The S-K transmission equations:
The currently accepted passive neural conduction equations
are the standard conduction equations of electrical engineering, usually
written in a contracted form that discards the terms in L that are negligible
in this equation.
Robert Showalter and Stephen Jay Kline have found that these equations lack crossterms because special restrictions on the use of dimensional parameters in coupled finite increment equations have not been understood(11)
. The crossterms, which can also be derived (within a scale constant)
by standard linear algebra based circuit modelling, are negligible in most
engineering applications. But these crossterms are very large in the neural
context. The Showalter-Kline (S-K) equations are isomorphic to the standard
conduction equations of electrical engineering and are written as follows.
For set values of resistance R, inductance L, membrane
conductance per length G, and capacitance C, these equations
have the solutions long used in electrical engineering. The hatted values
are based on a notation adapted to crossproduct terms. In this notation,
the dimensional coefficients are divided into separate real number parts
(that carry n subscripts) and dimensional unit groups, as follows.
For wires,, the crossproduct terms are negligible, and the two kinds
of equations are the same. But under neural conditions the crossproduct
terms are LARGE. For instance, effective inductance is more than 1012
times what we now assume it to be. The S-K equation predicts two modes
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.