Reentrant and retracting waves of cortical spreading depression in migraine

Reentrant and retracting waves of cortical
spreading depression in migraine

Markus A. Dahlem

Research group: Nonlinear Dynamics in Physiology and Medicine

engulfing localized fragmented

berlin

Cardiovascular Research Lecture, UCLA, June 18, 2012
Markus A. Dahlem, TU Berlin

Confined 2D wave segments A pattern formation problem

Spatio-temporal patterns in the heart and brain

”Conduction in nervous tissue resembles that in somatic striated in
cardiac muscle. The laws which apply to the muscle fibers are also
applicable to the nerve fibers”

Wiener & Rosenblueth (1946)


Conﬁned 2D wave segments A pattern formation problem

Spatio-temporal patterns in the heart and brain

(a) (b)
CS

LS

temporarily
affected area
(c) (d)

Dahlem et al. Physcia D 239, 2010

A. Garﬁnkel et al. PNAS 97, 2000


Outline

Normal neuron in healthy brain
K+
Iin Iout ECV 20%
+
2+ Na Ca
2+ Na+
Ca

Ca
2+ Ca2+
+
Na
+ DR
K
SK
–70 mV hypoxic tissue
Recovery in ischaemic stroke
Swollen neuron during spreading depolarization K+ n−gate deactivation Hopf
[K +]o Ipump eletrogenic pump Hopf
Na+
Iin H2O Iout
SNIC
ECV 5% Ca2+
Spreading depression
Fold
Seizure−like
–10 mV activity [K + ]o = 10mM
(ceiling level)

AMPA/ NMDAR SNIC
kainate
2+

1. Spreading depression: from
Ca
+ 2+
Na Ca
+
Ca
2+ Na n −gate
Nonspecific
V membrane
voltage
Insufficient cation channels
sodium pump

molecules to cell to tissue

2. Reentrant and retracting
waves in the cortex

3. Migraine as a dynamical gyral crowns
positive (fender)

disease: Towards therapy
transient and
slow dynamics

entrance to sulci

negative (saddle)


Outline

K+
Iin Iout ECV 20%
+
2+ Na Ca
2+ Na+
Ca

Ca
2+ Ca2+
+
Na
+ DR
K
SK
–70 mV hypoxic tissue
Swollen neuron during spreading depolarization K+ n−gate deactivation Hopf
[K +]o Ipump eletrogenic pump Hopf
Na+
Iin H2O Iout
SNIC
ECV 5% Ca2+
Fold
Seizure−like
–10 mV activity [K + ]o = 10mM
(ceiling level)

AMPA/ NMDAR SNIC
kainate
2+

1. Spreading depression: from
Ca
+ 2+
Na Ca
+
Ca
2+ Na n −gate
Nonspecific
V membrane
voltage
sodium pump

molecules to cell to tissue


Spreading depression From molecules to cell to tissue

Migraine Types and subforms

from Migraine Aura Foundation

”Cortical spreading depression is key to the genesis of migraine”
Michael Moskowitz (Harvard Medical School)



SD: From molecules to entire brain

Functional mutations Spreading depression (SD)

(e.g. FHM2: sodium-potassium pump)
cf.: Maagdenberg, et al., Ann. Neurol., 67 2010

Tottene, et al., Neuron, 61 2009

Freilinger, et al. Nature Genetics online 2012



SD: From molecules to entire brain

Electrophysiology Thermodynamics
break down of ion grandients massive release of Gibbs free energy
cell swelling
K+
Iin Iout ECV 20%
+
2+ Na Ca
2+ Na+
Ca

Ca
2+ Ca2+
+
Na
+ DR
K
SK
–70 mV

Swollen neuron during spreading depolarization K+

Na+
Iin H2O Iout

ECV 5% Ca2+

–10 mV
AMPA/ NMDAR
kainate
2+
Ca
+ 2+
Na Ca
+
Ca
2+ Na
Nonspecific
sodium pump

J.P. Dreier Nature Medicine 17 2011 J.P. Dreier et al. Neuroscientist accepted


Spreading depression Local dynamics

Spreading depression wave: ion gradients breakdown

(mM)
Ve
+
Na 150
60
50
log [cat] , M +
Na
+
-1 K 3
1.5
Ca++
0.08
-2
+ 0 10 20 30 s
K

-3 Ca++

-4
-7 +
H

-8
Ve
20 mV

unit
act.

