(oxygen)isinfinitelyavailable.

assumesionconcentra(onsareconstant.

CurrentModels:

Bazhenovetal.,2004,Krishnanetal.,2011:Seizuremodel

8

52%33%

6%

5%4%

0.25%

TheCostofCor(calComputa(onLennie,2003,CurrentBiology

Distribu(onofEnergyUsedinSpikes:

1.RestoreIonGradient

2.TransmihersReleaseandRecycling

2/14/18 9

OutIn

2K+

3Na+

Na+ K+ Cl-

Na/Kpump

ATP

C6H12O6+6O2à

6CO2+6H2O+36ATP

Energyconsump?oninbrain

2/14/18 10

PartI:OxygenandSeizure

ExperimentalO2Observa?ons1)  Seizure=>O2Deficit

2) NarrowBandO2=>Seizure

3)E&IInterplay=>O2Interplay

2/14/18

-60

20

V (m

V)

40 60 80 100 120 140 160 22

28

Time (s)

[O 2 ] o

(mg/

L)

11

Ingrametal.,2014JNeurophys

Ziburkusetal.2006JNeurophysiol.

Mathema?calModel

IonConcentra?onDynamics:[ ] 2K pump glia diff

od Kd

I It

I Iγβ β− − −=

[ ] 3Na pumpid

tINa

dIγ− −=

MembranePoten?alDynamics:

[ ] 140 (18 [ ] )i iK Na= + −

[ ] 144 ([ ] 18)o iNa Naβ= − −

Cressmanetal.2009

2/14/18

(HodgkinHuxleyEqua(ons)

1 ( )Na K CldV I I Idt C

= − − −

[ ][

26.64ln( ), , ,]o

xi

xE

xx Na K Cl= =

3

4

( ) ( )

( ) ( )( )

Na Na Na NaL Na

K K K KL K

Cl Cl Cl

I G m h V E G V EI G n V E G V EI G V E

= − + −

= − + −

= −

12

Neuron

ExtracellularSpace

Glia

3Na+

2K+

O2K+

3Na+

2K+

Na+Cl- K+

K+

Na+

Cl-

K+

OxygenDynamics:

1.01.0 exp((25 [ ] ) / 3) 1.0 exp(5.5 [ ] )pump

i o

INa K

ρ= ×

+ − + −

1.01 exp((25 [ ] ) / 3) 1.0 exp(5.5 [ ] )

21.0 exp((18 [ ] ) / 2.5)

13gliapump

ig o

glia gliapumpo

INa K

gliaI I

K

ρ= ×

+ − + −

= ++ −

Diffusion,PumpandGlialUptakeCurrents:

([ ] [ ] )diff k o bathI K Kε= −

Ahwell&Laughlin2001JournalofCerebralBloodFlowandMetabolism:AnEnergyBudgetforSignalingintheGreyMaheroftheBrain

Mathema?calModel

2/14/18

2 22 ( ) ([[ ] [ ]] )pump gliapump o bath ood I O O

dtIO

αλ ε− −= + +

13

Pumps Diffusion

Mathema?calModel

OxygenDynamics:

2 22 ( ) ([[ ] [ ]] )pump gliapump o bath ood I O O

dtIO

αλ ε− −= + +

1.01.0 exp((25 [ ] ) / 3) 1.0 exp(5.5 [ ] )pump

i o

INa K

ρ= ×

+ − + −

1.01 exp((25 [ ] ) / 3) 1.0 exp(5.5 [ ] )

21.0 exp((18 [ ] ) / 2.5)

13gliapump

ig o

glia gliapumpo

INa K

gliaI I

K

ρ= ×

+ − + −

= ++ −

Diffusion,PumpandGlialUptakeCurrents:

([ ] [ ] )diff k o bathI K Kε= −max

21 exp((20 3)] /[ )oOρ

ρ =+ −

10 20 30

0.2

0.4

0.6

0.8

[O2]o (mg/L)

Petrushankoetal.2007

2/14/18 14

Pumps Diffusion

theNa/KATPaseasaFunc(onOfO2

Potassium-inducedSpontaneousSeizure

2/14/18

-50

0

50

V (m

V)

