The modified Luo-Rudy I model

The full Luo-Rudy I model [30] includes 6 voltage-dependent transmembrane ionic currents and a single variable accounting for the intracellular Ca²⁺ level. The inward currents include a spike-producing Na⁺ current (I_Na), an L-type Ca²⁺ current (I_Ca-L), and a constant conductance background current (I_b). The outward currents include a delayed rectifier K⁺ current (I_K), an extracellular [K⁺]-dependent K⁺ current (I_K1), and a high-threshold K⁺ current (I_Kp). Together, the Luo-Rudy I model contains 8 coupled nonlinear ordinary differential equations.

Our analysis, however, utilizes a reduced Luo-Rudy I model that only contains elements for the electrical component. This facilitates the mathematical analysis, and allows us to demonstrate that even a simple model can account for the findings of the dynamic clamp experiments [14, 15] that are the focus of this study. The modified model does not include equations for the intracellular Ca²⁺ concentration, because in the dynamic clamp experiments Ca²⁺ influx was pharmacologically blocked. Also, since the model Na⁺ current rapidly inactivates for V > −40 mV, i.e., I_Na ≈ 0 when EADs occur, this current is also excluded.

The modified model contains the following differential equations for the membrane electrical dynamics: (1) with ionic currents given by (2) Here, C_m is membrane capacitance and I_stim is a time-dependent mollified square-wave stimulus current with amplitude 70 μA/cm² and 2 ms duration. Each transmembrane ionic current is formulated using the standard Hodgkin-Huxley formalism for excitable membranes [31, 32]. For example, in the expression for the Ca²⁺ current (I_Ca-L), g_Ca is the maximal conductance, a parameter; the dynamic variables d and f are the open fraction of activation and inactivation gates, respectively, of all voltage-gated Ca²⁺ channels; and (V − V_Ca) is the driving force for ion flux, where V_Ca is the reversal potential for Ca²⁺.

The x variable, which appears in the expression for I_K, denotes the (slow) activation of this current. Each of the steady-state activation and inactivation functions, j_∞(V) for j = d, f, x, X₁, K₁ and K_p, are increasing and decreasing sigmoids, respectively. We use upper-case letters to denote quantities that adjust instantaneously to variation in V and thus remain at quasi-equilibrium. The time constants, τ_d(V) and τ_x(V), are bell-shaped, while τ_f(V) is strictly increasing. The magnitudes of the time constants govern how quickly the companion gating variable adapts to changes in V. Small (large) values of τ_j(V), j = d, f, x represent rapid (slow) adaptation. We refer the reader to [30] for the full model formulation.

All parameter values are identical to those used in [30], with the exception of the default maximal I_Ca-L conductance, g_Ca, which is set at 0.112 mS/cm² to facilitate EAD production. Some parameter values are varied to examine robustness of behaviors, and this is stated explicitly in the text of figures. Under all relevant parameter variations, the model (1) (absent I_stim) possesses a stable equilibrium, E₁, which functions as the cell rest state. Under parameter sets that are capable of producing EADs, (1) possesses two additional equilibria, E₂ and E₃, which are located at elevated membrane potentials. The equilibrium E₂ can be either an unstable or stable spiral in parameter regions that produce APs with EADs, while E₃ is always an unstable saddle point. The computer programs used to generate the results herein are available at: www.math.fsu.edu/∼bertram/software/cardiac.

Model I_Ca-L and modifications of its “window region”

The manuscript focuses primarily on model responses to translations in the steady-state I_Ca-L activation and inactivation functions, d_∞(V) and f_∞(V), respectively. The region where these two curves overlap has been termed the “window region” [9] (see Fig 2a) and it has been implicated in the generation of EADs. Fig 2 shows plots of d_∞(V) and f_∞(V) under the default parameter set (black curves). In Fig 2a, the window region is increased by either (or both) translating d_∞(V) leftward or translating f_∞(V) rightward. In Fig 2b, the window region is reduced by translating d_∞(V) rightward or translating f_∞(V) leftward.

Both d_∞(V) and f_∞(V) are sigmoidal in V, and are parameterized by their steepness and by the value, V, of half-activation and half-inactivation, respectively. Translation of each curve is accomplished by varying its half-activation/inactivation value. For clarity and consistency with experimental works, we discuss variation in the half-activation/inactivation values of the curves with reference to the default parameter set and denote the direction and magnitude of variation in the half-activation value of d_∞(V), for instance, by ΔV_1/2(d_∞). We similarly denote translations in f_∞(V) by ΔV_1/2(f_∞). We also note that the enlargement of the window region in Fig 2a and the narrowing of the window region in Fig 2b are symmetric with respect to the direction and magnitude of the translation in each curve. That is, the translations of both curves in each panel are equal in magnitude, but opposite in sign (i.e., for Fig 2a, ΔV_1/2(f_∞) = -ΔV_1/2(d_∞)).

Symmetric enlargement of the model window region can produce EADs

Previous experimental and mathematical studies of EADs have concluded that most EADs occur while voltage is within the interval where the activation and inactivation curves (d_∞(V)and f_∞(V), respectively, in our model) of I_Ca-L overlap, termed the “window region”. The experimental work [15] showed that symmetric enlargement of the window region can lead to EADs as well as the inability of the cell to repolarize (see Fig 5 of [15]) in response to low-frequency periodic pacing.

