Download - Migraine: A dynamical disease
![Page 1: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/1.jpg)
Migraine: A dynamical disease
Markus A. Dahlem
HU Berlin, Research group: Dynamics and Neuromodulation of Migraine
23 min
21
19
17
17
15
1311
975
10°
1 cm
Visual hemifield Primary visual cortex
nucleationcritical
“Theorie-Seminar”, Johann Wolgang Goethe University of Frankfurt
13. Mai 2014
![Page 2: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/2.jpg)
Outline
Preface: Dynamical disease — Migraine — Models fill a gap
Mathematical migraine models: two (out of four) examplesModeling macroscopic fMRI dataModeling pain / central sensitisation
Towards therapeutic intervention
![Page 3: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/3.jpg)
Outline
Preface: Dynamical disease — Migraine — Models fill a gap
Mathematical migraine models: two (out of four) examplesModeling macroscopic fMRI dataModeling pain / central sensitisation
Towards therapeutic intervention
![Page 4: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/4.jpg)
“Dynamical disease”
Leon Glass and Michael Mackey coined the termdynamical disease to identify diseases that occur dueto an abrupt change in the natural rhythms of thebody and rhythms become abnormal. In particular,chronic disorders with episodic manifestations.
“The significance of identifying a dynamical disease isthat it should be possible to develop therapeuticstrategies based on our understanding of dynamicscombined with manipulations of the physiologicalparameters back into the normal ranges.”
(Belair, Glass, an der Heiden, & Milton, Chaos, 5, 1995)
![Page 5: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/5.jpg)
Migraine 6= Headache
![Page 6: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/6.jpg)
The International Headache Classification – All types
1.
1.1. 1.2. 1.4. 1.5. 1.6.1.3.
1.2.1. 1.3.1. 1.5.1. 1.6.1.
Sub
form
s
Migraine
Subtypes
2 symptom, 3 combinations: both or either of them(more detailed: 64% only MWoA, 18% only MWA, 13% MWoA and MWA, 5% MxWA (without pain))
![Page 7: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/7.jpg)
The International Headache Classification – Major types
with aura
w/o aura
typical aura
without headache
1.
1.1. 1.2.
1.2.1.
Subfo
rms
Subtypes
Migraine
1.1.
1.2.1.
1.2.3.
MWoA 70%
MWA 30%
2 symptom, 3 combinations: both or either of them(more detailed: 64% only MWoA, 18% only MWA, 13% MWoA and MWA, 5% MxWA (without pain))
![Page 8: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/8.jpg)
What is a migraine aura?
based on: • Dahlem & Muller Biol. Cybern. 88,419 (2003)
based on:
• Dahlem et. al. Eur. J. Neurosci. 12,767 (2000).
![Page 9: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/9.jpg)
Multiscale disease: From molecules to entire brain
Functional mutations Spreading depression (SD)
(e.g. FHM2: sodium-potassium pump)
Maagdenberg, et al., Ann. Neurol., 67 (2010)
Tottene, et al., Neuron, 61 (2009)
Freilinger, et al. Nature Genetics 44 (2012),
Dahlem, et al PeerJ, 2,379 (2014)
during a migraine attack
Atlas of Migraine and Other Headaches, Silberstein
et al (Editors)
![Page 10: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/10.jpg)
Models fill the ‘gaps’ in the multiscale framework
insid
e ce
ll
outsid
e ce
ll
a e
sensory aura (15min)
visual aura (0min)
behavior, perceptionsensory processing
(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) From local circuits to reaction-diffusion and larger networks(migraine generator network).
(e) Throbbing pain, aura symptoms, mental dysfunctions,impared sensory and cognitive processing.
![Page 11: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/11.jpg)
Models fill the ‘gaps’ in the multiscale framework
0 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
+
mole
cula
r le
vel &
g
eneti
cs
b
insid
e ce
ll
outsid
e ce
ll
a e
sensory aura (15min)
visual aura (0min)
behavior, perceptionsensory processing
(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) From local circuits to reaction-diffusion and larger networks(migraine generator network).
(e) Throbbing pain, aura symptoms, mental dysfunctions,impared sensory and cognitive processing.
![Page 12: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/12.jpg)
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 e
sensory aura (15min)
visual aura (0min)
behavior, perceptionsensory processing
balanced excitation and inhibition in ion-based models
(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) From local circuits to reaction-diffusion and larger networks(migraine generator network).
