A Modified Equation for Neural Conductance and Resonance

M. Robert Showalter

University of Wisconsin, Madison

A modified equation, the S-K equation, fits data that the current neural conduction equation, the K-R equation, does not. The S-K equation is a modified Heaviside equation, based on a new interpretation of cross terms.

Recently, a Japanese television cartoon broadcast bright and repetitive T.V. screen flashes. Many hundreds of children and adults who saw the cartoon had epileptic seizures that resulted in hospital visits(1). More than ten thousand children, and many adults, seem to have been affected(2). The cartoon makers accidentally repeated experiments that are discussed below(3). None of the children who watched the cartoon seems to have been permanently hurt. Even so, this event underscored again the need to reconsider the differential equation now used to describe passive neural conduction.

Neurons acting together in brains seem to have enormous inductance. That has been recognized and discussed for more than thirty years(4) (5). Even so, the equation now used to model these neurons, the Kelvin-Rall (K-R) equation, has no inductance.

                       (x is position on the conductor line, v is voltage, i is current, R, C and G are resistance, capacitance,
                                           and membrane leakage conductance, per unit length.)

Reasons to doubt K-R empirically have been accumulating(6) . K-R has never fit data on brain scale neurons well. However, K-R has been accepted because it has seemed to be based on a "mathematically unquestionable"derivation(7). Perhaps because of difficulties with K-R, neurophysiology is in a divided state. The part of neurophysiology and neural medicine that can be connected to biochemistry and genetics is in a rapidly advancing state, and has many impressive achievements that accumulate year by year. On the other hand, the part of neurophysiology and neural medicine that depends on an understanding of neural electricity (the part that depends on K-R) is in a primitive state, with little progress to show for years of hard effort. As reviewed below, the K-R equation does not fit essential facts about the electrical activity of brain. Nonetheless, the K-R derivation is based on an "inescapable" assumption about mathematical representation. This "inescapable" assumption has no provable basis from axioms, and fails some consistency tests. Based on another assumption, which is also unprovable but that is consistent (Appendix 2), a new transmission equation with large effective inductance is derived. Let's call the equation based on the new assumption the Showalter-Kline (S-K) equation after myself and Professor Stephen J. Kline, of Stanford, who worked with me on the derivation of this equation and its justification for many years before his death.

Numerical parts of R, L, G, and C are denoted by subscripts. The dimensions of the quantity are the dimensions (voltage, v; time, t; charge, Q; and distance along the line, x) in a compatible unit system (MKS).

The S-K equation is the standard electrical engineering transmission equation, the Heaviside equation(8), with the hatted values of resistance, inductance, membrane conductance, and capacitance substituted for the usual ones. For the values of R, L, G, and C that are common in engineering experience (that correspond to wires), numerical predictions of S-K are indistinguishable from the predictions of the Heaviside equation. For the very different values of R, L, G, and C that apply to neurons, the effective inductance of S-K can be 1015 or more larger than the Heaviside equation predicts. S-K fits much neural data well. Let's review some interesting data that motivates the S-K equation, and then review S-K's derivation, and how that derivation differs from the derivation of K-R. Then the very good fit between neural data and S-K will be illustrated.


Last December, a popular Japanese television cartoon broadcast bright and repetitive T.V. screen flashes. According to The New York Times, many hundreds of children and adults who saw the cartoon had epileptic seizures that resulted in hospital visits throughout Japan. More than ten thousand children, and many adults, seem to have been affected(9). Tests since have shown that the flashing TV screen stimulus does produce seizures and other ill effects in many people.

The flashing TV screen the cartoon employed was the same stimulus used by electrophysiologist David Regan a decade ago. Reagan studied brain responses to these flashes (and other repetitive stimuli) using very sensitive zoom FFT electroencephalogram (EEG) and magnetoencephalogram (MEG) tests. EEG records voltage fluctuations of electrodes on scalp, face and head. MEG records fluctuations in the head's magnetic field.     Figs 1(10) shows some of Regan's EEG measurements using his zoom FFT technique.    The behavior shown is typical of that measured, magnetically or electrically, in response to many kinds of repetitively presented stimuli (light, touch, sound, etc.) For Fig 1 the stimulus was a TV screen flashing at two superimposed frequencies F1 and F2.

Fig. 1

The data show sharp peaks at sums and differences of excitation frequency. This kind of signature characterizes resonant systems, with very low damping, that include many passive resonators. It is "as if" the brain is resonating. Resonance would require inductance that the K-R equation lacks.

Fig 1 shows much-repeated results from Reagan's tests, for several kinds of stimuli, for both EEG and MEG. In all these tests, Regan was measuring scalp electrodes in EEG, or whole brain magnetic fields in MEG. He was measuring activity integrated over substantial volumes of brain. Large volumes of brain, including at least millions of electrically coordinated cells, had to be "resonating."

