What Is an Eigenmode

An important key to this approach for studying and controlling alternans at the ion channel level lies in a mathematical construct called eigenmodes. Eigenmode theory is common in applied math, physics, and engineering, but is seldom seen in cardiac electrophysiology. In brief, the theory provides us with a method for separating the complicated behavior of certain types of systems into a set of simpler, characteristic behaviors called eigenmodes. Each of these eigenmodes then provides a complete description of the corresponding characteristic behavior in pure form. One ordinary example of eigenmodes comes from the theory of electromagnetic waves. Electromagnetic waves come in all frequencies, from radio waves to X-rays and beyond. Each of these frequencies is a separate eigenmode. These eigenmodes have vastly different properties from one another - radio waves, microwaves, visible light, and X-rays have different wavelengths and differ widely in the degree to which they are absorbed, scattered, reflected by, and/or propagate through various materials. These properties may all be diagnosed once the structure of the eigenmode in the given material is known. Other examples of eigenmodes include sound waves of every frequency, the vibrational modes of entities as diverse as stars, bridges, and molecules, and the quantum wave functions of electron orbitals in atoms.

In order for eigenmode theory to be applicable, the behavior of the system under study must generally be divisible into two components: a steady state component, and a perturbation, which is the term that generally refers to any disturbance to the steady state. Each of these two components must have certain properties. The steady state component, as the name suggests, must be unchanged when viewed at regularly spaced time intervals. There are two variations here: either this time interval is infinitesimal, meaning that the state may be viewed at any time, which requires that the steady state be exactly the same all the time, or the time interval is finite, in which case the state of the system is allowed to change, provided it then changes back in time for the next viewing time. The condition on the perturbation is simply that it must be small in some sense.

If the behavior of the system is divisible in this way, it can be shown mathematically (with a couple of extra technical conditions not discussed here) that there always exists a set of special perturbations, each of which grows, decays, and/or oscillates by a constant factor from one viewing time to the next, and each of which is, apart from this factor, identical from one viewing time to the next. These special perturbations are the eigenmodes, and the constant factor associated with each of these eigenmodes is called its eigenvalue. These eigenvalues each have two components: one that gives the frequency at which the corresponding eigenmode oscillates, and one that gives the exponential rate of growth or decay of the eigenmode in time. When the eigenvalue indicates decay, the corresponding eigenmode is called stable. It will thus effectively disappear over time. If the eigenmode is growing exponentially, it is called unstable. These are the modes that are normally of concern.

As an example, consider the case of sound waves bouncing around inside a room. The steady state for this case is simply the room with no sound in it. The perturbations are the sound waves. All sound waves of normal listening volume are small enough to satisfy the "smallness" condition for the perturbation. The eigenmodes are then simply the sound waves at each frequency. The eigenvalues are the frequencies themselves together with the rate at which each eigenmode wave exponentially decreases in amplitude. The eigenmode itself consists of the special set of changes, at every point in space, in air pressure, density, and fluid velocity relative to their steady state values, which together embody the propagating sound wave at the given frequency. Each of these quantities oscillates with this same (eigenvalue) frequency and decreases exponentially in amplitude with the same (eigenvalue) decay rate. Consequently, the proportionality of the amplitude of all these quantities with one another remains identical for all times, thus satisfying the requirement that the eigenmode remains unchanged apart from an overall scaling factor given by the eigenvalue.

There exists a nice, geometric method for visualizing the workings of the eigenmode formulation. The idea is to represent any general perturbation as a single vector living in a very high dimensional space. This vector is the sum of component vectors, each of which represents the amplitude of one of the dynamical variables, as shown schematically in Fig. 2a. (The two component vectors shown in each panel are meant to represent the usually much larger number of dynamical variables actually present.) Examples of dynamical variables are the variations in density, pressure, and fluid velocity at every point in space, as in the sound wave example, or variations in the membrane potential, gating variables, and ion concentrations and/or ion currents, for the case discussed later in this chapter. For most perturbations, this vector continuously changes length and direction as time passes, as illustrated by the successive panels in Fig. 2a. Eigenmode theory says that any general perturbation that is small enough may be written as the vector sum of so-called eigenvectors, as illustrated in Fig. 2b. Again, the number of eigenvectors is usually much larger than two; but only two are shown here for clarity. As depicted in Fig. 2b-d, each of these eigenvectors has the properties that (1) its direction remains unchanged and (2) its length increases or decreases by a constant factor given by the eigenvalue, from one viewing time to the next. From Fig. 2c, d, we see that this invariance in direction forces the amplitudes of the dynamical variables to maintain the same proportionality relative to one another at every viewing time and also forces each of the dynamical variables associated with the eigenmode to grow or decay with same constant factor as the eigenvector itself.

The usefulness of the eigenmode technique should be clear from Fig. 2. The behavior of the dynamical variables and vector representation of a typical perturbation can appear to be very complicated, with the dynamical variables changing sign and fluctuating in a complicated way, and the vector changing direction, and shortening and lengthening seemingly at random, as suggested by Fig. 2a. In contrast, as shown in Fig. 2b-d, when the perturbation is decomposed into eigenmodes, arbitrary perturbations are seen to be composed of a set of very simple behaviors, each of which grows or decays exponentially in amplitude and whose dynamical variables maintain constant proportionality to one another.

In particular, the eigenmode decomposition makes it clear that, in order to control arbitrary perturbations, it suffices to control the unstable eigenmodes, either by modifying the system so that all modes are rendered stable, or by adding a "control" disturbance that zeroes out the unstable mode(s). Both of these approaches are potentially relevant to the problem; thus both are currently under study. The former falls within the discipline of control theory and is used extensively in several areas of engineering. When applied to the

Legend

Dynamical variable vector

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General perturbation vector

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Eigenvector #1

-►

Eigenvector #2

----^

Control perturbation vector

Viewing time #1

Viewing time #2

Viewing time #3

Viewing time #4

(a) Vector representation of a general perturbation:

(b) The same general perturbation decomposed into eigenmodes:

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