A new passive neural equation.
Part b: neural conduction properties
M. Robert Showalter(1)
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 K-R equations are the standard conduction equations of electrical engineering, usually written in a contracted form that discards the terms in L that are negligible compared to R, C, and G in neural cases. The S-K equations have the same form, and are written below.
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 1012 times what we now assume it to be under K-R. The S-K equation predicts two modes of behavior.
When G is high (some channels are open) behavior similar to that of the K-R model is predicted.
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.
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.
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/cm2 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/cm2. 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)
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
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/cm2 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
sigma= 50 ohm-cm capacitance 10-6 farads/cm2 g=10-9 omega=900
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
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.
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 .