In tests similar to those of Fig. 1, Regan measured peak bandwidths narrower than .0019 Hz.    Bandwidths of the peaks measured in Fig 1 may have been as sharp as this. These are astonishingly sharp peaks, orders of magnitude too sharp to be accounted for by membrane channel activity. The peaks are so sharp that elements "resonating" together must have been coupled with very small lags. The coupling Reagan measured is much sharper than any possible if there were significant conduction time lags between reacting elements. This rules out coupling of elements by conduction along neural lines. According to K-R, conduction is the only source of coupling between neurons.

Reagan's results have recently been seen again, under less controlled circumstances. We now know that such repetitive stimuli, delivered at higher intensity than Reagan used for shorter times, can evoke epileptic seizures in populations of children and adults.

Regan's data shows electrical coordination over volumes of brain with special clarity. Electrical coordination of large volumes of brain may be THE central result of brain electrophysiology, and has been known for a century. When one looks at the coordinated waveforms that happen over volumes of brain in health or disease, they almost always seem to happen impossibly fast (if you ask them to happen by conduction, that is, if you rule out inductance.) K-R has no inductance.

Epilepsy resembles electrical instability of the brain in resonance. That is medically interesting. Another reason to be interested in brain resonance is information processing. If one follows analogies between human engineering and brain, resonance may also be connected with how the brain processes information. Resonance is logically interesting. Enormous resonant magnifications of tightly selected signals are possible. In this sense, resonant systems can function as highly selective amplifiers. This fact is a foundation of communication technology. Radio and television offer familiar examples of resonant selectivity. Radio and television receivers exist in an electromagnetic field consisting of a bewildering and undescribable variety of electromagnetic fluctuations. Reception occurs because the resonant receiver is selective for a specific frequency at a high degree of phase coherence. Signals off frequency are not significantly detected, and "signals" of random phase that are on frequency cancel rather than magnify in resonance. Radar receivers also operate on the principle of resonance. Other examples are our telephone system and cable television system, each organized so that a multiplicity of different signals can be carried in physically mixed form over the same conduits. These "mixed" signals can be separated and detected with negligible crosstalk by resonant means. Based on S-K and anatomy I've speculated that high sensitivity passive resonant brain elements might work as compact passive memory elements, consuming no energy until excited. Membrane channel means to switch resonant sensitivity on and off would provide much logical capacity. Reagan's measurements did not measure any such information processing in brain, but perhaps he did measure resonant responses consistent with such an information processing arrangement of brain.

Many neurophysiologists have thought about these matters for years. Few seem to doubt that inductance would explain a great deal "if only the inductance was there.(11)" Nonetheless, the K-R equation is justified by "trusted mathematics." (The derivation was carefully done, under the aegis of the N.I.H., in response to questions about brain inductance. No plausible source of brain inductance was found.) K-R was also derived in careful analogy with electrical engineering experience. According to K-R, neural line inductance is just the same electromagnetic inductance any conducting line would have. In comparison to the huge values of resistance and capacitance that neural lines have, this inductance is negligible, so K-R is written without inductance.

Matching anatomy, one can choose inductances that would produce the behavior Reagan measured by resonant action. The inductances needed are immense - more than 1016 times electromagnetic inductance. The S-K equations have these large effective inductances at neural scales with neural values of R, L, G, and C, and also have other characteristics needed to account for much neurophysiological data, including Regan's. S-K fits the other data Kline and I have looked at, as well. S-K is consistent with the electrical engineering data that we've checked, and is the same as the standard Heaviside equation for usual wire values of R, L, G and C.


Both the K-R equation and the S-K equation derive from the same basic physical model. The derivation of equations from the model differs in two ways:

The S-K derivation writes down crossterms that are not usually written, that the K-R derivation does not treat.

The S-K derivation interprets these crossterms using a new assumption that defines them consistently at finite scale, before taking the limit of these crossterms.

As a result of these differences, S-K includes some physical crosseffect terms that K-R lacks, including an effective inductance term the right size to account for neurophysiological data.

The conductance equations that apply to a line conductor, such as a wire, are the Heaviside equations, standard in electrical engineering:

            v is voltage, i is current, x is distance along the line, and R, L, G, C are resistance, inductance,
                        membrane conductance, and capacitance respectively, all per unit length.

The Heaviside equation is derived by cross substituting dv/dx and di/dx equations (4) that represent the physical definitions of R, L, G, and C. In Kelvin-Rall, the same cross substitution is done.

It is then noticed that in the neural regime, the Heaviside equation (2) so formed has an electromagnetic inductance that is minuscule compared to R, C, and the changeably valued G. So the terms in L are deleted, and the K-R equation remains:

The K-R equations lack the LC term passive electrical resonance requires. Conduction velocity under K-R goes as the square root of waveform fourier frequency component. This means that K-R shows large phase distortion - complex waveforms rapidly change shape, spead out, and lose information as they conduct down a line under K-R.


