lashleyFig2Mapped.eps

Migraine aura: retracting particle-like waves in weakly susceptible cortex

M. A. Dahlem

and N. Hadjikhani

Institut f¨

ur Theoretische Physik, Technische Universit¨

at Berlin, Hardenbergstraße 36, 10623 Berlin, Germany,

Klinik f¨

ur Neurologie II, Otto-von-Guericke-Universit¨

at Magdeburg, 39120 Magdeburg, Germany,

Leibniz Institute f¨

ur Neurobiologie, 39118 Magdeburg, Germany

Martinos Center for Biomedical Imaging, Massachusetts General

Hospital, Harvard Medical School, Charlestown, MA 02129, USA,

Brain Mind Institute, EPFL, Lausanne, Switzerland

(Dated: July 11, 2008)

Cortical spreading depression (SD) has been suggested to underlie migraine aura. Despite a

precise match in speed, the spatio-temporal patterns of SD and aura symptoms on the cortical
surface ordinarily differ in aspects of size and shape. We show that this mismatch is reconciled
by utilizing that both pattern types bifurcate from an instability point of generic reaction-diffusion
models. To classify these spatio-temporal pattern we suggest a susceptibility scale having the value

= 1 at the instability point. We predict that human cortex is only weakly susceptible to SD ( < 1),

and support this prediction by directly matching visual aura symptoms with anatomical landmarks
using fMRI retinotopic mapping. We discuss the increased dynamical repertoire of cortical tissue
close to = 1, in particular, the resulting implications on migraine pharmacology that is hitherto
tested in the regime ( 1), and potentially silent aura occurring below a second bifurcation point
at = 0 on the susceptible scale.

INTRODUCTION

Migraine aura is a collection of transient neurological

symptoms characterized by a gradual onset as the dis-
tinctive clinical feature. It may be classified into sen-
sory and cognitive modalities. Visual aura predominate,
usually consisting of a distortion in the visual field of-
ten characterized by an expanding zigzag pattern at the
leading front and a scotoma in the back [16] (Fig. 1
(a)). Direct correlations between aura percepts and neu-
ral properties have been demonstrated, e. g., the typical
zigzag patterns are reflected in reversed cortical feature
maps[2, 7, 8]. While the pseudohallucinatory percept
during the aura (visual or other) is specific from the af-
fected sensory modality and is independent of etiology[9],
the spatio-temporal course of aura progression is a clear
signature of the underlying pathological process.

Reverse retinotopic mapping of aura symptoms reveals

a constant propagation speed of about 3 mm/min on the
cortical surface [1, 6] (Fig. 1 (b)). The remarkable slow
velocity fits with the pace of spreading depression (SD), a
profound but transient all-or-none type process charac-
terized by redistribution of ions across cell membranes
and nearly complete neuronal depolarization[10, 11].
This suggests that both phenomena rely on the same
propagation mechanism[12].

Despite the precise match in speed of mapped aura

symptoms and SD, both processes ordinarily differ in as-
pects of size and shape on the cortical surface. While SD
waves usually invade the entire gray matter region and
stop only at the border to white matter--at least if ob-
served in the most prone brain regions, the hippocampus
and neocortex of nonprimate mammals--migraine aura
symptoms, in contrast, seem to be more spatially con-
fined. This can be deduced from the fact that visual

symptoms often last not longer than 20 min correspond-
ing to a distance of 60 mm, the length of the early visual
areas located along the sulcus calcarine (Fig. 1). It is the
central result of this article to show that this mismatch in
size and shape between mapped aura symptoms and SD
propagation may be reconciled by utilizing that both pat-
tern types occur in a generic reaction-diffusion model but
are separated by a bifurcation, that is, a sudden qualita-
tive change in the spatio-temporal SD pattern after only
a small smooth change made to cortical susceptibility to
SD. Our predictions are supported by directly matching
visual aura symptoms with anatomical landmarks using
fMRI retinotopic mapping.

