Transient Localized Wave Patterns and Their Application to Migraine

2.1 Spatiotemporal Behavior of Excitable Systems

An excitable system can either be only time-dependent, which will be called in this study an “excitable element,” or excitable systems can be time- and space-dependent, which we call “excitable medium.” These systems are described by ordinary differential and by partial (integro)differential equations, respectively. The episodic migraine attacks, which we model in this study as transient responses of the cortex described by an excitable medium, remind us more of the transient behavior of excitable elements than of the persistent excited state obtained in excitable media. Therefore, we review the general spatiotemporal behavior of excitable elements and spatially extended excitable media in this section.

Excitability was first described for neurons in the original conductance-based membrane model by Hodgkin and Huxley [2]. This and many more refined versions of neural excitability to date contain four or more dynamic variables, but fortunately this is not essential for excitable systems. In fact, it turned out for excitable elements that the main two classes of excitability are actually amenable to direct analysis in a two-dimensional phase plane by identifying in the conductance-based model fast and slow processes and grouping these into dynamics of just two lump variables [4, 5]. Using such a geometrical approach and partly analytical theory, the original empirical classification of excitability was further pursued with bifurcation analysis [6], explaining class I by identifying its threshold as a stable manifold of a saddle point on an invariant cycle and the threshold of class II as a trajectory from which nearby trajectories diverge sharply (called canard trajectory). Extensions to these principal mechanisms involve codimension 2 bifurcations and lead also to bursting in three-variable models, which have been investigated in great detail [7]. However, the two-variable models of a fast activator and slow inhibitor and their phase portraits of class I and II became qualitative prototypes for excitable elements in various biological [8], chemical [9], and physical contexts [10].

Distinct from excitable elements and their classification are spatially extended excitable media. Already the original work by Hodgkin and Huxley [2] described spatially extended, tube-like membranes (axons) and introduced the cable equation as a parabolic partial differential equation, which is in the same class as the diffusion equation. In this reaction-diffusion framework, an excitable medium is the continuum limit of a locally coupled chain of excitable elements. Even in reaction-diffusion media with infinite-dimensional phase space, we can again apply geometrical approaches, simply because excitable media are not defined by—in contrast to excitable elements—transient dynamics but traveling wave solutions. The quiescent state is the nonexcited homogeneous steady state. And like the quiescent state, excited states of a medium are usually stationary states in some appropriate comoving frame, for example, along the x-direction with $\xi =x-ct$. Furthermore, the threshold is related to an unstable stationary state, the critical nucleation solution, usually in another comoving frame including $c=0$. The existence of the nucleation solution is a simple consequence of multistability; see Fig. 1a, but note that even monostable excitable elements have similar unstable stationary states in class I.

In this study, we propose a spatially extended model for wave-like patterns with a characteristic shape, size, and duration. These patterns are (spatially confined) transient responses to confined, spatially structured perturbations of the globally stable homogeneous steady state. These transient responses—after propagating for a while—will always eventually approach the homogeneous state. This leads to a new type of local excitability in a spatially extended medium involving transient traveling wave solutions as invariant sets. Both the model and the initial conditions are motivated by the pathophysiology of migraine and clinical observations [11–13].

2.2 Cortical Activity Pattern and Neurovascular Coupling During Migraine

Migraine is characterized by recurrent episodes of head pain, often throbbing and unilateral. In migraine without aura (MO), headache attacks are usually only associated with nausea, vomiting, and sensitivity to light, sound, and even movement [14]. Migraine with aura (MA) involve in addition, but also rarely exclusively, neurologic symptoms (aura) that are caused by waves of cortical spreading depression [12, 15–17].

Spreading depression (SD) is a massive but temporary perturbation of ion homeostasis due to seizure-like discharges of neurons. The ion concentrations are usually kept with a narrow range of acceptable limits, while during SD ion concentrations can change by over one order of magnitude to a nearly complete depletion of transmembrane chemical gradients. The ignition of this perturbed ionic balance can spread by diffusion of ions in the extracellular space. Essentially, SD is a slow (about 3 mm/min = 50 μm/sec) reaction-diffusion wave in the approximate 2D cortical sheet of gray matter.

