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An ionic model for rhythmic activity in small clusters of embryonic chick ventricular cells

¹Department of Physiology and ²Centre for Nonlinear Dynamics in Physiology and Medicine, McGill University, Montreal, Quebec, Canada; and ³Center of Physiological Medicine, Institute of Biophysics, Medical University Graz, Graz, Austria

Submitted 12 July 2004 ; accepted in final form 3 February 2005

	ABSTRACT

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

 
We recorded transmembrane potential in whole cell recording mode from small clusters (2–4 cells) of spontaneously beating 7-day embryonic chick ventricular cells after 1–3 days in culture and investigated effects of the blockers D-600, diltiazem, almokalant, and Ba²⁺. Electrical activity in small clusters is very different from that in reaggregates of several hundred embryonic chick ventricular cells, e.g., TTX-sensitive fast upstrokes in reaggregates vs. TTX-insensitive slow upstrokes in small clusters (maximum upstroke velocity 100 V/s vs. 10 V/s). On the basis of our voltage- and current-clamp results and data from the literature, we formulated a Hodgkin-Huxley-type ionic model for the electrical activity in these small clusters. The model contains a Ca²⁺ current (I_Ca), three K⁺ currents (I_Ks, I_Kr, and I_K1), a background current, and a seal-leak current. I_Ca generates the slow upstroke, whereas I_Ks, I_Kr, and I_K1 contribute to repolarization. All the currents contribute to spontaneous diastolic depolarization, e.g., removal of the seal-leak current increases the interbeat interval from 392 to 535 ms. The model replicates the spontaneous activity in the clusters as well as the experimental results of application of blockers. Bifurcation analysis and simulations with the model predict that annihilation and single-pulse triggering should occur with partial block of I_Ca. Embryonic chick ventricular cells have been used as an experimental model to investigate various aspects of spontaneous beating of cardiac cells, e.g., mutual synchronization, regularity of beating, and spontaneous initiation and termination of reentrant rhythms; our model allows investigation of these topics through numerical simulation.

pacemaker; seal-leak current; rapid delayed rectifier potassium current block; slow inward calcium current block; bifurcation analysis

After a couple of days in culture, the electrical activity in an isolated embryonic chick ventricular cell, in a small cluster of a few such cells, or in a sparse monolayer is very different from that in situ or in a reaggregate of hundreds or thousands of cells isolated from the ventricle. For example, when trypsin-dispersed ventricular cells from 7-day embryonic chick hearts are used, the upstroke velocity is much lower in single cells, in small clusters of cells, and in sparse monolayers (17, 49, 51, 95) than in reaggregates (14, 16, 19) or in the intact ventricle (19, 97, 98, 118). Spontaneous beating can be abolished by addition of tetrodotoxin (TTX), a blocker of the fast inward Na⁺ current (I_Na), to the medium bathing reaggregates of trypsin-dispersed 7-day ventricular cells (16, 70), but spontaneous activity continues in single cells and monolayers (57, 70, 81, 95). However, spontaneous activity in some TTX-insensitive preparations can be abolished by addition of either of the Ca²⁺ channel blockers D-600 or diltiazem (49, 51).

We carried out an experimental study to characterize the electrical properties of spontaneously beating clusters of cells isolated from the 7-day embryonic chick ventricle and then assembled a mathematical model of this activity. The goal is to use this model to investigate phenomena seen in experiments, such as irregularity of beating in small clusters (10), mutual synchronization of pairs of cells (18), phase resetting and phase locking (9, 11, 52, 53), current-pulse-induced annihilation of spontaneous activity (90, 94), and spontaneous initiation and termination of spiral-wave reentrant motions in monolayers (6). Although ionic models of reaggregates of embryonic chick atrial cells have been described (9, 13, 52, 89, 90), we are not aware of any models of small clusters of isolated embryonic chick ventricular cells.

	METHODS

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

 
Cell Isolation

Ventricular myocytes were isolated from embryonic chick hearts by means of techniques previously described (50) with modifications (79). The hearts of 7-day embryos were removed, and the ventricles were chopped off, minced, and transferred to flasks containing 0.25% trypsin (bovine pancreas; Boehringer Mannheim, Deisenhofen, Germany) in a nominally Ca²⁺- and Mg²⁺-free Hanks' balanced salt solution (HBSS; in mM: 137 NaCl, 5.4 KCl, 0.34 Na₂HPO₄, 0.44 KH₂PO₄, 4.2 NaHCO₃, and 5 glucose, pH 7.4). The flasks were placed in a shaker bath at 37°C for 7 min. The resulting cell suspension was gently agitated with a pipette and filtered through a 100-µm mesh. HBSS, supplemented with fetal calf serum (5% final concentration), was added to stop trypsin activity. The cell suspension was centrifuged at 100 g for 5 min at 4°C, the supernatant was discarded, and the cell pellet was resuspended in fresh trypsin-free HBSS. The centrifugation and resuspension processes were then repeated. The solution was centrifuged for a third time and resuspended in cell culture medium [M199 (Sigma) supplemented with 4% fetal calf serum, 2% horse serum, and 0.7 mM glutamine, pH 7.4] to yield a density of 5 x 10⁵ cells/ml.

The cell suspension was separated into aliquots on nonadhesive plastic culture dishes that were incubated at 37°C in a water-saturated atmosphere of 95% air-5% CO₂. To obtain small clusters of cells, 0.6-ml aliquots of cell suspension were removed after 0.5–2 days and placed into the lumen of flexiPERM silicone rings (Heraeus, Hanau, Germany), which were attached to microscope slide coverslips. Coverslips with attached flexiPERM rings were placed in cell culture dishes (Greiner) and stored in the incubator. This procedure allowed the myocytes to adhere to the glass surface, where they could divide and form small clusters of cells. However, we can offer no guarantee that a particular cluster is made up entirely of cells that divided in culture.

Experiments were performed 2–36 h after the cells were plated, on clusters containing two to four cells. Recordings were made in the whole cell mode from clusters that were spontaneously beating before they were patched. We use clusters of a few cells, rather than single cells, because we were unable to obtain recordings of spontaneous activity from single cells. [It is also easier to successfully impale a cell in a cluster, rather than a single cell, if a conventional sharp microelectrode is used (17, 26).] In addition, the effect of the seal-leak current (I_seal; see below) on spontaneous activity is expected to be considerably smaller for a cluster than for a single cell (but see Ref. 78). For electrophysiological recording, the coverslip with attached myocytes was used to form the bottom of the experimental chamber, which was placed on the stage of an inverted microscope (Zeiss, Axiovert). The experimental chamber was perfused with extracellular solution (in mM: 137 NaCl, 5.4 KCl, 1.8 CaCl₂, 1.1 MgCl₂, 2.2 NaHCO₃, 0.4 NaH₂PO₄, 10 Na-HEPES, and 5.6 glucose, with pH adjusted to 7.4 with NaOH) at 36–37°C with a flow rate of 1.5 ml/min.

Electrophysiological Recording

The transmembrane potential (V) was recorded using the whole cell recording mode of the patch-clamp technique. Patch pipettes (2 M resistance) were pulled from glass capillary tubes and filled with pipette solution (in mM: 110 KCl, 4.3 K₂-ATP, 2 MgCl₂, 1 CaCl₂, 11 EGTA, and 10 K-HEPES, with pH adjusted to 7.4 with KOH) with estimated free Ca²⁺ concentration ([Ca²⁺]) < 10^–8 M. Electrode potentials were zeroed before seal formation. After the patch was broken, the transmembrane potential was recorded with an amplifier (model EPC-7, List, Darmstadt, Germany). The membrane capacitance was measured by integration of the capacitive transient in response to a voltage-clamp step from –50 to –60 mV. After capacity compensation, we compensated for series resistance by turning up the series-resistance compensation control (which controls the amount of positive feedback) to just below the value where ringing in the current monitor signal would occur. Usually, compensation could be made for >50% of series resistance. For generation of voltage-clamp protocols and for recording voltage and current, a personal computer equipped with pCLAMP version 5.5.7 software (Axon) and a DigiData 1200 interface (Axon) were used. Signals were also digitized at 44 kHz, pulse-code modulated, and stored on video cassette tape for offline analysis. The signal was played back, sampled at 1 kHz (Axotape, Axon Instruments), and stored on a disk file for computerized analysis.

Action Potential Parameters

Several parameters are measured to characterize spontaneous electrical activity. Interbeat interval (IBI) is the time between consecutive crossings of 0 mV on the upstroke of the action potential. The maximal diastolic potential (MDP) is the most negative voltage recorded during an action potential and the overshoot potential (OS) the most positive. The action potential amplitude (APA) is the difference between MDP and OS. Action potential duration (APD) is the time from the crossing of 0 mV on the upstroke to the time of 50% repolarization (APD₅₀) or 100% repolarization (APD₁₀₀). The diastolic depolarization rate (DDR) is the slope of the chord joining the point where (MDP + 1 mV) is crossed and the point 70 ms later (108, 114). The maximum rate of rise of the upstroke (_max) is computed using the greatest voltage difference between two consecutive samples on the upstroke (1-ms sampling interval).

Simulation Methods

Numerical integration of the Hodgkin-Huxley-type ionic model was carried out using a forward Euler scheme, with V at time t + t calculated as follows

where C_m is membrane capacitance and I_i represents the individual ionic currents described below. (A 4th-order Runge-Kutta integration scheme produced virtually superimposable results.) With our choice of time step (t = 0.1 ms), the change in voltage from time t to t + t was kept to <1 mV. The value of each activation or inactivation variable (_i) at time t + t was obtained from its value at time t using the analytic formula

where _i() is the steady-state or asymptotic value of _i at V(t) and _i is the time constant of _i at V(t). L'Hôpital's rule was used to calculate _n when V came to within ±0.1 mV of the value producing an indeterminate form. The numerical integration routine was written in C, and all variables were double precision (16 significant decimal places).

Bifurcation analysis was carried out using AUTO, as incorporated in XPPAUT (25). The model equations file for use with XPPAUT is available as supplemental material (supplemental data for this article may be found at http://ajpheart.physiology.org/cgi/content/full/00683-4.2004.DC1).

Formulation of the Model

Our model consists of six currents: a slow inward Ca²⁺ current (I_Ca), a slow delayed K⁺ current (I_Ks), a rapid delayed rectifier K⁺ current (I_Kr), an inward rectifier K⁺ current (I_K1), a linear time-independent background current (I_b), and a linear nonspecific seal-leak current (I_seal), generated by the leakage of ions through the gigaohm seal of the recording pipette. We now give the rationale for including each of these currents and for the particular formulation that we employ for each of these currents, as well as reasons for using a "first-generation," rather than a "second-generation," model.

Difficulties with second-generation models. In the earlier Hodgkin-Huxley-type ionic models of cardiac cells, all the concentrations of the various ionic species were held fixed, so that no provision had to be made for pumps and exchangers to regulate these concentrations. We refer to models that incorporate both of these refinements as second-generation models, in contrast to the earlier first-generation models. We formulate our model below as a more primitive first-generation model, because there are two major problems with the more physiologically realistic second-generation models: 1) drift, with very slow long-term trends in some of the variables, particularly some ionic concentrations (3, 21, 23, 24, 37, 45, 103, 116), and 2) degeneracy, with nonuniqueness of equilibrium solutions such as steady states and limit cycles (23, 24, 30, 37, 102).

Drift has been managed in several ways: 1) by finely adjusting parameters to achieve flux balance across the membrane (21), 2) by adding an electroneutral Na⁺ current of a precise size to produce stability of concentrations (74), 3) by monitoring the stimulus current in a paced quiescent cell (37, 45), and 4) by ensuring that certain ionic concentrations remain fixed (3, 55, 103, 121). It is not clear whether strategies 1 and 2 are robust, because a change in some parameter in the model might require further fine adjustment of the stabilizing parameters. (This is reminiscent of a neutrally stable equilibrium.) Strategy 3 is, of course, of no use in an unpaced pacemaker cell. Strategy 4 defeats, at least in part, the initial intent in formulating the model as a second-generation model; e.g., when all the internal and external iconic concentrations are held constant, the Na⁺-K⁺ pump current (I_NaK) and the Na⁺/Ca²⁺ exchange current (I_NaCa) are effectively background currents, and one is left essentially with a first-generation model, in which activity-dependent effects due to changes in certain ionic concentrations are not manifest.

