dynamical network biomarkers in migraine

Dynamical network biomarkers in migraine

Markus A. Dahlem (HU Berlin)attack-fre

e

headache

prodrome

postdrome

postdrome

aura

magnetic

elec

tric

al

HY,TH

SPG

SSN

TCC

PAG

LC

RVMTG

cortex

cranial circulationbone

ON OFF

cranial innervation 1

day 5-60min

4-7

2h

day1

days to m

onth

s (a

vg 2

wee

ks

)

dynamical network biomarkers

migraine cycle

(a)

(b)

(c)

Berlin Center for Studies of Complex Chemical Systems, Jan 24, 2014

Outline

Electrical & magnetic stimulation of the brain: Neuromodulation

Causality confusion in migraine: Dynamical network biomarkers

Mathematical migraine models: From genotype to phenotype

Towards therapeutic intervention

Conclusion

Outline

Electrical & magnetic stimulation of the brain: Neuromodulation

Causality confusion in migraine: Dynamical network biomarkers

Mathematical migraine models: From genotype to phenotype

Towards therapeutic intervention

Conclusion

History of electrical & magnetic stimluation

Non-drug treatment for headaches (AD 47)

Scribonius Largus, court physician to the Roman emperor Claudius 47 AD usedthe black torpedo fish (electric rays) to treat migraine.

History of electrical & magnetic stimluation

Non-drug treatment for headaches (1788)

P. J. Koehler and C. J. Boes, A history of non-drug treatment in headache,particularly migraine. Brain 133:2489-500. 2010

History of electrical & magnetic stimluation

Non-drug treatment for headaches (1887)

P. J. Koehler and C. J. Boes, A history of non-drug treatment in headache,particularly migraine. Brain 133:2489-500. 2010

History of electrical & magnetic stimluation

Non-drug treatment for headaches (1896)

History of electrical & magnetic stimluation

Non-drug treatment for headaches (1961)

(1985) Zeitschrift EEG-EMG, Georg Thieme Verlag Stuttgart

History of electrical & magnetic stimluation

Non-drug treatment for headaches (2013)

Disclosure: Conflict of interest

Consulting services for Neuralieve Inc. (trading as eNeuraTherapeutics)

Modern neuromodulation (invasive)

Modern neuromodulation (invasive)

Modern neuromodulation (invasive)

Modern neuromodulation (invasive)

Modern neuromodulation (invasive)

Neuromodulation in migraine

I hypothalamic deep brain stimulation (hDBS),

I sphenopalatine ganglion stimulation (SPGS)

I occipital nerve stimulation (ONS),

I cervical spinal cord stimulation (cSCS),

I hypothalamic deep brain stimulation (hDBS),

I vagus nerve stimulation (VNS),

I transcutaneous electrical nerve stimulation (TENS),

I transcranial magnetic stimulation (TMS),

I transcranial direct current stimulation (tDCS),

I transcranial alternate current stimulation (tACS).

Long history in non-drug migraine treatment

Long history in non-drug migraine treatment

Old problems remain

“Uber die physiologischen Wirkungen der elektrischen Bader liegen eine Reihe von Angaben [...] vor.[...]Im allgemeinen haben faradische Bader einen erfrischenden Einfluß, galvanische sollen mude machen.Es kommt fur die Wirkung entschieden auf die Dauer der Bader an, kurzere werden mehr anregend, langere mehrerschlaffend wirken.Durchsichtig ist jedenfalls die physiologische Begrundung dieser Bader durchaus nicht, man wird sich vorstellen,daßsie im allgemeinen die eines indifferenten Bades, mit dem ein milder Hautreiz verbunden ist, haben.Es mogen dadurch Aenderungen in unseren Allgemeingefuhlen, also Wohlbehagen, Erfrischung oder Mudigkeitbedingt werden. Nach meiner Ansicht liegt aber die Hauptwirkung dieser elektrischen Bader in erster Linie aufsuggestivem Gebiete, und das rechtfertigt ihre Anwendung und ihre unleugbaren Erfolge auf dem Gebiete dernervosen Allgemeinleiden, wie Hysterie, Neurasthenie etc.”

(Lehrbuch der klinischen Hydrotherapie, Max Matthes)

Outline

Electrical & magnetic stimulation of the brain: Neuromodulation

Causality confusion in migraine: Dynamical network biomarkers

Mathematical migraine models: From genotype to phenotype

Towards therapeutic intervention

Conclusion

Migraine Generator Network & Dynamical Network Biomarker

attack-free

headache

prodrome

postdrome

postdrome

aura

magnetic

elec

tric

al

HY,TH

SPG

SSN

TCC

PAG

LC

RVMTG

cortex

cranial circulationbone

ON OFF

cranial innervation 1

day 5-60min

4-7

2h

day1

days to m

onth

s (a

vg 2

wee

ks

)

dynamical network biomarkers

migraine cycle

(i)

(ii)

(iii)

• M. A. Dahlem, Migraine generator network and spreading depression dynamics as neuromodulation targets inepisodic migraine. Chaos, 23, 046101 (2013).

Karatas H et al., Spreading depression triggers headache by activating neuronal Panx1 channels. Science,339:1092-5 (2013).• M. A. Dahlem, S. Rode, A. May, N. Fujiwara, Y. Hirata, K. Aihara, J. Kurths, Towards dynamical networkbiomarkers in neuromodulation of episodic migraine, Translational Neuroscience, 4,282-294 (2013).

Migraine Generator Network & Dynamical Network Biomarker

attack-free

headache

prodrome

postdrome

postdrome

aura

TCC

TG

cortex

cranial circulationbone

cranial innervation 1

day 5-60min

4-7

2h

day1

days to m

onth

s (a

vg 2

wee

ks

)

dynamical network biomarkers

migraine cycle

(i)

(ii)

(iii)

• M. A. Dahlem, Migraine generator network and spreading depression dynamics as neuromodulation targets inepisodic migraine. Chaos, 23, 046101 (2013).

