FUNCTIONAL TAYLOR SERIES

EXPANSION OF A FUNCTIONAL AND SUSCEPTIBILITIES OF A SYSTEM

Continuous, infinitely-derivable functions can be expanded in polynomial series, known as Taylor series. Functional derivatives make it possible to generalize this idea to functionals. The analogue of the reference point in regular series (zero in MacLaurin series) is a reference function. Hence, we define a reference state consisting of the pair of functions

(1)

These signals need not be causal. For example, the origin of times may have been shifted to minus infinity, so that they would represent an asymptotic state of a system. It is convenient to refer the excitations and responses to the reference state, so we consider, for a one-input system, the variations

(2)

and

(3)

Although it is not required, we assume from now on that the reference state is a steady state.

The usual Taylor series relates finite variations of a function to finite variations of its variable. Infinitesimal variations, i.e. differentials, are related through a derivative. The functional case is completely analogous. Thus, functional derivatives connect differentials of functionals and functions [eq. (4) in the page on functional derivatives], whereas increments of functionals can be expanded in terms of increments of functions as

(4)

where causal signals are assumed and therefore the integration limits are taken from zero to real time. The functions are the susceptibilities of the system. They are defined as follows.

Definition of the susceptibilities

Susceptibilities are functional derivatives, normalized by the factorial of the order, , and evaluated at the reference state of the system they represent:

(5)

Knowing the susceptibilities of a system, or a few of them, makes it possible to describe and predict the response of a nonlinear system without any information on its composition, internal structure and mechanism. Furthermore, eq. (4) is linear in the susceptibilities, so impulse perturbations of the reference state lead to linear algebraic equations.

SUSCEPTIBILITIES AND RESPONSES TO IMPULSES

A helpful way of understanding the properties of susceptibilities is discretizing them. Let us consider the simplest nonlinear case, i.e. only two time-points and two susceptibilities.

The discrete first susceptibility is a matrix:

(6)

The second susceptibility is a tensor:

(7)

We reserve the first index for real time. There are only three possible impulse-perturbation experiments:

1) Perturb at the first time point

2) Perturb at the second time point

3) Perturb at both time points

Let us consider them one by one, and in order.

Perturbation at the first time point

In this case, the variation of the excitation is

(8)

The variation of the response is

(9)

These are linear equations with two unknowns per time point, and , and hence the experiment has to be repeated twice for different values of the perturbation magnitude, .

Perturbation at the second time point

Analogously, for the variation of the excitation

(10)

the response is

(11)

Again, this extracts from the susceptibilities, eqs. (6) and (7), two new unknowns (per time point), and , which can be determined from two different experiments of different magnitude.

Perturbation at both time points

If the excitation consists of two consecutive impulses,

(12)

the response yields equations of the form

(13)

where terms already determined from single-impulse experiments are grouped in the left-hand member of the equation and the last two undetermined elements of the susceptibilities (per time point), and , appear in the right-hand member.

However, in this case it is not possible to obtain the unknowns. If the experiment is repeated for two different variations of the excitation, labeled and , the matrix of coefficients of the system is

(14)

which is singular, i.e. its determinant is zero. Hence, the system of equations is undetermined. To solve this problem, several properties must be taken into account, which are described below.

PROPERTIES OF THE SUSCEPTIBILITIES OF BIOCHEMICAL SYSTEMS

Symmetry

The most general property of susceptibilities is their symmetry. Due to the singularity of matrix (14), it is not possible to determine and unless they are equal. Theoretically, they could be different, as long as they satisfy eq. (13). However, the response of the system they would characterize would be identical, for any excitation, to that of a system with symmetrical susceptibilities. Hence, it can be assumed, without loss of generality, that susceptibilities are always symmetrical, in the sense that they are invariant to permutations of the perturbation times. For example,

(15)

Furthermore, symmetrical susceptibilities are consistent with the structure of eq. (4): Permuting the perturbation times, which act as dummy variables, means relabeling them. This can be done arbitrarily without changing the meaning of the Taylor series.

Causality

In causal systems, eq. (11) is equal to zero for , because at this time the impulse has not been introduced yet. Since this is true for any value of , we have that

(16)

In addition, eq. (13) for represents the response to the first impulse and must coincide with eq. (9). Substituting eq. (16) in eq. (13), and taking into account the symmetry of the susceptibilities, we see that the only possibility is

(17)

Clearly, these ideas can be generalized for more than one time point and for higher-order susceptibilities. Therefore, as a result of causality, susceptibilities are zero before the last perturbation time.

Time-invariance

Not all biochemical systems are time-invariant. However, for those that are, susceptibilities are redundant. This is very helpful in reducing the amount of work required for determining them. Consider the above example. One expects that the response to an impulse at the second time point of the discretization will be the same as the response to an impulse at the first time point, but shifted (provided that the intensity of both perturbations is also the same, say ). That is

(18)

Since this must be true for any value of , it follows that

(19)

or, in general, that the value of a time-invariant susceptibility depends on the delays of all times with respect to the first impulse-perturbation time, instead of on real time. In other words, the reference time for the response of a time-invariant system is the perturbation time of the excitation, and not real time.

Copyright © 2003-2013 Antonio Sánchez Torralba
Last modified:
Monday, 04-Mar-2013 11:58:25 CET