The other major problem noted with second-generation models is degeneracy. In second-generation models of several different types of cardiac cells, the system of differential equations could be rewritten as a system of N – 1 equations in N unknowns (30, 102). The Jacobian is then singular, and there is a continuum of equilibrium points, rather than isolated equilibrium point(s), so that, e.g., the resting potential of a quiescent system depends on the initial conditions (30, 102). A similar finding of degeneracy holds for the limit cycle that corresponds to spontaneous activity (24, 30). It has been suggested that the original N-variable fully differential model should be recast as a differential-algebraic system, with the equation controlling voltage being algebraic and the remaining (N – 1) equations being differential (23, 24, 30, 37, 45, 102). In one report in a sinoatrial (SA) node model in which the differential-algebraic formulation was used, it was stated that there was no long-term drift (23). In earlier work in which drift was abolished by making some ionic concentrations fixed, this also had the unintended benefit of removing the degeneracy, thus allowing the bifurcation analysis of isolated equilibria by means of continuation techniques (103; see also Refs. 55 and 121).

Finally, in situations such as ours in which cells are studied using patch micropipettes, a more realistic model of the experimental situation is one in which internal concentrations are kept fixed as a result of Ca²⁺ buffering with EGTA and dialysis of the cell contents of a very small cell volume against the much larger pipette volume (55, 121). Making concentrations fixed then also removes degeneracy and drift. Given all the above uncertainties and complications and given that very little information is available about the control of intracellular ionic concentrations in our cells, we decided to use a first-generation model, as have some authors of other quite recent studies (3).

Capacitance. Unless stated otherwise, ventricular and atrial cells will refer to embryonic chick ventricular and atrial cells, respectively, and n-day will refer to a cell isolated from the embryo after n days of incubation. Because the capacitance of a single 7-day ventricular cell in our laboratory is 8–9 pF, we set the capacitance in our three-cell model cluster to 25.5 pF. Our value of the single-cell capacitance agrees with that reported in several whole cell voltage-clamp studies employing single 7-day ventricular cells [e.g., 5–10 pF (29) and 4–7 pF (44)]. We model the cluster, which is a mutually synchronized population oscillator, as an isopotential preparation (27).

I_Ca. I_Ca has been described in 7-day reaggregates of ventricular cells (72), in small clusters of ventricular cells (28, 79), in single ventricular cells (13, 28, 42, 44), and at the single-channel level (43, 68). In one laboratory, L-type I_Ca (I_Ca,L) was found in 11 of 12 6-day ventricular cells, and T-type I_Ca (I_Ca,T) was found in the remaining cell (13). In another laboratory, only I_Ca,L was found in 7-day ventricular cells (68). In a third study, both currents were found, but in only 45% of 7-day ventricular cells was I_Ca clearly separable into I_Ca,L and I_Ca,T on the basis of the voltage threshold for activation (44). In these cells, I_Ca,T is half-inactivated at –49 mV, whereas I_Ca,L is half-inactivated at –27.5 mV. In the 55% of cells with nonseparable I_Ca, half-inactivation occurs at –42.8 mV.

We have chosen to use a nonseparable description of I_Ca in our model

where g_Ca is the maximal conductance, d is the activation variable, f is the inactivation variable, and E_Ca is the reversal potential, which we set to 40 mV on the basis of our own voltage-clamp results (79). The equations governing d and f are given in the APPENDIX.

The steady-state inactivation (f) curve of the nonseparable current in 7-day ventricular cells, which lies between the inactivation curves for I_Ca,L and I_Ca,T, is taken from Fig. 9 of Ref. 44. Because the steady-state activation (d) curve of the nonseparable current was not reported, we assume that it lies between the I_Ca,L and I_Ca,T activation curves of Fig. 10 of Ref. 44 and that it has an intermediate slope factor. We have chosen the values of parameters so that the d curve is closer to the I_Ca,L than to the I_Ca,T activation curve, in order that the peak-current current-voltage (I-V) curve has a maximum at 0 mV, which is close to the experimental value for the nonseparable current (44). Figure 1A shows the steady-state d and f curves.

View larger version (23K): [in this window] [in a new window]   Fig. 9. Block of ICa (experiment and model). A and B: block of ICa with D-600 in experiments [adapted from Koidl and Tritthart (49)]. This preparation is thus not one of the 17 shown in Fig. 5; a sharp microelectrode was used. C and D: block of ICa in the model. Maximal conductance of ICa (gCa) was decreased linearly starting at 45 s at a rate of 0.36 nS/s. At 120 s, when spontaneous activity stops, 90% of gCa is blocked. E: bifurcation diagram with gCa as bifurcation parameter. For a steady state, the bifurcation variable is the voltage coordinate (V) of that point: solid line, stable [resting membrane potential (RMP)]; dashed line, unstable. For a limit cycle, maximum [i.e., overshoot potential (OS)] and minimum [i.e., maximal diastolic potential (MDP)] values of V are plotted: , stable; , unstable; HB, Hopf bifurcation.

View larger version (38K): [in this window] [in a new window]   Fig. 10. Effect of almokalant (experiment and model). A: effect of almokalant on cluster 9 in Fig. 5 (solid line) vs. control (dashed line). B: effect of block of 50% (solid line) and 100% (dashed-dotted line) of IKr in the model. C: IKs during 100% block of IKr in the model (solid line). D: effect of 50% block of IKr and 20% block of IKs in the model (solid line; latter to simulate rundown).

View larger version (30K): [in this window] [in a new window]   Fig. 1. Characteristics of slow inward Ca2+ current (ICa) in the model. A: steady-state activation (d) and inactivation (f) curves. B: time constants of activation (d) and inactivation (f). C: peak ICa in voltage-clamp steps from a holding potential of –80 mV to various test potentials (protocol of Ref. 44). D: simulated current during voltage-clamp steps from a holding potential of –80 mV to 0, –10, and –20 mV (protocol of Ref. 44).

We have set g_Ca so that _max is 9.5 V/s in the model, which is close to our mean experimental result (8.5 V/s). The peak-current I-V curve then has a maximum value of 30 pA/pF at 0 mV (Fig. 1C), which is within the range seen in our cells [79; see also Fig. 4C of Ref. 44 and Fig. 1 of Ref. 13, scaled for differences in capacitance and temperature (Q₁₀ 3 for peak current amplitude for guinea pig ventricular cells (7))]. Figure 1D shows that the current traces from a voltage-clamp protocol in the model are similar, in magnitude and time course, to the corresponding experimental traces in Ref. 79 (see also Fig. 7B of Ref. 44, scaled for temperature and capacitance).

View larger version (13K): [in this window] [in a new window]   Fig. 4. Characteristics of inward rectifier K+ current (IK1) in the model. A: steady-state activation (K1) curve. B: time constant of activation (K1).

View larger version (32K): [in this window] [in a new window]   Fig. 7. Steady-state current-voltage (I-V) curves. A: steady-state I-V relation of total current (Itotal) in experiments (Exp, *; n = 5) and model (curve). B–D: I-V curves of individual currents in the model. Iscal, seal-leak current; Ib, background current.

Delayed K⁺ currents. The delayed K⁺ currents I_Ks and I_Kr have been described in reaggregates of ventricular cells (13, 14, 88) and atrial cells (9, 88, 89), in small clusters of ventricular cells (79), and in single ventricular (8) and atrial (12) cells. These currents have also been observed at the single-channel level in ventricular cells (8, 67).

Two components of the delayed current, initially termed I_x1 and I_x2, have been seen in atrial reaggregates (89). These two currents correspond to those more recently termed I_Kr and I_Ks, respectively, in isolated adult mammalian ventricular cells. I_x2, or I_Ks, has been described in 7-day ventricular reaggregates (14, 15), in single ventricular cells (8), and in small clusters of such cells (79). However, although I_x1 is robust in experiments carried out using sharp microelectrodes on atrial (89) and ventricular (13) reaggregates, it has not been seen in whole cell clamp experiments carried out using patch pipettes on isolated 7- to 10-day ventricular cells (8) or on single 6- to 11-day atrial cells or small clusters of such cells (12). This might be due to an intrinsic absence of the current [e.g., there is good evidence for cell-cell contact-dependent regulation of expression of two different K⁺ channels in cultured adult rat ventricular cells (35)] or to rapid washout of this current in the whole cell ruptured-patch recording mode. Indeed, it has been suggested that the main difference in the action potentials of atrial reaggregates and small clusters of atrial cells can be accounted for by the absence of I_x1 in the latter (see Fig. 9 of Ref. 12 and Fig. 17 of Ref. 13). In contrast to the above-mentioned reports, in our 7-day ventricular clusters, the envelope-of-tails test shows two components (79), and application of the specific I_Kr blocker almokalant (113) removes the more rapidly deactivating component of the tail current. In addition, almokalant produces changes in the action potential consistent with I_Kr block (see below). At the single-channel level, a K⁺ channel that activates over a voltage range similar to that over which I_Kr activates has been described (8). We thus incorporate the I_Ks and I_Kr components into our model.

For I_Ks, we use the formulation previously used in an atrial reaggregate model (52)

where n is the activation variable. We set the reversal potential (E_Ks) equal to –75 mV, the value found in our cells (79; see also Fig. 1C of Ref. 8). This value is depolarized to the equilibrium potential for K⁺ of –85 mV, computed from the Nernst equation at 37°C, external [K⁺] of 5.4 mM, and pipette [K⁺] of 129 mM. This relatively depolarized value of E_Ks has been found in many studies on I_Ks and has been attributed to a slight permeability of the channel to Na⁺ (65, 121). In obtaining the formulas for the rate constants _n and _n (see APPENDIX), we have first divided the original equations (52) by a factor of 3, to obtain a _n curve consistent with the experimental values in single ventricular cells at room temperature (see Fig. 2C of Ref. 8), and then multiplied _n and _n by a factor of 2, in correspondence with the reported Q₁₀ (111), to obtain values appropriate for our experimental temperature of 36–37°C (Fig. 2B). The maximal conductance (g_Ks) was set to give a fully activated I-V curve (Fig. 2C), similar to that seen experimentally (see Fig. 1C of Ref. 8), scaled for capacitance and temperature (Q₁₀ 2) (111). The steady-state activation curve (Fig. 2A) and the voltage-clamp currents (Fig. 2D) are similar to those previously reported from our laboratory at 37°C (79), as well as those reported elsewhere (8), when compensated for temperature.

View larger version (27K): [in this window] [in a new window]   Fig. 2. Characteristics of slow delayed K+ current (IKs) in the model. A: steady-state activation (n) curve. B: time constant of activation (n). C: fully activated (act) IKs. D: simulated current during voltage-clamp steps from a holding potential of –60 mV to –20, 0, and +20 mV (protocol of Ref. 8).

where s is the activation variable, and we introduce z(V) to provide inward rectification so that the fully activated I-V relation (Fig. 3C) is similar in shape to that seen in atrial reaggregates (see Fig. 8C of Ref. 89). Our form of z(V) (see APPENDIX; Fig. 3A) gives more current at depolarized voltages than the fit originally used in Fig. 8C of Ref. 89, which corresponds to the fact that we observe a maintained almokalant-sensitive current at positive voltages (see RESULTS). The function z(V) represents the very rapid inactivation described for the human ether-a-go-go-related gene (HERG) subunit of I_Kr (92, 93) and for I_Kr in single SA node cells (54, 76). The reversal potential (E_Kr) was set to –81 mV on the basis of observations in our laboratory. (This closeness of E_Kr to the Nernst potential of –85 mV for K⁺ is characteristic of I_Kr in many other cardiac preparations.) Our setting of the maximal conductance (g_Kr) yields a maximum value of the fully activated current (Fig. 3C) comparable to that seen in atrial reaggregates (89).

