Am J Physiol Heart Circ Physiol
279: H1869–H1879, 2000.
Dynamics of action potential head-tail interaction
during reentry in cardiac tissue: ionic mechanisms
THOMAS J. HUND,1 NIELS F. OTANI,1 AND YORAM RUDY1–3
Cardiac Bioelectricity Research and Training Center, and Departments of
1
Biomedical Engineering, 2Physiology and Biophysics, and 3Medicine,
Case Western Reserve University, Cleveland, Ohio 44106-7207
Received 29 November 1999; accepted in final form 21 April 2000
reentry; ion accumulation; alternans; restitution
34, 42). Such oscillations often precede spontaneous
termination of reentry (14, 17, 32) or breakup of the
reentry loop into multiple pathways resulting in fibrillation.
Theoretical studies of reentry (8, 21, 33, 42) have
used relatively simple models of the cardiac AP (4, 15,
29, 30) that lack realistic elements such as intracellular calcium handling and dynamic intracellular ion
concentration changes during excitation. Changes in
the intracellular concentrations of ions such as Ca2⫹
([Ca2⫹]i) and Na⫹ [Na]i) occur during rapid pacing,
drug application, and ion-channel dysfunction (35). Importantly, intracellular ions accumulate during reentrant tachyarrhythmias due to the rapid repetitive
excitation of the cells. These ionic changes affect the
AP through various membrane currents and modify
the dynamics of reentry. In this study, we examine
reentry in a one-dimensional closed pathway using the
Luo-Rudy dynamic (LRd) model of a mammalian ventricular myocyte (28, 39, 43, 46). This model accounts
for dynamic intracellular concentration changes of
Na⫹, Ca2⫹, and K⫹ during excitation. It also incorporates the effects of such changes on transmembrane
currents that determine the AP, including currents
through ion channels, pumps, and exchangers. The
aims of this study are to examine: 1) the response of the
reentrant AP to varying degrees of head-tail interaction and 2) the effect of ion accumulation on this
relationship.
REENTRY IS THE UNDERLYING mechanism of many common
cardiac arrhythmias (11, 32, 36), which are a major
cause of death and disability. Effective diagnosis, prevention, and treatment of these life-threatening arrhythmias require an in-depth understanding of the
reentrant action potential (AP).
During reentry in a sufficiently short pathway, the
excitation wave front (head) propagates into tissue
that is still refractory (tail) from the previous excitation. Interaction between the head and tail of an AP is
a common phenomenon in the heart. It has been observed in an anatomically defined reentry pathway
(18), along the arm of a spiral wave (21), and in the
leading circle of functional reentry (2). Head-tail interaction is also known to have a profound effect on the
dynamics of the reentrant AP. For example, a significant degree of head-tail interaction produces oscillations in key AP properties such as AP duration (APD),
conduction velocity (␪), and cycle length (CL) (8, 18, 21,
The reentry pathway. By connecting individual LRd ventricular cell models (28, 39, 43, 46) with passive resistances
to represent gap junctions (38), a ring of cells is created (33)
that represents a closed pathway in the heart (a schematic is
shown in Fig. 1). Simulations were conducted for gap-junction conductances (gj) of 0.076 and 0.285 mS. The simulated
behavior of interest, namely the effect of ion accumulation on
reentry dynamics, did not depend on the particular value of
gj. Results for gj ⫽ 0.076 mS are presented unless otherwise
stated. With this choice, we could use a shorter pathway and
significantly reduce computing time. Also, gj ⫽ 0.076 mS
produced slowed conduction, which is a prerequisite for sus-
Address for reprint requests and other correspondence: Yoram
Rudy, Cardiac Bioelectricity Research and Training Center, 319
Wickenden Bldg., Case Western Reserve Univ., Cleveland, OH
44106-7207 (E-mail: [email protected]).
The costs of publication of this article were defrayed in part by the
payment of page charges. The article must therefore be hereby
marked ‘‘advertisement’’ in accordance with 18 U.S.C. Section 1734
solely to indicate this fact.
http://www.ajpheart.org
METHODS
0363-6135/00 $5.00 Copyright © 2000 the American Physiological Society
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Hund, Thomas J., Niels F. Otani, and Yoram Rudy.
Dynamics of action potential head-tail interaction during
reentry in cardiac tissue: ionic mechanisms. Am J Physiol
Heart Circ Physiol 279: H1869–H1879, 2000.—In a sufficiently short reentry pathway, the excitation wave front
(head) propagates into tissue that is partially refractory (tail)
from the previous action potential (AP). We incorporate a
detailed mathematical model of the ventricular myocyte into
a one-dimensional closed pathway to investigate the effects of
head-tail interaction and ion accumulation on the dynamics
of reentry. The results were the following: 1) a high degree of
head-tail interaction produces oscillations in several AP
properties; 2) Ca2⫹-transient oscillations are in phase with
AP duration oscillations and are often of greater magnitude;
3) as the wave front propagates around the pathway, AP
properties undergo periodic spatial oscillations that produce
complicated temporal oscillations at a single site; 4) depending on the degree of head-tail interaction, intracellular [Na⫹]
accumulation during reentry either stabilizes or destabilizes
reentry; and 5) elevated extracellular [K⫹] destabilizes reentry by prolonging the tail of postrepolarization refractoriness.
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DYNAMICS OF ACTION POTENTIAL HEAD-TAIL INTERACTION
RESULTS
Fig. 1. Dynamic Luo-Rudy (LRd) cell models are connected by intercellular resistive pathways (gap junctions) with conductance (gj) to
form a closed pathway of reentry. Head and tail of reentrant action
potentials (AP) are indicated. Below the reentry pathway a schematic of the LRd cell model is shown with various ion channels,
pumps, and exchangers represented in the model: Na⫹ channel
current (INa); L-type Ca2⫹ current [ICa(L)]; Na⫹-Ca2⫹ exchange current (INaCa); Na⫹-K⫹ pump current (INaK); slow delayed-rectifier K⫹
current (IKs); junctional sarcoplasmic reticulum (JSR); and network
SR (NSR). LRd model details are provided in the literature (28, 39,
43, 46). This model code can be downloaded from the research section
of Web site http://www.cwru.edu/med/CBRTC. See this Web site for
additional definitions of the current abbreviations.
tained reentry and is observed in the border zone of a healing
infarct (10) and during ischemia (38). Computations converged for the entire range of parameter values used with the
chosen spatial and temporal discretization steps [⌬x ⫽ 100
␮m; ⌬t varies from 0.005 to 1.0 ms (38)].
