Vulnerability of cardiac dynamics

Figure 1: Each propagating excitation wave is trailed by a region of recovering cells. Within the recovery wave is a vulnerable region, within which excitation produces a front that propagates only in the retrograde direction. The extend of the vulnerable region depends on both the test stimulus amplitude as well as medium properties. For a two dimensional medium, the test wave can evolve into counter-rotating spirals

Vulnerability is an asymmetric response of an excitable medium to a stimulus, when the stimulus successfully initiates propagation some directions and fails in other directions. The asymmetric response, a discontinuous wave, is the result of propagation into regions of spatially non-uniform excitability. Propagation of discontinuous waves can evolve to spiral waves as shown in Figure 1 and result in life threatening alterations in cardiac rhythm.

## Contents

* 1 Introduction
* 2 Exploring Excitation and the Refractory period of a Fitzhugh-Nagumo cell
* 3 Stationary Vulnerability
* 4 Dynamic Vulnerability: A traveling vulnerable region trails each activation front
* 5 The Cardiac Vulnerable Period
* 6 Vulnerability in a 2D medium:
* 7 Discussion
* 8 References
* 9 See Also

## [edit] Introduction

Coordinated activation of cardiac cells is essential for effective pumping. Altering the excitation sequence of the heart often produces life-threatening arrhythmias such as [ventricular fibrillation] and its mechanical correlate, loss of effective pumping. Critically timed (premature) excitation can alter the cardiac excitation sequence. When the site of premature excitation falls within a region of vulnerability, the resulting discontinuous excitation wave can lead to potentially fatal reentrant arrhythmias.

Propagation is an intrinsic property of an excitable medium. Exciting a region larger than a threshold region (see Initiation of excitation waves) in a uniformly excitable medium initiates a wave propagating in all directions (a continuous wave). A region is considered vulnerable when excitation produces a front that propagates in some directions and fails to propagate in other directions (a discontinuous wave). Excitability gradients trail every propagating front and represent one mechanism for establishing a vulnerable region. In cardiac tissue, membrane repolarization trailing a propagating wave produces a spatial gradient of excitability due to time-dependent restoration of sodium channel availability (non-inactivated sodium channels). Vulnerability is also observed in other excitable systems such as the [Belousov-Zhabotinsky] chemical reagent (Gomez-Gesteira, M. et. al, 1994). Vulnerable regions also appear when excitability gradients are generated from external excitation such as defibrillation. Such extrinsic vulnerability is critical for developing defibrillation waveforms that minimize creating vulnerable regions. For insights into induced or extrinsic vulnerability see Efimov, Cheng, Van Wagoner, Mazgalev and Tchou, 1998; Qu, Li, Nikolski, Sharma and Efimov, 2005.

Although the heart is a structurally complex organ, experimental studies indicate that the intrinsic vulnerability observed in a homogeneous medium of identical cells is qualitatively similar to that seen in cardiac muscle (see Chen et al, 1988; Starmer et al 1992 and Nesterenko et al, 1992). To fix ideas, here we demonstrate vulnerability with a simple model of an excitable cell comprised of single excitatory and single inhibitory current.

## [edit] Exploring Excitation and the Refractory period of a Fitzhugh-Nagumo cell

Figure 2: Action potential

Figure 3: Altered excitability associated with the time course of an action potential is readily demonstrated by using paired excitation. A conditioning stimulus, s1, initiates an action potential. After a delay, the threshold amplitude of a test stimulus, s2, is determined. As the s1-s2 delay is increased, the threshold s2 amplitude decreases, reflecting greater excitability.

We use the FitzHugh-Nagumo model, a two current description of an excitable media to demonstrate temporal alterations in excitability associated with any action potential, an essential requirement for vulnerability. As shown in Figure 2 a suprathreshold disturbance triggers an action potential (red, Figure 2). Trailing the increase in membrane potential is an increase in the inhibitory current (blue) which reduces the excitability of the cell. As the membrane potential returns to the rest potential, the slower inhibitory current decays to its rest state. With a propagating action potential, the variation in inhibitory current produces a spatial gradient of excitability.

