NETWORKS OF COUPLING OSCILLATORS:

The case for a "Math" Psychiatry

Henrique Schützer Del Nero, M.D.

Lucia Maria Argollo Maciel, M.D.

ABSTRACT

The problem of mental emergence in complex architectures has dominated the field of Cognitive Science and of Neural Networks in the past decades. Several algorithms and ways for processing information have been invented, but it still remains a problem concerning the gap between mental function and dysfunction: are neural networks an adequate way to

explain psychiatryc disorders? Talking about MathPsychiatry means a common search for meta-foundations, theoretical inspirations that might enlighten the frontiers and limitations of such "discipline". Freud wrote his Metapsychology, but nowadays Psychiatry is closer to the brain than to the mother. Brain mechanisms, however, are meaningless if one doesn't scaffold the way information is codified in semantical nets, subserving function and deviance.

Dynamical nets of van der Pol's oscillators can display lot of topologycal variations, due mainly to non-linearities and sensitivity to the initial conditions. This doesn't want to a model of the brain a but a source of inspiration to a Math Psychaitry that uses Mathematics as founadtion and not as tool to cope with data.

I. Psychiatry

It is difficult to characterize in very few words what is the very object of Psychiatry. Common-sensical approaches confuse the clinical and pharmacologycal practice with psychoterapies, mainly psychoanalysis. The Babel that results from this confusions are due mainly to a strict difficulty to define the object, the relations and their deviances. In this way, Cognitive Science, Neuropsychology and trhe formalisms of Artificial Intelligence can be very welcome, if the are preceeded by a careful conceptual analysis of the potential mistakes that crowd the field.

Psychology can be seen in different ways:

a) as doctrine of intentional states: beliefs, intentions, desires, goals, etc. This usage of the psychic, mental, etc is common to a certain field that culminated with the logical style of seeing the mind: traditional Artificial Intelligence. Mind here is a program with its own primitives and rules. Inference rules will guide the formation of strings of primitive atoms. These atoms are not to be confused with the neuronal atoms that underlie mental activity.

b) as a doctrine of faculties: memories, judgments, imagination, perception, consciousness, critics, conation, affects, humor, temperament, personality, etc.

The history of Psychiatry, of Psychotherapeutical approaches(e.g. Psychoanalysis), of Cognitive Science in its reaction against Behaviorism and of Neural Networks has much more in common as one can imagine at first sight. Mind has been the Meca of the models and searching for Meca one may, sometimes, abbandon meta-models, i.e theoretical foundations.

Psychology as intentionality pervaded a lot of arguments of the traditional models based upon rules that grasp inspiration from Logics and that manipulates semantical blocks that are supposed to exist. Two extreme reactions supervened: either common-sensical knowledge was elected as the source of mental categories (e.g. theories of epistemic utility using programs that contain "beliefs", "desires, etc. [1] ) or mental blocks were denied even to exist (eliminative materialism of Churchland among others [2])

Psychology as faculties pervaded Neruropsychology and still impresses this kind of mitigated-modern phrenology. Localization of brain function, being mental derivates epiphenomenal, is the root of today's sophisticated models based upon PET-scan (positron emission tomography), MRT (magnet ressonance tomography) etc. Neurophysiology in another extreme through careful dissecation of ionic channels through intracelular patch-clamp tecniques seeks the very nature of the flow of energy in the Central Nervous System (CNS). Models try to grasp at the same time ionic plausible biologycal candidates for back-propagation algorithms (e.g. micro-tubulular antidromical movement) and for memories and learning and purpose. Everything sometimes has the influence of well-known behavioral "laws", allowing conditioning to hold and reward and punishment to complete the mess.

May one be confident about the future of such theoretically-empty strategies? Or is data something that must be accumulated until it reaches a critical value and allows phase-transition to occur? Until many of us search for the Graal or wait till glasses become cristals, let us put more ingredients in the field.

Psychiatry deals with mental disfunction. The mental is a certain kind of processes that happen in highly complex structures like the human-neocortex. The most important traits of the mental are: consciousness, language, "free-will", judgement and responsibility. Then: psychaitry deals with disfunctions of consciousness in its phenomenal form (and not mechanical like stupor, coma, drowsiness, etc), of language in its "symbolic" form (and not in the mechanical aphasic form), of "free-will" in its compulsive-obsessive forms (and not with bad choices), of judgement in its psychopatic and psychotic deviances (amoral disorders and delusions for example) and of responsibility (aging and mental retardation with preclusion of imputability from the juridical point of view).

Emotions are a very special part of the mental subserving mood disorders and anxiety but the commitment of these syndromes with more primitive neuronal systems might suggest that there equivalents in other animals.

It may look easy to define things as above done and then persecute, without any caution principle, models of very sort. Hybrid models that have neural nets and expert systems, carefully handled by external heuristics and exausting supervised trainning sets sometimes relax unto stable configurations that mimic a deviant episode. In this syncretic run, biochemistry and will, neurons and subjects, bifurcations and choices, all treated at the same level. In Psychiatry the situation is no more gratifying: drugs that act at the synaptic cleft are used conjoinly with hugs to establish a rich "cross-talk" between signals and symbols. Is the Freudian mother still present, are the primitives of mankind so simple, is it love that cures or is there something still hidden from our simple minded brains?

II. Semantics and Mathematics

In the very core of the problems quickly mentioned above stays the place semantical categories must accupy in formal systems. There are several conceptions of formalisms. One states that categories only sustitute bounded-variables, being dissociated from them. Others defend that there is a more essential relationship between mathematical structures and the domains of interpretation of the varaibles and parameters.

