Molecular Ethology, An Immodest Proposal for Semantic Clarification - Understanding Understanding: Essays on Cybernetics and Cognition - page 145

FIGURE 4. Two finite state machines ( E ) (W) connected via delays (black
semicircles).

v ( t ), z ( t ), z( t ), i.e., expressing these variables explicitly as functions of time,
is purely mathematical. In other words, the “meta-system” ( E W) composed
of the subsystems E and W, is physically as well as mathematically “closed,”
and its behavior is completely determined for all times. Moreover, if at a
particular time, say t = 0 (initial condition), the values of all variables u (0),
v (0), z (0), z(0) are known, it is also completely predictable. Since this meta-
system is without input, it churns away according to its own rules, coming
ultimately to a static or dynamic equilibrium, depending on the rules and
the initial conditions.
In the general case the behavior of such systems has been extensively
studied by computer simulation (Walker, 1965; Ashby and Walker, 1966;
Fitzhugh, 1963), while in the linear case the solutions for Eqs. (28) can be
obtained in straight-forward manner, particularly if the recursions can be
assumed to extend over infinitesimally small steps:

dw
dt

( ) = () +

wwt ¢= + D wt D (29)

Under these conditions the four Eqs. (28) become

4

 a
1

w

- w

(30)

i i j j

j =

where the w i ( i = 1, 2, 3, 4) are now the real numbers and replace the four

variables in question, w . represents the first derivative with respect to time,
and the 16 coefficients a ij ( i , j = 1, 2, 3, 4) define the four linear functions
under consideration. This system of simultaneous, first-order, linear differ-
ential equations is solved by