TX 72514Circulation Research is published by the American Heart Association. 7272 Greenville Avenue, Dallas,
1992, 71:260-270Circulation Research
http://circres.ahajournals.org/content/71/2/260located on the World Wide Web at:
The online version of this article, along with updated information and services, is
http://www.lww.com/reprintsReprints: Information about reprints can be found online at [email protected]. E-mail:
Fax:Kluwer Health, 351 West Camden Street, Baltimore, MD 21202-2436. Phone: 410-528-4050. Permissions: Permissions & Rights Desk, Lippincott Williams & Wilkins, a division of Wolters http://circres.ahajournals.org//subscriptions/Subscriptions: Information about subscribing to Circulation Research is online at
by guest on July 10, 2011http://circres.ahajournals.org/Downloaded from
Reflection After Delayed Excitation in aComputer Model of a Single Fiber
Candido Cabo and Roger C. Barr
Reflection (reflected reentry) is a case of reentry in a one-dimensional structure, divided into proximaland distal segments, in which tissue excited by a wave front propagating in a forward direction is reexcitedby electrical activity coming backward from the original direction of propagation. Cases of reflection havebeen demonstrated in Purkinje fibers and in ventricular muscle preparations containing multiple fibers.Several mechanisms possibly responsible for reflected reentry have been proposed. However, the difficultyin the interpretation of the experimental results, as well as the limited number of different conditions inwhich reflection was obtained, has kept open the question about conditions and mechanisms for reflection.We have developed a computer model in which reflection occurs. The model involves a single fiber anduses the DiFrancesco-Noble equations for the Purkinje fiber to model the ionic currents. The results showthat reflection is possible in a single fiber and that diastolic depolarization (automaticity) is not arequirement for reflection. Active membrane responses to a just-above-threshold stimulus were importantfor achieving the necessary time delay. Systematic simulations showed further that reflection occurredonly when the right coupling conditions linked a short or long proximal fiber to a short distal segment.(Circulation Research 1992;71:260-270)KEY WoRDs * reflection * delayed excitation * reentry * DiFrancesco-Noble model
R eflection, or "reflected" reentry, is a specialtype of reentrant circuit in a one-dimensionalstructure, in which tissue excited by a wave
front propagating in a forward direction is reexcited byelectrical activity (reflected wave front) coming backfrom the direction of propagation.1-7When propagationin both directions is over the same fibers (rather thanover different fibers in a bundle), reflection is calledtrue reflection.Even though experiments on reflection started early
in the century,8-12 the recent approach to the study ofreflection was first initiated in the early seventies byCranefield and colleagues13-15 and Wit and col-leagues,16'7 who studied the propagation characteristicsof Purkinje fibers with depressed excitability. In 1971,Cranefield, Hoffman, and collaborators13-5 showed thatsegments of Purkinje fibers, with normal conductionvelocities of 2-4 m/sec, can conduct with apparentvelocities of 0.01-0.1 m/sec when encased in high K'agar. In 1972, Wit, Cranefield, and Hoffman'7 demon-strated reentry in small loops (12-35 mm) of canine andbovine Purkinje fibers depressed by a solution contain-
This manuscript was sent to Brian F. Hoffman, ConsultingEditor, for review by expert referees, editorial decision, and finaldisposition.From the Departments of Biomedical Engineering and Pediat-
rics, Duke University, Durham, N.C.Supported in part by US Public Health Service awards HL-
11307 and HL-33637. C.C. was supported in part with a fellowshipfrom Ministerio de Educaci6n y Ciencia (Becas para Doctores yTecn6logos) of the Government of Spain and by NSF/ERC grantCDR-8622201.Address for correspondence: Roger C. Barr, Professor, Depart-
ment of Biomedical Engineering, Duke University, 136 Engineer-ing Building, Durham, NC 27706.
Received September 16, 1991; accepted March 6, 1992.
ing a very high concentration of K'. Using unbranchedbundles of Purkinje tissue with a central segment withdepressed excitability, they observed'6 a phenomenonsimilar to the one reported by Schmitt and Erlanger10 inwhich an impulse (stimulated) propagating in one di-rection of the bundle is followed by an impulse (reflect-ed, nonstimulated) traveling in the opposite direction(return extrasystole). They proposed that the reflectedresponse was caused by reentry at the level of thesyncytial interconnections (microreentry), an explana-tion identical to the one used by Schmitt and Erlangerto interpret their observations. They also suggested thepossibility of true reflection based on the observationthat a fast action potential upstroke occurred in thedepressed segment after the action potential in thetissue beyond the depressed segment."3"16
In trying to demonstrate true reflection, a series ofremarkable experiments were performed by Antzelev-itch, Jalife, Moe, and associates in Purkinje,'8-21 ven-tricular,22 and atrial23 muscle. They used a three-com-partment tissue bath in which a zone of conductionblock was created in the central segment either byperfusion with a solution containing a high concentra-tion of potassium (central zone inexcitable) or by per-fusing the central compartment with isotonic sucrose(electrical insulation of the extracellular space). Theessential feature of both experimental conditions is thatthe central compartment can transmit only electrotonicpotentials. The proposed mechanism for reflection wasthat the electrotonic potential transmitted through theinexcitable gap excites the distal tissue after a delay; theactive response in the distal tissue causes an electro-tonic potential that is transmitted over the same inex-citable gap and reexcites the proximal tissue. The delayacross the inexcitable gap was long enough to allow the
by guest on July 10, 2011http://circres.ahajournals.org/Downloaded from
Cabo and Barr Reflection After Delayed Excitation 261
proximal tissue to repolarize. More recently, reflectionhas been suggested as the mechanism underlying extra-systolic activity in ventricular tissue excised from a1-day-old infarcted canine heart24 and in a clinical caseof incessant ventricular bigeminy in a patient with noevidence of organic heart disease.25
Despite the elegance of the experiments performedby Antzelevitch, Jalife, Moe, and associates,18-23 inter-pretation of the experimental results is complicated bytwo factors.2 First, it is well known that subthresholdpotentials affect the automatic pacemaker of Purkinjefibers.26,27 When they occur early in the cycle, thespontaneous discharge will be delayed, and when theyoccur late in the cycle, the spontaneous discharge isaccelerated. Therefore, the reflected propagation mightbe instead a consequence of the automatic pacemakerof Purkinje fibers in the distal segment, modified byelectrotonic interaction. This premise was supported bya computer modeling study of reflection28 in which, forreflection to occur, it was necessary to incorporatepacemaking properties in the distal elements. On theother hand, the demonstration of reflection in strips ofventricular muscle (no diastolic depolarization) byRozanski et a122 and in thin strands of Purkinje fibershomogeneously bathed with a solution containing a highconcentration of potassium7 indicated that diastolicdepolarization is not a requirement for reflection.The second complicating factor is that inhomogene-
ities in the inexcitable gap or at the boundaries betweenthe depressed segment with the normally excitabletissue may affect the interpretation of the experimentalresults. When asymmetric depression is present in theinexcitable gap, microreentry of the type described bySchmitt and Erlanger10 may occur.2 When inhomogene-ities exist at the boundaries, transmission between theproximal and distal excitable segments might not bepurely electrotonic but a combination of electrotonicpropagation and slow conduction.2Another important question relates to the implica-