(e) Throbbing pain, aura symptoms, mental dysfunctions,impared sensory and cognitive processing.
![Page 13: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/13.jpg)
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) From local circuits to reaction-diffusion and larger networks(migraine generator network).
(e) Throbbing pain, aura symptoms, mental dysfunctions,impared sensory and cognitive processing.
![Page 14: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/14.jpg)
Outline
Preface: Dynamical disease — Migraine — Models fill a gap
Mathematical migraine models: two (out of four) examplesModeling macroscopic fMRI dataModeling pain / central sensitisation
Towards therapeutic intervention
![Page 15: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/15.jpg)
Reseach interests – overview
Cortical architecture Ion-based cellular models
a
b cd
e
fgI
II
III
IV
V
VI
Api
cald
endr
ite
Current distribution
Soma
Glia
Extrac
ellula
r
Osmotic force
Pump
I Na,P
I K ,DR
I K ,A
I NM DA
I Na,T
K +
[K+]o
[Na+]i
N
C
inside cell
outside cellI
IIIII
IVG
P
P PP P
SS
SS
II
II
P
fast
(ms)
Voltage(mV)
–120 –110 –100 –901
10
100
Nor
mal
ised
I Na
Time(ms)
1 10 100
0·0
0·2
0·4
0·6
0·8
1·0Wild-typeMutant
Mutant GLN1478Lys
2mS
1nA
Wild-typeSCN5A
2mS
0·5nA
Macroscopic patterns in migraine Pain pathways & modulation
�����������������������������������������������������������������������
����������������������������������������������������������������������
SSN
TNC
C1
C2
PAG
LC
MRN Amyg
S1S2
ACC
PFC ThPPC
Insula
SMA
PPC
PAG
![Page 16: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/16.jpg)
From
HH-type conductance-based
toconductance- & ion-based models (2nd generation model)
C∂V
∂t= −INa − IK − Ileak
−Ipump
+Iapp (1)
INa = gNam3h(V − ENa)
IK = gKn4(V − EK )
Ileak = gleak(V − Vrest)
∂n
∂t= αn(1− n)− βn,
∂h
∂t· · · (2)− (4)
∂[ion]e∂t
= − A
FVoloIion
∂[ion]i∂t
=A
FVoliIion (5)− · · ·
HH: Hodgkin-Huxley
![Page 17: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/17.jpg)
From HH-type conductance-based toconductance- & ion-based models (2nd generation model)
ion reservoirs
isolated boundary
syst
em
surroundingsenergysource
extracellular
intracellular
C∂V
∂t= −INa − IK − Ileak−Ipump+Iapp (1)
INa = gNam3h(V − ENa)
IK = gKn4(V − EK )
Ileak = gleak(V − Vrest)
∂n
∂t= αn(1− n)− βn,
∂h
∂t· · · (2)− (4)
∂[ion]e∂t
= − A
FVoloIion
∂[ion]i∂t
=A
FVoliIion (5)− · · ·
HH: Hodgkin-Huxley
![Page 18: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/18.jpg)
The neuron in analogy to a heat engineFour processes make up a spreading depression “cycle”
−60 −40 −20 0 20 400
20
40
60 resting state
ion gain through reservoire / (mM)
ion c
onc.
in (
e)
/ (m
M) ion
reservoirs
isolated boundary
syst
em
surroundingsenergysource
extracellular
intracellular
• Dreier et al., Neuroscientist 19, (2012)• Hubel et al., PLOS Comp. Biology. 10, e1003551 (2014)
• Hubel & Dahlem, arXiv:1404.3031 (under review in PLOS Comp. Biology )
![Page 19: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/19.jpg)
The neuron in analogy to a heat engineFour processes make up a spreading depression “cycle”
−60 −40 −20 0 20 400
20
40
60m fluxes across membrane
resting state
m
ion gain through reservoire / (mM)
ion c
onc.