View larger version (29K): [in this window] [in a new window]   Fig. 3. Characteristics of rapid delayed rectifier K+ current (IKr) in the model. A: steady-state activation (s) curve and rectification variable (z). B: time constant of activation (s). C: fully activated IKr; note pronounced rectification. D: simulated current during voltage-clamp steps from a holding potential of –26 mV to –34, –43, –51, and –61 mV (protocol of Ref. 89).

View larger version (29K): [in this window] [in a new window]   Fig. 8. Spontaneous activity in the model. A: transmembrane potential. B: total current. C and D: individual currents. Peak value of ICa is –262 pA (off-scale deflection in C). E and F: activation and inactivation variables in the model. Also shown is the function z(V), describing rectification of IKr.

View larger version (25K): [in this window] [in a new window]   Fig. 5. Transmembrane potential recorded during spontaneous activity from clusters 1–17. Tic marks indicate 0 mV.

where we set E_K1 = –81 mV, which is, as commonly observed, slightly depolarized to the calculated Nernst potential for K⁺ (–85 mV). Although a time-independent description of I_K1 (i.e., K1 = K1) was originally used (65), we employ the time-dependent description to allow us to later formulate a model with stochastic gating kinetics to investigate beat-to-beat fluctuations in IBI (unpublished observations). However, the time constant of this current is so small (_K1 < 0.2 ms over the operative range of voltage; Fig. 4B) that the current is virtually identical in the time-dependent and time-independent descriptions. The maximal conductance (g_K1) is reduced from the guinea pig value to reflect the smaller I_K1 earlier in development (15, 41). The steady-state I-V curve for the total current (see curve in Fig. 7A) is then very flat between –70 and –30 mV, which agrees with our experimental results (see symbols in Fig. 7A). The I_K1 I-V curve (see Fig. 7C) is the main contributor to the positive slope of the total-current I-V curve at very hyperpolarized potentials (see Fig. 7A) and is similar to the Ba²⁺-sensitive current at hyperpolarized potentials (4, 79).

I_b. In addition to I_K1, which is outward at potentials depolarized to –81 mV, there is inward background current in 7-day ventricular reaggregates (14). This component has been modeled as a Na⁺ current

where E_b = 40 mV and g_b is obtained from Fig. 11 of Ref. 14, scaled for capacitance. Figure 7D gives the I-V relation for this linear current.

View larger version (33K): [in this window] [in a new window]   Fig. 11. Effect of Ba2+ (experiment and model). A: effect of 0.1 mM Ba2+ on spontaneous activity in a cluster (solid line) vs. control (dashed line). Cluster is not among those (clusters 1–17) in Fig. 5. B: effect of 100% IK1 block (solid line) vs. control (dashed line) in the model. C: steady-state I-V curve with 0.5 mM Ba2+ () and in control (). Values are means of 9 clusters, none of which are shown in Fig. 5. D: steady-state I-V curve with 100% block of IK1 and 33% block of IKs and IKr (solid line) vs. control I-V (dashed line) in the model.

with g_seal corresponding to a nominal seal-leak resistance of 5 G (20) and E_seal = 0 mV. Figure 7D gives the I-V relation of I_seal.

Currents Not Included in the Ionic Model

I_Na. There is voltage-clamp evidence for the existence of I_Na in reaggregates of 7- to 11-day ventricular cells (22, 72), in single 2- to 18-day ventricular cells (29, 40, 82, 83, 112), and at the single-channel level in 7-day ventricular cells (64, 112). Voltage-clamp studies of 7-day ventricular clusters in our laboratory show a fast inward current upon a depolarizing clamp step from potentials more hyperpolarized than about –60 mV. However, our clusters have a very low upstroke velocity (8.5 V/s), suggesting that I_Na might not contribute appreciably to the upstroke phase, especially because Ca²⁺ channel blockers abolish spontaneous activity (see Fig. 9, A and B). The MDP (–60 mV in the clusters and –67 mV in the model) is sufficiently depolarized to essentially render I_Na fully inactivated, because the foot of the I_Na steady-state inactivation curve lies at about –50 to –60 mV in 7-day ventricular reaggregates (22) and 7-day ventricular cells (29, 82). Indeed, addition of I_Na to our model, on the basis of the conductance and the activation and inactivation curves from single 7-day ventricular cells (29) and the time constants from 11-day reaggregates (22), slightly increases _max from 9.5 to 10.2 V/s. In contrast, reaggregates of trypsin-dissociated 7-day ventricular cells have a TTX-sensitive upstroke velocity of 120 V/s in 1.3 mM K⁺ and 91 V/s in 4.5 mM K⁺, presumably due to the more hyperpolarized MDP of about –90 and –76 mV, respectively (16, 19).

I_Na can also be involved in generating the pacemaker potential. There is indeed evidence that I_Na is necessary in some isolated embryonic cells for the generation of spontaneous activity. After 24 h in culture, 37% of single cells dissociated using trypsin from 7-day hearts (whole hearts, atria, or ventricles) stop beating after the addition of 10^–5 g/ml TTX (70), showing that I_Na is crucial in generating spontaneous activity in these cells. However, the percentage of TTX-insensitive cells increases with time spent in culture: 43% at 4 h, 64% at 24 h, and 100% at 48 h (57). In contrast, reaggregates of trypsin-dissociated 7-day ventricular cells that are cultured for 24–72 h stop beating when exposed to TTX (16, 70). This difference in the response to TTX almost certainly indicates the importance of cell-to-cell interactions (19, 69). In newborn rabbit SA node, a TTX-sensitive current, which gradually disappears within the first 30 days postnatum, has been implicated in the generation of diastolic depolarization (2). This contribution is not due to the I_Na window current but, rather, is a consequence of relatively slow inactivation of I_Na in the pacemaker range of potentials. Recently, modeling work has suggested a role for a persistent component of a mutated I_Na in the generation of diastolic depolarization in long Q-T (LQT3) syndrome (105).

A third role for I_Na is maintenance of the plateau of the action potential, e.g., via a window current contribution. Application of TTX results in a shortening of the APD in some 7-day ventricular cells before they stop beating (64). Single Na⁺ channels occasionally (1 of 100 beats) stay open throughout the action potential plateau (64) and burst for >150 ms in 16% of trials during a long voltage-clamp step (40). Because these long openings do not persist into diastole (see Fig. 1 of Ref. 64), they would not contribute to diastolic depolarization. Incorporation of our standard Hodgkin-Huxley-type I_Na into the model, as described above, results in a slight 3-ms increase in APD₅₀ and a slight 5-ms increase in APD₁₀₀.

Pacemaker current. The pacemaker current (I_f) has been reported in ventricular reaggregates (4, 14, 15, 87, 88) as well as in single atrial and ventricular cells and small clusters of such cells (4, 5, 85). The midpoint of the activation range of I_f is 30 mV more negative in single ventricular cells and small clusters than in reaggregates, with the foot of the activation curve of this hyperpolarization-activated current lying at –70 mV in single cells and small clusters (5). In our clusters, we find I_f activated at potentials negative to –70 mV (79). On the basis of the conductance, reversal potential, kinetics, and activation curve described in Ref. 5, we find that addition of I_f to the model causes only a very slight decrease in IBI from 392 to 390 ms. The MDP in our cells is, hence, too depolarized for I_f to activate and contribute significantly to pacemaking activity; therefore, we do not include it in our model.

Transient outward current. Although the size of the transient outward current (I_to) increases with development, the number of isolated ventricular cells possessing I_to is extremely low: 7 of 300 cells at 3 days, 5 of 200 cells at 10 days, and 5 of 100 cells at 17 days (84). At the single-channel level, an early outward channel appeared in only 1 of 80 patches from 7-day ventricular cells (67). I_to was not seen in single atrial cells (12), nor was it "clearly observed" in 7- to 12-day atrial reaggregates (9). Moreover, in our own voltage-clamp experiments, we have also found no evidence of I_to in the clusters. In the SA node, the I_to density is smaller in cells with a smaller capacitance (58, 109). For these reasons, we do not include I_to in our model.

Cl^– current. A time-independent Cl^–-sensitive current (I_Cl) has been described in isolated 11-day ventricular cells (63). We do not include I_Cl explicitly in the model, but we consider it to be a component of I_b.

I_NaK and I_NaCa. Currents provided by ion pumps and exchangers, e.g., I_NaK, I_NaCa, and the Ca²⁺ pump, also contribute to V. We employ a first-generation model, which does not have an Na⁺-K⁺ pump, an Na⁺/Ca²⁺ exchanger, a Ca²⁺ pump, internal Ca²⁺ dynamics, and variable ionic concentrations. However, I_NaK and I_NaCa are present in 7-day ventricular cells. Although these currents are included in several recent ionic models of cardiac tissue, we do not include them in our model, because this would result in a second-generation model.

In our laboratory, 10 µM ouabain has been used to block I_NaK in 7- to 10-day ventricular cells or in small clusters of such cells (48). After 1 min of superfusion, the IBI first decreases, due to an increase in DDR, OS and MDP gradually fall, APD rises, and IBI increases, so that within a few minutes, spontaneous activity ceases, with the membrane coming to rest at about –30 mV. (Sharp microelectrodes were used in these experiments, so that artifact due to dialysis and current rundown was minimal.) A similar result is seen in 11-day cells cultured as a confluent layer or polystrand, except the initial effect is seen immediately, presumably because of the use of a perfusion system with rapid perfusate changeover (half-time of 5 s), with the membrane coming to rest at about –40 mV (see Fig. 1 of Ref. 39). It has been estimated that I_NaK contributes 0.35 pA/pF at –70 mV in spontaneously beating 11-day reaggregates (100). In another report on 11-day reaggregates at an internal [Na⁺] of 41 mM, the ouabain-sensitive current amounts to 1.7 pA/pF and is independent of voltage over the operating range effective in our clusters (from –60 to +20 mV) (see Fig. 1 of Ref. 99). This value scales to 0.5 pA/pF at an internal [Na⁺] of 10 mM (see Fig. 7 of Ref. 99), which agrees with SA node modeling work (54).

In our model, we can thus mimic the effect of blocking the electrogenic component of I_NaK by adding a constant depolarizing current of 10.2 pA (i.e., 0.4 pA/pF). This has the effect of decreasing IBI from 392 to 337 ms and depolarizing the MDP from –67 to –61 mV; both effects are seen immediately upon block of I_NaK in an experiment (see Fig. 1B of Ref. 39). Because the electrogenic component of I_NaK is removed within a few seconds of the start of block (Fig. 1 of Ref. 100), some secondary change must be responsible for the cessation of activity that occurs some minutes later (see Fig. 1 of Ref. 39). The most likely candidates are the rises in internal [Na⁺] and [Ca²⁺] after I_NaK block, which occur with a time course on the order of minutes (38, 39).

There is clear evidence for I_NaCa in 11-day ventricular reaggregates (100) and 11-day cells cultured as a confluent layer or polystrand (38). The maximum amplitude of I_NaCa in embryonic chick cells is about the same as in guinea pig ventricular cells (62). However, because of the difficulties inherent in interpreting experiments attempting to characterize I_NaCa, many of its fundamental properties (e.g., stoichiometry) remain uncertain in embryonic chick ventricular cells (62). The extent to which I_NaCa is involved in generating diastolic depolarization in SA node cells is controversial (56) and is very different in different SA node models, to the extent that although I_NaCa is inward in most models, it is outward in at least one (see Fig. 7 of Ref. 54). In addition, in a model of spontaneous activity induced by suppression of I_K1 in guinea pig ventricular cells, I_NaCa is inward during the pacemaker potential (91). We are not aware of any studies of the Ca²⁺ pump in embryonic chick ventricular cells.

Given the above problems, as well as other problems described earlier involving degeneracy and drift in models where pumps and exchangers have been added, we have chosen not to include these currents in our model. Rather, I_NaK can be thought of as being incorporated into I_b, whereas the time course of I_Ca in our model very closely resembles the action potential clamp record (i.e., sum of I_Ca and I_NaCa and any Ca²⁺-activated currents) obtained in the SA node when Ca²⁺ entry is blocked (120).