Karatas H et al., Spreading depression triggers headache by activating neuronal Panx1 channels. Science,339:1092-5 (2013).• M. A. Dahlem, S. Rode, A. May, N. Fujiwara, Y. Hirata, K. Aihara, J. Kurths, Towards dynamical networkbiomarkers in neuromodulation of episodic migraine, Translational Neuroscience, 4,282-294 (2013).

Migraine Generator Network & Dynamical Network Biomarker

attack-free

headache

prodrome

postdrome

postdrome

aura

TCC

TG

cortex

cranial circulationbone

cranial innervation 1

day 5-60min

4-7

2h

day1

days to m

onth

s (a

vg 2

wee

ks

)

dynamical network biomarkers

migraine cycle

(i)

(ii)

(iii)

• M. A. Dahlem, Migraine generator network and spreading depression dynamics as neuromodulation targets inepisodic migraine. Chaos, 23, 046101 (2013).

Karatas H et al., Spreading depression triggers headache by activating neuronal Panx1 channels. Science,339:1092-5 (2013).• M. A. Dahlem, S. Rode, A. May, N. Fujiwara, Y. Hirata, K. Aihara, J. Kurths, Towards dynamical networkbiomarkers in neuromodulation of episodic migraine, Translational Neuroscience, 4,282-294 (2013).

Migraine Generator Network & Dynamical Network Biomarker

attack-free

headache

prodrome

postdrome

postdrome

aura

HY,TH

TCC

TG

cortex

cranial circulationbone

cranial innervation 1

day 5-60min

4-7

2h

day1

days to m

onth

s (a

vg 2

wee

ks

)

dynamical network biomarkers

migraine cycle

(i)

(ii)

(iii)

• M. A. Dahlem, Migraine generator network and spreading depression dynamics as neuromodulation targets inepisodic migraine. Chaos, 23, 046101 (2013).

Karatas H et al., Spreading depression triggers headache by activating neuronal Panx1 channels. Science,339:1092-5 (2013).• M. A. Dahlem, S. Rode, A. May, N. Fujiwara, Y. Hirata, K. Aihara, J. Kurths, Towards dynamical networkbiomarkers in neuromodulation of episodic migraine, Translational Neuroscience, 4,282-294 (2013).

Migraine Generator Network & Dynamical Network Biomarker

attack-free

headache

prodrome

postdrome

postdrome

aura

HY,TH

TCC

TG

cortex

cranial circulationbone

cranial innervation 1

day 5-60min

4-7

2h

day1

days to m

onth

s (a

vg 2

wee

ks

)

dynamical network biomarkers

migraine cycle

(i)

(ii)

(iii)

• M. A. Dahlem, Migraine generator network and spreading depression dynamics as neuromodulation targets inepisodic migraine. Chaos, 23, 046101 (2013).Karatas H et al., Spreading depression triggers headache by activating neuronal Panx1 channels. Science,339:1092-5 (2013).

• M. A. Dahlem, S. Rode, A. May, N. Fujiwara, Y. Hirata, K. Aihara, J. Kurths, Towards dynamical networkbiomarkers in neuromodulation of episodic migraine, Translational Neuroscience, 4,282-294 (2013).

Migraine Generator Network & Dynamical Network Biomarker

attack-free

headache

prodrome

postdrome

postdrome

aura

HY,TH

SPG

SSN

TCC

TG

cortex

cranial circulationbone

cranial innervation 1

day 5-60min

4-7

2h

day1

days to m

onth

s (a

vg 2

wee

ks

)

dynamical network biomarkers

migraine cycle

(i)

(ii)

(iii)

• M. A. Dahlem, Migraine generator network and spreading depression dynamics as neuromodulation targets inepisodic migraine. Chaos, 23, 046101 (2013).Karatas H et al., Spreading depression triggers headache by activating neuronal Panx1 channels. Science,339:1092-5 (2013).

• M. A. Dahlem, S. Rode, A. May, N. Fujiwara, Y. Hirata, K. Aihara, J. Kurths, Towards dynamical networkbiomarkers in neuromodulation of episodic migraine, Translational Neuroscience, 4,282-294 (2013).

Migraine Generator Network & Dynamical Network Biomarker

attack-free

headache

prodrome

postdrome

postdrome

aura

HY,TH

SPG

SSN

TCC

TG

cortex

cranial circulationbone

cranial innervation 1

day 5-60min

4-7

2h

day1

days to m

onth

s (a

vg 2

wee

ks

)

dynamical network biomarkers

migraine cycle

(i)

(ii)

(iii)

• M. A. Dahlem, Migraine generator network and spreading depression dynamics as neuromodulation targets inepisodic migraine. Chaos, 23, 046101 (2013).Karatas H et al., Spreading depression triggers headache by activating neuronal Panx1 channels. Science,339:1092-5 (2013).

• M. A. Dahlem, S. Rode, A. May, N. Fujiwara, Y. Hirata, K. Aihara, J. Kurths, Towards dynamical networkbiomarkers in neuromodulation of episodic migraine, Translational Neuroscience, 4,282-294 (2013).

Migraine Generator Network & Dynamical Network Biomarker

attack-free

headache

prodrome

postdrome

postdrome

aura

magnetic

elec

tric

al

HY,TH

SPG

SSN

TCC

PAG

LC

RVMTG

cortex

cranial circulationbone

ON OFF

cranial innervation 1

day 5-60min

4-7

2h

day1

days to m

onth

s (a

vg 2

wee

ks

)

dynamical network biomarkers

migraine cycle

(i)

(ii)

(iii)

• M. A. Dahlem, Migraine generator network and spreading depression dynamics as neuromodulation targets inepisodic migraine. Chaos, 23, 046101 (2013).Karatas H et al., Spreading depression triggers headache by activating neuronal Panx1 channels. Science,339:1092-5 (2013).

• M. A. Dahlem, S. Rode, A. May, N. Fujiwara, Y. Hirata, K. Aihara, J. Kurths, Towards dynamical networkbiomarkers in neuromodulation of episodic migraine, Translational Neuroscience, 4,282-294 (2013).