in (
e)
/ (m
M) ion
reservoirs
isolated boundary
syst
em
surroundingsenergysource
extracellular
intracellular
• Dreier et al., Neuroscientist 19, (2012)• Hubel et al., PLOS Comp. Biology. 10, e1003551 (2014)
• Hubel & Dahlem, arXiv:1404.3031 (under review in PLOS Comp. Biology )
![Page 20: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/20.jpg)
The neuron in analogy to a heat engineFour processes make up a spreading depression “cycle”
−60 −40 −20 0 20 400
20
40
60m fluxes across membrane
r fluxes from reservoir
resting state
r
m
ion gain through reservoire / (mM)
ion c
onc.
in (
e)
/ (m
M) ion
reservoirs
isolated boundary
syst
em
surroundingsenergysource
extracellular
intracellular
• Dreier et al., Neuroscientist 19, (2012)• Hubel et al., PLOS Comp. Biology. 10, e1003551 (2014)
• Hubel & Dahlem, arXiv:1404.3031 (under review in PLOS Comp. Biology )
![Page 21: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/21.jpg)
The neuron in analogy to a heat engineFour processes make up a spreading depression “cycle”
−60 −40 −20 0 20 400
20
40
60m fluxes across membrane
r fluxes from reservoir
resting state
r
m
m
ion gain through reservoire / (mM)
ion c
onc.
in (
e)
/ (m
M) ion
reservoirs
isolated boundary
syst
em
surroundingsenergysource
extracellular
intracellular
• Dreier et al., Neuroscientist 19, (2012)• Hubel et al., PLOS Comp. Biology. 10, e1003551 (2014)
• Hubel & Dahlem, arXiv:1404.3031 (under review in PLOS Comp. Biology )
![Page 22: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/22.jpg)
The neuron in analogy to a heat engineFour processes make up a spreading depression “cycle”
−60 −40 −20 0 20 400
20
40
60m fluxes across membrane
r fluxes from reservoir
resting state
r
r
m
m
cyclic process
ion gain through reservoire / (mM)
ion c
onc.
in (
e)
/ (m
M) ion
reservoirs
isolated boundary
syst
em
surroundingsenergysource
extracellular
intracellular
• Dreier et al., Neuroscientist 19, (2012)• Hubel et al., PLOS Comp. Biology. 10, e1003551 (2014)
• Hubel & Dahlem, arXiv:1404.3031 (under review in PLOS Comp. Biology )
![Page 23: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/23.jpg)
The neuron in analogy to a heat engineFour processes make up a spreading depression “cycle”
−60 −40 −20 0 20 400
20
40
60m fluxes across membrane
r fluxes from reservoir
resting state
r
r
m
m
cyclic process
ion gain through reservoire / (mM)
ion c
onc.
in (
e)
/ (m
M)
folded
fixed point
branch
ion reservoirs
isolated boundary
syst
em
surroundingsenergysource
extracellular
intracellular
• Dreier et al., Neuroscientist 19, (2012)• Hubel et al., PLOS Comp. Biology. 10, e1003551 (2014)
• Hubel & Dahlem, arXiv:1404.3031 (under review in PLOS Comp. Biology )
![Page 24: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/24.jpg)
Reaction-diffusion model
The Hodgkin-Grafstein modelof SD (1963)
(cf. Zeldovich-Frank-Kamenetskii,
Schlogl, ...)
u = [K +]e
−60 −40 −20 0 20 400
20
40
60m fluxes across membrane
r fluxes from reservoir
resting state
r
r
m
m
cyclic process
ion gain through reservoire / (mM)
ion c
onc.
in (
e)
/ (m
M)
folded
fixed point
branch
u = (u − urest) (u − uceiling ) (u − umax) + D∇2u (1)
‘Obvious’ extension (add FitzHugh-Nagumo inhibitor equations)
u =
(u − u3
3− v
)+ D∇2u (2)
ε−1v = (u + β − γv) (3)
![Page 25: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/25.jpg)
Reaction-diffusion modelThe Hodgkin-Grafstein modelof SD (1963)
(cf. Zeldovich-Frank-Kamenetskii,
Schlogl, ...)
u = [K +]e −60 −40 −20 0 20 400
20
40
60m fluxes across membrane
r fluxes from reservoir
resting state
r
r
m
m
cyclic process
ion gain through reservoire / (mM)
ion c
onc.