Representative responses of the model cell to symmetric broadening of the I_Ca-L window region are shown in Fig 3. Fig 3a shows a sequence of symmetric translations of both the steady-state activation and inactivation curves, which enlarge the window region. The green curves denote the default state of the model window region (ΔV_1/2(d_∞) = ΔV_1/2(f_∞) = 0 mV), while the black curves denote the largest translation depicted (ΔV_1/2(d_∞) = -3.12 mV and ΔV_1/2(f_∞) = +3.12 mV). Fig 3b shows color-coded voltage traces of the corresponding model responses to a single stimulus pulse under each translation condition from Fig 3a. The green voltage trace shows the standard cardiac action potential without alteration. The orange trace shows a slightly prolonged action potential in response to a small symmetric enlargement of the window (ΔV_1/2 = 1.04 mV), but no EADs. The red trace shows that a larger translation (ΔV_1/2 = 2.08 mV) elicits two EADs, which prolong the duration of the action potential dramatically. Finally, the black trace shows that a sufficiently large increase in the size of the window region (ΔV_1/2 = 3.12 mV) leads to repolarization failure, where the cell remains at a depolarized voltage.

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Fig 3. A sufficiently large symmetric broadening of the window region can lead to EADs and repolarization failure in response to a stimulus pulse.

(a) An equally-spaced sequence of three color-coded symmetric window broadening translations in d_∞(V) and f_∞(V) (orange, red, and black curves) are shown alongside the default curves d_∞(V) and f_∞(V) (green). The magnitudes of each of the simultaneous changes to both ΔV_1/2(d_∞) and ΔV_1/2(f_∞) are shown in the legend. (b) The color-matched model responses correspond to the manipulations in panel (a).

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Left shifts in the activation curve are more effective at facilitating EADs than right shifts in the inactivation curve

Using the dynamic clamp technique to inject a model Ca²⁺ current into a cardiomyocyte, it was shown that simultaneous broadening of the window region by shifting both the Ca²⁺ current activation and inactivation curves facilitates EAD production and repolarization failure [14, 15]. Translations in either the activation or inactivation curves, but not both, were also examined. It was determined that left-translations in the activation curve alone were a more potent driver of EADs and repolarization failure than right-translations in the inactivation curve alone [15]. That is, using equal-in-magnitude translations of each curve in separate trials, left-translations in d_∞(V) more often led to EADs and repolarization failure than did right-translations of f_∞(V).

To test this experimental finding with the modified Luo-Rudy model, we first applied left-shifts of the Ca²⁺ activation curve, d_∞(V), of magnitudes such that the first shift (ΔV_1/2(d_∞) = −1.8 mV) resulted in a longer action potential, the second (ΔV_1/2(d_∞) = −3.6 mV) resulted in an action potential with two EADs, and the third shift (ΔV_1/2(d_∞) = −5.4 mV) resulted in repolarization failure. That is, the magnitude of the shifts were chosen so that the responses mimicked those of Fig 3. These are shown in Fig 4a and 4b. We then applied right shifts of the same magnitude to the Ca²⁺ inactivation curve, f_∞(V). These translations and the responses are shown in Fig 4c and 4d. In this case, EADs are only produced with the largest translation (ΔV_1/2(f_∞) = 5.4 mV), and none of the translations result in repolarization failure. Thus, the left shifts in d_∞(V) are more potent than equal right shifts in f_∞(V) at evoking EADs and repolarizaiton failure, as was shown experimentally in [15].

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Fig 4. Left shifts in the Ca²⁺ current activation curve are more effective at inducing EADs and repolarization block than right shifts in the inactivation curve.

(a) Three equally-spaced left shifts in d_∞(V) (ordered orange, red, then black) are shown, while leaving f_∞(V) (dashed, black) unchanged. As in Fig 3, green denotes the default. The shifts are given in the legend. (b) The model responses to the left-translations shown in (a) mirror those of Fig 3b: sufficiently large translation induces two EADs (red trace) and the largest translations lead to repolarization failure (black trace). (c) Right shifts in f_∞(V) of equal size to those of (a). (d) The model responses to increasing ΔV_1/2(f_∞) are less severe than those of equally-sized changes in ΔV_1/2(d_∞): the largest change in ΔV_1/2(f_∞) produces EADs (black trace) instead of repolarization failure.

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Enlarging the model window region generically leads to EADs and repolarization failure

In this section, we quantify the effectiveness of activation/inactivation curve shifts in inducing pathological behavior by examining combinations of the shifts, ΔV_1/2(d_∞) and ΔV_1/2(f_∞), that produce EADs or repolarization failure. This is organized using a two-dimensional grid in ΔV_1/2(d_∞) and ΔV_1/2(f_∞), noting that left-shifts in d_∞(V) induce EADs, while right-shifts in f_∞(V) induce EADs. Moving leftward along the ΔV_1/2(d_∞)-axis (to negative values) in Fig 5 corresponds to left shifts in d_∞(V), while moving upward along the ΔV_1/2(f_∞)-axis (to positive values) corresponds to right shifts in f_∞(V). To determine model behavior at each point in the 300 × 300 grid of parameter values, the model was integrated for 10,000 ms at each point using the stable rest state as initial condition. In each case, a supra-threshold pulse of current of amplitude 70 μA/cm² was applied for 2 ms to initiate an AP.

