Résumé.

Un système complexe auto-organisé, tel un système biologique ou social, est caractérisé par le fait que sa dynamique est engendrée par un réseau de régulations compétitives, chacune opérant comme un système simple (au sens newtonien) à un certain niveau de complexité et à sa propre échelle de temps. Une dialectique dépendant des contraintes temporelles s'instaure entre elles, ponctuée par des fractures locales avec changement de régime. Ces systèmes sont capables d'anticipation et d'adaptation grâce au développement d'une mémoire. Les Systèmes Evolutifs avec Mémoire en donnent un modèle mathématique, basé sur la Théorie des catégories, permettant l'étude des phénomènes d'émergence. On montre que ce modèle répond à la définition donnée par Rosen pour un "organisme".

1. Introduction

In a preceding paper [*InterSymp'95*, Baden-Baden 1995], we have compared two models for complex natural systems, the C8 theory of Chandler [*WESS Comm.* 1, 1991; 2, 1992] and the mathematical model called Memory Evolutive Systems (MES) of Ehresmann & Vanbremeersch [EV: *Bull. Math. Biol.* 49, 1987; *J. of Systems Ana., Mod., Simul.*, 1996]. Here we don't intend to recall them (reprints are available), but we propose some reflections on the prospect these models offer to characterize *self-organized hierarchical complex systems*, e.g., living systems. The thesis we propose to defend is that these systems can be characterized by the following properties:

1. The general dynamics is generated by a *web of* overlapping, possibly collaborative or conflicting *regulations*, each one operating as a simple (newtonian) system, at a specific complexity level, with its own time-scale. The equilibration process between them is not directed from above, but depends on *structural temporal constraints*. It may lead to the emergence of more complex structures.

2. An internal memory, which may develop in time, allows for a selection of local operations based on *anticipation* of their results, and for possible repairs in case such an operation cannot be performed in due time.

3. Higher level structures have several different lower level implementations (*symmetry-breaking* in the passage from higher to lower levels), thus ensuring structural stability and plasticity.

We'll analyse these properties and their consequences and we'll indicate how they are modeled in MES.

2. Characteristics of complexity

The newtonian paradigm, valid for "simple" systems (or "mechanisms"), represents a system by its successive instantaneous states (defined through observables in the phase space) and the rules of change between them (physical laws). However the relation between Newtonian systems and emergent natural systems such as living systems is unknown (several incommensurable partial descriptions).

We propose a different approach to study complex natural systems (say, biological, social or cultural systems). These systems are evolutionary autonomous systems which have emerged into a hierarchy of components, they have organized exchanges with the sustaining environment, and are able to adapt to changing conditions.

a) Formally, the *organisation* of the system at a given instant would be determined by its actual components of any level and the relations between them. A higher level component in the hierarchy corresponds to the aggregation of specific patterns of lower level components. For instance in a cell, we distinguish its components of increasing levels, from atoms, to molecules, macromolecules, organelles, with chemical and topological relations between them.

The *change of organisation* comes from the archetypal operations: "birth, death, scission, collusion": adjunction of new elements and destruction or rejection of some components (exchanges with the environment), binding together of patterns of elementary components into a more complex one (synthesis of a protein), decomposition of higher order components.

b) However this description is purely formal, in the sense that no observer could give such a complete description of the system. In fact, the dynamics is generated by a *web of coordinated and possibly conflictual local regulations*, and an observer could only describe separately these regulations.

We consider (in MES) that each local regulation is directed by a sub-system, called a *Center of Regulation* (CR). A CR consists of a small number of components of the system, called its actors, which belong to a particular complexity level and act cooperatively through some distinguished links. The CR operates a stepwise process, *with a specific time-scale*. It can only perform some kinds of operations (we call them its "strategies"), resulting in specific energy generating and energy consuming reactions which make and break binding relationships. The usual result of a strategy (if there is no external obstruction) can be anticipated, and the CR can recognize if it has been obtained or not.

The strategies available to the CRs and their results form the "*memory*". Lower CRs may have only one strategy available, in which case a cyclic process is developed and sustained. But for higher CRs, there may exist a large choice of strategies allowing for more flexibility. For instance in the cell the promoter of a gene is a lower CR, whose unique strategy consists in switching on and off the gene depending if the operator of the gene is occupied or not. Higher CRs (e.g. in a neural system) can learn new strategies, thus developing the memory.