Once reentry has been initiated in the closed pathway, the
path length (L) is reduced by removing 10 cells in the plateau
phase of their respective APs, similar to the protocol used by
Vinet and Roberge (42). Propagation is allowed to reach a
steady state, after which 10 additional cells are removed.
This process is repeated until the pathway is sufficiently
The regime of stable AP behavior. Figure 2 shows
APD (A), [Ca2⫹]i,peak (B), and CL (C) as functions of L
for L ⬍ 150 cells. The data shown are the steady-state
values recorded at each L during the short steady-state
protocol. Below a certain L (the bifurcation point, Lcriti,
marked with an arrow in Fig. 2A), significant head-tail
interaction occurs, and the nonlinear properties of the
system become apparent as all AP properties (APD,
[Ca2⫹]i,peak, CL, DI, and ␪) display complex oscillatory
behavior (DI and ␪ are not shown). The shaded region
in Fig. 2 indicates the region of oscillatory behavior.
For L ⬎ Lcrit, reentry is stable (oscillations in AP
properties are transient and subside within 10 revolutions). We begin by discussing the dependence of AP
properties on L in the stable regime.
CL is the time it takes for the reentrant wave front to
propagate once around the pathway. As L becomes
shorter (for L ⬎ Lcrit), the wave front takes less time to
travel around the reentry pathway and CL decreases
(Fig. 2C). A smaller CL implies that every cell in the
reentry pathway is being stimulated at a more rapid
rate, which results in reduced APD (Fig. 2A) and in-
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small such that oscillations in APD are persistent (last more
than 10 revolutions). After the onset of oscillations in AP
properties, cells are removed two at a time to achieve greater
spatial resolution. Close to the point of termination, cells are
removed one at a time. In the case of transient oscillations,
the system is considered to be in steady state once APD
changes by less than 1%. In the case of persistent oscillations, two different protocols are used to determine steady
state. In the first protocol, propagation is defined to be in
steady state when APD oscillations maintain the same pattern for at least 10 revolutions (“short steady state”). In the
second protocol, steady state is declared when an oscillatory
pattern repeats itself for at least 100 revolutions (“long
steady state”).
The APD is the time between the AP upstroke and
90% repolarization. The peak Ca2⫹ transient concentration
([Ca2⫹]i,peak) is defined as the maximum value of the Ca2⫹
transient following the AP upstroke. Diastolic interval (DI) is
measured as the time between 90% repolarization and the
next upstroke. In the presence of APD oscillations, DI and
APD are recorded around the pathway for several revolutions
to create the APD restitution curve (APD as a function of the
previous DI). In the absence of oscillations, the restitution
curve is created by applying a premature stimulus on the tail
of the reentrant AP, which induces transient oscillations in
APD and DI (18). These data are then used to create the APD
restitution curve. CL is measured as the time between two
successive upstrokes at a particular cell. During oscillations,
intermediate APD (APDmed), DI (DImed), and [Ca2⫹]i,peak
([Ca2⫹]i,peak,med) for a single cell are the midpoints between
the maximum and minimum values.
In certain simulations, we use a [Na⫹]i clamp protocol in
which [Na⫹]i is held at a fixed value and prevented from
accumulating as reentry continues. In another simulation,
we uncouple cells (set gj ⫽ 0) away from the AP upstroke. The
region of coupled cells and the AP wave front propagate
together in the reentry pathway during this simulation. This
allows us to minimize the effects of electrical loading on the
dynamics during sustained reentry.
DYNAMICS OF ACTION POTENTIAL HEAD-TAIL INTERACTION
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Fig. 2. A: AP duration (APD); B: peak intracellular Ca2⫹ transient
concentration ([Ca2⫹]i,peak); C: cycle length (CL) as functions of
length of reentry pathway (L). These data are the result of a protocol
requiring an oscillatory pattern to repeat itself for at least 10 revolutions before being defined as steady state (“short steady state”).
Shaded region corresponds to where oscillations in AP properties
occur. Bifurcation point (A) (Lcrit) ⫽ 80 cells in this protocol and is
marked with an arrow. In the case of transient oscillations (L ⬎
Lcrit), last value observed before reducing L is plotted. For persistent
oscillations (L ⬍ Lcrit), values during last 10 cycles before reducing L
are plotted.
creased [Ca2⫹]i,peak (Fig. 2B). These rate-dependent AP
changes are manifestations of rate-adaptation cellular
processes that are well understood (43, 46).
Time dependence of the critical L for bifurcation. For
L ⬍ Lcrit, APD, DI, [Ca2⫹]i,peak, CL, and ␪ begin to
oscillate. In Fig. 2, corresponding to the short steadystate protocol, Lcrit ⫽ 80 cells. However, if reentry in an
80-cell pathway continues for another 400 revolutions,
the stable beat-to-beat APD alternans (2:2 stimulusresponse ratio) observed after 50 revolutions disappears. In Fig. 3 (shown on a scale of only 70 cells to
expand the region of oscillatory behavior), the long
steady-state protocol is implemented. In this figure,
Lcrit ⫽ 64 cells compared with 80 cells in Fig. 2. Note in
Fig. 3 the division of the oscillatory region into subregions with distinct oscillatory patterns. (These pat-
Fig. 3. AP oscillatory region in greater detail: APD (A); [Ca2⫹]i,peak
(B); and CL (C) as functions of L during a protocol that requires an
oscillatory pattern to repeat itself for at least 100 revolutions before
steady state is declared (“long steady state”). This protocol produces
oscillations in AP properties at L ⫽ 64 cells. Regions of periodic and
quasi-periodic behavior are labeled in A.
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terns are discussed in Figs. 7 and 8.) With models that
do not account for dynamic changes in intracellular ion
concentrations (e.g., Beeler-Reuter; see Ref. 4), AP
properties begin oscillating at a well-defined (fixed)
Lcrit that does not decrease with time (8, 21, 34, 42).
The LRd model used in this study accounts for dynamic
intracellular ion concentration changes that occur
when cells are stimulated rapidly during reentry. With
these physiological processes, Lcrit shows strong time
dependence (compare Figs. 2 and 3).
Time dependence of oscillations for a relatively large
L (weak head-tail interaction) and ionic mechanism.