Altered excitability in cardiac and nerve tissue is evaluated using paired stimulation of a cell. A suprathreshold stimulus, s1, is used to trigger an action potential in a resting medium. As the membrane potential depolarizes, the inhibitory current increases (blue trace in Figure 2) which reduces medium excitability. The amplitude of a test stimulus, s2, characterizes the altered excitability as shown in Figure 3.

## [edit] Stationary Vulnerability

Figure 4: In a spatially uniform medium, we identified the threshold for sustained propagation by varying the stimulation amplitude and duration. The length of the excited region was 2.5 arbitrary units.

Vulnerability is typically demonstrated in the setting of a propagating wave. The essential characteristic of a vulnerable medium is spatial non-uniform excitability. To fix ideas, here we create a static gradient of excitability and demonstrate symmetric responses to excitability when the gradient is zero and an asymmetric response when the gradient is non-zero. Central to understanding propagation and vulnerability is the threshold relationship between an excited region and either propagation or collapse of a front. As the stimulus amplitude is increased, the spatial region exposed to suprathreshold excitation increases. Figure 4 illustrates the relationship between stimulus amplitude and duration in a uniformly excitable medium that separates successful propagation from decaying propagation. Excitation at values above the curve initiate symmetric responses while excitation using values below the curve collapse, (see Initiation of excitation waves).

To explore responses to excitation in both uniform and non-uniform excitability, we alter the slow current, \\(W\ ,\\) to include a linear gradient\\[ \frac{\partial W}{\partial t} = \epsilon (\gamma U + \beta - W + \alpha (x - x_{stim}))\\] where \\(x_{stim}\\) is the location of the stimulation site. For the case of uniform excitability, \\(\alpha\\) = 0 ( Figure 5 and Figure 5). To illustrate vulnerability (asymmetric front formation), a small gradient is added to the slow current ( Figure 7) with \\(\alpha\\) = 0.005. After excitation of both a uniformly excitable medium ( Figure 5 and Figure 5) and a medium exhibiting an excitability gradient ( Figure 7), the time-dependent development of the excited region was captured as shown below. The stimulus site was located at \\(x = 40\\) and the stimulus duration was 1.5 time units. The stimulus amplitude was either slightly below the strength-duration curve (0.691) and slightly above the strength-duration curve (0.692).

Figure 5 illustrates the fate of subthreshold excitation in a uniformly excitable medium. The stimulus amplitude is slightly below (0.691) the strength-duration curve. After the initial disturbance, the membrane potential expands to approximate that of a stationary wave (critical nucleus), hesitates and then collapses. Figure 5 illustrates the fate of a suprathreshold excitation slightly above (0.692) the strength duration curve. Following the initial disturbance, the pulse rapidly expands and approximates that of a stationary wave, hesitates and then resumes expansion. As the wave expands it splits into two fronts that propagate away (symmetric response) from the site of excitation. Figure 7 illustrates the fate of a suprathreshold excited region in a medium exhibiting a gradient excitability. The gradient was defined such that the excitability at the stimulus site was equal to that of the uniformly excitable medium as shown in Figure 5 and Figure 5. Using the same amplitude stimulus as in Figure 5, 0.692, the initial disturbance expands, approximates that of a stationary wave and after some hesitation, the excited region continues to expand, eventually splitting into two oppositely directed fronts but asymmetric fronts. Expansion in the direction of increasing excitability (to the left) continues while the front expansion in the less excitable direction displays decremental propagation and collapses as the front is no longer able to excite adjacent medium.

Figure 5: Propagation failure in a medium with uniform excitability. The stimulus current is 0.691, below the stimulus-duration curve ( Figure 4. The blue trace represents the inhibitory current and the red trace is the membrane potential. Following disturbing the region with a stimulus slightly below the strength-duration curve, the disturbed region initially expands and approaches the profile of a stationary wave and then decrementally propagates away from the stimulation site.

