Limits...
On propagation of excitation waves in moving media: the FitzHugh-Nagumo model.

Ermakova EA, Shnol EE, Panteleev MA, Butylin AA, Volpert V, Ataullakhanov FI - PLoS ONE (2009)

Bottom Line: At high stream velocities, the wave does not span the channel from wall to wall, forming isolated excited regions, which we called "restrictons".In the subcritical region of parameters, a sufficiently fast stream can result in the survival of excitation moving, as a rule, in the form of "restrictons".These findings clarify mechanisms of wave propagation and survival in flow.

View Article: PubMed Central - PubMed

Affiliation: Semenov Institute of Chemical Physics, Russian Academy of Sciences, Moscow, Russia.

ABSTRACT

Background: Existence of flows and convection is an essential and integral feature of many excitable media with wave propagation modes, such as blood coagulation or bioreactors.

Methods/results: Here, propagation of two-dimensional waves is studied in parabolic channel flow of excitable medium of the FitzHugh-Nagumo type. Even if the stream velocity is hundreds of times higher that the wave velocity in motionless medium (), steady propagation of an excitation wave is eventually established. At high stream velocities, the wave does not span the channel from wall to wall, forming isolated excited regions, which we called "restrictons". They are especially easy to observe when the model parameters are close to critical ones, at which waves disappear in still medium. In the subcritical region of parameters, a sufficiently fast stream can result in the survival of excitation moving, as a rule, in the form of "restrictons". For downstream excitation waves, the axial portion of the channel is the most important one in determining their behavior. For upstream waves, the most important region of the channel is the near-wall boundary layers. The roles of transversal diffusion, and of approximate similarity with respect to stream velocity are discussed.

Conclusions: These findings clarify mechanisms of wave propagation and survival in flow.

Show MeSH

Related in: MedlinePlus

Effect of stream velocity on the evolution of (a–g) downstream waves and (h–j) upstream waves into restrictons, as calculated for (a–e)  (above-critical value for which plane waves exist in the absence of flow), and for (f–j)  (subcritical value for which no plane waves exist in still medium).Note that, for subcritical , restrictons arise near the channel wall at low stream velocities (panels h–j). Activator  () is shown on the same gray scale as in Fig. 1. In calculations,  and . The x and y axes are scaled differently: y axis is eightfold expanded relative to the x axis.
© Copyright Policy
Related In: Results  -  Collection


getmorefigures.php?uid=PMC2636873&req=5

pone-0004454-g004: Effect of stream velocity on the evolution of (a–g) downstream waves and (h–j) upstream waves into restrictons, as calculated for (a–e) (above-critical value for which plane waves exist in the absence of flow), and for (f–j) (subcritical value for which no plane waves exist in still medium).Note that, for subcritical , restrictons arise near the channel wall at low stream velocities (panels h–j). Activator () is shown on the same gray scale as in Fig. 1. In calculations, and . The x and y axes are scaled differently: y axis is eightfold expanded relative to the x axis.

Mentions: The critical velocity depends on the “chemical” parameters of the system. With our choice of and (see (3)), one-dimensional pulses exist for . Near this critical value, the one-dimensional waves are “weaker” and more susceptible to external disturbances. In Figs. 4a–4c, one can see how the wave shape varies with increasing stream velocity for . Waves are more complex in shape (cf. Fig. 1d with Figs.4a, 4b. In the leading region in the mid-channel, something like a nucleus develops: a zone of large values (activator) surrounded on all sides with large values (inhibitor). For , restrictons emerged in a rather wide channel at and existed throughout the stream velocity range used (Figs. 4c–4e). We also observed restrictons for H = 20, L = 800, and the stream velocity as large as 2000.


On propagation of excitation waves in moving media: the FitzHugh-Nagumo model.

Ermakova EA, Shnol EE, Panteleev MA, Butylin AA, Volpert V, Ataullakhanov FI - PLoS ONE (2009)

Effect of stream velocity on the evolution of (a–g) downstream waves and (h–j) upstream waves into restrictons, as calculated for (a–e)  (above-critical value for which plane waves exist in the absence of flow), and for (f–j)  (subcritical value for which no plane waves exist in still medium).Note that, for subcritical , restrictons arise near the channel wall at low stream velocities (panels h–j). Activator  () is shown on the same gray scale as in Fig. 1. In calculations,  and . The x and y axes are scaled differently: y axis is eightfold expanded relative to the x axis.
© Copyright Policy
Related In: Results  -  Collection

Show All Figures
getmorefigures.php?uid=PMC2636873&req=5

pone-0004454-g004: Effect of stream velocity on the evolution of (a–g) downstream waves and (h–j) upstream waves into restrictons, as calculated for (a–e) (above-critical value for which plane waves exist in the absence of flow), and for (f–j) (subcritical value for which no plane waves exist in still medium).Note that, for subcritical , restrictons arise near the channel wall at low stream velocities (panels h–j). Activator () is shown on the same gray scale as in Fig. 1. In calculations, and . The x and y axes are scaled differently: y axis is eightfold expanded relative to the x axis.
Mentions: The critical velocity depends on the “chemical” parameters of the system. With our choice of and (see (3)), one-dimensional pulses exist for . Near this critical value, the one-dimensional waves are “weaker” and more susceptible to external disturbances. In Figs. 4a–4c, one can see how the wave shape varies with increasing stream velocity for . Waves are more complex in shape (cf. Fig. 1d with Figs.4a, 4b. In the leading region in the mid-channel, something like a nucleus develops: a zone of large values (activator) surrounded on all sides with large values (inhibitor). For , restrictons emerged in a rather wide channel at and existed throughout the stream velocity range used (Figs. 4c–4e). We also observed restrictons for H = 20, L = 800, and the stream velocity as large as 2000.

Bottom Line: At high stream velocities, the wave does not span the channel from wall to wall, forming isolated excited regions, which we called "restrictons".In the subcritical region of parameters, a sufficiently fast stream can result in the survival of excitation moving, as a rule, in the form of "restrictons".These findings clarify mechanisms of wave propagation and survival in flow.

View Article: PubMed Central - PubMed

Affiliation: Semenov Institute of Chemical Physics, Russian Academy of Sciences, Moscow, Russia.

ABSTRACT

Background: Existence of flows and convection is an essential and integral feature of many excitable media with wave propagation modes, such as blood coagulation or bioreactors.

Methods/results: Here, propagation of two-dimensional waves is studied in parabolic channel flow of excitable medium of the FitzHugh-Nagumo type. Even if the stream velocity is hundreds of times higher that the wave velocity in motionless medium (), steady propagation of an excitation wave is eventually established. At high stream velocities, the wave does not span the channel from wall to wall, forming isolated excited regions, which we called "restrictons". They are especially easy to observe when the model parameters are close to critical ones, at which waves disappear in still medium. In the subcritical region of parameters, a sufficiently fast stream can result in the survival of excitation moving, as a rule, in the form of "restrictons". For downstream excitation waves, the axial portion of the channel is the most important one in determining their behavior. For upstream waves, the most important region of the channel is the near-wall boundary layers. The roles of transversal diffusion, and of approximate similarity with respect to stream velocity are discussed.

Conclusions: These findings clarify mechanisms of wave propagation and survival in flow.

Show MeSH
Related in: MedlinePlus