(a)

(b)

FIG. 1: (a) Right visual hemifield (dotted polar grid) with
five subsequent sketched "snapshots" of a traveling visual
migraine aura symptom in the shape of a crescent pattern.
Numbers inside the scotom gives the time passed (in minutes)
since first occurrence. (b) Visual field disturbance shown by
reversed retinotopic mapping.

Our results lead us to the conclusion that SD in hu-

mans is much closer to a bifurcating instability point of
pattern formation than in nonprimate mammals. From a
synergetics point of view, the brain is in general viewed
as a self-organizing pattern forming system that oper-
ates close to instability points[13]. In the case of mi-
graine aura, the crucial instability point separates tran-
sient from sustained wave propagation. Being close to
the instability point dramatically changes the dynami-
cal repertoire. This factor, as will be discussed, should
have severe implications on the design of migraine drug
tests. Moreover, it may favor the hypothesis of the oc-
currence of silent aura in diagnosed forms of migraine
without aura

[14].

RESULTS

The visual aura symptoms typically affect only a part,

albeit large part, of a visual hemifield (VH). The affected
area forms an expanding circular arc often centred close
to the fovea. In Fig. 1 (a), a sequence of subsequent mi-
graine aura "snapshots" visualizes the typical course of a
visual disturbance in the right VH. The perimetric data
is taken from Lashley[1] and the corresponding spatio-
temporal pattern in the primary visual cortex (V1) is ob-
tained by reversed retinotopic mapping (Fig. 1 (b)). The
crescent pattern in the VH translates into a wave segment
resembling a "particle-like wave"[15], as described in the
next subsection. From this pattern, we can estimate an
average length of the wave front of about 35 mm and a
propagation speed of 3mm/min. Therefore, such a wave
segment temporarily recruits a total of about 2100mm

cortical surface within 20 min into the depolarized SD
state, that is, only approximately 1.7% of the surface of
one human cortical hemisphere.

Susceptibility scale based on wave instabilities

Transient and spatially confined waves were first sug-

gested to cause aura symptoms in a descriptive mathe-
matical model considering the motion of curves with free
ends[16]. These curves represent segments of excitation
fronts with two open ends, as shown in Fig. 1 (b). Fur-
thermore, unstable--and thus also transient and spatially
confined--waves, termed particle-like waves have been
found and studied in the chemical Belousov-Zhabotinskii
(BZ) reaction and their spatio-temporal dynamics are de-
scribed by reaction-diffusion (RD) equations [15, 17, 18].
Particle-like wave propagation differs significantly from
the current view of SD as a pattern engulfing posterior
cortex (Fig. 1 (a)).

We suggest to introduce a macroscopic susceptibility

scale to classify such spatio-temporal RD patterns in
excitable media that are weakly susceptible to SD wave

propagation. A two-point definition is used for calibrat-
ing this scale, whereby = 1 represents particle-like
waves and = 0 the propagation boundary (see Meth-
ods). The two points are defined each by such a bifur-
cation point. The value = 1 separates excitable me-
dia with capacity to propagate growing waves segments

-1

-2

(d)

(c)

(b)

(a)

FIG. 2: Schematic view of the spatio-temporal course of a
reaction-diffusion wave for different tissue susceptibility val-
ues : wave front (red), recovery phase (yellow), blue arrows
indicate normal velocity, future location is dashed (red). (a)
sustained wave, (b) retracting wave, indicated by green arrow
heads, (c) collapsing wave, (d) no spread. The gray interval
is defined as weakly susceptible.

( > 1) from those where only retracting waves segments
( < 1) occur. When susceptibility changes to a value be-
low = 0, the amplitude of the wave decreases so that a
wave segment not only retracts from its sides (decreasing
length as indicated by green arrow heads in Fig. 2 (b))
but also its wave profile collapses (decreasing width).