The cortical tissue SD traverses is functionally impaired causing the neurological migraine aura symptoms, like visual hallucinations [11]. Whether SD is also a key to the subsequent headache phase is an open question, in particular, in cases of migraine without aura (MO). If SD occurs in MO, the massive ionic imbalance must remain clinically silent [18, 19] or—by definition of diagnostic criteria—neurological symptoms must last less than 5 min.

The aura is usually, though not always [20], before the headache phase and last usually less than one hour [21]. The transient nature of SD (and thus the migraine aura) poses challenging problems in clinical observability. Objective measures by means of noninvasive imaging are in particular difficult to access when clinical symptoms do not indicate the aura phase, i.e., if SD stays silent. Attacks observed with noninvasive imaging are usually triggered, which also could cause a trigger-specific bias. Only one well-documented case of a spontaneous migraine headache supports the still contested notion of “silent aura” [22]: Blood-flow changes where observed that were most likely the result of SD.

SD in the cortex is accompanied by a pronounced increased regional cerebral blood flow for about 2 min and a long lasting (∼2 h) decrease [23]. This naturally raises the question of the physiological relevance of these blood-flow changes. Are they just an epiphenomenon that can be used to indirectly measure SD or do these changes participate in the pattern forming mechanism of SD? We suggest that the initial increase in cerebral blood flow for about 2 min is effectively an inhibitory feedback mechanism for SD. The hyperemic phase (increased blood flow) engulfs large regions of the human cortex [16], while, as we further investigate in this study, the massive ionic imbalance directly due to SD is much more limited in extent [12, 13]. This suggests that the inhibitory feedback by neurovascular coupling is a global control mechanism of the neural reaction-diffusion system.

The pattern forming interactions are described in the next section from a mathematical perspective. We end this section by providing some background of the physiological bases of the spatial mismatch between the more globally increased blood flow (hypermia) and more spatially confined ionic imbalance during SD.

The neural activity and subsequent changes in blood flow are closely coupled called neurovascular coupling. Not only the magnitude but also the spatial location of blood flow changes reflect neural activity, a fact that is used in noninvasive brain imaging techniques such as functional magnetic resonance imaging (fMRI). SD, however, is a pathological state in which this coupling is to some degree impaired, in particular evoking long lasting decreases phase (oligemia). To measure the spatial confinement of SD in human cortex during a migraine attack with fMRI, not merely hypermia, but also several other characteristic neurovascular events were used that resemble SD, among others [12]: the initial hyperemia with a characteristic duration followed by oligemia with recovery to baseline, the characteristic velocity of these events, and a concurrent recovery of stimulus driven activation (see Fig. 2).

While the pathological activity of neurons during SD is mainly characterized by a depressed state with literally no activity—hence the name, in the tissue ahead of SD, high-frequency activity and increased synaptic noise has been recorded [24]. The tissue surrounding the current location of SD is functionally connected through lateral neural networks with neurons that undergo seizure-like discharging at the rising front of SD. This provides an electrical signal transmission pathway several orders faster than SD, that is, in a first approximation an instantaneous connection. As firstly suggested by Wilkinson [25], the feed-forward and feedback cortical circuitry can also explain a more global hypermia by neural and synaptic activation in adjacent cortical areas, which might be mistaken as the area SD traverses, when only hypermia is used to estimate the spatial extend of SD in human cortex during a migraine attack with noninvasive imaging.

Increased neural activity and the resulting hypermia in an extended area surrounding SD can make the tissue that is yet to be recruited into the depressed SD state less susceptible to it and possible completely protect it from SD. Therefore, this mechanism could provide a neuroprotective mechanism that we mimic by the inhibitory mean field feedback. It is reasonable to assume that the larger SD spreads out, the large the increase in neural activity in the surrounding tissue and subsequent hypermia with neuroprotective effect. The coupling between increased neural activity and subsequent changes in cerebral blood flow has a significant time delay in the order of seconds, which we ignore for the sake of simplicity in our model.

3.1 Model Equations

The parameter ε separates the timescales of the dynamics of the activator u and the inhibitor v, and ε is taken to be small. In the present work, we use a value of $\epsilon =0.04$. The parameter β is a threshold value which determines from which activator level on the inhibitor concentration is rising. The local dynamics of Eq. (1) (i.e., without the diffusion term) is oscillatory for $|\beta |<1$ and excitable for $|\beta |>1$. At $|\beta |=1$, the local dynamics undergo a supercritical Hopf-bifurcation. We choose a value of $\beta =1.1$ throughout this work. To integrate Eq. (1), we used a simulation based on spectral methods [27] and adaptive timestepping.

where H is the Heaviside function and we chose $u_0=0$.