10 20 30 40 50 60 70 80 90 100 110 120 5

10

15

[K + ]

o (mM

)

Time (s)

20 s

10 min Timeline Spont. Seizures

Traynelis&Dingledine1988

Experiment

Model

15

100µm

Electrode

350µmBrainSlice

optode

electrodecell

optode

OxygenDynamicsaroundSingleCellExperiment

2/14/18

-60 -40 -20

0 20

V (m

V)

40 60 80 100 120 140 160

22 24

26 28

Time (s)

[O 2 ]

o (m

g/L)

16

Ingrametal.,2014JNeurophys

Time (s)

-50

0

50

5

10

15

20 22 24 26 28

10 20 30 40 50 0.5

1 1.5

2 2.5

- by pump + from bath

V (m

V)

[K + ] o (m

M)

[O 2 ] o (m

g/L)

[O

2 ] rate

(mg/

L/s)

O2 flux

OxygenDynamicsaroundSingleCellExperiment

2/14/18

-60 -40 -20

0 20

V (m

V)

40 60 80 100 120 140 160

22 24

26 28

Time (s)

[O 2 ]

o (m

g/L)

Model

17

Ingrametal.,2014JNeurophys

Wei,Ingram,Ullah,Schiff,2014JNeurophys

Bifurca?onAnalysiswithFixedIonConcentra?ons

18

5 10 15 20 25 30 35 -100

-50

0

50

HB

SN

5 10 15 20 25 30

5

10

15

20

25

30

35

HB SN

[K + ] o (mM)

V (m

V)

[Na +

] i (mM

)

[K + ] o (mM)

Rest

Spiking

DB

HopfBifurca?on(HB):alimitcycledecreasesun(litisreducedtoapointanddisappears.

SaddleNodeBifurca?on(SN):twoequilibriumpointscollideanddisappear.

Bazhenovetal2004JNeurphysiol;KrishnanandBazhenov,2011JNeurosci;Barretoetal2011JBiolPhys;Wei,Ingram,Ullah,Schiff,2014JNeurophys

Bifurca?onAnalysis

2/14/18

-100

50

-100

50

0 20 40 60 80 100 120 -100

50

Time (s)

[Na +

] (m

M)

i

V (mV)

5 10 15 20 25 30

o [K + ] (mM)

5

10

15

20

25

30

35

[O 2 ] bath = 31

[O 2 ] bath = 18

[O 2 ] bath = 10

o [K + ] (mM)

>

22 2[( ) ( [ ] )[ ]]

pump gliapump o ath obod OOI

dtIO

αλ ε− −= + +

Rest

Spiking

DB

19Wei,Ingram,Ullah,Schiff,2014JNeurophys

Ko,Nai,constrainedKi,Nao

Time (s) 1300 0

5

[O 2 ] ba

th (m

g/L)

15

10

25

20

35

30

Experiment Model

[O2]bathBifurca?oninNormal[K+]bath

Seizure

2/14/18

[O 2 ] bath (mg/L)

Time (s) [K

+ ] o (mM

)

50 100

150 200

250 300

350 400 10

15 20

25 30

5

10

20

Ingrametal.,2014JNeurophys Wei,Ingram,Ullah,Schiff,2014JNeurophys

[O2]bathBifurca?oninNormal[K+]bath

2/14/18

0.05 0.1 0.15 0.2 0.25 0.3

-60

-40

-20

0

20

V (m

V)

Time (s)

[O 2 ] bath = 31 mg/L

10 12 14 16 18 20 22 24 26 28 30 32 2

4 6

8 10

12 14 16

[O 2 ] bath (mg/L)

[K + ] o

(mM

)

20 40 60 80 100 120

-60

-40

-20

0

20

40

V (m

V)

Time (s)

[O 2 ] bath = 10 mg/L

50 100 150 200 250 300 350 -80

-60

-40

-20

0

20

V (m

V)

Time (s)

[O 2 ] bath = 27 mg/L

21Wei,Ingram,Ullah,Schiff,2014,JNeurophys

Experiment

2/14/18

ExcitatoryandInhibitoryInterplay

22Ziburkusetal.2006JNeurophysiol.Ziburkusetal.2013JNeurophysiol.