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Fig 5. Model responses to a single depolarizing pulse over a uniform grid in the (ΔV_1/2(d_∞), ΔV_1/2(f_∞)) parameter plane (units in mV).

The green region, labelled “No EADs”, denotes solutions that do not exhibit EADs before returning to rest. The white region, labelled “Repolarization Failure”, denotes solutions that can exhibit EADs around an elevated membrane potential, but remain depolarized. The red region, labelled “EADs”, contains solutions that exhibit EADs and return to rest at the end of the action potential. Darker shades of red in this region denote increasing numbers of EADs in response to the pulse. The dashed blue line segment gives the path in parameter space that corresponds to symmetric window-broadening. Green, red, orange, and black disks along this path correspond to the specific parameter values that produce the color-matched window regions and model responses shown in Fig 3. Blue ⋄ markers labeled 7a, 7b, 7c and 9a, 9b, 9c are parameter sets whose solutions are viewed in (f, x, V) phase space in Figs 7 and 9, respectively. The slope (>1) of the green curve, which marks the boundary between the “No EADs” and “EADs” regions, explains why left shifts in d_∞(V) are a more reliable source of EAD production than right shifts in f_∞(V).

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The light green region in Fig 5, labeled “No EADs”, shows parameter values that produce action potentials without EADs. These solutions may, however, exhibit prolonged action potentials (e.g., orange trace, Fig 4b). The white region, labeled “Repolarization Failure”, denotes the region of parameter combinations that produce solutions that remain in the depolarized state in response to the stimulus pulse (e.g., black trace, Fig 4b). The red region denotes those parameter combinations that produce solutions that contain EADs, but return to rest following the pulse (e.g., red trace, Fig 4b). A dashed curve is superimposed on the figure denoting the path in the (ΔV_1/2(d_∞), ΔV_1/2(f_∞))-plane used to produce Fig 3. The sequence of parameter sets shown in Fig 3 are marked with color-matched disks: the green disk signifies the default parameter set, the red disk (within the blue “⋄” marker labeled “(b)”) lies within the “EADs” region, and the black disk lies in the “Repolarization Failure” region.

The red “EADs” region possesses finer structure than the light green or white regions. Increasingly darker shades of red are used to indicate incremental increases in the number of EADs produced: 6 or more EADs are produced within the darkest shade of red, and some parameter combinations in this region produce solutions with as many as 40 EADs. The diagram shows that variation in the number of EADs elicited in this red region is organized into bands that gradate the transition from “No EADs” to “Repolarization Failure” and that the size of the bands declines corresponding to more EADs. That is, the red “EADs” region is dominated by solutions exhibiting few, rather than many, EADs. This finding predicts that action potentials with relatively few EADs should be more readily observed in experimental settings, as does indeed seem to be the case in published voltage traces from isolated myocytes [7, 14, 15].

The finding (both in the model and experimentally) that EADs are produced more effectively by left shifts in d_∞(V) than right shifts in f_∞(V) is evident in Fig 5. The curve that separates the “No EADs” region from the “EADs” region (green line) is approximately linear with slope s ≈ 1.34. Because the slope is greater than 1, it takes a larger change in ΔV_1/2(f_∞) than in ΔV_1/2(d_∞) to move from a parameter combination producing a pure action potential to one producing an action potential with EADs.

We can also use the slope of the green EAD boundary curve to make predictions about the potential therapeutic effects of window-shrinking shifts in either d_∞(V) or f_∞(V). Because the slope is greater than 1, the horizontal (rightward) distance from any point in either the “EADs” (red) or “Repolarization Failure” (white) regions to the green boundary between the “EADs” and “No EADs” regions is always smaller than the vertical (downward) distance. Thus, small window-shrinking translations in d_∞(V) should be a more reliable therapeutic target than small window-shrinking translations of f_∞(V) for the elimination of pathological rhythms (EADs or repolarization failure) induced by an enlarged window region.

An additional feature of the diagram that would not be readily discernible from either experiments or simulations is that the “EADs” region (bounded between the green and black curves) grows in width for increasing values of ΔV_1/2(f_∞) but, shrinks in width for decreasing values of ΔV_1/2(d_∞), even though both of these manipulations enlarge the window region. This feature of the diagram arises from the fact that the slope of the (almost linear) black curve, marking the boundary between the “EADs” and “Repolarization Failure” regions, has an even larger average slope than that of the green boundary curve. This feature of the grid makes the experimentally testable prediction that the transition of a cell from EADs to repolarization failure should also occur for smaller window-enlarging shifts in d_∞(V) than f_∞(V). That is, given a cell exhibiting EADs due to an enlarged window region, small increases in the magnitude of ΔV_1/2(d_∞) should be more likely to lead to repolarization failure than small increases in ΔV_1/2(f_∞). In addition, this predicted disparity between the effects of ΔV_1/2(d_∞) and ΔV_1/2(f_∞) in producing repolarization failure should be more pronounced than the disparity observed for the production of EADs shown in Fig 4.