Insight into the time-dependent processes responsible
for damping of oscillations and reduction of Lcrit is
provided in Figs. 4 and 5. Figure 4, left, shows APD
restitution curves during reentry in a pathway with
L ⫽ 80 cells. Figure 4, right, shows corresponding APD
values recorded at every cell in the pathway during one
revolution of reentry. Oscillations in APD persist after
150 revolutions (Fig. 4A, right). The corresponding
APD restitution curve (Fig. 4A, left) has a slope (m)
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DYNAMICS OF ACTION POTENTIAL HEAD-TAIL INTERACTION
Evidence for this is given in Fig. 4C, where [Na⫹]i
during reentry after 950 revolutions is reset to its
value after 150 revolutions. When [Na⫹]i is decreased
in such a manner, APDmed increases from 71.8 to 76.2
ms, and DImed decreases from 30.4 to 26.7 ms. Consequently, mmax ⬎ 1 (left), and oscillations in APD resume (right).
To provide insight into the ionic mechanism by
which APD decreases with time and oscillations cease,
we compare APs (Fig. 5A) and ionic currents (Fig. 5,
B–E) observed at a single cell after 150 revolutions
(solid line) to those observed after 950 revolutions
(dashed line). Accumulation of [Na⫹]i after 950 revolutions increases the reverse-mode activity (Na⫹ extruDownloaded from http://ajpheart.physiology.org/ by 10.220.33.5 on July 12, 2017
Fig. 4. Time dependence of oscillations in a relatively large pathway
(L ⫽ 80 cells). APD restitution curves (left) and APD along reentry
pathway for one revolution (right) are calculated after 150 revolutions (A), 950 revolutions (B), and 980 revolutions with intracellular
[Na⫹] ([Na⫹]i) reset to value after 150 revolutions (C). A line with
slope (m) ⫽ maximal slope value (mmax) is shown adjacent to each
restitution curve (left).
whose maximum value (mmax) is ⬎1, which agrees with
theoretical and experimental observations that mmax ⱖ
1 for oscillations in APD to occur (8, 18, 21, 24). After
950 revolutions (Fig. 4B), the oscillations in APD cease
(right), and mmax ⬍ 1 (left). In addition, [Na⫹]i (not
shown) accumulates from 18 to 21 mM, in agreement
with experimental findings that [Na⫹]i increases by
about 30% during rapid pacing (7). Furthermore, APDmed decreases from 75.6 to 71.8 ms, and DImed increases from 26.4 to 30.4 ms over the course of 800
revolutions.
[Na⫹]i accumulation dampens oscillations after 950
revolutions through the following mechanism. The
point about which APD and DI oscillate (the operating
point, discussed further in Fig. 10) is determined by
the intersection of the line APD ⫽ ⫺DI ⫹ CL and the
APD restitution curve (31). [Na⫹]i accumulation decreases APD (shifts the restitution curve downward,
Fig. 4B) relative to control (Fig. 4A). Assuming CL
remains constant (a good assumption because CL
changes are very small relative to changes in APD and
DI), a downward displacement of the restitution curve
(decrease in APDmed) shifts the operating point to a
larger DI where mmax ⬍ 1, and oscillations disappear.
Fig. 5. Ionic mechanism of time dependence of APD restitution curve
and Lcrit: APs (A); INaCa (B); INaK (C), IKs (D), and ICa(L) (E). Data
were recorded after 150 revolutions (solid line) when oscillations
were present and after 950 revolutions (dashed line) when APD was
constant from beat to beat (same conditions as in Fig. 4, A and B,
respectively). [Na⫹]i increased from 18 mM after 150 revolutions to
21 mM after 950 revolutions (not shown). Arrow in E indicates larger
ICa(L) corresponding to shorter APD in A. Vm, membrane potential.
DYNAMICS OF ACTION POTENTIAL HEAD-TAIL INTERACTION
Fig. 6. Temporal oscillations in APD (A), diastolic interval (DI) (B),
and CL (C) recorded at a single site in a 154-cell reentry pathway.
Quasi-periodic temporal behavior is observed that agrees with experimental recordings shown at right (18). Maximum degree of CL
oscillations (indicated with arrow in C) occurs where oscillations in
APD and DI are at a minimum (arrows, A and B, respectively).
Conductance gj ⫽ 0.285 mS was used in these simulations.
repeats itself every five beats) is shown in Fig. 7B.
These oscillatory patterns agree with patterns observed experimentally (Fig. 7, A and B, bottom) (14,
18). An example of quasi-periodic behavior, which is
characterized by a pattern that repeats itself only
approximately, is shown in Fig. 7C.
The 2:2 pattern in Fig. 7A is an example of periodic
behavior observed during the short steady-state protocol (Fig. 2). The 5:5 pattern in Fig. 7B comes from the
“periodic window” observed during the long steadystate protocol (Fig. 3A). Figure 7C (also from the long
steady-state protocol) contains a quasi-periodic pattern
from the “quasi-periodic region” located to the left
(smaller L) of the periodic window in Fig. 3A. As L is
reduced, the oscillatory behavior passes through a
quasi-periodic regime that is interrupted by a small
window of periodicity at L ⫽ 61 cells (Fig. 3). We
discuss the factors that contribute to the formation of
these different patterns in the next section.
Interestingly, for a relatively low degree of head-tail
interaction (Fig. 7A) the maximal percent change in
[Ca2⫹]i,peak (94%, right) during oscillations is much
greater than the maximal percent change in APD
(17%, left). This difference disappears as the degree of
head-tail interaction increases (Fig. 7, B and C). Furthermore, [Ca2⫹]i,peak oscillations appear only in the
presence of APD oscillations, and in general the two
are in phase with the larger Ca2⫹ transient corresponding to the longer AP. An exception to this inphase relationship is observed during the 5:5 periodic
pattern shown in Fig. 7B. The AP labeled A (left) is
slightly longer than the AP labeled B (77.0 ms com-
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sion with 3Na⫹-1Ca2⫹ stoichiometry) of the Na⫹-Ca2⫹
exchanger INaCa (Fig. 5B). This results in an increase of
an outward repolarizing current. Similarly, elevated
[Na⫹]i increases the repolarizing Na⫹-K⫹ pump current (INaK), which has a 3Na⫹-2K⫹ stoichiometry (Fig.
5C). As a consequence of this increase in the total
repolarizing current secondary to [Na⫹]i accumulation,
APDmed decreases (downward shift of the APD restitution curve).
The slow component of the repolarizing delayed rectifier K⫹ current (IKs) (Fig. 5D) is reduced after 950
revolutions and therefore does not contribute to APD
shortening with time. The reduction of IKs is the result
of an increase in DI secondary to shortening of APD,
which allows IKs more time to deactivate after its
activation. The depolarizing L-type Ca2⫹ current
[ICa(L)] (Fig. 5E) does not contribute to the downward
shift of the restitution curve either. This is apparent in
Fig. 5E, which shows after 950 revolutions a larger
ICa(L) (arrow) during the plateau of a shorter AP (Fig.
5A).