Ingrametal.,2014JNeurophys

2/14/18

OSt.Oriens

St.Pyramidale

St.Radiatum

St.Lacunosummoleculare

P

ε k ratio

G gl

ia ra

tio

0 0.2 0.4 0.6 0.8 1 0

1

2

3

4

5

6

BroadRegion

Time (s)

Ve (m

V)

Vo (m

V)

[K + ] o

(mM

) [O

2 ] o (m

g/L)

-50

0

50

-50

0

50

5

10

15

50 100 150 200 250 300 20

25

Gradient,NotSpikes,DrivesO2UseModel

23Wei,Ingram,Ullah,Schiff,2014JNeurophys

Time (s)

-50

0

50

5

10

15

20 22 24 26 28

10 20 30 40 50 0.5

1 1.5

2 2.5

- by pump + from bath

V (m

V)

[K + ] o (m

M)

[O 2 ] o (m

g/L)

[O

2 ] rate

(mg/

L/s)

[O 2 ] bath

(mg/L)

Time (s)

[K + ] o (m

M)

100

200

300

400 10 15

20 25

30

5

10

ShortSummaryq  O2Deficit

aherSeizureq  Neuronvs.

OxygenInterplay

q  HypoxiaInducesSeizures

Time (s) Ve

(mV)

Vo

(mV)

[K

+ ] o (

mM

) [O

2 ] o (m

g/L)

-50 0

50

-50 0

50

5 10 15

50 100 150 200 250 300 20

25

2/14/18 24

PartII:OxygenandSpreadingDepression

2/14/18 25

SDModel-AddingChlorideandVolume

2/14/18

Pi=[Na+]i+[K+]i+[Cl-]i+Ai

Po=[Na+]o+[K+]o+[Cl-]o+Ao

Osmo?cPressure:

VolumeRegula?on:

dVol/dt~(Pi-Po) Dreier,2011NatureMedicine

26

OutIn

2K+

3Na+K+ Cl-

K+2Cl- Na+ Na+ K+ Cl-

DynamicsforAllIons–KeeptrackofN

The activation and inactivation variables m, h, and n vary between 0 and 1 and represent the

fraction of ion selective channels in the closed and open states. The parameters ↵m, �m, ↵h, �h,

↵n, �n are opening and closing rates of the ion channel state transitions that are dependent on

V . The equations of these rates are from a pyramidal cell model31, originally derived from a

model of hippocampal neurons32, shown as follows:

↵m = 0.32(54 + V )/(1� exp(�(V + 54)/4))

�m = 0.28(V + 27)/(exp((V + 27)/5)� 1)

↵h = 0.128exp(�(50 + V )/18)

�h = 4/(1 + exp(�(V + 27)/5))

↵n = 0.032(V + 52)/(1� exp(�(V + 52)/5))

�n = 0.5exp(�(57 + V )/40).

(9)

The reversal potentials of Na+ (ENa) , K+ (EK ) and Cl� (ECl) are given by Nernst equations:

ENa = 26.64ln([Na+]o[Na+]i

)

EK = 26.64ln([K+]o[K+]i

)

ECl = 26.64ln([Cl�]i[Cl�]o

)

(10)

where [.]i and [.]o represent concentrations inside and outside the cell, respectively. The units

and description of all parameters are summarized in Table 1.

Ion concentration dynamics. Unlike the Hodgkin-Huxley equations where various ion concen-

trations are fixed, we have incorporated potassium, sodium, and chloride ion concentration

dynamics. The ion concentrations are calculated by the ion number over the volume within the

21

compartment, such as [.]i = Ni/V oli, [.]o = No/V olo.

