If equations derived according to Maxwell's 1st method are right, inferences from experiment are only valid over a RESTRICTED range.


This paper offers a numerical example in support of two propositions. The first is that our new pattern for deriving coupled equations(1) (2) (3) from physical models fits some evidence(4) (5). The second is that, according to our method, some inferences from experiment that have been generalized in an unrestricted fashion are only valid over RESTRICTED ranges of the dimensional parametric spaces on which they are defined.

We are taking a position that goes against "common knowledge" in mathematics and mathematical physics. We believe our formal point could be easily taught to fresh students. Our point fits evidence. But our point is psychologically hard for people of established experiences and habits of thought, however able they may be(6)

. All of us make assumptions in our patterns of thought. The more well trained and automatic our thought patterns become, the harder it is to question (or even identify) our assumptions(7). Most scientists would be quick to agree to the following admonition by George Johnson(8), taken as an expression of an ideal:

                  " Scientists must constantly remind themselves that the map is not the territory, that the                    models might not be capturing the essence of the problem, and that the assumptions built                    into a simulation might be wrong.(9) "

Even so, most scientists would make exceptions, including some unconscious exceptions(10), and would elevate some "universally acknowledged" ideas and patterns to the status of unquestionable articles of faith, no longer subject to reconsideration. For instance, our basic mapping patterns at the interface between measurable physics and mathematical representation seem beyond doubt. Who can imagine, after so much work and experience, that these usages could be wrong? One might as well question arithmetic. Even so, we have found that the mapping usages between physical map and analytical territory do contain a mistake, and in particular an error in the usage of arithmetic. A rule restricting arithmetic with dimensional parameters has not been understood. As a consequence, invalid arithmetic has sometimes been accepted, and has sometimes misled us.

We have faced the following strongly felt but false inference repeatedly:

"If the procedural change that Showalter and Kline suggest is right, wouldn't it invalidate many results that everyone has good reason to trust?"

The question is reasonable. An affirmative answer seems self-evident to many. Nonetheless, the answer is no. The error that we have found can be important (under conditions where our current models do not work well) and yet seems totally insignificant in the cases where our models already work well.

This paper offers an example, the conduction equation. We've derived a new conduction equation, with new terms. In the cases where the old derivation works well empirically, the new procedure works just as well. The new terms are too small to matter. In the case of neural function, where the old derivation fails, the new derivation has the properties needed to describe behavior. The same equation, with the same terms, has enormously different properties depending on what the numerical values of the dimensional parameters in it happen to be.

We have shown that differential equations derived from coupled physical models may have crossterms, now thought to be infinitesimal, that are finite(11). The transmission line equations, derived elsewhere(12), are scientifically interesting examples. A partial expansion of one of these equations follows:

R is resistance per length, L is inductance per length, G is membrane conductance per length, and C is capacitance per length.

For many values of R-L-G-C, all the terms in (1) that include products of R, L, G, and C will be very small. However, some of these crossproduct terms can be enormous for different values of R, L, G, and C. Comparison of a wire case and a neural dendrite case shows the contrast in values that can occur.

For a 1 mm copper wire with ordinary insulation and placement, typical values of the dimensional parameters would be:

          R = .14 x 10-2 ohm/m                 C = 3.14 x 10-7 farads/m

          G = 3.14 x 10-8 mho/m                L = 5 x 10-7 henries/m

Here is (1) with the numerical value of terms set out below the symbolic expression of the terms:

Here are the corresponding dimensional parameter values for a 1 micron diameter neural dendrite, assuming accepted values of axolemma conductivity, capacitance per membrane area, and inductance per unit length (volume conductivity 110 ohm-cm g = 3.18 x 10-5 mho/cm2 c=10-6 farads/cm2.)

                        R = 1.4 x 1012 ohm/m                        L = 5 x 10-7 henries/m

                        C = 3.14 x 10-8 farads/m                   G = 3.14 x 10-6 mho/m

Note that R is 1011 larger than in the previous case of the wire.

Crossproduct terms that were trivial in the wire case become dominant in the neural dendrite case.

For the wire case, the numerical values of the primary terms (the numerical values of the dimensional parameters R, L, G, and C) are compared with the numerical values of the numerically most important crossproduct terms in the voltage gradient transmission equation below.

None of the crossproduct terms are big enough to attend to in a practical model equation. The sensible formula to use for this particular set of values of R, L, G, and C is the one Lord Kelvin derived in 1855:

In the wire case the crossproduct terms are finite, but operationally negligible, and undetectably small.