If equations (4) are exact, then K-R follows. So Kline and I looked again at how these equations were derived from a physical model. We set up equations that corresponded to our physical model at finite scale, including crosseffect terms, that plainly existed at finite scale, but that were not defined in a way that was consistent with the requirements of physical representation. When we defined these crossterms so that they were consistent with physical representation (Appendix 2), the crossterms were finite, and the S-K equation followed.

In a physical model of conduction along a line over an interval, the conduction and voltage equation, set out in terms of the physical laws represented by R, L, G, and C, are each implicitly defined by the other. (Appendix 1) Centering our interval at x, so that our interval goes from x-x/2 to x+x/2, we have the following voltage equation (5):

The current change equation (6) is exactly symmetric:

Equation (5) includes i(x+x/2,t) and its time derivative. i(x+x/2,t) is defined by equation (6).

Equation (6) includes v(x+x/2,t) and its time derivative. v(x+x/2,t) is defined by equation (5).

Equations 5 and 6 each contain the other. Each requires the other for full specification.

If the cross-substitutions implicit in these equations are explicitly made, each of the resulting equations will contain the other in turn. Expression of current, voltage, and their time derivatives at x, the midpoint of the interval, truncates the series at the stage desired.

7 is a voltage difference equation representing three stages of this substitution. The analogous current difference equation would result from swapping of parameters and variables as follows: v-i, R-G, L-C. The length increments (x) within the curly brackets of 7 are set out at subscript level to make eye hesitate. The question "do these increments multiply?" will be raised below. For a consistent physical representation, they cannot.

Deleting crossproducts that happen to be are too small to consider for neural values of R, L, G, and C, for x<=1, 7 reduces to:

the corresponding current equation is:

Suppose we make the following KEY ASSUMPTION:

"the expressions within the curly brackets of equations (7), (8) and (9) are physical interpretations of natural laws that happen to have been "effectively measured" at scale x. To compute a natural law coefficient that fits the term, we convert x to

(1 length unit)

and algebraically simplify on that basis."

If we proceed according to this KEY ASSUMPTION (8 and 9) correspond to the following differential equations:

In the more compact hatted notation, these equations can be written


The S-K equation is isomorphic to the standard electrical engineering transmission equation (3), and follows if (12 ) and (13 ) are crossubstituted according to the usual Heaviside derivation.

For wires and other familiar engineering conductors, the K-R and S-K equations are identical within the limitations of measurement. But under neural conditions S-K and K-R are very different. Effective inductance in S-K is more than 1012 times inductance in K-R. S-K fits much data, and predicts two modes of behavior. When G is high (some channels are open) behavior similar to that of the current model is predicted. When G is low, transmission has very low dissipation, and the system is adapted to inductive coupling effects including resonance. Switching between these two modes is sharp. Considering resonance, we'll look at the case where G is negligible. In this case, to be expected when membrane channels are closed, S-K reduces to:

Equations (14) are familiar textbook equations used in the study of the resonance of electrically conductive lines.

A Question of mathematical modeling:

The S-K equation is dimensionally consistent, and seems to fit data. Many equations used in the sciences have no more foundation than that. Even so, the derivation of K-R was carefully and publically done, according to standard conventions. K-R has been accepted for decades in the neurosciences

To question K-R, one must question how implicit crosseffect terms, that must be finite at finite scale, are to be calculated. The S-K derivation sets out these crosseffect terms explicitly, and calculates them according to a new assumption that interprets their symbols differently prior to an algebraic simplification..

Neither the implicit calculation assumption of the K-R derivation nor the explicit calculation assumption of the S-K derivation can be proved from established axioms of abstract mathematics. The assumption used in K-R is the usual one, and the assumption used in S-K is new. Even so, the assumption used in the S-K derivation survives consistency tests that the K-R assumption does not survive. The derivation of S-K is set out in more detail in Appendix 1, and the key assumption behind it is discussed in more detail in Appendix 2.


The very numerous spiny cells in brain should behave in a manner that generates the kind of behavior that Regan measured if S-K is assumed. If K-R is assumed, brain resonance cannot occur.

Details of resonance that are important to connecting S-K to anatomy are set out in Appendix 3. Some calculations based on these details are set out below.

Assuming S-K, predicted behavior of brain corresponds to what Regan measured. We consider Regan's frequencies of 7-46 Hz, and set peak bandwidth at .0019 Hz. We calculate resonant magnification factors, Q's, of 3680 to 24,200 for the ensembles that represent frequency peaks. These are very high calculated ensemble Q's. Individual resonator Q's cannot be less, but could be higher still. Regan's measurements give upper bounds on bandwidths, lower bounds on Q's.

