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FUNDER - Theoretical Background |
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FUNCTIONAL DERIVATIVES DEFINITION
In much the same way
as functionals
are generalizations of multivariate
functions, functional derivatives are generalizations of partial
derivatives. The total differential of
which can be rewritten as
where
eq. (2) becomes
where the time interval has been defined as in the page on functionals [eq. (5)]. Note that, for the sake of simplicity, the differential of time has been omitted in the notation of functional derivatives. There exists an alternative, more useful way of defining functional derivatives. First, consider the fact that
or, introducing the change of variable
The figure below shows how, as the time increment tends to zero, the variation of the k-th variable in partial derivative eq. (6) [dashed lines] becomes a Dirac's delta [arrow].
Hence, functional derivatives can also be defined as
which gives a convenient way of calculating the functional derivative from a partial derivative, even in the continuous limit:
In addition, this shows that functional derivatives quantify by how much a functional varies when its argument (the excitation) varies infinitesimally at a given time. However, this time is not fixed and, as a consequence, a new variable is introduced in the derivative that was not present in the functional itself (see examples). This variable can obviously be interpreted as a pertubation time corresponding to impulse perturbations. DIMENSIONS It is worth noting that the dimensions of functional derivatives are those of the functional over those of the function over those of the variable of the function. The latter is hidden in standard notation. However, analysis of eqs. (2), (3) and (4) makes it clear.
In a typical situation, the functional has the same dimensions as its
argument and thus its functional derivative has dimensions of time to
the minus one. This has computational implications. Consider
eq. (6)
for a biochemical system. If both
the input and
the output
are fluxes, i.e. they have units of concentration over time, then
EXAMPLES Functional derivative of a function Let
From eq. (8), we have
In other words, the functional derivative of a function is
a Dirac's delta.
Note that, although the original
functional
is a function of
real time
only, its derivative also depends on the
perturbation time
Functional derivative of a the integral of a function Let
Taking into account that the area under a delta is one,
This is analogous to the usual derivative of a first-degree polynomial. Functional derivative of a linear superposition law Remember that the response of any linear system is
Substituting in eq. (8),
changing the dummy variable under the integral to
Functional derivative of quadratic functionals Let
Operating on
eq. (8),
eliminating quadratic and independent terms on
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,
eq. 
is the time increment between points of
the 
and
.
As the number of discrete-time points increases, total differentials
tend to functional
differentials, 
and
,
the time increment tends to the differential of time and, after introducing
the notation
,
must be a concentration. Therefore, in order to calculate the functional
derivative of the response (a flux) with respect to the excitation
(another flux), we must perturb a concentration.
.
,
eliminating terms that do not depend on
