Excito-oscillatory dynamics as a mechanism of ventricular fibrillation

Abstract

Background: The instabilities associated with reentrant spiral waves are of paramount importance to the initiation and maintenance of tachyarrhythmias, especially ventricular fibrillation (VF). In addition to tissue heterogeneities, there are only a few basic purported mechanisms of spiral wave breakup, most notably restitution.

Objective: We test the hypothesis that oscillatory membrane properties act to destabilize spiral waves.

Methods: We recorded transmembrane potential (V(m)) from isolated rabbit myocytes using a constant current stimulation protocol. We developed a mathematical model that included both the stable excitable equilibrium point at resting V(m) (-80 mV) and the unstable oscillatory equilibrium point at elevated V(m) (-10 mV). Spiral wave dynamics were studied in 2-dimensional grids using variants of the model.

Results: All models showed restitution and reproduced the experimental values of transmembrane resistance at rest and during the action potential plateau. Stable spiral waves were observed when the model showed only 1 equilibrium point. However, spatio-temporal complexity was observed if the model showed both excitable and oscillatory equilibrium points (i.e., excito-oscillatory models). The initial wave breaks resulted from oscillatory waves expanding in all directions; after a few beats, the patterns were characterized by a combination of unstable spiral waves and target patterns consistent with the patterns observed on the heart surface during VF. In our model, this VF-like activity only occurred when the single cell period of V(m) oscillations was within a specific range.

Conclusion: The VF-like patterns observed in our excito-oscillatory models could not be explained by the existing proposed instability mechanisms. Our results introduce the important suggestion that membrane dynamics responsible for V(m) oscillations at elevated V(m) levels can destabilize spiral waves and thus may be a novel therapeutic target for preventing VF.

Figure 1

Schematic representation of excito-oscillatory dynamics. Schematic of nullclines corresponding to Eqns. [1] and [2]; W = f(V)) is the fast variable nullcline (solid line), and W = g(V)) is the slow variable nullcline (dashed line). Filled circles represent stable equilibrium points; half-filled circles represents marginally stable equilibrium points; and open circles indicate unstable equilibrium points. Asterisks indicate initial conditions, and the thin arrows with lines represent the corresponding “trajectories” in state space. A: When no bias current (I_bias) is applied, a brief S1 stimulus initiates an all-or-none action potential characteristic of excitable cells. The cardiac action potential time series is characterized by four phases (1–4) as shown on the right, while its “state-space” representation is shown by the arrows in the left panel. B: The model cell demonstrates “bistability” when a large value of I_bias is applied, resulting in a stable equilibrium either in the plateau or near rest. C: The model cell exhibits both a stable equilibrium point near rest and an oscillatory equilibrium at elevated V_m when a small value of I_bias is applied resulting in either V_m oscillations in the plateau if I_bias was turned on during the action potential or equilibration near resting V_m if I_bias was turned on during diastole. See text for more details.

Figure 2

Isolated cardiac myocyte dynamics. A: Stimulus current protocol consisted of a brief 3 ms pulse to stimulate the cell (S1) followed by a 4 sec bias current. B: The V_m response for various values of bias current, I_bias (B). C: The steady state V_m at the end of the 4 sec (V_m.end) as a function of I_bias; two steady states can be seen, a low (l) and high (h) branch. The range of bistability and oscillations are shown as thick solid horizontal bars. Open circles with error bars represent the range of V_m at the onset of oscillations. D: Two-state variable excitable model; the fast (solid line) and slow (dashed line) variable nullclines are shown. The filled circle at the nullclines intersection represents the stable excitable equilibrium point. The resting and threshold values of V_m are shown graphically as well as action potential amplitude (APA).

Figure 3

Excito-oscillatory dynamics. A: Sustained oscillations in an isolated myocyte at elevated V_m for intermediate I_bias. B: Fast variable nullcline (solid line) and slow variable nullcline (dashed line) and trajectory evolution away from upper equilibrium point. C: Experimental data were used to identify the characteristics (i.e., growth and frequency of oscillations) of the unstable equilibrium point. D: Bifurcation diagram of excito-oscillatory model eo5 illustrating the relationship between V_m,end and I_bias. The range of bistability and oscillations are shown as thick solid horizontal bars. The two branches in the oscillation region indicate the maximum and minimum values of V_m during oscillations.

Figure 4

Stable spiral waves (excitable models, left; oscillatory models, right). Snapshots of V_m along with trajectory of spiral wave tip with time series of V_m plotted beneath each image (horizontal scale bars indicate 500 ms). All square images (including subsequent figures) represent snapshots of a 10 cm × 10 cm area of V_m represented as greyscale with black representing −80 mV and white representing +50 mV. Model parameter sets are A: e7; B: e1; C: o6; D: o2.

Figure 5

Spiral wave stability. Period of 2-D spiral wave activity (T_2D) versus the corresponding eigenvalue period (T_eigen) for oscillatory (open circles) and excito-oscillatory (filled squares) models. The corresponding model parameters are shown in Table 1. The images represent snapshots of V_m five seconds after spiral wave initiation via cross-field gradients.

Figure 6

Evolution of target pattern. Snapshot of V_m indicating the emergence of a target pattern; white arrows indicate the direction of wave front movement. Time space plots of both V_m and W along the vertical (horizontal) cross sections indicated by dashed lines are shown to the right (bottom). The time series of both variables at the site indicated by intersection of dashed lines is shown at the bottom. The 500 ms scale bar indicates the beat illustrated in the time-space plots and the asterisk indicates the time of V_m the snapshot.

Figure 7

Fibrillation (model, left; experiment, right) evolution from initiation. Phase map (top). Single cell dynamics during fibrillation (middle) and its representation in state space (bottom). Phase maps for the model were computed from both state variables (true state space) with the location of the unstable equilibrium point as the state space origin. Phase maps for the experiment (snapshot area = 4 cm × 4 cm) were computed from reconstructed state space; τ = 10 ms and the state space origin was set equal to 63% of the maximum fluorescence (F) value.[3, 35] Asterisks indicate the origin of target patterns. V_m signal from the model (left) and fluorescence, F, from experiment (V_m surrogate) (right) both exhibit elevated take-off potential and rapid oscillations (horizontal scale bar = 1 sec). True state space for the model (bottom left) and reconstructed state space from the experimental data (bottom right) both reflect oscillations around an upper unstable equilibrium point. The experimental data was taken from an isolated pig heart used in experiments described in Ref #[55].