Fast-slow analysis reveals a mechanism for EAD generation

We have seen that broadening the I_Ca-L window region can lead to EADs and repolarization failure. Here we explore why, using a fast-slow analysis. Fast-slow analysis splits a model into (simpler) lower-dimensional subsystems in order to analyze these subsystems semi-independently and stitch together the results. In [21], we showed that (1) possesses a multi-timescale structure. This structure is reflected by the rapid upstrokes and downstrokes of the AP, with long depolarized plateau (Fig 1b). Specifically, we showed that the 4-dimensional model contains fast variables V and d (voltage and I_Ca-L activation), and slow variables f and x (I_Ca-L inactivation and I_K activation). The parameter C_m approximately characterizes the timescale separation, with C_m → 0 (termed the singular limit) yielding the decomposition of (1) into separate fast and slow subsystems (see [21] for details).

With our (2,2)–fast-slow splitting, the 2-dimensional fast subsystem (3) is an approximation of the fast motions of (1) (see Fig 6, double arrows) in which the slow variables, f, and x, are treated as parameters. The time-dependent forcing, I_stim, is dropped from the V-equation because I_stim ≈ 0 after the stimulus has been applied. The equilibria of (3) (traced out by independent variation in f and x) form a 2-dimensional surface, called the critical manifold. Fig 6 shows two views of the EAD-containing voltage trace from Fig 3b in (f, x, V) phase space and superimposed on the critical manifold. The critical manifold is comprised of attracting ( and , blue) and saddle-type (, red) sheets that are connected by curves of fold points. Only the upper fold, L (green), falls within the physiologically relevant domain (the lower curve is out of the frame of the figure, so not visible). The stability properties of the critical manifold are determined by linear stability analysis of the fast subsystem. The true equilibria, E₁, E₂, and E₃ of the full system (1) persist as equilibria of the fast subsystem (3). While E₂, under this parameter set, is a stable spiral of the full flow (1) (i.e., for C_m = 1 μF/cm²), it becomes a saddle point (located on ) of the fast subsystem (3) (i.e., for C_m = 0 μF/cm²). We note that there are no Hopf bifurcations in the fast subsystem, so EADs do not arise as oscillations in the fast subsystem as they do in previous works (e.g., [22]).

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Fig 6. Two views of the EAD-containing voltage trace from Fig 3b superimposed on the critical manifold in (f, x, V) phase space.

(a) (x, V)-dominant view. (b) (f, V)-dominant view. The superimposed solution (black) is comprised of: 1) a fast upstroke (cyan double arrows) caused by a stimulus pulse applied at rest (stable equilibrium E₁), 2) slow evolution (single arrow) along the upper attracting sheet of the critical manifold, (upper blue surface), during the plateau phase, 3) oscillatory EADs (unfilled arrows) near the fold curve, L (green), 4) fast transition (double arrows) toward the lower attracting sheet, , and 5) slow return (single arrow) to E₁ along . The folded node singularity (FN, purple marker), a pseudo-equilibrium of the slow subsystem, lies within L; its associated singular strong canard, (magenta), a special solution of the slow subsystem, together with L, bounds the region of solutions of the slow subsystem, that are funneled through the folded node. Parameter values: ΔV_1/2(d_∞) = -ΔV_1/2(f_∞) = -2.08 mV.

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The 2-dimensional slow subsystem (4) is an approximation of the slow motions of (1) (see Fig 6, solid single arrows) in which V and d are assumed to be at quasi-equilibrium. Hence, solutions of the slow subsystem (4) are slaved to the critical manifold.

To understand the trajectory of the full model (1), one can concatenate orbit segments from the fast and slow subsystems. This is only an approximation, however, and as we see below neither the fast nor the slow dynamics independently explain the EADs. The fast and slow motions are denoted using single and double arrows, respectively. A sufficiently strong stimulus pulse applied to the rest state, E₁(on ), triggers a rapid excursion toward (cyan double arrows denote that this motion is the result of a depolarizing pulse). Once near , the solution moves slowly as it follows closely during the plateau phase, toward the fold, L. The oscillations that occur near L are the EADs. Once several of these have occurred, the trajectory moves rapidly toward . It then follows closely as it moves slowly back towards the rest state, E₁.

The unfilled arrows along the oscillatory EAD portion of the solution indicate that this motion is neither strictly fast nor slow. Indeed, it is precisely at the fold curve L where the fast-slow approximation breaks down. That is, the fold marks the transition boundary between the non-overlapping regions of validity for the fast and slow subsystem approximations.

Without a fast subsystem mechanism for the generation of EADs, we turn to further inspection of the slow subsystem. The general procedure for this analysis can be found in the review article [33] and the details for the particular case of the slow subsystem (4) can be found in [21]. Here, we summarize the key elements. Solutions of the slow subsystem, when initiated on , flow toward the fold curve. Upon reaching the fold, these solutions typically transition to the fast subsystem dynamics, so the trajectory quickly moves from the top sheet to the bottom sheet . However, there may exist distinguished points on the fold curve called folded node singularities [34] (Fig 6; purple marker, “FN”) at which solutions can cross from to , remain governed by the slow subsystem dynamics, and follow for long times. Such solutions are known as singular canards. Given the presence of a folded node singularity, there is a special singular canard that acts as a boundary along between solutions that, upon reaching the fold, either funnel through to the folded node or transition to the fast dynamics. This special singular canard is called the singular strong canard (Fig 6; , magenta).