The two bifurcation at = 0 and = 1 are generic in

the sense that they apply to excitable media based on
RD mechanisms irrespective of the particular model. In
Fig. 2, generic spatio-temporal RD patterns are classi-
fied into four intervals based on a linear scale between
the points = 0 and = 1. The linear scale and the locus
of further bifurcation points on this scale depend on the
specific RD model. We used the FitzHugh-Nagumo sys-
tem (FHN) and fixed all parameters but the threshold
such that the experimentally observed re-entrant pattern
of retinal SD, which performs a complex meandering[19],
is obtained at > 2 (Methods)[20, 21].

Four susceptibility intervals are relevant for reconciling

the mismatch in size and shape between mapped aura
symptoms and SD propagation. They are ordered by
decreasing susceptibility: (a) ( > 1): sustained waves,
(b) (1 > > 0): retracting waves, (c) ( < 0): collapsing
waves, (d) ( < 2.2): no spread.

The regime (a) has the highest susceptibility to spread-

ing phenomena due to the lowest threshold values among
the four intervals.

The spatio-temporal patterns ob-

tained in (a) show the typical course of SD waves ob-
served in animal experiments. In particular, an SD wave,
initiated at the occipital pole and propagating in ante-
rior direction, will eventually engulf the whole cortical
surface. In this regime, wave segments with free open
ends curl in to form rotors (spiral shaped waves), there-
fore the lower bound = 1 is called the rotor boundary
(RB)[20]. RB is marked by the occurrence of particle-like
waves. Adhering strictly to the definition of particle-like
waves as a wave form with natural length and shape that
will either grow or decay when perturbed, RB is wave
size dependent[17]. In the limit of large wave segments
(critical fingers[22, 23]) susceptibility approaches a lower
bound that is used as the defining point for calibrating
.

In both the intervals (b) and (c) transient waves forms

occur.

In (b), the interval with higher susceptibility

among (b) and (c), 2D wave segments with free open
ends, such as shown in Fig. 1 (b), eventually disappear
because open ends retract and thereby constantly reduce
the instantaneous size of excitation. In susceptibility in-
terval (c) the RD equations describe rather a process of
facilitated diffusion than travelling wave processes in ex-
citable media. For this reason, the boundary between
(b) and (c) is called propagation boundary (PB)[20, 23].
In (c), the pulse evolution, following an initial increase
in the activator, is similar to passive diffusion only that
the spatially elevated distribution collapses slower than
without the activator reaction and this process can be

directed, if the spatial distribution of activator and in-
hibitor are not symmetric. In regime (d), an initially
imposed localized spatial distribution of the activator
collapses without broadening (spreading boundary (SB))
because the reaction part provides a sink that decreases
the activator faster than it is transported outwards by
diffusion[24].

Effects of gyrification on RD waves

Critical properties of RD waves such as retracting

particle-like wave propagation in the weakly susceptibil-
ity domain 1 > > 0 are modulated by the bending of the
cortical surface. This can be deduced from experimental
and theoretical[2528] studies of the chemical BZ model
systems of RD waves on curved surfaces in the regime of
weakly excitable media. Weak excitability is not strictly
defined but usually refers to values close to = 1. In
these systems, it is shown that propagation depends cru-
cially on the geometric properties of the surface. As a
consequence, we can predict that a correlation must ex-
ist between anatomical landmarks and the course of aura
symptoms if migraine aura is caused by a RD process.

In this subsection, we consider the gross gyral mor-

phology in relation to the typical aura onset, course and
ending. But before, we refer to a particular curvature-
induced phenomenon that provided a mechanism how
wave segments can emerge in the first place.

It was

shown that the wave front can undergo a critical defor-
mation above which propagation is blocked[29]. A broken
wave front is needed to distinguish spatio-temporal pat-
tern obtained in the susceptibility intervals ( > 1) and
(1 > > 0). The evolution of closed wave fronts does
not differ much until the front breaks open, for instance
due to a local curvature-induced excitation block. Then
the resulting open ends will either grow or retract if the
susceptibility is in the interval ( > 1) and (1 > > 0),
respectively.

