**The Glial membrane-fluid cleft-neural membrane arrangement
cuts effective neural capacitance, greatly increasing signal conduction
velocity and greatly reducing the energy requirement per action potential.**

July 14, 1997 M.R.
Showalter

**Unmyelinated axons and dendrites are not bathed by open volumes of
grounded fluid. Instead, they are closely surrounded by the membranes of
other cells, separated by thin fluid clefts that are 2-4 times the neural
membrane thickness. It has been assumed that the fluid clefts act as perfect
neural membrane capacitor grounds. The assumption has been that the clefts
have such high effective conductance that they act in the same way as an
open volume of grounded fluid would do. For very slow voltage fluctuations,
this would be true enough. However, for voltage fluctuation frequencies
of neural interest, the fluid clefts are ungrounded. **

**Considering cleft resistance alone, conduction in the plane of the
fluid cleft would be badly and inconsistently grounded. Cleft resistance
is typically very much higher than neural line conductance. When cleft
R ^{2}C effective inductance is also taken into account, and clefts
are seen to be ungrounded for signal propagation purposes. Voltage fluctuation
propagation velocities in the plane of the cleft are about 1/(R^{2}C^{2})^{.5}
meters/sec (about .0001 meters/sec) - much slower than action potentials
and other signals in the neuron. The fluid cleft acts as a dielectric layer,
surrounded by two dielectric membranes, not as a ground. The R^{2}C
effective inductance does not reduce smooth cleft current flows, which
are dominated by cleft resistance, and does not reduce molecular diffusion
through the clefts.**

**Analysis indicates that ion flows in and out of a fluid cleft during
action potentials are capacitive, and do not involve significant conductance
along the plane of the fluid cleft. The membrane-fluid cleft-membrane assembly
has a much larger effective capacitance with respect to charged ion flows
between cleft and neuron than it does between neuron and effective neuron
capacitance ground. The effective capacitance for charged ion flows across
the cleft is about the same as the capacitance of a membrane surrounded
by a large volume of grounded fluid would be. This means that much less
ion flow is required with the membrane-cleft-membrane assembly than would
be required otherwise for a given change in axolemma voltage. At the same
time the capacitance between neural membrane and cleft provides an ample
source of ions for the action potential at a nearly steady voltage. **

**A neural membrane-fluid cleft-glial membrane assembly reduces the
effective capacitance of the neural membrane by about two orders of magnitude,
compared to the capacitance of the neural membrane surrounded by a large
volume of grounded fluid. A neural membrane-fluid cleft-neural membrane
assembly in the cerebral parallel fibers and similar structures reduces
effective capacitance for the same reasons and in similar proportion. **

**The effectively ungrounded fluid clefts have strong physiological
effects and advantages. **

**The energy requirement (ion flow) per action potential is proportional
to the effective neural membrane capacitance. Therefore, the fluid cleft
assembly reduces energy consumption per action potential about 100 fold.
**

**Ion flow required to produce a unit of axolemma voltage change is
proportional to the effective neural membrane capacitance. Therefore, the
fluid cleft assembly cuts the time to equilibration for a set number of
channels and a set initial voltage difference across the membrane. **

**The velocity of action potential conduction and passive conduction
is inverse with assembly capacitance (according to Showalter-Kline theory),
so that the fluid cleft increases conduction velocity about 100 fold. **

**Thin fluid clefts essentially always surround unmyelinated neural
membrane. This seems to be consistently true for all animals that have
nerves. It is suggested that the important energy conservation and conduction
facilitation that these clefts provide is the biological reason for their
consistent occurrence through phylogeny. **

**BACKGROUND:**

Ideal capacitor grounds have high conductance to ground on the time
scales where they are used. Visual inspection of the fluid clefts shown
between neural membranes in electron micrographs makes the ideality of
these clefts as grounds at least questionable. Current paths through the
clefts are generally restrictive and appear to vary over a wide range from
cleft to cleft. Nor can the clefts be simple resistive conductors. Charges
in the clefts are strongly coupled with the capacitance of the neural and
glial membranes that are the walls of the cleft. Even so, the idea of the
cleft as ideal ground has been accepted.

I was forced to question this ideality when my theoretical work calculated
conduction velocities about 100-fold too slow. According to the Showalter-Kline
conduction theory, conduction velocity for frequencies of neural interest^{(1)},
for the case of low membrane conductance, is predicted by equation (1).
Conduction velocities at higher membrane conductances are proportional
to (1).