It is important to note that in addition to [Na⫹]i
accumulation, there is also [Ca2⫹]i accumulation
([Ca2⫹]i,peak,med increases from 2.24 to 2.62 ␮M after
800 revolutions, not shown). When [Na⫹]i accumulation is prevented, [Ca2⫹]i does not accumulate either,
making it difficult to state whether [Na⫹]i accumulation leads to [Ca2⫹]i accumulation or vice versa. However, Ca2⫹-sensitive currents in the model [IKs, ICa(L)],
and forward-mode INaCa do not contribute to the downward shift of the restitution curve. Furthermore, resetting [Na⫹]i is sufficient to reverse the time-dependent
change in dynamics. These results suggest that [Na⫹]i
accumulation is primarily responsible for the downward shift of the restitution curve with time.
Temporal oscillations in AP properties at a single
site. Once bifurcation occurs (L ⬍ Lcrit), oscillations in
AP properties measured at a single cell (the temporal
domain) may assume a number of complicated patterns. Figure 6, left, shows temporal oscillations in
APD (Fig. 6A), DI (Fig. 6B), and CL (Fig. 6C). The
theoretical oscillatory patterns agree with experimental measurements (Fig. 6, right) conducted by Frame
and Simson (18) in the canine tricuspid orifice ring as
well as with theoretical studies performed in other ring
models (8, 21, 42). Importantly, the maximal percent
changes in APD and DI (33% and 153%, respectively)
during oscillations are much greater than the maximal
percent change in CL (6%). Furthermore, APD and DI
oscillations are 90° out of phase relative to the CL
oscillations (i.e., APD and DI oscillations are maximal
where CL oscillations are minimal, as indicated by
arrows in Fig. 6).
Examples of temporal oscillations in APD from the
short steady-state (Fig. 2) and the long steady-state
(Fig. 3) protocols are shown in Fig. 7 and may be
classified as either periodic or quasi-periodic. Only
APD (left) and [Ca2⫹]i,peak (right) are shown, but other
AP properties (DI, ␪, and CL) display a similar behavior. A periodic 2:2 behavior (stable beat-to-beat alternans) appears in Fig. 7A, and a 5:5 periodicity (pattern
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DYNAMICS OF ACTION POTENTIAL HEAD-TAIL INTERACTION
pared with 76.8 ms); however, A has a smaller peak
Ca2⫹ transient (right) than B (3.43 ␮M compared with
3.65 ␮M). Larger variations in APD and [Ca2⫹]i,peak
during the same 5:5 sequence are in phase.
Spatial oscillations of AP properties. To understand
the different temporal oscillation patterns presented in
Fig. 7, it is helpful to consider how AP properties
change in space along the entire pathway (spatial domain) during reentry. Figure 8 shows CL as a function
of location for consecutive revolutions of the reentrant
wave front under the same circumstances as in Fig. 7.
Importantly, Fig. 8 reveals that spatial oscillations in
CL are sinusoidal during reentry. Spatial oscillations
in APD, DI, ␪, and [Ca2⫹]i,peak are also sinusoidal (not
shown). The wavelength of spatial sinusoidal variation
of AP properties is referred to as the oscillation wavelength (⌳). The 2:2 temporal behavior in Fig. 7A is the
result of a spatial oscillation pattern with ⌳/L ⫽ 2/7
(Fig. 8A). In this case, two revolutions accommodate an
integer multiple of ⌳ (2L ⫽ 7⌳), which gives rise to
beat-to-beat alternans observed temporally at a single
site (in Fig. 8, values recorded at a single site for
consecutive revolutions are marked with an x).
Figure 8 also shows spatial oscillatory patterns from
the 5:5 periodic (Fig. 8B) and quasi-periodic (Fig. 8C)
Fig. 8. Spatial CL pattern along the reentry pathway during consecutive revolutions in a 70-cell (A), 61-cell (B), and 60-cell (C) pathway
corresponding to conditions and protocols in Fig. 7. L is marked by
circles at bottom of each figure, and wavelength (⌳) is shown above
each spatial CL oscillatory pattern. X, CL values at a single site for
consecutive revolutions.
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Fig. 7. Different temporal oscillatory behavior observed at a single
site during reentry. APD (left) and [Ca2⫹]i,peak (right) oscillations
were recorded from pathways with three different L values: A:
beat-to-beat AP alternans in a 70-cell reentry pathway; B: 5:5 periodic behavior (pattern repeats every five beats) in a 61-cell pathway
[*APA (APD ⫽ 77 ms and [Ca2⫹]i,peak ⫽ 3.4 ␮M) and APB (APD ⫽
76.8 ms and [Ca2⫹]i,peak ⫽ 3.6 ␮M)]; and C: quasi-periodic behavior
in a 60-cell pathway. Short steady-state protocol is used in A; long
steady-state protocol is used in B and C. Experimental CL oscillations are shown for comparison in A (18) and B (14). In these
simulations gj ⫽ 0.076 mS was used.
oscillatory regions in Fig. 3. In Fig. 8B, ⌳/L ⫽ 5/3 such
that 5 revolutions accommodate an integer multiple of
⌳ (5L ⫽ 3⌳), which produces the 5:5 periodic pattern at
a single site (Fig. 7B). For a shorter L (Fig. 8C), the
magnitude of CL oscillation increases, and ⌳/L decreases to slightly less than 5/3. Now five revolutions
fit slightly more than 3⌳, producing a shift in the
spatial oscillation pattern every revolution, which generates the quasi-periodic pattern at a single site (Fig.
7C).
Two factors are responsible for the formation of different temporal oscillatory patterns. First, increased
head-tail interaction increases CL oscillations and the
complexity of the spatial oscillation pattern (which
accounts for the difference between Fig. 7, B and C).
Second, ⌳ changes with time (the difference between
Fig. 7, A and B), a phenomenon that is discussed in the
next section.
Time dependence of oscillations for relatively small L
(strong head-tail interaction). The ⌳ of the spatial oscillation pattern from a 70-cell pathway (relatively
small L) increases from 0.29L (Fig. 9A, shown previously in Fig. 8A) after 150 revolutions to 0.39L after
another 760 revolutions (Fig. 9B). Simultaneously,
[Na⫹]i increases from 19.7 to 21.8 mM (not shown), and
the APD restitution curve shifts downward (Fig. 9B).