Migraine Generator Network & Dynamical Network Biomarker

attack-free

headache

prodrome

postdrome

postdrome

aura

magnetic

elec

tric

al

HY,TH

SPG

SSN

TCC

PAG

LC

RVMTG

cortex

cranial circulationbone

ON OFF

cranial innervation 1

day 5-60min

4-7

2h

day1

days to m

onth

s (a

vg 2

wee

ks

)

migraine cycle

(i)

(ii)

(iii)

• M. A. Dahlem, Migraine generator network and spreading depression dynamics as neuromodulation targets inepisodic migraine. Chaos, 23, 046101 (2013).Karatas H et al., Spreading depression triggers headache by activating neuronal Panx1 channels. Science,339:1092-5 (2013).• M. A. Dahlem, S. Rode, A. May, N. Fujiwara, Y. Hirata, K. Aihara, J. Kurths, Towards dynamical networkbiomarkers in neuromodulation of episodic migraine, Translational Neuroscience, 4,282-294 (2013).

Migraine Generator Network & Dynamical Network Biomarker

attack-free

headache

prodrome

postdrome

postdrome

aura

magnetic

elec

tric

al

HY,TH

SPG

SSN

TCC

PAG

LC

RVMTG

cortex

cranial circulationbone

ON OFF

cranial innervation 1

day 5-60min

4-7

2h

day1

days to m

onth

s (a

vg 2

wee

ks

)

dynamical network biomarkers

migraine cycle

(i)

(ii)

(iii)

• M. A. Dahlem, Migraine generator network and spreading depression dynamics as neuromodulation targets inepisodic migraine. Chaos, 23, 046101 (2013).Karatas H et al., Spreading depression triggers headache by activating neuronal Panx1 channels. Science,339:1092-5 (2013).• M. A. Dahlem, S. Rode, A. May, N. Fujiwara, Y. Hirata, K. Aihara, J. Kurths, Towards dynamical networkbiomarkers in neuromodulation of episodic migraine, Translational Neuroscience, 4,282-294 (2013).

Early-warning signals for sudden deterioration of diseases

Dynamical Network Biomarkers

Early-warning signals for sudden deterioration of diseases

prodrome aura headache postdrome

1 day 5-60min 4-72h 1 day

spreading depression

(SD)

Dynamical network biomarker (DNB) & Migraine generator network (MGN)

DNB

DNB

highly variable pattern of free

energy stravation

large fluctuations in cerebreal blood flow control

cerebral blood flowbone

ON OFF

cran

ial in

nerv

atio

n

cortex

subcortical

bra

inst

em

mod

ula

tory

netw

ork control

para-sympathetic

Which subnetworks

drive the transtions into

the pain phase?

?

• M. A. Dahlem, S. Rode, A. May, N. Fujiwara, Y. Hirata, K. Aihara, J. Kurths, Towards dynamical network

biomarkers in neuromodulation of episodic migraine, Translational Neuroscience, 4,282-294 (2013).

Unitary hypothesis for multiple natural migraine triggers

I light

I exercise

I foods (chocolate, wine, ...)

I menstrual cycle

I olfactory stimuli

I stress

I sleep deprevation

I ...

All triggers originate or activate subnetwork: pre- and

postganglionic parasym- pathetic neurons in the

superior salivatory nucleus (SSN) and sphenopalatine

ganglion (SPG).

cerebral blood flowbone

ON OFF

cranialinnervatio

n

cortex

subcortical

brainstem

modulatory

netw

ork control

para-sympathetic

SPG

SSN

R. Burstein and M. Jakubowski, Unitary hypothesis for multiple triggers of the pain and strain of migraine. J Comp

Neurol. 493,9-14 (2005).

Causality confusion in migraine

τ

attack-free interval

prodromal phase

pain phase

tim

e

•

Hougaard et al, Provocation of migraine with aura using natural trigger factors, Neurology 80,428-431 (2013)

• M.A. Dahlem, J. Kurths, A. May, K. Aihara, M. Scheffer, and M. D. Ferrari, Causality confusion in migraine: ,

Cephalalgia accepted

Outline

Electrical & magnetic stimulation of the brain: Neuromodulation

Causality confusion in migraine: Dynamical network biomarkers

Mathematical migraine models: From genotype to phenotype

Towards therapeutic intervention

Conclusion

Modeling pain initiation and sensitization

off on

HY,TH

SPG

SSN

TCC

PAG

LC

RVMTG

cortex

cranial circulation

cranial innervation

bone

off on

HY,TH

SPG

SSN

TCC

PAG

LC

RVMTG

cortex

cranial circulation

cranial innervation

bone

off on

HY,TH

SPG

SSN

TCC

PAG

LC

RVMTG

cortex

cranial circulation

cranial innervation

bone

SDignition

d e f

SD SDcortico-thalamicaction

releasenoxiousagents

cortical tissue

I

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

tow

ard

s g

eneti

cs

b c

insid

e ce

ll

outsid

e ce

ll

a

g

sensory aura (15min)

visual aura (0min)

behavior, sensory processing

• Dahlem et al. (2013) Translational Neuroscience, 4, 282

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)

INa = gNam3h(V − ENa)

IK = gKn4(V − EK )

Ileak = gleak(V − Vrest)

∂n

∂t= αn(1− n)− βn, ∂h

∂t· · · (2)− (4)

∂[ion]o∂t

= − A

FVoloIion

∂[ion]i∂t

=A

FVoliIion (5)− · · ·

• N. Hubel, E. Scholl, M. A. Dahlem, Bistable dynamics underlying excitability of ion homeostasis in neuronmodels under review

name value & unit description

Cm 1µF/cm2 conductanceα 0.2 volume fraction of ECSvoltot 4/3π53µm3 total volume

0.5236× 10−15m3 total volume in m3

rn 5 3√

1− αµm≈4.6416µm radius neural compartmentA 4πr 2

n ≈ 270.73424µm2 surface area≈ 2.7× 10−6cm2 surface are in cm2

Γ A (αvoltot)−1 F−1 conversion factor

0.06698725858 num. value (for given α, rn)R 8.3144621 JK−1 mol−1 gas constantT 293.5K temperaturF 96 485.3415 sA/mol Faraday constantIpmax,1 5.14µA/cm2 pump rateIpmax,2 matched Ipmax,1 at rest pump rategNamax 120 mS/cm2 conductancegKmax 36 mS/cm2 conductancegNal 0.12 mS/cm2 leak conductancegKl 0.001 mS/cm2 leak conductancegCl 0.5 mS/cm2 leak conductanceECl -65mV initial reversal potential