tions of these in vitro studies for the whole heart. Thepreparations in which reflection was demonstrated wereisolated bundles of Purkinje fibers shorter than 10 mm,with the inexcitable zone (sucrose gap) in the middle.The fact that the size of the excitable segments of tissuewere of the order of one or two (resting) space constants(2-4 mm) raises questions as to whether reflection ispossible in long fibers. Furthermore, the use of singletransmembrane recordings in the excitable segmentsmakes it difficult to decide if the reflected responses areindeed propagated or just electrotonic responses.To deal with some of the questions above, we devel-
oped a computer model in which reflection occurs underspecific conditions. The model consists of a single fiberwhose ionic currents are represented by the Di-Francesco-Noble equations.29 The DiFrancesco-Noblemodel was chosen because it accurately reproduces thebehavior of real Purkinje tissue during the refractoryperiod and with respect to electrical stimulation.30
Materials and Meths-Propagation and Membrane Models
If propagation is planar, a fiber in a multifiber prepa-ration can be modeled satisfactorily by a one-dimensional
core-conductor model in which the extracellular imped-ance depends on the depth of the fiber considered.31,32The model used was a continuous cable, described by
the following equation:
a dx 1 aVm(xt)2 ax Ri(x) ax
aVm(x,t)=Iion+Cm' at (1)
where I.. is the transmembrane current (,gA/cm2), a isthe fiber radius (0.001 cm), Ri(x) is the intracellularresistance (0.250 kflcm), Vm is the transmembranepotential (mV), lio, is the DiFrancesco-Noble ioniccurrent (,gA/cm2), and Cm is the specific membranecapacitance (1.2 uF/cm2).The cable equation was made discrete with a space
step of 100 gm and a time step of 10 ,usec. The numberof nodes depended on the simulation. The ends of thefiber were considered sealed (i.e., there was no intra-cellular current), so at both ends the boundary condi-tion was aVm/ax=0. The method to numerically solveEquation 1 has been described elsewhere.30
In this model, the intracellular and extracellularimpedances are purely resistive, and the membraneimpedance is a simple capacitance (as opposed to twocapacitances33'34) in parallel with the ionic currentsdescribed by the DiFrancesco-Noble model of the mem-brane.29 Despite the discontinuous nature of propaga-tion in cardiac muscle35-37 and because of the lowresistance of the gap junctions,38'39 we lumped thejunctional resistance in with the intracellular axial re-sistance. The presence of capacitative effects in the gapjunction38'40 were neglected. The maximum conduc-tance for the sodium current was set to 16.87 mS/cm2,one and a half times the standard conductance in theDiFrancesco-Noble model, so that the maximum rate ofmembrane depolarization fell into the normal range.
Reflection ModelTo obtain reflection, incomplete isolation of fiber
segments is needed. In the experimental studies,18-22this zone was created by perfusing the central segmentof a three-compartment chamber with either isotonicsucrose (sucrose gap model) or a solution containing avery high concentration of potassium (ischemic model).The effect of the isotonic sucrose is the conversion ofthe extracellular space in the sucrose region into anonconducting zone. Propagation between the excitablesegments can be restored by connecting the extracellu-lar spaces by an external resistance.
In our simulations, the sucrose gap was representedby a segment with high axial resistance (gap resistance).The proximal segment was the one where the externalstimulus was applied (unless otherwise stated). Thedelay between the proximal and distal segments (P-Ddelay) was controlled in one of two ways: first, byadjusting the resistance of the segment simulating thesucrose gap; or second, by changing the coupling inter-val of a premature beat.
ResultsMechanism of Reflection in a Single FiberTo study whether reflection was possible in a single
fiber, we simulated a fiber that was near the size ofexperimental preparations. It had a length of 8 mm (80nodes) and proximal and distal segments both 4 mm (40
by guest on July 10, 2011http://circres.ahajournals.org/Downloaded from
A B c D E delay required for reflection in the experimental prep-arations.18-21 A still further decrease in coupling (withrespect to that required for reflection) resulted in an
action potential in the distal segment with no reflectedresponse in the proximal segment (Figure 1, column D).This phenomenon might be the same as the "silent"period observed in studies of reflection in ventricularmuscle.22 A further slight decrease in electrical couplingresulted in block, where no action potential was pro-duced in the distal segment (Figure 1, column E). Notethat in this case the electrotonic effect of the distalsegment caused a shortening of the action potential innodes of the proximal segment close to the gap.Some conclusions can be extracted from the results in
Figure 1 about the possibility of reflection. First, clearlyit was possible to evoke a (true) reflected response in asingle fiber (Figure 1, column C), since the modelstructure precluded microreentry. Second, diastolic de-polarization was not a requirement for reflection, sincethe diastolic depolarizing current was inactivated duringthe simulations. Some conclusions can also be extracted
5WmvV 2 about the mechanisms for reflection. First, the P-Ddelay had to be long enough to allow the tissue in theproximal segment to recover its excitability, or no
Recordings showing transmembrane voltages cal- reflection occurred (Figure 1, column B). Second, thefiber with a length of8 mm divided into proximal distal segment had to be able to (re)excite the proximalegments, each 4 mm (40 nodes) long, for increas- segment. P-D delay alone was not enough to obtainof the resistance between the proximal and distal reflection (Figure 1, column D). It is also interestingWolumn A, 0.250 kfQcm; column B, 10.5 kfQcm; that for long delays between action potentials in the10.9438 kfQcm; column D, 10.94381 kfQcm; proximal and distal sites (Figure 1, columns C and D)
10.9439 kfQcm. Rows show transmembrane volt- the action potential in the distal site actually originatedbeginning of the proximal segment (row 1), at the away from the gap, as seen from the fact the action,roximal segment (row 2), at the beginning of the potentials at nodes close to the gap (e.g., node 41)ent (row 3), and at the end of the distal segment occurred later than at nodes far away from the gap (e.g.,)r each resistance, a stimulus was applied at time node 80). This phenomenon also was observedbeginning of the proximal segment (row 1). experimentally.22
nodes) long. In all the simulations the diastolic depo-larization current was set to zero so that there was no
automaticity. The initial conditions for each node of thesimulated fiber correspond to those occurring 600 msec
after an action potential. In Figure 1, a progressiveincrease in the intracellular resistance between nodes40 and 41 (to simulate the sucrose gap) is shown in fivesteps. Each column shows the Vm at the beginning of theproximal segment (node 1, row 1), at the end of theproximal segment (node 40, row 2), at the beginning ofthe distal segment (node 41, row 3), and at the end ofthe distal segment (node 80, row 4). Column A shows a
control fiber in which the gap resistance is equal to thestandard intracellular resistance and all action poten-tials (rows 1-4) are almost the same. As uncouplingincreased between the proximal and distal segments,P-D delay increased (Figure 1, column B). In this case,
the P-D delay was not enough to allow the proximalsegment to recover its excitability, but the action poten-tial in the distal segment caused an electrotonic prolon-gation of the action potential in the proximal segment.