Figure 6: Propagation success in a medium with uniform excitability. The stimulus current is 0.692. The blue trace represents the inhibitory current and the red trace is the membrane potential. Disturbing the region with a stimulus slightly above the strength-duration curve produces a wave that slowly develops, approximating the profile of a stationary wave, hesitates, then resumes growing as it propagates bidirectionally away from the stimulation site.

Figure 7: Propagation in a medium with a linear gradient of excitability. Following initial expansion that approximates the profile of a stationary wave, pulse expansion hesitates and then continues to develop. Eventually the pulse splits into two fronts. The left front incrementally increases in amplitude as it propagates into the more excitable direction while the right front, propagating into a linearly less excitable region, decrementally propagates and eventually collapses.

## [edit] Dynamic Vulnerability: A traveling vulnerable region trails each activation front

A propagating action potential is trailed by a gradient of excitability, reflecting the dynamics of the inhibitory current, W. Within the wake of each propagating front is a vulnerable region within which excitation results in an asymmetric response. As shown above, unidirectional propagation arises when a properly configured stimulus is placed within a stationary vulnerable region. Here we explore a traveling vulnerable region and determine the boundaries of the vulnerable region by first initiating a conditioning wave, S1, and then after a suitable delay, use a test stimulus, S2 to test for excitability and propagation. We determined the boundaries of the vulnerable region by timing a test stimulus, S2, to occur before, within or after passage of the vulnerable region over the stimulus site. Figure 8, Figure 9, Figure 10 and Figure 11 illustrate the transition from a symmetric response to the S2 stimulus, no propagation for short S1-S2 delays, to asymmetric responses, unidirectional propagation for S1-S2 delays within the vulnerable region, to symmetric responses, sustained bidirectional propagation, for S1-S2 delays beyond the vulnerable region.

In this example, waves initiated by test stimuli timed to occur between 105 and 125 time units produce asymmetric responses where the antegrade directed front propagates decrementally while the retrograde directed front enjoys sustained propagation as it moves into progressively higher more excitable medium. In these numerical experiments, the amplitude of the test stimuli is constant so that variations in the extent of the vulnerable region reflects only variations in the excitability at the test site. The red curve is the potential, U, and the blue curve is the inhibitory current, W. Following the front, excitability is decreased as W increases. As W continues to increase, U eventually reaches a peak and starts to decrease. This results in a slow reduction in inhibitory current which increases excitability until both U and W return to their equilibrium values. The response of W to changes in U creates a non-uniform distribution of excitability

Figure 8: A symmetric response to excitation: decremental propagation of the antegrade and retrograde waves following excitation before the most premature boundary of the vulnerable period: T = 104.

Suprathreshold excitation of the left end of the cable produces a propagating action potential directed from left to right. Following the initial excitation, the repolarizing current (W, blue) increases, rendering the region inexcitable. As the repolarizing current continues to increase, a point is reached when the net transmembrane current is zero after which, the membrane potential decreases and is followed by a reduction in the repolarizing current. The wave back follows the wave front and can be viewed as an excitability wave. When a second, S2, stimulus is initiated the fate of the disturbance depends on the relationship between the S2, the threshold for establishing a propagating front. the S2 wave velocity the excitability at the S2 site and the gradient of excitability at the S2 site. In this example, the S2 timing and amplitude produce a disturbance below the threshold for sustainable propagation. Consequently the front decrementally propagates in both retrograde and antegrade directions.

Figure 9: Rapid decay of the antegrade wave following S2 excitation early within the vulnerable period: T = 105.

Delaying the S2 stimulus by 1 time unit allows additional recovery of excitability (reduction in W) at the S2 site. Stimulation with the same amplitude as in Figure 8 produces an excited region that exceeds the threshold for sustainable propagation for medium in the retrograde direction (left of the stimulus site) but fails to exceed the propagation threshold for the less excitable medium to the right of the stimulus site. The rapid reduction in the antegrade wave amplitude reflects the progressive mismatch between front charge available to diffuse into adjacent medium and the decrease in excitability of adjacent medium excitability. The result is sustained retrograde propagation and decremental antegrade propagation.