Other currents. Other currents, such as I_st [a sustained inward current, carried by Na⁺, insensitive to TTX, and sensitive to Ca²⁺ channel blockers (71, but see Ref. 107)] and I_K(Ca) [a Ca²⁺-activated K⁺ current, for which evidence is found only in the perforated-patch configuration (120)], exist in the SA node. Because there are no reports of these currents in ventricular cells, we do not include them in our model.

	RESULTS

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

 
Spontaneous Activity

Action potentials recorded from 17 small clusters clearly show considerable cluster-to-cluster variability (Fig. 5) (17). Figure 6A shows a recording of V obtained from one small cluster (cluster 7 in Fig. 5), whereas Fig. 6B shows the phase-plane trajectory, in which the rate of change of V () is plotted vs. V. For this cluster, the mean values of the parameters, averaged over 100 cycles of activity, were as follows: IBI = 458 ms, MDP = –57 mV, APA = 89 mV, _max = 7.5 V/s, DDR = 85 mV/s, APD₅₀ = 124 ms, and APD₁₀₀ = 224 ms. The action potential parameters (means ± SD) of the 17 clusters are given in Table 1. Because of beat-to-beat variability, the action potential parameters for each cluster were averaged over 100 beats before the population average was taken. Figure 6, C and D, gives the voltage-time series and the phase-plane trajectory for the ionic model, and Table 1 gives the action potential parameters in the model, which are quite close to the mean experimental values.

View larger version (32K): [in this window] [in a new window]   Fig. 6. A: transmembrane potential (V) recorded during spontaneous activity from 1 cell in a cluster (cluster 7 in Fig. 5). B: phase-plane trajectory of 8 cycles in the cluster [dV/dt or maximum rate of rise of upstroke () vs. V]. C: V during spontaneous activity in the model. D: phase-plane trajectory in the model (dV/dt or vs. V). Initial conditions (see APPENDIX) closely approximate a point on the limit cycle corresponding to spontaneous activity. In C and D, a transient of 2 s was removed.

The curve in Fig. 7A gives the steady-state I-V relation for the total current in the model. This curve corresponds closely to the mean I-V data points obtained from five 7-day ventricular clusters in our laboratory (Fig. 7A, symbols); see also Ref. 4. The steady-state I-V relations of the individual currents in the model are shown in Fig. 7, B–D.

Currents Underlying the Action Potential

We previously mentioned that the spontaneous activity in the model (Fig. 6, C and D) has action potential parameters that compare well with the mean experimental values (Table 1). Figure 8 shows V during approximately one cycle of spontaneous activity in the ionic model (A) and the various currents (B–D) and the various activation and inactivation variables (E and F). The upstroke phase is clearly generated by I_Ca (Fig. 8C), which rapidly activates (d in Fig. 8E, see also Fig. 1B). During the first third of the action potential, the slow activation of I_Ks (n in Fig. 8F, see Fig. 2B) contributes increasingly to repolarization (Fig. 8C); I_Kr is small (Fig. 8C), despite rapid activation (s in Fig. 8F, see Fig. 3B) because of its strong inward rectification (z in Fig. 8F, see Fig. 3C); there is also a smaller contribution from I_seal, which is outward but becomes less outward with time (Fig. 8D); I_b is inward and gradually becomes more inward with time (Fig. 8D); I_K1 plays no role here (Fig. 8D) because of its strong inward rectification (K1 in Fig. 8E, see Fig. 7C). There is also a secondary increase of I_Ca (Fig. 8C), which serves to maintain the plateau phase of the action potential, despite decreased activation and increased inactivation of I_Ca (d and f in Fig. 8E); this is due to an increase in driving force. The overall shape of the waveform of I_Ca during the action potential resembles that seen in action potential-clamp studies on SA node cells (120) and in some models of such cells (see Fig. 6 of Ref. 54). During the middle part of repolarization, I_Ca, after its secondary peak (Fig. 8C), falls as a result of inactivation and deactivation (f and d in Fig. 8E), which would per se promote repolarization. Because of a decrease in driving force, I_Ks decreases (Fig. 8C) and later starts to deactivate (n in Fig. 8F). There is also a fall in the outward I_seal, which eventually becomes an inward current (Fig. 8D). During the final stage of repolarization, there are contributions from I_Kr (Fig. 8C) and I_K1 (Fig. 8D), which are no longer completely rectified (K1 and z in Fig. 8, E and F).

Currents Underlying the Pacemaker Potential

Because DDR in a three-cell cluster is 100 mV/s (Table 1), the net current during diastolic depolarization is tiny (2.6 pA); it is not even appreciable on the scale of Fig. 8B. I_Ca is inward and gradually becomes more inward throughout phase 4 depolarization (Fig. 8C), which agrees with the results from ruptured-patch action potential clamp studies on single SA node cells (120). I_b and I_seal are also inward throughout phase 4 depolarization but gradually become less inward (Fig. 8D). Although I_Ks, I_Kr, and I_K1 are outward during phase 4 depolarization, I_K1 becomes much less outward (Fig. 8D), I_Ks gradually becomes slightly less outward (not visible on the scale of Fig. 8C) but does not contribute much current, and I_Kr contributes increasingly less outward current as a result of slow deactivation (Fig. 8, C and F).

I_Kr deactivates slowly during diastolic depolarization (Fig. 8, C and F), because the time constant for activation (_s) is several hundred milliseconds over the pacemaker range of potentials (Fig. 3B). Hence, I_Kr is not fully deactivated by the beginning of the upstroke of the action potential. However, the increase in voltage during the upstroke rapidly abolishes I_Kr (Fig. 8C) because of its profound inward rectification (Figs. 3C and 8F). As the membrane then repolarizes, fast recovery from the inactivation of I_Kr is responsible for its rectification (Fig. 8F). The time course of I_Kr during spontaneous activity is very different from that seen in an atrial reaggregate model (see Fig. 15 of Ref. 89), where I_Kr deactivates much more rapidly because of its shorter time constant at the more hyperpolarized MDP of the reaggregate model: about –90 mV (89) vs. –67 mV (present study). However, action potential clamp studies of rabbit SA cells, which are more depolarized than the chick atrial reaggregate, show a time course of I_Kr very similar to that in our model (see Fig. 1C of Ref. 76), as do SA node models that incorporate a sharply rectifying I_Kr component (see Fig. 6, D and E, of Ref. 54).

Effect of Ca²⁺ Channel Blockers on Spontaneous Activity

Application of D-600, a Ca²⁺ channel blocker, on 7- to 10-day cells and small clusters in our laboratory results in the abolition of spontaneous activity (Fig. 9A), with mean resting potential of –36.2 mV (n = 14) (49). We observed similar results with another Ca²⁺ channel blocker, diltiazem (51). In Fig. 9A, a sharp microelectrode is used so that the cessation of spontaneous activity is not due to dialysis of the pipette contents against the intracellular medium, leading to effects such as current rundown.

Gradually increasing block of I_Ca in the model, starting at 45 s in Fig. 9C, gives a time course of the voltage that is similar to the experimentally observed effect of D-600, with the membrane eventually coming to rest at –37 mV at 120 s when 90% of I_Ca is blocked. Because D-600 blocks I_Ca,L and because our nonseparable I_Ca is close to I_Ca,L, the modeling intervention is similar to the experimental intervention of applying an I_Ca,L blocker. In the experiment and the model, loss of OS initially proceeds at a slow rate (from just after arrow 1 to just after arrow 3 in Fig. 9C); then the rate of loss accelerates just before spontaneous activity is extinguished (i.e., just after arrow 3 in Fig. 9C). The MDP initially drifts slowly positive and then suddenly depolarizes much more quickly (starting at arrow 2 in Fig. 9C) before spontaneous activity ceases. In the experiment and the model, the phase of more rapid loss of MDP precedes the phase of more rapid loss of OS. The upstroke velocity gradually decreases throughout the course of the block, and APD₁₀₀ increases (Fig. 9, B and D). Effects in many ways opposite to those described above are seen in our laboratory with administration of a Ca²⁺ channel agonist (BAY K 8644): there are increases in _max, OS, DDR, and APD, as well as a hyperpolarization of MDP and the threshold or take-off potential, and a fall in IBI (28); these changes are also seen in the model.

Figure 9E gives the bifurcation diagram for I_Ca block, computed using XPPAUT (25). The bifurcation parameter is g_Ca, and the bifurcation variable is V. The periodic activity of the model corresponds to the existence of a stable limit cycle in the six-dimensional phase space of the system. As g_Ca is reduced from its nominal value of 30 nS, the limit cycle decreases in size, so that the APA falls: the maximum value of the V coordinate of the limit cycle (i.e., the OS) and its minimum value (i.e., the MDP) are shown in Fig. 9E. The locus of the V coordinate of the unstable steady state, which is also present in the phase space of the system and corresponds to the zero-current crossing of the total-current I-V curve in Fig. 7A, is also shown in Fig. 9E (dashed line). At g_Ca = 4.2 nS, a subcritical Hopf bifurcation (Fig. 9E) produces an unstable limit cycle oscillation, which grows in amplitude as g_Ca is reduced further, until the stable and unstable limit cycles collide at g_Ca = 3.7 nS in a reverse saddle-node bifurcation of limit cycles (31).

Relatively slow ("quasi-static") reduction in g_Ca from its control value of 30 nS (Fig. 9C) corresponds to moving from right to left along the stable limit-cycle branch of the bifurcation diagram in Fig. 9E. Eventually, at g_Ca 3.7 nS, the state point will leave the stable periodic branch and move toward the stable steady state produced in the subcritical Hopf bifurcation. (The resting membrane potential corresponding to this stable steady state is shown in Fig. 9E.) This agrees with the simulation of Fig. 9C, where spontaneous activity is abolished with 90% block of I_Ca. This "falling off" is responsible for the rapid phase of decline in the OS after arrow 3 just before spontaneous activity is abolished in Fig. 9C. [This is also seen in the experiment (Fig. 9A).] In the model (Fig. 9C), the more rapid phase of decline of MDP starts earlier, before the falling off (just after arrow 2), at g_Ca 6 nS, which agrees with the change in the slope of MDP in the bifurcation diagram (Fig. 9E). [This pattern is also seen in the experiment (Fig. 9A).] When a trace such as that shown in Fig. 9C, with distinct phases of change of MDP and OS, is seen experimentally, one should begin to think that abolition of spontaneous activity might involve a subcritical, rather than a supercritical, Hopf bifurcation.

The coexistence of a stable limit cycle and a stable steady state for 3.7 nS < g_Ca < 4.2 nS in Fig. 9E implies that, over this range, one should be able to trigger activity from the resting state by injecting a stimulus and that this activity should be annihilated by injection of a single well-timed stimulus (31). We have indeed observed single-pulse triggering and annihilation in the model at g_Ca = 3.9 nS. Annihilation has been seen in isolated ventricular cells (94) and in reaggregates of atrial cells exposed to TTX (90). We do not know whether the bistable range in Fig. 9E would be wide enough in the experiment to allow observation of single-pulse triggering and annihilation in these clusters, because this would necessitate adjustment of the D-600 concentration to a value within a rather narrow range, which will be different from cluster to cluster and will be unknown a priori. However, other experimental evidence supports the scenario of Fig. 9E.

The existence of a saddle-node bifurcation in Fig. 9E is consistent with three prior observations from our laboratory: 1) During washout of D-600, transient flurries of action potentials occur spontaneously before spontaneous activity is permanently reestablished. The amplitude of the first action potential in each flurry is relatively large, with the amplitude of the following action potentials gradually declining during the course of each episode of transient triggered activity (see Fig. 2C of Ref. 49). 2) Once beating has stopped under the influence of diltiazem, injection of a hyperpolarizing bias current can provoke an episode of transient triggered activity, with the first action potential being an anodal-break response (see Fig. 2 of Ref. 51). As time proceeds and the degree of block continues to increase during quiescence, the number of action potentials in an episode decreases. This "critical slowing-down" behavior is consistent with the existence of a saddle-node bifurcation of limit cycles and can be seen in simulations with the model. 3) In some cells that are initially found to be not spontaneously active, injection of a single hyperpolarizing current pulse again elicits a flurry of triggered action potentials, with the action potential amplitude gradually decreasing during each flurry (see Figs. 4 and 6 of Ref. 49). During ongoing superfusion of these cells with D-600, the number of nondriven action potentials in any one trial gradually decreases from tens of action potentials, then the membrane does not produce triggered action potentials, and finally the membrane becomes inexcitable (see Fig. 4 of Ref. 49).

