Biomedical Engineering Reference
In-Depth Information
Fig. 1. Denition of eigenslope with an example of multiexponential decay of voltage
across a membrane. The decay rate of the exponentials (inverse of time constant) are
equal to the eigenvalue, and the initial voltages (at t = 0) are proportional to the inverse
of the eigenslope. A: The voltage decay () is composed of the sum of simple exponential
decays (1, 2, 3, ...). B: Example of transcendental eigenvalue function associated with
multiexponential decay in A. Eigenvalues are the roots (zero crossings) and the eigenslope
is the slope of the curve at the zero crossing. Vertical lines show where eigenvalue function
goes to1. From Glenn and Knisley
12
by permission.
the eigenslope is dened to be the slope of the eigenvalue function at the
given eigenvalue
12
.
The eigenslope is shown in Fig. 1. We consider a Fourier series solu-
tion common in electrical engineering and neurobiology of the form:
1
X
C
i
e
i
t
:
V (t) =
i=1
The solution for V (which later will represent membrane voltage) for any
set of initial value and boundary conditions is an innite series of simple
exponentials, each with a time constant of 1=
i
, the rst three terms of
which are plotted in Fig. 1A. The eigenvalues
i
are obtained from an
eigenvalue function
f() = 0
plotted in Fig. 1B, where the eigenvalues are the points where the eigenvalue
function intercepts the abscissa. The eigenslope is then dened as the rst
derivative of the equation, or
df()=d
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