in (
e)
/ (m
M)
folded
fixed point
branch
u = (u − urest) (u − uceiling ) (u − umax) + D∇2u (1)
‘Obvious’ extension (add FitzHugh-Nagumo inhibitor equations)
u =
(u − u3
3− v
)+ D∇2u (2)
ε−1v = (u + β − γv) (3)
![Page 26: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/26.jpg)
Reaction-diffusion modelThe Hodgkin-Grafstein modelof SD (1963)
(cf. Zeldovich-Frank-Kamenetskii,
Schlogl, ...)
u = [K +]e −60 −40 −20 0 20 400
20
40
60m fluxes across membrane
r fluxes from reservoir
resting state
r
r
m
m
cyclic process
ion gain through reservoire / (mM)
ion c
onc.
in (
e)
/ (m
M)
folded
fixed point
branch
u = (u − urest) (u − uceiling ) (u − umax) + D∇2u (1)
‘Obvious’ extension (add FitzHugh-Nagumo inhibitor equations)
u =
(u − u3
3− v
)+ D∇2u (2)
ε−1v = (u + β − γv) (3)
![Page 27: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/27.jpg)
What is Spreading Depression on macro-scale?
SD: massive perturbation of ion homeostasis in gray matter
-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
1506050
31.5
0.08
unitact.
M. Lauritzen, Trends in Neurosciences 10,8 (1987).
A controversial debate but
“...essential view of a reaction-diffusion process still holds ...”
Herreras (2005) J. Neurophysiol. 94:3656
Somjen & Strong (2005) J. Neurophysiol. 94:3656
![Page 28: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/28.jpg)
What is Spreading Depression on macro-scale?
SD: massive perturbation of ion homeostasis in gray matter
-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
1506050
31.5
0.08
unitact.
any textbook
from ~2004
many websites
M. Lauritzen, Trends in Neurosciences 10,8 (1987).
A controversial debate but
“...essential view of a reaction-diffusion process still holds ...”
Herreras (2005) J. Neurophysiol. 94:3656
Somjen & Strong (2005) J. Neurophysiol. 94:3656
![Page 29: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/29.jpg)
Migraine full-scale attack is more confined
(a) (b)
(c)
LS
CS
(d)
affected areatemporarily
• Dahlem et al. ”2D wave patterns ... ”. Physcia D 239 (2010) Special issue: Emerging Phenomena.
![Page 30: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/30.jpg)
Migraine visual field defects reported in 1941 by K. Lashley
visual field defect pattern on primary visual cortex
0
5
10
15
5min7min
9min
11min
15min
0 10 20 30 40 50mm
5min7min
9min11min
15min
Only about 2-10% but not 50% cortical surface area is affected!• Dahlem & Hadjikhani (2009) PLoS ONE 4: e5007.
![Page 31: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/31.jpg)
Tracking migraine aura symptoms
Vincent & Hadjikhani (2007) Cephalagia 27
![Page 32: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/32.jpg)
Confined spatial patterns of spreading depression
Hadjikhani et al. (2001) PNAS
![Page 33: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/33.jpg)
Confined spatial patterns of spreading depression
neighboring points
collapse
?
16 min
31 min
1 cm
nucleationrecordedslice not
Hadjikhani et al. (2001) PNAS
![Page 34: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/34.jpg)
Migraine full-scale attack is more confined
(a) (b)
(c)
LS
CS
(d)
affected areatemporarily
• Dahlem et al. ”2D wave patterns ... ”. Physcia D 239 (2010) Special issue: Emerging Phenomena.
![Page 35: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/35.jpg)
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)
![Page 36: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/36.jpg)
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
![Page 37: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/37.jpg)
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
nucleationcritical
![Page 38: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/38.jpg)
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
![Page 39: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/39.jpg)
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
![Page 40: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/40.jpg)
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
![Page 41: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/41.jpg)
Migraine waves look like unstable nucleation solutions
nucleationcritical
![Page 42: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/42.jpg)
What are we missing?
![Page 43: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/43.jpg)
What are we missing?
Kapitza’s pendulum
![Page 44: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/44.jpg)
Mapped visual symptoms on cortex via fMRI retinotopy
1 cm
10°
1 357
15
1719
2123
25
27 min
Visual hemifield Primary visual cortex
• Dahlem & Hadjikhani (2009) PLoS ONE 4: e5007.
![Page 45: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/45.jpg)
Mapped visual symptoms on cortex via fMRI retinotopy
23 min
21
19
17
17
15
1311
975
10°
1 cm
Visual hemifield Primary visual cortex
• Dahlem & Hadjikhani (2009) PLoS ONE 4: e5007.