A response similar to that shown in Fig. 9, A–D, is seen with I_Ca,L block in the SA node in experiments (see references in Ref. 31) and in several ionic models (31, 54, 55). However, in some of these SA node models, a supercritical Hopf bifurcation occurs, so that annihilation and single-pulse triggering cannot occur.

Effect of Almokalant on Spontaneous Activity

Addition of 1 or 2 mM almokalant, a specific blocker of I_Kr (113), to the bath results in a slowing of the terminal rate of repolarization, a small depolarization of the MDP, and a slight loss of OS (Fig. 10A).

In the model, 100% block of I_Kr (Fig. 10B shows 50%, as well as 100%, block of I_Kr) results in a marked depolarization of MDP, a slight slowing of the terminal rate of repolarization, a small increase in APD₁₀₀, a decrease in APD₅₀, a decrease in IBI, a fall in _max, and a decrease in OS. The slowing of the terminal rate of repolarization and the depolarization of the MDP are due to the absence of the I_Kr contribution normally present (see I_Kr trace in Fig. 8C during control activity). The resultant relative depolarization then causes I_Ks to remain considerably more outward during the entire pacemaker potential and even during the early part of the action potential (Fig. 10C shows I_Ks time course during 100% I_Kr block), which is the major cause of the decrease in APD₅₀ in Fig. 10B. This secondary increase in I_Ks also accounts for the relatively small effect of the loss of I_Kr on the rate of terminal repolarization in Fig. 10B. In cells that do not have I_Ks, this indirect effect of I_Kr block on I_Ks would not occur, and one would expect a prolongation of APD, as observed in SA node cells in which I_Ks was not found (106). [In later studies, however, I_Ks was clearly found in SA node cells (59, 110).] The fall in _max and OS is due to a decrease in I_Ca, with a fall in peak value from 263 to 207 pA during 100% I_Kr block as a result of greater inactivation of I_Ca during diastolic depolarization. Similarly, in experiments on the SA node, effects on action potential parameters due to selective block of I_Kr with the compound E-4031 have been found to be due to "a combination of direct and indirect effects on various ionic currents" (106).

The shortening of APD₅₀ in the model with I_Kr block (Fig. 10B) does not occur in the experiment (Fig. 10A). One reason that might account for this discrepancy is the rundown of I_Ks that occurs during the 10-min period between rupture of the patch (to enter whole cell recording mode) and initiation of the recording of the effect of almokalant (79). Figure 10D shows the combined effect in the model of 50% block of I_Kr (to simulate almokalant) and 20% block of I_Ks [to simulate the degree of rundown seen experimentally (79)]: APD₅₀ is no longer decreased.

To avoid the above-mentioned artifact due to dialysis of cytoplasm against the pipette contents and rundown of currents, we also used visual monitoring of mechanical beating to assess the effect of almokalant. Although three clusters stopped beating on exposure to almokalant (and the effect was reversible on washout of almokalant from the bath), another four clusters did not stop beating: IBI was increased in three of four clusters, and there was no change in the remaining cluster. In response to 0.1 µM E-4031, another specific I_Kr blocker, half of the single SA node cells stopped beating in one study (106), whereas none stopped in another study (76). When the concentration was raised to 1.0 µM, at which there are still negligible effects on other currents, all cells ceased spontaneous activity in both studies. A similar response was found in small balls of tissue isolated from the SA node: with 1.0 µM E-4031, activity was abolished in all balls; with 0.1 µM E-4031, activity was sometimes extinguished in balls from the central area, but not in those from more peripheral areas (46). The fact that 0.1 µM E-4031 abolishes spontaneous activity in smaller, but not in larger, single SA node cells agrees with this observation (59), provided that small cells do indeed stem from the central area of the node and larger cells stem from the periphery.

Complete block of I_Kr does not abolish spontaneous activity in the model (Fig. 10B). In contrast, cessation of spontaneous activity does occur with complete I_Kr block in some SA node models, sometimes via a supercritical Hopf bifurcation, sometimes via what is probably a subcritical Hopf bifurcation, and sometimes through an intermediary phase of "irregular dynamics" (see Fig. 13 of Ref. 54).

Effect of Ba²⁺ on Spontaneous Activity

In 7-day ventricular cells, a low concentration of Ba²⁺ (0.2 mM) blocks I_K1 without greatly affecting I_Ks and I_Kr (8). Figure 11A shows the effect of 0.1 mM Ba²⁺ in an experiment, and Fig. 11B shows the effect of blocking I_K1 completely in the model. In the experiment and the model, there is a depolarization of MDP, a decrease in _max, a loss of OS, and a decrease in IBI. With the average drift of action potential parameters in control recordings after 10 min (the time after which the effects due to Ba²⁺ were measured) taken into account, MDP depolarizes by 5%, _max decreases by 13%, APA decreases by 7%, and IBI decreases by 9% (n = 4) (79), whereas in the model, MDP depolarizes by 15%, _max decreases by 28%, APA decreases by 13%, and IBI decreases by 11%. However, the increases in APD and DDR in the experiments (12% increase in APD₅₀ and 28% increase in APD₉₀) are not replicated in the model. Because we have observed neither an increase in the peak I_Ca nor slowed inactivation kinetics of I_Ca with elevated Ba²⁺ (79), we attribute the increase in APD in the experiments at least partially to rundown of I_Ks, as we observed with almokalant (see Effect of Almokalant on Spontaneous Activity).

Increasing Ba²⁺ to 0.5–1.0 mM results in partial block of I_Ks (8, 79) and I_Kr (79), in addition to complete block of I_K1. Figure 11C shows the mean effect (n = 9) on the steady-state I-V relation when 0.5 mM Ba²⁺ is applied to small clusters. Figure 11D shows the effect on the steady-state I-V relation in the model of 100% block of I_K1, 33% block of I_Ks, and 33% block of I_Kr. (The degree of block of I_Ks and I_Kr in the model was chosen to make the blocked current comparable with the combined effects of rundown and Ba²⁺ in Fig. 11C). The steady-state I-V curve is no longer N-shaped. The effect on spontaneous activity in the experiment and the model is a more marked change from control in the action potential parameters than at the lower Ba²⁺ concentration (Fig. 11, A and B). Specifically, in the experiment, again with the average drift over a period of 10 min in action potential parameters in control recordings taken into account, the MDP depolarizes by 11%, _max decreases by 49%, APA decreases by 15%, and IBI decreases by 34% (n = 8) (79), whereas in the model, MDP depolarizes by 27%, _max decreases by 39%, APA decreases by 23%, and IBI decreases by 20%.

At an even higher concentration (1 mM), Ba²⁺ abolishes spontaneous activity in the clusters, with the membrane coming to rest at about –35 mV (79). The effect is reversible, in that spontaneous activity resumes with washout of Ba²⁺. Depolarization of MDP and cessation of spontaneous activity have previously been described in 6- to 7-day ventricular cells exposed to 5–10 mM Ba²⁺ (96). Cessation of spontaneous activity is also seen in the model if, in addition to complete block of I_K1, I_Ks and I_Kr are blocked by 68%, with the membrane coming to rest at –17 mV. In this case, the activity is abolished through a supercritical Hopf bifurcation. This modeling result agrees with there being only one distinct phase of change of MDP and OS in the experiment (79), in contrast to the result of I_Ca block (Fig. 9). Blocking the other background current in the model (I_b) results in the cessation of spontaneous activity via an infinite-period bifurcation.

Effect of I_seal on Spontaneous Activity

It is possible that I_seal is essential for the generation of spontaneous activity in our model. However, when I_seal is removed, spontaneous beating continues in the model, although at a considerably increased IBI of 535 ms and a more hyperpolarized MDP of –72 mV (Fig. 12). This result agrees with the fact that we record from clusters that are seen to be beating before they are subjected to patching; i.e., the depolarizing I_seal does not induce spontaneous beating in our clusters. However, patching onto a single cell can change the IBI or even abolish preexisting spontaneous activity (112). Indeed, if our model (with the nominal seal-leak resistance of 5 G) is reformulated for a single cell (i.e., capacitance = 8.5 pF), the MDP depolarizes to –51 mV and the IBI decreases to 293 ms; with a seal-leak resistance of 3 G, spontaneous activity is abolished in the single-cell model.

View larger version (19K): [in this window] [in a new window]   Fig. 12. Effect of removal of Iseal in the model.

	DISCUSSION

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

 
Spontaneous Activity in Isolated Chick Ventricular Cells

Electrical recordings of spontaneous activity in single embryonic chick ventricular cells, in small clusters of such cells, and in sparse monolayers have been made using conventional sharp microelectrodes (17, 26, 48, 49, 51, 78, 95) or patch pipettes in the whole cell recording mode (64, 67, 112). Among these studies, those that use 7-day ventricular cells at an external [K⁺] of 2.7–5.4 mM generally report that the cells have an MDP between –60 and –70 mV and a slow upstroke velocity of 10 V/s. One systematic study, which used sharp microelectrodes, reported MDP of –68.4 ± 1.0 mV and OS of 22 ± 0.9 mV (n = 30) at an external [K⁺] of 4.2 mM (17). Our findings in small clusters (Table 1) are in agreement with the findings of these previous studies.

How does the electrical activity of small clusters compare with that of the in situ ventricular muscle? The APA and APD of our small clusters (and in the other reports on single cells, small clusters, and sparse monolayers mentioned above) are quite similar to those of in situ ventricular muscle (98, 101, 118). In contrast, in situ ventricular muscle, with one exception (97), has not been reported to show spontaneous phase 4 depolarization (61, 101, 118) and has a much larger _max: 70–94 V/s at 5–7 days (19, 98, 118) vs. 8.5 V/s (present study). The absence of diastolic depolarization in the in situ 7-day chick ventricle could be due to overdrive suppression, because, in the only report in which pacemaker potentials were seen in intact 7-day ventricles, such potentials were seen in only 20–40% of intact ventricles but in 100% of cut 7-day ventricular fragments (97). Alternatively, the dissociation procedure or the time spent in culture (104) could change the electrophysiological properties of cultured cells. Adding the I_Na known to be present in the small clusters to the model (see METHODS) and increasing I_K1 to the point where spontaneous activity stops with a resting membrane potential of –73 mV give a _max of an induced action potential of 54 V/s, suggesting that the low value of _max in the clusters is indeed due to virtually complete inactivation of I_Na.

The action potential parameters of 7-day embryonic chick ventricular cells are very close to those of isolated SA node cells (compare data in Table 1 with data from the many experimental studies summarized in Table 1 of Ref. 54). This is not too surprising, given that a similar mix of currents seems to be present in the central part of the SA node and in our cells. For example, in the smallest SA node cells (presumably from the central part of the node), which have a capacitance about the same as one of our three-cell clusters (i.e., 20 pF), currents such as I_Ca,L, I_Kr, and I_Ks and background currents are present, but other currents such as I_Na, I_f, and I_Ca,T are absent or the membrane is too depolarized to allow activation (e.g., I_f) or to allow removal of inactivation (e.g., I_Na) (36). Indeed, a minimal SA node model with only I_Ca, I_K, and a background current produces very respectable-looking spontaneous activity (33).

Currents Underlying Diastolic Depolarization in the Model

A DDR of 100 mV/s (Table 1) in a three-cell cluster with a capacitance of 25.5 pF requires that the net current flowing during spontaneous diastolic depolarization be 2.55 pA. In our model, this net current is obtained from the algebraic sum of several much larger inward and outward currents (Fig. 8). In models of SA node cells, these individual currents contribute to various extents to produce this tiny net current (see Fig. 4 of Ref. 115 and Figs. 6 and 7 of Ref. 54). At present, we have no way of knowing exactly which currents are flowing and in what amounts during diastolic depolarization in a given cell; to make matters worse, this mix of currents almost certainly changes from cell to cell, especially in cells from structures as inhomogeneous as the SA node (47).

