dynamical network biomarkers in migraine
TRANSCRIPT
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)