Colloque sur le chaos

Organisée du 28 août au 8 septembre 1984, à l’initiative de Murray Gell-Mann (qui a dû annuler sa venue) et de Bernardo Huberman, la rencontre qui s’est déroulée aux Treilles était articulée en deux parties : la première a permis à chaque participant de présenter les résultats de ses propres recherches, la seconde de discuter des problèmes situés à la frontière de la neurobiologie et des systèmes dynamiques.

Communications

• Bernardo A. Huberman (XEROX Palo Alto Research Center), Understand biological computation
• Peter Kaus (University of California), Biological clocks
• Roy King (Stanford University Medical Center), Dopamine dynamics and schizophrenia
• Steven W. Matthysse (Mailman Research Center Belmont), Theory of Mood
• David Ruelle (I.H.E.S. Bures-sur-Yvette, France), Introductory remarks on dynamical systems
• René Thom (1923 – 2002) (I.H.E.S. Bures-sur-Yvette, France), Organs and Tools
• Christian Vidal (CNRS, Centre de Recherche Paul-Pascal), Chemical reactions

René Thom : The role of mathematics in the description of reality

1. The first type of mathematical models deals with situations where the model is built by applying known mechanical or physical laws. (For example: classical « celestial” mechanics). In such a case, prediction (and eventually control) are possible. Solving the system is always possible through numerical analysis computation. In that case, qualitative conside­rations obtained by Differentiable Dynamics theory may be interesting when trying to describe rapidly the global phase portrait of the system, but can always be dispensed with.
2. A second type of models is given by the case where the known laws of Mechanics or Physics do not suffice to determine the temporal evolution of the system, but have to be complemented by empirical laws of phenomenolo­gical origin. In that case, problems of fitting the model to the experimental data may frequently diminish the interest of the model, both from the theoretical and the pragmatic point of view.
3. A third type of model is found when one gives up the idea of writing altogether equations but one builds a qualitative Dynamical description, arising from observing the experimental morphology studied. Here, a strict quantitative control is no longer possible, but one may get clues about how to treat practically the system in view of obtaining a given result. This « qualitative » modelling is particularly interesting in « soft sciences » (as in Physiology), where an exact knowledge of the underlying system is in general out of reach.

To make this discussion concrete, consider the following three models of an acutely psychotic’s rapidly fluctuating levels of activity and motivation.

1. Patient’s Delusional Interpretation – « I feel surges of energy because radio transmitters from the sun influence my mind ».
2. Standard Dopamine Hypothesis – « The patient has too much dopamine which compels him to think many weird thoughts ».
3. Dynamic Dopamine Hypothesis -« The patient is in fact experiencing endogenously generated dynamical fluctuations in dopamine activity. This leads both to his behavioral fluctuations, as well as to his delusional thoughts. His delusions are stereotyped thoughts conditioned to the unpredictable surges in dopamine activity ».

As three schemas to describe both the client’s perceptions of the world and his behavior in it, all suffer from limitations. Model 1 has no utility value, at least for changing behavior. Model 2 has utility value, but ignores the fine structure of the behavioral data, namely its fluctuating nature. Taking into account these details, as Model 3 does, may further refine the temporal course of medication treatment, or suggest treatments of completely different modalities, such as behavior modification. It might then have an enhanced usefulness for psychiatric therapists.

We, therefore, see that alternate models which offer various styles for sorting the world can have radically differing utility value for a particular group of scientists.

Such qualitative models which frequently -as in Catastrophe theory- use the distinction between a fast and a slow dynamics- may sometimes duplicate a purely verbal description in ordinary language. In such a case, the model has interest only if it introduces some « imaginary » connection (or entity) which could not be found by empirical observation. An example is the statement: « the hungry predator is his prey » arising from the predation loop associated to cusp catastrophe.

Personally, I do believe that the most promising models are of the third type. As they allow in an easy way an articulation between the « local’ and the « global », they seem to be particularly interesting for the modelling of biological structures; they can help bridge the gap between a global atemporal « physiological-biochemical » schema of cybernetic nature and local spatio-temporal patterns as they appear in embryology and adult physiology.

