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.
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
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.
RESONANCE-LIKE RESPONSES IN BRAIN:
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.
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
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.
DERIVATION OF THE S-K AND K-R EQUATIONS:
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
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:
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.
DERIVATION OF S-K.
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
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.
S-K FITS DATA:
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
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
Regan's setup could have detected no tighter bandwidths than he did
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
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.