![Page 46: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/46.jpg)
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
![Page 47: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/47.jpg)
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.
![Page 48: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/48.jpg)
Effect of intrinsic curvature of the medium
1. Lower threshold for SD if cortex is (intrinsically) negatively curved.
2. Stable wave segments: center being at positive curvature while the open ends extend into negative curvature.
nucleationcritical
0
10
20
30
40
50
60
1.3 1.32 1.34 1.36 1.38 1.4
wav
esi
zeS
threshold β
torus outside
torus inside2
211
ring wave
1
2
∂P1D∂R∞
neg. curvaturelower threshold
bifucation analysison torus pos. curvature
stabilization
• Kneer, Scholl & Dahlem, New J Phys, 16 053010 (2014)
![Page 49: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/49.jpg)
Hot spots left visual field
23 min
21
19
17
17
15
1311
975
10°
1 cm
Visual hemifield Primary visual cortex
Cooperation with Andrew Charles, UCLA.
![Page 50: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/50.jpg)
Hot spots right visual field
1 cm
10°
1 357
15
1719
2123
25
27 min
Visual hemifield Primary visual cortex
![Page 51: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/51.jpg)
Labyrinth path in reverse direction
1 cm
10°
1 357
15
1719
2123
25
27 min
Visual hemifield Primary visual cortex
![Page 52: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/52.jpg)
Reaction-diffusion with augmented transmission
The extended Hodgkin-Grafstein model (1963) of SD(+ FHN inhibitor equations + nonlocal term)
u =
(u − u3
3− v
)+ D∇2u
ε−1v = (u + β) + KF [u] (4)
Global controlF [u] = Su(t − τ)− S0
Su(t) =
∫Θ(u(r, t)− ue) dr,
cf. K. Krischer and A. Mikhailov, (1994) PRL 73, 3165
Sakurai et al., (2002) Science 296, 2009
![Page 53: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/53.jpg)
Reaction-diffusion with augmented transmission
The extended Hodgkin-Grafstein model (1963) of SD(+ FHN inhibitor equations + nonlocal term)
u =
(u − u3
3− v
)+ D∇2u
ε−1v = (u + β) + KF [u] (4)
Global controlF [u] = Su(t
− τ
)− S0
Su(t) =
∫Θ(u(r, t)− ue) dr,
cf. K. Krischer and A. Mikhailov, (1994) PRL 73, 3165
Sakurai et al., (2002) Science 296, 2009
![Page 54: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/54.jpg)
Cortical homeostasis is excitable (bistabe)
Hypothesis: Cortical susceptibility to SD depends on the size ofthe momentarily affected tissue.
nucleationcritical
![Page 55: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/55.jpg)
Inhib. global feedback: long transient (ghost behavior)
Hypothesis: Cortical susceptibility to SD depends on the size ofthe momentarily affected tissue.
slow dynamicstransient and
![Page 56: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/56.jpg)
Confined spatial patterns of spreading depression
neighboring points
collapse
?
16 min
31 min
1 cm
nucleationrecordedslice not
![Page 57: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/57.jpg)
Confined spatial patterns of spreading depression
neighboring points
0
4
8
12
16
20
32
28
24
time
16 min
31 min
1 cm
recordedslice not
![Page 58: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/58.jpg)
Confined spatial patterns of spreading depression
neighboring points 16 min
31 min
1 cm
recordedslice not
![Page 59: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/59.jpg)
Confined spatial patterns of spreading depression
neighboring points 16 min
31 min
1 cm
recordedslice not
![Page 60: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/60.jpg)
Confined spatial patterns of spreading depression
neighboring points 16 min
31 min
1 cm
recordedslice not
![Page 61: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/61.jpg)
Confined spatial patterns of spreading depression
5cm
00
0 0
32 16
6 24
time / m
in
![Page 62: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/62.jpg)
Modeled spatio-temporal signatures
0
6.25
25
50 (1)
(2)
(3)
(4)
2.5 6.25to
tal
aff
ec
ted
are
a 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)
neighboring points
collapse
?
16 min
31 min
1 cm
nucleationrecordedslice not
with aura
w/o aura
typical aura
without headache
1.
1.1. 1.2.
1.2.1.
Su
bfo
rms
Subtypes
Migraine
1.1.