Equations (1) are a paradigmatic model of an excitable medium even beyond neuroscience [28]. They possess a stable homogeneous solution as well as stable excited states (pulses, spirals, or double spirals) cf. [29, 30]. The boundary separating the basins of attraction of these types of solution is given by the stable manifold of the so-called “nucleation-solutions” (NS) whose stability is of saddle-type with one unstable direction; see Fig. 1. These nucleation-solutions are localized areas of excitation, which are traveling at uniform speed without changing shape. The size of these solutions, in the sense of Eq. (2), depends on the parameters β and ε. In Fig. 3, the size S of these nucleation solutions is plotted against β. This solution branch is also called ∂R and the parameter value for which it diverges is called $\partial R_{\mathrm{\infty }}$ or the “rotor boundary” (see next section). Of course, one could also use a measure different from S for visualizing the branch ∂R, e.g., the propagation speed of the nucleation solution.

To obtain solutions lying on ∂R, we used a pseudo-continuation procedure, which is described below.

where K and ${\beta }_0$ are control parameters.

The aim of the present work is to shed light on the transient behavior, occurring when the control line, Eq. (3) is close to ∂R but does not intersect it as the ones depicted in Fig. 3.

With this procedure, it is not possible to obtain $(\beta -S)$-pairs below a certain value of S. The reason is that the system (with mean-field control) is undergoing a saddle-node bifurcation as visualized in the right of Fig. 3. For the purpose of visualization for this saddle-node bifurcation, we fitted the locus of the measured branch of nucleation solutions (in $\beta -S$ space) to a function of the form $\beta =a+\frac{b}{cS+S^2}$ (where $a,b$, and c are the fit parameters). We have not only used this function for visualizing the saddle-node bifurcation, but we also deduced an approximate β value for the rotor boundary $\partial R_{\mathrm{\infty }}$ by letting $S\to \mathrm{\infty }$.

3.2 Effect of Mean Field Inhibitory Feedback Control

The diversity of the behavior of traveling waves in two spatial dimensions was studied in canonical models depending on the two generic parameters β and ε in Eq. (1), which determine the parameter plane of excitability without mean field inhibitory feedback [32]. In those media, patterns of discontinuous (open ends) and spiral-shaped waves are used to probe excitability. These spiral patterns are closely related to the discontinuous, localized transient waves we propose in our model. In fact, the effect of mean field inhibitory feedback control can best be understood, if we compare these patterns in models with and without this control.

Before we further consider the effect of mean field inhibitory feedback control, we have to describe the behavior of continuous waves (closed wave fronts without open ends) when excitability is decreased, e.g., by increasing β, without mean field inhibitory feedback control. This will be important if we want to understand the fate of any solution, discontinuous or not, under mean field feedback control. Unbroken plane waves propagate persistently even if the parameters are chosen in the subexcitable regime until β reaches a value called the propagation boundary ∂P. At this boundary, the medium’s excitability becomes too weak for continuous plane waves to propagate persistently. The boundary ∂P in parameter space marks also a saddle-node bifurcation at which a planar traveling wave solution collides with its corresponding nucleation solution. Note, that the planar wave is essentially a pulse solution in 1D and the nucleation solution in 1D is called the slow wave [35].

In Fig. 3(left), both the rotor boundary $\partial R_{\mathrm{\infty }}$ and the propagation boundary ∂P are shown in a bifurcation diagram for the excitable medium described by Eqs. (1). We chose β as the bifurcation parameter and follow (see previous section) the branch ∂R of the unstable nucleation solution (NS) whose stable manifold separates the basins of attraction of the homogeneous state and a spiral wave (with two counter-rotating open ends). The unstable manifold of NS consists of the two heteroclinic connections, one to the stable homogeneous state and the other to the traveling wave solution (see Fig. 1). The order parameter on the ordinate in Fig. 3 is the surface area S inside the isoclines at $u=u_0=0$ of the traveling wave solutions; see Eq. (2).