At each step, the choice of a strategy by the CR depends on the particular information on the system observable by its actors at this instant, and on the anticipated memorized result of the strategy. These informations form what we call the *actual landscape* of the CR. Let us remark that this landscape is not a sub-system of the system, but the partial and more or less deformed description of the system which can be given by observers at the level of the CR. The dynamics of the landscape during one step (once the strategy has been chosen and up to its realization) can be modeled by a *simple *physical *system* (e.g. by systems of differential equations satisfied by appropriate observables).

c) However the selected strategy may not have the anticipated result, because of the competition with the other CRs which is characteristic of complex systems and make their behavior impredictible. Indeed, though their landscapes are incommensurable, the different CRs all depend on the same global resources, and there are direct or indirect interactions between them. At a given instant, their actual strategies are all relayed to the system, and there is need for an equilibration process between them.

In part organized complexity emerges, though there is no central controller, from an equilibration process between local complexifications, extremely sensitive to temporal constraints. As said, each CR operates at its own time-scale, but its operations rely on informations or reactions processed in due time by other CRs. The accuracy of the informations used to select and then realize the strategy depends on the propagation delays of the links which convey them to the actors, and on the stability spans of the components used in these operations. Thus some *structural temporal constraints* must be respected, connecting these propagation delays and stability spans to the period of the CR (mean length of its steps). Though these constraints leave some flexibility, there is a limit to the discrepancies (perturbations) which are tolerable without disruption of the process.

d) The *equilibration process* just discards the strategy of a CR whose constraints are not satisfied, so that a "*fracture"* occurs in its landscape, imposing a change of strategy to repair it. In the description of the landscape by a simple system, such a fracture may just correspond to the introduction of a singularity (bifurcation or chaotic behavior), or impose a complete change of representation. For instance, in a cell, if the replication of the DNA has not been completed in time, the protein p53 temporarily blocks the CR directing the entry in the M-phase of cell division.

The equilibration process, which can lead to the emergence of higher order objects and new properties, is facilitated by the robustness and plasticity of complex systems: The sustainability of a higher level component may not be affected by a progressive enough change of its lower level internal organization; and a strategy for a higher level operation is memorized as an invariance class of lower level operations, which may be implemented by any of the instances of the class, depending on the context (*symmetry-breaking from higher to lower levels*).

3. Memory Evolutive Systems

MES represent a mathematical model in which the characteristics of complexity which we have singled out can be represented. This model is based on category theory, enriched to take into account quantifications relative to energy and time.

a) Category theory is generally presented as a purely abstract domain, allowing for some general classification of structures, be they mathematical or else. A category can be described as an oriented graph with several arrows (called "links") between two vertices, in which we associate to a path a well-defined link which is its "composite". This *composition law* entails a classification of the paths into classes of invariance formed by paths with the same composite.

The aggregation process is modeled by the colimit operation: the colimit of a pattern of linked objects is an object whose links toward any object *A* are in 1-1 correspondence with the collective links of the pattern to *A*. Those collective links model the operations which can be performed by the objects of the pattern acting collectively through their distinguished links; and the colimit models a more complex object performing by itself these same operations; thus it represents an invariant for the class of patterns which perform the same actions. If this class contains "non-equivalent" patterns (in a well-defined sense), we say that the category satisfies the *Multiplicity Principle* (its interest will be seen later on). The study of colimits is linked to connectivity and coherence problems which become more rich and complex in categories than in simple graphs (generalizing Ramsey theory). And it is at the basis of our description of *hierarchy* (an object of level* n*+1 being a colimit of a pattern of level *n*) and of emergence.

b) In this general form, categories seem very abstract. But they are made amenable to *quantifications* by labelling the links with some observables (e.g., the force of the link, or its* propagation delay*), called their* weight*. In this case, the composition law is generally determined so that the weight of the composite of a path be a given function of the weights of its factors.

To model evolutionary systems, we need to introduce *Time*. The system will not be represented by a unique category but by what we call an *Evolutive System* (ES): it is a family of categories, each one representing the organisation of the system with its components and their weighted relations at a given instant *t*. The change of organisation from *t* to a later instant *t'* is represented by a "transition" functor. This functor can be generated by a "*complexification with respect to a strategy*" which describes the change in a category under the adjunction or suppression of objects and of colimits of patterns (modeling exchanges with the environment, internal operations of synthesis and scission, and possible emergence of new structures).

c) A *MES* is defined as an ES in which are distinguished: - a "Memory" represented by a hierarchical sub-ES, possibly developing in time by emergence of new objects; - and a net of CRs which are sub-ES with discrete time-scales.

This model allows to represent the stepwise process of a CR with, at each step: formation of the *landscape*, selection of a *strategy* (using the Memory) and its realization, and comparison of the anticipated landscape (obtained by the complexification process) with the effective landscape at the end of the step.