Predicted velocities were gratifyingly congruent with measurements when
calculations were done in centimeter-gram-second-coulomb-volt units. CGS
units are the customary units in neural studies. However, thanks to an
incisive question from a reviewer, it became clear that the CGS unit system
did not correctly account for crossterms in the Showalter-Kline theory.
The fully dimensionally consistent Meter-Kilogram-Second-Coulomb-Second
(MKS-Georgi) unit system was necessary instead. For the nonlinear terms
in the Showalter-Kline theory, a change of unit systems is an integration
scale correction that changes magnitudes. I found that in the MKS-Georgi
unit system, (1) was predicting conduction velocities about a factor of
100 too slow. Either the Showalter-Kline theory was wrong, or one or more
of the parameters used to calculate (1) had to be wrong, or (1) had to
be an incorrect representation of neural circumstances.

Resistivity could not be changed much. On first inspection, the calculation
for membrane capacitance seemed solid also.

I was therefore forced to look hard at the role of the fluid clefts
in determining __effective__ neural capacitance, for the neuron as it
actually was in tissue. If the cleft was part of a neural membrane-fluid
cleft-glial membrane assembly with a capacitance per unit area C_{a},
then (2) should be substituted for (1):

The cleft IS part of an assembly that is more complicated than the single
membrane capacitor we have assumed in our models of neural capacitance.

Fig. 1 shows how glial membrane surrounds almost all unmyelinated neural
membrane. (Parallel fibers in cerebellum, and some similar structures,
are an exception, but have fluid clefts that are similar.) The fluid in
the cleft is much like plasma or cerebrospinal fluid. Dielectric constant
of the membranes is about 2, and the dielectric constant of (pure) water
is about forty times higher, about 80.

Fig. 1 is a crossection of the membranes as a planar structure. Length
and depth of the planar membranes is very large with respect to 20 nanometers.

How does this assembly act as a capacitor, with respect to the neural
axolemma (with charge differentials perpendicular to the plane)? It is
useful to think of the two polar cases, one corresponding to effectively
infinite cleft conductance in the plane, and the other to effectively zero
cleft conductance in the plane.

Polar Case 1. For charge fluctuations so slow that the electrical conductance of the fluid cleft is asymptotically large, THE FLUID CLEFT IS THE CAPACITOR GROUND with respect to both neural and glial membrane. In this "infinite cleft conductance" case, the capacitance with respect to the neural membrane is no different than it would be for a single membrane - the potential the neural membrane "sees" does not depend at all on the glial membrane.

Polar Case 2. For charge fluctuations in the neuron so fast that the
electrical conductance of the fluid cleft is asymptotically small, THE
FLUID CLEFT IS UNGROUNDED. The effective capacitor ground of the neural
axolemma is the glial axolemma (or vice versa). The dielectric properties
of the ungrounded fluid cleft will be nearly the same as they would be
for pure water (an ion does not move more than half a water molecule length
because polar water molecules around it rotate with uniform statistics.)

(Example for polar case 2, for 30 nm cleft thickness, and dielectric
constants of 2 for the membrane and 80 for the cleft fluid. Capacitance
would be 1/152nd (.0066) of the capacitance for the unshielded single membrane.)

In terms of Showalter-Kline theory, the fluid cleft is effectively ungrounded
at action potential and neural signal frequencies. An exact quantitative
model of cleft conductance as a function of frequency and the (extremely
variable) cleft geometry that occurs in neural tissue would be daunting
to do and to read. No such model is offered here. It should suffice to
know that the conduction velocity of a voltage disturbance in the cleft
(along the plane of the cleft) depends on cleft thickness and cleft fluid
resistivity, and is about 10^{-5} meter/sec (or .001 cm/sec) or
less. (See Appendix.). (Velocity perpendicular to the plane of the cleft
will be a factor >10^{8} faster.) The planar cleft signal velocity
is far too low to carry much of the charge flux involved in

the action potential or passive signal propagation in the neuron. RAPID
charge equilibration across the neural membrane must come from assembly
capacitance, not from charge flows along the cleft plane. For dynamic purposes,
the cleft is ungrounded.

The assembly capacitance is amply large to supply the charge flows needed
for signal propagation and action potentials. To see this, it is useful
to review the electrochemistry of action potentials and other neural potentials.
Levitan and Kaczmarek phrase the central fact of electrochemistry nicely:

**The Nernst equilibrium potential. Assume the membrane is permeable
to a single charged ion, X+, which is asymmetrically distributed on the
two sides of the membrane. The counterion, Y-, cannot cross the membrane.
X+ will flow across the membrane from the side of high concentration to
the side of low concentration, until the buildup of charge is sufficient
to (generate a voltage that is sufficient to) oppose net ion flow. The
transmembrane voltage (V _{m}) is 0 before X+ begins to flow. At
equilibrium V_{m} = E_{x+}**

Suppose that the charge flow sufficient to buildup voltage E_{x}
changes by a factor of **N** (suppose that capacitance changes by a
factor of **N**). **Even if N is a large number, the equilibrium voltage
itself does not change.** The charge flow (and the energy consumption
that charge flow represents) changes in proportion to the capacitance.
Equilibration becomes faster as the charge flow required for equilibration
becomes smaller. From both these points of view, reducing the net neural
membrane capacitance is an advantage. Reduced capacitance cuts energy requirements
and speeds performance.