The current for which we have the least direct experimental evidence in our model is I_b (14). We have added this current to the model, because we know that an inward background current is needed to generate a steady-state I-V curve that agrees with the experiment (Fig. 7A) and to allow the membrane to come to rest at about –40 mV when I_Ca is blocked (Fig. 9). Background currents have also been described in SA node cells (34, 106). One or more inward background currents are incorporated into all models of atrial (9, 52, 88–90) and ventricular (11, 88) reaggregates, as well as into all models of SA node cells (see Fig. 4 of Ref. 115 and Fig. 7 of Ref. 54). As in many of the above-mentioned models, the inward background current is necessary for spontaneous activity in our model, in that deletion of the inward background current results in the cessation of spontaneous activity.

Very early during development (3 days), the steady-state inactivation curve of I_Na is shifted in the depolarizing direction (82); simulations suggest that the window component of I_Na might then contribute to diastolic depolarization (83). I_st has been reported during diastolic depolarization in spontaneously active single SA node cells (71, but see Ref. 107); intriguingly, I_st is not present in quiescent SA node cells (71). We know of no reports of I_st in embryonic chick cells.

I_seal injected through the gigaohm seal-leak resistance is a source of artifact in our experimental recordings. This is true even if the perforated-patch, rather than the ruptured-patch, technique were to be used. In contrast to the case in relatively large adult mammalian ventricular cells with input resistances on the order of tens of megohms, I_seal has significant effects when smaller cells (e.g., SA node cells and embryonic cells) with membrane resistances on the order of 1 G at –70 mV [as do our clusters; see also Ref. 29] are studied (20). It is thus conceivable that I_seal is causing depolarization to the point that I_Na and I_Ca,T, although present, become inactivated and I_f becomes deactivated. However, in our model, subtraction of I_seal hyperpolarizes the MDP by only 6 mV (resp. 2 mV) while increasing the IBI from 392 to 535 ms (resp. 472 ms) when the seal-leak resistance is 5 G (resp. 20 G). This extra hyperpolarization is not sufficient to activate significant amounts of I_Na, because when I_Na is added to the model (see METHODS), removal of the 5-G seal-leak resistance results in a decrease in IBI from 392 to 533 ms and an increase in _max from 10.2 to 10.9 V/s. Similarly, when I_f is added to the model (see METHODS), the extra hyperpolarization does not activate I_f significantly: in this case, removal of I_seal increases the IBI from 390 to 520 ms.

Spontaneous Activity: Cell-to-Cell Variability and Activity-Dependent Conductances

There is considerable variability in electrical parameters from cluster to cluster (Fig. 5). The extent to which this is a result of true inhomogeneity within the cellular population [e.g., the regional heterogeneities in the early embryonic chick heart (1, 101) and the apex-to-base and endocardial-to-epicardial ventricular gradients known to be present in adult muscle (77)] is not clear. There is almost certainly an artifact stemming from differences in dissociation conditions from one culture to another (and even from cluster to cluster within the same culture). I_seal is also different from cluster to cluster. Population inhomogeneity exists in ventricular cells isolated from the adult rat heart (77) and the adult guinea pig heart (119), as well as in cells isolated from the rabbit SA node (75). Although at first sight these cell-to-cell differences might appear to be so large as to be problematic for physiological function, experimental and modeling work indicates that cell-to-cell differences in APD would be largely smoothed out in situ by gap-junctional coupling (119); a similar conclusion holds for IBI (108).

The action potential parameters in the model are very close to the mean values obtained in the experiment (Table 1). However, because of the inhomogeneity of the cellular population, a particular action potential parameter in a given cluster can be quite different from that in the model. For example, in Fig. 5 the MDP in the model is more depolarized than in cluster 16, the IBI is much smaller than in cluster 17, and the APD is much shorter than in cluster 2. In each of these cases, the model could presumably be modified to replicate the electrical activity in the particular cluster by changing particular currents (e.g., increasing the conductance of one or more of the K⁺ currents to produce a more hyperpolarized MDP or increasing the conductance of I_Ca to extend the APD). However, this is an exercise in "theorizing in a vacuum," given that one does not know (and cannot know, with the available technology) the exact parameters describing all the ionic currents in a given cell.

The wide variety of shapes of the action potentials seen in the experiment and the corresponding cell-to-cell variability in the individual currents must mean that the currents influencing APD and DDR (and, hence, IBI) are mixed to various extents in different clusters. Because 40% of our clusters do not beat (50), it is perhaps not surprising that there is a great deal of cluster-to-cluster variability in IBI and in the waveform of the action potential. It has been suggested that activity-dependent conductances might provide a negative-feedback mechanism by which spontaneous activity can be regulated (for review see Ref. 80). It is quite conceivable that some such homeostatic process is occurring here, because, e.g., it is known that messenger RNA expression and protein level of one K⁺ channel are decreased as a result of KCl-induced depolarization in pituitary cells (60). It is also possible that the individual currents, through their voltage dependence, negatively feed back on each other to regulate spontaneous activity (73).

Limitations of the Model

Our first-generation model contains no pumps or exchangers and has fixed ionic concentrations. It thus cannot be used to realistically model phenomena such as the rise in internal [Na⁺] seen when I_NaK is blocked or the change in internal [Ca²⁺] that occurs should IBI be changed. Because data are not available for all the currents in 7-day embryonic chick ventricular cells, there are uncertainties in some of the currents: I_Kr is taken from experiments on atrial reaggregates, and we use a nonseparable I_Ca. Although we have used the classic Hodgkin-Huxley description for the gating of all the currents in our model, there are channels that do not always behave in this way, e.g., the "bursting" mode of I_Na, which occasionally results in a longer-lasting current (40, 64). As in all work on ionic models of spontaneously active cells, I_b, which in our case is a composite current (including, e.g., I_NaK), is titrated to produce a reasonable IBI. There is evidence from recent work on the SA node that internal Ca²⁺ cycling contributes to generating spontaneous diastolic depolarization (56). Because there is no firm experimental evidence for this mechanism in embryonic chick ventricular cells, we do not entertain this possibility in our model.

The deterministic Hodgkin-Huxley-type model we present here shows no beat-to-beat fluctuation in action potential parameters. In contrast, all of our experimental recordings show considerable beat-to-beat fluctuations in the various action potential parameters (e.g., IBI). Previous modeling studies of rabbit SA node cells showed that the beat-to-beat fluctuations observed experimentally can be accounted for by the stochastic opening and closing of membrane channels (32, 114). Development of a stochastic single-channel version of the deterministic model we have described above allows investigation of this hypothesis (unpublished observations).

	APPENDIX

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Model Equations

Some model parameters (maximal conductances, reversal potentials, and capacitance) are given in Table 2.

Slow Inward Ca²⁺ Current

Slow Delayed K⁺ Current

Rapid Delayed Rectifier K⁺ Current

Inward Rectifier K⁺ Current

Background Current

Seal-Leak Current

Voltages are in millivolts, currents in picoamperes, conductances in nanosiemens, capacitance in nanofarads, and time in seconds. The initial conditions are V = –66.5526 mV, d = 1.0932 x 10^–4, f = 0.6309, n = 0.0764, s = 0.8019, and K1 = 0.2953.

	GRANTS

TOP ABSTRACT METHODS RESULTS DISCUSSION APPENDIX GRANTS REFERENCES

 
This work was supported by Austrian Science Fund Operating Grant P 15403 (to B. Koidl) and Canadian Institutes for Health Research (CIHR) Operating Grant MOP-43846 (to M. R. Guevara). M. R. Guevara thanks the Medical Research Council of Canada for a Sabbatic Leave Research Grant (1992-93), and T. Krogh-Madsen thanks the CIHR, the Danish Research Agency, and Otto Mønsted's Foundation for predoctoral financial support.

	ACKNOWLEDGMENTS

 
We thank Petra Lang for expert technical assistance and Prof. Erik Mosekilde (Technical University of Denmark), Prof. Michael Mackey (McGill University), and Prof. Helmut Tritthart (Medical University Graz) for facilitating our collaborative work.

	FOOTNOTES

 

Address for reprint requests and other correspondence: M. R. Guevara, Dept. of Physiology, McGill University, 3655 Sir William Osler Promenade, Montreal, Quebec H3G 1Y6, Canada (E-mail: michael.guevara{at}mcgill.ca)

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.