Reflection occurred when P-D delay (-300 msec)was enough to allow the proximal segment to recover itselectrical excitability (Figure 1, column C). The toptracing shows that the reflected response caused prop-agation in the proximal segment. The P-D delay re-
quired for reflection in the model is similar to the P-D
Importance of the GeometryThe previous section showed that reflection occurred
between two nodes of a cable with proximal and distalsegments of 4 mm. The fact that most of the experimen-tal studies in which reflection was demonstrated18-22were preparations whose proximal and distal segmentswere short (one to two resting space constants forPurkinje fibers, 2-4 mm) points to a possible role ofsegment length in the occurrence of reflection. There-fore, to study the importance of segment length onreflection, we tried to induce reflection in simulatedfibers with proximal and distal segments of variouslengths. The results are shown in Figure 2, where lengthcombinations are grouped by regions.There is one region of length combinations where
reflection occurred (region II), and two regions where,despite the careful adjustment of the resistance betweenthe proximal and distal segments, reflection never oc-curred (regions I and III). For configurations in regionII, adjustment of the gap resistance caused enough P-Ddelay for the proximal tissue to recover its excitabilityand for the distal action potential to be able to (re)ex-cite the proximal segment. Therefore reflection oc-curred, as described in the previous section for a cablewith proximal and distal segments of 4 mm.
In region I, there was enough P-D delay for theproximal tissue to recover its excitability, but the distalaction potential was not able to (re)excite the proximal
1 i\
2
3
4
FIGURE 1.culated in aand distal sing values c
segments: ccolumn C,column E,ages at theend of the jdistal segm4(row 4). Fozero at the l
by guest on July 10, 2011http://circres.ahajournals.org/Downloaded from
Cabo and Barr Reflection After Delayed Excitation 263
5
4
.E3
2
o
. . . 3Ill. Proximal not sufficiently recovered
0 1 2 3 4 5 6 7 8 9 10
Proximal (mm)
Proximal
FIGURE 2. Graph showing the length of the proximal anddistal segments and the occurrence of reflection. Reflectionwas obtained (by adjusting the resistance between theproximaland distal segments) onlyforpreparations in the shaded region(region II). For preparations outside the shaded region (re-gions I and III), reflection was not obtained (despite adjust-ment of the resistance between the proximal and distalsegments). The initial conditions for each node correspond tothose occurring 600 msec after an action potential. Examplesof resistances forparticular length combinations identified onthe figure are as follows: a, 27.38 kflcm; b, 10.9438 kQlcm; c,10.234791 kflcm.
segment; therefore, reflection never occurred. In regionIII, there was not enough P-D delay for the proximaltissue to recover its excitability; therefore, reflectionnever occurred.For distal segments between 3 and 6 mm, reflection
was obtained for long lengths of the proximal segment,indicating that reflection is possible in a semi-infinitestructure. On the other hand, reflection in an infinitestructure (infinitely long proximal and distal segments)is not indicated, because reflection never occurred inlong distal segments.The resting membrane resistance of the DiFrancesco-
Noble membrane was 20,000 Qcm2 (as calculated by theratio of small increments of Vm over increments of ioniccurrent), and the intracellular resistance used for prox-imal and distal segments was 250 Q7cm. Consequently,the (resting) space constant was 2 mm. Using this valuefor the space constant, the lengths of proximal anddistal segments in Figures 2 (and other figures) can beexpressed in terms of space constants, with greatergenerality.
Reexcitation of the Proximal Segment bythe Distal Segment
Reflection did not always occur even with enough P-Ddelay, because the distal segment had to be able toreexcite the proximal segment. There were two cases inwhich there was enough delay but not reexcitation:lengths in region I (Figure 2) and lengths in region II
(Figure 2) at certain gap resistances (Figure 1, columnD).
Failure to reexcite in region I occurred when the gapresistance required to get enough P-D delay was largerthan the maximum resistance that allowed propagation
from the distal to the proximal segments, on stimulationof the distal segment. For any structure characterized byproximal and distal segments of fixed length, there wasa maximum gap resistance for which there was conduc-tion from the proximal to the distal segment on stimu-lation of the proximal segment; for larger resistances,there was block (such as Figure 1, column E). In regionI, consider the structure whose proximal and distalsegments were 6 and 2 mm, respectively. To get enoughP-D delay, a gap resistance of 27.687 kQ1cm was re-quired. In comparison, the maximum gap resistance thatallowed backward propagation was 9.907 kQlcm. There-fore, for this structure, reflection was not possible.
For structures in region II and gap resistances be-tween the ones required for reflection and the ones thatcause proximal-distal block, it was possible to obtainenough P-D delay for reflection but also distal-proxi-mal block. The failure to reflect (after the success of theforward conduction) was caused by several factors.With proximal and distal segments both 4 mm and a gapresistance of 10.94381 kQcm, there was proximal-distalconduction with a long enough delay for reflection butwith distal-proximal block (Figure 1, column D). Usingthe same gap resistance and initial conditions, on stim-ulation of the distal segment, there was distal-proximalconduction, as expected. Therefore, a cause of block ofthe reflected response was the partial inactivation of thesodium channels for tissue close to the gap (presentedbelow, Figures 4C and 4D). In contrast, when proximaland distal segments were 10 and 4 mm, respectively (gapresistance, 11.0718 kflcm), on stimulation of the proxi-mal segment, reflection was obtained, obviously includ-ing distal-proximal conduction. However, with the samegap resistance and initial conditions, on stimulation ofthe distal segment, there was distal-proximal block. Inthis case, the distal segment was able to stimulate theproximal segment when the proximal segment was closeto threshold (the case of reflection, when Vm is :-60mV) but not when it was fully repolarized (the case ofexternal distal stimulation, when Vm is =-90 mV).Therefore, another cause for distal-proximal block,even with enough P-D delay for reflection, was that theproximal segment was too nearly recovered (too faraway from threshold) to be stimulated by the distalsegment. That is, there was a window of "supernormal-ity" in phase 3 of the action potential in the proximalsegment that allowed reflection.