In contrast to the decrease of mmax observed in Fig. 4,
mmax increases from 1.31 to 1.51 in response to this
downward shift. Consequently, the difference (⌬APD)
between the maximum value of APD (APDmax) and the
minimum value of APD (APDmin) increases from 12.4
to 16.3 ms, reflecting an increase in the magnitude of
oscillations. Note that the data in Fig. 4 come from a
relatively long pathway (L ⫽ 80 cells) in the presence of
weak head-tail interaction. In contrast, Fig. 9 corre-
DYNAMICS OF ACTION POTENTIAL HEAD-TAIL INTERACTION
sponds to a relatively short pathway (L ⫽ 70 cells)
where there is strong head-tail interaction.
We provide the following hypothesis for the mechanism by which [Na⫹]i accumulation increases oscillations and ⌳ in a relatively short pathway. As discussed
previously, [Na⫹]i accumulation decreases APD at every DI (downward shift of the restitution curve; Fig. 9),
shifting the operating point to a larger DI. At very
short DIs (high degree of head-tail interaction), the
APD restitution curve begins to flatten (decreased m)
as DI decreases. Consequently, a slight shift of the
operating point to a larger DI increases m and the
magnitude of oscillations. The result is an increase in
⌬APD of the spatial APD oscillation pattern, which
increases spatial gradients of the membrane potential
(Vm) along the reentry pathway. However, electrotonic
interaction between neighboring cells limits the possible steepness of these gradients and therefore limits
⌬APD. As [Na⫹]i accumulation increases ⌬APD (Fig. 9)
and the limit on Vm gradients imposed by electrotonic
interaction is reached, ⌳ increases (Fig. 9B). This increase in ⌳ allows for APD to change from APDmax to
APDmin over a longer distance, reducing the steepness
of spatial gradients.
Support for our hypothesis is provided by the fact
that if [Na⫹]i is held constant during reentry, ⌳ and CL
oscillations do not change with time. If we allow [Na⫹]i
to accumulate but reduce the effect of electrotonic
interaction by reducing intercellular coupling (from
0.076 to 0 mS) away from the AP upstroke, CL oscillations increase slightly with [Na⫹]i accumulation (from
1.5 to 2.4 ms) but ⌳ does not change. It is important to
note that if we allow both [Na⫹]i and ⌳ to increase with
time (intercellular coupling increased back to 0.076
mS), then CL oscillations increase to 3.3 ms (compared
with 2.4 ms for shorter ⌳), suggesting that the increase
of ⌳ itself increases oscillations.
The increase in oscillations with [Na⫹]i accumulation for a high degree of head-tail interaction is dependent on flattening of the APD restitution curve for
short DIs. Experimental studies have shown a similar
flattening of the APD restitution curve for short DIs
and have attributed it to increased latency between the
stimulus and the AP upstroke (6, 24). In our study, the
restitution curve flattens because for very short DIs
the Ca2⫹ transient begins to decrease significantly,
which is in agreement with experimental observations
(26). A smaller peak Ca2⫹ transient results in reduced
Ca2⫹-dependent inactivation of ICa(L) (not shown) and
therefore an increase in depolarizing current during
the plateau of the AP, which acts to oppose APD shortening.
Elevated [K⫹]o. Experiments have shown that K⫹
accumulates in extracellular clefts during rapid pacing
(23). To investigate the effect of extracellular K⫹ concentration ([K⫹]o) accumulation on the dynamics of
reentry, we increase [K⫹]o from 4.5 mM (control) to 7
and 12 mM in a 78-cell pathway. In the control
([K⫹]o ⫽ 4.5 mM), oscillations during reentry disappear within 600 revolutions. When [K⫹]o is increased to
7 mM, the resting Vm depolarizes from ⫺83 to ⫺76 mV,
and oscillations persist for over 1,000 revolutions despite a decrease of APDmed from 72 to 63 ms (not
shown). If [K⫹]o is elevated instead to 12 mM, the
resting Vm depolarizes to ⫺65 mV, and reentry terminates within one revolution. Thus elevated [K⫹]o acts
to enhance oscillations and destabilize reentry.
DISCUSSION
Summary of findings. Important findings of this
study are: 1) a high degree of head-tail interaction
produces oscillations in AP properties and in the Ca2⫹
transient; 2) the Ca2⫹ oscillations are generally in
phase with APD oscillations; 3) the magnitude of Ca2⫹
oscillations is greater than the magnitude of APD oscillations for moderate degrees of head-tail interaction;
4) [Na⫹]i accumulation shifts the APD restitution
curve downward to smaller APD values which, depending on the degree of head-tail interaction, may either
stabilize or destabilize reentry; 5) for a relatively low
degree of head-tail interaction, accumulation of [Na⫹]i
tends to stabilize the dynamics of the reentrant AP; 6)
for a higher degree of head-tail interaction, [Na⫹]i
accumulation is accompanied by increased wavelength
of spatial oscillations, which augments CL oscillations;
and 7) elevated [K⫹]o increases oscillations and destabilizes reentry.
Changes in [Na⫹]i alter the dynamics of reentry. We
find that the behavior of reentry evolves in time due to
accumulation of [Na⫹]i. Previous experimental and
theoretical studies have shown that the APD restitution curve shifts downward (to smaller APD values) in
response to an increase in pacing rate (5, 46). It has
been proposed that the mechanism for this downward
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Fig. 9. Time dependence of oscillations in a relatively small reentry
pathway (L ⫽ 70 cells). APD restitution curves (left) and APD along
reentry pathway for 1 revolution (right) are calculated after 100
revolutions (A) and after another 760 revolutions (B).
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DYNAMICS OF ACTION POTENTIAL HEAD-TAIL INTERACTION
shift is either [Ca2⫹]i accumulation (5, 46) or [K⫹]o
accumulation (5). In this study we observe that while
[Ca2⫹]i accumulates along with [Na⫹]i, [Na⫹]i is the
primary cause of APD shortening and the downward
shift of the restitution curve. This downward shift may
shift the point about which APD oscillates (the operating point) to a less steep (Fig. 4) or more steep (Fig. 9)
portion of the restitution curve, depending on the degree of head-tail interaction. Therefore, this shift may
either suppress or augment AP oscillations.
A schematic is used in Fig. 10 to summarize the
effects of [Na⫹]i accumulation on the APD restitution
curve and the dynamics of reentry for low and high
degrees of head-tail interaction. The APD restitution
curve in the presence of [Na⫹]i elevation (dashed line)
is shifted downward relative to the control curve (solid
line). The operating point about which APD and DI
oscillate is determined by the intersection of the line
APD ⫽ ⫺DI ⫹ CL (dashed-dotted line) with the restitution curve. For a low degree of head-tail interaction
(CLA), the operating point occurs where m ⫽ 1 in the
control case but where m ⬍ 1 in the presence of elevated [Na⫹]i, which dampens oscillations. For a high
degree of head-tail interaction (CLB ⬍ CLA), the operating point in the presence of elevated [Na⫹]i occurs
where the slope is steeper than in the control case,
which increases the amplitude of oscillations. The
schematic in Fig. 10 illustrates the differential effect
[Na⫹]i accumulation has on the dynamics of reentry
depending on the degree of head-tail interaction in the
pathway.