Figure 10: Slow decay of the antegrade wave following excitation late within the vulnerable period: T = 125

At T = 125 arbitrary units, the S2 initiated antegrade threshold for sustainable propagation is smaller than that associated with shorter S1-S2 delays as W decreases. Consequently, following S2 excitation, the stimulus field initially exceeds the propagation requirements in both the antegrade and retrograde directions resulting in bidirectional propagation. However, as the antegrade front propagates into progressively more refractory medium, propagation is decremental (front amplitude decreases with propagation) until the antegrade front fails to propagate. Because the gradient of excitability at S2 and T = 125 is smaller than at T = 105, the rate of propagation collapse is smaller. The result is unidirectional propagation.

Figure 11: Bidirectional propagation after the S1-S2 delay extends beyond the least premature boundary of the vulnerable period: T = 126

Waiting an additional time unit enables additional recovery of antegrade excitability such that antegrade propagation is sustained. In this case the stimulus field in both the retrograde and antegrade directions exceeds the initial the propagation threshold requirements. As the antegrade wave propagates, the medium invaded has had sufficient time to recover such that propagation is sustained. Note however that even though the antegrade wave initially propagates decrementally, antegrade recovery is rapid compared with the antegrade wave velocity such that propagation switches from decremental to incremental propagation. The result is that both S2 wave propagation is sustained.

## [edit] The Cardiac Vulnerable Period

Figure 12: Shown is a cardiac action potential traveling from left to right with velocity, \\(v\ .\\) Following the conditioning wave is a gradient of excitability created as sodium channel inactivation gates slowly open. The (arbitrarily placed) transition point, P, marks the least premature boundary of the vulnerable period. The vulnerable period, VP, is the time required for P to pass across the vulnerable region, VR: VP = VR/v

Figure 13: The cardiac vulnerable period can be altered with drugs or sodium channel mutations. The Luo-Rudy ventricular model was used to determine the boundaries of the vulnerable period. The population of sodium channels, \\(G_{max Na}\\) and channel kinetics were altered as occurs with drugs or mutant sodium channels. As the sodium channel density was reduced, the propagation velocity decreased and the vulnerable period increased. The lower curve represents normal Na channel kinetics while the upper curve displays the variation of the vulnerable period associated with slowed Na channel kinetics

In cardiac tissue, excitability is primarily a reflection of the availability of sodium channels (or in specialized tissue, calcium channels) to open. The sodium channel exhibits two processes, activation and inactivation. Following exciting a cell, the activation gates rapidly open, depolarizing the cell, while the inactivation gates slowly close such that sodium ions flow into the cell for approximately 1 msec. As the cell repolarizes, the inactivation gates open while the activation gates close. With a propagating excitation wave, the relatively slow transition rates of the inactivation process produces a gradient of excitability trailing the front within which exists a vulnerable region. Of clinical interest is the vulnerable period, the time that a specific region is vulnerable, the result of the vulnerable region passing over a stimulus site. Figure 12 illustrates the intrinsic vulnerable region and associated vulnerable period (see Starmer, C.F., Colatsky, T.J. and Grant, A.O., 2003a). The spatial extent of the vulnerable region, VR = L + L*, depends on the spatial extent of the test stimulus field, L, and an intrinsic component, L*, reflecting the distance a front travels before it collapses and determined by the gradient of sodium channel availability. The vulnerable period is VP = VR/v where v is the velocity of the excitation wave.

The cardiac vulnerable period can be evaluated with paired stimulation where a conditioning stimulus, s1, initiates a propagating front and a delayed stimulus, s2, is used to probe excitability at the s2 site. The vulnerable period is bounded the shortest s1-s2 delay that produces a discontinuous wave and the longest s1-s2 delay that produces a discontinuous wave.

