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Restitution in mapping models with an arbitrary amount of memory
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10.1063/1.1876912
/content/aip/journal/chaos/15/2/10.1063/1.1876912
http://aip.metastore.ingenta.com/content/aip/journal/chaos/15/2/10.1063/1.1876912
View: Figures

Figures

Image of FIG. 1.
FIG. 1.

An RP produced by iteration of the mapping (A4) given in the Appendix. BCL was decreased from in increments. At each BCL, perturbations were applied at . CB–S responses do not fall on a single line. The line drawn through the CB–S responses, which is a least-squares fit to the responses, emphasizes that the CB responses do not fall on the S1–S2 RC segment. This can be seen more clearly in a close-up for a single BCL in Fig. 2 .

Image of FIG. 2.
FIG. 2.

Details of the RP shown in Fig. 1 for a single BCL .

Image of FIG. 3.
FIG. 3.

For mapping models of the form (4) , the duration of the action potential is a function of previous DIs and APDs as shown in the figure. The BCLs are shown above the action potentials.

Image of FIG. 4.
FIG. 4.

Responses during the application of a perturbation to determine the S1–S2 RC. All variables previous to are at the steady-state for the BCL . The S2 response is given by .

Image of FIG. 5.
FIG. 5.

Visualization of the dynamic RC. (A) The intersection of the surfaces and forms the 3D–DRC (light gray trace). Note that the surfaces and are at a shallow angle to one another. (B) The projection (black) of the 3D–DRC (gray) onto the plane gives the conventional dynamic RC. The vector is tangent to the 3D–DRC at the fixed point for the BCL . is scaled in magnitude for viewing purposes. (C) The surface intersected by the hatched plane defined by (25) , where . The stars in (A) and (B) indicate the steady-state response for the BCL as determined by this intersection. The particular form of used for this visualization is given as (A5) in the Appendix.

Image of FIG. 6.
FIG. 6.

Visualization of the S1–S2 RC. (A) The shaded surface is the surface . The outlined plane is the plane . The intersection of these two surfaces (light gray trace) forms the 3D–SRC. (B) The projection (black) of the 3D–SRC (gray) onto the plane yields the conventional S1–S2 RC. The stars indicate the value of the S1–S2 RC when . The specific form of is given as (A7) in the Appendix.

Image of FIG. 7.
FIG. 7.

Visualization of the CB responses. The shaded surface is the surface . The two planes outlined in black correspond to the two eigendirections of the matrix given in (33) . The intersection of each of these planes with the surface (gray curves) governs the behavior of the transient CB responses. The filled black circles show the CB–D responses that occur when the BCL is changed from . The filled white circles show CB–S responses that occur after perturbations are applied at BCL . The intersection of the two gray curves occurs at the steady-state value for the BCL . The specific form of is given as (A8) in the Appendix.

Image of FIG. 8.
FIG. 8.

Eigendirections and associated eigenvalues shown in the plane. The CB–D responses (filled circles) and CB–S responses (open circles) are also shown in this plot, illustrating how the directions of the two eigenvectors determine the CB responses.

Image of FIG. 9.
FIG. 9.

Superposition of projections of different RCs onto the plane. Responses to perturbations of BCL are shown on the S1–S2 RC. The CB–D responses show the transient that occurs in decreasing the BCL from . The CB–S responses show the recovery from the perturbations.

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/content/aip/journal/chaos/15/2/10.1063/1.1876912
2005-04-07
2014-04-24
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Restitution in mapping models with an arbitrary amount of memory
http://aip.metastore.ingenta.com/content/aip/journal/chaos/15/2/10.1063/1.1876912
10.1063/1.1876912
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