**A new passive neural equation.
Part b: neural conduction properties **

M.
Robert Showalter^{(1)}

** The now-accepted
model of passive neural conduction (the Kelvin-Rall or K-R model ^{(2)})
is based on differential equations of conduction that are incomplete. Characteristics
of a more correct set of conduction equations (that we call Showalter-Kline,
or S-K equations^{(3)}) have been set out in
Part a. Neural conduction consequences of the S-K equation
are graphed and discussed here. According to the S-K model, there is a
high G regime with characteristics qualitatively and quantitatively similar
to the characteristics of the K-R theory. There is also
a low G regime characterized by wave propagation with low attenuation and
small or negligible phase distortion. Very small neural lines seem well
adapted for information processing: they have an on and off state, and
in the on state are characterized by low enough dissipation for strong
resonance on micron scales. When groups of channels open
abruptly, when there is abrupt crossection change, or when anything else
produces abrupt changes in neural line impedance, S-K predicts reflection
effects in neural lines. ** The

The difference between the **S-K** and **K-R** equations is the difference
between the hatted values of **R**, **L**, **G**, and **C**,
that include significant crossproduct effects, compared to the unhatted
values of **K-R**. In the hatted notation, the dimensional coefficients
are divided into separate real number parts (that carry **n** subscripts)
and dimensional unit groups.

For wires, the crossproduct terms
are negligible, and **S-K** and **K-R** are the same. **But
under neural conditions the crossproduct terms are LARGE.
For instance, effective inductance is more than 10 ^{12} times what
we now assume it to be under K-R. **The

When G is high (some channels are open) behavior similar to that of the

When G is low, transmission has very low dissipation, and the system is adapted to inductive coupling and resonance.

Basic similarities and differences
between **S-K** and **K-R** are shown below. The biologically
important case of the limit where membrane conductance approaches **0**
has been emphasized.

In
**S-K**, attenuation of waves per wavelength or per unit distance varies
over a much larger range than occurs in **K-R**. In
**S-K**, there is a low membrane conductance regime where attenuation
of waves is small, and wave effects are predicted. However,
as channels open attenuation increases enormously, and waves may be damped
out in a few microns. Under **S-K** a neural passage can
be either sharply "on" or sharply "off" depending on
the degree of channel-controlled membrane conductance.
**In the high g regime, attenuation per wavelength values are qualitatively
similar for K-R and S-K.**

Figures 1, and 2 plot unit wave amplitude
after one wavelength (left axis) or damping exponent per wavelength (right
axis) as a function of membrane conductance, **g**, for both **K-R**
and **S-K**. The curves map functions that move rapidly - an exponent
is mapped in log-log coordinates.

Figure
1 plots calculated responses at the low frequency of 10 radians/second
for neural process diameters ranging over five decades (from 1000 microns
to .1 micron). For the 1000 and 100 micron cases the
**K-R** and **S-K** curves are almost identical.
Results for these large diameters and low frequencies are also nearly identical
on an attenuation per unit length and a phase distortion basis. These
conditions correspond to some of the most influential and carefully done
experimental tests of **K-R**.

For smaller diameters,
attenuation according to **S-K** is much less than that according to
**K-R** in Figure 1.

Figure 2 plots calculated
responses at 10,000 radians/second (1591 Hz) for the same diameters plotted
in Figure 1. Attenuation values are substantially less
for **S-K** than for **K-R** even for the 1000 micron diameter case.
For **K-R**, the value of attenuation exponent per wavelength never
falls below 2. This means that the **maximum** amplitude
of a wave after 1 wavelength of propagation is .00187 (about 1/535th) of
its initial value under **K-R** for any diameter neural process.
It makes little sense to talk of "wave propagation" and no sense
to talk about "resonant responses" under these conditions.
According **S-K**, as much as 99.995% of unit wave amplitude may remain
after a single wavelength. Under these very different
conditions, notions of wave propagation and resonance do make sense.

Figure
3 shows calculated conduction velocity versus membrane conductance, **g**,
plotting diameters .1, 10, and 1000 microns versus frequencies of 10, 1,000,
and 100,000 radians/second (1.59, 159.1, and 15,915 Hz respectively) .
Velocity is shown in centimeters/second and in diameters/second.
As before, membrane capacitance is 1 microfarad/cm^{2} and axolemma
resistance is 110 ohm-cm for all curves. A number of trends stand out.
In the low **g** range, velocity expressed in diameters/second is invariant
with diameter according to **S-K**. In the much-studied
(giant squid sized) 1000 micron diameter, low frequency case, the **K-R**
and **S-K** results are very similar over a wide range of **g**'s.
However, for any given diameter, propagation velocity in the low **g**
range is independent of frequency for **S-K** but not for **K-R**.

The **K-R** and **S-K**
velocity-**g** characteristics are most different in the low **g**
range. According to **K-R**, increases in **g**, beyond a certain
value, sharply increase conduction velocity. According to **S-K**, increases
in **g** beyond a threshold decreases conduction velocity.

