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The authors have declared that no competing interests exist.

Conceived and designed the experiments: NV IVK AVP. Performed the experiments: NV. Analyzed the data: NV IVK RP AVP. Contributed reagents/materials/analysis tools: NV IVK AN LDW AVP. Wrote the paper: NV LDW RP AVP.

Sudden cardiac death is often caused by cardiac arrhythmias. Recently, special attention has been given to a certain arrhythmogenic condition, the long-QT syndrome, which occurs as a result of genetic mutations or drug toxicity. The underlying mechanisms of arrhythmias, caused by the long-QT syndrome, are not fully understood. However, arrhythmias are often connected to special excitations of cardiac cells, called early afterdepolarizations (EADs), which are depolarizations during the repolarizing phase of the action potential. So far, EADs have been studied mainly in isolated cardiac cells. However, the question on how EADs at the single-cell level can result in fibrillation at the tissue level, especially in human cell models, has not been widely studied yet. In this paper, we study wave patterns that result from single-cell EAD dynamics in a mathematical model for human ventricular cardiac tissue. We induce EADs by modeling experimental conditions which have been shown to evoke EADs at a single-cell level: by an increase of L-type Ca currents and a decrease of the delayed rectifier potassium currents. We show that, at the tissue level and depending on these parameters, three types of abnormal wave patterns emerge. We classify them into two types of spiral fibrillation and one type of oscillatory dynamics. Moreover, we find that the emergent wave patterns can be driven by calcium or sodium currents and we find phase waves in the oscillatory excitation regime. From our simulations we predict that arrhythmias caused by EADs can occur during normal wave propagation and do not require tissue heterogeneities. Experimental verification of our results is possible for experiments at the cell-culture level, where EADs can be induced by an increase of the L-type calcium conductance and by the application of I

The mechanical pumping of the heart is initiated by electrical waves of excitation. The abnormal propagation of such waves may result in cardiac arrhythmias which disrupt the normal pattern of cardiac contraction and can, therefore cause cardiac arrest and sudden cardiac death

One of the challenges in the study of EADs is that they occur normally at the level of a single-cell, as a result of mutations or changes of properties of the individual ionic channels. Therefore, the first question is to find the conditions that are responsible for the onset of EADs. An action potential is generated by many different interacting ionic channels, hence, gradual controlled changes of the properties of an individual channel and studies of their effects on the action potential is a non-trivial problem for experimental research. The second important question is to find mechanisms for the progression of EADs to cardiac arrhythmias. Although EADs occur at a single-cell level, cardiac arrhythmias occur because of abnormal wave propagation at the tissue or whole-heart level. The relation of such abnormal propagation to single-cell behaviors is a complex problem. The solution of such a challenging problem requires that we complement experimental studies by alternative methods. One such method is multi-scale mathematical modeling, which is now widely used in studies of cardiac arrhythmias

The first question, namely, how changes in ionic currents may result in EADs, was addressed in earlier modeling studies. The first studies were performed in the group of Rudy

The second question, namely, how EADs can progress to cardiac arrhythmias, has been studied much less than the first one. Reference

Our study yields several new and interesting results, which we summarize here before we present the details of our work. We first find regions of existence of EADs in a single-cell in a modified TNNP-TP06 model and obtain three two-parameter portraits of their existence by varying

In this paper, we use the recent TNNP-TP06 model for human ventrical cells

The L-type Calcium current is given by

Here

Parameter | Value | Current |

G |
2 |
I |

G |
0.3923027 |
I |

G |
0.1532432 |
I |

k |
1000 mV/ms | I |

The slow delayed rectifier current

Next, we have the rapid, delayed rectifier current

Another important current in our simulations is the

Here,

For completeness, in

At the tissue level we used a standard mono-domain description for isotropic cardiac tissue:

For a more detailed description of the model, we refer to

In order to obtain an EAD, during the plateau phase of the action potential, the inward currents should exceed the outward currents. Therefore, we alter the conductances of the different ion channels, such as those associated with the currents

All our two-dimensional (2D) simulations have been carried out in an isotropic domain with

A wave is initiated at the left of the tissue by stimulating a region of 6

We induce a spiral wave by using the standard S1–S2 stimulation protocol. We first stimulated the left side of the domain (a), as in protocol P1, to induce a plane wave that propagates from the left side of the domain to its right boundary (b). Once this wave has passed over the first half of the domain, we applied a second stimulus in the first quarter of the domain (c), which induces the spiral (d)–(e).

We present the results of our extensive numerical studies here. We begin with single-cell studies and then discuss the results of our 2D simulations.