Mechanism of Delayed ExcitationReflection required a P-D delay long enough to allow
the tissue in the proximal segment to recover its elec-trical excitability. Long delays occurred when the prox-imal segment stimulated the distal segment close to thethreshold for excitation. Stimulation just above thresh-old caused, in short distal segments (shorter than 6mm), delays of hundreds of milliseconds. In contrast, inlong distal segments (larger than 6 mm), the delays wereof tens of milliseconds. To understand the differentmechanisms of delayed excitation, we studied two prep-arations with a proximal segment length of 6 mm anddistal segment lengths of 4 and 6 mm, respectively.
Short distal segment, just below threshold. Figure 3shows Vm, total ionic current, sodium activation param-eter m, and sodium deactivation parameter h in a distalsegment of length 4 mm at times 100, 200, 284, and 297
by guest on July 10, 2011http://circres.ahajournals.org/Downloaded from
nodeFIGURE 3. Graphs showing the just-below-threshold re-
sponse of a short distal segment (4 mm) after stimulation atthe beginning of the proximal segment (6 mm long). Thedifferentpanels show transmembrane voltage (Vm, panel A),total ionic current (I, panel B), sodium activation (m, panelC), and sodium deactivation (h, panel D) at times 100, 200,284, and 297 msec after the stimulus.
msec after stimulation in the proximal segment. Be-cause the gap resistance was adjusted to the just-below-threshold stimulus level, no action potential was elicitedin the distal segment; nonetheless, its response was notpurely passive. During the first 100 msec there was a netnegative (inward) total ionic current at nodes close tothe gap (e.g., node 61) (Figure 3B). The inward currentdiminished with time. At 200 msec the total ioniccurrent was positive at every node of the distal segmentbut less positive (more inward current) at nodes close tothe gap. At 284 msec the situation was reversed, and thetotal ionic current was less positive (more inward cur-rent) at nodes far away from the gap. This contrast canbe explained by the contribution of both the sodiumactivation and deactivation to the inward current. SinceVm was always more positive for nodes close to the gap(Figure 3A), the sodium activation parameter m wasalways more positive for nodes close to the gap (Figure3C), and the sodium inactivation parameter h wasalways more positive for nodes away from the gap(Figure 3D). The product m3h determines at whichnodes the inward current is larger (or at which nodesthe total ionic current is smaller). Even though theinward (depolarizing) current was not strong enough tocause an action potential, its effect was to cause aredistribution of charge with a length constant largerthan what would be expected from a passive responsealone, resulting in an almost uniform Vm at 297 msec(Figure 3A). Note that Vm across the length of the distalsegment neared the same value as time increased; thisnear equilibration was associated with long delays.
Short distal segment, just above threshold. With the gapresistance lowered slightly, the stimulus to the distalsegment was just above threshold, and an action poten-tial in the distal segment was elicited (Figure 4). Acombination of passive and active responses close to thegap tended to cause a uniform Vm along the wholesegment, as before. At 284 msec, Vm was approximately-60 mV for every node in the segment (Figure 4A). Atthis time, even though the Vm was the same for everynode, the state of activation of the sodium channels wasnot (Figure 4D). Therefore, the sodium current (in-ward) was larger for nodes far away from the gap. Thesodium activation parameter m was approximately con-stant along the segment because of the short timeconstant of m (:0.150 msec) (Figure 4C). On the otherhand, the sodium inactivation parameter h was muchsmaller at nodes close to the gap than at nodes far awayfrom the gap. This h gradient occurred because hdecreases as Vm increases (gets less negative) and has along time constant (~'-40 msec), and nodes closer to thegap were depolarized longer. Thereby, initiation of theexcitation began away from the gap (at 297 msec)because m3h was higher there.Long distal segment. With the longer distal segment
and just-above-threshold stimulus, an action potentialbegan in the distal segment (Figure 5) at the nodeclosest to the gap (node 61). At 100 msec, for a longsegment, Vm in nodes 60-70 varied between -53 and-60 mV (Figure 5A), whereas for a short segment, itvaried between -60 and -65 mV (Figure 4A). Therewas a crucial difference in the behavior of the sodiumactivation m in these two ranges: for potentials morepositive than - 60 mV, m has a higher value and changesmore rapidly with Vm than for potentials more negative
by guest on July 10, 2011http://circres.ahajournals.org/Downloaded from
Cabo and Barr Reflection After Delayed Excitation 265
Ao
-30
'297ms
----~~' 284ms200ms
0o 80 100node
g T 100mA__Oms- 284ms 2OOms_
- t~~2O7ms
0o 80 100node
/297ms
- ~
284m
0o 80 100node
60 80 100node
FIGURE 4. Graphs showingjust-above-threshold response ofa short distal segment (4 mm) after stimulation at thebeginning of the proximal segment (6 mm long). The differentpanels show transmembrane voltage (Vm, panel A), totalionic current (I, panel B), sodium activation (in, panel C),and sodium deactivation (h, panel D) at times 100, 200, 284,and 297 msec after the stimulus.
than -60 mV. For potentials below -60 mV, Vm canhover for a long time around a fixed value, but forpotentials more positive than -60 mV, the Vm risesrapidly. Therefore, further stimulation for long seg-ments caused an increase in sodium activation (Figure5C) that activated nodes close to the gap rapidly.Conversely, the electrotonic interaction with the rightportion of the distal segment, where Vm remained nearbaseline, meant that a smaller stimulus did not result inan action potential anywhere.
Importance of the Frequency of Stimulation (InitialConditions) on Reflection
In the previous simulations, the initial conditions foreach node of the simulated fiber corresponded to 600msec after an action potential. To investigate the effectof the frequency of stimulation on reflection, we createda graph in Figure 6 similar to the one in Figure 2. InFigure 6, the initial conditions for each node of thesimulated fiber correspond to 1,600 msec after an actionpotential (i.e., 1,000 msec after the previously usedinitial conditions). For the "late" initial conditions(1,600 msec), the gap resistances that cause the maxi-mum P-D delay are greater than the corresponding gapresistances for the "early" initial conditions (600 msec)for every preparation. With the late initial conditions,some of the preparations formerly in region II (Figure 2,reflection) moved to region I (Figure 6, no reflection),and some preparations formerly in region III (Figure 2,no reflection) moved to region II (Figure 6, reflection).