FHM3 and spreading depression

0 20 40 60 80 100t / s

140

100

60

20

20

60

V/mV

mutantV

EK

ENa

20%

100%

140

100

60

20

20

60V/mV

wild-typeV

EK

ENa

20%

100%

• M.A. Dahlem, J. Schumacher, N. Hubel, Linking a mutation in familial hemiplegic migraine type 3 to its

phenotype in spreading depression (in preparation)

FHM3 and action potential

0 50 100 150 200Iapp / µA cm

−2

70

60

50

40

V/mV

HB1

HB2

wild-type

stable FP

unstable FP0 3070

40excitatory

0 5070

40oscillatory

0 50 100 150 200Iapp / µA cm−2

70

60

50

40

V/mV

HB1

HB2

mutant

stable FP

unstable FP0 3070

40excitatory

0 5070

40oscillatory

• M.A. Dahlem, J. Schumacher, N Hubel, Linking a mutation in familial hemiplegic migraine type 3 to its

phenotype in spreading depression (in preparation)

Multiscale disease: 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 44 2012

during a migraine attack

Atlas of Migraine and Other Headaches, Silberstein

et al (Editors)

Multiscale disease: 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 44 2012

during stroke

Blood clot

Arteriole

Arachnoid

Subarachnoid space

Neocortex

Dura mater

Spreadingdepolarization

Energyfailure

GlutamateK+

O2

Glucose

VasoconstrictionMicroemboli

Delayed ischemic lesions

Rupturedaneurysm

Katie V

icari

Iadecola, ”Killer waves ...” Nature Medicine 15

(2009)

Neural dynamics during anoxia

mem

bra

ne p

ote

nti

al in

mV

-60

0

60

-60

0

60

time in s24 28 32 36 40

Wild-type

Mutant

van Rijn CM, et al. (2011) Decapitation in rats: latency to unconsciousness and the ‘wave of death’.PLoS One 6, e16514.

Zandt B-J, et al. (2011), Neural Dynamics during Anoxia and the “Wave of Death”.PLoS ONE 6, e22127.

Dichgans M, et al. (2005) Mutation in the neuronal voltage-gated sodium channel SCN1A in familial hemiplegicmigraine. Lancet 366, 371.

Neural dynamics during anoxia

mem

bra

ne p

ote

nti

al in

mV

-60

0

60

-60

0

60

time in s24 28 32 36 40

Wild-type

Mutant

van Rijn CM, et al. (2011) Decapitation in rats: latency to unconsciousness and the ‘wave of death’.PLoS One 6, e16514.

Zandt B-J, et al. (2011), Neural Dynamics during Anoxia and the “Wave of Death”.PLoS ONE 6, e22127.

Dichgans M, et al. (2005) Mutation in the neuronal voltage-gated sodium channel SCN1A in familial hemiplegicmigraine. Lancet 366, 371.

Outline

Electrical & magnetic stimulation of the brain: Neuromodulation

Causality confusion in migraine: Dynamical network biomarkers

Mathematical migraine models: From genotype to phenotype

Towards therapeutic intervention

Conclusion

Old problems remain

“Uber die physiologischen Wirkungen der elektrischen Bader liegen eine Reihe von Angaben [...] vor.[...]Im allgemeinen haben faradische Bader einen erfrischenden Einfluß, galvanische sollen mude machen.Es kommt fur die Wirkung entschieden auf die Dauer der Bader an, kurzere werden mehr anregend, langere mehrerschlaffend wirken.Durchsichtig ist jedenfalls die physiologische Begrundung dieser Bader durchaus nicht, man wird sich vorstellen,daßsie im allgemeinen die eines indifferenten Bades, mit dem ein milder Hautreiz verbunden ist, haben.Es mogen dadurch Aenderungen in unseren Allgemeingefuhlen, also Wohlbehagen, Erfrischung oder Mudigkeitbedingt werden. Nach meiner Ansicht liegt aber die Hauptwirkung dieser elektrischen Bader in erster Linie aufsuggestivem Gebiete, und das rechtfertigt ihre Anwendung und ihre unleugbaren Erfolge auf dem Gebiete dernervosen Allgemeinleiden, wie Hysterie, Neurasthenie etc.”

(Lehrbuch der klinischen Hydrotherapie, Max Matthes)

Feedback control of spreading depression (Talk 5. Feb!)

From bench to bedside

!

!"#$%&'$()'#"*(

+,-+,."(!"#$%&/*#(

0&1'2(3"'$#(

4#&5$1"(!"#$%&'$()'#"*( 6/&'7/2%1"(!"#$%&'$()'#"*(

Cooperation with Stephen Schiff & Bruce Gluckman

Dept. Biomedical Engineering, Penn State (CRCNS)

Courtesy Neuralieve

Feedback control with Kalman filter TMS (external forcing)

Migraine Generator Network & Dynamical Network Biomarker

attack-free

headache

prodrome

postdrome

postdrome

aura

magnetic

elec

tric

al

HY,TH

SPG

SSN

TCC

PAG

LC

RVMTG

cortex

cranial circulationbone

ON OFF

cranial innervation 1

day 5-60min

4-7

2h

day1

days to m

onth

s (a

vg 2

wee

ks

)

dynamical network biomarkers

migraine cycle

(i)

(ii)

(iii)

• M. A. Dahlem, Migraine generator network and spreading depression dynamics as neuromodulation targets inepisodic migraine. Chaos, 23, 046101 (2013).Karatas H et al., Spreading depression triggers headache by activating neuronal Panx1 channels. Science,339:1092-5 (2013).• M. A. Dahlem, S. Rode, A. May, N. Fujiwara, Y. Hirata, K. Aihara, J. Kurths, Towards dynamical networkbiomarkers in neuromodulation of episodic migraine, Translational Neuroscience, 4,282-294 (2013).

