**REASONS TO DOUBT THE KELVIN-RALL TRANSMISSION EQUATION**

**M. Robert Showalter ** (1320
words, including footnotes )

Sir:

The Kelvin-Rall (**K-R**) neural conduction equations have been "proved"
mathematically, but reasons to doubt them have been accumulating. The "proof"
of **K-R** includes a common but unjustified assumption^{(1)}
^{(2)}. The correct neural conduction equation
contains additional terms. I review empirical-theoretical reasons to doubt
**K-R** here.

The **K-R** equation is the standard electrical line conduction equation
stripped of terms in electromagnetic inductance. The **RC** line that
remains has conduction velocity that varies with the square root of (fourier
component) frequency. Action potential waveforms have widely distributed
fourier components. Under **K-R** there is no way to propagate the coherent
action potential waveforms observed, even with uniform channel populations
boosting the wave. The heterogeneous and lumpy channel distributions observed
by Zecevic^{(3)} in axons showed conventional
action potentials. Action potential dissipation along the axon segments
with few channels, and dissipative reflections between low and high channel
density segments would be expected under **K-R**.

Large populations of neurons in the brain fire in synchrony. The coordinated
firing occurs over such long distances that the synchrony observed cannot
be due to conduction. Coordinated firing of neurons over (apparently impossibly
large) distances seems to be associated with logically coordinated function^{(4)}
^{(5)} ^{(6)} of
the neurons involved. Oscillation patterns shift as logic shifts. The epileptic
diseases involve especially intense, uncontrolled, rapidly varying synchronies.
The diverse and rapidly changing synchronies observed seem beyond the capacity
of conductive networks that have been proposed^{(7)}
under the most ideal theoretical circumstances that might be proposed for
an ideal "brain". But anatomy and physiology are not ideal in
this sense. By the logic of **K-R**, information flows only move through
connected axons and dendrites. These connected lines have a wide range
of conduction speeds, usually below 1 meter/second, and the lines trace
various, more or less convoluted paths. Different paths have different
numbers of synapses with different synaptic lags. Such a system seems designed
to produce dispersion, not the synchrony that is observed.

**K-R** also predicts that individual neurons are slow. From resistance
measurements, **K-R** predicts (1/e) time constants of 20-100 msec or
more for retinal, olfactory, and other cells^{(8)}
^{(9)}. These predicted time constants are
far too long to match behavior. 1 millisecond time constants are about
the fastest the **K-R** model will generate for any single cell. However,
people discriminate sounds less than .020 msec apart. Bats discriminate
sound waves less than .0005 millisecond apart^{(10)}.

By the logic of **K-R**, **all** the perceptual and logical performances
that people and animals accomplish must be controlled by information flows
moving through connected axons and dendrites only. For complicated tasks,
this can be implausibly slow. Moreover, if the brain is such a connection
constrained system, it is subject to the combinatorial limitations that
have been intractable for the much smaller "neural networks"
modelled on (much faster cycle time) computers.

**K-R** is particularly dissonant with our knowledge of brain's ability
to handle information in temporal and frequency form. Electrical oscillations
in brain are significant, and can carry information^{(11)}.
People are so adapted to high information content oscillatory signals that
we remember and can generate words and popular songs. No way has been suggested
under **K-R** to send, receive or encode such detailed oscillatory information.

Advances in visualization and measurement^{(12)}
in the last few years have greatly increased our knowledge of neurons.
Sejnowski says that "more has been learned about the secret life of
dendrites in the past year than in all previous years^{(13)}."
Koch speaks of an ongoing "revolution" in neuroscience^{(14)}.
As information accumulates, the theoretical burden on dendrite channels
and synapses has become very heavy. Until a few years ago, channels were
understood to augment a propagating signal by the stereotyped, all-or nothing
action potential mechanism. This is complicated. But to explain dendritic
conduction within the framework of **K-R** requires more. One must posit
high and uniform densities of very special channels that somehow proportionately
boost the analog signals propagating along the dendritic lines. The existence,
high density, and uniformity^{(15)} of these
__special__ channels remain subject to question.

However, low dissipation dendritic conduction has been measured by many
workers. An especially clear example is Fig. 1 of Haag and Borst^{(16)},
that shows high fidelity dendritic conduction of an analog signal in a
dendrite. This is "not compatible with an electrically passive neural
membrane" under the **K-R** model. In Fig 3b they show a frequency
dependent curve with an amplified peak. They infer very complex channel
behavior from this technically impressive data. **However, both the low
dissipation conduction and the resonance-like amplification of the dendritic
(antenna shaped structure) follow directly according to the differential
equation we derive ^{(17)} ^{(18)},
without any need for channel activity.** (Haag and Borst's Fig 3a is
consistent with both

An additional reason to doubt **K-R** compels me more than any other.
David Regan^{(19)} used the zoom FFT technique
on human and animal electroencephalographs and magnetoencephalographs.
He used repeated evoked stimuli of several modalities. His data showed
peaks organized in the integer multiple sums and differences characteristic
of resonance. His EEG and MEG data measured neural population behavior
over millions of neurons. Even so the integrated effect peak bandwidths
he shows are as tight as he can measure (.002 Hz or less). Kline and I
concluded that brain had to be an assembly including large populations
of very high Q resonant structures coupled by the waves that the EEG was
measuring. Consulting anatomy, we had to assume that either some short
dendritic sections or some dendritic spines (or both) were switched elements
with sharply resonant states. Regan's data implied that the effective inductance
predicted by the presently accepted **K-R** equation was too small by
factors of 10^{10}-10^{18}:1. These were the same factors
that the commonly observed coherent propagation of action potentials required.
They are the factors that our new conduction equation predicts at dendritic
scales.

**K-R** has been doubted before. Lieberstein also suggested that
inductance was important in neural transmission^{(20)},
largely because many neural waveforms look like reflection traces on electrical
transmission lines^{(21)}. The matter was
investigated^{(22)} ^{(23)},
and no plausible source of the inductance required was seen. The derivation
procedure used by Kelvin and Rall is a standard one, and agrees closely
with experiment in wires, muscle fibers, and large axons. Kline and I show
that this derivation is incomplete, and that the proper derivation of the
neural conduction equation includes new terms, including a large new inductance,
that vary as the inverse cube of diameter. Predicted conduction has two
modes, a high **g** mode very much like **K-R**, and a low **g**
mode with low dissipation that is adapted to resonant effects.

Madison, Wisconsin

showalte@macc.wisc.edu

**NOTES:**

1. Showalter, M.R., Kline, S.J. __Modelling of Physical
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2. Showalter, M.R. and Kline, S.J. __Equations from
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Form__ available FTP
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__NATURE__ **383** 621-624 (1996).

8. Woolf T.B., Shepherd, G.M., & Greer, C.A.
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10. Moss, C.F. & Simmons, J.A. in ** NEUROETHOLOGICAL
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17. Showalter, M.R. __A New Passive Neural Equation.
Part a: derivation__. available FTP
.

18. Showalter, M.R. __A Passive Neural Equation:
Part b: neural conduction properties__ available FTP

19. Regan, D. ** HUMAN BRAIN ELECTROPHYSIOLOGY**
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20. Lieberstein, H.M. __Mathematical Biosciences__
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21. Stephenson, D.T. __"Transmission Lines"__
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22. Kaplan S. & Trujillo D. __Mathematical
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23. A.C. Scott __Mathematical Biosciences__ **11**,
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