Doi: 10.1145/1467247.1467271
Learning and Detecting
Emergent Behavior in Networks
of Cardiac Myocytes
By Radu Grosu, Scott a. Smolka, Flavio Corradini, anita Wasilewska, emilia entcheva, and ezio Bartocci
abstract
We address the problem of specifying and detecting emergent
behavior in networks of cardiac myocytes, spiral electric waves
in particular, a precursor to atrial and ventricular fibrillation.
To solve this problem we: ( 1) apply discrete mode abstraction
to the cycle-linear hybrid automata (clha) we have recently
developed for modeling the behavior of myocyte networks; ( 2)
introduce the new concept of spatial superposition of clha
modes; ( 3) develop a new spatial logic, based on spatial superposition, for specifying emergent behavior; ( 4) devise a new
method for learning the formulae of this logic from the spatial
patterns under investigation; and ( 5) apply bounded model
checking to detect the onset of spiral waves. We have implemented our methodology as the Emerald tool suite, a component of our eha framework for specification, simulation,
analysis, and control of excitable hybrid automata. We illustrate the effectiveness of our approach by applying Emerald
to the scalar electrical fields produced by our CellExcite
simulation environment for excitable-cell networks.
1. intRoDuction
One of the most important and intriguing questions in systems biology is how to formally specify emergent behavior in
biological tissue, and how to efficiently predict and detect its
onset. A prominent example of such behavior is electrical
spiral waves in spatial networks of cardiac myocytes (heart cells).
Electrical impulses regularly circulate through cardiac tissue
and cause the heart’s muscle fibers to contract. In a healthy
heart, these electrical impulses travel smoothly and unobstructed, like a water wave that ripples gently in a pond. These
waves can, however, sometimes develop into troublesome,
whirlpool-like spirals of electrical activity. Spiral waves of this
nature are a precursor to a variety of cardiac disturbances,
including atrial fibrillation (af), an abnormal rhythm originating in the upper chambers of the heart. af afflicts two–three
million Americans alone, putting them at risk for clots and
strokes. Moreover, the likelihood of developing af increases
with age.
In this paper, we address this question by proposing a
simple and efficient method for learning, and automatically detecting the onset of, spiral waves in cardiac tissue.
See Figure 1 for an overview of our approach. Underlying our
method is a linear spatial-superposition logic (lssl) we have
developed for specifying properties of spatial networks. lssl
is discussed in greater detail below. Our method also builds
upon hybrid automata, image processing, machine learning,
and model-checking techniques to first learn an lssl formula
that characterizes such spirals. The formula is then automatically checked against a quadtree representation of the sca-
20
lar electrical field (sef) produced at each discrete time step
by a simulation of a hybrid-automata network modeling the
myocytes. A scalar field is a function that associates a scalar
value, which in our case is an electric potential, to every point
in space. The quadtree representation is obtained via discrete
mode abstraction and hierarchical superposition of the elementary units within the sef.
The electric behavior of cardiac myocytes is hybrid in nature:
they exhibit an all-or-nothing electrical response, the so-called
action potential (ap), to an external excitation. An ap can thus
be viewed as triggering a discrete mode transition from the
cell’s resting mode of continuous behavior to its excited mode
of continuous behavior. Despite their discrete-continuous
hybrid nature, networks of myocytes have traditionally been
modeled using nonlinear partial differential equations. 13, 17
While highly accurate in describing the molecular processes
underlying cell behavior—nonlinear differential equations
allow one to closely match the values of a multitude of state
variables to their actual physical values—these models are not
particularly amenable to formal analysis and typically do not
scale well for the simulation of complex cell networks.
In Grosu et al., 11 we showed that it is possible to automatically learn a much simpler hybrid automaton (HA) 12 model for
cardiac myocytes, which explicitly captures, up to a prescribed
error margin, the mixed discrete and continuous behavior of
the ap. To highlight its cyclic structure and its linear dynamics, which may vary in interesting ways from cycle to cycle, we
called it a cycle-linear hybrid automaton (clha). Moreover, one
can use a variant of this clha model to efficiently (up to an
order of magnitude faster) and accurately simulate the behavior of myocyte networks, and, in particular, induce spirals and
fibrillation. 2, 24, 25
A key observation concerning our simulations, see Figure 3,
is that mode abstraction, in which the ap value of each clha in
the network is abstracted to its corresponding mode, faithfully
preserves the network’s waveform and other spatial characteristics. Hence, for the purpose of learning, and detecting the
onset of, spirals within clha networks, we can exploit mode
abstraction to dramatically reduce the system state space. A
An earlier version of this paper appeared in Proc. 11th
International Conference on Hybrid Systems: Caomputation
and Control (HSCC’08), Springer, LNCS 4981, April 2008.