Two tipping points

prodrome aura headache postdrome

1 day 5-60min 4-72h 1 day

spreading depression

(SD)

Dynamical network biomarker (DNB) & Migraine generator network (MGN)

DNB

DNB

highly variable pattern of free

energy stravation

large fluctuations in cerebreal blood flow control

cerebral blood flowbone

ON OFF

cran

ial in

nerv

atio

n

cortex

subcortical

bra

inst

em

mod

ula

tory

netw

ork control

para-sympathetic

Which subnetworks

drive the transtions into

the pain phase?

?

Modeling pain initiation and sensitization

off on

HY,TH

SPG

SSN

TCC

PAG

LC

RVMTG

cortex

cranial circulation

cranial innervation

bone

off on

HY,TH

SPG

SSN

TCC

PAG

LC

RVMTG

cortex

cranial circulation

cranial innervation

bone

off on

HY,TH

SPG

SSN

TCC

PAG

LC

RVMTG

cortex

cranial circulation

cranial innervation

bone

SDignition

d e f

SD SDcortico-thalamicaction

releasenoxiousagents

cortical tissue

I

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

tow

ard

s g

eneti

cs

b c

insid

e ce

ll

outsid

e ce

ll

a

g

sensory aura (15min)

visual aura (0min)

behavior, sensory processing

• Dahlem et al. (2013) Translational Neuroscience, 4, 282

Twist: A ‘ghost’ phathological state (waves)

interictal

prodrome

ictal

postdrome

normal state

ictu

s

ictu

s

ictu

s

ictu

s

ictu

s

disease state episodic manifestation

a

b

c

d

e

+ aura

chronification

ictu

sic

tus

f

g

• Dahlem et al. (2013) Translational Neuroscience, 4, 282

Phase-dependend neuromudulation

0

1

2

3

4

5

0 5 10 15 20 25 30 35time

dep

lete

d s

urf

ace

are

aslow dynamics

a

b

c

d e f

AP: acute phaseIP: ignition phase

MIA

xy

arachnoid

blood

arachnoid

blood

bone

sensory innervation

dura

small MIA

large MIA

SD

SD

dural sinuses

cortex

pia

cortex

pia

cf. Karatas H et al., Spreading depression triggers headache by activating neuronal Panx1 channels. Science,339:1092-5 (2013).cf. Charles AC, Baca SM., Cortical spreading depression and migraine. Nat Rev Neurol. 9:637-44, (2013)• M. A. Dahlem and T. Isele: Transient localized wave patterns and their application to migraine. J. Math.Neurosci. 3,7 (2013).

Phase-dependend neuromudulation

0

1

2

3

4

5

0 5 10 15 20 25 30 35time

dep

lete

d s

urf

ace

are

aslow dynamics

a

b

c

d e f

AP: acute phaseIP: ignition phase

MIA

xy

IP Stim.

AP Stim.

arachnoid

blood

arachnoid

blood

bone

sensory innervation

dura

small MIA

large MIA

SD

SD

dural sinuses

cortex

pia

cortex

pia

cf. Karatas H et al., Spreading depression triggers headache by activating neuronal Panx1 channels. Science,339:1092-5 (2013).cf. Charles AC, Baca SM., Cortical spreading depression and migraine. Nat Rev Neurol. 9:637-44, (2013)• M. A. Dahlem and T. Isele: Transient localized wave patterns and their application to migraine. J. Math.Neurosci. 3,7 (2013).

Phase-dependend neuromudulation

0

1

2

3

4

5

0 5 10 15 20 25 30 35time

dep

lete

d s

urf

ace

are

aslow dynamics

a

b

c

d e f

AP: acute phaseIP: ignition phase

MIA

xy

IP Stim.

AP Stim.

blood

arachnoid

blood

bone

sensory innervation

dura

large MIA

dural sinuses

cortex

pia

off on

CC hypermia

CTX w loc SD

HY,TH

SPG

SuS

TCC

vlPAG

LC

RVMTG

aura

off on

CC oligemia +

CTX

HY,TH

SPG

SuS

TCC

vlPAG

LC

RVMTG

pain

a b

any "node" in themigraine generator network is potential target for neuromodulation

cf. Karatas H et al., Spreading depression triggers headache by activating neuronal Panx1 channels. Science,339:1092-5 (2013).cf. Charles AC, Baca SM., Cortical spreading depression and migraine. Nat Rev Neurol. 9:637-44, (2013)• M. A. Dahlem and T. Isele: Transient localized wave patterns and their application to migraine. J. Math.Neurosci. 3,7 (2013).

Single pulse stimulation (current TMS strategy)

0

5

10

15

20

25

0 5 10 15 20 25 30 35

noise sample 1 k=0.010noise sample 1 k=0.100noise sample 1 k=0.300noise sample 2 k=0.010noise sample 2 k=0.100noise sample 2 k=0.300

without noise

time

noise on

wav

e si

ze

Constant noise stimulation

0

5

10

15

20

25

0 5 10 15 20 25 30 35

noise sample 1 k=0.030noise sample 1 k=0.040noise sample 1 k=0.050noise sample 2 k=0.030noise sample 2 k=0.040noise sample 2 k=0.050

without noise

time

noise on

wav

e si

ze

Various SD patterns cause different migraine types?

MA

aura

MIA

MIA

MO

TAA

b

apain onset

< 5 min ~20 min ~60 min (full-scale aura)

TAA

• M. A. Dahlem et al., Spreading depression in migraine without aura: a model-based hypothesis. (in preparation)

Various SD patterns cause different migraine types?