Accumulation of [Na⫹]i is known to promote spontaneous activity, which may trigger an arrhythmia (12,
35). Our results illustrate how [Na⫹]i accumulation
may affect the time course of an arrhythmia once
⌳k ⬇
2L
4L 2 ␣
⫺
,
2k ⫹ 1 共2k ⫹ 1兲 3 ␲ 2
k ⫽ 0, 1, 2, . . .
(1)
where ␣ is proportional to the slope of the dispersion
relation (␪ as a function of DI) and k is the mode
number. Because k can assume any positive integer
value, an infinite number of oscillatory modes can
occur in the delay-difference model. The spatial oscillation patterns with different ⌳ shown in Fig. 8 correspond to different eigenmodes in this equation. The
implication of the possible existence of many oscillatory modes is discussed in the following text.
In our simulations, we observe that a longer ⌳/L
promotes CL oscillations (Fig. 9 and corresponding
text). Fig. 11 presents two hypothetical spatial oscillation patterns in 1/␪ (the reciprocal of conduction velocity) to explain this behavior. The pattern in Fig. 9A has
a relatively long ⌳ relative to L, whereas that in Fig.
9B has a short ⌳. It is assumed that the magnitude of
Fig. 11. Mechanism by which a spatial oscillation pattern with a
longer ⌳ yields greater CL oscillations. Reciprocal of conduction
velocity, 1/␪, is plotted around the reentry pathway for one revolution. Spatial oscillation patterns with long ⌳ (A) and short ⌳ (B) are
presented. CL in each panel is area beneath the curve, and shaded
area represents the amount by which CL is greater than the intermediate CL value.
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Fig. 10. Effect of [Na⫹]i accumulation on dynamics of reentry in the
presence of weak head-tail interaction (CLA) and strong head-tail
interaction (CLB) (CLB ⬍ CLA). Control APD restitution curve (solid
line) and restitution curve in the presence of elevated [Na⫹]i (dashed
line) are shown. Operating point is defined as the intersection of line
APD ⫽ ⫺DI ⫹ CL (dashed-dotted line) and the restitution curve.
Circle on each restitution curve indicates where slope ⫽ 1. For an
operating point located right (larger DI) of the circle, no oscillations
occur because the slope of the restitution curve is ⬍1.
initiated. This effect can be stabilizing or destabilizing,
depending on the degree of head-tail interaction of the
circulating AP. For relatively slow tachycardias (weak
head-tail interaction), the effect may be stabilization of
reentry due to a shift of the operating point to a less
steep portion of the restitution curve. In contrast,
[Na⫹]i accumulation may act to destabilize reentry
during rapid tachycardias (strong head-tail interaction) leading to either its termination or transition to
fibrillation.
Longer ⌳ leads to greater CL oscillations. We observe
that [Na⫹]i accumulation with time produces spatial
oscillation patterns with different ⌳s. The existence of
different ⌳s has been predicted by Courtemanche,
Glass, and Keener (8), who proposed a delay-difference
equation to describe the evolution of AP properties as
the reentry wave front propagates. They solved for all
⌳ values (eigenmodes, Eq. 1), which yield solutions to
the delay-difference equation once bifurcation occurs
DYNAMICS OF ACTION POTENTIAL HEAD-TAIL INTERACTION
the oscillations in ␪ in Fig. 9A and B, is the same and
that the only difference is in ⌳. CL for one revolution is
given by
CL ⫽
兰
x
x⫺L
1
䡠 ds
␪
(2)
[Ca2⫹]i oscillations has been suggested as a mechanism
(22, 25). Some studies have hypothesized that mechanical alternans generates electrical alternans (20, 25,
41), and others have proposed the converse to be true
(27). The hypothesis that mechanical alternans causes
electrical alternans is supported by observations of
mechanical alternans in the absence of electrical alternans (41). In our study oscillations in APD and
[Ca2⫹]i,peak always appear together. However, over a
certain range of head-tail interaction, percent changes
in [Ca2⫹]i,peak during oscillations are much larger than
percent changes in APD, which may explain why mechanical alternans has been observed without detection of electrical alternans. In general, APD and the
Ca2⫹ transient oscillate in phase (a large Ca2⫹ transient corresponds to a long APD). However, occasionally an out-of-phase relationship (a large Ca2⫹ transient with a short APD) may occur (Fig. 7B).
Study limitations. The limitations of this study were
the following. The ring model in this study is used to
investigate an important property of reentrant excitation, namely the interaction between the head and tail
of the reentrant AP. The center (core) of the reentry
pathway in this model of “anatomical reentry” (16) is
inexcitable and represents an anatomical obstacle. In
functional forms of reentry, such as leading-circle or
anisotropic reentry (2, 10) or during spiral-wave activity (13), the core is excitable. In the leading-circle
concept, the core is created by the collision of centripetal wavelets generated by the reentrant wave front
propagating along the smallest possible pathway (the
“leading circle”) (2). Although the nature of the core is
different, important properties of leading-circle reentry are represented in the ring model. For example,
there is no excitable gap in leading-circle reentry because of a high degree of head-tail interaction (2), as is
the case for reentry in a small pathway in our simulations.
Another limitation of our model is the absence of
heterogeneities in cell properties [e.g., midmyocardial
M cells (43)] or tissue architecture [e.g., anisotropy due
to fiber orientation (10)]. Although such inhomogeneities will affect the oscillatory patterns observed during
reentry, the principles established here regarding the
dynamics of head-tail interaction and its time dependence can provide insight into these phenomena in the
inhomogeneous myocardium.
In the spiral-wave concept, a high degree of curvature at the tip of the reentrant wave front creates a
source-sink mismatch that leaves a central core of
unexcited but excitable tissue (13). The absence of an
unexcited but excitable core in our model is significant
because such a core influences the reentrant AP via
electrotonic interaction (3). Nevertheless, the phenomenon of head-tail interaction explored here occurs
along the arm of a spiral wave (21). Furthermore, in
many instances, a drifting spiral wave anchors to an
anatomical obstacle, such as a small artery (9). In such
situations, the distinction between anatomical and spiral-wave reentry becomes less obvious.