1 min

M. Lauritzen TINS 10:8 (1987)



Spreading depression wave: ion gradients breakdown

(mM)
Ve
+
Na 150
60
“SD is such a drastic and
50
log [cat] , M
Na
+

+
extensive phenomenon that
-1 K 3
Ca++
1.5
almost any speculation is safe
0.08
-2
K
+ 0 10 20 30 s concerning probable involvement
-3 Ca++
of all cortical elements”
Marshall (1959) Physiol. Rev. 39:239.
-4
-7 +
H

-8
Ve
20 mV

unit
act.

1 min
SD: spreading depression
M. Lauritzen TINS 10:8 (1987)



Le˜o original demonstration of SD
a

Le˜o AAP (1944) J. Neurophysiol. 7:359
a



SD in human cortex shown by ECoG

Fabricius et al. (2006) Brain 129:778



Mathematical models cells, circuits, and to tissue

Current distribution

I

Apical dendrite
IN a,P
II
IK,DR IN MDA
III IK,A

IV
Glia K+

V Soma
Osmotic force
IN a,T
VI Pump
[N a+ ]i

r
la
llu
ce
tra
[K + ]o

Ex



Local Dynamics during SD

∂V
C = −INa − IK − ICl + I pump + Iapp
∂t

Current distribution
Apical dendrite

IN a,P
IK,DR IN MDA
IK,A

Glia K+

Soma
Osmotic force
IN a,T
Pump
[N a+ ]i
r
la
llu
ce
tra

[K + ]o
Ex




∂V
∂t
3
INa = −m∞ h(ENa − V )
Current distribution IK = −n4 (EK − V )
Apical dendrite

IN a,P
IK,DR IN MDA
IK,A

Glia K+

Soma
Osmotic force
IN a,T
Pump
[N a+ ]i
r
la
llu
ce
tra

[K + ]o
Ex




∂V
∂t
3
∂n ∂h
= αn (1 − n) − βn, ···
∂t ∂t
Apical dendrite

IN a,P
IK,DR IN MDA
IK,A

Glia K+

Soma
Osmotic force
IN a,T
Pump
[N a+ ]i
r
la
llu
ce
tra

[K + ]o
Ex




∂V
∂t
3
∂n ∂h
= αn (1 − n) − βn, ···
∂t ∂t
Apical dendrite

IN a,P
IK,DR IN MDA ∂[ion]o Iion A
= + Idiﬀ
IK,A ∂t FVolo
∂[ion]i Iion A
K+ =
Glia
∂t FVoli
Soma
Osmotic force
IN a,T
Pump
[N a+ ]i
r
la
llu
ce
tra

[K + ]o
Ex




∂V
∂t
3
∂n ∂h
= αn (1 − n) − βn, ···
∂t ∂t
Apical dendrite

IN a,P
= + Idiﬀ
IK,A ∂t FVolo
∂[ion]i Iion A
K+ =
Glia
∂t FVoli
Soma
Osmotic force
IN a,T −2 −3
Pump pump KmK KmNa
[N a+ ]i
Iion (V ) = βion Imax 1+ 1+
r
la
llu

[K ]o [Na]i
ce
tra

[K + ]o
Ex




∂V
∂t
3
∂n ∂h
= αn (1 − n) − βn, ···
∂t ∂t
Apical dendrite

IN a,P
= + Idiﬀ
IK,A ∂t FVolo
∂[ion]i Iion A
K+ =
Glia
∂t FVoli
Soma
Osmotic force
IN a,T −2 −3
Pump pump KmK KmNa
[N a+ ]i
Iion (V ) = βion Imax 1+ 1+
r
la
llu

[K ]o [Na]i
ce
tra

[K + ]o
Ex

Alternatively (GHK currents)

[ion]i − [ion]o e −αV
Iion = V αF Pion
1 − e −αV


Tissue properties & engery state change time scales . . .