For C_m > 0, singular canards become solutions of the full model (1) with similar properties, i.e., they remain near for long times on the slow time scale [33, 35]. These solutions are called canards and they are the basis for EADs, as demonstrated in [21].

Canards explain the emergence and number of EADs

Many features of the slow flow persist in the flow of the full system of equations provided there is sufficient timescale separation between fast and slow variables. Theoretical justification for this persistence is provided by Fenichel theory [36, 37]. Specifically, Fenichel theory guarantees that the attracting and saddle-type sheets of the critical manifold, outside the vicinity of the fold curve, perturb smoothly to nearby slow manifolds under the flow of the full system, with their local attraction properties perturbing smoothly as well. In turn, the (slow) flow on these sheets is a smooth perturbation of the slow subsystem flow.

Near the folded node, the relationship between the slow subsystem flow and that of the full system is more intricate, and is described by canard theory [33–35, 38]. In particular, canard theory holds that in the neighborhood of the folded node, under the full system flow, the attracting and saddle-type sheets perturb to slow manifolds that (approximately) twist around the weak eigendirection of the folded node [33, 39]. This twisting allows the slow manifolds to be partitioned into rotational sectors, each of which oscillates around the weak eigendirection of the folded node a fixed number of times. The boundaries between different rotational sectors are curves called maximal canards. The first maximal canard, the boundary between the rotational sector that does not oscillate near the folded node (the left half of the upper attracting sheet) and the sector that oscillates once, is called the primary maximal canard.

Maximal canards have been shown to be objects of key importance in determining whether, and what kinds of potentially erratic, EAD rhythms are evoked in low-dimensional variants of the Luo-Rudy model in response to changes in ion channel expression and chemical composition of the cellular environment [21, 24, 25]. The primary maximal canard (γ₀) is the perturbed analog of the slow subsystem singular strong canard () and is, therefore, the boundary between standard action potentials—to its left—and those that exhibit EADs or repolarization failure—to its right. A solution that enters the rotational sector between the primary maximal canard, γ₀, and the maximal canard, γ₁, exhibits one canard-induced EAD; a solution that enters the rotational sector between maximal canards γ₁ and γ₂ exhibits two canard-induced EADs; so, in general, a solution that enters the rotational sector between γ_n and γ_n+1 exhibits n canard-induced EADs.

Fig 7 shows key structures in phase space for responses that exhibit no EADS (Fig 7a), EADs (Fig 7b), and repolarization failure (Fig 7c). Parameter values for these behaviors are marked with ⋄ in Fig 5 labeled 7a, 7b, and 7c. Each panel shows the critical manifold and its stability properties along with the first three maximal canards (γ₀, magenta; γ₁, cyan; γ₂, orange), computed using numerical continuation and bifurcation software AUTO [40] and methods developed in [41] which are described for this system in [21]. Also superimposed are portions of the solution segment of the full system (Γ, black) following an impulse-producing stimulus.

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Fig 7. Maximal canard locations in (f, x, V) phase space mediate the transition from standard action potentials to repolarization failure and determine EAD number under symmetric I_Ca-L window region enlargement.

(a) Local phase space for marker 7a in the “No EADs” region of Fig 5 (ΔV_1/2(d_∞) = -ΔV_1/2(f_∞) = -1.83 mV). The pulse-induced solution segment, Γ (black), lies to the left of the primary maximal canard, γ₀ (magenta), and does not exhibit EADs. (b) Local phase space of marker 7b in the 2 EAD band of the “EADs” region of Fig 5 (ΔV_1/2(d_∞) = -ΔV_1/2(f_∞) = -2.08 mV). The solution segment lies within the rotational sector between maximal canards γ₁ (cyan) and γ₂ (orange) and exhibits two EADs. (c) Local phase space for marker 7c in the “Repolarization Failure” region of Fig 5 (ΔV_1/2(d_∞) = -ΔV_1/2(f_∞) = -2.33 mV). The solution segment spirals toward stable equilibrium E₂, failing to return to rest. Attracting (, blue) and saddle-type (, red) sheets of the critical manifold meet at the fold curve, L (green). Parameter values used are listed in each panel.

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In Fig 7a, the solution segment (Γ, black) evolves closely along the critical manifold, and since it lies to the left of the primary maximal canard it does not exhibit EADs. Instead, it returns to the repolarized rest state to complete the action potential. However, the close proximity of Γ to γ₀ extends the duration of the plateau phase of the action potential evident in the orange traces of Figs 3b and 4b. We note that the equilibrium, E₂, is unstable for this choice of parameters (ΔV_1/2(d_∞) = -ΔV_1/2(f_∞) = -1.83 mV).

