An ap is an externally triggered event (with duration): an ec fires an ap as an all-or-nothing response to a suprathreshold stimulus, and each ap follows the same sequence of phases (described below) and exhibits roughly the same waveform

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regardless of the applied stimulus. During most of the ap no re-excitation can generally occur (the ec is in a refractory

0

period).

Despite differences in duration, morphology, and underlying ion currents, the following major ap phases can be identi-

Voltage (mv) Voltage (mv)

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fied across different species and ec types: resting, stimulated, upstroke, early repolarization, plateau, and final repolariza- S1

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tion. We abbreviate them as r (resting and final repolariza-

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(a)

tion), s (stimulated), u (upstroke), and p (plateau and early repolarization).

Intuitively, a hybrid automaton¹² is an extended finitestate automaton, the states of which encode the various phases of continuous dynamics a system may undergo, and the transitions of which are used to express the switching logic between these dynamics. The clha we obtained are fairly compact in ¹⁴⁰ 120

Using the ap phases as a guide, we have developed, for several representative ec types, clha models that approximate the ap and other bioelectrical properties with reasonable accuracy. Their derivation was first performed manually. ^{24, 25} We subsequently showed in Grosu et al. ¹¹ how to fully automate this process by learning various biological aspects of the ap of different cell types.

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nature, employing two or three continuous state variables and
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four to six modes. The term cycle-linear is used to highlight the (c)
cyclic structure of clha, and the fact that while in each cycle
they exhibit linear dynamics, the coefficients of the corre-
sponding linear equations and mode-transition guards may
vary in interesting ways from cycle to cycle.

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Figure 2 presents one of our clha models. To understand the model, first note that when an ec is subjected to repeated stimuli, two important time periods can be identified: ap duration (apd), the time the cell is in an excited state, and diastolic interval (di), the time between the “end” of the ap and the next stimulus. Figure 2(a) illustrates the two intervals. The function relating apd to di is called the apd restitution function. As shown in Figure 2(b), the relationship is nonlinear and captures the phenomenon that a longer recovery time is followed by a longer apd. A physiological explanation of a cell’s restitution is rooted in the ion-channel kinetics as a limiting factor in the cell’s frequency response.

The clha model itself, superimposed over the image of a typical ap, is given in Figure 2(c). Each mode has an associated

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linear dynamics x = Ax + Bu, v = Cx, where x is the clha state, u is the input, and v (for voltage) is the output. A mode also has an associated invariant in v, forcing the outgoing transition to be eventually taken. The concept of mode dynamics and invariant is illustrated in Figure 2(c) for mode p (plateau and early repolarization); see that mode’s callout. Transition labels are of the form e ∧ g/a, where e is an (optional) event, g is a guard, and /a is an optional set of assignments. The only events in the model, representing the start and end of stimulation, are denoted by s and s , respectively. Observe the per-

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mode and transition-guard dependence on the di, which is measured with the help of clock variable t.

The dynamics of excitable-cell networks play an important

figure 2: AP duration, restitution function, and CLHA model of cardiac myocytes.

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APD (ms)

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DI APD 90

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S2

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Time (ms)

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100 150 200 DI (ms)

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(b)

Early repolarization
and
Plateau

Upstroke

Stimulated

Final repolarization

and Resting

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role in the physiology of many biological processes. For cardiac cells, on each heart beat, an electrical control signal is generated by the sinoatrial node, the heart’s internal pacemaking region. Electrical waves then travel along a prescribed path, exciting cells in atria and ventricles and assuring synchronous contractions. Of special interest are cardiac arrhythmias: disruptions of the normal excitation process due to faulty processes at the cellular level, single ion-channel level, or at the level of cell-to-cell communication. The clinical manifestation is a rhythm with altered frequency, tachycardia (rapid heart beat) or bradycardia (slow heart beat), or the appearance of multiple frequencies, polymorphic ventricular tachycardia, with subsequent deterioration to a chaotic signal, ventricular fibrillation (vf). vf is a typically fatal condition in which there is uncoordinated contraction of the cardiac muscle of the ventricles in the heart. As a result, the heart fails to adequately pump blood and hypoxia (lack of oxygen) may occur.

In order to simulate the emergent behavior of cardiac tissue, we have developed CellExcite, ² a clha-based simulation environment for excitable-cell networks. CellExcite allows the user to sketch a tissue of excitable cells, plan the stimuli to be applied during simulation, and customize the arrangement of cells by selecting the appropriate lattice. Figure 3 presents our simulation results for a 400 × 400 clha network of cardiac myocytes. Nine 50-ms simulation steps are shown, during which (steps 1 and 4) the network was stimulated twice, at different regions. The results we obtain demonstrate the feasibility of using clha networks to capture and mimic different spatiotemporal behavior of wave propagation