
MBW:Traveling Waves in Excitable MediaFrom MathBioContentsIntroductionWaves in excitable media can be modeled under the framework of reaction diffusion models. These models have been found to characterize the essential qualitative behavior of traveling waves in nerve fiber, chemical reactions, atrial flutter and fibrillation, and even the formation of galaxies. Gaining insight into wave generation and propagation under these models can be done with boundary layer theory of asymptotic analysis. Reaction diffusion models inherently contain two different scales: an rapid scale of small size during the abrupt changes on the wave fronts and backs, and the larger scale defining the relaxation times following the abrupt changes. Analysis in the onedimensional case leads to a condition for wave propagation that holds in all dimensions. Analysis in higher dimensions is difficult, especially as when the curvature of the wave front is large. Most of this wiki is primarily a summary of the results for the one dimensional case covered in Keener ^{[1]}, but further development of the geometrical theory in higher dimensions can be found in subsequent papers by Keener^{[2]},^{[3]}. OutlineTraveling wave solutions to the reaction diffusion equation can be applied generally.^{[4]} Math used  PDEs, specifically the reaction diffusion equation.(look at model equations for an example of PDEs used in biofilm modelling and its solution for 1D case). Another example of PDEs in mathematical biology can be seen here Gravitational Effects on Blood Flow, with the use of the NavierStokes equations. Population type  general. Chemical reactions are common example Biological systems studies  Tumor growth^{[4]}, chemical waves, spot formation on animal hides. MBW:PrePattern Formation Mechanisms for Animal Coat Markings HistoryOne of the first contexts within which reaction diffusion models arose was in the study ofnerve impulse conduction equations. In the early 1950s, Hodgkin and Huxley ^{[5]}(for more information on the research of Hodgkin and Huxley please read: APPM4390:HodgkinHuxley, for another model that uses reaction diffusion models see APPM4390:Evolution and Distribution of Species Body Size or APPM4390:Spatial Genetics) were the first to propose a model that quantitively described nerve conduction. Their model and corresponding equations were accurate and complex; it consisted of four states and was highly nonlinear. In an effort to tease out the essential characteristics of wave propagation in nerve tissue, Fitzhugh ^{[6]} found that a twostate reaction diffusion model sufficed as a simplification capable of predicting observed experimental behavior. Nagumo ^{[7]} found a corresponding circuit model analogy that provided a physical explanation of FitzHugh's mathematical model. The simplicity of the FitzHughNagumo model afforded insight that wasn't possible with Hodgkin and Huxley's model. With two states, it became possible to visualize phase trajectories in their native dimension rather than projections of the phase trajectories. Reaction Diffusion models were found to be applicible in a number of different applications including oscillating chemical reactions (of the BesoulovZhabotinskii type) and muscle tissue. Mathematical ModelAn excitable medium can be vaguely described as one in which, given a sufficient initial stimulus, a pulse can be initiated which propagates throughout the medium ^{[8]}. We consider mediums that can be characterized by two different reagents: a propagator and a controller. What the propagator and controller are depend on the context (see Table 1), but their dynamics can be modeled studied in a setting independent of the specific context of interest. (For another project involving waves see APPM4390:Timescales in Metabolic Pathways.)