If migraine aura is caused by retracting RD waves

(1 > > 0) that are guided by anatomical landmarks, the
main course of the neurological symptoms within differ-
ent people can be similar, because many studies of hu-
man cytoarchitecture show that sensory and motor areas
have some relationship to the gross sulcal and gyral mor-
phology. In some cases very precise correlations between
sulci and functional entities could be demonstrated, most
prominent is the calcarine sulcus (CS) as a landmark of
the primary visual cortex (V1)[30]. Furthermore, the pri-
mary auditory cortex has a clear spatial relationship with
Heschl's gyrus[31, 32], and the motor cortex can be iden-
tified by the position of the central sulcus[33]. Yet a
substantial interindividual and interhemisphere variabil-
ity in both size and location of anatomical landmarks is
observed[34], and major sulci and gyri are individually
composed of smaller gyral folds and sulci indents, which

provides a variability for individual local characteristics
of the spatio-temporal aura symptoms.

Due to CS's precise landmark position of V1[30] its

geometric properties are best suited for comparison with
visual aura symptoms. Furthermore, its retinotopic map-
ping of visual input is well studied in human[3538].
We therefore consider the gross morphology of CS and
the relative position of V1 in relation to the typical on-
set, course and ending of crescent shaped visual aura as
shown in Fig. 1.

Onset

Most of the crescent shaped aura pattern start

in one VH close to the fovea (center of gaze). The neu-
ral representation of the fovea is located at the occipital
pole often extending about 10 mm onto the lateral con-
vexity. CS is formed by the cuneus and lingual gyrus on
the medial surface and runs forward to the corpus cal-
losum. Approximately two-thirds of V1 lies within the
CS walls [30]. A difference of 1

visual angle between

the onset of aura symptoms and the fovea corresponds
to a cortical distance of about 1cm (see Fig. 1) because
of the large linear cortical magnification factor M (see
Methods) close to the fovea. Therefore, the crescent aura
symptoms start near the entrance of CS.

Course

Typical crescent pattern propagate along the

horizontal hemimeridian (HM) towards the visual periph-
ery. The pattern extends into both quadrants of the VH,
which is a clear sign that it is caused in V1. V1 is the
only of the early visual areas where the two quadrants
of the VH are not split along HM[39]. Extrastriate vi-
sual cortical areas represent the two quadrants of VH in
dorsal and ventral areas that are connected only close
to the fovea. And the pattern is caused in early visual
areas because orientation selective cells with moderate
receptive field sizes are only found there. They represent
the individual edges of the zigzag aura percept at the
propagating front[8].

The locus of the neural representation of HM in V1 is

near the fundus of CS. Individual aura reports show an
asymmetric propagation to either the upper or lower vi-
sual field quadrant [13, 6]. If the visual field defect falls
behind in one visual quadrant, this could indicate that
M is larger in this quadrant. In deed, anatomical data
suggest that V1 proceeds farther anteriorly in the lingual
gyrus[30, 40], which suggests that more cortical surface
is devoted to upper quadrant, however, fMRI data show
that the dorsal and ventral compartments of V1 are at
least similar in absolute extent measured in geodesic dis-
tance [38].

Ending

Visual aura symptoms stop in the periphery

of the VH. The extreme periphery of the VH is repre-
sented at the anterior boundary of V1 close to the T-
shaped or sometimes Y-shaped junction of the CS and
the medial part parieto-occipital sulcus. Such a junction
might act as a diode being transparent for wave propaga-
tion only in one direction, but not in the other. Critical
properties of excitation waves on curved surfaces that

lead to a curvature-dependent loss of excitability have
been studied in BZ system[28].