The *structural temporal constraints* of a CR are translated into inequalities between the propagation delays of the links to the actors, the period of the CR and the stability spans of the objects in the landscape. We have analysed their rôle in the equilibration process, and how their non-respect can lead to different kinds of fractures and dyschronies, more or less easily repaired. In particular, we have studied more explicitly the *"dialectics" between two heterogeneous CRs*, say a higher level CR and a lower level CR with a much shorter period. (Cf. [EV].)

d) The equilibration process also benefits from the Multiplicity Principle (cf. above). This principle (which entails the symmetry-breaking in the passage from a higher to a lower level) has the following important consequences, at the basis of the robustness and plasticity of a MES: - A complex object may emerge as the colimit of a pattern and then take an identity and sustainability of its own, with a well-measured stability span. - The complexification process leads to the emergence of new properties and interactivities, represented by well-defined "complex links" (obtained as composites of "simple links" binding non-adjacent clusters of lower level links). - A sequence of complexifications is not always reducible to a unique complexification with respect to a strategy englobing the successive strategies. (We'll come back on the meaning of this property later on.)

The evolution of the system under these operations can lead to the emergence of objects of strictly increasing order, in particular developing the Memory. For instance, in a neural system there is formation of synchronous assemblies of neurons (in the sense of Hebb) which will emerge as stable new higher order objects, of which new assemblies may be formed, and so on.... Higher order cognitive processes will be memorized under this form.

4. Causality attributions in MES

Rosen has proposed to distinguish simple and complex systems by their causal behavior: in simple systems ("mechanisms"), Aristotelian material causation can be split off from efficient or formal causation, and final causation is rejected. In complex systems, the causal categories are mingled, and some "anticipation" is possible. This distinction is reflected in the model proposed here.

a) First, let us examine the situation relative to a particular CR. During one of its regular (i.e., without fracture) steps, we have seen that the dynamics of its landscape, after a strategy has been chosen, can be modeled by a simple system; its material cause is the initial state, its formal cause the chosen strategy and the efficient cause the realization of this strategy.

Now the choice of the strategy and the equilibration process between the different CRs rely on several factors depending on the system as a whole and not only on the information accessible to the CR. We can classify the relationships between these causal factors according to the description of causality within complex systems proposed by [Chandler 1995, unpublished], from the source of the effect on rate constants of processes: bottom-up, with emergence of higher level structures, interactivities or behaviors through the complexification process; top-down, when higher CRs impose a strategy on lower CRs; inside-outward by recourse to the memory and observance of the structural temporal constraints of the CR; outside-inward, with external constraints and exchanges (e.g., of energy).

b) If we consider the situation during several successive steps of the CR, the dynamics of the landscape evolves by a sequence of complexifications, which generally cannot be replaced by a unique complexification of the first landscape (cf. above). It means that we cannot directly select a strategy on the first landscape leading to the last of the landscapes, but we must first construct the second landscape, then choose a strategy on it leading to the third landscape, and so on up to the last landscape. In terms of causality, it follows that the material, formal and efficient causes have to be "updated" at each step, thus are completely mingled in the global transition from the first to the last landscape, during which the CR does no more function as a simple system. In particular, the emergence of new objects increasing the dimensionality of the system may require the introduction of new observables. Thus the *long-term* behavior of the landscape, even restricted at its level, is no more simple and may reveal an apparent a-causality.

c) However if the system is considered as a whole, the causal interactions between all the levels are continuously merged into the dynamic flow. We can say that a MES is closed under efficient causality, in the sense that the choice of the strategies in response to external constraints, and the equilibration between them, are internal processes, essentially controlled by temporal conditions. Moreover we can attribute some "finality" to the choice of strategies, since their result can be anticipated from the memory. Hence *a MES can be qualified as an "organism" in the terminology of Rosen*, with local and temporal anticipatory behavior due to the memory, and adaptability coming from the Multiplicity Principle.

This result could surprise: we say that the MES offer a relational model, incorporating time, and in which final causation and function are present; now Rosen has said that there cannot be such a model. Is there a contradiction? No, because in MES time is not just a parameter (as in Physics), but intervenes as a complex multifold dynamical process: each CR operates at its own time-scale; the equilibration process plays on the differences and constraints introduced by these time-scales; finally the Memory, which in some sense subsumes the past and the present, allows for some - more or less exact - anticipations of the future, so that a future state may influence the present state. Thus there is no need of new science to study life, only more thorough reflection on the nature of Time and Organization.

5. Conclusion

Mechanisms and Organisms satisfy the same physical laws. But the complexity of the latter comes from the fact that their dynamics is modulated by the coordinated and/or competitive interactions between a net of incommensurable simple systems, with different time-scales entailing specific structural temporal constraints, each one acting as an efficient agent with the help of a central memory which develops by memorizing past experiences.