... otherwise robust!

50 V 50 V
EK EK
ENa ENa
0

voltage (mV)
0 Iapp Iapp
voltage (mV)

50
50

100
100

0 1 2 3 4 5 6 0 5 10 15 20 25 30 35
time (s) time (s)

Parameters relevant for migraine aura–ischemic stroke continuum.



Bifurcations involved in local dynamics of SD
50 V
EK
ENa
0 Iapp
voltage (mV)

50

100

0 1 2 3 4 5 6
time (s)

hypoxic tissue
+]
n−gate deactivation Hopf
[K o I
pump eletrogenic pump Hopf
SNIC
Fold
Seizure−
like activ [K + ]o = 10mM
ity (ceiling level)

SNIC

V n −gate
membran
e voltage

M. A. Dahlem, Models of cortical SD, Scholarpedia



Migraine full-scale attack is more conﬁned

(a) (b)
CS

LS

temporarily
affected area
(c) (d)

Dahlem et al. ”2D wave patterns ... ”. Physcia D 239 (2010) Special issue: Emerging Phenomena.



Macroscopic RD with augmented nonlocal transmission

neurovascular coupling The extended Hodgkin-Grafstein model (1963) of SD

activator−inhibitor dynamics
u3 2
ion gradient u =
˙ u− −v +D u
diffusion

ion
pumps
3
out in

(+ FHN inhibitor equations + nonlocal term)

firing rate
ε−1 v
˙ = u + β − γv + KF [u]
ion neural network activity
depolarization
currents
Global control

ion F [u] = Su (t) − S0
conductance

Su (t) = H(u(r, t) − ue ) dr,



Outline

2. Reentrant and retracting
waves in the cortex


Spreading depression Spiral waves

Re-entrant SD waves with anatomical block

Reshodko, L. V. and Bureˇ, J Biol. Cybern. 18,181 (1975)
s



Experiments: open wave segments are unstable

Dahlem & M¨ller Exp. Brain Res. 115 (1997)
u



Spreading Depression: Reaction-diﬀusion in the brain

A nearly complete discharge and
recharge of chemical batteries in
neurons and glial cells

Dahlem & M¨ller (1997) Exp. Brain Res. 115:319
u



Spreading Depression: Reaction-diﬀusion in the brain

Z-type rotation causes a wave break in the spiral core.

Dahlem & M¨ller (1997) Exp. Brain Res. 115:319
u



Drugs adjust excitability:retracting & collapsing waves
a b c

d e f

g h i

j k l

Dahlem et al. 2D wave patterns ... . (2010) Physcia D



Drugs adjust excitability:retracting & collapsing waves

What happens if SD wave fragments with open ende occur in
human pathophysiology during migraine?

Do they form spirals?
Do fragments quickly retract?
Or: can wave fragments propagte some distance?



Engulﬁng RD-wave: current paradigm of migraine attack

M. Lauritzen (1987) Trends in Neurosciences 10:8.




(a) (b)
CS

LS

temporarily
affected area
(c) (d)




What is a migraine aura?



Visual migraine aura model

a e

b c
d

Dahlem et al. (2000) Eur. J. Neurosci. 12:767.
Dahlem and Chronicle (2004) Prog. Neurobiol. 74:351.



Migraine visual field defects reported in 1941 by K. Lashley
visual field defect pattern on primary visual cortex

15 11min15min
9min
7min
10 5min

5

0
5min
7min
9min
11min

15min

0 10 20 30 40 50
mm

Only about 2-10% but not 50% cortical surface area is affected!
Dahlem & Hadjikhani (2009) PLoS ONE 4: e5007.


