From epilepsy to migraine to stroke: A unifying framework

From epilepsy to migraine to stroke:A unifying framework

(Or: Act neurons like steam engines?)

Markus A. Dahlem (HU Berlin) &Niklas Hubel (TU Berlin)

Joint Focus Session DY/BP: Dynamical Patterns in Neural Systems: From

Brain Function to Dysfunction, April 1, 2014

Outline

Introduction

Unifying ion dynamics in the brain

Application: From genotype to phenotype

Summary

Outline

Introduction



Summary

Top three with respect to costs & burden

In Europe

27 Migraines

22 Strokes

15.5 Epilepsy

billion Euro each year.

Balak and Elmaci (2005) European

Journal of Neurology 12

“What is particularly interesting tonote is that the most recent reportsstate that migraine alone is responsibleof almost 3% of disability attributableto a specific disease worldwide, also inconsideration of its comorbidity. Thisplaces migraine as the 8th mostburdensome diseases, the 7th amongnon-communicable diseases and the 1st

among the neurological disordersranked in the GBD report.”

Models fill in the ‘gaps’ in clinical obervability

insid

e ce

ll

outsid

e ce

ll

a e

sensory aura (15min)

visual aura (0min)

behavior, perceptionsensory processing

(a) genetics defects, e.g. FHM, CADASIL, multifactorial (GWAS)

(b) Hodgkin-Huxley type, single cell electrophysiology models.

(c) Neural mass/fields population models, with subpopulations havingspecific synaptic receptor distribution.

(d) Local circuits, including the migraine generator network in the brainstem

(e) Pain, loss of function, seizing/convulsions, mental dysfunctions, imparedsensory and cognitive processing

• MAD, Migraine generator network and spreading depression dynamics as neuromodulation targets in episodicmigraine. Chaos, 23, 046101 (2013)• MAD, Migraines and Cortical Spreading Depression, Encyclopedia of Computational Neuroscience, (in press)

CADASIL: Cerebral Autosomal-Dominant Arteriopathy with Subcortical Infarcts and Leukoencephalopathy;

FHM: Familial hemiplegic migraine; GWAS: genome-wide association study

Models fill in the ‘gaps’ in clinical obervabilityI

II

III

IV

V

VI0 5 10 15 20 25 30 35

time (s)

100

50

0

50

volt

age

(mV

)

V

EK

ENa

Iapp

seizure-likeafterdischarges

depolarization block

dominance pump current

m-gatedeactivation

begin I -drivenrepolarizationNa

+

tran

smem

bra

ne &

cellu

lar

level

mole

cula

r le

vel &

g

eneti

cs

b c

insid

e ce

ll

outsid

e ce

ll

a

off on

HY,TH

SPG

SSN

TCC

PAG

LC

RVMTG

cortex

cranial circulation & innervation

bone

dSD

cortico-thalamicaction

release noxious agentse

sensory aura (15min)

visual aura (0min)

behavior, perceptionsensory processing

balanced excitation and inhibition in ion-based models

org

an level

(a) genetics defects, e.g. FHM, CADASIL, multifactorial (GWAS)

(b) Hodgkin-Huxley type, single cell electrophysiology models.

(c) Neural mass/fields population models, with subpopulations havingspecific synaptic receptor distribution.

(d) Local circuits, including the migraine generator network in the brainstem

(e) Pain, loss of function, seizing/convulsions, mental dysfunctions, imparedsensory and cognitive processing

• MAD, Migraine generator network and spreading depression dynamics as neuromodulation targets in episodicmigraine. Chaos, 23, 046101 (2013)• MAD, Migraines and Cortical Spreading Depression, Encyclopedia of Computational Neuroscience, (in press)

CADASIL: Cerebral Autosomal-Dominant Arteriopathy with Subcortical Infarcts and Leukoencephalopathy;

FHM: Familial hemiplegic migraine; GWAS: genome-wide association study

Outline

Introduction



Summary

From

HH-type conductance-based

toconductance- & ion-based models (2nd generation model)

C∂V

∂t= −INa − IK − Ileak

−Ipump

+Iapp (1)

INa = gNam3h(V − ENa)

IK = gKn4(V − EK )

Ileak = gleak(V − Vrest)

∂n

∂t= αn(1 − n) − βn,

∂h

∂t· · · (2) − (4)

∂[ion]e∂t

= − A

FVoloIion

∂[ion]i∂t

=A

FVoliIion (5) − · · ·

HH: Hodgkin-Huxley

From HH-type conductance-based toconductance- & ion-based models (2nd generation model)

