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MBW:Traveling Waves in Excitable Media

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Introduction

Waves 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 one-dimensional 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].

Outline

Traveling 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 Navier-Stokes equations.

Population type - general. Chemical reactions are common example

Biological systems studies - Tumor growth[4], chemical waves, spot formation on animal hides. MBW:Pre-Pattern Formation Mechanisms for Animal Coat Markings

History

One 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:Hodgkin-Huxley, 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 non-linear. In an effort to tease out the essential characteristics of wave propagation in nerve tissue, Fitzhugh [6] found that a two-state 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 FitzHugh-Nagumo 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 Besoulov-Zhabotinskii type) and muscle tissue.

Mathematical Model

An 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.)

SystemPropagatorController
Neuromuscular tissue membrane potential ionic conductance
Belousov-Zhabotinskii reaction bromous acid ferroin
Dictyostelium discoideum (slime mold) cyclic AMP membrane receptor
Epidemics infectious agent level of immunity
Spiral galaxies density of molecular cloud temperature of molecular cloud

Table 1: State variables of some representative excitable media

Figure 1: Examples of traveling waves

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,

Romero excitable eq.png

Table 2: Classical Models

where the functions f(u,v) and g(u,v) describe the non-linear kinetics of the system and Romero epsilon.png is a ratio of the rates of reactions of u and v. For excitable dynamics, Romero epsilon.png 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 N-shaped 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 Romero epsilon.png 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.

Figure 2: Typical phase plane for an excitable system. The colored lines represent the nullclines of u and v and the directed lines represent sample trajectories.

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,

Romero reactiondiffusion eq.png

where \delta ={\frac  {D_{v}}{D_{u}}} is the ratio of the diffusion coefficients of the two components. Because Romero epsilon.png is small, we note that this is really just a singular perturbation in Romero epsilon.png. 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 1-D media

Outer Solution

In the slowly varying regions, u varies slowly so the time derivatives and Laplacian of u are small are small. We can set Romero epsilon.png=0 to find leading order behavior of outer solution,

Romero outersol eq.png

Since f(u,v)=0 has a cubic shape f(u,v) admits three solutions for most values of v of interest: u=\{U_{+}(v),U_{0}(v)U_{+}(v)\}.

Inner time region

To 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 Romero tau.png 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 Romero epsilon.png, by

Romero rapidtime eq.png

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 the outer region. Highlighted are the two stable solution branches of f(u,v)=0 for v between the two local extrema of f.

Dynamics of v in outer region

Then, on the time scale t, u is effectively acts like a switch: Romero slowscaleswitch.png and v evolves according to the following differential equation, Romero vevolution.png where the +/- depends on which branch u lies on.

Inner Solution: Boundary Layer

Introduce a moving, scaled change of variables, Romero innervar.png to investigate the boundary layer between U+(v) and U-(v). After substitution, we find that to first order in Romero epsilon.png we have, Romero fastscale eq.png 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, Romero eigenvalueproblem.png

Dynamics of u within the Boundary Layer. This Boundary Layer is of size O(Romero epsilon.png) in 'both' time and space.

We can integrate the above equation to find a relation that the speed of propagation c(v0) must satisfy, Romero relation for c.png

So the sign of c(v0) is given by,Romero proposition1 eq.png. Recalling that our change of variables was Romero innervar.png, then for forward wave propagation we require that c<0. Formally, Keener[1] concludes with the following two propositions,


Proposition 1

Suppose v* is the unique value of v where

Romero proposition1 eq.png

Then if v0<v*, the medium is excitable; there are stable pulses which propagate throughout the medium.


Proposition 2

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)|'


The proposition sheds light on how a pulse can form but fail to propagate. If two pulses moving in opposite directions meet, they will annihilate each other as the trail edge of either pulse consists of a high wave of v which prevents either edge from reversing direction when then meet. This behavior is supported in simulations later (see Figure \ref{annihilation}). Furthermore, if v>v* on the leading edge of a pulse, then the sign of c(v) is reversed and pulse collapses on itself.

Recent Work

The mathematical techniques discussed here have been used in many situations. A recent paper models tumor growth with reaction-diffusion 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 reaction-diffusion equation, as shown below in reduced units.

u_{{t}}=u(1-u)-auw

v_{{t}}=d[(1-u)v_{{x}}]_{{x}}+bv(1-v)

w_{{t}}=w_{{xx}}+c(v-w)

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

  1. 1.0 1.1 James P. Keener, "Waves in Excitable Media". 'SIAM J. Appl Math' 39 (1980) 528-548.
  2. James P. Keener, ``A Geometrical Theory for Spiral Waves in Excitable Media. 'SIAM J. Appl Math' Vol 46, No 6 (1986) 1039-1055.
  3. John J. Tyson and James P. Keener, “Singular Perturbation Theory of Traveling Waves in Excitable Media (A Review)”. Physica D. 32 (1988) 327-361.
  4. 4.0 4.1 4.2 Antonio Fasano, Miguel A. Herrero, Marianito R. Rodrigo, Slow and fast invasion waves in a model of acid-mediated tumour growth, Mathematical Biosciences, Volume 220, Issue 1, July 2009, Pages 45-56, ISSN 0025-5564, DOI: 10.1016/j.mbs.2009.04.001.
  5. A. L. Hodgkin and A. F. Huxley, "A quantitative description of membrane current and its application to conduction and excitation in nerve" 'J. Physiol.', 117 (1952) 500-544.
  6. R. FitzHugh, "Impulses and physiological states in theoretical models of nerve membrane." 'Biophysical J.', 1 (1961) 445-466.
  7. J. Nagumo and S. Arimoto and S. Yoshizawa, "An active pulse transmission line simulating nerve axon." 'Proc IRE', 50 (1962) 2061-2070.
  8. H.R. Karfunkel, "Excitable chemical reaction systems,I. Definition of excitability and simulation of model systems 'J. Math. Biol.' 2 (1975) 123-132.