This must be stressed because:

a) every mature science gets more and more formalisms in its stetements

b) Psychiatry and Psychology are, in this sense, imature sciences

then : either more data must be waited until formalisms can hold or formalisms must preceed data, orienting research.

This way of looking things is not universal. Many people deny that Psychology and Psychiatry can be formalized, hence they are not imature. The very nature of these disciplines preclude formalisms to do a lot of job because they are firmly tied to first person's experiences: acquaintance, inner-feelings, etc. This wuold be a realm of phenomenology and Mathematical and Physical tools would be useless here.

I assume that this is not true, because phenomenal experience, in spite of being direct must be a mistake from the realistic point of view: as our senses may deceave us ( e.g. phantom limb pain) so may happen with our inner phenomenologycal experiences. In other words, the way our mental screen seems to see the world is only a perspective, theoretically loaded, of the world. It assumes a lot of boundaries that dont't really exist, or at least that are denied in several branches of hard sciences.

What characterizes the mental blocks, be in their semantical boundaries, be in their rules of connection, must be the new facts that appear with the neocortex in humans. What kind of novelty could have happened that allowed mind to emerge and everything else, like neurotic and psychotic disorders?

The cue must stay in the very nature of neuronal computations that at the same time create a rich syntax and a rich semantics. Postulating a dissotiation between semantical blocks and rules of connecting them must be the last mistake of a dualistic tradition. This still pervades our models, creating a fragile scheme that is not apt to answer problems like the Searle's Chinese room. (Briefly, if a computer manipulates chinese symbols with the approprieate rules, it can give right outputs that look like a chinese-speaker, but that in fact don't mean comprehension of chinese.) [3]

Searle proposes that the genuine model of a cognitive architecture must mimic the real powers of the human brain. This must seem elliptical and obscure if one doesn't pay attention to the basic proposals that stay beneath:

a) consider computation as the very essence of mindful arqchitectures

b) consider mathematical analysis as the broad sense of computation

c) consider what kind of highly complex events that happen in the CNS and mainly in the neocortex areas that can endorse a physically complex system to exist

d) consider the mathematical description of such complexity

e) try to translate the complexity at the processing level to the complexity at the semantical level, trying to find a homeomorphims between them (bijective, sobrejective inversible function).

Of course these statements are very general and vacuous in a certain sense. How can Psychaitry be firmly impressed by them?

Psychaitry stays exactly in the priviledged place: it has a branch in the biochemistry of the brains and another in the subject's experience, personal story and phenomenal experience. Moreover, consciousness must be the very essence of humankind, be in its physiology be it in pathology.

Then one may look at a model that shows:

a) real dynamical properties

b) self organizing algorithms (insofar as there is a lot of supervised learning)

c) that shows the capacity to change dramatically from past conducts and habits.

d) that can exibit learning through the presence of only one case

e) that have a way to process quantities and to analize qualities.

What does it mean? It means that the conscious mode of compuation and the emergence of the menatl must be a kind of reprocessing of information in new domains. There is a primitive chaim of information processing that is apt to give all animals, but humans, fantastic strategies of survival. In humans however, these signals must be converted unto symbols what means a kind of reinterpreation or second codification. Quantities at the firts level must become qualities at the new levels. What is the plausibility of this proposal?

1. The hipoccampus process information at the first level and the neocortex areas repreocess giving new kinds of interpretation withna highly semantical character.[4] But what means to have a highly semantical character? It means to have a kind of analyzer that grasps only topoogycal variations in the first module.

Then, mind would be a kind of codifying-manipulating engine that analyses states in space of states and not he structure of the equatiosn that generated these states.

There would be a kind of logic in the system: inputs must be converted and analysed through the "first layer" regarding state-variables and parameters. Solutions of the first level would constitute the flux, the phase portrait, the power spectrum, etc. The second interpreter would be interested only in the saliences from the topologycal point of view that came from the firts level. Suppose just for example that an linear equation is itearted at the first level. The second level would have only one state that could define the first: in other words the richer the non-linear system in the first level the more information (in Shannon's terms) the second level would grasp. Each bifurcation, i.e. a qualitative change in the system's behavior regarding a quantiattive variations of a parameter woud double the information in bits of the system at the second level.

Mind and consciousness, and alsso its deviances would be in this scheme a way to map and rename, using for it the blocks of experience with its natural intersections and ambiguities, topologycal variations at the first level;

1) primitive low-level system ; first level

2) intermidiate systems : first level with non-linear systems

3) mental systems: fisrts level with non-linear structures sensitive to the initial conditiosn: then full of bifurcations and even chaos to hold [5]

The following prescriptions would complete a way to see Psychiatry as a branch of Cognitive Science in its highly and broad compuational (hence formal0 character.

a) consider neurons as oscillators (e.g. van der Pol)

b) connect them through a kind of hebbian syanpse

c) build an architecture of three layers: the first receives the signal, the second processes it the third analyses the topologycal character of the processing mechanisms of the second (through Fourier series, flux, phase portraitetc)

d) consider this interplay when dealing with drugs and hugs

e) relate qualiatative variability from the sematical point of view with topologycal varaibility in the equations (non-linearities, bifurcations, chaos, etc)

f) consider learning, memories and categorization (dream algorithmns, for example) as modification of fricction coefficients, connections, and natural frequencies of each oscillator.

g) consider the source for crativity and for deviance that such systems have and mainly, the possibility that, in sopite of the imense difficulties from the analytycal point of view, this is the "geometry of mind", what may reallocate mind as kind of complex "two layers computation"and not a kind of dissotiaion between processing and encoding, syntax and semantics, drugs and hugs.