	REFERENCES

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1. Barry A. The intrinsic pulsation rates of fragments of the embryonic chick heart. J Exp Zool 91: 119–130, 1942.[CrossRef]
2. Baruscotti M, DiFrancesco D, and Robinson RB. Na⁺ current contribution to the diastolic depolarization in newborn rabbit SA node cells. Am J Physiol Heart Circ Physiol 279: H2303–H2309, 2000.[Abstract/Free Full Text]
3. Bernus O, Wilders R, Zemlin CW, Verschelde H, and Panfilov AV. A computationally efficient electrophysiological model of human ventricular cells. Am J Physiol Heart Circ Physiol 282: H2296–H2308, 2002.[Abstract/Free Full Text]
4. Brochu R. Pacemaking in Embryonic Chick Heart (Master's dissertation). Montreal, PQ, Canada: McGill University, 1990.
5. Brochu RM, Clay JR, and Shrier A. Pacemaker current in single cells and in aggregates of cells dissociated from the embryonic chick heart. J Physiol 454: 503–515, 1992.[Abstract/Free Full Text]
6. Bub G, Glass L, Publicover NG, and Shrier A. Bursting calcium rotors in cultured cardiac myocyte monolayers. Proc Natl Acad Sci USA 95: 10283–10287, 1998.[Abstract/Free Full Text]
7. Cavalié A, McDonald TF, Pelzer D, and Trautwein W. Temperature-induced transitory and steady-state changes in the calcium current of guinea pig ventricular myocytes. Pflügers Arch 405: 294–296, 1985.[CrossRef][Web of Science][Medline]
8. Clapham DE and Logothetis DE. Delayed rectifier K⁺ current in embryonic chick heart ventricle. Am J Physiol Heart Circ Physiol 254: H192–H197, 1988.[Abstract/Free Full Text]
9. Clay JR, Brochu RM, and Shrier A. Phase resetting of embryonic chick atrial heart cell aggregates. Experiment and theory. Biophys J 58: 609–621, 1990.[Medline]
10. Clay JR and DeHaan RL. Fluctuations in interbeat interval in rhythmic heart-cell clusters. Role of membrane voltage noise. Biophys J 28: 377–389, 1979.[Medline]
11. Clay JR, Guevara MR, and Shrier A. Phase resetting of the rhythmic activity of embryonic heart cell aggregates. Experiment and theory. Biophys J 45: 699–714, 1984.[Medline]
12. Clay JR, Hill CE, Roitman D, and Shrier A. Repolarization current in embryonic chick atrial heart cells. J Physiol 403: 525–537, 1988.[Abstract/Free Full Text]
13. Clay JR, Kristof AS, Shenasa J, Brochu RM, and Shrier A. A review of the effects of three cardioactive agents on the electrical activity from embryonic chick heart cell aggregates: TTX, ACh, and E-4031. Prog Biophys Mol Biol 62: 185–202, 1994.[CrossRef][Web of Science][Medline]
14. Clay JR and Shrier A. Analysis of subthreshold pace-maker currents in chick embryonic heart cells. J Physiol 312: 471–490, 1981.[Abstract/Free Full Text]
15. Clay JR and Shrier A. Developmental changes in subthreshold pace-maker currents in chick embryonic heart cells. J Physiol 312: 491–504, 1981.[Abstract/Free Full Text]
16. Colizza D, Guevara MR, and Shrier A. A comparative study of collagenase- and trypsin-dissociated embryonic heart cells: reaggregation, electrophysiology, and pharmacology. Can J Physiol Pharmacol 61: 408–419, 1983.[Web of Science][Medline]
17. DeHaan RL and Gottlieb SH. The electrical activity of embryonic chick heart cells isolated in tissue culture singly or in interconnected cell sheets. J Gen Physiol 52: 643–665, 1968.[Abstract/Free Full Text]
18. DeHaan RL and Hirakow R. Synchronization of pulsation rates in isolated cardiac myocytes. Exp Cell Res 70: 214–220, 1972.[CrossRef][Web of Science][Medline]
19. DeHaan RL, McDonald TF, and Sachs HG. Development of tetrodotoxin sensitivity of embryonic chick heart cells in vitro. In: Developmental and Physiological Correlates of Cardiac Muscle, edited by Lieberman M and Sano T. New York: Raven, 1975, p. 155–168.
20. DiFrancesco D. The contribution of the "pacemaker" current (i_f) to generation of spontaneous activity in rabbit sino-atrial node myocytes. J Physiol 434: 23–40, 1991.[Abstract/Free Full Text]
21. Dokos S, Celler BG, and Lovell NH. Modification of DiFrancesco-Noble equations to simulate the effects of vagal stimulation on in vivo mammalian sinoatrial node electrical activity. Ann Biomed Eng 21: 321–335, 1993.[CrossRef][Web of Science][Medline]
22. Ebihara L and Johnson EA. Fast sodium current in cardiac muscle. A quantitative description. Biophys J 32: 779–790, 1980.[Medline]
23. Endresen LP, Hall K, Hoye JS, and Myrheim J. A theory for the membrane potential of living cells. Eur Biophys J 29: 90–103, 2000.[CrossRef][Web of Science][Medline]
24. Endresen LP and Skarland N. Limit cycle oscillations in pacemaker cells. IEEE Trans Biomed Eng 47: 1134–1137, 2000.[CrossRef][Web of Science][Medline]
25. Ermentrout B. XPP/XPPAUT [Online]. Dept. of Mathematics, University of Pittsburgh. http://www.math.pitt.edu/bard/xpp/xpp.html [2003].
26. Fänge R, Persson H, and Thesleff S. Electrophysiologic and pharmacological observations on trypsin-disintegrated embryonic chick hearts cultured in vitro. Acta Physiol Scand 38: 173–183, 1956.[Web of Science][Medline]
27. Fischmeister R, Ayer RK Jr, and DeHaan RL. Some limitations of the cell-attached patch clamp technique: a two-electrode analysis. Pflügers Arch 406: 73–82, 1986.[CrossRef][Web of Science][Medline]
28. Flaschberger P. Der Einfluß des Ca²⁺-Agonisten BAY-K-8644 auf die Spontanaktivität embryonaler Hühnchenventrikelzellen (Doctoral dissertation). Graz, Austria: Karl-Franzens University, 1994.
29. Fujii S, Ayer RK Jr, and DeHaan RL. Development of the fast sodium current in early embryonic chick heart cells. J Membr Biol 101: 209–223, 1988.[CrossRef][Web of Science][Medline]
30. Guan S, Lu Q, and Huang K. A discussion about the DiFrancesco-Noble model. J Theor Biol 189: 27–32, 1997.[CrossRef][Web of Science][Medline]
31. Guevara MR and Jongsma HJ. Three ways of abolishing automaticity in sinoatrial node: ionic modeling and nonlinear dynamics. Am J Physiol Heart Circ Physiol 262: H1268–H1286, 1992.[Abstract/Free Full Text]
32. Guevara MR and Lewis TJ. A minimal single-channel model for the regularity of beating in the sinoatrial node. Chaos 5: 174–183, 1995.[CrossRef][Web of Science][Medline]
33. Guevara MR, van Ginneken ACG, and Jongsma HJ. Patterns of activity in a reduced ionic model of a cell from the rabbit sinoatrial node. In: Chaos in Biological Systems, edited by Degn H, Holden AV, and Olsen LF. New York: Plenum, 1987, p. 5–12.
34. Hagiwara N, Irisawa H, Kasanuki H, and Hosoda S. Background current in sino-atrial node cells of the rabbit heart. J Physiol 448: 53–72, 1992.[Abstract/Free Full Text]
35. Hershman KM and Levitan ES. Cell-cell contact between adult rat cardiac myocytes regulates Kv1.5 and Kv4.2 K⁺ channel mRNA expression. Am J Physiol Cell Physiol 275: C1473–C1480, 1998.[Abstract/Free Full Text]
36. Honjo H, Boyett MR, Kodama I, and Toyama J. Correlation between electrical activity and the size of rabbit sino-atrial node cells. J Physiol 496: 795–808, 1996.[Abstract/Free Full Text]
37. Hund TJ, Kucera JP, Otani NF, and Rudy Y. Ionic charge conservation and long-term steady state in the Luo-Rudy dynamic cell model. Biophys J 81: 3324–3331, 2001.[Web of Science][Medline]
38. Jacob R, Lieberman M, and Liu S. Electrogenic sodium-calcium exchange in cultured embryonic chick heart cells. J Physiol 387: 567–588, 1987.[Abstract/Free Full Text]
39. Jacob R, Lieberman M, Murphy E, and Piwnica-Worms D. Effect of sodium-potassium pump inhibition and low sodium on membrane potential in cultured embryonic chick heart cells. J Physiol 387: 549–566, 1987.[Abstract/Free Full Text]
40. Josephson IR and Sperelakis N. Tetrodotoxin differentially blocks peak and steady-state sodium channel currents in early embryonic chick ventricular myocytes. Pflügers Arch 414: 354–359, 1989.[CrossRef][Web of Science][Medline]
41. Josephson IR and Sperelakis N. Developmental increases in the inwardly-rectifying K⁺ current of embryonic chick ventricular myocytes. Biochim Biophys Acta 1052: 123–127, 1990.[Medline]
42. Kawano S and DeHaan RL. Low-threshold current is major calcium current in chick ventricle cells. Am J Physiol Heart Circ Physiol 256: H1505–H1508, 1989.[Abstract/Free Full Text]
43. Kawano S and DeHaan RL. Analysis of the T-type calcium channel in embryonic chick ventricular myocytes. J Membr Biol 116: 9–17, 1990.[CrossRef][Web of Science][Medline]
44. Kawano S and DeHaan RL. Developmental changes in the calcium currents in embryonic chick ventricular myocytes. J Membr Biol 120: 17–28, 1991.[CrossRef][Web of Science][Medline]
45. Kneller J, Ramirez RJ, Chartier D, Courtemanche M, and Nattel S. Time-dependent transients in an ionically based mathematical model of the canine atrial action potential. Am J Physiol Heart Circ Physiol 282: H1437–H1451, 2002.[Abstract/Free Full Text]
46. Kodama I, Boyett MR, Nikmaram MR, Yamamoto M, Honjo H, and Niwa R. Regional differences in effects of E-4031 within the sinoatrial node. Am J Physiol Heart Circ Physiol 276: H793–H806, 1999.[Abstract/Free Full Text]
47. Kodama I, Honjo H, and Boyett MR. Are we lost in the labyrinth of the sinoatrial node pacemaker mechanism? J Cardiovasc Electrophysiol 13: 1303–1305, 2002.[CrossRef][Web of Science][Medline]
48. Koidl B and Tritthart HA. The effects of ouabain on the electrical and mechanical activities of embryonic chick heart cells in culture. J Mol Cell Cardiol 12: 663–673, 1980.[CrossRef][Web of Science][Medline]
49. Koidl B and Tritthart HA. D-600 blocks spontaneous discharge, excitability and contraction of cultured embryonic chick heart cells. J Mol Cell Cardiol 14: 251–257, 1982.[CrossRef][Web of Science][Medline]
50. Koidl B, Tritthart HA, and Erkinger S. Cultured embryonic chick heart cells: photometric measurement of the cell pulsation and the effects of calcium ions, electrical stimulation and temperature. J Mol Cell Cardiol 12: 165–178, 1980.[CrossRef][Web of Science][Medline]
51. Koidl B, Tritthart HA, and MacLeod RS. Different effects of calcium-antagonists on automaticity in single pacemaker cells and in synchronized networks of cultured embryonic heart muscle cells. J Mol Cell Cardiol 18: 207–217, 1986.[CrossRef][Web of Science][Medline]
52. Kowtha VC, Kunysz A, Clay JR, Glass L, and Shrier A. Ionic mechanisms and nonlinear dynamics of embryonic chick heart cell aggregates. Prog Biophys Mol Biol 61: 255–281, 1994.[CrossRef][Web of Science][Medline]
53. Krogh-Madsen T, Glass L, Doedel EJ, and Guevara MR. Apparent discontinuities in the phase-resetting response of cardiac pacemakers. J Theor Biol 230: 499–519, 2004.[CrossRef][Web of Science][Medline]
54. Kurata Y, Hisatome I, Imanishi S, and Shibamoto T. Dynamical description of sinoatrial node pacemaking: improved mathematical model for primary pacemaker cell. Am J Physiol Heart Circ Physiol 283: H2074–H2101, 2002.[Abstract/Free Full Text]
55. Kurata Y, Hisatome I, Imanishi S, and Shibamoto T. Roles of L-type Ca²⁺ and delayed-rectifier K⁺ currents in sinoatrial node pacemaking: insights from stability and bifurcation analyses of a mathematical model. Am J Physiol Heart Circ Physiol 285: H2804–H2819, 2003.[Abstract/Free Full Text]
56. Lakatta EG, Maltsev VA, Bogdanov KY, Stern MD, and Vinogradova TM. Cyclic variation of intracellular calcium: a critical factor for cardiac pacemaker cell dominance. Circ Res 92: E45–E50, 2003.[Medline]
57. Le Douarin G, Renaud JF, Renaud D, and Coraboeuf E. Influence of insulin on sensitivity to tetrodotoxin of isolated chick embryo heart cells in culture. J Mol Cell Cardiol 6: 523–529, 1974.[CrossRef][Web of Science][Medline]
58. Lei M, Honjo H, Kodama I, and Boyett MR. Characterisation of the transient outward K⁺ current in rabbit sinoatrial node cells. Cardiovasc Res 46: 433–441, 2000.[Abstract/Free Full Text]
59. Lei M, Honjo H, Kodama I, and Boyett MR. Heterogeneous expression of the delayed-rectifier K⁺ currents i_K,r and i_K,s in rabbit sinoatrial node cells. J Physiol 535: 70–74, 2001.