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where x is the current position of the wave front. The
CL calculated in Fig. 9A and B, is therefore greater
than the intermediate CL (corresponding to no oscillations with a fixed ␪ ⫽ ␪med) by an amount equal to the
area of the shaded region. The larger shaded area in
Fig. 9A implies that a spatial oscillation pattern with a
longer ⌳ augments CL oscillations.
Experiments have shown that the existence of CL
oscillations favors spontaneous termination of reentry
(14, 17, 18, 32). In our study, we find that [Na⫹]i
accumulation with time increases ⌳ of a spatial oscillation pattern, which in turn augments CL oscillations
(Fig. 9). This result suggests that lengthening of ⌳ with
time (transition from one eigenmode to another) could
be one mechanism by which reentry spontaneously
terminates. Furthermore, it has been shown that AP
oscillations in a one-dimensional ring occur for the
same rotation periods as does spiral-wave breakup in
two-dimensional models, a phenomenon that produces
a disordered fibrillatory state (21, 44). This suggests
that lengthening of ⌳ with time and the consequent
increase in oscillations may promote spiral-wave
breakup and could be one mechanism for the transition
from ventricular tachycardia to fibrillation.
Effect of elevated [K⫹]o on the dynamics of reentry. A
previous study from our laboratory (39) showed that
elevated [K⫹]o, by depolarizing the resting membrane
potential, caused prolonged postrepolarization refractoriness as a result of prolonged recovery from inactivation of INa following an AP. During reentry, this
phenomenon acts to prolong the tail of refractoriness
and reduced excitability that follow the AP; thus the
degree of head-tail interaction increases despite APD
shortening. In this study, we observe that a moderate
elevation of [K⫹]o promotes AP oscillations, although a
more severe elevation leads to termination of reentry.
The simulated values of elevated [K⫹]o (7 and 12 mM)
are representative of hyperkalemia during acute ischemia (39). The results suggest that hyperkalemia may
have a destabilizing effect on reentry and may contribute to its termination or its disintegration into multiple
reentrant circuits and a fibrillatory state.
Electrical and mechanical oscillations. The LRd
model computes both the AP and the Ca2⫹ transient.
This enables us to study oscillations in APD and
[Ca2⫹]i for different degrees of head-tail interaction.
Studies suggest that T wave alternans in the electrocardiogram reflects APD oscillations and indicates an
increased risk for cardiac arrhythmia and sudden
death (20, 37, 40). Similar to APD alternans, beat-tobeat alternation in the force generated by cardiac muscle (mechanical alternans) has also been observed at
rapid rates (25), and intracellular Ca2⫹ cycling causing
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DYNAMICS OF ACTION POTENTIAL HEAD-TAIL INTERACTION
This study was supported by Grants R01-HL-49054 and R37-HL33343 (to Y. Rudy) from the National Heart, Lung, and Blood
Institute and by a Whitaker Foundation Development Award.
REFERENCES
1. Aggarwal R and Boyden PA. Diminished Ca2⫹ and Ba2⫹
currents in myocytes surviving in the epicardial border zone of
the 5-day infarcted canine heart. Circ Res 77: 1180–1191, 1995.
2. Allessie MA, Bonke FI, and Schopman FJ. Circus movement
in rabbit atrial muscle as a mechanism of tachycardia. III. The
“leading circle” concept: a new model of circus movement in
cardiac tissue without the involvement of an anatomical obstacle. Circ Res 41: 9–18, 1977.
3. Beaumont J, Davidenko N, Davidenko JM, and Jalife J.
Spiral waves in two-dimensional models of ventricular muscle:
formation of a stationary core. Biophys J 75: 1–14, 1998.
4. Beeler GW and Reuter H. Reconstruction of the action potential of ventricular myocardial fibres. J Physiol (Lond) 268: 177–
210, 1977.
5. Boyett MR and Jewell BR. A study of the factors responsible
for rate-dependent shortening of the action potential in mammalian ventricular muscle. J Physiol (Lond) 285: 359–380, 1978.
6. Chialvo DR, Michaels DC, and Jalife J. Supernormal excitability as a mechanism of chaotic dynamics of activation in
cardiac Purkinje fibers. Circ Res 66: 525–545, 1990.
7. Cohen CJ, Fozzard HA, and Sheu SS. Increase in intracellular sodium ion activity during stimulation in mammalian cardiac
muscle. Circ Res 50: 651–662, 1982.
8. Courtemanche M, Glass L, and Keener JP. Instabilities of a
propagating pulse in a ring of excitable media. Physiol Rev 70:
2182–2184, 1993.
9. Davidenko JM, Pertsov AV, Salomonsz R, Baxter W, and
Jalife J. Stationary and drifting spiral waves of excitation in
isolated cardiac muscle. Nature 355: 349–351, 1992.
10. Dillon SM, Allessie MA, Ursell PC, and Wit AL. Influences of
anisotropic tissue structure on reentrant circuits in the epicardial border zone of subacute canine infarcts. Circ Res 63: 182–
206, 1988.
11. El-Sherif N, Scherlag BJ, Lazzara R, and Hope RR. Reentrant ventricular arrhythmias in the late myocardial infarction period. 1. Conduction characteristics in the infarction zone.
Circulation 55: 686–702, 1977.
12. Faber GM and Rudy Y. Action potential and contractility
changes in [Na⫹]i overloaded cardiac myocytes: a simulation
study. Biophys J 78: 2392–2404, 2000.
13. Fast VG and Kleber AG. Role of wave front curvature in
propagation of cardiac impulse. Cardiovasc Res 33: 258–271,
1997.
14. Fei H, Yazmajian D, Hanna MS, and Frame LH. Termination of reentry by lidocaine in the tricuspid ring in vitro. Role of
cycle-length oscillation, fast use-dependent kinetics, and fixed
block. Circ Res 80: 242–252, 1997.
15. FitzHugh RA. Impulses and physiological states in theoretical
models of nerve membrane. Biophys J 1: 445–466, 1961.
16. Frame LH, Page RL, Boyden PA, Fenoglio JJ Jr, and
Hoffman BF. Circus movement in the canine atrium around the
tricuspid ring during experimental atrial flutter and during
reentry in vitro. Circulation 76: 1155–1175, 1987.
17. Frame LH and Rhee EK. Spontaneous termination of reentry
after one cycle or short nonsustained runs. Role of oscillations
and excess dispersion of refractoriness. Circ Res 68: 493–502,
1991.
18. Frame LH and Simson MB. Oscillations of conduction, action
potential duration, and refractoriness. A mechanism for spontaneous termination of reentrant tachycardias. Circulation 78:
1277–1287, 1988.
19. Gaspo R, Bosch RF, Bou-Abboud E, and Nattel S. Tachycardia-induced changes in Na⫹ current in a chronic dog model of
atrial fibrillation. Circ Res 81: 1045–1052, 1997.