The dependence of the vulnerable period on excitation wave velocity is readily demonstrated in models of cardiac tissue where the excitation wave velocity is altered by varying the maximum Na conductance. Figure 13 depicts the relationship between the vulnerable period and maximum Na channel availability for normal and retarded transition rates for channel inactivation. The symbols represent observed values of the VP using a model of a cardiac ventricular cell. The lines represent the least squares fit of the \\(VP = \frac{\alpha L}{v} - \frac{\beta}{\nabla G_{Na}h v}\\) where v is the conditioning wave velocity, h is the Na channel availability, \\(G_{Na}\\) is the sodium channel density. Shown are two conditions, normal inactivation gate transition rates (green) and retarded inactivation gate transition rates caused by a mutation of the Na channel (blue). Such mutations are frequently associated with patients with a high risk of sudden cardiac death (Long QT syndrome).

## [edit] Vulnerability in a 2D medium:

As with the 1D case described above, there is a vulnerable region that trails each excitation wave in 2D and 3D media. Allessie, Bonk and Shopman (1973) demonstrated vulnerability as a mechanism for initiating reentry in isolated rabbit atrium. As shown in Figure 1, exciting a region near its threshold within the vulnerable region produces a wave fragment that propagates in the more excitable retrograde direction (away from the conditioning front) and decrementally propagates in the less excitable antegrade direction (toward the conditioning front).

## [edit] Discussion

Potentially fatal cardiac arrhythmias (externally induced see Wiggers and Wegria, 1939; atrial reentry see Allessie et. al. 1974; drug associated see Whitcomb et. al, 1989) are often associated with premature excitation within a vulnerable region of the heart. The mechanistic basis of vulnerability is poorly understood due to the complex anisotropic structure of cell types and connectivity. However the essential features observed in cardiac media can be observed with simple models such as the FitzHugh-Nagumo model.

A vulnerable region is identified by its asymmetric response to excitation. Within a vulnerable region, a front established by excitation will display asymmetric propagation, propagating in some directions and failing to propagate in other directions. In cardiac media, a propagating wave is trailed by a gradient of excitability that reflects transient changes in cellular excitability, typically associated with inactivation of sodium channels. Though the sodium channel had yet to be discovered, Wiener and Rosenblueth (1946) predicted vulnerability using a finite state model of a cardiac cell.

In cardiac media, vulnerability is not limited to that of sinus initiated excitation waves but is associated with any propagating excitation wave. For example, Glass and Josephson (1995) and Nomura and Glass (1996) explored vulnerability as a tool for terminating a reentrant wave in a ring. In these studies, equivalent to reentry initiated around an obstacle, stimuli timed to occur within the vulnerable region propagated in the retrotegrade direction and collided with the initial circulating conditioning wave, thus terminating the reentrant wave. Stimuli timed to occur after passage of the vulnerable region over the stimulus site propagated in both the antegrade as well as retrograde directions. The retrograde wave annihilated the circulating reentrant wave while the antegrade wave continued to circulate, thus effectively changing the relative phase of the circulating wave.

