A new passive neural equation. Part a: derivation
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 lack significant terms(3) (4). Characteristics of a more correct set of conduction equations (that we call Showalter-Kline, or S-K equations) are derived, discussed, and compared to the K-R equations according to a long-established electrical engineering analysis. For very low values of membrane conductance, g, and small diameters, the S-K and K-R equations predict very different neural properties discussed in detail in companion papers.
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.
We have found that these equations lack crossterms because special restrictions on the use of dimensional parameters in coupled finite increment equations have not been understood2, 3. The crossterms, some of which can also be derived (within a scale constant) by standard linear algebra based circuit modelling (Showalter, in preparation), are negligible in most engineering applications. But the crossterms shown here are very large in the neural context. The S-K equations are of the same form as the standard conduction equations of electrical engineering and are written as follows.
For set values of R, L, G, and C, these equations have the standard electrical engineering solutions. The hatted values are based on a notation adapted to crossproduct terms. In this notation, the dimensional coefficients are divided into separate real number parts (that carry n subscripts) and dimensional unit groups, as follows.
For wires,, the crossproduct terms are negligible, and the two kinds of equations 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.
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.
Under S-K, attenuation of waves per wavelength or per unit distance varies over a much larger range than occurs under K-R. 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. 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.
The standard engineering
analysis of a transmission line will be used in this paper for both K-R
and S-K, to provide direct comparisons between the predictions of these
different neural conduction models. The parabolic
equations of K-R are subsets of the usual hyperbolic equations of
conduction that occur when L=0 and the terms in L are deleted.
Setting L=0, the customary hyperbolic equations can be applied
to the K-R as well as S-K.
Transmission line analysis describes the propagation of a fluctuating voltage (or current) source along a line in terms of two variables, attenuation per unit distance, , and a propagation number, , and radian frequency . These variables determine propagation velocity, wavelength, attenuation in one wavelength, and other values. For example, in the standard electrical engineering formality, voltage proceeding in the positive and negative directions as a function of distance and time is
where Vo is maximum voltage at distance zero, and
plusses and minuses denote conduction up and down the line.
In this standard electrical engineering analysis, and are the real and imaginary parts of a complex characteristic number , as follows:
as a polar expression in the imaginary plane this is:
The radical part of the expression corresponds to a radius. The exponent on e, raised to an imaginary power, corresponds to a rotation in the imaginary plane. The quantity
is an angle in the complex plane. In K-R the range of is from
0 to 45 degrees. In S-K, the range of is from 0 to 90
The attenuation per unit distance, is
An n neper decay is a decay factor of e-n. When
approaches /2 (90 degrees) cos(), and hence attenuation,
approaches 0. Wave phenomena become important under these circumstances,
which occur in S-K but not K-R.
The wave propagation number, , is
Important relations are:
Attenuation in one wave length is
Consider a neural (dendritic) passage in terms of its defining variables, conductor diameter, membrane thickness, membrane dielectric constant, and axolemma conductivity,
Including electromagnetic inductance per length, lm, the radical is:
in the neural case, lm is negligible compared to the other parameters, and is neglected. Setting lm to 0, the radical is
The case where membrane conductance, g, is nearly zero has special, logically interesting properties. Setting g to 0, the radical becomes:
The S-K formula for angle is
theta can approach a maximum value of /2, a value that corresponds to 0 attenuation.
These relations are different from those of K-R, which are set
out in terms of the intensive variables below.
ThetaK-R approaches a maximum of /4, so that attenuation in one wavelength is always 2 nepers or more.
For S-K in the low g regime, the following are very useful and quantitatively accurate simplifications. For wavelength:
For attenuation per wavelength:
When the fourth root C3 term in (16c) dominates the radical, conduction velocity becomes independent of frequency(5), and simplifies to
Dendritic neural passages will have impedance
in the electrical engineering sense. The standard models
for reflection at impedance mismatches that apply in electrical transmission
lines also apply to neural transmission under S-K.
Impedance may be defined in the usual way, but with hatted values substituted
for the unhatted ones.
Impedance has the units of lumped resistance, as in coax and other wave carrying lines. The interface between two sections of transmission line will generate reflections unless their impedances are matched. For a line with an impedance Zo terminated in another line (or lumped resistance) called the "load", and having an impedance Zl, the reflection coefficient Kl (the ratio of reflected to incident voltage at the load) will be
`Reflection coefficient will vary from -1 to +1. When Zl>>>Zo then Kl is approximately 1; when Zl<<<Zo then Kl is approximately -1; when Zl = Zo then Kl is 0, and there is no reflection. Similar reflection rules are familiar in optics and acoustics.
1. Department of Curriculum and Instruction, School of Education, University of Wisconsin, Madison Wi. USA email: firstname.lastname@example.org
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, Md. (1977).
3. Showalter, M.R and Kline, S.J. Modeling of Physical Systems according to Maxwell's First Method available FTP and on the web
4. Showalter, M.R. and Kline, S.J. Equations from Coupled Finite Increment Physical Models Must be Simplified in Intensive Form available FTP and on the web
5. Showalter, M.R. A New Passive Neural Equation. Part a: derivation Fig. 4. available FTP angus.macc.wisc.edu/pub2/showalt