The logic of the action potential, and its dependency on the opening
and closing of potassium and sodium channels, does not change when the
neuron is surrounded by a fluid cleft (rather than by an infinite bath
of grounded fluid) except that energy efficiency and dynamic response are
much improved by the capacitive reduction of the cleft. The effective capacitance
of a neuron in the course of the action potential has two parts:

a: First, the capacitance from neural axolemma to glial axolemma (from
1-4 on Fig 1) for the full voltage difference v_{1} - v_{4}.

b: Second, the capacitance across the neural membrane from neural axolemma
to cleft, for a voltage difference of about 1/100th v_{1} - v_{4}.

The net result is an effective capacitance about half of the capacitance
from 1 to 4 in Fig. 1.

According to Showalter-Kline theory, in the very low **g** state,
signal forms can propagate passively down neural lines. For such propagations,
the effective capacitance is the capacitance between 1 and 4 in Fig. 1.

**A DISCUSSION OF THE ROLE OF GLIA**

History shows that it is dangerous to suggest an essential role for
glial cells^{(3)} ^{(4)}.
Even so, the role of glia in forming the fluid clefts around neurons seems
apparent from these calculations. The clefts reduce energy consumption
(and increase conduction velocity) by about two orders of magnitude - enough
to make them biologically worthwhile, enough to make them as conserved
as they are throughout taxonomy and phylogeny.

Perhaps, with benefit of hindsight, this is not so surprising. From
an engineer's perspective, the ability of glia to wrap around neurons and
form fluid clefts is wonderful. It is nearly impossible to model the geometry
of single neurons in three dimensional detail. There are billions of neurons,
and they lie together in an ordered but still much tangled three dimensional
array. How would you design a cell type that would wrap all the neurons,
wrap them everywhere, and wrap them in a way that forms such consistent
fluid clefts? Glia do wrap the neurons, excepting myelinated neural lines,
and some bundles such as the parallel fibers. In electron micrographs,
which are crossections, one simply expects a thin fluid cleft surrounding
axons and dendrites.

Glial geometries seem haphazard or semichaotic in many other ways, but
glia wrap neurons with stark precision. Thin fluid clefts essentially always
surround unmyelinated neural membrane. This seems to be consistently true
for all animals that have nerves. Since the glia do such a difficult thing
when then they form the clefts so consistently, must not the cleft geometries
formed be very important? The calculations here show that the clefts ARE
very important in neural transmission.

The fluid clefts between parallel fibers in cerebellum and elsewhere
should act similarly to increase conduction velocity and reduce neural
energy consumption.

This paper has assumed the correctness of the Showalter-Kline neural
conduction model, and found that predicted characteristics of the fluid
cleft strongly reinforce that model. The combination of Showalter-Kline
with the clefts forms a functionally neat package, well adapted for precision
neural transmission and resonance, and energetically efficient.

In contrast, the clefts make Kelvin-Rall, already very implausible,
more implausible still. Strong reasons to doubt Kelvin-Rall have been reviewed
elsewhere^{(5)} Without considering the clefts,
Kelvin-Rall has the following problems. In Kelvin-Rall, electrical propagation
velocity goes as the square root of fourier frequency, and so a coherent
waveform such as an action potential waveform would be impossible. Some
unexplained mechanism whereby channel-generated voltage changes travel
down the neuron at a speed different from other voltage changes has to
be posited. Velocities are also much too high. Kelvin-Rall lacks any mechanism
for the coordinated firing of many neurons over large distances that is
observed. Kelvin-Rall predicts that neurons are much slower than they are.
For these reasons, the sophisticated information processing brains do seems
impossible under Kelvin-Rall. When the effects of clefts are added, under
the Kelvin-Rall assumption of no effective inductance in the clefts, the
model becomes much more complicated, and more statistically varied, because
of wide ranges of resistivity from cleft to cleft, or along a single cleft.
Such a muddy system does not fit the requirements of know patterns of information
processing.

Under Showalter-Kline theory, the fluid clefts ubiquitously observed
around neurons are part of an efficient neural conduction assembly capable
of precision function.

**APPENDIX 1: Cleft Velocity, Showalter-Kline
theory**

Conduction velocity of fluctuating voltage in the cleft is much smaller
than line action potential velocity or passive conduction velocity, so
that conduction along the plane of the cleft cannot carry a significant
fraction of the current flow necessary to support either sort of neural
line conduction.