3Na+

2K+

K+

Na+

K+

K+

Na+

Extracellular Space

Bath/Vasculature

Neuron

Cl

Diff

usi

on

Cl-

-

C∂V

∂t= −INa − IK − Ileak−Ipump+Iapp (1)




∂n

∂t= αn(1 − n) − βn,

∂h

∂t· · · (2) − (4)

∂[ion]e∂t

= − A

FVoloIion

∂[ion]i∂t

=A


HH: Hodgkin-Huxley

From HH-type conductance-based toconductance- & ion-based models (2nd generation model)

ion reservoirs

isolated boundary

syst

em

surroundingsenergysource

extracellular

intracellular

C∂V

∂t= −INa − IK − Ileak−Ipump+Iapp (1)




∂n

∂t= αn(1 − n) − βn,

∂h

∂t· · · (2) − (4)

∂[ion]e∂t

= − A

FVoloIion

∂[ion]i∂t

=A


HH: Hodgkin-Huxley

Unifying ion dynamics in epilepsy, migraine, and stroke

Some terminology is due:heterogenous open system

ion reservoirs

isolated boundary

syst

em


extracellular

intracellular

• P. Dreier, ... MAD ... Is spreading depolarization characterized by an abrupt, massive release of Gibbs free energyfrom the human brain cortex? The Neuroscientist 19,25-42 (2012)


Some terminology is due:heterogenous “closed” system

ion reservoirs

isolated boundary

syst

em


extracellular

intracellular



Some terminology is due:heterogenous isolated “plus” system

ion reservoirs

isolated boundary

syst

em


extracellular

intracellular


Many, many, parameters, but most fixed by experiments

Table: Parameters for ion–based model – Part 2

Name Value & unit Description

Cm 1 µF/cm2 membrane capacitanceφ 3/msec gating time scale parameterg lNa 0.0175 mS/cm2 sodium leak conductance

g gNa 100 mS/cm2 max. gated sodium conductance

g lK 0.05 mS/cm2 potassium leak conductance

g gK 40 mS/cm2 max. gated potassium conductance

g lCl 0.02 mS/cm2 chloride leak conductance

Na0i 25.23 mM/l intracell. sodium conc.

Na0e 125.31 mM/l extracell. sodium conc.

K 0i 129.26 mM/l intracell. potassium conc.

K 0e 4 mM/l extracell. potassium conc.

Cl0i 9.9 mM/l intracell. chloride conc.

Cl0e 123.27 mM/l extracell. chloride conc.

E 0Na 39.74 mV sodium Nernst potential

E 0K -92.94 mV potassium Nernst potential

E 0Cl -68 mV chloride Nernst potential

And still more parameters, but most fixed by experiments

Table: Parameters for ion–based model – Part 2

Name Value & unit Description

ωi 2.16 µm3 intracell. volumeωe 0.72 µm3 extracell. volumeF 96485 C/Mol Faraday’s constantAm 0.922 µm2 membrane surface

γ 9.556e–6 µm2MolC

conversion factorρ 6.8 µA/cm2 max. pump currentk1 5e–5/sec/(mM/l) buffering ratek1 5e–5/sec backward buffering rateλ 3e–2/sec diffusive coupling strengthKbath 4 mM/l potassium conc. of extracell. bathB0 500 mM/l buffer conc.

Fixed points in 2nd generation HH

I “closed” system & leaky membrane

I “closed” system & voltage–gated membrane

I open system & voltage–gated membrane

(This will help to understand periodic solutions)

Model without voltage-gating: Pump establishes polarizedstate (beyond a Gibbs-Donnan equilibrium)

ion reservoirs

isolated boundary

syst

em


extracellular

intracellular

only leak currents

0 5 10 15 20Imax in µA/ cm2

100

80

60

40

20

0

Vmax

inmVolt

polarized physiological state

• N. Hubel et al., Bistable dynamics underlying excitability of ion homeostasis in neuron models (in press PLOSComp. Biology)

Model with voltage-gating: Bistability!

ion reservoirs

isolated boundary

syst

em


extracellular

intracellular

gated currents

0 5 10 15 20Imax in µA/ cm2

100

80

60

40

20

0

Vmax

inmVolt HB

HBHB


depolarized pathophysiological state


Choices: Current and pump equations, ions, ...Two pump types

Iion,pumped ,1([K ]o , [Na]i ) = Imax

(1 +

KmK

[K ]o

)−2(1 +

KmNa

[Na]i

)−3

Iion,pumped ,2([K ]o , [Na]i ) = Imax1

1 + e(25−[Na]i/3)

1

1 + e(5.5−[K ]o)

HH current or GHK currents

Iion = gion(V − Eion)

Iion = V αF Pion[ion]i − [ion]oe

−αV

1 − e−αV

With or without chloride dynamics

d[Cl−]

dt= ... or 0

cf. Krogh-Madsen et al. Am. J. Physiol. Heart Circ. Physiol., 289,398-413 (2005).