Identifying the underlying dependence of the vulnerable region on media excitability provides new insights into the proarrhythmic potential of supposedly antiarrhythmic drugs. For example, use-dependent drugs that bind to cardiac sodium channels reduce the density of sodium channels and retard the recovery of excitability (Starmer, C.F., Lastra, A.A., Nesterenko, V.V. and Grant, A.O. 1991; Starmer, C.F., Lancaster, A.R., Lastra, A.A. and Grant. A.O., 1992)resulting in slowed conduction. Similarly sodium channel mutations that alter cardiac excitability have an enhanced proarrhythmic potential (Grant, A.O., et. al, 2002; Starmer, C.F., Colatsky, T.J. and Grant, A.O., 2003a). Clinical trials of antiarrhythmic drugs have uniformly revealed that use-dependent drugs that reduce excitability by binding to cardiac sodium channels display proarrhythmic behavior, i.e. there is an increase in the rate of sudden cardiac death associated with the use of drugs classified as antiarrhythmic (see Cardiac Arrhythmia Suppression Trial (CAST) Investigators, 1989; The Cardiac Arrhythmia Suppression Trial-II Investigators, 1992). The mechanism of failure of these clinical trials is consistent with the concept of the vulnerable region described above. Specifically any drug that reduces the excitability will also slow propagation of a wave thereby enlarging the vulnerable region and the probability that spontaneous oscillation of cells within the vulnerable region will trigger a reentrant arrhythmia (Wiener, N. and Rosenblueth, A, 1946; Starmer et. al, 1991, Starmer et. al, 1993, Starobin, J., Zilberter, Y.I. and Starmer, C.F., 1994). Vulnerability is a generic property of any excitable medium. These studies demonstrate how simple models of an excitable medium can be used develop insights into the genesis of cardiac arrhythmias in a more complex setting, in this case demonstrating that use-dependent sodium channel blocker's antiarrhythmic potential (reducing excitability) cannot be divorced from its proarrhythmic potential (slowing the passage of an intrinsic vulnerable region over a site of spontaneous oscillation).

## [edit] References

* Wiggers, C.F. and Wegria, R. Ventricular fibrillation due to single localized induction in condenser shock supplied during the vulnerable phase of ventricular systole. Am. J. Physiol 1939, 128:500-505.

* Wiener, N. and Rosenblueth, A. The mathematical formulation of the problem of conduction of impulses in a network of connected excitable elements, specifically in cardiac muscle. Archives of the Institute of Cardiology, Mexico, 1946; 16:205-265.

* Hodgkin, A.L. and Huxley, A.F. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. (London) 1952; 117:500-544.

* Allessie, M.A., Bonke, F.I.M., and Schopman, F.J.G: Circus movement in rabbit atrial muscle as a mechanism of tachycardia. Circ. Res. 1973, 33:54-62.

* Rinzel, J. Excitation dynamics: insights from simplified membrane models. Fed Proc. 1985; 44:2944-6.

* Chen, P.S., Wolf, P.D., Dixon, E.G., Danieley, N.D., Frazier, D.W., Smith, W.M. and Ideker, R.E. Mechanism of ventricular vulnerability to single premature stimuli in open-chest dogs. Circ. Res. 1988; 62:1191-1209.

* Cardiac Arrhythmia Suppression Trial (CAST) Investigators. Preliminary report: effect of encainide and flecainide on mortality in a randomized trial of arrhythmia suppression after myocardial infarction. N Engl J Med. 1989; 321:406-412.

* Whitcomb, D.C., Gilliam, F.R., Starmer, C.F. and Grant, A.O. Marked QRS complex abnormalities and sodium channel blockade by propoxyphene reversed with lidocaine. Journal of Clinical Investigation 1989; 84:1629-1636.

* Starmer, C.F., Lastra, A.A., Nesterenko, V.V. and Grant, A.O. A proarrhythmic response to sodium channel blockade: Theoretical model and numerical experiments. Circulation 1991; 84:1364-1377.

* Starmer, C.F., Lancaster, A.R., Lastra, A.A. and Grant. A.O. Cardiac instability amplified by use-dependent Na channel blockade. American Journal of Physiology 1992; 262:H1305-H1310.

* Nesterenko, V.V., Lastra, A.A., Rosenshtraukh, L.V. and Starmer, C.F. A proarrhythmic response to sodium channel blockade: The influence of antiarrhythmic drugs on the window of vulnerability in guinea-pig myocardium. J. Cardio Pharm 1992; 19:810-820.

* The Cardiac Arrhythmia Suppression Trial-II Investigators. Effect of antiarrhythmic agent moricizine on survival after myocardial infarction: the Cardiac Arrhythmia Suppression Trial-II. N Engl J Med. 1992; 327:227-233.

