In this study, we are interested not only in the pattern forming mechanism of transient localized traveling waves, but also in the generic spatio-temporal properties of such waves. To this end, we design a canonical reaction-diffusion model (Sect. 3.1) augmented by an inhibitory mean-field feedback control (Sect. 3.2). The use of this model is motivated as described in the previous section. The phenomena our attention will be focused on in the next section are nucleaction, growth, and subsequent shrinking of the transient wave segments. To study statistical properties of these events, we must consider the set on initial conditions (see Sect. 3.3) over which statistical analysis are performed (Sect. 4), because these are features of transient dynamics.
3.1 Model Equations
The canonical model for excitable media are the well-known FitzHugh–Nagumo equations [26] with diffusion in the activator variable:
(1)
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 . 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 and excitable for . At , the local dynamics undergo a supercritical Hopf-bifurcation. We choose a value of throughout this work. To integrate Eq. (1), we used a simulation based on spectral methods [27] and adaptive timestepping.
We define the (instantaneous) wave size as the area with activator level u over a certain threshold :
(2)
where H is the Heaviside function and we chose .
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 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.
Making the parameter β dependent on the wave size S adds a mean field control to the system.
where K and are control parameters.
This equation defines a straight line in -space, we call the control-line. If the control line intersects ∂R (in -space), the point of intersection with higher S is stabilized, cf. [31]. This is visualized in a movie; see Supplementary Material Video 1. This can be understood intuitively, as the stable manifold of a point on ∂R separates the attraction basin of the homogeneous solution for which S shrinks until and the attraction basin of rotating spirals with growing . Imagine the system to be on ∂R, that is, showing a nucleation solution as discussed above. If the current state is perturbed to have slightly smaller S, in the uncontrolled system we would have entered the attraction basin of the homogeneous solution (cf. Fig. 3). The control however forces the system to stay on the line defined by Eq. (3). If this line intersects ∂R, a slightly smaller S makes the control adjust the value of β to smaller values, taking the system into the attraction basin of the spiral waves, where S will grow. The same process happens with different signs if the current state is perturbed to have slightly smaller S thus in effect stabilizing the (one unstable direction of) the nucleation solution.
Additional file 1: Transient localized wave patterns and their application to migraine. (AVI 2 MB)
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.
To account for the imprecision in ∂R of the simulation and the exact ∂R, we measured the ∂R in our simulation using a pseudo-continuation procedure. For this, we set the control such that it intersects ∂R, and thus stabilize an otherwise unstable solution on it. Letting the simulation run until the system has stopped fluctuating, saving the -pair, changing the control slightly and doing things over yields points of ∂R in our system. From this measured ∂R and the propagation boundary ∂P, inferred from continuation in 1D, we chose 3 suitable control lines which were used for simulations in this work:
(4)
With this procedure, it is not possible to obtain -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 space) to a function of the form (where , 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 by letting .
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.
In the design of our model, we make use of the fact that in a model without mean field inhibitory feedback control spiral waves do not curl-in anymore, but become half plane waves at a low critical excitability, called the rotor boundary [33, 34]. Beyond the rotor boundary lies the subexcitable regime in which discontinuous waves start to retract at their open ends and any discontinuous wave is transient and will eventually disappear (see Videos 2 and 3). In other words, spirals do not exist beyond . The boundary marks a saddle-node bifurcation at which discontinuous spiral waves collide with their corresponding nucleation solution. This leads to the key idea of our model, namely to introduce mean field inhibitory feedback control. A linear mean field feedback control moves this saddle-node bifurcation toward distinct localized wave segments with a characteristic form (shape, size) and behind this bifurcation these waves become transient objects; see Fig. 1, Fig. 3, and Video 4.
Additional file 2: Transient localized wave patterns and their application to migraine. (AVI 2 MB)
Additional file 3: Transient localized wave patterns and their application to migraine. (AVI 464 KB)
Additional file 4: Transient localized wave patterns and their application to migraine. (AVI 958 KB)
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 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 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 , the threshold parameter for the medium without an excited state (). Note that the parameter 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 and K, at the same time β becomes dependent upon the control, so that we have a total of three free parameters.
We chose 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 of the bifurcation diagram in a to the new axes 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, 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 until they collide and annihilate each other at a finite value of for . For the fixed value of , 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).
3.3.2 Sampling of Patterns in Cortical Feature Maps
In [43], the authors analyzed the design principles that lie behind the columnar organization of the visual cortex. The precise design principles of this cortical organization is governed by an annulus-like spectral structure in Fourier domain [36, 43], which is determined by mainly one parameter (scaling), that is, the annulus width. The parameter depth reflects the tuning properties of orientation preference or we can also interpret this as the range of orientation angles that we consider within the iso-orientation domain. The third parameter reflects the distance long-range coupling ranges before it significantly attenuates.
These design principles can be exploited and a procedure can be designed to construct maps with the same properties. The constructed maps come very close to the maps found in brains of macaque monkeys (see [43] and references therein).
To construct initial conditions from these maps we used a procedure that uses four control parameters and is visualized in Fig. 4. The details are as follows: A pinwheel map is a function that maps our two-dimensional plane to the interval . We construct such a map using the procedure in [43]. During construction, we can choose the scaling of the map. This is our first parameter. After constructing this map, by means of a Gaussian, we choose a range of orientations that is excited. Mathematically speaking, this is the concatenation of the Gaussian distribution with the pinwheel map. This gives the next parameter, namely the width of the Gaussian that selects the angles, we call that parameter the depth. The next step is to constrain the generated pattern spatially by multiplication with another Gaussian, which is defined on the plane P and chosen to be rotationally symmetric. The width of this Gaussian gives rise to the third parameter, the size of the pattern. Finally, we multiply the pattern by a certain amplitude, which is chosen such that the integral of the pattern over the plane gives a chosen number, which constitutes the fourth parameter, we called the excess.
Finally, initial conditions are generated by setting the plane to the (stable) homogeneous state and then adding the generated pattern, which represents increased activity like in neural avalanche, to the activator variable u, which represents the ionic imbalance most notably the extracellular potassium concentration.
In a first run, we scanned the space spanned by the four parameters coarsely. We used the marginal distributions of the number of solutions with an excitation duration (ED) >0 with respect to the parameters to decide how densely to sample the parameter space in the final run.