As EADs occur because of an increase in inward currents and/or a decrease in the outward current, we have first performed studies in which we progressively increased

The channel conductivity of the L-type Ca is enhanced, while the slow delayed rectifier channel conductivity is reduced as indicated in the figures.

The numbers on the

The parameters of the red curves are

In

1 For other parameter values, we obtain qualitatively similar results.

To address the important issue of the manifestations of the aforementioned single-cell patterns in 2D tissue, we have performed extensive, 2D simulations by using initial-point stimulation (protocol P1,

Our study leads to a classification of excitation patterns into the following three different types, see

This figure shows a phase diagrams, in a two-dimensional parameter space, of the different types of excitation patterns, obtained by using the stimulation protocols (a) P1 (56 simulations) and (b) P2 (134 simulations). The yellow, blue, and red colors indicate, respectively, no EAD, EAD, and oscillatory AP single-cell behaviors, as in

Protocol 1 is used with parameters

Protocal 2 is used with parameters

We turn now to a characterization of the spatial patterns in our simulations.

In general, spiral fibrillation of type a consists of multiple interacting spiral waves. A spiral wave always occurs because of the formation of wave-breaks. The initiation of the first spiral, in protocol P2, is straightforward, as we create it by the S2 stimulus. Interestingly enough, we are also able to obtain initial breaks, after the point-stimulation protocol P1. We have found multiple ways by which this breaks occur. We show a few examples, by gray-scale plots of

The left column- spatial pattern of voltage along the red line of 2D excitation pattern shown on the right. The parameter values are

All notations the same as in

In many cases, EAD waves do not result in the formation of an initial break. This explains why we do not have spiral fibrillation for all the parameter values in

Although the first break can occur at different locations, the final state for all patterns, which we have obtained via protocol P1, is qualitatively similar. In

For the protocol P2, we get similar patterns. Moreover, when we compare patterns for the same parameter values for the protocols P1 and P2, we find that the final state does not depend sensitively on the initial conditions (of course, for those conditions for which P1 leads to spiral fibrillation patterns of type a). We do see some differences in the patterns, when we use different parameter values. The closer our system is to the blue region in

Finally, notice that these types of patterns corresponds roughly to the single-cell behavior in which the AP shows an EAD. This spiral pattern of type a is the only type of spiral pattern that we have been able to obtain with protocol P1. Protocol P2 allows us to obtain a second type of spiral fibrillation, which we discuss in the next subsection.

We illustrate spiral fibrillation of type b by the representative gray-scale plots of

All notations the same as in

We show another type of pattern in

Protocol P1 is used with the following parameters:

The patterns in this oscillatory regime have various unusual manifestations, e.g., in addition to point sources, lines of point sources can also emerge. In

(a) A line source with the parameter values

In summary, then, we have found the following three different types of spatial patterns: (a) spiral fibrillation of type a, (b) spiral fibrillation of type b, and (c) oscillatory fibrillation. Other wave patterns, shown in

The initial conditions P1 represents one of the standard conditions that can be used in numerical simulations; these conditions are easily reproducible; and they can be viewed as a limiting case of low-frequency stimulation of cardiac tissue and thus can be related to arrhythmias that occur at low heart rates. However, it is also interesting to see if similar phenomena can occur at other frequencies, as arrhythmias can occur at various heart rates. We have, therefore, used the stimulation protocol P1 again; but during the simulation, we have applied P1 with a certain rate ranging from 300 ms to 1000 ms.

We used protocol P1 for the parameters:

In normal conditions, excitation waves in cardiac tissue are driven by Na currents. However, in some situations, waves driven by Calcium can also occur

If the gates are fully opened, the value on the picture is equal to 1 (white), if the gates are closes, the value is 0 (black). Ca-waves dominate in spiral fibrillation type a (Protocol: P1, parameters:

We now contrast several characteristics of the three main types of patterns we have mentioned above. In particular, we compare the shapes of the APs in the middle of our simulation domains, the electrocardiogram (ECG) generated by a pattern, and finally the temporal Fourier Transform of

We show plots for the spiral fibrillation of type a (protocol P1 and the parameters

For a spiral pattern of type a, a typical AP recording has

The AP shape for the SF

For the oscillatory pattern (Osc), each cell exhibits a stable, low-amplitude oscillation (

It is interesting to characterize the pattern by the average of the temporal Fourier transform of the voltage

