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 K-R and the high dissipation, high g, mode of our equation.) As information accumulates, K-R requires the attribution of fancier and fancier, less and less plausible, channel and synaptic function. Our equation explains the same results more simply.

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 10¹⁰-10¹⁸:1. 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:

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