These quantitative models may also have the interest of making a global situation intelligible, even if they do not allow precise control, nor experimental verification. Concerning the last point, let me repeat here that the role of experimentation in modern science has been grossly overem­phasized; not unfrequently the interplay of human free-will with the dynamics of natural phenomena creates a wealth of new phenomena, which -as mere artefacts- may not be really meaningful.

Bernardo Huberman : Understanding complexity

Complex systems are situated between the dynamics of few degrees of freedom and statistical mechanics. As such they lack the simplicity implied by low dimensional geometry (i.e. attractors) or the simplifications brought about by the laws of large numbers. They also underlie the interes­ting properties of self-organization which appear in biological structures, collective computor behavioral neurosciences and social organizations. Therefore, progress in understanding complexity is bound to have important implications in a variety of disciplines which have been represented at this meeting.

In spite of its importance, little can be said about complexity which is of relevance to its unfolding in time or quantification. I believe that a research program with an experimental flavor and a playful, heuristic attitude, will be necessary in order to uncover new phenomena and general paradigms. Dynamical systems theory might provide the required language, but one is quite aware of its limitations when pushed into descri­bing spatio-temporal structures.

Complexity often manifests itself in hierarchical fashion. By this, I mean that at a given level of description there always exists subunits where characteristic spatio-temporal frequences are high compared to larger units and slow compared to smaller ones. Therefore, the effects of perturba­tions at one level of the hierarchy diffuse very slowly into the overall structures. This in turn suggests a possible measure of complexity since self-organization appears to be associated to a contraction process whereby the system selects dynamics whose characteristic periods are much shorter than those implied by its combinatorial capacity.

Roy King : Dopamine dynamics and schizophrenia

The application of theoretical techniques to the analysis of behaviour is often impeded by prior conceptions of scientific proof. The natural variability of an organism’s interpretation of the world forces us to seek novel, precausal interpretations of its behaviour. Models in psycho­biology thus must always remain closely linked to experimental and phenomeno­logical data. This dictum needs not, however, force us to be gloomy about theory. On the contrary, like the animal immersed in its perceptions that develops schemas for organizing its world, we, too, must channel our comprehension of experimental data into well-defined conceptual clusters. Mathematics, in general, and dynamical systems theory, in particular, does offer methods for performing these reductions. Although clearly non-unique, we can through interpersonal communication find overlap among our conceptual categories.

An important process remains in our scientific prescriptions; namely the rejection of competing “schemas”. Since behaviour is so complex, it is not difficult to reject all hypotheses concerning behaviour. This is a stultifying but common approach in biological research. Again, we can borrow from research in perception that suggests that an organism not only fits its perceptions to its innate or learned schemas, but modifies those schemas to account for the unexpected. In such a case the animal is concerned solely with events that may have a pronounced effect on its biological needs: hunger, affiliation, curiosity, individual survival. We can also shift the process determining the validity of models from absolute to aesthetic-utilitarian criteria: how easily can a model accommodate the new or previously ignored empirical evidence, that some scientist who clamors loudly enough, compels us to notice.

David Ruelle

The background of the present meeting has been largely provided by the current new interest in conceptual problems of complexity in natural phenomena, seen in their scientific and philosophical aspects.

This includes as topics biological problems like those of functioning of the brain, or self- organization, or philosophical problems like that of hydrodynamic turbulence. From a philosophical point of view, the concept of time is central, as is that of spatial structure (or form). The technical tools and notions are provided by ergodic theory, differential geometry, the logics of decidability and the theory of algorithmic complexity. The approach is qualitative rather than quantitative (but a qualitative theory like that of differentiable dynamical systems can provide a quantitative prediction like that of the FEIGENBAUM exponents). Many model systems are investigated (cellular automata, ordinary and partial differential equations). They provide an interpretation of « chaotic » experimental data which were yesterday not considered part of science, and which can now be analyzed and understood in terms of characteristics exponents, entropy and HAUSDORFF dimension.