Recovery of Excitability During DiastoleTo understand the results in the previous section, the
excitability of the fiber at 600 msec was compared withthat at 1,600 msec after the onset of an action potential.The fiber was 10 mm long. The threshold for excitationwas tested with the fiber not separated into segments. Atest pulse of a duration of 100 msec was applied at oneend of the fiber. A long stimulus was used, sincestimulation across the gap leads to a long stimulationpulse. The current strength required for activation was10% higher at 600 msec than at 1,600 msec. Slowrecovery of excitability has previously been shown inventricular myocytes4' and in a modified Beeler-Reutermodel of the membrane (Delmar et a142). The lowerthreshold explains why the gap resistances that causedmaximum P-D delay (i.e., stimulation close to thresh-old) were greater for late initial conditions for everystructure: higher resistance between the excitable seg-ments meant less current between the segments. Fur-thermore, if higher gap resistances are required to causethe appropriate P-D delay for reflection, the distalsegments (during phase 3 of the action potential in theproximal site) see a higher input impedance when tryingto reexcite the proximal site. The higher resistancecaused some of the preparations formerly in region II(Figure 2, reflection) to move to region I (Figure 6, noreflection).
Reflection Using Premature StimulationIn the previous sections, reflection was obtained
(when possible) by adjusting the gap resistance betweenthe proximal and distal segments. To get enough P-Ddelay for reflection, the threshold gap resistance had tobe adjusted in some cases with a precision of seven
EE
-60
c'JE
< -25-d-H--
-501
C 1
E 0.5
0
D
by guest on July 10, 2011http://circres.ahajournals.org/Downloaded from
FIGURE 5. Graphs showingjust-above-threshold response ofa long distal segment (6 mm) after stimulation at the begin-ning of the proximal segment (6 mm long). The differentpanels represent transmembrane voltage (Vm, panel A), totalionic current (I, panel B), sodium activation (m, panel C),and sodium deactivation (h, panel D) at times 50, 100, 146,and 149 msec after the stimulus.
8
7
6
5
4
3
2
00 1
Ill. Proximal not sufficiently recovered
1. Distal not able to re-excite Proximal
2 3 4 5 6 7 8 9 10
Proximal (mm)
imal istal
FIGURE 6. Graph showing the length of the proximal anddistal segments and the occurrence of reflection for initialconditions different from those in Figure 2. Reflection wasobtained (by adjusting the resistance between theproximal anddistal segments) only for preparations in the shaded region(region II). For preparations outside the shaded region (re-gions I and III), reflection was not obtained (despite adjust-ment of the resistance between the proximal and distalsegments). The initial conditions for each node correspond tothose occurring 1,600 msec after an action potentiaL Exam-ples of resistances for particular length combinations identi-fied on the figure are as follows: a, 38.995 kflcm; b, 15.205kfQcm; c, 11.0712 kflcm.
significant figures. On the other hand, in real tissue,18-22the P-D delay required for reflection was obtained by acombination of adjusting the gap resistance and prema-ture stimulation. How does the gap resistance interactwith premature stimulation? To investigate, the proce-dure we used was always the same: First, we chose abelow-threshold gap resistance giving proximal-distalconduction. Second, we stimulated the proximal sitewith a basic stimulus, P1, at time zero and with apremature stimulus, P2 (these two stimuli caused tworesponses in the distal segment Dl and D2, respective-ly). Third, we adjusted the P1-P2 coupling to change theP2-D2 delay to get reflection, if possible.
In this first set of simulations, we used late initialconditions. For a preparation whose proximal and distalsegments are both 4 mm long, a gap resistance of 15.205kfQcm (threshold gap resistance) causes a Pl-Dl delayof 367 msec, enough to obtain reflection. If instead agap resistance of 15 kflcm was used, a Pl-Dl delay of242 msec was obtained, which was not enough forreflection. However, the P2-D2 delay can be adjustedby changing the P1-P2 coupling interval. For a P1-P2interval of 1,365 msec or greater, the P2-D2 delay wasnot enough for reflection to occur (Figure 7A). For aP1-P2 interval between 1,361 and 1,364 msec, enoughP2-D2 delay (=350 msec) can be obtained for reflectionto occur (Figure 7B). For a P1-P2 interval of 1,360 msecor less, there is P2-D2 block (Figure 7C). Therefore, fora gap resistance of 15 kfcm, there is a time window (forP1-P2 coupling) in the order of milliseconds for reflec-tion to occur. These simulations agree with the experi-mental results.18-22
-80
B
chJ
C.).: -120
-4
-240-
C
S 0.5
0
D
-a
.in
1
by guest on July 10, 2011http://circres.ahajournals.org/Downloaded from
Cabo and Barr Reflection After Delayed Excitation 267
C
P1 P2
FIGURE 7. Recordings showing transmembranevoltages calculated in a fiber with a length of 8 mmand proximal and distal segments, both 4 mm (40nodes) long, for a resistance of 15 kflcm betweenthe proximal and distal segments and decreasingvalues of the coupling interval for stimuli 1 and 2 atthe proximal site (P1-P2): column A, 1,365 msec;column B, 1,362 msec; column C, 1,360 msec. D]and D2 are responses in the distal segment to stimuli1 and 2. Each panel shows transmembrane voltagesat the beginning of the proximal segment (row 1), atthe end of the proximal segment (row 2), at thebeginning of the distal segment (row 3), and at theend of the distal segment (row 4). For each panel,stimuli P1 and P2 were applied at the beginning ofthe proximal segment (row 1).
1 2V 3 4
Using early initial conditions and integer kfcm value,gap resistances (resulting from the truncation of thethreshold gap resistance) showed a sharp contrast in theeffectiveness of premature stimuli. The width of theP1-P2 coupling time window for reflection to occur was
in the order of microseconds (instead of milliseconds).A possible explanation is given in the discussion.
Gap Resistance Range for ReflectionIn the previous section we have shown that gap
resistances close to the threshold gap resistance andpremature stimulation can be combined to obtain re-
flection. For gap resistances further apart from thresh-old, the time window in which reflection can be ob-tained narrows, and the P1-P2 coupling that causes themaximum P2-D2 delay decreases. Therefore, the more
favorable the gap resistance the less favorable must bethe P1-P2 coupling, and vice versa.
What is the maximum decrease in the gap resistance(from the threshold gap resistance) such that, with a
precision of 1 msec in the P1-P2 coupling interval, reflec-tion is still possible? We evaluated a few examples, usinga precision of 1 msec in the P1-P2 coupling interval. Fora preparation whose proximal and distal segments are
both 4 mm long (threshold gap resistance, 15.205 kQcm),the gap resistance can be reduced to 12 kfQm, andreflection still can be obtained by premature stimulation.
This much change gives a gap resistance range for reflec-tion of approximately 20%. For a preparation whoseproximal and distal segments are 10 and 5 mm long,respectively (threshold gap resistance, 12.0065 kQcm), thegap resistance can be reduced to 11.5 kfQm. This changeis approximately 5%. In both cases, note how markedly theprecision required for the coupling resistance diminisheswhen premature stimuli are allowed.
