The situation of the visual system of the brain, namely that the system is organized in a reticular pattern, and there is a convergence or coherence among all the parts concerned, is not specific of this issue but is generalizable to all the brain areas and in general to all the nervous system: the flow of process/information occurs in a global network with multiple interconnections which works at any time generating an internal coherence state according to a cooperative process.
The focus for the description of such set of processes is no longer to establish the information flow, which is practically impossible and also useless, but the specific modalities where the internal coherence states occur in the area of this network which defines itself.
This requires a change of the paradigmatic general principle to describe the systems by stimulus/reaction, input/output, and so on, characteristic of the heteronomous systems.
Such a scheme is valid when dealing with computers or control circuits or cybernetic system, but no longer when dealing with complex systems such the nervous system.
Maturana and Varela have defined a key concept (just apparently tautological) for the latter as Operational Closure:
the consequences of the operations of the system
are the operations of the system
where closure is not closeness. The Operational Closure defines some eigenbehaviors where the operations of a complex system, made by interconnected elemets, have as result
an operation which falls again into the system domain itself and in its internal dynamics.
Closure is refers to the fact that the result of an operation is still inside the system itself this does not mean thta the system has not interactions with the external, since that - as any living system - it is an open system: the system is organizationally closed but open for what energy and environment exchange are concerned. The Operational Closure defines the stability and autonomy points, namely where the relations and interactions which defines the overall system are determined only by the sustem itself, and finally defines the system homeostasis, a condition of complementary interaction stability/change which has a consequence the persistence of the system following changes: to be always itself the system must continuously change, and at the same time to change it must remain itself.
An example of an operational closure for a complex system is the one between sensory-motor system and nervous system:
In the figure can be noted three levels of circular processes: that of the sensori-motor system with operational closure which defines the state of the brain and of the body, that of the nervous system ad closed dynamical system and finally the closed circular interaction between the two. The autonomous system so defined responds to the external perturbations adjusting itself and producing effects outwards.
The same example is valid for a cell scheme:
The metabolic cell internal network produces a cell membrane such that allows at the metabolic network to produces the metabolites which form it, and so on. Through the cell membrane then there is the interchange of energy, molecules and so typical of the open systems. the system has clear characteristics of operational closure, stability, autonomy and homeostasis.
Generally the operational closure as recursive circular process in living systems links an automous system which generates a process network which produces some system components which in turn determine the closure/autonomy of the system, and so on:
Maturana and Varela have schematically symbolized the structure of any living system as:
where the operational closure part defines, and is in turn defined, by the organization of the living system which exchanges interaction with the environment as an open system.
The following table summarizes the characteristics of the heteronomous and autonomous systems:
heteronomous systems autonomous systems
operations logic correspondence coherence
organizational type input/output operational closure
interaction mode instructive-representational creating a world
The concept of operational closure, defined as the fact that a system has coherence states, - and one may say of
existence - in the case where the operations made by the system fall within the system domain itself, is quite general.
At the level 0 and 1 physical-chemical, where a formal system is available this is represented by a fundamendal class of equations called
eigenvalues equations, in the form:
Hf=af
where
H is a functional
operator,
f some functions defined in a
functional space S(f) and
a are generally real numbers.
If the equation, given a specific operator
H, and specified the
boundary conditions, has solutions
fi and
ai, with
i a discrete or continuos index depending from the boundary conditions, then these are called
eigenfunctions and
eigenvalues of the equation.
The equation entirely expresses the operational closure concept, since that made an operation H over a function
fi the result is still the function
fi a unless a number
ai, that is the operation lies always in the functional space
S(f).
In physics some of the most important equations are of this kind, in particular (for the stationary sytates - i.e. time invariant) the Newton motion equations of classical mechanics, expresses in the
hamiltonian form, and those of quantum mechanics in the two dual representations wavefunction/particle expresses in the first case by the
Schrödinger equation and in latter by the
Heisenberg equation, where
H is a
hamiltonian operator associated to the energy of the system.
The solutions in both representations give the system eigenfunctions and the energy eigenvalues, for example in the case of the simplest physical-chemical system, that of the
hydrogen atom, the eigenfunctions are of the type:
while the energy eigenvalues give a set of possible discrete
quantum levels for the electron:
In the same way the solution of the energy eigenvalue equation for higher elements, molecules and molecules chains by the atomic orbital model poses the basis for the chemical bond and therefore for the existence of any chemical compound.
Another fundamental class is the one whwre the operators H are linear, and therefore define a linear system; in this case any function f is an eigenfunction and, depending on wether the eigenvalues are greater or lesser then 1, there are characteristics of amplification or attenuation.
At levels higher then the 2-3 biological/organism the operational closure concept continues to be meaningful as guideline to define or establish which are the stable states of a system.
There cab be cases of casi di meta-operational closure; a typical example by Von Foerster is the coupling between the nervous system and the endocrine system:
Both the nervous and the endocrine systems are operationally closed, represented by closed circles, and interact with each other, in particular the endocrine stabilize the nervous and vice versa. The resulting system is representable in a three-dimensional way by a torus, where the longitudinal rings represent a system and the trasverse ones the other system. T'he overall effect is that of a meta-regulation, i.e. a regulation of a regulation.
The general treatment of the operational closure as limit of recursive operations which lie in the same domain has been developed by Heinz Von Foerster in the following way:
let a variable x0: x0 is quite general, it can be a function, a numerical value, an arrangement (number lists, vectors, geometrical configurations), behaviors described by functions, behaviors described by propositions and so on. We define an operation over x0 symbolized by Op. Op can be an operator, a functional, un algorithm and so such that applied to x0 transforms to x1:
x1=Op(x0)
subsequently applying the operation Op we have:
x1=Op(x0)
x2=Op(Op(x1))
......
xn=Op(n)(x0)
and repeating infinitly times the application of Op:
x∞=Op(∞)(x0)
that is
x∞=Op(Op(Op(Op(Op(Op( ...
in the last expression we note that the initial variable x0 is disappeared, and that any infinite sequence of Op can be substituted by Op(∞):
x∞=Op(x∞)
x∞=Op(Op(x∞))
x∞=Op(Op(Op(x∞)))
...
if this system of equation has solutions of the form Ei=x∞i then they are called eigenvalues, eigenoperators, eigenalgorithms, eigenbehaviors and so.
The operational closure is expressed as the limit of a recursive process of applications of Op:
lim (n→∞) Op(n) = OP →
↑←↓
and in particular the operator Op implies its own eigenvalues Ei, and is implied by these; operators and eigenvalues are complementary:
Op↔Ei
besides, since the Ei self produce themself, through the Op(n) complementary to them, they are self-reflexive.
Some examples given by Von Foerster are the operator H=SQRT, the square root of a number; starting from any real positive number x0 and applying infinite times the operation SQRT the eigenvalue is x∞=1 and SQRT(1)=1 is an eigenvalue.
Another example is the phrase (in english):
THIS PHRASE HAS ... LETTERS
where instead of ... should be substituted a number by letters which make true the phrase; In an ontological sense the phrase exists, namely becomes logically true, only for its eigenvalues, otherwise is false.
In the case of levels 2-3 studied by Maturana and Varela lthe operational closure of the operations Op of the systems becomes:
ORG →
↑ ← ↓
that is the operational closure defines the organization of the system and vice versa the organization defines its closure.
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