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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.

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Related in: MedlinePlus

Dependence of wall shear rate, at which upstream wave is immobile with respect to the channel walls, on the  parameter. H = 16, .At , the excitation wave is always carried away by flow, and there is no such flow velocity, when the wave velovity is zero. Interestingly, at , the dependence begins not at zero, but at a large gradient of flow velocity.
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pone-0004454-g007: Dependence of wall shear rate, at which upstream wave is immobile with respect to the channel walls, on the parameter. H = 16, .At , the excitation wave is always carried away by flow, and there is no such flow velocity, when the wave velovity is zero. Interestingly, at , the dependence begins not at zero, but at a large gradient of flow velocity.

Mentions: Upstream waves demonstrate some similarity with the dowstream waves upon changes in the “chemical” parameters, but this similarity is not strong (Fig. 5). The flow stabilizes upstream waves, as well as downstream ones. Flow results in the formation of stable steadily moving excitation waves at the same values of the parameter ., at which excitation in the immobile medium rapidly disappears. The border between the region where excitation disappears and the excitable region (Fig. 5, bottom part, border between region I and regions II, III), is achieved at higher flow velocities with the decrease of . As for the downstream waves, restrictons appear with the increase of flow velocity at subcritical values of . These small excitation regions near the borders do not resist the flow well and cannot move upstream, although their velocity is much smaller than maximal flow velocity. With the increase of flow velocity, the restrictons are stronger carried away by flow. At , excitation cannot exist in the form of two restrictons and rapidly disappears in the middle of the channel. For , restrictons do not appear at all, and stable excitation wave appears (III). The ability of the wave to move against the current increases with the increase of . This demonstrates the dependence of the wall shear rate when the upstream wave is immobile with regard to the wall (Fig. 7). At values close to , this dependence is strongly non-linear (Fig. 7); however, when exceeds 8, it becomes almost proportional to .


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)

Dependence of wall shear rate, at which upstream wave is immobile with respect to the channel walls, on the  parameter. H = 16, .At , the excitation wave is always carried away by flow, and there is no such flow velocity, when the wave velovity is zero. Interestingly, at , the dependence begins not at zero, but at a large gradient of flow velocity.
© Copyright Policy
Related In: Results  -  Collection

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

pone-0004454-g007: Dependence of wall shear rate, at which upstream wave is immobile with respect to the channel walls, on the parameter. H = 16, .At , the excitation wave is always carried away by flow, and there is no such flow velocity, when the wave velovity is zero. Interestingly, at , the dependence begins not at zero, but at a large gradient of flow velocity.
Mentions: Upstream waves demonstrate some similarity with the dowstream waves upon changes in the “chemical” parameters, but this similarity is not strong (Fig. 5). The flow stabilizes upstream waves, as well as downstream ones. Flow results in the formation of stable steadily moving excitation waves at the same values of the parameter ., at which excitation in the immobile medium rapidly disappears. The border between the region where excitation disappears and the excitable region (Fig. 5, bottom part, border between region I and regions II, III), is achieved at higher flow velocities with the decrease of . As for the downstream waves, restrictons appear with the increase of flow velocity at subcritical values of . These small excitation regions near the borders do not resist the flow well and cannot move upstream, although their velocity is much smaller than maximal flow velocity. With the increase of flow velocity, the restrictons are stronger carried away by flow. At , excitation cannot exist in the form of two restrictons and rapidly disappears in the middle of the channel. For , restrictons do not appear at all, and stable excitation wave appears (III). The ability of the wave to move against the current increases with the increase of . This demonstrates the dependence of the wall shear rate when the upstream wave is immobile with regard to the wall (Fig. 7). At values close to , this dependence is strongly non-linear (Fig. 7); however, when exceeds 8, it becomes almost proportional to .

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