20. Hirayama Y, Saitoh H, Atarashi H, and Hayakawa H.
Electrical and mechanical alternans in canine myocardium in
vivo. Dependence on intracellular calcium cycling. Circulation
88: 2894–2902, 1993.
21. Karma A. Spiral breakup in model equations of action potential
propagation in cardiac tissue. Physiol Rev 71: 1103–1106, 1993.
22. Kihara Y and Morgan JP. Abnormal Cai2⫹ handling is the
primary cause of mechanical alternans: study in ferret ventricular muscles. Am J Physiol Heart Circ Physiol 261: H1746–
H1755, 1991.
23. Kline RP and Morad M. Potassium efflux in heart muscle
during activity: extracellular accumulation and its implications.
J Physiol (Lond) 280: 537–558, 1978.
24. Koller ML, Riccio ML, and Gilmour RF Jr. Dynamic restitution of action potential duration during electrical alternans
and ventricular fibrillation. Am J Physiol Heart Circ Physiol 275:
H1635–H1642, 1998.
25. Lab MJ and Lee JA. Changes in intracellular calcium during
mechanical alternans in isolated ferret ventricular muscle. Circ
Res 66: 585–595, 1990.
26. Lipsius SL, Fozzard HA, and Gibbons WR. Voltage and time
dependence of restitution in heart. Am J Physiol Heart Circ
Physiol 243: H68–H76, 1982.
27. Lu HH, Lange G, and Brooks CM. Comparative studies of
electrical and mechanical alternation in heart cells. J Electrocardiol 1: 7–17, 1968.
28. Luo CH and Rudy Y. A dynamic model of the cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes. Circ Res 74: 1071–1096, 1994.
29. Luo CH and Rudy Y. A model of the ventricular cardiac action
potential. Depolarization, repolarization, and their interaction.
Circ Res 68: 1501–1526, 1991.
30. Nagumo J, Arimoto S, and Yoshizawa S. An active pulse
transmission line simulating nerve axon. Proc IRE 50: 2061–
2070, 1962.
31. Nolasco JB and Dahlen RW. A graphic method for the study
of alternation in cardiac action potentials. J Appl Physiol 25:
191–196, 1968.
32. Ortiz J, Igarashi M, Gonzalez HX, Laurita K, Rudy Y, and
Waldo AL. Mechanism of spontaneous termination of stable
atrial flutter in the canine sterile pericarditis model. Circulation
88: 1866–1877, 1993.
33. Quan W and Rudy Y. Unidirectional block and reentry of
cardiac excitation: a model study. Circ Res 66: 367–382, 1990.
Downloaded from http://ajpheart.physiology.org/ by 10.220.33.5 on July 12, 2017
The use of a one-dimensional model of reentry allowed us to use a detailed model of the cardiac cell that
accounts for dynamic changes of ionic concentrations
(e.g., [Na⫹]i accumulation) over many reentrant cycles.
With this approach, we could characterize the temporal evolution of the AP and the dynamics of reentry.
The focus of this study is on head-tail interaction, AP
oscillations (alternans), and stability of reentrant activity. Other properties of reentrant excitation, such as
the ionic activity in the core of a spiral wave or the
effects of anisotropy, must be studied using higherdimensional models. The insights obtained from the
present study can guide such simulations where the
use of a detailed cell model during many reentry cycles
still constitutes a major computational challenge.
Moreover, the time scale investigated in this study is
still relatively short. It allows for ion accumulation to
occur but does not take into account changes due to
electrical remodeling (1, 19, 45) that occur over a much
longer time frame. Remodeling involves changes in ion
channels, gap junctions, and tissue structure. Many of
these changes can be incorporated within the framework of the detailed cell model used here and will be an
important subject of future modeling studies.
DYNAMICS OF ACTION POTENTIAL HEAD-TAIL INTERACTION
41. Spear JF and Moore EN. A comparison of alternation in
myocardial action potentials and contractility. Am J Physiol 220:
1708–1716, 1971.
42. Vinet A and Roberge FA. The dynamics of sustained reentry
in a ring model of cardiac tissue. Ann Biomed Eng 22: 568–591,
1994.
43. Viswanathan PC, Shaw RM, and Rudy Y. Effects of IKr and
IKs heterogeneity on action potential duration and its rate dependence: a simulation study. Circulation 99: 2466–2474, 1999.
44. Weiss JN, Garfinkel A, Karagueuzian HS, Qu Z, and Chen
PS. Chaos and the transition to ventricular fibrillation: a new
approach to antiarrhythmic drug evaluation. Circulation 99:
2819–2826, 1999.
45. Wijffels MC, Kirchhof CJ, Dorland R, and Allessie MA.
Atrial fibrillation begets atrial fibrillation. A study in awake
chronically instrumented goats. Circulation 92: 1954–1968,
1995.
46. Zeng J, Laurita KR, Rosenbaum DS, and Rudy Y. Two
components of the delayed rectifier K⫹ current in ventricular
myocytes of the guinea pig type. Theoretical formulation and
their role in repolarization. Circ Res 77: 140–152, 1995.
Downloaded from http://ajpheart.physiology.org/ by 10.220.33.5 on July 12, 2017
34. Quan WL and Rudy Y. Termination of reentrant propagation
by a single stimulus: a model study. Pacing Clin Electrophysiol
14: 1700–1706, 1991.
35. Ravens U and Himmel HM. Drugs preventing Na⫹ and Ca2⫹
overload. Pharmacol Res 39: 167–174, 1999.
36. Rogers JM, Huang J, Smith WM, and Ideker RE. Incidence,
evolution, and spatial distribution of functional reentry during
ventricular fibrillation in pigs. Circ Res 84: 945–954, 1999.
37. Rosenbaum DS, Jackson LE, Smith JM, Garan H, Ruskin
JN, and Cohen RJ. Electrical alternans and vulnerability to
ventricular arrhythmias. N Engl J Med 330: 235–241, 1994.
38. Shaw RM and Rudy Y. Electrophysiologic effects of acute
myocardial ischemia. A mechanistic investigation of action potential conduction and conduction failure. Circ Res 80: 124–138,
1997.
39. Shaw RM and Rudy Y. Electrophysiologic effects of acute
myocardial ischemia: a theoretical study of altered cell excitability and action potential duration. Cardiovasc Res 35: 256–272,
1997.
40. Smith JM, Clancy EA, Valeri CR, Ruskin JN, and Cohen
RJ. Electrical alternans and cardiac electrical instability. Circulation 77: 110–121, 1988.
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