A solution segment with two EADs is shown in Fig 7b (red). The solution segment (Γ, black) lies to the right of γ₀ (magenta) and between γ₁ (cyan) and γ₂ (orange), so that two small oscillations are produced, as predicted by canard theory. The equilibrium, E₂, is stable for this parameter set (ΔV_1/2(d_∞) = -ΔV_1/2(f_∞) = -2.08 mV), but Γ simply does not enter its basin of attraction. However, E₂ possesses a pair of complex conjugate eigenvalues (λ ± ωi) which, in the vicinity of E₂, predict an oscillatory period (2π/ω) of ≈ 340 ms. The duration of the first and second EADs are ≈ 386 ms and ≈ 340 ms, respectively.

Fig 7c shows a case in which there is repolarization failure since the trajectory enters the basin of attraction of E₂ and remains depolarized. The spiraling reflects the fact that E₂ is a stable spiral equilibrium of the full system.

This analysis suggests that the responses of the model cell to window-enlarging manipulations are determined by how the manipulations affect the maximal canards in phase space. Pathological oscillatory dynamics are brought about by manipulations that translate the maximal canards leftward (in the increasing x-coordinate direction) relative to the solution trajectory, so that the solution trajectory enters the funnel region to the right of the primary maximal canard. Enlargement of the I_Ca-L window region can make this happen, leading to EADs or repolarization failure.

Why left shifts of the I_Ca-L activation curve are more effective than right shifts of the inactivation curve at evoking EADs

We have shown that maximal canards mediate the transition from standard action potentials, through EADs, to repolarization failure in phase and parameter space under symmetric window enlargement. We now examine why left-shifts in the I_Ca-L activation curve are more effective than right shifts in the inactivation curve at producing EADs and repolarization failure. This should be explainable in terms of the primary maximal canard, which is the border (in phase space) of the funnel region for EADs. What effects do equally sized shifts of the activation curve d_∞(V) and inactivation curve f_∞(V) have on the primary maximal canard?

Fig 8a shows a phase-space view with the critical manifold and the primary maximal canard γ₀ (magenta) prior to a shift in the activation/inactivation curves. When the Ca²⁺ channel activation curve is left shifted by 3.6 mV (ΔV_1/2(d_∞) = −3.6 mV) the primary maximal canard moves leftward in phase space, as indicated in the figure. An equal right shift in the inactivation curve (ΔV_1/2(f_∞) = 3.6 mV) also moves γ₀ leftward, but not as far. The figure also includes a portion of the trajectory during the action potential plateau (Γ, black) with and without a shift in either the activation or inactivation curve. It is apparent that the shift in these curves has very little effect on this portion of the trajectory (the three black segments are very close together), however with the shift in the activation curve the trajectory enters the funnel and will exhibit EADs, while with the equal shift of the inactivation curve it will not. Thus, the reason that EADs are facilitated more by left shifts in the activation curve than right shifts in the inactivation curve is that the primary maximal canard is affected more by the former maneuver than the latter.

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Fig 8. Left shifts in the Ca²⁺ channel activation curve move the primary maximal canard further than equal right shifts in the inactivation curve.

(a) Three primary maximal canards corresponding to default (γ₀, right, magenta), right-shifted f_∞(V) (middle, magenta), and left-shifted d_∞(V) (left, magenta) conditions are superimposed on the critical manifold of the default parameter set. Also shown is a portion of the trajectory during the plateau phase of an action potential (Γ, black) for each condition. These three trajectory segments are almost identical, but the one corresponding to left-shifted d_∞(V) enters the funnel and will subsequently exhibit EADs. (b) The distance, δ, between Γ and γ₀ declines faster with left shifts in d_∞(V) than with right shifts in f_∞(V).

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To make these arguments more precise, in Fig 8b we introduce a quantity, δ, that measures the signed distance between a point on the pulsed solution Γ (that also lies on the slow manifold corresponding to ) and the primary maximal canard, γ₀, as a function of the shift magnitude, |ΔV_1/2|, in either d_∞(V) (purple curve) or f_∞(V) (orange curve). Positive values of δ indicate that Γ lies to the left of γ₀ (no EADs), while negative values of δ indicate that Γ lies to the right of γ₀ (EADs or repolarization failure). Zeros of δ indicate that Γ coincides with γ₀ and is the boundary between action potentials with and without EADs; zeros correspond to points on the green boundary curve in Fig 5. The locations of the zeros of δ are unaffected by the point on Γ (that coincides with the slow manifold) from which the measurements are made.

In agreement with Fig 8a (and Fig 5), we find that δ decreases more rapidly for left shifts in d_∞(V) (purple curve) than for right shifts in f_∞(V) (orange curve), corresponding to more rapid leftward movement of γ₀ toward Γ under left-shifts in d_∞(V). As a result, δ crosses zero (near |ΔV_1/2| ≈ 3.45 mV) as |ΔV_1/2| increases toward 3.6 mV for d_∞(V), while δ remains greater than 0 over the same range of |ΔV_1/2| for f_∞(V).

A left shift in the I_Ca-L activation curve narrows the parameter range for EADs by constricting the maximal canards

One peculiar observation from Fig 5 is that the EAD sector (in red) is narrow at the bottom and wider at the top. This means that with a large left-shift in d_∞(V) the range of right-shifts in f_∞(V) that can produce EADs becomes smaller. Why is this? To address this question, we examine the maximal canards in phase space for three values of ΔV_1/2(d_∞) (⋄ markers in Fig 5). The first panel of Fig 9 shows the situation when the left-shift in d_∞(V) is not large enough to evoke EADs. In this case, the trajectory segment lies to the left of γ₀ and thus outside the funnel. In the second panel, with a larger left shift, the trajectory lies between γ₁ (cyan) and γ₂ (orange), so two EADs are produced. In the third panel, the trajectory spirals into the equilibrium E₂ and there is repolarization failure.