* Starmer, C.F., Biktashev, V.N., Romashko, D.N., Stepanov, M.R., Makarova, O.N. and Krinsky, V.I. Vulnerability in homogeneous excitable media: Analytical and numerical studies of unidirectional propagation, Biophysical Journal, 1993; 65:1775-1787.

* Gomez-Gesteira, M., Fernandez-Garcia, G., Munuzuri, A.P., Perez-Munuzuri, V., Krinsky, V.I., Starmer, C.F. and Perez-Villar, V. Vulnerability in an excitable Belousov-Zhabotinsky medium: from 1D to 2D. Physica D 1994; 76:359-368.

* Starobin, J., Zilberter, Y.I. and Starmer, C.F. Vulnerability in one-dimensional excitable media. Physica D. 1994; 70:321-341.

* Glass, L., and Josephson, M.E. Resetting and annihilation of reentrant abnormally rapid heartbeat. Phys Rev Letters 1995; 75:2059-2063.

* Starmer, C.F.,Romashko, D.,Reddy, R.S., Zilberter, Y.I., Starobin, J., Grant, A.O. and Krinsky, V.I. Proarrhythmic Response to Potassium Channel Blockade: Numerical Studies of Polymorphic Tachyarrhythmias. Circulation 1995; 92:595-605.

* Nomura T, Glass L. Entrainment and termination of reentrant wave propagation in a periodically stimulated ring of excitable media. Phys Rev. E. 1996; 53:6353-6360.

* Efimov, I.R., Cheng, Y., Van Wagoner, D.R., Mazgalev, T. and Tchou, P.J. Virtual electrode-induced phase singularity: A basic mechanism of defibrillation failure. Circ. Res. 1998; 82:918-925.

* Grant, A.O., Carboni, M.P., Neplioueva, V., Starmer, C.F., Memmi, M., Napolitano, C. and Priori, S. Long QT, Brugada and conduction system disease linked to a single Na channel mutation. J. Clin Invest. 2002; 110:1201-1209.

* Starmer, C.F., Colatsky, T.J. and Grant, A.O. What happens when cardiac Na channels lose their function? 1: Numerical studies of the vulnerable period in tissue expressing mutant channels. Cardiovascular Research 2003a; 57:82-91.

* Starmer, C.F., Grant, A.O. and Colatsky, T.J. What happens when cardiac Na channel function is compromised? 2: Numerical studies of the vulnerable period in tissue altered by drugs. Cardiovascular Research 2003b; 57:1062-1071.

* Qu F, Li L, Nikolski VP, Sharma V, Efimov IR. Mechanisms of superiority of ascending ramp waveforms: new insights into mechanisms of shock-induced vulnerability and defibrillation. Am J Physiol Physiol. 2005; 289:H569-77.

Internal references

* Vadim N. Biktashev (2007) Drift of spiral waves. Scholarpedia, 2(4):1836.
* James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
* Eugene M. Izhikevich (2007) Equilibrium. Scholarpedia, 2(10):2014.
* Eugene M. Izhikevich and Richard FitzHugh (2006) FitzHugh-Nagumo model. Scholarpedia, 1(9):1349.
* James M. Bower and David Beeman (2007) GENESIS. Scholarpedia, 2(3):1383.
* C. Frank Starmer (2007) Initiation of excitation waves. Scholarpedia, 2(2):1848.
* Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
* Gregoire Nicolis and Anne De Wit (2007) Reaction-diffusion systems. Scholarpedia, 2(9):1475.

## [edit] See Also

Alternans, Barkley Model, Cardiac Arrhythmia, Drift of Spiral Waves, FitzHugh-Nagumo Model, Initiation of Excitation Waves, Noble Model, Models of Cardiac Cell, Models of Heart, Reaction-Diffusion Systems, Restitution, Scroll Wave, Scroll Wave Turbulence, Spiral Breakup, Spiral Waves, Symmetry Breaking in Reaction-Diffusion Systems, Traveling Wave Vulnerability of Cardiac Dynamics