A: (From left to right) Examples of spatial and spatio-temporal models. Turing patterns with spots and stripes, both in coupled Schnakenberg elements; Formation of target waves from a central pacemaker and an established spiral wave, both in coupled FitzHugh-Nagumo oscillators. B: Snapshots of spiral wave patterns of different biological systems: (left to right) cAMP signaling in a colony of Dictyostelium discoideum, local contraction in cardiac monolayer cultures of newborn rats, MinD protein density in a lipid bilayer, and simulated cytokine levels in a two-dimensional cellular network. See Acknowledgments for image sources. C (top row): The rules for updating the minimal model of cellular automata in three steps result in the formation of spiral waves when applied to an open wavefront (composed of a layer of excited cells, shown in black, and an adjacent layer of refractory cells, shown in gray). Bottom row: a similar digital experience for the model of [6,7]. The influence of variability on spiral wave patterns is best visualized in an event-driven perspective in which target wave centers and spiral wave peaks are considered “spatio-temporal events”. The temporal sequences and spatial distributions of these model events can then be compared to the spatial distribution of cellular properties. A group of oscillating elements is a likely candidate for the emission of target waves. The additional pattern events emerging from this region are then a consequence of the proximity of other pacemaker regions and the “roughness” of the propagating wavefront, which is the result of the excitability sequences encountered by the propagating wavefront – to the point where the target waves decay into spirals in a possible mechanism. In the work of the authors of , the strength of a regulatory feedback loop is related to the spatial density of spiral wave patterns in dictyostelium cell colonies. By studying mutants in key components of the regulatory feedback loop, the authors systematically varied this intrinsic parameter and observed how spatio-temporal patterns changed accordingly.
In order to illustrate more clearly the systematics of their results, the authors of  simulate models with a simple model based on cellular automata in which the feedback force (i.e. the increase in excitability dependent on the cAMP pulse) appears as an explicit parameter. Their observability is the spatial frequency of spiral waves, which they obtain by counting phase singularities in their spatio-temporal models. Remarkably, the intermediate feedback force found in wild-type cells turns out to be an optimal (i.e. minimal) number of phase singularities compared to mutants with higher and lower feedback strength, respectively. In extreme cases, where feedback is constantly lacking, no stable spiral wave pattern develops. This observation has two implications. First, the size of the aggregation area is optimized by changing the feedback force as a function of the pulse. For the wild type, this allows optimally sized attraction basins for the process of successive aggregation of cells, which allows the multicellular organism to form spores, thus completing the development cycle of the dictyostelium. Second, the geometry of the wave is determined by the feedback force. Spiral waves seem to be preferred by the system to target waves, although both wave geometries produce fruiting bodies for the experimental system.
The biological advantage of spiral-based signaling is that spirals are self-sustaining continuous structures that maintain their stability to some extent outside of the excitable regime. This allows the aggregation process to be maintained under developmental and environmental changes. In contrast, target waves require periodic activity of oscillating regions (i.e., pacemakers). As in many of these modeling situations, one now has the choice to explore the biological details of these spatio-temporal models or, alternatively, to work towards understanding the generic characteristics. The second model, which will be discussed in detail below, is therefore an arrangement of FitzHugh-Nagumo coupled oscillators as a generic model for spatio-temporal models resulting from excitable dynamics [53,54]. Spatio-temporal models often arise from self-organization from local interactions. In biology, the resulting models are also subject to the influence of systematic differences between system components (biological variability). This regulation of spatio-temporal models by biological variability is the subject of our review. We discuss some examples of correlations between cellular properties and self-organized spatio-temporal patterns and their relevance to biology. Our main illustrative example will be cAMP spiral waves in a colony of Dictyostelium discoideum cells. Similar processes take place in different situations (e.g., in heart tissue, where spiral waves occur in life-threatening ventricular fibrillation), so a deeper understanding of this additional layer of self-organizing pattern formation would be beneficial for a variety of applications. One of the most striking differences between pattern formation systems in physics or chemistry and those in biology is the potential importance of variability.
In the first case, the components of the system are essentially identical to the random fluctuations that determine the details of the self-organization process and the resulting patterns. In biology, due to variability, the properties of potentially very few cells can have a decisive influence on the collective asymptotic state of the colony. Variability is a way to implement a control control with few elements in collective mode. Control architectures, signaling cascade parameters, and properties of structure formation processes can be “reverse engineered” from observed spatio-temporal models, as different types of regulation and forms of interaction between components can lead to significantly different correlations. The power of this biologically inspired vision of pattern formation lies in the construction of a bridge between two scales: the patterns as a collective state of a very large number of cells on the one hand and the internal parameters of individual cells on the other. A variety of spatio-temporal data has often been modeled as dynamic model realizations. Examples are dynamic textures , human joint angle trajectories  and silhouettes . A well-known dynamic model for these time series data is the Autoregressive Moving Average Model (ARMA).
Let f(t) be a sequence of characteristics extracted from a video, indexed by time t. The ARMA model parameterizes the evolution of the characteristics f(t) using the following equations: In recent work on excitable dynamics and spiral wave models, there is a strong tendency to consider realistic structural geometries in order to better understand observed spatio-temporal models (see for example ).