Appendix 1: Derivation of a finite increment equation
from a coupled physical model, showing combined effect terms.
Most mathematical models of physical circumstances start by matching
one or more differential equations from a handbook with a physical case.
Thereafter, analysis is carried out according to the rules of abstract
mathematics. In this usual procedure, the differential equation is not
derived from a physical model, but is applied to that model and taken as
an operationally perfect or good enough match.
However, the differential equations themselves are sometimes inferred
directly from physical models in a step by step way from a sketch-model.
This process is called "derivation," and is taught in engineering
To derive a differential equation from a physical model
finite scale model that sets out the laws and geometry to be represented;
then derive (one or more) finite increment equation(s) that map that finite model.
After a finite increment equation that represents our model has been defined, we may
pass that finite equation to the infinitesimal limit to yield a differential equation.
Equation definition can take special attention if our finite model includes coupled effects. In such a case, two equations are implicitly defined in terms of each other.
One of the simplest and most important examples is current change and
voltage change along a length of conductive line (such as a wire or a length
of neural tube.) Current drops i are coupled to voltage and voltage
drops according to logic like the following:
i over the interval is a function of v at x and x+x
which is a function of i at x and x+x
which is a function of v at x and x+x
which is a function of i at x and x+x
Voltage drops are coupled to current and current drops in the same nested way.
We have to be able to represent the coupled effects that occur correctly at finite scales, before we can take the limit of those terms, at successively smaller finite scales, to the infinitesimal limit.
Fig. 1 shows a conducting line that could be neural conductor.
v = voltage i = current x
= position along the line alpha= arbitrary length interval
R = resistance/length L=electromagnetic inductance/length
G= membrane leakage conductance/length C=capacitance/length
Fig. 1 shows an arbitrarily chosen length , of arbitrary magnitude, which we will call deltax.
R, L, G, and C are natural law operators (Appendix 2 .) They represent physical laws, and are defined as the ratio of one measurable to another under particular measurement circumstances. The natural law operators, which implicitly represent much measurement detail, are our interface between the detailed measurement procedures of physical reality and abstract equation representations of physical circumstances. Natural law operators are justified by inductive generalization, not axiomatic proof. We have no provable reason to think R, L, G, and C have exactly the properties of numbers. In the derivation below, we'll operate on terms including the symbols R, L, G, and C in the usual algebraic way, stopping short of algebraically simplifying the terms. We'll not interpret these terms numerically or physically here in appendix 1, leaving that for Appendix 2 .
To derive line differential equations in dv/dx and di/dx, we first need
finite difference equations, dv/dx and di/dx. For the finite
equations, we'll be writing out terms that have usually been understood
to exist, but that have been called infinitesimal and neglected. Let's
consider the coupled effects physically.
Voltage drop over deltax
depends on current. Current over length deltax varies if charge
carriers are stored or
discharged in capacitance over the interval deltax. The capacitive
effect will depend on dv/dt. Current
over deltax will also be lost from membrane leakage over delatx.
That loss will depend on v. Voltage
drop depends not only on current, through R, and L, but also depends
on interactions between voltage
and G and between dv/dt and C.
In an exactly symmetric way
Current drop over deltax depends on voltage. Voltage over length deltax varies with inductive reactance, L di/dt. Voltage over deltax will also be lost from interaction between resistance and current, Ri. Current drop depends not only on voltage, through G and C, but also depends on interactions between current and R, and between di/dt and L.
The idea that di/dt depends ONLY on G and C neglects effects that act over deltax. The idea that dv/dt depends ONLY on R and L neglects crosseffects over deltax.
Let's derive voltage and current equations that include crosseffects. We'll write our voltage and current functions as v(x,t) and i(x,t). We assume homogeneity and symmetry for our conductor. We assume that, for the small lengths of interest, the average voltage (average current) across the interval deltax is the average value of voltage (current) at the endpoints of the interval deltax.
Writing down voltage change as a function of the natural law operators and variables that directly affect voltage, and centering our interval at x, so that our interval goes from x- deltax/2 to x+deltax/2, we have:
The current change equation is isomorphic:
Note that equation (1) includes i(x+deltax/2,t) and its time
derivative. i(x+deltax/2,t) is defined by equation (2).
Equation (2) includes v(x+deltax/2,t) and its derivative. v(x+deltax/2,t)
is defined by equation (1).
Each of these equations requires the other for full specification:
each contains the other.
If the cross-substitutions implicit in these equations are explicitly
made, each of the resulting equations will also contain the other. So will
the next generation of substituted equations, and the next, and so on.
This is an endless regress. Each substitution introduces new functions
with the argument (x+deltax/2), and so there is a continuing need
for more substitutions. To achieve closure, one needs a truncating approximation.
Expression of current, voltage, and their time derivatives at x, the midpoint
of the interval, truncates the series.
Let's proceed with these substitutions, associating symbols without
interpreting them numerically or physically. For example
which expands algebraically to
These terms would be simpler if voltage averages and derivative averages
were taken at the interval midpoint, x, as follows:
How may terms like those of (6) be interpreted, physically and numerically,
at finite scale? In these expressions, two natural law operators are EACH
associated with the SAME interval of length. Do the lengths multiply? On
what authority do we say that the lengths multiply? If the lengths do multiply,
what does this represent physically? Does the multiplication make numerical
sense, and is that multiplication consistent with tests the expression
must pass? It turns out that if we apply standard arithmetical rules to
these crossterms, we are led to mathematical and physical inconsistencies.
(Appendix 2 )
The equation below shows voltage change over an interval of length deltax, divided by the length deltax to produce a gradient form analogous to a derivative. Terms derived from three stages of cross substitution are shown. Symbols are grouped together and algebraically simplified up to the point where the meaning of further algebraic simplification of relations in the dimensional parameters R, L, G, C, and deltax becomes unclear. Expresssions in curly brackets are NOT YET DEFINED.
The current gradient equation over the same interval is isomorphic to
7 with swapping of v for i, R for G,
and L for C.
Whenever coupled physical effects act over an interval of space, combined
effect terms are to be expected. Rules for their interpretation must be
found. Those rules are beyond the authority of the axioms of pure mathematics,
but consistent rules for interpreting these expressions can be inferred
from mathematical experiments. (Appendix
Combined effect terms such as those shown here are seldom derived, because
they are thought to always vanish in the limit. However, that interpretation
involves basic inconsistencies (Appendix 2). Expressions such as those
in the curly brackets, interpreted in a consistent way, are finite, and
yield finite terms in differential equations. Often such combined effect
terms are negligible, but sometimes they are important.