dNK+o

dt=

1

⌧(��(IK � 2Ipump)� Idiff � Iglia � 2�Igliapump + �Ikcc2 + �Inkcc1) ⇤ V olo

dNK+i

dt=

1

⌧(��(IK � 2Ipump)� Ikcc2 � Inkcc1) ⇤ V oli

dNNa+odt

=1

⌧(��(INa + 3Ipump) + �Inkcc1) ⇤ V olo

dNNa+idt

=1

⌧(��(INa + 3Ipump)� Inkcc1) ⇤ V oli

dNCl�odt

=1

⌧(���IClL + �Ikcc2 + 2�Inkcc1) ⇤ V olo

dNCl�idt

=1

⌧(�IClL � Ikcc2 � 2Inkcc1) ⇤ V oli

(11)

where ⌧ = 1000 is used to convert second to millisecond. � = V oli/V olo is the ratio of intra-

/extracellular volume. � = S/(FV oli) is a conversion factor from the current unit (µA/cm2) into

the concentration unit (mM/s), where S is the surface area of the cell, and F is the Faraday

constant.

The neuronal Na/K pump current (Ipump), glial Na/K pump current (Igliapump), glial buffering

current (Iglia), and potassium diffusion current (Idiff ) are modified based on our previous work7.

The Na/K pump strength is modified by 1/3 for glia in our model because the relative resting

energy consumption in neurons versus glia is about 3:133.

Ipump =⇢F1([O2]o)

�

1.0

1.0 + exp((25� [Na+]i)/3)⇥ 1.0

1.0 + exp(3.5� [K+]o)

Igliapump =1

3

⇢F1([O2]o)

�

1.0

1.0 + exp((25� [Na+]gi)/3)⇥ 1.0

1.0 + exp(3.5� [K+]o)

Iglia =GgliaF2([O2]bath)

1.0 + exp((18� [K+]o)/2.5)

Idiff = ✏kF2([O2]bath)F3(�)⇥ ([K+]o � [K+]bath)

(12)

where ⇢, Gglia, ✏k, and [K+]bath represent maximum pump strength, maximum glial uptake

strength, potassium diffusion coefficient, and bath potassium concentration, respectively. Here,

22

Wei,Ullah,Schiff,2014JNeuroscience2/14/18

SeizureandSpreadingDepression

2/14/18

10 20 30 40 50 60 70 -100

0

100

V (m

V)

10 20 30 40 50 60 70 5

10

15

[K + ] o (m

M)

10 20 30 40 50 60 70 7

7.5

Time (s)

10 20 30 40 50 60 70 -100

0

100

V (m

V)

10 20 30 40 50 60 70 0

20

40

[K + ] o (

mM

)

10 20 30 40 50 60 70 10

20

30

Time (s)

SeizuresSpreadingDepression

Voli/

Vol o

Voli/

Vol o

28Wei,Ullah,Schiff,2014JNeuroscience

2/14/18 29

50 100 150 -100

50 V

(mV)

50 100 150 10 20 30

β

50 100 150 10

30

o [K

+ ] (m

M)

50 100 150 120

130 140

[K + ] i (

mM

)

50 100 150 50

150

[Na + ] o (m

M)

50 100 150 20

30

[Na + ] i (m

M)

50 100 150 50

100

[Cl - ] o (m

M)

50 100 150 10 15 20

[Cl - ] i (m

M)

50 100 150 20

25

[O 2 ] o (m

g/L)

Time (s) 50 100 150

5

10

I pum

p (uA/

cm 2 )

Time (s)

ModelVariablesduringSD

Wei,Ullah,Schiff,2014JNeuroscience

20 40 60 80 100 -80

20

V (m

V)

20 40 60 80 100

20 40 60

[K + ] o (m

M)

20 40 60 80 100 10 20 30

β

Time (s)

HypoxicSDModel

Hypoxia

Hypoxia

Czehetal.,1993BrainResearchWei,Ullah,Schiff,2014JNeuroscience

Experiment

WaveofDeath

2/14/18

Voli/

Vol o

20 40 60 80 100

-50 0

50

V (m

V)

V E Na E K E Cl

20 40 60 80 100 0

50

100

[K + ] o (

mM

)

20 40 60 80 100 10

20

30

Time (s)

WaveofDeath(Zandtetal,2011PLOSONE) 31

IncreasedO2Availability[O2]DeterminesDura(onSD

Takanoetal2007NatNeurosci 0 5 10 15 20 25 30

5

10

15

20

25

30

35

0

[K + ] o (m

M)

[K + ] bath (mM)