The mean field control that we introduce by Eqs. (2)–(3) establishes a linear feedback signal of the wave size S to the threshold β. With this linear relation, we introduce two new parameters, the coupling constant K and ${\beta }_0$, the threshold parameter for the medium without an excited state ($S=0$). Note that the parameter ${\beta }_0$ can be also seen as the sum of two threshold values, the former β in Eq. (1) and an offset coming from the new control scheme. While the introduction of the control introduces two new parameters ${\beta }_0$ and K, at the same time β becomes dependent upon the control, so that we have a total of three free parameters.

We chose ${\beta }_0$ as the new bifurcation parameter in the bifurcation diagram for the completed reaction-diffusion model with mean field coupling described by Eqs. (1)–(3); see Fig. 3(right). This diagram is a sheared version of the one without mean field coupling in Fig. 3(left). While it is a trivial fact that the linear relation in Eq. (3) describes an affine shear of the axes $(\beta ,S)$ of the bifurcation diagram in a to the new axes $({\beta }_0,S)$ in b, the fact that the branch ∂R of the nucleation solutions can be mapped this way is not. Firstly, this relies on the way we introduce the feedback term. It just adds a constant value to the old bifurcation parameter β, if the solution under consideration is stationary. Therefore, any stationary solution must exist in both diagrams being just sheared branches. The same still holds true for traveling wave solutions that are stationary in some appropriate comoving frame, for instance, $\xi =x-ct$ with speed c. However, not much can be said about the stability of such solutions, when we introduce the mean field feedback term.

The branch ∂R of the formerly unstable nucleation solutions NS (Fig. 3(left)) folds in Fig. 3(right) such that two solutions coincide for a given value of ${\beta }_0$ until they collide and annihilate each other at a finite value of $S\approx 5.5$ for $K=0.003$. For the fixed value of $K=0.003$, the upper branch consists of stable traveling wave solutions in the shape of a wave segment, while the lower branch belongs to the corresponding nucleation solutions of these wave segments, as schematically shown in Fig. 1b. The fact that the upper branch is stable was confirmed by numerical simulations (cf. Sect. 3.1 and Video 1). Larger K, that is, a less steep control line in Fig. 3(left) can be seen as a “harder” control, because a small given change in S leads to larger variations in the effective parameter β. As a consequence, it is difficult to stabilize lower part of the branch corresponding to small traveling wave segments in numerical simulations by means of this control.

The choice of the parameter regime given by Eq. (4), which shows only transient localized waves for this model and leads to a globally stable homogeneous state as the only attractor, is straightforward given the branch ∂R. In this sense, we designed the model to exhibit transient localized waves due to a bottleneck—or ghost behavior—after the saddle-node bifurcation.

3.3 Initial Conditions

We need an appropriate sampling of initial conditions for Eqs. (1)–(3), ideally being equidistantly sampled in some distribution. The set of all initial conditions for this system does not—to our knowledge—carry a helpful mathematical structure which allows us to achieve this aim easily. In order to attack this problem, we turned to the physiological motivation of the chosen model explained in Sect. 2.

A set of initial conditions should naturally reflect plausible spatial perturbations of the homogeneous steady state of the cortex. This can be achieved by defining localized but spatially structured activity states on large scales of the order of millimeters. Such pattern are obtained from cortical feature maps (see Fig. 4) by sampling three parameters (scaling, depth, and size) that define patches of lateral coupling in theses maps. A fourth parameter (excess) determines the amplitude of the perturbation. In the following, we first describe the rational behind using a cortical feature map and then the sampling.

3.3.1 Rational to Use Cortical Feature Map

We focus on a cortical feature map in the primary visual cortex (V1) called the pinwheel map. V1 is located at the occipital pole of the cerebral cortex and is the first region to process visual information from the eyes. Migraine aura symptoms often start there or nearby where similar feature maps exist.

In V1, neurons within vertical columns (through the cortical layers) represent by their activity patterns edges, elongated contours, and whole textures “seen” in the visual field. This representation has a distinct periodically microstructured pattern: the pinwheel map. Neurons preferentially fire for edges with a given orientation and the preference changes continuously as a function of cortical location, except at singularities, the pinwheel centers, where the all the different orientations meet [36, 37].