Fig.
4 plots conduction velocity versus frequency for a 1 micron dendrite or
spine neck versus frequency in the limiting case where membrane conductance,
**g**, is approximately zero. The **S-K** model is very different
from the **K-R** theory. In **K-R**, conduction velocity
is a rapidly increasing function of frequency. In the **S-K** model,
conduction velocity is a strong function of frequency at extremely low
frequencies, but rapidly and asymptotically approaches a peak velocity,
so that above a frequency threshold conduction speed is substantially invariant
with frequency. For large diameter neural processes,
this threshold is so high that the velocity-frequency relation is similar
to that of **K-R**. But for small neural processes, velocity is essentially
constant above quite low threshold frequencies. The following
chart is based on axolemma conductance of 110 ohm-cm and a membrane capacitance
of 1 microfarad/cm^{2}. Note that for a .1 micron dendrite,
99.99% of peak velocity occurs at .0511 cycles/second.

**diameter
Frequencies for the following fractions of**

**(microns) Peak
velocity (radians/second)**

** 95%
99% 99.99%**

**1000 1,320
3,160
32,120**

** 100 132
316
3,212**

** 10 13.2
31.6
321.2**

** 1 1.32
3.16
32.12**

** .1
.132
.316 3.212**

The velocity-frequency
relation is important because of its connection to phase distortion. Phase
distortion occurs when different frequency components of a signal move
at different speeds. Phase distortion degrades the information content
of a signal, and limits the signal processing techniques that are possible.
"The 99% velocity cutoff frequency according to the **S-K** model
offers a good basis for comparing phase distortion predictions between
**K-R** and **S-K**. For **S-K**, in the low **g** limit, a
dendrite or axon will have a characteristic frequency _{99} that
has 99% of maximum propagation velocity. Above _{99}, propagation
will be almost free of phase distortion. _{99} correlates with
diameter, conductivity, and membrane capacitance according to the relation

The **S-K** model is most interesting in its low attenuation
mode. This may be described as an "on" state,
in contrast to the high **g** high attenuation "off" state.
In the "on" state, system characteristics are close to those
predicted in the limit as **g** approaches 0. In this
"on" condition, important relationships are simple, particularly
for small neural diameters.

In low attenuation mode,
impedance behaves as it does in other transmission lines and transmission
media. Impedance jumps in neural passages due to changes
in **g** are analogous to similar sharp impedance jumps that occur in
wind musical instruments. Wind instruments work because
they are abruptly switchable impedance mismatch devices driven by cyclic
energy input means that adapt a fluctuating input flow to match air column
load. A sharply defined quasi-steady resonance is established
in the musical instrument's air column. Because the reflection
coefficient at the discontinuity is not perfect, some of the sound from
this column radiates to listeners of the music. Analogies between
the acoustical function of wind instruments and neural lines are useful
when considering two linked issues.

1) A sharp change in membrane conductance, **g**, (whether
"inhibitory" or "excitatory") can produce reflection
as well as attenuation. If the change is relatively large, this reflection
can be both sharp and strong.

2. Because reflecting discontinuities can be set up by
synaptic or dendritic spine activity, the possibility that a neural structure
may be an "abruptly switchable impedance mismatch device" with
variable resonance properties under some conditions should be considered.

Consider a dendrite of 2 microns diameter, with a group
of channels of one hundred 200 picosiemen channels over a 1 micron axial
distance. Conditions are:

sigma=110 ohm-cm capacitance/area=10^{-6}
farads/cm^2

**g _{closed channel}**=10

When the channels
are closed, they do not effect conduction. Suppose that
the channels open so that **g** goes from 10^{-12} to some much
higher value. See Fig 5, which plots a change for reflection
coefficient from 1.0 to less than .0025. Fig 5 was calculated
for** **= 10, but values for much higher or substantially lower frequencies
would be about the same. Note that the difference in reflection
coefficient that occurs between 10^{-12} and 10^{-6} mhos/cm^{2}
is small, but that very large changes in reflection coefficient are calculated
for higher conductances. Opening a large number of channels in the side
of a dendritic passage can change the channel from a transmission line
to a sharply reflecting discontinuity. The physics is analogous to that
which occurs when a clarinettist opens or covers a finger hole.

Abrupt changes in crossection are also calculated to produce
analogous reflecting impedance mismatches. Fig 6 shows reflection coefficient
at a discontinuity between a smaller and a larger diameter. Calculated
conditions are:

sigma= 50 ohm-cm capacitance 10^{-6}
farads/cm^{2 }
**g**=10^{-9} omega=900

**d**_{1}=1 micron

The shape and slope of this function is insensitive to
changes in frequency, initial diameter,** **, **g** and **c**.
Abrupt changes in crossection can produce strongly reflecting impedance
mismatches.

When changes in
crossection or impedance are gradual, they can occur with little or no
reflection, for reasons exactly analogous to those that permit smooth transitions
in the bells of wind instruments or the gradual impedance (refraction)
transitions that can be arranged in optics and in waveguide practice.

**NOTES:**

1. Department of Curriculum and Instruction, School of Education, University of Wisconsin, Madison, USA

2. Rall, W. "__Core conductor
theory and cable properties of neurons__" ** Handbook of Physiology
- The Nervous System v 1**, Ch. 3. Williams and Wilkens, Baltimore,
(1977).

3. Showalter, M.R. __A New Passive
Neural Equation. Part a: derivation__. available FTP
and on
the web .