The CS as a major sulci is composed of smaller gy-

ral folds and sulci indents resulting in a complex sur-
face. While the gross morphology of CS can determine
the basic course of particle-like wave propagation, it is
this individually complex surface that needs to be consid-
ered if precisely recorded perimetric data of visual aura
progression are compared with anatomy. Furthermore,
only rather sharp deformations of the cortical surface can
directly induce a critical deformation in the wave front
above which propagation is then blocked[29]. The effect
of smaller gyral folds and sulci is considered in the next
subsection.

fMRI retinotopy and perimetric aura data

To investigate the effects of small gyrification pattern

on RD waves, the 3D form of V1 and its retinotopic map
was obtained by fMRI from a migraineur (PVV) who has
made precise perimetric recordings of his visual aura[41].
In Fig. 3 (a), the right V1 is shown. Its color codes the az-
imuthal angle of the contralateral left VH by a half HVS
(hue, value, saturation) color wheel, in counterclockwise
direction from red (upper hemimeridian) via light green
(HM) to cyan (lower hemimeridian). The rostral/caudal
(r, c) and dorsal/ventral (d, v) directions are indicated
by a cross.

The dorsal bank of the right CS is noticeable heavily

ramified with small gyral folds and sulci indents. The
progression pattern of the visual field defect in the lower
visual quadrant shows accordingly a rather complex pat-
tern. The spatial progression is marked in Fig. 3 (b)
by drawing with white lines the current position of the
propagating field defect at one minute intervals within
24 minutes. The wave runs from minute 4 to 13 in the
lower visual quadrant and this quadrant is mapped, as
can be seen by the color code, onto the dorsal bank of
CS. Partly the wave pulsates back and forth between 11-
13 minute and eventually terminates in the lower end of
this visual quadrant in an excitation block, but continues
to propagate within the upper quadrant.

In Fig. 3 (c), the left V1 is shown with the color coding

azimuthal angles of the contralateral right VH by the
other half HVS color wheel, in counterclockwise direction
from cyan (lower hemimeridian) via dark magenta (HM)
to red (upper hemimeridian). Marked with white lines at
one minute intervals, the spatial progression of the visual
aura in the right VH is shown in Fig. 3 (d). As can be
seen by the color code, the wave runs from minute 1 to
8 over a gyral crown (gc) as part of the cuneus. Between
minute 8 to 15 the wave disappeared, but reappeared at
minute 15 propagating upwards in the visual field for a
duration lasting 12 minutes being approximately parallel
to visual hemimeridians, i. e., running from the dorsal to

the ventral bank of CS and ending on the anterior edge
of the lingual gyrus.

DISCUSSION

The crescent shaped aura pattern, as shown in Fig. 1

(a), is often reported [16] but the phenomenology of mi-
graine aura is much richer as documented by the variety
of illustrations and descriptions collected on the Migraine
Aura Foundation website (www.migraine-aura.org). In
a single migraine aura attack, migraineurs can also ex-
perience diverse visual, as well as sensory, motor and
language disturbances[42, 43]. This variety clearly indi-
cates that other areas beside early visual cortex can be
affected, even cortical areas outside the occipital lobes,
and it therefore seemingly supports the idea that the pro-
cess causing the aura can engulf all of posterior cortex on
its course, like a cortical SD wave observed in animal ex-
periments.

Schematic drawings similar to Fig. 2 (a) illustrate en-

gulfing spatio-temporal wave patterns.

Such illustra-

tions are found in modern textbooks of headache[44] and
appeared first in Lauritzen's seminal paper spearhead-
ing the SD theory of migraine aura[45]. They became
paradigmatic for migraine with aura.

However, they

might need to be revised, as we show.

The activity pattern causing crescent shaped aura is

remarkably similar to a particle-like wave segment on the
cortical surface (Fig. 1), a pattern that exists only in cor-
tex being weakly susceptible to SD. Other factors also
support the concept that human cortex is only weakly
susceptible to SD, maybe foremost that susceptibility be-
comes the lower the higher up the species is in the phy-
logenetic tree. Another clear indication is that SD prop-
agation is modulated by cortical morphology, as can be
seen in Fig. 3. Similar pattern were also observed for the
gyrencephalic feline brain[46], but there the primary SD
wave engulfed the hemisphere and only succeeding sec-
ondary waves remained within the originating gyrus and
were more fragmented. Since secondary waves run into
partly refractory tissue, susceptibility to SD is decreased.