For the neural dendrite, some of these crossterms become dominant terms. The numerical values of the primary terms (the numerical values of the dimensional parameters R, L, G, and C) are compared with the numerical values of the numerically most important crossproduct terms in the voltage gradient transmission equation below.

For these neural values of R, L, G, and C, two crossproduct terms that were too small to attend to in the wire case have become dominant terms. Magnetic inductance, L, has become too small to include, because for these values of R, L, G, and C, the R2C crossproduct term is 3 x 1018 times bigger than the first order L.

Current neurological modelling of dendrites uses the same equation that models the wire.  This radically understates effective inductance. Sensible modelling formula for the dendrite involves the following quite different choice of terms. (Note that this equation is a selection of important terms among others; it is an approximation that approximates reality well enough in a particular regime of R-L-G-C.)

The effective inductance is 1016 times larger than in the Kelvin (or Kelvin-Rall) equation. would be

Lines that have been thought devoid of inductance, and incapable of inductive effects such as resonance, have very large effective inductance.

The notion that solutions are parameter dependent is well established in some fields, for example viscous fluid flow. That notion applies to these new terms. In the neural transmission case and elsewhere, crossterms now dismissed as infinitesimal are finite, and can be large. Electrical measurements testing the conduction equations have been carefully and accurately done, over a limited range of parametric values. They have been "experimentally perfect" over that range. Nevertheless, too large an extrapolation beyond that tested range of parametric values can be treacherous. Because the dimensional parameters that we use in our physical representations operate with type-restricted arithmetic rules, terms we have thought to be zeros are finite. We must take this into account when we extrapolate an equation that may be "a perfect match to experiments" in one range of parameters into some very different range of parameters.

M. Robert Showalter

Madison, Wi. USA

Stephen Jay Kline

Stanford, Ca. USA




3. Showalter, M.R. A (1997) A new passive neural equation. Part a: derivation.

4. Showalter, M.R., and Kline, S.J. C (1997) Equations derived by Maxwell's first method restrict the range of applicability of inferences from experiments.


6. Mill, J.S. (1869) THE SUBJECTION OF WOMEN from HISTORY OF IDEAS ON WOMAN, R. Agonito, ed. Putnam, 1977. Here are interesting excerpts, that still seem fresh today, that do not establish the rightness of any particular questioning of "common knowledge," but that do set out difficulties that go along with that questioning:

" In every respect the burden is hard on those who attack an almost universal opinion. They must be very fortunate as well as unusually capable if they obtain a hearing at all. They have more difficulty in obtaining a trial, than any other litigants have in obtaining a verdict. If they do extort a hearing, they are subjected to a set of logical requirements totally different from those exacted from other people." . .

" The difficulty . . . exists in all cases in which there is a mass of feeling to be contended against. So long as an opinion is strongly rooted in the feelings, it gains rather than loses in stability by having a . . . weight of argument against it. For if it were accepted as the result of argument, the refutation of the argument might shake the solidity of the conviction; but when it rests . . . on feeling, the worse it fares in argumentative contest, the more persuaded its adherents are that their feeling must have some deeper ground, which the arguments do not reach; and while the feeling remains, it is always throwing up fresh entrenchments of argument to repair any breach made in the old.

". . . a cause supported on the one hand by universal usage, and on the other hand by so great a preponderance of popular sentiment, is supposed to have presumption in its favour, superior to any conviction which any appeal to reason has power to produce in any intellects but those of a high class.

" I do not mention these difficulties to complain of them, first, because it would be useless; they are inseparable from having to contend with people's understandings against the hostilities of their feelings and practical tendencies: and truly the understandings of the majority of mankind would need to be much better cultivated than has ever yet been the case, before they can be asked to place such reliance in their own power of estimating arguments, as to give up practical principles in which they have been born and bred and which are the basis of much of the existing order of the world, at the first argumentative attack which they are not capable of logically resisting. I do not quarrel with them for having little faith in argument, but for having too much faith in custom and the general feeling."

7. Kuhn, T.S. (1962) The Structure of Scientific Revolutions U.of Chicago Press, Chicago Chapter 6.

8. See http://www.santafe.edu/~johnson and Science Forums for THE NEW YORK TIMES at http://www.nytimes.com/.

9. Johnson, G. (1997) Proteins Outthink Computers in Giving Shape to Life NEW SCIENTIST March 25 1997.

10. Johnson, G. (1996) FIRE IN THE MIND: Science, Faith, and the Search for Order Vintage, N.Y. p. 200.

11. Showalter & Kline A pp 11-16.

12. Showalter A.