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
50
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)
− 50
fied across different species and ec types: resting, stimulated,
upstroke, early repolarization, plateau, and final repolariza- S1
− 100
tion. We abbreviate them as r (resting and final repolariza-
0 50
(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.
100
80
60
40
20
0
nature, employing two or three continuous state variables and
0 50
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.
− 20
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
.
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-
–
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.
130
120
110
APD (ms)
100
DI APD 90
80
S2
100 150 200 250
Time (ms)
300
70
0 50
100 150 200
DI (ms)
250 300
(b)
Early repolarization
and
Plateau
Upstroke
Stimulated
Final repolarization
and
Resting
100 150 200
Time (ms)
250
300
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