We gave it a fair shake. It’s robust

0

20

40

60

80

100

120

140

160

180

ρinµ

A/cm

2

I p,B

,exc

l.Cl−

,Ner

nst

I p,B

,inc

l.Cl−

,Ner

nst

I p,B

,exc

l.Cl−

,GH

K

I p,B

,inc

l.Cl−

,GH

K

I p,A

,exc

l.Cl−

,Ner

nst

I p,A

,inc

l.Cl−

,Ner

nst

I p,A

,exc

l.Cl−

,GH

K

I p,A

,inc

l.Cl−

,GH

K

1 2 3 4 5 6 7 8

Stability Regimes of Ion-Based Modelsstbl. depol. fixed pointbistablestbl. pol. fixed point

0

5 1 2 3 4 5 6 7 8

0.1 2.0 4.0 6.0 8.0 10.0

χA

0

5

10

15

20

25

30

35

ρinµ

A/cm

2

HB1

LP2 HB2

HB3

2 10 20 30 40 50

f in %

0

5

10

15

20

25

30

35

ρinµ

A/cm

2 HB1

LP2HB2

HB3

stbl. depol. fixed pointbistablestbl. pol. fixed point

0.1

2.0

LP10.1

2.0

LP1


HH 2nd -generation “closed” systems are bistable

C∂V

∂t= −

ion∑(Iion,gated + Iion,pumped)

fixed point!= 0

current, pump, and gating equations . . .

∂[K+]e∂t

=A

FVole(IK+,gated + IK+,pumped)

+λ([K+]bath − [K+]e)︸︷︷︸buffer to bath

fixed point!= 0

f.p.!= 0

ion reservoirs

isolated boundary

syst

em


extracellular

intracellular

0 5 10 15 20Imax in µA/ cm2

100

80

60

40

20

0

Vmax

inmVolt HB

HBHB



Open system: Diffusion (buffering) of potassium

C∂V

∂t= −

ion∑(Iion,gated + Iion,pumped)

fixed point!= 0

current, pump, and gating equations . . .

∂[K+]e∂t

=A

FVole(IK+,gated + IK+,pumped)+λ([K+]bath − [K+]e)︸︷︷︸

buffer to bathfixed point!

= 0

f.p.!= 0

ion reservoirs

isolated boundary

syst

em


extracellular

intracellular

0 5 10 15 20Imax in µA/ cm2

100

80

60

40

20

0

Vmax

inmVolt HB

HBHB



Periodic solutions in 2nd generation HH

I open system & voltage–gated membrane

I full bifurcation analysis

I slow–fast analysis

Time scales in ion dynamics

1st generation Hodgkin–Huxley model

0.01ms RC membrane time constant

1ms ion gating

2nd generation Hodgkin–Huxley model has in addition

1s volume–to–surface–area ratio / permeability

100s potassium regulation

Unified minimal (4D) model ofspiking, seizures and spreading depression

5 10 15 20

Kbath / (mM/l)

−150

−100

−50

0

50

mV

HB1

LP1

LP2 HB2 HB3 HB4

LP1lc

LP2lc

TR1

TR2TR3TR4PD

membrane potential

5 10 15 20

Kbath / (mM/l)

0

10

20

30

40

50

60

70

80

90

mM

/l

HB1

LP2 LP1HB2

HB3HB4

extrac. potassiumstable FPunstable FPstable LCunstable LCstable torus

0 100 200 300 400 500

t / sec.

−100

−80

−60

−40

−20

0

20

40

60

mV

potentialEK

ENa

ECl

V

0.0 0.5 1.0 1.5 2.0

t / sec.

−80

−60

−40

−20

0

20

40

60

mV

0 500 1000 1500 2000

t / sec.

−100

−80

−60

−40

−20

0

20

40

60

mV

120130140

ion conc.

0 100 200 300 400 500

t / sec.

10

20

120

130

0.0 0.5 1.0 1.5 2.0

t / sec.

15

20

0 500 1000 1500 2000

t / sec.