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Fig 9. Maximal canards shift leftward and constrict with increasing left shifts in d_∞(V).

(a) At a value of ΔV_1/2(d_∞) (= -3.35 mV) where no EADs are produced (corresponding to ⋄ marker 9a in Fig 5) the trajectory lies outside the funnel region for EADs. (b) With a somewhat greater shift in d_∞(V) (ΔV_1/2(d_∞) = -3.6 mV), corresponding to the ⋄ marker 9b in Fig 5, the trajectory enters the region between γ₁ (cyan) and γ₂ (orange) and two EADs are produced. The maximal canards have shifted leftward and are closer together than in the first panel. (c) With an even greater left-shift in d_∞(V) (ΔV_1/2(d_∞) = -3.85 mV) the trajectory is attracted to equilibrium E₂ and there is repolarization failure. With this greater shift the maximal canards are even more constricted.

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What is important to observe in Fig 9 is that the spacing between the maximal canards gets smaller for large left shifts in d_∞(V). Thus, there is a constriction of the region in phase space where EADs, rather than repolarization failure, are evoked. Constriction of the phase space region where EADs are evoked also occurs with right shifts in f_∞(V), but the rate and severity are less pronounced. This too corroborates a prediction from canard theory. In the singular limit, the ratio of the eigenvalues of the folded node, μ ≔ λ_w/λ_s < 1, can be used to estimate how densely the secondary maximal canards (γ₁, γ₂, etc.) accumulate near the primary maximal canard (γ₀) in the full system flow (see Propositions 3.5 and 3.6 of [38]). We find that μ decreases more rapidly for left shifts in d_∞(V) than for right shifts in f_∞(V), which predicts that the maximal canards will accumulate more densely on the primary maximal canard under left shifts d_∞(V), as we observe. It is for this reason that the EAD region in Fig 5 is narrow at the bottom and wider at the top.

Decreasing the size of the window region can compensate for pathological conditions that promote EADs

While broadening the I_Ca-L window can lead to pathological electrical rhythms, it is also plausible that pathological conditions can be compensated for by narrowing the window. In vitro experiments with isolated cardiomyocytes and cardiac tissue have shown that simulating hypokalemia by reducing the extracellular K⁺ concentration in the bath reliably elicits EADs [8, 11, 42, 43]. In [21], we showed that simulating hypokalemia (by reducing the parameter [K⁺]_o) in the model (1) also elicits EADs, due to a canard mechanism similar to that described above. In [14] it was shown that narrowing the I_Ca-L window in dynamic clamp experiments can overcome the effects of low extracellular K⁺ and eliminate the EADs. Can this also be explained by the model?

To investigate, we reduced the extracellular K⁺ concentration parameter [K⁺]_o over a range of values, which has the effect of increasing the K⁺ Nernst potentials, V_K and V_K1, while decreasing the maximal conductances, g_K and g_K1. We also translated the Ca²⁺ activation curve d_∞(V) over a range of values so as to evaluate the combined effects of these maneuvers. The top panels of Fig 10 show the result. The green marker labelled b1 (Fig 10a) shows that with the default [K⁺]_o(= 5.4 mM) and no shift in d_∞(V) a standard action potential is produced (Fig 10b). In fact, for any shift in d_∞(V) a standard action potential is produced. For lower values of [K⁺]_o (simulating hypokalemia), EADs become possible if d_∞(V) is left shifted. For a sufficiently low value of [K⁺]_o, EADs occur even with no left-shift in d_∞(V). This is the case with [K⁺]_o = 2.0 mM shown with the red marker labelled b2 in Fig 10a. With this parameter combination two EADs are produced, greatly extending the duration of the action potential (Fig 10b). However, if d_∞(V) is then right shifted (ΔV_1/2(d_∞) = 0.75 mV), to the orange point labelled b3 (Fig 10a) the EADs are eliminated, yielding an action potential of almost-normal duration (Fig 10b). Thus, right shifts in d_∞(V) can eliminate the EADs brought about by hypokalemia in model simulations.

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Fig 10. Narrowing the Ca²⁺ current window by shifting d_∞(V) or f_∞(V) eliminates hypokalemia-induced EADs in the model.

(a) Model responses in the (ΔV_1/2(d_∞), [K⁺]_o) parameter plane. The green marker (b1) denotes the default condition, the red marker (b2) denotes the hypokalemia condition, and the orange marker (b3) denotes the d_∞(V)-shifted hypokalemia condition. (b) Voltage time courses for the color-matched markers (b1), (b2), and (b3) of panel (a). (c) Model responses in the (ΔV_1/2(f_∞), [K⁺]_o) parameter plane. The green marker (d1) denotes the default condition, the red marker (d2) denotes the hypokalemia condition, and the orange marker (d3) denotes the f_∞(V)-shifted hypokalemia condition. (d) Voltage time courses for the color-matched markers (d1), (d2), and (d3) of panel (c).