We show in

These power spectral densities are averaged over

Finally, we plot the ECGs for each type of the excitation pattern. For SF

We now illustrate another feature of our patterns. This is related to wave propagation and the distinction between real and phase waves. This requires some explanation as it has not been used widely in studies of wave propagation in cardiac tissue. By real waves we mean conventional electrical-activation waves, which arise because of an interplay of the excitability of the medium and the diffusion. Such waves travel with a velocity that is proportional to the square root of the diffusion constant. They are absorbed at impermeable boundaries and do not go through regions in which the medium is in a refractory state

To do this we use barriers, which are organized into a grid that is a square lattice. It has been shown that, if such a mesh is stimulated by an external current that makes it a refractory barrier, it removes wave patterns that are generated by real waves

Our simulation is divided (see text) into blocks of

In this paper we have presented a comprehensive numerical study of 2D wave patterns, which are generated by cells that produce EAD responses. In our study we have used a state-of-the-art mathematical model for human ventricular cells

It is well established that changes, similar to those used in our modeling studies, also promote the onset of EADs in experimental and clinical studies. An increase of I

We have shown that the mechanism of EAD generation, in our model, is the reactivation of I

At the cellular level we have obtained phase diagrams in two-dimensional parameter spaces, that show the parameter ranges in which EAD activity occurs (we use parameters such as I

We have then generated 2D patterns, which occur in our simulation domain with cells that show EADs and obtained therefrom phase diagrams, in two-dimensional parameter spaces, that help us to find the regions of stability of these patterns and leads to a natural classification for them. We have shown that all non-trivial spatial patterns can be classified into 3 main classes: spiral wave fibrillation, of types a and b, and oscillatory spatial patterns. The tissue size used in our simulations is rather large, as we want to study the wave dynamics with minimal effects of boundaries. However, we have also performed some simulations on patches with smaller sizes (5×5 cm) and we observed patterns of the same type for the same parameter values as for the tissue of larger size in most of our calculations. We have also studied the onset of the wave breaks in 2D. We show in

Parameters for which we obtain oscillatory patterns are located in the range for which we find oscillatory patterns in our single-cell simulations. We demonstrate that (a) spiral fibrillation of type a occurs normally in the parameter region with a single or multiple EADs and (b) spiral fibrillation of type b is observed at the boundary between the no-EAD and EAD regions and also in the region where a single stimulus in the single-cell does not show any EADs. Note that there is no clear boundary between these two latter regimes: we observe a smooth, gradual transition from one regime to another.

To the best of our knowledge, our study is the first one that has obtained phase diagrams, in two-dimensional parameter spaces, which lead to regions of stability for the different spatial patterns we find in our simulation domain that consists of EAD cells. However, a few examples of complex spatial patterns, originating from EADs, have been reported in Refs.

We find that spiral patterns of type b and oscillatory patterns are mediated by Ca waves, whereas spiral type a patterns are formed by Sodium mediated waves. In the case of oscillatory patterns, we have observed phase waves. As far as we aware, we have given the first demonstration of the existence of phase waves in a detailed model of cardiac tissue. Our study also suggests a clear method for identifying phase waves in such systems. This can be used in experimental research too. In our method we investigate wave propagation in the presence of oscillatory EADs. We demonstrate that phase waves cannot be eliminated by propagation barriers. We note that phase waves do not propagate because of local interactions of adjacent cells. Such behavior in single cells has also been found in other modeling studies (see, e.g.,

The effects, which we have found, can be studied experimentally. The most relevant experimental setup is a cell culture of the type used in Ref.

Finally, let us consider some limitations of this study. We have only studied a monodomain idealized, homogeneous and isotropic tissue model in 2D. Generally, studies on EAD dynamics have usually been carried out in 2D tissue

Important ionic currents related with EAD formation. We show all the currents which directly contribute to the potential when no EAD is present (green line with parameters:

(TIFF)

Important ionic currents related with EAD formation part 2. Parameters left:

(TIFF)

An example of a spiral fibrillation of type a. The parameters are

(AVI)

An example of a spiral fibrillation of type b. The parameters are

(AVI)

An example of a spiral fibrillation of type b. The parameters are

(AVI)

This video shows voltage (left), opening of Ca-gates (middle), and opening of sodium gates (right) in the SF

(AVI)

This video shows Voltage voltage (left), opening of Ca-gates (middle), and opening of sodium gates (right) in the SF

(AVI)

This video shows Voltage voltage (left), opening of Ca-gates (middle), and opening of sodium gates (right) in the oscillatory pattern. The parameters are

(AVI)

Starting from the pattern spiral fibrillation of type a (with parameters

(AVI)

Starting from the pattern spiral fibrillation of type b (with parameters

(AVI)

Starting from the pattern oscillatory fibrillation (with parameters

(AVI)

We would like to thank Daniel Pijnappels for useful discussions.