Here is the standard formula for line propagation velocity. We can treat
a strip of cleft as a line.

we interpret **R**, **L**, **G**, **C** including crossproducts,
taking hatted values as below^{(6)}

It is useful to consider the case where G is negligibly small, so that
only the R^{2}C crossterm has to be included in the modelling.
Then we can write

n subscripts emphasize that we are dealing with numerical values (and
keeping track of units implicitly.) In the neural range this simplifies^{(7)}
to

including units, this equation is

Consider a cleft one unit of length deep (1 meter deep). In MKS-Georgi
units resistance and capacitance are

Capacitance is proportional to width, and resistance is inversely proportional
to width: the RC product is independent of width, as it must be.

We can now calculate conduction velocity in a cleft for reasonable values
of the dimensional parameters. We'll choose

**c** = .01 farad/meter^{2} (1 microfarad/cm^{2})
= .55 ohm-meter

**t** = 3 x 10^{-8} meter

The velocity is **3.86 x 10 ^{-6} meters/second**, or

**This result does not effect molecular diffusion in
the clefts. It does not effect the steady current flow capacity of the
clefts, which is controlled by resistance alone, not by effective inductance.
HOWEVER, THE RESULT SHOWS THAT THE CLEFTS ARE UNGROUNDED
TO GOOD APPROXIMATION SO FAR AS ACTION POTENTIALS OR OTHER RAPID LINE CONDUCTANCES
ARE CONCERNED. **

**APPENDIX 2: Cleft Conduction Velocity - standard
theory**

According to Showalter-Kline conduction theory, conduction velocity
of fluctuating voltage in the cleft is much smaller than line action potential
velocity or passive conduction velocity. According to conventional conduction
theory, calculated cleft conduction velocity is orders of magnitude faster.

Here again is the standard formula for line propagation velocity. We
can treat a strip of cleft as a line.

In Appendix 1, we interpreted **R**, **L**, **G**, **C**
including crossproducts. In this appendix, we neglect the crossproducts,
and compute cleft velocity according to the old standard theory.

Again, we consider the case where G is negligibly small, and electromagnetic inductance is also negligible, so that we can write

n subscripts emphasize that we are dealing with numerical values (and
keeping track of units implicitly.) R and C are the same as in Appendix
1. As before, capacitance is proportional to width, and resistance is inversely
proportional to width, so that the RC product is independent of width,
as it must be, since velocity must not vary with width.

We can now calculate conduction velocity according to the standard theory
using the same parameters as in Appendix 1. The calculated old-theory velocity
varies with the square root of frequency, and would be

for a fourier component of 5,000 hz, this would be .29 meters/sec, or
29 cm/sec, about 75,000 times faster than velocity calculated accoding
to the Showalter-Kline theory. This old theory velocity (for a 30 nm fluid
cleft) is about twice as fast as the conduction velocity Sakmann et al
measured for a 6 micron (200 x 30 nm) diameter dendrite.

**NOTES: **

1. Showalter, M.R. __A Passive Neural Conduction
Equation: Part a: derivation__ p. 10 and __A Passive Neural Conduction
Equation: Part b. neural conduction properties.__ p. 7 available FTP
angus.macc.wisc.edu/pub2/showalt

2. Levitan, I.B. and Kaczmarek, L.K. ** THE NEURON:
Cell and Molecular Biology** Oxford U. Press (1991) pp 54, 55

3. "There are between 10 and 50 times more glial cells than neurons in the central nervous system of vertebrates. Glial cells are probably not essential for processing information . . . "

Kandel, E.R. __Nerve Cells and Behavior__ p 23 in ** Principles
of Neural Science 3d ed** Kandel, E.R., Schwartz, J.H., and Jessell,
T.M., eds. Elsevior, 1991

4. "**Studies of glial cells are in a peculiar
state. . . . It is remarkable that one should have to discuss the performance
of the brain in terms of neurons only, as if glial cells did not exist.
Even if in the end this neglect should turn out to be correct, which we
doubt, one ought to be able to justify the reasons for ignoring the most
numerous cell types in the nervous system.**"

(the "peculiar state" Kuffler et al speaks of persists.)

in Kuffler, S.W. Nicholls, J.G. and Martin, A.R. ** FROM NEURON TO
BRAIN: A Cellular Approach to the Function of the Nervous System 2d ed**
Sinauer Associates, Sunderland, Mass 1984 p 324

5. Showalter, M.R. __Reasons to Doubt the Current
Neural Conduction Model__ available FTP angus.macc.wisc.edu\pub2\showalt