The engulfing wave pattern is originally motivated by

SD propagating in the smooth cortex of rats and rabbits.
It has been debated whether SD can occur in the highly
convoluted cortex of humans, until spatial and tempo-
ral events were followed using high-field functional MRI
[47] demonstrating that at least eight characteristics of
SD are present and the events are time-locked to percept
onset of the aura in human cortex. However, the precise
spatio-temporal course of the events is still ambiguous.
Much of posterior cortex, including several retinotopi-
cally organized visual areas, showed simultaneous acti-
vation during much of the period of the aura, while the
percept in the VHs is reported to be more spatially con-
fined.

As already noted by Wilkinson[48] this mismatch in

fMRI data and aura percept can be explained by at least
two alternatives: (i) either SD engulfs all of posterior cor-
tex. Then only a subset of this activation results in sen-
sory awareness. Or (ii), the spread of the SD wave is, in
contrast to the fMRI data, more limited in extent. Then
the rest of the observed activation in adjacent cortical ar-
eas represents synaptic activation through feed-forward
and feedback circuitry. While (i) is in agreement with
observed cortical SD wave patterns in animals, it opens
up questions about the nature of the often reported lim-
itation to spatially confined crescent-shaped visual field
defects. In (ii) spatially confined SD waves causing cor-
responding field defects are simply postulated[48].

If SD in human is more limited in extent, the mis-

match with animal data needs to be addressed. To rec-
oncile this, we provide a theoretical framework, which is,
moreover, of practical use to both experimental neurosci-
entists and clinicians. We propose a susceptibility scale
based on nonlinear bifurcation analysis. Not unlike the
Celsius temperature scale, the term susceptibility to SD
is made a precise scale by a two-point definition, i. e.,
two macroscopically observable cortical states at which
a phase transition in SD pattern formation occurs. The
relevance and applicability of this scale is described in
the following.

The weakly susceptible state (1 > > 0) of human cor-

tex to SD can be achieved in experimental migraine mod-
els if the tissue is preconditioned reducing excitability
towards the gray marked regime in Fig. 2. The proce-
dure to find this regime experimentally is described in
the Methods section for retinal SD. Retinal SD is accom-
panied with an intrinsic optical signal that makes precise
spatio-temporal recordings of the evolutionary SD pat-
tern possible. Similar precise spatio-temporal recordings
have been made in cortex using a fluorescent, voltage-
sensitive dye [49].

We predict that effects of antimigraine drugs depend

on the susceptibility range they are tested in, because the
dynamical behavior of a nonlinear system changes dras-
tically when crossing a bifurcation point. Antimigraine
drugs tests and tests to unravel the mechanism of SD
in retina[5052] have been performed far away from the
regime (1 > > 0). This can be shown, by precisely mea-
suring in this system the complex meandering pattern of
spiral SD[19]. On the scale, obtained from the generic
FHN model, these pattern occur above > 2 and are sep-
arated by two further bifurcations [21]. In general, SD
experiments are performed in the most prone tissue re-
gions where SD can more easily be observed. This might
remind one at Watzlawick's man searching for his keys
under the streetlight rather than where he lost them[53].

Furthermore, our results supports the idea that SD

could activate the trigeminovascular system that gener-
ates and maintains migraine pain [54] even in diagnosed
forms of migraine without aura. For susceptibility val-

(a)

(b)

(c)

(d)

(e)

t=0s

t=4s

t=6s

t=12s

t=43s

1mm

FIG. 4: Creation of an SD wave segment with free open ends in submerged retina. (a) Mechanical stimulation with sharp glass
needle s, (b) circular SD wave evolves, (c)-(d) local application of Mg

via pipette p, (e) wave propagation is locally blocked

and consequently SD front brakes open and curls in to form a spiral at the lower open end, while the upper open end is guided
by the Mg

-pipette to the border of the retina where it attaches.

ues below the weakly susceptible regime, the model pre-
dicts spatio-temporal SD pattern that do not break away
from an initially restricted focus. We can draw a direct
analogy to clinically silent epilepsy caused by interictal
activity that does not break away from a focus. Like-
wise, previously proposed silent aura, in which "some
migraineurs exhibit blood flow 'fingerprints' of CSD [cor-
tical SD] and aura but are subjectively unaware that the
phenomenon is propagating"[14], may be explained by
localized SD patterns occurring at the one end of the in-
creased dynamical repertoire that emerges if being close
to a bifurcation.

