
MBW:Timescales in Metabolic PathwaysFrom MathBioContentsSummaryThe characteristic time over which nontrivial changes happen to a system is determined by the 'soundspeed', or more generally, the speed of the fastest signal in the medium. For example, if one has a water tank at the top of a hill connected via pipes to a house in a valley, nontrivial changes  like punching a hole in the pipe  causes changes to the system in the timescale of the sound speed in water. Specifically, if one actually were to punch a hole, the highspeed water spout only forms after a characteristic time given by the distance from the source divided by the sound speed. In much the same way, biological systems also evolve on timescales determined by the speed of the fastest wave that can propagate through them  a muscle contraction is limited by the speed of the signal through the axon. In addition to this, the propagation of waves through the medium can have many interesting dynamical effects, a good example of this is the generation of nonlinear chemical oscillators in the BelousovZhabotinsky (BZ) reaction. The following writeup compiles information on the propagation of waves in excitable media in one and two dimensions, exploring the dynamics of said systems by using perturbation techniques outlined in Tyson and Keener^{[1]} IntroductionWaves in a variety of contexts can be modeled under a generic framework. Gaining insight into wave generation and propagation can be done by dividing the analysis into two time scales: an small time scale during the abrupt changes on the wave fronts and backs, and the larger time scale relating the frequency of passing waves to the speed at which they travel. Tyson and Keener ^{[1]} show that methods of perturbation analysis applied to a governing singular partial differential equation are adept at offering insight into this division of time scales. The onedimensional case offers no serious challenges, but analysis in two dimensions presents challenges that have no clear unique solution. In Tyson and Keener^{[1]}, several methods are compared and contrasted. Connections are made between the theories and they identify some fundamental relationships that underlie all approaches. They also show how the results can offer insight into the three dimensional case. Mathematical ModelGenerically, a wave can be characterized by two different reagents: a propagator and a controller. What the propagator and controller are depend on the context (Table 1), but their dynamics can be modeled studied by a setting independent of the speciﬁc context of interest. These two reagents exist in a medium that nominally allows the two to coexist in a stable relaxed state. However, given a sufficient perturbation to this resting steady state, a cascade of effects characterizing a wave will occur: an abrupt rise in the levels of the propagator triggers a rise in the levels of the controller, which in turn drives the levels of the propagator back to the relaxed steady state.
Table 1: State variables of some representative excitable media
Mathematically, let u(x,t) represent the levels of the propagator and v(x,t) the levels of the controller. They are defined in space (over some domain D as a subset of the ndimensional reals) and evolve in time (t). The dynamics between the propagator and controller can be framed under a reactiondiffusion model,
Figure 1: From ^{[1]} For a detailed example of modeling under the framework of reaction diffusion models, see APPM4390:Traveling Waves in Excitable Media. Example in 1D mediaIn one dimension, there are two possible behaviors: a single solitary pulse or a periodic wave train. In the case of the wave train, the properties are summarized by the dispersion relation, omega = H(k;epsilon,delta) which relates the frequency (cycles per second: omega/(2 pi)) to the wave number (k= 2 pi/ wavelength) of the traveling wave train. For k small, this dispersion relation is essentially linear in k: H ~= c k. (For more information on traveling waves, please see APPM4390:Traveling Waves in Excitable Media). Higher Dimensions: 2D and 3D mediaIn two dimensions, the principle difference to consider is that the curvature of the wave front affects wave propagation. The effect of curvature leads to difficulties in the study of wave propagation emanating at from a point: near a point, the curvature of a wave front is large. Because of this limitation, the jumping point for the study of waves in two dimensions is in an annular region. The extension of analysis to the whole plane, including every small neighborhood of the origin, is difficult and has no good answer. There are two canonical forms of wave propagation: Target patterns An emanating series of concentric circular waves; here as the circular waves emanate outward, the curvature becomes negligible and the problem reduces to the onedimensional case. Spiral Waves Rotating waves that spiral out; here curvature plays an important role in limiting possible solutions. The authors show that their singular perturbation analysis leads them to describing waves in 2D as subject to two relations:
where the dispersion relation depends on the specific excitable medium and the curvature relation is universally applicable to all mediums. Previous ApproachesZykov ^{[2]} parametrizes the wave differently than Tyson and Keener and arrives at a relationship that combines the dispersion relationship with a curvature relation. Greenberg ^{[3]}introduces a coordinate system and that allows the two dimensional case to approximately be separated into one dimensional cases. The approximation is valid when not to close to the origin and for large enough wavelengths, but leads to a dispersion relation of a different form than Tyson and Keener's, but qualitatively similar. Fife^{[4]} improves on Greenberg's approach by using his coordinate system and not making simplifying approximations. Then, a dispersion relation identical up to a constant to Tyson and Keener's is derived. Analysis of ResultsSingular perturbation analysis leads to simple relations that summarize the properties of wave propagation. In the one dimensional case, the fundamental relationship is a dispersion relation; that is a relation between the frequency and speed of the traveling waves. In two dimensions, curvature plays a role, and in addition to the dispersion relation, the eikonal equation plays a role, N = c + DK where N is the normal velocity of the wave front, K is its curvature and c is the velocity of the plane waves (that is, when the wave front has no curvature: K=0). The type of possible solutions in each dimension can be summarized as follows. In one dimension, there exists a family of one parameter wavetrain solutions.In two dimensions, there exists a oneparameter family of expanding concentric circle waves but only one possible spiral pattern. In three dimensions, the situation is analogous: there exists a oneparameter family of spherical wave solutions, but the number of possible scroll waves is limited. In all cases, the singular perturbation analysis of Keener and Tyson agrees well with previous theories and numerical results. Recent WorkThe original paper by Tyson and Keener ^{[1]} was written back in 1988 and is widely used as a foundational citation in several studies on dynamical systems in biology. The most recent paper that included a citation of Tyson and Keener ^{[1]} was a study by Burglund and Kuehn ^{[5]} proving the local existence of solutions for the FitzHughNagumo system  a model that describes the evolution of action potentials for large ensembles of neurons. In essence, the paper proved the existence of local solutions for a system of stochastic PDEs when driven by white noise. References