0 0.2 0.4 0.6 0.8−1

0

10

0.5

1

0

1k

2k

0 0.2 0.4 0.6 0.80

1k

2k

smooth

histo

gra

m (so

lid lin

es)

mean

& S

TD

corr

ela

tion

(dott

ed)

total area affected (TAA) exct. duration (ED) classification

maximal instantaneous area (MIA)1 1

3

2

1

4

21

3

2

14

2

1 MWoA

MWAMxWAw/o head-ache

aura threshold

pain

thre

shold

IA

1cm

MWAMWoA

MWA

MWoA

total affected area (TAA)

"aura"

large TAA

#

MIA"pain"

• M. A. Dahlem et al., Spreading depression in migraine without aura: a model-based hypothesis. (in preparation)

Individual ‘hot spots’ and ‘labyrinth’ determine attack

positive

negative

Principles• simulations on simpler shapes• analytical results with isothermal

coordiantes (toroidal coordinates)

Validate• uploading patient’s MRI scanner readings• finite element analysis

• polygon mesh processing

Excitation waves on curved surfaces

Paradigmatic SD model on gyrified cortex.

∂u

∂t= u − 1

3u3 − v + Du

1√g

∂

∂αi

(gij√g∂u

∂αi

)∂v

∂t= ε(u + β)

SD in weakly excitable cortex posses critical properties.

First approximation: localized SD follows shortest path.

Outline

Electrical & magnetic stimulation of the brain: Neuromodulation

Causality confusion in migraine: Dynamical network biomarkers

Mathematical migraine models: From genotype to phenotype

Towards therapeutic intervention

Conclusion

Modern neuromodulation

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

Modern neuromodulation

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

Modern neuromodulation

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

Modern neuromodulation

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

Modern neuromodulation

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

Modern neuromodulation

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

Models fill the ‘gaps’ in the multiscale framework

I

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) Functional mutations, either FHM, CADASIL, ... or GWAS.

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

(c) Neural mass/fields population models, with subpopulationshaving specific synaptic receptor distribution.

(d) Local circuits, in particular including the migraine generatornetwork in the brainstem

(e) Aura symptoms, mental dysfunctions, impared sensory andcognitive processing

Recent papers

I M.A. Dahlem, J. Kurths, A. May, K. Aihara, M. Scheffer, and M. D.Ferrari, Causality Confusion in Migraine Cephalalgia, (accepted).

I M. A. Dahlem, Migraines and Cortical Spreading Depression,Encyclopedia of Computational Neuroscience, Springer. (accepted)

I M. A. Dahlem, S. Rode, A. May, N. Fujiwara, Y. Hirata, K. Aihara, J.Kurths, Towards dynamical network biomarkers in neuromodulation ofepisodic migraine, Translational Neuroscience, 4,282-294 (2013).

I M. A. Dahlem: Migraine generator network and spreading depressiondynamics as neuromodulation targets in episodic migraine. Chaos, 23,046101 (2013).

I M. A. Dahlem and T. Isele: Transient localized wave patterns and theirapplication to migraine. J. Math. Neurosci. 3,7 (2013)

I J. P. Dreier, T. Isele, C. Reiffurth, N. Offenhauser, S. A. Kirov, M. A.Dahlem, and O. Herrars: Is spreading depolarization characterized by anabrupt, massive release of Gibbs free energy from the human braincortex? The Neuroscientist 19,25-42 (2012).

I M. A. Dahlem and J. Tusch: Cortical magnification tensor predicts avirtual visual streak in humans waves, J. Math. Neurosci. 2,14 (2012)

Manuscripts in preparation

I N. Hubel, E. Scholl, and M. A. Dahlem, Bistable dynamics underlyingexcitability of ion homeostasis in neuron models (revised and underreview, 14 pages, 7 figures, 3 tables, ms on arxiv)

I F. Kneer, E. Scholl, and M.A. Dahlem, Nucleation of reaction-diffusionwaves on curved surfaces. (submitted, 23 pages, 11 figures, 4 movies)

I M. A. Dahlem, F. Kneer, E. Scholl, B. Schmidt, I. Bojak, and J. Kurths,Personalized treatment strategies in episodic migraine byneuromodulation devices. (12 pages, 5 figures, 1 movie)

I M. A. Dahlem, Julia Schumacher, N. Hubel, Linking a mutation infamilial hemiplegic migraine type 3 to its phenotype in spreadingdepression, 4 figures, 8 pages

I F. Kneer, K. Obermayer, M. Dahlem, Analyzing critical propagation in areaction-diffusion-advection model using unstable slow waves.(submitted)

I M. A. Dahlem, et al. Spreading depression in migraine without aura: amodel-based hypothesis. (in preparation)

Conclusions

I We need more non-invasiveimaging data of migraine with aurato test predictions.

I Sef-organizing patterns provide aunifying concept including silent aura,migraine w or w/o headache/aura

I Dynamical cocepts may refineneuromodulation strategies:

I Being close to a saddle-nodebifurcation (”ghost” plateau)

I Design (feedback) control tointelligently target certain propertiesof SD in migraine

1 cm

10°

1 357

15

1719

2123

25

27 min

Visual hemifield Primary visual cortex

Conclusions

I We need more non-invasiveimaging data of migraine with aurato test predictions.

I Sef-organizing patterns provide aunifying concept including silent aura,migraine w or w/o headache/aura

I Dynamical cocepts may refineneuromodulation strategies:

I Being close to a saddle-nodebifurcation (”ghost” plateau)

I Design (feedback) control tointelligently target certain propertiesof SD in migraine

0

6.25

25

50 (1)

(2)

(3)

(4)

2.5 6.25

tota

la

ffe

cte

da

rea T

AA

maximal instantaneous area MIA 0 3 6 9 12 15 18 21 24 27

(1)

(2)

(3)

(4)

MWA

MWoA

no attack

cortextop view

x

y

time

abovepainthreshold

(a) (b)

Conclusions

I We need more non-invasiveimaging data of migraine with aurato test predictions.