Model

Wei,Ullah,Schiff,2014JNeuroscience

TheUnifica?onofSeizuresandSD

2/14/18

[K+]bath(mM)

[K+ ]O(m

M)

Localpotassiumchangesasafunc(onofbathpotassiumandbathoxygen

[K+]batharound8-12mM:burs(ng(Rutecki1985)andseizures(Traynelis1988)

[K+]batharound26mM:spreadingdepression(Anderson2002)

33

[K+ ]O(m

M)

[K+]bath(mM)[O2]bath(mg/L)

0 5 10 15 20 25 30 0

5

10

15

20

25

30

35

[K + ] o (m

M)

[K + ] bath (mg/L) -100 -50 0 50 -150

-100

-50

0

I Na

V (mV)

-100 -50 0 50 0

50

100

150

V (mV)

HH GHK ,P/G=3e-6 GHK ,P/G=2e-6

a" b" c"

(uA/

cm 2 )

I K (u

A/cm

2 )

DoubleBifurca?onPreservedwithGoldman-Hodgkin-Katzcurrentequa?on

The bifurcation analysis of the simplified model were performed using XPPAUT42.

Goldman-Hodgkin-Katz (GHK) model. The Hodgkin-Huxley model uses linear relationships

for current and voltage (Ohms law). Nevertheless, it is well known that the actual current flow-

ing through ion-selective permeability channels in membranes is nonlinear and rectifying43,

and can be more accurately accounted for using the Goldman-Hodgkin-Katz equations44. The

equations modeling the current due to the flow of a given ion across the membrane given by the

Hodgkin-Huxley and GHK formalism respectively are:

IHH = G⇥ (V � E) = G⇥ (V � RT

zFln(

Co

Ci

))

IGHK = P ⇥ z2F 2V

RT

Ci � Coexp(� zFVRT

)

1� exp(� zFVRT

)

(21)

where E,R, T, z, and F , represent the Nernst potential, gas constant, absolute temperature, ion

valence, and Faraday’s constant respectively. Ci and Co are the concentrations of a specific ion

outside and inside the cell. P is the permeability of the membrane to the flow of a specific ion,

which depends only on the types and numbers of ion channels present in the membrane. G is

the ion conductance that measures the ability of the membrane to carry electrical current. The

detailed calculations for the conversion between P and G are as follows:

1) G (in units of S/m2), is equal to �/L, where � (in units of S/m) is electrical conductivity and

L (in units of m) is thickness of the cell membrane.

2) P (in units of m/s), is equal to D/L, where D (in units of m2/s) is diffusion coefficient.

3) The relation between ionic mobility µ (in units of m2/(Vs)) and electrical conductivity �

(in units of S/m) is � = neµ, where n (in units of m�3) is the number density of monovalent

ions and e is the electronic charge. For any substance, n can be expressed in terms of molar

28

Wei,Ullah,Schiff,2014JNeuroscience

20 22 24 26

-50

50

V (m

V)

Time (s)

20 40 60 80 100 120 -100

50 V

(mV)

Time (s) 20 40 60 80 100 120 -100

50

V (m

V)

Time (s)

20 40 60 80 100 120 -100

50

V (m

V)

Time (s) 20 40 60 80 100 120 -100

50

V (m

V)

Time (s)

b c d

g h i

e a f

5 10 15 20 25 30 0

5

10

15

20

25

30

[O 2 ] ba

th (m

g/L)

[K + ] (mM) bath

SZ

SD

WoD

SS TF

0 1 2 3

-50

0

50

V (m

V)

Time (s)

20 40 60 80 100 120 -100

50

V (m

V)

Time (s)

0 1 2 3 -100

50

V (m

V)

Time (s)

Traynellis&Dingledine1988 Anderson&Andrew2002Brisson&Andrew2012

Zhouetal2010

Czehetal1993Zandtetal2011

HH1952

2/14/18 35

Acknowledgements

Justin M. Ingram Steven J. Schiff Ghanim Ullah

US-GermanCollabora?veResearchinComputa?onalNeuroscience(CRCNS)

2/14/18 36Code Archive: https://scholarsphere.psu.edu/!

2/14/18 37