Tracking migraine aura symptoms

Vincent & Hadjikhani (2007) Cephalagia 27


Spreading depression Open wave segments - fMRI evidence & retinal SD

Conﬁned spatial patterns of spreading depression

Hadjikhani et al. (2001) PNAS




collapse

?
nucleation slice not
recorded

31 min

neighboring points 1 cm 16 min





28 min.
23 min
18 min.





28 min.
23 min
18 min.

Open wave fronts move along
a rather straight line
preventing a reentry of SD





28 min.
23 min
18 min.

Open wave fronts move along
a rather straight line
preventing a reentry of SD

Dahlem & Hadjikhani (2009) PLoS ONE




28 min.

33 min.

23 min
38 min.
18 min.

1 mm

Spiral waves (reentry) observed in retinal SD
with a rotation period of 2.45 min

Dahlem & Hadjikhani (2009) PLoS ONE
Dahlem & M¨ller (1997) Exp. Brain Res.
u



Clinical evidence



Mapped visual symptoms on cortex via fMRI retinotopy

Visual hemifield Primary visual cortex

1 cm
27 min
10°
25
23
21
1
3
5 19
17
7

15




Mapped visual symptoms on cortex via fMRI retinotopy

Visual hemifield Primary visual cortex

23 min
10°
21
19
5
17 7 9
11
13
15
17
1 cm



Outline

3. Migraine as a dynamical gyral crowns
positive (fender)

disease: towards therapy
transient and
slow dynamics

entrance to sulci

negative (saddle)


Dynamical disease Towards therapeutical approaches


(a) (b)
CS

LS

temporarily
affected area
(c) (d)




Cortical homeostasis is stable



Yet, too big a perturbation triggers SD



Yet, too big a perturbation triggers SD

critical
nucleation



But a global negative feedback keeps SD conﬁned

Hypothesis: Cortical susceptibility to SD depends on the size of
the momentarily aﬀected tissue.

transient and
slow dynamics



Threshold surface separates attractor basins
phase space
u i+2(x) traveling wave
ui+1 (x)

u i (x)

ra
. sup
m
sti
. sub
m
sti
threshold

homo. steady state
Excitable media.



Solution on threshold surface
phase space
u i+2(x) traveling wave
ui+1 (x)

u i (x)

ra
. sup
m
sti
. sub
m
sti
threshold

homo. steady state
Excitable media.



Nonlinear delayed transitions: saddle-node ghosts

st
fa

w
te slo
sta
ady
ste st
o. fa
m
ho



Bottleneck due to saddle-node bifurcation

(a) (b) stable wave segment
e
w av
ra ng
m
. sup
veli
sti tra
. sub
m
sti
te
sta Ø Ö × ÓÐ
y
ad te
ste sta ∂R
o. ad
y
m te
ho .s
o
ho
m (a) ∂R (b)
(c)
st
(d) traveling wave
fa (c)

Û Ú ×Þ S
w
te slo
sta
y
ad
ste st
o. fa homo. steady state
h om
Ø Ö × ÓÐ β



Simulation of transient SD wave segment

gray = cortical surface; red = SD wave



Typical trajectory: fast growth and collapse & bottleneck

nucleation model−based
cortical surface area invaded by SD

25
therapeutic TMS
stimulation strategies
20
CSD break−up

15
long transient propagation

10

5
collapse
0
0 5 10 15 20 25 30 35
time




collapse

?
nucleation slice not
recorded

31 min





time
32
28
24

20
16
slice not
12 recorded

8 31 min
4
0





slice not
recorded

31 min





5cm

32 16

time / min
0 0

6 24

0 0



Varying contact to the ghost

# Occurrences
240
(2) (3)
160
β0 = 1.32
80
0
450
400 (1) (1)
total aﬀected area (TAA)

350 (2)
300 (3)
250
(4)
200
150
(4)
100
50 0 30 60 90 120 150 180 210 240 270
time
0
0 80 160240
300
80
(1) (1)
250 70
excitation duration (ED)