I Sef-organizing patterns provide aunifying concept including silent aura,migraine w or w/o headache/aura

I Dynamical cocepts may refineneuromodulation strategies:

I Being close to a saddle-nodebifurcation (”ghost” plateau)

I Design (feedback) control tointelligently target certain propertiesof SD in migraine

Cooperation & Funding

Frederike Kneer, Niklas Hubel, JuliaSchumacher, Thomas Isele

Paul Van Valkenburgh, Bernd Schmidt

Nouchine Hadjikhani(Martinos Center for Biomedical Imaging, MGH)

Steven Schiff(Penn State Center for Neural Engineering)

Andrew Charles(Headache Research and Treatment Program, UCLA School of

Medicine)

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

Klaus Podoll(University Hospital Aachen)

berlin

Migraine Aura Foundation

Mainly two neural theories of migraine

”Migraine generator”-theory

S1

PFCTh

PPC

PAG

Amyg Insula

SMA

ACC

”Spreading depression”-theory

SD: Wave of massive ionic imbalance

-1

-2

-3

-4-7

-8

1 min

20 mV

log [cat] , M

(mM)

VeNa+

Na+

K+

Ve

K+

Ca++

Ca++

H+

0 10 20 30 s

150

6050

31.5

0.08

unitact.

Lauritzen (1994) Brain 117:199.

SD does not curl-in in human cortex

1cm

10 min

Only about 2-10% but not 50% cortical surface area is affected!

•

right: modified from Hadjikhani et al. PNAS 98:4687 (2001).• Dahlem & Hadjikhani, PLoS ONE, 4: e5007 (2009).

• Dahlem & Muller, Exp. Brain Res. 115,319, (1997).

SD does not curl-in in human cortex

1cm

10 min

1cm

SD curls in to formspirals with T=2.45min!

spiralcore

Only about 2-10% but not 50% cortical surface area is affected!

•

right: modified from Hadjikhani et al. PNAS 98:4687 (2001).• Dahlem & Hadjikhani, PLoS ONE, 4: e5007 (2009).

• Dahlem & Muller, Exp. Brain Res. 115,319, (1997).

Re-entrant SD waves with functional block

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

Dahlem & Muller (1997) Exp. Brain Res. 115:319

Re-entrant SD waves with anatomical block

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

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

Nucleation failure on torus

Transient times in flat and curved geometry

0

10

20

30

40

50

1.3 1.32 1.34 1.36 1.38

S

β

with controlwithout control

torus outside

flat

torus inside

ring wave

∂R∞

0

10

20

30

0 10 20 30 40 50 60 70 80

S

t

outside

inside

outside

inside

torus, without controltorus, with control

flat, without control

Simulation of transient SD wave segment

gray = cortical surface; red = SD wave

Simulation of an engulfing SD wave

In cooperation with Bernd Schmidt,

MagdeburgIn cooperation with Jens Dreier &

Denny Milakara, Charite

SD triggers trigeminal meningeal afferents, ie, headache

see e.g.: Bolay et al. Nature Medicine 8, 2002Review: Eikermann-Haerter & Moskowitz, Curr Opin Neurol. 21, 2008

Figure: Dodick & Gargus SciAm, August 2008

Organic Physics

Organic Physics

Organic Physics

Organic Physics

Cerebral blood flow in migraine

Radionuclide xenon 133 method, used to image brain’s blood flow

Olesen, J. , Larsen, B. and Lauritzen, M., Focal hyperemia followed by

spreading oligemia and impaired activation of rCBF in classic migraine, Ann.

Neurol. 9, 344 (1981)

Tracking migraine aura symptoms

Vincent & Hadjikhani (2007) Cephalagia 27

Tracking migraine aura symptoms

Vincent & Hadjikhani (2007) Cephalagia 27

fMRI patterns is more diffuse than SD patterns

reference (min 0)

start (min 20)

end (min 30)

What if the the blood flow provides along-range or global negative feedback?

modified from Hadjikhani et al. (2001) PNAS 98

fMRI patterns is more diffuse than SD patterns

reference (min 0)

start (min 20)

end (min 30)

What if the the blood flow provides along-range or global negative feedback?modified from Hadjikhani et al. (2001) PNAS 98

”Migraine generator” in the brainstem

70% rarely (but: unreported cases)

SD

aura

trigger

”Migraine generator” in the brainstem

70% rarely (but: unreported cases)

?

trigger A

SD

trigger B

?

trigger C

?

trigger D

postdromeprodrome aura headache

mysterious conductor

about 1 day about 1 day4−72h< 60 min

A conductor of a neural orchestra playing migraine

70%

rarely (but: unreported cases)

?

trigger A

?

trigger C

?

trigger D

postdromeprodrome headache

mysterious conductor

trigger B

SD

aura

about 1 day about 1 day4−72h< 60 min

A conductor of a neural orchestra playing migraine

70%

rarely (but: unreported cases)

?

trigger A

?

trigger D

postdromeprodrome

mysterious conductor

headache

trigger C

?SD

trigger B

aura

about 1 day about 1 day< 60 min 4−72h

A conductor of a neural orchestra playing migraine

70% rarely (but: unreported cases)

?

trigger A

SD

trigger B

?

trigger C

?

trigger D

postdromeprodrome aura headache

mysterious conductor

about 1 day about 1 day4−72h< 60 min

SD is playing jazz – self-organizing dynamics

70% rarely (but: unreported cases)

SD

postdromeaura headache

about 1 day about 1 day4−72h< 60 min

delaytime

trigger

prodrome

heightened susceptibility

cort

ical

hom

eost

asis

prodrome

Common etiology or 2 mechanisms in MWoA and MWA?

SD

headacheprodrome aura

trigger

heig

hten

edsusceptibility

delayed trigger

1.Only one upstream trigger?2.MWoA & MWA share same pain phase? 3. Silent aura? 4.Even prevalent?

5.Delayed headache link? 6.Missing the pain phase?

SD: Spreading Depression, see next slide

Unified Minimal (4D) Model ofSpiking, Seizures and Spreading Depression

4 6 8 10 12 14 16 18 20 22Kbath in mMol/ l

0

10

20

30

40

50

60

70Kex,maxinmMol/l

HB

HB

LPLP

LP?