There are more than 1010 neurons in brain, and roughly 1013 dendritic spines, many shaped like Fig 2 (and many more with calculated frequency responses such as those of Fig 2.) Figure 2 shows an electrical model of a thin spine. In the scaled figure, 4/5 of spine capacitance is in the bag section.

The "spine" of Fig. 2 can be modeled as an LRC resonant system. Capacitance is the capacitance of the "bag" section, plus half the capacitance of the shaft section. The shaft has resistance of R, and an effective inductance of Lex. The Le/R ratio is inverse with diameter. Different bag sizes for the same shaft size yield different LC products, and different resonant frequencies. In the model, bandwidth is proportional to diameter. Let's arbitrarily chose a shaft diameter of .1 µ, shaft length of .5 µ, interspine medium conductance of 110 ohm-cm, membrane capacitance of 1 microfarad/cm2, and zero membrane leakage, g. Holding these values, and varying bag size, yields the following relation between frequency o and Q.

Q = o(910)

For Regan's measured frequency range of 7-45 hz (44-283 radians/sec) Q's between 40,000 and 257,000 are estimated. Regan's data correspond to Q's about a decade smaller, between 3,680 and 24,000 over that same 7-45 Hz frequency range. This is an acceptable fit because:

Regan must have measured ensemble properties, not properties of single neural elements.

Regan's setup could have detected no tighter bandwidths than he did detect.


Within the constraints of biological knowledge, we could have guessed other values of the parameters to come closer to Regan's values (or even to match them.)

The figures Ap3-2 below (from appendix 3) show steady-state magnification of a signal as a function of frequency calculated for the LRC spine model of Fig 13. The peak magnification factor is about 70,000. Note the sharpness of the magnification as a function of frequency.

In addition to LRC resonance, the model spine of Fig 13 would also have a column resonance mode. Because of end effects, the column calculation is harder, but spine column resonant frequency will be about 220 cycles/sec, column resonant bandwith about .007 Hz.

Anatomy is complicated, and these calculations are necessarily inexact models of brain. But according to S-K, resonant behavior similar to that Reagan observed follows. The brain has about 1013 spines. If spine resonant frequencies are widely distributed, and some reasonable fraction of the dendritic spines are in the high Q state, one would expect fixed frequency stimuli, such as Regan supplied, to yield the sort of excitation curves that Regan observed. Coupling of the spines would be via the very rapid conduction of the extracellular medium, not via conduction along dendrites or axons. Excessive stimulation could generate the eplileptiform seizures that occurred in Japan.

The S-K conduction equation theory fits Regan's data and other neuroscience data well. The predictions of S-K theory, including those related to information processing and memory, have seemed plausible to experts, and are discussed in Appendix 4.


Dedication: Professor Stephen J. Kline, of Stanford University, author of SIMILITUDE AND APPROXIMATION THEORY(12) and one of the great mathematical and experimental fluid mechanicians of this century, was my partner in the work leading up to this paper. We worked together on this for almost ten years, up to his death in November of 1997. Steve's contributions were many and indispensible. Steve thought hard about the problems of physical representation, and was completely clear about the need to find and fix an error at the interface between the representation of coupled physical models at the level of a sketch, and representation by a differential equation.



Appendix 1   Derivation of crossterms that represent combinations of physical laws for a line conductor of finite length.

Appendix 2:   Evaluation of crossterms that represent combinations of physical laws according to consistency arguments.

Appendix 3:   Some background on resonance

Appendix 4:   Web access to correspondence with NATURE, including some detailed brain modeling.


1. Sheryl WuDunn "TV Cartoon's Flashes send 700 Japanese into Seizures" The New York Times, December 18, 1997.

2. Sheryl WuDunn "Japan TV to Act Against Seizure-Causing Cartoon Flashes" The New York Times, December 20, 1997.

3. Regan, David HUMAN BRAIN ELECTROPHYSIOLOGY: Evoked Potentials and Evoked Magnetic Fields in Science and Medicine Elsevior, 1989.

4. Lieberstein, H.M. Mathematical Biosciences, 1 pp 45-69 (1967)

5. Scott, A.C. Mathematical Biosciences 11, 277-290, (1971).

6. REASONS TO DOUBT THE CURRENT NEURAL CONDUCTION MODEL by M.R. Showalter at http://www.wisc.edu/rshowalt/doubt/ (See also Appendix 4.)

7. 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).

8. Stephenson, D.T. "Transmission Lines" McGraw-Hill Encyclopedia of Science and Technology, 7th ed (1992).

9. Sheryl WuDunn, op. cit.

10. Regan, David op. cit. , Fig 1.70A , p 106.

11. Rall, op. cit.

12. Kline, S.J. SIMILITUDE AND APPROXIMATION THEORY McGraw-Hill, 1967; Springer-Verlag, 1984.