Biomedical Engineering Reference
In-Depth Information
capacitance of approximately 1F per cm
213;14;15;16
. The existence of an
electrical capacitance across cell membranes is vital to cell life
17;18
. Ca-
pacitance is used as a mechanism for rapid biological signaling and for in-
tegration of electrical signals over time and space
3
. Capacitance-dependent
synaptic integration is also the basis of central nervous system function
(thought, sensation, perception, behavior, etc.). Until recently, models of
cellular integration have assumed a constant membrane capacitance over
the cell, usually of 1F per cm
2
. Although there is a wide agreement that
cell membranes have an average membrane capacitance near 1F per cm
2
,
recent experimental work has led to questions about the assumption that
membrane capacitance is constant over the surface of a cell
19
. Membrane
capacitance is dependent on ion channel density
20;21;22;23
(but see Gentet
et al
24
.), and evidence has been mounting that ion channels are often dis-
tributed unevenly in cells
25;26;27;28;29
. The consequences of capacitative non
uniformities in the passive propagation of electrical potentials in cylindrical
membrane processes (such as axons, dendrites, and muscle bers) are not
known. In fact, the simplifying assumption that membrane capacitance is
xed has been made in almost all analytic and computational models of
neurons and other cells to date (see reviews by Rall
3
, Lindsay et al
30
., and
Glenn and Knisley
9
). No systematic studies have been conducted to test
this assumption, so one of us (J.K.) used the eigenvalue method to develop
models in space and time of cylinder-shaped brain cells with longitudinally
graded membrane capacitance specically with an exponential increment
with distance from the end of the cylinder. The hypothesis tested was that
there are no biologically-signicant dierences between membrane cylinders
with a homogeneous membrane capacitance and those with a exponentially-
graded capacitance, provided that the spatial variation in capacitance is
within the range of that estimated in experimental studies.
4.1. Denition of the General Model
A membrane cylinder with spatially-graded capacitance can be modeled
by the cable equation
@
2
V
@X
2
R
m
C
m
(X)
@V
@t
V = 0;
0 < X < L
(3)
where V (X; t) is membrane potential, R
m
is the specic membrane resis-
tance in ohms, C
m
is the specic membrane capacitance in farads, L is the
electrotonic length of the equivalent cylinder, and X is the electronic dis-
tance from the origin (electronic units are dimensionless and by convention,
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