TR

TR?

extrace llular potass iumperiodic SDunstable lim it cycles table equilibrium

Unified Minimal (4D) Model ofSpiking, Seizures and Spreading Depression

4 6 8 10 12 14 16 18 20 22Kbath in mMol/ l

80

60

40

20

0

20

40

60

Vmax

inmVolt

LPLP

HB

HB HB HB

LP

LP

LP?

TR

TR?

membrane potential

1min

0

-100

E

E

VNa

K

Additional slides

Excitable media – Traveling wave solutionsCanonical RD eqs.

(in weak limit, β large but not too large)

∂tu = f (u)− v +∇2u

∂tv = ε(u + β)

Excitable media – Traveling wave solutionsCanonical RD eqs.

(in weak limit, β large but not too large)

∂tu = f (u)− v +∇2u

∂tv = ε(u + β)

Schenk et al. Phys. Rev. Lett. 78, 3781 (1997)

Excitable media – Traveling wave solutions

Canonical RD eqs.

(in weak limit, β large but not too large)

∂tu = f (u)− v +∇2u

∂tv = ε(u + β)

0

10

20

30

40

50

60

1.3 1.32 1.34 1.36 1.38 1.4

wa

ve

siz

e S

threshold β

∂R∞

super-threshold stim

ulation

sub-thresholdstim

ulation

threshold

homogeneous steady state

traveling wave

critical nucleactionsolution

Excitable media – Traveling wave solutions

Canonical RD eqs.

(in weak limit, β large but not too large)

∂tu = f (u)− v +∇2u

∂tv = ε(u + β)

0

10

20

30

40

50

60

1.3 1.32 1.34 1.36 1.38 1.4

wa

ve

siz

e S

threshold β

su

b-e

xcita

ble

∂R∞

super-threshold stim

ulation

sub-thresholdstim

ulation

threshold

homogeneous steady state

traveling wave

critical nucleactionsolution

nucleationcritical

Excitable media – Traveling wave solutions

Canonical RD eqs.

(in weak limit, β large but not too large)

∂tu = f (u)− v +∇2u

∂tv = ε(u + β)

0

10

20

30

40

50

60

1.3 1.32 1.34 1.36 1.38 1.4

wa

ve

siz

e S

threshold β

no

t e

xcita

ble

su

b-e

xcita

ble

∂P1D∂R∞

super-threshold stim

ulation

sub-thresholdstim

ulation

threshold

homogeneous steady state

traveling wave

critical nucleactionsolution

nucleationcritical

Excitable media – Traveling wave solutions

Canonical RD eqs.

(in weak limit, β large but not too large)

∂tu = f (u)− v +∇2u

∂tv = ε(u + β)

0

10

20

30

40

50

60

1.3 1.32 1.34 1.36 1.38 1.4

wa

ve

siz

e S

threshold β

no

t e

xcita

ble

su

b-e

xcita

ble

∂P1D∂R∞

super-threshold stim

ulation

sub-thresholdstim

ulation

threshold

homogeneous steady state

traveling wave

critical nucleactionsolution

nucleationcritical

Current and pump equations

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)

0

0.2

0.4

0.6

0.8

1

5 10 15 20 25 30 35 40Nai

K0 fixed 3.5mM

12

0

0.2

0.4

0.6

0.8

1

5 10 15 20 25 30 35 40Nai

K0 fixed 8mM

12

0

0.2

0.4

0.6

0.8

1

5 10 15 20 25 30 35 40Nai

Ko fixed 10mM

12

0

0.2

0.4

0.6

0.8

1

5 10 15 20 25 30 35 40Nai

K0 fixed 55mM

12

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14Ko

Nai fixed 10mM

12

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14Ko

Nai fixed 20mM

12

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14Ko

Nai fixed 30mM

12

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10 12 14Ko

Nai fixed 40mM

12

0

0.02

0.04

8 10 12

0

0.02

0.04

2.5 3 3.5 4 4.5

0 1 2 3 4 5

5 10 15 20 25 30 35 40

0

0.1

0.2

0.3

0 2 4 6 8 10 12 14

Minimum threshold in a flat geometry

0

10

20

30

40

50

60

1.3 1.32 1.34 1.36 1.38 1.4

wave s

ize S

threshold β

torus inside1

∂P1D∂R∞

Minimum threshold in a flat geometry

0

10

20

30

40

50

60

1.3 1.32 1.34 1.36 1.38 1.4

wave s

ize S

threshold β

torus outside

torus inside

1

1

∂P1D∂R∞

Minimum threshold in a flat geometry

0

10

20

30

40

50

60

1.3 1.32 1.34 1.36 1.38 1.4

wave s

ize S

threshold β

torus outside

torus inside

1

1

∂P1D∂R∞

Minimum threshold in a flat geometry

0

10

20

30

40

50

60

1.3 1.32 1.34 1.36 1.38 1.4

wave s

ize S

threshold β

torus outside

torus inside

1

1

ring wave

1

∂P1D∂R∞

Minimum threshold in a flat geometry

0

10

20

30

40

50

60

1.3 1.32 1.34 1.36 1.38 1.4

wave s

ize S

threshold β

torus outside

torus inside

221

1

ring wave

1

2

∂P1D∂R∞

Migraine scotoma reveal functional properties

Pattern matching

”Curved” retinotopic mapping

47

913

A B

C

• Dahlem & Tusch, J. Math Neuroscie. 2,14 (2012)

Migraine scotoma reveal functional properties

Pattern matching ”Curved” retinotopic mapping

47

913

A B

C

a d

b ce

m

m lv u10Ælingual gyrus uneus CS

• Dahlem & Tusch, J. Math Neuroscie. 2,14 (2012)

Migraine scotoma reveal functional properties

Pattern matching ”Curved” retinotopic mapping

47

913

A B

C

2 4 6 8 10 12 14

0.1

0.2

0.3

2 4 6 8 10 12 14

20406080

100120140

2 4 6 8 10 12 14

0.2

0.4

0.6

0.8

1a 60Æ6Æ M=(a�1 )

HM�=(%)K=(mm2 ) �=(rad)a�2 00:20:40:60:8

b d

• Dahlem & Tusch, J. Math Neuroscie. 2,14 (2012)