Iso-orientation domains form continuous bands or patches around pinwheels and, on average, a region of about 1 mm² (hypercolumn) will contain all possible orientation preferences. This topographical arrangement allows one hypercolumn to analyze all orientations coming from a small area in the visual field, but as a consequence, the cortical representation of continuous contours in the visual field is depicted in a patchy, discontinuous fashion [38]. In general, spatially separated elements are bound together by short- and long-range lateral connections. While the strength of the local short-range connection within one hypercolum is a graded function of cortical distance, mostly independent of relative orientation [39], long-range connections over several hypercolumns connect only iso-orientation domains of similar orientation preference [40, 41]. Even nearby regions, which are directly excitatory connected, have an inhibitory component through local inhibitory interneurons and this is likely be used to analyze angular visual features such as corners or T junctions [39].

Given the arguments above, we can now obtain localized yet spatially structured activity states on the scale we aim for as initial conditions by using iso-orientation domains that form continuous patches around pinwheels and extend in a discontinuous fashion over larger areas. In these patches, neural activity can get into a critical mode, like neural avalanches [42] that would locally perturb the ionic homeostasis as exemplary shown in Fig. 5 (lower left).

5.1 Canonical Model and Free Parameters for Weakly Excitable Media

Central to our approach is the localization resp. spatial confinement of the transient traveling waves. Reaction-diffusion waves would engulf all of the medium, if formed in a two-variable system with only one activator and one inhibitor with the system’s parameters in the appropriate regime. In contrast, localized traveling waves indicate a demand-controlled excitability. Similar ideas to obtain localized traveling, though not transient, waves have been introduced in various contexts, for instance, an integral negative feedback or a third, fast diffusing inhibitory component for moving spots in semiconductor materials, gas discharge phenomena, and chemical systems [31, 44–46]. Furthermore, in neural field models [47], localized two-dimensional bumps are studied [48–50] in integrodifferential equations (without diffusion) in the context, for example, of memory formation [51]. Localized structures have also been discussed in the context of cortical spreading depression (SD) in migraine before, in particular a model with narrowly tuned parameters that shows transient waves [13, 52, 53] and a model with mean field feedback control that allows for localized waves [30]. But it is for the first time now that a model is presented in which wave phenomena occur that are both localized and transient, so that a variety of new questions that are controversially discussed in migraine research [18, 54–56] can be addressed. A central question is of course, which level of detail a model of SD needs to investigate localization of SD and the transient response properties.

Physiological detailed models of SD are given by conductance-ion-based models with 9 to 29 dynamical variables in various (∼200) electrically coupled neural compartments [57–59]. They usually do not include lateral space—the compartments extend in the vertical direction to model the apical dendritic tree, that is, these models are not spatially extended to describe an excitable medium. In fact, this lateral extension is far from straightforward. Naively adding diffusion to the extracellular dynamical ion concentrations is possible but does not reflect the necessary detail that needs to be considered to take the spatial continuum limit. Neural field models, for example, describe this limit [47]. The first part of the global inhibitory feedback by neurovascular coupling, i.e., the fast spread of neural hyperactivity that initiates hypermia—as we suggested in this study—can be modeled by neural fields. The second part of this neurovascular coupling, the actual feedback signal needs also some physiological detail. This cannot easily be incorporated because neural field models are rate-based or activity-based while the detailed SD models [57–59] start with a conductance-based Hodgkin–Huxley approach and include ion-based dynamics but do not describe the dynamics on a rate-based foundation.

Even if a SD detailed tissue-model, that is, an excitable medium with the same physiological detail as in the single cell models of SD [57–59], were available, it would require enormous computer capabilities to calculate from the activity of single cells on the scale of milliseconds the macroscopic patterns on the spatial scale of several centimeters and on a temporal scale of up to one hour. Quite apart from the fact that five orders of magnitude in both the spatial and temporal scales suggest that the macroscopic phenomena require their own level of description in some form of effective medium theory.

We suggested that the primary objective in research relating SD to migraine should be to obtain a measure of the noxious signatures that are transmitted into the meninges during SD [60, 61]. Canonical reaction-diffusion models seem to be at least one way to approach this objective. A further advantage of such canonical models lies in the fact that they allow insight in the phase space structure of the whole class of models they represent, as schematically shown in Fig. 1. Therefore, in the following, we will argue in which sense our model is canonical for the problem we attack.