(2) (2)
60
200 (3) (3)
50
150 (4) (4) 40

100 30
20
50
10
0 1
0 10 20 30 40 50 60 0 50 100 150 200 250 300 350 400 450 0 80 160240
maximal instantaneous area (MIA) total aﬀected area (TAA) # Occurrences




# Occurrences
240
160 (3)
β0 = 1.33
80
0 (2)
450
400 (1)

350 (2)
300 (3)
(1)
250
(4)
200
150
(4)
100
50 0 20 40 60 80 100120140160180
time
0
0 100 200
300 90
(1) (1) 80
250

(2) (2) 70
200 (3) (3) 60

(4) (4) 50
150
40
100 30
20
50
10
0 1
0 10 20 30 40 50 60 0 50 100 150 200 250 300 350 400 450 0 100200300




# Occurrences
240
160 (3)
β0 = 1.34
80
0
(2)
450
400 (1)

350 (2)
300 (3)
(1)
250
(4)
200
150
(4)
100
50 0 10 20 30 40 50 60 70 80 90
time
0
0 250 500
300 130
(1) (1) 120
250 110

(2) (2) 100
200 90
(3) (3)
80
150 (4) (4) 70
60
50
100
40
30
50 20
10
0 1
0 10 20 30 40 50 60 0 50 100 150 200 250 300 350 400 450 0 150 300



IHS Classiﬁcation ICHD-II – All Types

Migraine
1.
Subtypes

1.1. 1.2. 1.3. 1.4. 1.5. 1.6.
Subforms

1.2.1. 1.3.1. 1.5.1. 1.6.1.



IHS Classiﬁcation ICHD-II – Major Types

Migraine
1.
Subtypes

1.1. 1.2. 1.1. without aura
Subforms

1.2.1.
1.2.1. with aura

typical aura
1.2.3.
without headache

2 symptom, 3 combinations: both or either of them



Model-based hypothesis testing

1.1. 1.2.1

Affected cortical area
Sub−
threshold 1.2.3

SD in migraine attack

Survival time



Neuromodulation



Homo Neuromodulandus
”The headache future is bright for neuromodulation techniques ... if we
manage to understand how they work” (Jean Schoenen)

ﬁgure courtesy of Jean Schoenen Dahlem, TU Berlin
Markus A.


Control of spreading depression
From bench to bedside

!

Cooperation with Stephen Schiﬀ Bruce Gluckman Courtesy of Neuralieve



Typical trajectory: fast growth and collapse bottleneck


25
therapeutic TMS
20
CSD break−up

15

10

5
collapse
0
0 5 10 15 20 25 30 35
time



Typical trajectory: fast growth and collapse bottleneck


25
therapeutic TMS
20
CSD break−up

15
noise!
10

5
collapse
0
0 5 10 15 20 25 30 35
time



Single-pulse transcranial magnetic stimulation

Lipton et al. Lancet Neurology 9,373, 2010



Single-pulse transcranial magnetic stimulation



Double pulse stimulation (current TMS strategy)

25
wave size
noise sample 1 k=0.010
20 noise sample 2 k=0.010
15 without noise

10
noise on

5

0
0 5 10 15 20 25 30 35
time



Permanent noise stimulation

25
wave size
20 noise sample 2 k=0.030
15 without noise

10
noise on

5

0
0 5 10 15 20 25 30 35
time



Single pulse vs. constant noise stimulation

0.5

0.4

0.3
probability

0.2

0.1

0.00 5 10 15 20 25 30 35
survival time of unstable solitons


Single pulse vs. constant noise stimulation

Migraine aura duration
0.5
without noise
on t=5, k = 0.050
0.4 on t=5, k = 0.100
noise 0.050
pulse t=5, k = 0.100
pulse t=5, k = 0.500
0.3
probability