60. Levitan ES, Gealy R, Trimmer JS, and Takimoto K. Membrane depolarization inhibits Kv1.5 voltage-gated K⁺ channel gene transcription and protein expression in pituitary cells. J Biol Chem 270: 6036–6041, 1995.[Abstract/Free Full Text]
61. Lieberman M and Paes de Carvalho A. The spread of excitation in the embryonic chick heart. J Gen Physiol 49: 351–363, 1965.[Abstract/Free Full Text]
62. Liu S, Stimers JR, and Lieberman M. Whole-cell current associated with Na-Ca exchange in cultured chick cardiac myocytes. Ann NY Acad Sci 639: 468–470, 1991.[Web of Science][Medline]
63. Liu S, Stimers JR, and Lieberman M. A novel Cl^– conductance in cultured chick cardiac myocytes: role of intracellular Ca²⁺ and cAMP. J Membr Biol 141: 59–68, 1994.[Web of Science][Medline]
64. Liu YM, DeFelice LJ, and Mazzanti M. Na channels that remain open throughout the cardiac action potential plateau. Biophys J 63: 654–662, 1992.[Medline]
65. Luo CH and Rudy Y. A model of the ventricular cardiac action potential. Depolarization, repolarization, and their interaction. Circ Res 68: 1501–1526, 1991.[Abstract/Free Full Text]
66. 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.[Abstract/Free Full Text]
67. Mazzanti M and DeFelice LJ. K channel kinetics during the spontaneous heart beat in embryonic chick ventricle cells. Biophys J 54: 1139–1148, 1988.[Medline]
68. Mazzanti M, DeFelice LJ, and Liu YM. Gating of L-type Ca²⁺ channels in embryonic chick ventricle cells: dependence on voltage, current and channel density. J Physiol 443: 307–334, 1991.[Abstract/Free Full Text]
69. McDonald TF and Sachs HG. Electrical activity in embryonic heart cell aggregates. Developmental aspects. Pflügers Arch 354: 151–164, 1975.[CrossRef][Web of Science][Medline]
70. McDonald TF, Sachs HG, and DeHaan RL. Development of sensitivity to tetrodotoxin in beating chick embryo hearts, single cells, and aggregates. Science 176: 1248–1250, 1972.[Abstract/Free Full Text]
71. Mitsuiye T, Shinagawa Y, and Noma A. Sustained inward current during pacemaker depolarization in mammalian sinoatrial node cells. Circ Res 87: 88–91, 2000.[Abstract/Free Full Text]
72. Nathan RD and DeHaan RL. Voltage clamp analysis of embryonic heart cell aggregates. J Gen Physiol 73: 175–198, 1979.[Abstract/Free Full Text]
73. Noble D, Denyer JC, Brown HF, and DiFrancesco D. Reciprocal role of the inward currents i_b,Na and i_f in controlling and stabilizing pacemaker frequency of rabbit sino-atrial node cells. Proc R Soc Lond B Biol Sci 250: 199–207, 1992.[Medline]
74. Nygren A, Fiset C, Firek L, Clark JW, Lindblad DS, Clark RB, and Giles WR. Mathematical model of an adult atrial cell. Circ Res 82: 63–81, 1998.[Abstract/Free Full Text]
75. Oei HI, van Ginneken ACG, Jongsma HJ, and Bouman LN. Mechanisms of impulse generation in isolated cells from the rabbit sinoatrial node. J Mol Cell Cardiol 21: 1137–1149, 1989.[CrossRef][Web of Science][Medline]
76. Ono K and Ito H. Role of rapidly activating delayed rectifier K⁺ current in sinoatrial node pacemaker activity. Am J Physiol Heart Circ Physiol 269: H453–H462, 1995.[Abstract/Free Full Text]
77. Pandit SV, Clark RB, Giles WR, and Demir SS. A mathematical model of action potential heterogeneity in adult rat left ventricular myocytes. Biophys J 81: 3029–3051, 2001.[Web of Science][Medline]
78. Pappano AJ and Sperelakis N. Low K⁺ conductance and low resting potentials of isolated single cultured heart cells. Am J Physiol 217: 1076–1082, 1969.[Free Full Text]
79. Pelzmann B. Die Wirkung von Bariumionen auf Spontanaktivität und Ionenströme isolierter embryonaler Hühnerherzventrikelzellen (Doctoral dissertation). Graz, Austria: Karl-Franzens University, 1996.
80. Rosati B and McKinnon D. Regulation of ion channel expression. Circ Res 94: 874–883, 2004.[Abstract/Free Full Text]
81. Sachs HG, McDonald TF, and DeHaan RL. Tetrodotoxin sensitivity of cultured embryonic heart cells depends on cell interactions. J Cell Biol 56: 255–258, 1973.[Free Full Text]
82. Sada H, Ban T, Fujita T, Ebina Y, and Sperelakis N. Developmental change in fast Na channel properties in embryonic chick ventricular heart cells. Can J Physiol Pharmacol 73: 1475–1484, 1995.[Web of Science][Medline]
83. Sada H, Ban T, Fujita T, Ebina Y, and Sperelakis N. Role of the steady-state Na⁺ channel current in pacemaker depolarizations in young embryonic chick ventricular myocytes. Jpn J Pharmacol 69: 159–166, 1995.[Medline]
84. Satoh H. Identification of and developmental changes in transient outward current in embryonic chick cardiomyocytes. Reprod Fertil Dev 7: 1369–1374, 1995.[CrossRef][Medline]
85. Satoh H and Sperelakis N. Hyperpolarization-activated inward current in embryonic chick cardiac myocytes: developmental changes and modulation by isoproterenol and carbachol. Eur J Pharmacol 240: 283–290, 1993.[CrossRef][Web of Science][Medline]
86. Schanne OF, Qu J, Haddad GE, and Ruiz-Petrich E. Membrane dysfunction and abnormal spontaneous activity: a study in explanted cardiac cells. In: Membrane Physiopathology, edited by Bkaily G. Boston: Kluwer, 1994, p. 47–70.
87. Shrier A and Clay JR. Pacemaker currents in chick embryonic heart cells change with development. Nature 283: 670–671, 1980.[CrossRef][Medline]
88. Shrier A and Clay JR. Comparison of the pacemaker properties of chick embryonic atrial and ventricular heart cells. J Membr Biol 69: 49–56, 1982.[CrossRef][Web of Science][Medline]
89. Shrier A and Clay JR. Repolarization currents in embryonic chick atrial heart cell aggregates. Biophys J 50: 861–874, 1986.[Medline]
90. Shrier A, Clay JR, and Brochu RM. Effects of tetrodotoxin on heart cell aggregates. Phase resetting and annihilation of activity. Biophys J 58: 623–629, 1990.[Medline]
91. Silva J and Rudy Y. Mechanism of pacemaking in I_K1-downregulated myocytes. Circ Res 92: 261–263, 2003.[Abstract/Free Full Text]
92. Smith PL, Baukrowitz T, and Yellen G. The inward rectification mechanism of the HERG cardiac potassium channel. Nature 379: 833–836, 1996.[CrossRef][Medline]
93. Spector PS, Curran ME, Zou A, Keating MT, and Sanguinetti MC. Fast inactivation causes rectification of the I_Kr channel. J Gen Physiol 107: 611–619, 1996.[Abstract/Free Full Text]
94. Sperelakis N. Electrical properties of embryonic heart cells. In: Electrical Phenomena in the Heart, edited by De Mello WC. New York: Academic, 1972, p. 1–61.
95. Sperelakis N and Lehmkuhl D. Insensitivity of cultured chick heart cells to autonomic agents and tetrodotoxin. Am J Physiol 209: 693–698, 1965.[Abstract/Free Full Text]
96. Sperelakis N and Lehmkuhl D. Ionic interconversion of pacemaker and nonpacemaker cultured chick heart cells. J Gen Physiol 49: 867–895, 1966.[Abstract/Free Full Text]
97. Sperelakis N and McLean MJ. The electrical properties of embryonic chick cardiac cells. In: Fetal and Newborn Cardiovascular Physiology, edited by Longo LD and Reneau DD. New York: Garland STPM, 1978, p. 191–236.
98. Sperelakis N and Shigenobu K. Changes in membrane properties of chick embryonic hearts during development. J Gen Physiol 60: 430–453, 1972.[Abstract/Free Full Text]
99. Stimers JR, Liu S, and Lieberman M. Apparent affinity of the Na/K pump for ouabain in cultured chick cardiac myocytes. Effects of Na_i and K_o. J Gen Physiol 98: 815–833, 1991.[Abstract/Free Full Text]
100. Stimers JR, Shigeto N, and Lieberman M. Na/K pump current in aggregates of cultured chick cardiac myocytes. J Gen Physiol 95: 61–76, 1990.[Abstract/Free Full Text]
101. Van Mierop LHS. Location of pacemaker in chick embryo heart at the time of initiation of heartbeat. Am J Physiol 212: 407–415, 1967.[Free Full Text]
102. Varghese A and Sell GR. A conservation principle and its effect on the formulation of Na-Ca exchanger current in cardiac cells. J Theor Biol 189: 33–40, 1997.[CrossRef][Web of Science][Medline]
103. Varghese A and Winslow RL. Dynamics of abnormal pacemaking activity in cardiac Purkinje-fibers. J Theor Biol 168: 407–420, 1994.[CrossRef][Web of Science][Medline]
104. Veldkamp MW, de Jonge B, and van Ginneken ACG. Decreased inward rectifier current in adult rabbit ventricular myocytes maintained in primary culture: a single-channel study. Cardiovasc Res 42: 424–433, 1999.[Abstract/Free Full Text]
105. Veldkamp MW, Wilders R, Baartscheer A, Zegers JG, Bezzina CR, and Wilde AAM. Contribution of sodium channel mutations to bradycardia and sinus node dysfunction in LQT3 families. Circ Res 92: 976–983, 2003.[Abstract/Free Full Text]
106. Verheijck EE, van Ginneken ACG, Bourier J, and Bouman LN. Effects of delayed rectifier current blockade by E-4031 on impulse generation in single sinoatrial nodal myocytes of the rabbit. Circ Res 76: 607–615, 1995.[Abstract/Free Full Text]
107. Verheijck EE, van Ginneken ACG, Wilders R, and Bouman LN. Contribution of L-type Ca²⁺ current to electrical activity in sinoatrial nodal myocytes of rabbits. Am J Physiol Heart Circ Physiol 276: H1064–H1077, 1999.[Abstract/Free Full Text]
108. Verheijck EE, Wilders R, Joyner RW, Golod DA, Kumar R, Jongsma HJ, Bouman LN, and van Ginneken ACG. Pacemaker synchronization of electrically coupled rabbit sinoatrial node cells. J Gen Physiol 111: 95–112, 1998.[Abstract/Free Full Text]
109. Verkerk AO and van Ginneken ACG. Considerations in studying the transient outward K⁺ current in cells exhibiting the hyperpolarizing-activated current. Cardiovasc Res 52: 517–518, 2001.[Free Full Text]
110. Verkerk AO, Wilders R, Coronel R, Raveslot JH, and Verheijck EE. Ionic remodeling of sinoatrial node cells by heart failure. Circulation 108: 760–766, 2003.[Abstract/Free Full Text]
111. Walsh KB, Begenisich TB, and Kass RS. -Adrenergic modulation of cardiac ion channels. J Gen Physiol 93: 841–854, 1989.[Abstract/Free Full Text]
112. Wellis DP, DeFelice LJ, and Mazzanti M. Outward sodium current in beating heart cells. Biophys J 57: 41–48, 1990.[Medline]
113. Wettwer E, Grundke M, and Ravens U. Differential effects of the new class III antiarrhythmic agents almokalant, E-4031 and d-sotalol, and of quinidine, on delayed rectifier currents in guinea pig ventricular myocytes. Cardiovasc Res 26: 1145–1152, 1992.[Abstract/Free Full Text]
114. Wilders R and Jongsma HJ. Beating irregularity of single pacemaker cells isolated from the rabbit sinoatrial node. Biophys J 65: 2601–2613, 1993.[Medline]
115. Wilders R, Jongsma HJ, and van Ginneken ACG. Pacemaker activity of the rabbit sinoatrial node. A comparison of mathematical models. Biophys J 60: 1202–1216, 1991.[Medline]
116. Xu A and Guevara MR. Two forms of spiral-wave reentry in an ionic model of ischemic ventricular myocardium. Chaos 8: 157–174, 1998.[CrossRef][Web of Science][Medline]
117. Yasui K, Liu W, Opthof T, Kada K, Lee JK, Kamiya K, and Kodama I. I_f current and spontaneous activity in mouse embryonic ventricular myocytes. Circ Res 88: 536–542, 2001.[Abstract/Free Full Text]
118. Yeh BK and Hoffman BF. The ionic basis of electrical activity in embryonic cardiac muscle. J Gen Physiol 52: 666–681, 1968.[Abstract/Free Full Text]
119. Zaniboni M, Pollard AE, Yang L, and Spitzer KW. Beat-to-beat repolarization variability in ventricular myocytes and its suppression by electrical coupling. Am J Physiol Heart Circ Physiol 278: H677–H687, 2000.[Abstract/Free Full Text]
120. Zaza A, Micheletti M, Brioschi A, and Rocchetti M. Ionic currents during sustained pacemaker activity in rabbit sino-atrial myocytes. J Physiol 505: 677–688, 1997.[Abstract/Free Full Text]
121. Zhang H, Holden AV, Kodama I, Honjo H, Lei M, Varghese T, and Boyett MR. Mathematical models of action potentials in the periphery and center of the rabbit sinoatrial node. Am J Physiol Heart Circ Physiol 279: H397–H421, 2000.[Abstract/Free Full Text]

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