1.2.1.
1.2.3.
MWoA 70%
MWA 30%
• Dahlem & Isele: J. Math. Neurosci. 3,7 (2013).
![Page 63: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/63.jpg)
Outline
Preface: Dynamical disease — Migraine — Models fill a gap
Mathematical migraine models: two (out of four) examplesModeling macroscopic fMRI dataModeling pain / central sensitisation
Towards therapeutic intervention
![Page 64: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/64.jpg)
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.
![Page 65: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/65.jpg)
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
![Page 66: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/66.jpg)
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
![Page 67: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/67.jpg)
History of electrical & magnetic stimluation
Non-drug treatment for headaches (1896)
First steps in miniaturization
![Page 68: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/68.jpg)
History of electrical & magnetic stimluation
Non-drug treatment for headaches (1961)
(1985) Zeitschrift EEG-EMG, Georg Thieme Verlag Stuttgart
![Page 69: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/69.jpg)
History of electrical & magnetic stimluation
Non-drug treatment for headaches (2013)
![Page 70: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/70.jpg)
Disclosure: Conflict of interest
Consulting services for Neuralieve Inc. (trading as eNeuraTherapeutics)
![Page 71: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/71.jpg)
Modern neuromodulation (invasive)
![Page 72: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/72.jpg)
Modern neuromodulation (invasive)
![Page 73: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/73.jpg)
Modern neuromodulation (invasive)
![Page 74: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/74.jpg)
Modern neuromodulation (invasive)
![Page 75: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/75.jpg)
Modern neuromodulation (invasive)
![Page 76: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/76.jpg)
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).
![Page 77: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/77.jpg)
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)
![Page 78: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/78.jpg)
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).
![Page 79: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/79.jpg)
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).
![Page 80: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/80.jpg)
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).
![Page 81: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/81.jpg)
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).
![Page 82: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/82.jpg)
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).
![Page 83: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/83.jpg)
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).
![Page 84: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/84.jpg)
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).
![Page 85: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/85.jpg)
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).
![Page 86: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/86.jpg)
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).
![Page 87: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/87.jpg)
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).
![Page 88: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/88.jpg)
Cortical homeostasis is excitable (bistabe)
Hypothesis: Cortical susceptibility to SD depends on the size ofthe momentarily affected tissue.
nucleationcritical
![Page 89: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/89.jpg)
Inhib. global feedback: long transient (ghost behavior)
Hypothesis: Cortical susceptibility to SD depends on the size ofthe momentarily affected tissue.
slow dynamicstransient and
![Page 90: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/90.jpg)
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
MIA
xy
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).
![Page 91: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/91.jpg)
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
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).
![Page 92: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/92.jpg)
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).
![Page 93: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/93.jpg)
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).
![Page 94: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/94.jpg)
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
![Page 95: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/95.jpg)
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
![Page 96: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/96.jpg)
Spatio-temporal waves need spatio-temporal control
Old paradigm�����������������������������������������������������������������������
����������������������������������������������������������������������
New paradigm: opens up new strategies, eg, transcranial randomnoise stimulation (tRNS) at special locations
![Page 97: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/97.jpg)
Feedback control of spreading depression
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)
![Page 98: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/98.jpg)
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
![Page 99: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/99.jpg)
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)
![Page 100: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/100.jpg)
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
![Page 101: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/101.jpg)
Cooperation & Funding
Niklas Hubel, Julia Schumacher,Thomas IseleBernd Schmidt
Steven Schiff(Penn State Center for Neural Engineering)
Jens Dreier(Department of Neurology, Charite; University Medicine, Berlin)
berlin
Migraine Aura Foundation
![Page 102: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/102.jpg)
Additional slides
![Page 103: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/103.jpg)
Many, many, parameters, but most fixed by experiments
Table: Parameters for ion–based model – Part 2
Name Value & unit Description
Cm 1 µF/cm2 membrane capacitanceφ 3/msec gating time scale parameterg lNa 0.0175 mS/cm2 sodium leak conductance
g gNa 100 mS/cm2 max. gated sodium conductance
g lK 0.05 mS/cm2 potassium leak conductance
g gK 40 mS/cm2 max. gated potassium conductance
g lCl 0.02 mS/cm2 chloride leak conductance
Na0i 25.23 mM/l intracell. sodium conc.