Generally speaking, an excitable medium is a spatially extended system with a stable homogeneous steady state being the quiescent state and one or many excited states that develop after a sufficient perturbation from the quiescent state (Fig. 1a). The excited states are traveling wave solutions that propagate with a stable profile of permanent shape (possibly with some temporal modulation, such as breathing or meandering). To study generic features of an excitable medium, the simulations are often carried out in the reaction-diffusion system given by Eqs. (1), the popular FitzHugh–Nagumo kinetics. Originally, the FitzHugh–Nagumo kinetics were a caricature of the electrophysiological properties of excitable membranes [62, 63], but these equations with $D=0$ became a canonical model of local excitability of type II (based on Hopf bifurcation, either supercritical with subsequent extremely fast transition to a large amplitude limit cycle, named canard explosion [64], or subcritical [65]). For $D\ne 0$, the FitzHugh–Nagumo kinetics became also a canonical model for spatial excitability [32]. Sometimes diffusion in the second inhibitory species is included, which we do not consider here. Because we investigate transient behavior originating from a high threshold regime (toward weak excitability), the classification of local excitability in types I and II (based on the transition at vanishing threshold, i.e., into the oscillatory regime) is not relevant. Furthermore, it is not clear whether this classification carries over in a meaningful way to the dynamics of spatially extended systems.

We consider the set of Eqs. (1) as canonical for two reasons. First, because the u (activator) equation of Eq. (1) has the simplest polynomial form of bistability. Note that for this reason this activator equation was originally suggested by Hodgkin and Huxley as the first mathematical model of the potassium dynamics in SD. It was published by Grafstein, who also provided experimental data supporting such a simple reaction-diffusion scheme for the front dynamics in SD [66]. Second, the inhibitor equation of Eq. (1) has a linear rate function, in fact, the rate function is only a function of the activator u. This is the simplest inhibitor dynamics needed for pulse propagation. By neglecting an additional linear term $-\gamma v$ in the inhibitor rate function, we limit the origin of excitability to the case of a supercritical Hopf bifurcation with subsequent canard explosion and avoid the bistable regime that exits in the subcritical case. The subcritical Hopf bifurcation occurs only in a narrow regime when γ is close to 1 and β close to 0. We have tested some simulations with $\gamma =0.5$ with similar results.

As a consequence of our assumptions about the model being in this canonical form, only two free parameters exist, β which is associated with the threshold and the ε, the time scale separation of activator and inhibitor dynamics. Of course, the choice of parameters can be quite different, a common choice is α in the cubic rate function $f(u)=u(u-\alpha )(u-\alpha )$ but there are only two free parameters or two equivalent groups of parameters. So, there are the same bifurcations in the parameter planes $(\epsilon ,\beta )$ or $(\epsilon ,\alpha )$, but to map the dynamics between equivalent groups of parameters might involve changes in time, space, and concentrations scales.

In particular, the question of how the incidence of MA is reflected in the distance to the saddle-node bifurcation, involves a measure on the parameter space whether is $(\epsilon ,\beta )$, $(\epsilon ,\alpha )$, or any other parameter plane. We have previously suggested to get such measures from pharmacokinetic-pharmacodynamic models [53].

5.2 Application to Migraine Pathophysiology

We suggest a qualitative congruence between the prevalence of MO and MA with the statistical properties we found in the transient response properties. We do not suggest that all migraine attacks are related to SD nor that pain formation in MA is exclusively caused by SD. Rather that SD is one pathway of pain formation in both symptom-based subtypes MO and MA. We refer to this pathway as the “spreading depression”-theory of migraine [17]. The “migraine generator”-theory (MG) is for various reasons not less plausible [54]. It assumes a dysfunction in a central pattern generator in the brainstem that modulates the perception of pain. Some of the seemingly conflicting and controversially discussed evidence is probably resolved when one considered the basis of the classification of migraine subforms. We currently have a symptom-based classification for migraine with possibly overlapping etiologies for individual subforms. In the light of an etiology-based classification with possibly overlapping symptoms the conflicts seem less puzzling to us. To further resolve this, we investigated the interplay of SD and MG and suggested to unify these approaches within a network theory [61].

In the remainder of this section, we focus first on migraine pain and then on the migraine aura.

5.2.1 Migraine Pain

The cortex is not pain sensitive. Therefore, SD in the cortex cannot explain the headache phase in migraine. There are detailed investigations how SD in the cortex can cause pain via pain sensitive intracranial tissues and subsequent activation in the trigeminal nucleus caudalis in the brainstem [67, 68], but cf. [69, 70]. The qualitative congruence between the prevalence of MO and MA with the statistical properties we found in the transient response properties is based on the following assumption on the geometrical layout of the cortex and the pain sensitive cranial tissues (Fig. 10): In the initial phase of cortical SD, with increased blood flow (hyperemic phase), a local release of noxious substances (ATP, glutamate, K⁺, H⁺) are thought to diffuse outward in the direction perpendicular to the cortex into “the leptomeninges resulting in activation of pial nociceptors, local neurogenic inflammation, and the persistent activation of dural nociceptors, which triggers the migraine headache” [71], but for issues concerning the blood brain barrier system cf. [72].

