Transient Localized Wave Patterns and Their Application to Migraine

Markus A. Dahlem 0 1 Thomas M. Isele 0 1 0 T.M. Isele Institute of Theoretical Physics, Technische Universitt Berlin , Berlin, Germany 1 M.A. Dahlem ( ) Department of Physics, Humboldt-Universitt zu Berlin , Berlin, Germany Transient dynamics is pervasive in the human brain and poses challenging problems both in mathematical tractability and clinical observability. We investigate statistical properties of transient cortical wave patterns with characteristic forms (shape, size, duration) in a canonical reaction-diffusion model with mean field inhibition. The patterns are formed by ghost behavior near a saddle-node bifurcation in which a stable traveling wave (node) collides with its critical nucleation mass (saddle). Similar patterns have been observed with fMRI in migraine. Our results support the controversial idea that waves of cortical spreading depression (SD) have a causal relationship with the headache phase in migraine and, therefore, occur not only in migraine with aura (MA), but also in migraine without aura (MO), i.e., in the two major migraine subtypes. We suggest a congruence between the prevalence of MO and MA with the statistical properties of the traveling waves' forms according to which two predictions follow: (i) the activation of nociceptive mechanisms relevant for headache is dependent upon a sufficiently large instantaneous affected cortical area; and (ii) the incidence of MA is reflected in the distance to the saddle-node bifurcation. We also observed that the maximal instantaneous affected cortical area is anticorrelated to both SD duration and total affected cortical area, which can explain why the headache is less severe in MA than in MO. Furthermore, the contested notion of MO attacks with silent aura is resolved. We briefly discuss model-based control and means by which neuromodulation techniques may affect pathways of pain formation. 1 Introduction The undoubtedly most fundamental example of transient dynamics is the phenomenon of excitability, that is, all-or-none behavior. Shortly after transient response properties of excitable membranes were classified into two classes [1], it was also explained in a detailed mathematical model how excitability emerges from electrophysiological properties of such membranes in the ground-breaking work by Hodgkin and Huxley [2]. Two features are central and are by no means exclusive to biological membranes but shared by all excitable elements. Firstly, the inevitable threshold in any all-or-none behavior requires nonlinear dynamics. Secondly, the transient response of the system to a super-threshold stimulation eventually has to lead back to a globally stable steady state after some large phase space excursion. This indicates global dynamics, that is, dynamics involving not only fixed points and their local bifurcations but more complex invariant sets, for instance periodic orbits that collide with fixed points. An excitable element is in some sense the washed-up brother of the relaxation oscillator: When the threshold vanishes, a single excitable element usually becomes a simpler behavedand much longer knownrelaxation oscillator [3]. Vice versa, when a saddle-node disrupts a limit cycle and introduces a threshold, the sustained oscillations are reduced to long transient responses after perturbations, that is, the dynamics becomes excitable. In this study, we utilize a similar scenario to disrupt sustained traveling wave solutions in a spatially extended medium such that only transient waves occur. We investigated statistical properties of these transient waves to gain a dynamical understanding of spontaneous episodes in migraine. We will briefly introduce concepts of excitable elements and excitable media in two-variable reaction-diffusion systems. While we also introduce migraine, the view of migraine as a dynamical disease is more elaborated in the discussion in Sect. 5. A particular focus is set on the idea to introduce an global inhibitory feedback that is also studied in various other systems outside the neurosciences and also in neural field models. Section 2 sets the stage for our canonical model introduced in Sect. 3 from were we proceed to our results on the statistical properties of transient waves in Sect. 4. 2 Motivation of a Macroscopic Model for Migraine Aura 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 threevariable models, which have been investigated in great detail [7]. However, the twovariable 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 inf (...truncated)