DiscussionRelation of Gap Resistances and Coupling Intervals
Consider a hypothetical "strength-interval" curve forthe distal segment (Figure 8). Suppose the preparationresponds according to the solid line in the figure.Further, suppose that stimulation just above thresholdis essential for a delay long enough for reflection.Finally, suppose that the strength of the current (pro-vided by the proximal segment) stimulating the distalsegment is proportional to the reciprocal of the gapresistance. The P-D delay depends on the relation ofthe stimulation current to the threshold current. Byusing the late initial conditions and a structure in whichreflection is possible (i.e., from Figure 6, region II), thegap resistance can be adjusted carefully to a thresholdgap resistance (Rthr, point A, Figure 8) to obtain enoughPl-Di delay for reflection. If the gap resistance isdecreased (R,, point B, Figure 8), the Pl-Dl delay is
A
P1 P2
B
P1 P2
1
2
D1 D2
c0,3
Dl D2
c
Dl
4
50 mV
1000 ms
by guest on July 10, 2011http://circres.ahajournals.org/Downloaded from
FIGURE 8. "Strength-interval" curvefor a generic structurewith proximal (P) and distal (D) segments connected by a gap
resistance (Rgap). P1 and P2 are stimuli 1 and 2, respectively,at the proximal site; D] and D2 are responses in the distalsegment to stimuli 1 and 2, respectively. Rgap can be adjustedto a threshold gap resistance (Rthr, point A) to obtain enoughPJ-DJ delay for reflection. If Rgap is decreased (R], point B),the PJ-DJ delay is less because point B is further away fromthe threshold than point A, so reflection will not occur. Theonly way to get close to threshold (for a fixed Rgap) is bypremature stimulation (point C); as the strength comes closerto threshold, the P2-D2 delay is increased until reflectionoccurs. If the Rgap is reduced still more (R2, point D), thePJ-D] delay is further decreased (D is stillfurther awayfromthe threshold than B). Premature stimulation brings thestimulus close to threshold (point E) at a shorter P1-P2coupling time and a narrower range oftimes than before. (Seetext for further discussion.)
less than in the previous case because point B is furtheraway from the threshold than point A, so reflection willnot occur. The only way to get close to threshold (for a
fixed gap resistance) is by premature stimulation (pointC, Figure 8); as the strength comes closer to threshold,the P2-D2 delay is increased, until reflection occurs. Ifthe gap resistance is reduced still more (R2, point D,Figure 8), the Pl-Dl delay is further decreased (D isstill further away from the threshold than B). Prematurestimulation brings the stimulus close to threshold (pointE, Figure 8) at a shorter P1-P2 coupling time, and witha narrower range of times, than before. That is, possiblythe gap resistance used determines the (threshold)point of the strength-interval curve, and prematurestimulation determines how closely the threshold isapproached. The slope of the strength-interval curve atthe threshold point determines the width of the windowfor reflection (for getting enough P2-D2 delay): thesteeper the curve, the narrower the window. Therefore,with the late initial conditions, the width of the reflec-tion window is in milliseconds, whereas with the earlyinitial conditions, the width is in microseconds, becausethe strength-interval curve is flatter for the late initialconditions than for the early initial conditions. Evenwhen using the late initial conditions, however, if thegap resistance is much smaller than the threshold gapresistance, premature stimulation might lead us to a
point in the strength-interval curve with a big slopeand, hence, a narrow reflection window.
P-D Delay, Hovering Voltage in the Distal Segment,and the Occurrence of ReflectionOne of the requirements for reflection to occur was
enough P-D delay for the proximal segment to recoverits excitability. The simulations using the late initialconditions (1,600 msec after the onset of the actionpotential) allowed us to further refine this requirement.For structures in region III at the early initial conditions(Figure 2), the maximum P-D delay obtained by adjust-ing the gap resistance was 150 msec. For structures inregion III (distal too long) at the late initial conditions(Figure 6), P-D delays around 400 msec can be ob-tained, but still reflection did not occur. With the lateinitial conditions, less current was required for proxi-mal-distal conduction (see previous section), resultingin stimulation closer to threshold and hence longerdelays. For long delays, nodes of the distal segmentclose to the gap hover around a constant Vm for a longperiod of time. Reflection did not occur because thehovering voltage in nodes of the distal segment close tothe gap was more positive than the refractory voltage inthe proximal segment. Despite the long P-D delay,electrotonic coupling to the distal segment, togetherwith the distal segment's high hovering voltage, pre-vented the proximal segment from repolarizing andrecovering its electrical excitability. For example, for astructure whose proximal and distal segments were both7 mm, by using the late initial conditions and a gapresistance of 11.09885 krcm, a P-D delay of 391 msecwas obtained. Even so, the node of the proximal seg-ment closest to the gap repolarized only to -51 mV.Reflection did not occur. Therefore, long P-D delaysalone do not ensure recovery of excitability of theproximal segment: a hovering voltage in the distalsegment low enough to be close to the threshold forexcitation in the proximal segment also is required;otherwise, the proximal segment does not repolarizeenough to be restimulated. For long delays, the hoveringpotential depends on the length of the distal segment.This requirement is satisfied by distal segments shorterthan 6 mm, for both initial conditions (Figures 2 and 6).
Relation to Experimental FindingsThe results obtained with experimental models of
reflection18-22 (proximal and distal segments, <5 mm;basic pacing cycle length, between 1,000 and 2,000 msec;P-D delays caused by premature stimulation) have beenreproduced accurately with the single-fiber model ofreflection described above. As a consequence, eventhough we cannot rule out the possibility that reflectionin multiple-fiber preparations is caused by syncytialreentry, the simulation results support the hypothesisthat true reflection was the mechanism in the experi-mental models of reflection.Because the model is constructed with an elevated
axial resistivity in one segment of the cable, it is clearthat reflection was achieved in the model by pureelectrotonic transmission between the segments; i.e.,there were no active responses in the gap. In experi-ments in which the proximal and distal segments wereuncoupled by perfusing the central compartment (thegap) with a high concentration of extracellular potassi-um,22 the origin of the distal activation far away fromthe gap was explained by the possibility that potassium
by guest on July 10, 2011http://circres.ahajournals.org/Downloaded from
Cabo and Barr Reflection After Delayed Excitation 269
leaking from the gap would render inexcitable tissue ofthe distal segment close to the gap. In the simulationspresented here, for long P-D delays, the origin ofactivation in the distal segment was also far away fromthe gap. The mechanism, however, was partial inactiva-tion of the sodium channels of nodes of the distalsegment close to the gap (Figure 4). The mechanism ofthe simulation also could be what happened in theexperiments.An excellent earlier computer modeling study of re-
flection28 suggested that the long delay required forreflection to occur might require a combination of elec-trotonic transmission and modulation of the automaticpacemaker of Purkinje fibers.26,27 In the simulationspresented here, reflection was obtained with the diastolicdepolarization current of the Purkinje fiber model inhib-ited, showing that indeed diastolic depolarization is not arequirement for reflection. The different nature of themodels that led to different conclusions makes difficult amore detailed comparison of the results. The demonstra-tion of reflection in ventricular muscle,22 however, sup-ports the idea that reflected activity can occur indepen-dent of a pacemaker mechanism.