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Fig 10c and 10d show a similar scenario, but in this case left-shifts in f_∞(V) are used to narrow the Ca²⁺ current window. Starting from the default value of [K⁺]_o and with no shift (green point d1), simulated hypokalemia brings the system into the EAD region (red point d2). Applying a left-shift to f_∞(V) of ΔV_1/2(f_∞) = −0.75 mV eliminates the EADs (orange point d3). Thus, both window-narrowing maneuvers produce the desired result of eliminating hypokalemia-induced EADs. Because the EAD region is smaller in Fig 10a than in Fig 10c, it would generally be more successful in the model to eliminate EADs in conditions of hypokalemia with shifts in d_∞(V) than with shifts in f_∞(V), as observed experimentally in [14].

Given the importance of excess I_Ca-L in the production of EADs, it is not surprising that when the Ca²⁺ current conductance was increased during dynamic clamp experiments there was an increase in EAD production and repolarization failure. These effects were eliminated when the I_Ca-L window was symmetrically narrowed [15]. We demonstrate that the model (1) recapitulates both the increase in propensity of repolarization failure with an increase in g_Ca and the rescue of a standard action potential with appropriate symmetric narrowing of the I_Ca-L window.

In Fig 11, the conversion of an action potential (green) to repolarization failure (red) in response to an increase in g_Ca (to 0.18 mS/cm²) is illustrated. By symmetrically narrowing the I_Ca-L window with ΔV_1/2(d_∞) = 1 mV and ΔV_1/2(f_∞) = −1 mV, there is recovery of an action potential response to the stimulus. In a physiological setting, this and the previous result suggest that dynamic regulation of the I_Ca-L window can be very effective at overcoming pathological conditions leading to EADs and repolarization failure.

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Fig 11. Symmetric narrowing of the model window region abolishes I_Ca-L amplitude-induced repolarization failure.

Repolarization failure is promoted by increasing the conductance of the I_Ca-L current (red). Narrowing the window recovers the action potential response (orange). Green: (g_Ca, ΔV_1/2(d_∞), ΔV_1/2(f_∞)) = (0.112, 0, 0); orange: (g_Ca, ΔV_1/2(d_∞), ΔV_1/2(f_∞)) = (0.18, 0, 0); red: (g_Ca, ΔV_1/2(d_∞), ΔV_1/2(f_∞)) = (0.18, 1, -1).

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Changes in Ca²⁺ channel time constants are predicted to eliminate hypokalemia-induced EADs

We have shown that the model reproduces many of the experimental results obtained with dynamic clamp in [14] and [15]. We have also shown that the EADs induced under these manipulations can be explained mathematically as canard-induced oscillations. We now extend our analysis by using the model to make predictions about the anti-arrhythmic effects of altering kinetic properties of the Ca²⁺ current. Specifically, we examine model responses to changes in the time constants of I_Ca-L activation, τ_d(V), and inactivation, τ_f(V), under simulated hypokalemia.

To examine the effects of changing Ca²⁺ current time constants we multiply the voltage-dependent timescale functions by scaling parameters, α and β. Then the activation and inactivation variables change in time according to: (5) Values of a scaling parameter larger than 1 make the corresponding time constant larger and thus slow the rate of adjustment of the corresponding gating variable to the variations in V; values of a scaling parameter less than 1 hasten this adjustment.

The model responses to independent variation in α and β are shown in Fig 12. For reference, the blue ⋄ marker in the two EADs band of the red “EADs” region of Fig 12 denotes the baseline hypokalemia condition ([K⁺]_o = 2.0 mM) in the absence of time constant manipulations. Two dashed blue arrows, one pointing leftward toward decreases in α alone and the other pointing upward toward increases in β alone, show separate manipulations that predict the elimination of hypokalemia-induced EADs. The EAD-eliminating decreases in α correspond to more rapid activation of I_Ca-L in response to a depolarizing stimulus while the EAD-eliminating increases in β correspond to delayed inactivation of I_Ca-L during an actio potential. These results seem counterintuitive, since the first manipulation makes I_Ca-L turn on faster and the second makes it turn off slower in response to a stimulus. Why would manipulations that are expected to prolong the influence of a depolarizing current shorten action potentials and reduce the likelihood of EADs?

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Fig 12. Model responses to variation in scaling parameters of Ca²⁺ channel activation (α) and inactivation (β) timescales under simulated hypokalemia.

The blue ⋄ marker denotes the hypokalemia condition of Fig 10 and the blue dashed arrows highlight two separate dynamic clamp manipulations predicted to eliminate hypokalemia-induced EADs.

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The answer again lies in the fast-slow analysis and, in particular, the location of the primary maximal canard γ₀ with respect to the location of the pulsed solution Γ in phase space. As we discussed earlier, and showed in detail in [21], the primary maximal canard moves far to the left of the singular strong canard as parameters are changed that move the system away from the singular limit. When the time constant for d is decreased or that for f is increased, this has the effect of further separating the timescales of fast and slow variables. That is, it moves the system closer to the singular limit. As a result, γ₀ moves rightward towards , and in the process crosses Γ, so that Γ now falls outside of the funnel region so no EADs are produced.