METHODS

Susceptibility scale in experimental and

mathematical models

A two-point definition is used for calibrating the newly

introduced susceptibility scale . These two points are
macroscopically observable states. We shortly describe
an experimental procedure to measure such states. A
precise determination of these two states in an animal
model of SD is, however, beyond the scope of our proof
of concept. RB ( = 1) can be obtained by changing the
tissue excitability until open wave segments stop curling
in to form reentrant SD waves with freely rotating open
ends forming two centers (spiral SD)[55]. For instance,
to obtain an SD wave segment in submerged retina (for
details, see Ref. [19]), an initially closed circular SD wave
front can be broken (at a diameter of about 0.75 mm)
by local application of 0.5 ml Ringer solution through a
pipette (tip diameter 0.5 mm) containing a tenfold raised
Mg

concentration (10 mM) (Fig. 4).

In Fig. 5 a retinal SD wave segment is shown that

evolves into a double spiral ( > 1). The mathemati-
cal model (see below) predicts that after crossing = 1
(RB), the open ends of the wave segments retract (direc-
tion indicated by green arrows in Fig. 5) and the SD wave
eventually vanishes. The Mg

concentration in Ringer

at which this transition occurs is in this experimental
set-up difficult to determined, because the initial raise in
Mg

needed to break the circular wave front cannot suf-

ficiently fast be washed out. However, it is known that
lowering calcium concentration to 0.5 mM and increasing
magnesium to 2.0 mM turns the tissue absolute refrac-
tory to SD[56], which corresponds to the regime < 0
and giving a lower bound of = 0.

The locus of RB as a function of excitability repre-

sents a critical perturbation threshold[17], separating an
attractor characterized by spiral waves from an attractor
characterized by the uniform physiological steady state
of the cortex. Such a threshold (RB) must exist if spiral
SD waves occur in the tissue and therefore the existence
of RB is independent of the particular model that de-
scribes the pattern formation process. The locus of PB
can be obtain similarly by decreasing further the tissue
excitability until reentrant SD waves collapse even if their

1 mm

FIG. 5: Retinal SD wave segment propagating (blue arrows)
with free open ends that grow (red arrow) and therefore curl
in to form a double spiral. At lower susceptibility values, RD
models of SD predict that open ends retract (green arrows)
and the wave vanishes.

open ends are attached to either the border of the retina
or a lesion (circling SD [57]).

In mathematical models of SD, the critical points RB

and PB are found by bifurcation analysis. Some SD
models investigate the local ignition of SD by math-
ematical models of single cells and their surrounding
compartments[58, 59]. Those models lack a spatial ex-
tension beyond the cell size. They cannot yet address
the clinically relevant question whether a local ignition
stays confined or breaks away but such microscopic mod-
els help to understand the pathophysiological mechanism
of SD and if they will be extended by a spatial coupling,
such as a diffusion term, also those models become ac-
cessible to the bifurcation analysis described in the fol-
lowing.

We exemplify with a standard RD scheme of activator-

inhibitor type the determination of the location of RB
and PB in the parameter space of this model and how
to obtain from a parameter value the susceptibility scale
. By choosing an activator-inhibitor type SD model, we
assume that all quantities with a positive feedback loop
can be lumped together, such as extracellular potassium
concentration [K

]

and inward currents[60, 61]. They

become a single activator variable u. The rate of change
in u is given by a single nonlinear reaction rate f . Like-
wise, a single inhibitor variable v represents the recovery
processes with reaction rate g. Processes represented by
inhibitor kinetics are, amongst others, the effective regu-
lation of [K

]

by the neuron's N a

-K

ion pump and

the glia-endothelial system. The general form of a RD
equation is then

= f (u, v) + D

(1)

= g(u, v),

where the term D

u represents the spatial coupling of

the local dynamics by diffusion of u with diffusion coef-
ficient D.