0.2

0.1

0.00 5 10 15 20 25 30 35
survival time


Noise sensitivity of transient wave segments

25
wave size

without noise
noise k=0.010
noise k=0.015 How to escape quickly
20 noise k=0.020 from the ”ghost” plateau?
noise k=0.025
noise k=0.030
15 noise k=0.035
noise k=0.040

10

5

0
0 5 10 15 20 25 30 35
time



From bifurcation bench to bedside


Conclusion

Spatio-temporal waves need
spatio-temporal control methods
Old paradigm

New paradigm: opens up new strategies, eg, transcranial random
noise stimulation (tRNS) at special locations


Conclusion

(i) Persistent migraine w/o infarction, (ii) Migrainous
infarction, (iii) ischemia-induced migraine

Dahlem et al. Physica D 239, 889 (2010)


Conclusion

2D patterns with laser speckle-contrast imaging

(a) KCL
MG
SG

EG

(b) 5 min 37s (c) 9 min 07s

Dahlem et al. 239, 889 (2009) Physica D


Conclusion

Costs of disorders of the brain in Europe

hypoxic tissue
55 Dementia n−gate deactivation Hopf
[K +]o I
pump eletrogenic pump Hopf
27 Migraines SNIC
22 Strokes Fold
Seizure−l
ike activ
15.5 Epilepsy SNIC
ity [K + ]o = 10mM
(ceiling level)

10.5 Parkinson’s
V n −gate
membran
e voltage
billion Euro a year!

Balak and Elmaci (2005) European Journal of Neurology 12


Conclusion

Conclusions

The predicted plateau theory can be
tested clinically with non-invasive
imaging
Unifying concept including silent aura, Visual hemifield Primary visual cortex

migraine with or without 1 cm

headache/aura 10°
27 min
25
23
Insights into the self-organized pattern 1
3
21

5 19
formation may reﬁne neuromodulation 7
17

strategies: 15

Being close to a saddle-node
bifurcation (”ghost” plateau)
Design (feedback) control to
intelligently target certain properties
of SD in migraine


Conclusion

Conclusions

imaging

# Occurrences
240
160 (3)
β0 = 1.34
80

Unifying concept including silent aura, 0
450
400 (1)
(2)

migraine with or without 350
300
250
(2)
(3)
(1)
(4)

headache/aura
200
150
(4)
100
50 0 10 20 30 40 50 60 70 80 90
time

Insights into the self-organized pattern 0
300
(1) (1)
0 250 500
130
120

formation may reﬁne neuromodulation
250 110

(2) (2) 100
200 90
(3) (3)
80
(4) (4) 70

strategies: 150

100
60
50
40
30
50 20

Being close to a saddle-node 0
0 10 20 30 40 50 60
maximal instantaneous area (MIA)
0 50 100 150 200 250 300 350 400 450
10
1
0 150 300
# Occurrences

of SD in migraine


Conclusion

Conclusions

imaging
Unifying concept including silent aura,
migraine with or without
headache/aura
Insights into the self-organized pattern
formation may reﬁne neuromodulation
strategies:
Being close to a saddle-node
of SD in migraine


Conclusion ¡

Cooperation Funding

Nouchine Hadjikhani
(EPFL Martinos Center for Biomedical Imaging, MGH)

Paul Van Valkenburgh

Jens Dreier berlin
(Department of Neurology, Charit´; University Medicine, Berlin)
e

Steve Schiﬀ
(Penn State Center for Neural Engineering)

Klaus Podoll
(University Hospital Aachen)

Migraine Aura Foundation

Thomas Isele


Conclusion ¡

Peri-infarct depolarizations

Dahlem et al. Physica D 239, 889 (2010)


Reentrant and retracting waves of cortical spreading depression in migraine

Reentrant and retracting waves of cortical spreading depression in migraine

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Reentrant and retracting waves of cortical spreading depression in migraine