Reflection and Segment LengthThe simulations predict that reflection is possible in
fibers with long or short proximal segments and shortdistal segments but not in fibers with long distal seg-ments. This result is suggestive in extrapolating conclu-sions about reflection obtained in isolated (short) fibersto the whole heart. As far as we know, reflection hasnever been specifically documented in the literature infibers with long segments whether proximal, distal, orbulk.
In all the simulations presented above, the diastolicdepolarization current was set to zero to avoid automa-ticity. Setting this current to zero was desirable becauseit made possible a clear judgment as to whether truereflection occurred. Nonetheless, real Purkinje fibersinclude a diastolic depolarization current. In the smallernumber of simulations that we have performed thatinclude this current, longer P-D delays occur in fiberswith long distal segments. It might be the case that thepresence of a reduced diastolic depolarization currentwould allow reflection in fibers with long distal segmentsindependent of automatic depolarization. A systematicevaluation of the interaction between electrotonic ef-fects from the proximal segment and the diastolicdepolarization of the distal segment will be complex,since there will be many degrees of interaction depend-ing on segment length, current intensity, and degree ofcoupling.
Delayed ExcitationA crucial aspect of reflection is obtaining a long delay
of excitation in the distal segment. One mechanism forobtaining long delays makes use of the long delaysassociated with series resistor-capacitor circuits, wherea high (axial) resistance causes a long delay in charging(membrane) capacitance to a threshold voltage. Thismechanism was the basis for only a part of the delay thatoccurred in the simulations here. The failure to reflectwith late initial conditions (even with long P-D delay)and long distal segment suggested that resistor-capaci-tor delay alone was insufficient to produce reflection.
Another crucial aspect was stimulation of the distalsegment to a degree that was just above threshold,initiating active changes in membrane conductances andthereby maintaining near-threshold Vms for hundreds ofmilliseconds. A distal segment length of approximatelytwo space constants allowed the entire segment toequilibrate in a near-threshold state. Longer fibers werenot equilibrated, so some portion moved to excitationmore quickly; shorter fibers could be equilibrated withlong delays but were unable to reexcite the proximalsegment. It appears to be the case that similarly longdelays could occur by this mechanism during propaga-tion down a fiber with an incompletely isolated interiorsegment as well as in the context of reflection.
Model ParametersIt is clear that the results of computer simulations are
only as accurate as the parameters used to represent thefiber structure and membrane behavior. In "Materialsand Methods," a number of simplifications used torepresent the fiber structure were enumerated. Amongthese are the representation of the actual structure ofmany interconnected cells by a single cylindrical struc-ture and the representation of the complex grid ofactual resistances and capacitances, including the actualstructure of the narrow clefts and gap junctions, bythose of the linear core-conductor model. It must bethat at some level of microscopic detail these differencesbecome highly significant, and we cannot rule out thepossibility that a more accurate representation of theanatomic complexities of Purkinje fibers would have aneffect on the results presented here. Even so, it hasconsistently been true that experimental-theoreticalcomparisons at a macroscopic level have shown closecorrespondence between Purkinje models using a cylin-drical representation and experimental results for mac-roscopic propagation. Examples include such earlierreports as those of Spach et a143 as well as more recentcomparisons involving premature stimulation.30 Sincethe electrotonic interactions of reflection occur at amacroscopic rather than microscopic scale, we think thecylindrical representation is a good starting point for amore detailed quantitative analysis.The simulations reported here used a value for the
maximum conductance of the sodium current 1.5 timesthe standard value in the DiFrancesco-Noble model forthe Purkinje fiber.29,30 With the standard value, reflec-tion still was obtained, but the reflected action potentialhad lower amplitude and slope than the action poten-tials reported in the experimental studies. With therevision, depolarization rates were closer to reportedexperimental depolarization rates. This raises somequestions about the accuracy of the representation ofthe fast sodium current in the DiFrancesco-Noblemodel, which is based on experimental data collected byColatsky.44 Even though Colatsky's studies have pro-vided more reliable information on the kinetics of thesodium currents in Purkinje fibers than previous studies,the major disadvantage of his data is that it was ob-tained in cooled fibers and the speed of the gate kineticshad to be adjusted to 37°C.
Conditions for ReflectionThe results of the simulations show that reflection
occurred in a single fiber when three conditions were
by guest on July 10, 2011http://circres.ahajournals.org/Downloaded from
met: the delay between the proximal and distal seg-ments was long enough for the proximal segment torecover its excitability, the hovering voltage in the distalsegment was close to the threshold for excitation in theproximal segment, and the distal segment was able toreexcite the proximal segment. Diastolic depolarizationwas not a requirement for reflection, and microreentrywas impossible. Just-above-threshold stimulation of ashort distal segment was an essential component of theprocess; it led to the long delays needed for the proxi-mal segment to repolarize. Moreover, the simulationssuggested that true reflection may occur not only inthose experimental preparations in which it has beenreported but also in many other circumstances in whichshort incompletely isolated fiber segments are present.