The variety of macroscopic RD pattern in u and v,

such as spirals and retracting waves, is largely indepen-
dent of the specific reaction rates f (u, v) and g(u, v), as
long as the local dynamics (D = 0) show all-or-none type
behavior. To obtain the scale shown in Fig. 2 we chose
the FitzHugh-Nagumo equations f (u, v) = u - u

/3 - v

and g(u, v) = (u + )/25, where is a threshold parame-
ter that selects the pattern. In this system, particle-like
waves can be stabilized with a control term that changes
as a linear function of wave size, a feedback mechanism
first proposed for chemical BZ waves[17]. At

= 1.34

the limit of large particle-like waves is reached (critical
fingers [22, 23]). The propagation boundary is found by
transforming Eqs. 1 into a co-moving frame and deter-
mine the largest value

P B

= 1.39 for which bounded

profile solutions exist [24]. The susceptibility scale, as a
linear function of with the defining points = 1 and

= 0 corresponding to RB and PB, respectively, is then
obtained by

() =

P B

+ 1.

(2)

This formula is independent on the specific choice of as
a parameter to change excitability, e. g., in an experimen-
tal system

and

P B

could be taken as the concentra-

tion of Mg

as described above. If more then on param-

eter is accessible to change tissue excitability a shortest,
i.e., metrical, distance between RB and PB can be de-
fined via pharmacokinetic-pharmacodynamic models[24].

Perimetric recordings and retinotopic mapping

The perimetric data shown in Fig. 1 were taken from

Lashley's precise drawings published in 1941[1]. The ra-
dial coordinate (eccentricity) of the crescent shaped aura
pattern in the right VH was calibrated assuming the blind
spot (not shown in Fig. 1) at 10

. The eccentricity is

then obtained assuming the percept is projected to a flat
tangent plane with respect to the center of the spherical
visual field. This tangent plane serves as the canvas to
draw the aura percept. The azimuthal coordinate can be
taken directly form the drawing in the tangent plane.

The flat retinotopic map in Fig. 1 (b) was created by

using the monopole map, that is, the complex logarithm
w = A log(z/E

+ 1) with the cortical magnification pa-

rameter E

= 0.75 and A = 17.3 adjusted to human data

[62]. The complex coordinates z and w describe locations
in the visual field and in the cortical domain, respectively.
The magnitude of z is the visual eccentricity and its ar-
gument is the azimuth (z = e

). The real and complex

parts of w are Cartesian coordinates on the cortical sur-
face. From the monopole map it follows that the linear
cortical magnification factor along HM is M () =

The perimetric data shown in Fig. 3 were provided

by a participant (PVV) who fulfills the International
Headache Society criteria for the diagnosis of migraine
with aura. As a research engineer he trained himself to
make precise recordings during his migraine with aura
attacks and documented over 350 aura episodes over
10 years [41]. To compare the topography of the vi-
sual aura with anatomical landmarks of the cortex, the
retinotopic organization in the visual cortex was obtained
with functional magnetic resonance imaging (fMRI). The
data were acquired in a 3-Tesla scanner, using echopla-
nar imaging as described in Refs. [47, 63]. All procedures
were approved by MGH IRB.

Acknowledgement

The authors want to acknowledge PVV for his ac-

tive participation in data collection and are indebted to

Jochem Rieger, Jan Tusch, Roland Aust, and Josh Sny-
der for technical assistance, and to Gerald Hiller for com-
ments on the manuscript. MAD was supported by the
Deutsche Forschungsgemeinschaft (DA 602/1-1 and SFB
555). NH was supported by NIH grant 5PO1NS 35611-
09.

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