Na0e 125.31 mM/l extracell. sodium conc.
K 0i 129.26 mM/l intracell. potassium conc.
K 0e 4 mM/l extracell. potassium conc.
Cl0i 9.9 mM/l intracell. chloride conc.
Cl0e 123.27 mM/l extracell. chloride conc.
E 0Na 39.74 mV sodium Nernst potential
E 0K -92.94 mV potassium Nernst potential
E 0Cl -68 mV chloride Nernst potential
![Page 104: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/104.jpg)
Including cell swelling
Electrophysiology Thermodynamics
break down of ion grandients
cell swelling
Iin
Iin
Iout
Iout
Normal neuron in healthy brain
ECV 20%
ECV 5%
AMPA/kainate
–70 mV
–10 mV
SK
Na+
Na+
K+
Na+
DR
Ca2+
Ca2+
Ca2+
Ca2+
Ca2+ Na
+Na
+
Ca2+
K+
Na+
Ca2+
Ca2+
K+
Swollen neuron during spreading depolarization
H2O
Nonspecific
cation channelsInsufficient
sodium pump
NMDAR
J.P. Dreier Nature Medicine 17 2011
massive release of Gibbs free energy
J.P. Dreier et al. Neuroscientist 19 2012
![Page 105: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/105.jpg)
Modeling the migraine aura–ischemic stroke continuum
0 1 2 3 4 5 6time (s)
100
50
0
50
volt
age (m
V)
V
EK
ENa
Iapp
−gate deactivation Hopf
−gatemembrane voltage
SNIC
Hopf
SNIC
Recovery
eletrogenic pump
FoldSeizure−like activity
in ischaemic stroke
hypoxic tissue
Spreading depression
(ceiling level)
n
nV
Ipump[K+]o
[K+]o = 10mM
![Page 106: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/106.jpg)
Modeling the migraine aura–ischemic stroke continuum
I Ischemia-inducedmigraine,
I Migrainous infarction,
I Persistent migraine w/oinfarction (see below).
Dahlem et al. Physica D 239, 889 (2010)
0 1 2 3 4 5 6time (s)
100
50
0
50
volt
age (m
V)
V
EK
ENa
Iapp
−gate deactivation Hopf
−gatemembrane voltage
SNIC
Hopf
SNIC
Recovery
eletrogenic pump
FoldSeizure−like activity
in ischaemic stroke
hypoxic tissue
Spreading depression
(ceiling level)
n
nV
Ipump[K+]o
[K+]o = 10mM
![Page 107: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/107.jpg)
Mainly two neural theories of migraine
”Migraine generator”-theory
S1
PFCTh
PPC
PAG
Amyg Insula
SMA
ACC
”Spreading depression”-theory
![Page 108: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/108.jpg)
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
![Page 109: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/109.jpg)
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
![Page 110: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/110.jpg)
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).
![Page 111: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/111.jpg)
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).
![Page 112: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/112.jpg)
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
![Page 113: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/113.jpg)
Re-entrant SD waves with anatomical block
Reshodko, L. V. and Bures, J Biol. Cybern. 18,181 (1975)
![Page 114: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/114.jpg)
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
![Page 115: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/115.jpg)
Nucleation failure on torus
![Page 116: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/116.jpg)
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
![Page 117: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/117.jpg)
Simulation of transient SD wave segment
gray = cortical surface; red = SD wave
![Page 118: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/118.jpg)
Simulation of an engulfing SD wave
In cooperation with Bernd Schmidt,
MagdeburgIn cooperation with Jens Dreier &
Denny Milakara, Charite
![Page 119: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/119.jpg)
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∞
![Page 120: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/120.jpg)
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∞
![Page 121: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/121.jpg)
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∞
![Page 122: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/122.jpg)
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∞
![Page 123: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/123.jpg)
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∞
![Page 124: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/124.jpg)
Migraine scotoma reveal functional properties
Pattern matching
”Curved” retinotopic mapping
47
913
A B
C
• Dahlem & Tusch, J. Math Neuroscie. 2,14 (2012)
![Page 125: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/125.jpg)
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)
![Page 126: Migraine: A dynamical disease](https://reader033.vdocuments.us/reader033/viewer/2022052600/55837787d8b42a88578b463f/html5/thumbnails/126.jpg)
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)