References1. Cranefield PF: The Conduction of the Cardiac Impulse. Mt Kisco,
NY, Futura Publishing Co, Inc, 19752. Janse MJ: Reentry rhythms, in Fozzard HA, Haber E, JenningsRB (eds): The Heart and Cardiovascular System. New York, RavenPress, Publishers, 1986, pp 1203-1238
3. Antzelevitch C: Clinical applications of new concepts of parasys-tole, reflection and tachycardia. Cardiol Clin 1983;1:39-50
4. Jalife J, Antzelevitch C, Moe GK: Models of parasystole andreflection, in Rosenbaum MB, Elizari MV (eds): Frontiers in Car-diac Electrophysiology. Amsterdam, Martinus Nijhoff Publishing,1983, pp 217-238
5. Antzelevitch C: Reflection as a mechanism of reentrant cardiacarrhythmias. Prog Cardiol 1988;1:3-16
6. Antzelevitch C, Lukas A, Litovsky S: Reflection as a subclass ofreentrant atrial cardiac arrhythmias, in Coumel P, Janse MJ (eds):The Atrium in Health and Disease. Mt Kisco, NY, Futura Publish-ing Co, Inc, 1989, pp 43-64
7. Antzelevitch C: Electrotonus and reflection, in Rosen MR, JanseMJ, Wit AL (eds): Cardiac Electrophysiology: A Textbook. MtKisco, NY, Futura Publishing Co, Inc, 1990, pp 491-516
8. Bethe A: Allgemeine Anatomie und Physiologie des Nervensystems.Leipzig, FRG, George Thieme, 1903
9. Bethe A: Rhythmik und periodik besonders im hinblick auf diebewegungen des herzens und der meduse. Pflugers Arch 1937;239:41-73
10. Schmitt FO, Erlanger J: Directional differences in the conductionof the impulse through heart muscle and their possible relation toextrasystolic and fibrillary contractions. Am J Physiol 1928;87:326-347
11. Segers M: La reaction repetitive du coeur. Compt Rend Soc Biol1940;133:460-461
12. Scherf D, Cohen J: Return extrasystoles, in The AtrioventricularNode and Selected Cardiac Arrhythmias. New York, Grune & Strat-ton, Inc, 1964, pp 226-279
13. Cranefield PF, Klein HO, Hoffman BF: Conduction of the cardiacimpulse: I. Delay, block, and one-way block in depressed Purkinjefibers. Circ Res 1971;28:199-219
14. Cranefield PF, Hoffman BF: Conduction of the cardiac impulse: II.Summation and inhibition. Circ Res 1971;28:220-233
15. Cranefield PF, Wit AL, Hoffman BF: Conduction of the cardiacimpulse: III. Characteristics of very slow conduction. J Gen Physiol1972;59:227-246
16. Wit AL, Hoffman BF, Cranefield PF: Slow conduction and reentryin the ventricular conducting system: I. Return extrasystole incanine Purkinje fibers. Circ Res 1972;30:1-10
17. Wit AL, Cranefield PF, Hoffman BF: Slow conduction and reentryin the ventricular conducting system: II. Single and sustained circusmovement in networks of canine and bovine Purkinje fibers. CircRes 1972;30:11-22
18. Antzelevitch C, Jalife J, Moe GK: Characteristics of reflection as amechanism of reentrant arrhythmias and its relationship to para-systole. Circulation 1980;61:182-191
19. Antzelevitch C, Moe GK: Electrotonically mediated delayed con-duction and reentry in relation to slow responses in mammalianventricular conducting tissue. Circ Res 1981;49:1129-1139
20. Jalife J, Moe GK: Excitation, conduction, and reflection ofimpulses in isolated bovine and canine Purkinje fibers. Circ Res1981;49:233-247
21. Antzelevitch C, Bernstein MJ, Feldman HN, Moe GK: Parasystole,reentry, and tachycardia: A canine preparation of cardiac arrhyth-mias occurring across inexcitable segments of tissue. Circulation1983;68:1101-1115
22. Rozanski GJ, Jalife J, Moe GK: Reflected reentry in nonhomoge-neous ventricular muscle as a mechanism of cardiac arrhythmias.Circulation 1984;69:163-173
23. Lukas A, Antzelevitch C: Reflected reentry, delayed conduction,and electrotonic inhibition in segmantally depressed atrial tissue.Can J Physiol Pharmacol 1989;67:757-764
24. Rosenthal JE: Reflected reentry in depolarized foci with variableconduction impairment in 1 day old infarcted canine cardiac tissue.JAm Coll Cardiol 1988;12:404-411
25. van Hemel NM, Swenne CA, de Bakker JMT, Defauw JJAM,Guiraudon GM: Epicardial reflection as a cause of incessant ven-tricular bigeminy. PACE 1988;11:1036-1044
26. Jalife J, Moe GK: Effect of electrotonic potentials on pacemakeractivity of canine Purkinje fibers in relation to parasystole. Circ Res1976;39:801-808
27. Jalife J, Moe GK: A biological model of parasystole. Am J Cardiol1979;43:761-772
28. Janse MJ, van Capelle FJL: Electrotonic interactions across aninexcitable region as a cause of ectopic activity in acute regionalmyocardial ischemia: A study of intact porcine and canine heartsand computer models. Circ Res 1982;50:527-537
29. DiFrancesco D, Noble D: A model of cardiac electrical activityincorporating ionic pumps and concentration changes. Philos TransR Soc [Biol] 1985;307:353-398
30. Cabo C, Barr RC: Propagation model using the DiFrancesco-Noble equations: Comparison to reported experimental results.Med Biol Eng Comput (in press)
31. Plonsey R, Barr RC: Interstitial potentials and their change withdepth into cardiac tissue. Biophys J 1987;51:547-555
32. Plonsey R, Henriquez CS, Trayanova N: Extracellular (volumeconductor) effect on adjoining cardiac muscle electrophysiology.Med Biol Eng Comput 1988;26:126-129
33. Fozzard HA: Membrane capacity of the cardiac Purkinje fibre.JPhysiol (Lond) 1966;182:255-267
34. Fozzard HA: Conduction of the action potential, in Berne RM,Sperelakis N, Geiger SR (eds): Handbook of Physiology, Section 2:The Cardiovascular System. Bethesda, Md, American PhysiologicalSociety, 1979, pp 335-356
35. Spach MS, Miller WT III, Geselowitz DB, Barr RC, Kootsey JM,Johnson EA: The discontinuous nature of propagation in normalcanine cardiac muscle: Evidence of recurrent discontinuities ofintracellular resistance that affect the membrane currents. Circ Res1981;48:39-54
36. Spach MS, Kootsey JM: The nature of electrical propagation incardiac muscle. Am J Physiol 1982;244:H3-H22
37. Spach MS: The discontinuous nature of electrical propagation incardiac muscle: Consideration of a quantitative model incorporat-ing the membrane ionic properties and structural complexities(ALZA lecture). Ann Biomed Eng 1983;11:209-261
38. Chapman RA, Fry CH: An analysis of the cable properties of frogventricular myocardiaum. J Physiol (Lond) 1978;283:263-282
39. Weingart R: Electrical properties of the nexal membrane studiedin rat ventricular cell pairs. J Physiol (Lond) 1986;370:267-284
40. Freygang WH, Trautwein W: The structural implications of thelinear electrical properties of cardiac Purkinje strands. J Gen Phys-iol 1970;55:524-547
41. Delmar M, Michaels DC, Jalife J: Slow recovery of excitability andthe Wenckebach phenomenon in the single guinea pig ventricularmyocyte. Circ Res 1989;65:761-774
42. Delmar M, Glass L, Michaels DC, Jalife J: Ionic basis and analyt-ical solution of the Wenckeback phenomenon in guinea pig ven-tricular myocytes. Circ Res 1989;65:775-788
43. Spach MS, Barr RC, Serwer GA, Kootsey JM, Johnson EA: Extra-cellular potentials related to intracellular action potentials in thedog Purkinje system. Circ Res 1972;30:505-519
44. Colatsky TJ: Voltage clamp measurements of sodium channelproperties in rabbit cardiac Purkinje fibers. J Physiol (Lond) 1980;305:215-234
by guest on July 10, 2011http://circres.ahajournals.org/Downloaded from