0

20

40

60

80

100

120

140

mM

/l

Ki

Nai

Cli

Ke

Nae

Cle

a)

b)

c)

mM

/lm

M/l


5 10 15 20

Kbath / (mM/l)

−150

−100

−50

0

50

mV

HB1

LP1

LP2 HB2 HB3 HB4

LP1lc

LP2lc

TR1

TR2TR3TR4PD

membrane potential

5 10 15 20

Kbath / (mM/l)

0

10

20

30

40

50

60

70

80

90

mM

/l

HB1

LP2 LP1HB2

HB3HB4

extrac. potassiumstable FPunstable FPstable LCunstable LCstable torus

5 10 15 20

Kbath / (mM/l)

20

40

60

80

100

120

140

160

mM

/l

LP1

LP2

HB1

HB2HB3

HB4

LP1lc

LP2lc

TR1

TR2TR3

TR4

extrac. sodium

5 10 15 20

Kbath / (mM/l)

−80

−60

−40

−20

0

20

40

mM

/l

LP1

LP2

HB1

HB2 HB3HB4

LP1lc

LP2lc

TR1

TR2

TR3

TR4

potassium gain/loss Ke

stable FPunstable FPstable LCunstable LCstable torus

6.7 6.9 7.1

HB1 TR4

PD


−60 −40 −20 0 20 40

Ke / (mM/l)

−80

−60

−40

−20

0

20

40

mV

HB1

LP1HB2

LP2

HB3HB4

LP1lc

LP2lc

LP3lcLP4lc

membrane potential for simple model

−60 −40 −20 0 20 40

Ke / (mM/l)

0

10

20

30

40

50

60

mM

/l

HB1

LP1HB2

LP2

HB3HB4

LP1lc

LP2lc

LP3lc

LP4lc

extrac. potassium for simple modelstable FPunstable FPstable LCunstable LC

−49 −47 −45 −43

HB3

LP1lc

27 29 31

HB1

0.00 0.03

+2.87×101

HB1

PD LP5lc

LP6lc

−49 −47 −45 −43

HB3LP1lc

27 29 31

LP1

LP3lc

5 10 15 20

Kbath / (mM/l)

20

40

60

80

100

120

140

160

mM

/l

LP1

LP2

HB1

HB2HB3

HB4

LP1lc

LP2lc

TR1

TR2TR3

TR4

extrac. sodium

5 10 15 20

Kbath / (mM/l)

−80

−60

−40

−20

0

20

40

mM

/l

LP1

LP2

HB1

HB2 HB3HB4

LP1lc

LP2lc

TR1

TR2

TR3

TR4

potassium gain/loss Ke

stable FPunstable FPstable LCunstable LCstable torus

6.7 6.9 7.1

HB1 TR4

PD

Slow–fast analysis using K+ gain–and–loss

−80

−60

−40

−20

0

HB1

LP1HB2

LP2

HB3HB4

LP1lc

LP2lc

LP3lcLP4lcp

ote

nti

al

(tra

nsm

em

.)

−60 −40 −20 0 20 400

10

20

30

40

50

60

HB1

LP1HB2

LP2

HB3HB4

LP1lc

LP2lc

LP3lc

LP4lc

particle exchange (potassium ions)

pota

ssiu

m

(extr

ace

ll.)

mM

mM

mV

Open systems yield someKe dynamics:

dKedt

= ...

Ke can be subsituted by Ke

(alternative formulation)

dKedt

= λ(Kbath − Ke)

with

Ke = K 0e + ωi

ωe(K 0

i − Ki ) + Ke

The “migraine–aura–ischemic–stroke” cycle

−80

−60

−40

−20

0

HB1

LP1HB2

LP2

HB3HB4

LP1lc

LP2lc

LP3lcLP4lcp

ote

nti

al

(tra

nsm

em

.)

−60 −40 −20 0 20 400

10

20

30

40

50

60

HB1

LP1HB2

LP2

HB3HB4

LP1lc

LP2lc

LP3lc

LP4lc


pota

ssiu

m

(extr

ace

ll.)

mM

mM

mV


−80

−60

−40

−20

0

HB1

LP1HB2

LP2

HB3HB4

LP1lc

LP2lc

LP3lcLP4lcp

ote

nti

al

(tra

nsm

em

.)