Table 1: State variables of some representative excitable media Mathematically, we represent excitable mediums as a dynamical system defined as follows. Let u(x,t) represent the levels of the propagator and v(x,t) the levels of the controller. They are defined in space and evolve in time (t) according to the following differential equations, where the functions f(u,v) and g(u,v) describe the nonlinear kinetics of the system and is a ratio of the rates of reactions of u and v. For excitable dynamics, is a small and its small size is responsible for the appearance of different time and spatial scales. Excitable dynamics also stipulates that the nullclines of f and g have a particular shape (Figure \ref{uvnullclines}); f(u,v)=0 is Nshaped and g(u,v) is monotone and intersects f(u,v)=0 only once (and thus there exists only one possible steady state solution). Examples of f, g, and with respect to nerve conduction and oscillating chemical reactions are given in Table 2. The phase space in Figure 2 is characteristic of excitable dynamics. There exists one unique stable steady state that allows the two reagents to nominally coexist in a stable relaxed state. Small perturbations to this resting steady state follow a short return trajectory in phase space to steady state. However, given a sufficient perturbation to this resting steady state, the state variables follow a long detoured trajectory around phase space before returning to rest at the steady state. This trajectory consists of a cascade of effects characterizing a wave: 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. If we allow the reagents in the excitable medium to diffuse as well as react, then we allow for the possibility for excitations in the medium (shocks or pulses) to travel in space. Then the governing equation becomes, where is the ratio of the diffusion coefficients of the two components. Because is small, we note that this is really just a singular perturbation in . The change from an ordinary differential equation to a partial differential equation opens up several different possibilities for scalings. In the analysis below, we find that there is a boundary layer in both time and space, and choose a scaling such that we can track that boundary layer in the (x,t) plane. Perturbation Analysis in 1D mediaOuter SolutionIn the slowly varying regions, u varies slowly so the time derivatives and Laplacian of u are small are small. We can set =0 to find leading order behavior of outer solution, Since f(u,v)=0 has a cubic shape f(u,v) admits three solutions for most values of v of interest: . Inner time regionTo examine which of the branches u lies on, we note that when the time derivative of u is large the approximating equations in the outer region are no longer valid. We conclude that there must be a boundary layer for small t. We find that provides a distinguished limit: applying this change of variable to our reaction diffusion equation to find that u,v are governed, to leading order in , by From this, we learn that u rapidly tends to U(v) if u is to the left of f(u,v) (where f(u,v)<0), or to U+(v) if u is to the right of f(u,v) (where f(u,v)>0). Hence the curve U0(v) acts like a threshold curve that dictates on which of the two branches of f=0 u will lie on. Dynamics of v in outer regionThen, on the time scale t, u is effectively acts like a switch: and v evolves according to the following differential equation, where the +/ depends on which branch u lies on. Inner Solution: Boundary LayerIntroduce a moving, scaled change of variables, to investigate the boundary layer between U+(v) and U(v). After substitution, we find that to first order in we have, The result is an eigenvalue problem where the speed of propagation (c) is the eigenvalue that depends on v0. Restating the eigenvalue problem we have, We can integrate the above equation to find a relation that the speed of propagation c(v0) must satisfy, So the sign of c(v0) is given by,. Recalling that our change of variables was , then for forward wave propagation we require that c<0. Formally, Keener^{[1]} concludes with the following two propositions,
Suppose v* is the unique value of v where Then if v0<v*, the medium is excitable; there are stable pulses which propagate throughout the medium.
Suppose that a shock is formed for some v<v^*, and that ahead of the shock the initial data also satisfy v0(x)<v*. Then the shock which forms will propagate through the medium with asymptotic speed c(v0)'
Recent WorkThe mathematical techniques discussed here have been used in many situations. A recent paper models tumor growth with reactiondiffusion equations (for more information see example of equation used in tumor growth modeling). In the early stages of tumor growth, the driving force is primarily cell division. As the tumor grows, it must do work on the surrounding cells, and acid mediated traveling waves may be the mechanism behind continued growth. The authors argue that traveling wavefronts, with certain parameters can predict certain features of the tumor. Antonio Fasano, Miguel A. Herrero, Marianito R. Rodrigo, 2009 ^{[4]}, model tumor growth as a reactiondiffusion equation, as shown below in reduced units.
The variables u,v and w correspond to healthy tissue, neoplastic tissue and excess positive hydrogen ions. The constants all have physical meaning. Particularly interesting is the parameter a, which is how damaging the hydrogen ions are on healthy tissue. The authors refer to it as aggressivity. For certain values this can predict the size of the interstitial gap, which is a layer of dead cells between the tumor and healthy cells. References