If diffusion vertical to the affected cortical area is critical, size and shape of this area should play a critical role; see Fig. 10. This suggests that SD waves activate nociceptive mechanisms dependent upon a sufficiently large instantaneously affected cortical area, i.e., large MIA and, as stated before, the primary objective should be to obtain a measure of the noxious signatures that are transmitted into the meninges during SD.

5.2.2 Migraine Aura

The aura phase, on the other hand, must clearly correlate with long duration of SD and a large enough cortical surface area being affected during the course of SD to notice neurological deficits. In particular, because the very noticeable visual symptoms often start where the cortical magnification factor is large, so that only if they move into regions of lower magnification they get significantly magnified by the reversed topographic mapping [52]. The seemingly contested notion of MO (migraine without aura) with silent aura can also be resolved when the spatiotemporal development of SD is taken into account.

The connection between MIA and TAA as well as MIA and ED in our model is shown in Fig. 11. It shows that in the range of high MIA the average values for TAA and ED are becoming smaller. From Fig. 11, we can also read off that the range with the most events is in the regime of relatively high MIA (around 30) and significantly after the peak of ED resp. TAA. Moreover, in the range with most events, the correlation coefficient $\mathrm{r}(\mathrm{MIA},\mathrm{ED})$ is always negative and the correlation coefficient $\mathrm{r}(\mathrm{MIA},\mathrm{TAA})$ is mostly negative. All these effects are stronger, the closer the control line is located to the saddle-node bifurcation.

From these statistical correlations between MIA and TAA resp. ED and the distribution of the number of events, one could speculate that cases of MA are more rare and the quality of the headache in these cases might be less severe. This is exactly what has been reported in the medical literature [73].

While the number of events with high ED and high TAA is influenced by the distance of the control line to the saddle node bifurcation, the number of events with high MIA is much lesser affected. So, in a way, the distance to the saddle node bifurcation controls the prevalence of MA in our model, while the prevalence of MO is not much affected.

5.3 Model-Based Control by Neuromodulation

We briefly discuss model-based control and means by which neuromodulation techniques may affect pathways of pain formation and the aura phase.

The emerging transient patterns and their classification according to size and duration offer a model-based analysis of phase-dependent stimulation protocols for noninvasive neuromodulation devices, e.g., utilizing transcranial magnetic stimulation (TMS) [74], to intelligently target migraine. For instance, noise is a very effective method to drive the system back into the homogeneous steady state more quickly; see Fig. 12. In general, responses of nonlinear systems to noise applied when the system is just before or past a saddle-node bifurcation are well studied. Before the saddle-node on limit cycle bifurcation, the phenomenon of coherence resonance (CR) describes that a certain amount of noise makes responses most coherent [75]. Behind the saddle-node bifurcation on a limit cycle, the time the flows spend in the bottleneck region of the ghost is shortened [76]. However, noise would, according to our model, mainly positively affect ED and TAA, that is, the aura, while it could even worsen the headache, if applied early during the nucleation and growth process. Therefore, TMS with noise stimulation protocols, which are currently investigated, should be applied only some time after first noticing aura symptoms.

Headaches are not generally considered appropriate for invasive neurosurgical therapy, but when all else fails—preventives, abortives, and pain management—invasive brain stimulation techniques are also considered, e.g., occipital nerve stimulation (ONS) [77, 78]. So model-based control will become increasingly important. Also, the importance of modeling related epileptic seizure dynamics as spatiotemporal transient patterns has been suggested in a recent paper [79]. Model-based control of Parkinson’s disease, is already considered, yet Schiff remarks quite correctly [80]: “It seems incredible that the tremendous body of skill and knowledge of model-based control engineering has had so little impact on modern medicine. The timing is now propitious to propose fusing control theory with neural stimulation for the treatment of dynamical brain disease.”

We suggest to consider migraine as a dynamical disease that could benefit from model-based control therapies.