−60 −40 −20 0 20 400

10

20

30

40

50

60

HB1

LP1HB2

LP2

HB3HB4

LP1lc

LP2lc

LP3lc

LP4lc


pota

ssiu

m

(extr

ace

ll.)

mM

mM

mV

trans-membraneevents


−80

−60

−40

−20

0

HB1

LP1HB2

LP2

HB3HB4

LP1lc

LP2lc

LP3lcLP4lcp

ote

nti

al

(tra

nsm

em

.)

−60 −40 −20 0 20 400

10

20

30

40

50

60

HB1

LP1HB2

LP2

HB3HB4

LP1lc

LP2lc

LP3lc

LP4lc


pota

ssiu

m

(extr

ace

ll.)

mM

mM

mV

+

+

iso-intracellularconcentration


−80

−60

−40

−20

0

HB1

LP1HB2

LP2

HB3HB4

LP1lc

LP2lc

LP3lcLP4lcp

ote

nti

al

(tra

nsm

em

.)

−60 −40 −20 0 20 400

10

20

30

40

50

60

HB1

LP1HB2

LP2

HB3HB4

LP1lc

LP2lc

LP3lc

LP4lc


pota

ssiu

m

(extr

ace

ll.)

mM

mM

mV

+

+


−80

−60

−40

−20

0

HB1

LP1HB2

LP2

HB3HB4

LP1lc

LP2lc

LP3lcLP4lcp

ote

nti

al

(tra

nsm

em

.)

−60 −40 −20 0 20 400

10

20

30

40

50

60

HB1

LP1HB2

LP2

HB3HB4

LP1lc

LP2lc

LP3lc

LP4lc


pota

ssiu

m

(extr

ace

ll.)

mM

mM

mV

+

+

release ofGibbs freeenergy

Open question:Can we treat this cycle inanalogy to a steam engine?

The “ceiling level” of [K+]e in seizure activity

−80

−60

−40

−20

0

HB1

LP1HB2

LP2

HB3HB4

LP1lc

lc

lcLP4lcp

ote

nti

al

(tra

nsm

em

.)

−60 −40 −20 0 20 400

10

20

30

40

50

60

HB1

LP1HB2

LP2

HB3HB4

LP1lc

LP4lc


pota

ssiu

m

(extr

ace

ll.)

mM

mM

mV

LP2

LP3

LP2lc

LP3lc

seizure-likeactivity

"ceiling level"

Outline

Introduction



Summary

From genotype to cellular phenotype (just the recipe)

Tail currents to HH parameters:

120 80 40 0 40V / mV

5

10

15

20

τ/ms

wild-type

mutant

0

1−1e

1

h

deinactivation

0 10t / ms120

10V / mV

0

1e

1

h

inactivation

0 10t / ms120

10V / mV

τ ∗hτh τ ∗

hτh

Reduced firing rate!

0 20 40 60 80 100 120 140 160 180Iapp / µA cm−2

0

50

100

150

200

wild-type

mutant

lower fire rate =hypoexcitablein rate-basedpopulation models

F(

) /

Hz

I app

More susceptible to migraine

0 20 40 60 80 100t / s

140

100

60

20

20

60

V /

mV

mutantV

EK

ENa

20%

100%

0 20 40 60 80 100t / s

140

100

60

20

20

60

V /

mV

wild-typeV

EK

ENa

20%

100%

13.6s

7.2s

• M.A. Dahlem, J. Schumacher, N Hubel, Linking a genetic defect in migraine to spreading depression in acomputational model (submitted arXiv 1403.6801)

Outline

Introduction



Summary

Conclusions

I Including ion dynamics into a Hodghin-Huxley frameworkyields slow quasiperodic dynamics:

I important bifurcation parameter is gain–and–loss of ions,I explain the “ceiling level” of [K+]e in seizure activity,I explain the “migraine–aura–ischemic–stroke” contiuum.

I No synaptic currents needed for slow dynamics, in particular,no metabotropic receptor that acts through a secondarymessenger, like GABAB .

I Remark : Ultra–slow (or near–DC (direct current)) activitythat cannot be observed by electroencephalography (EEG),because it is susceptible to uncontrollable artifacts such aschanges in the resistance of the dura.However : subdural electrode recordings provided recentlydirect and unequivocal evidence that such dynamics occurs inabundance in people with structural brain damage.

Cooperation & Funding

Niklas Hubel, Julia Schumacher,Thomas Isele

Steven Schiff(Penn State Center for Neural Engineering)

Jens Dreier(Department of Neurology, Charite; University Medicine, Berlin)

berlin

Migraine Aura Foundation

Health & Medicine

From epilepsy to migraine to stroke: A unifying framework