ABSTRACT. The dynamics of an organism is modulated by the interactions with its context. How does he learn to recognize the main features of his environment and to respond in an adequate manner? How can he acquire emergent capacities, such as higher cognitive processes up to consciousness?

These questions are analyzed in the frame of a mathematical model for natural complex systems, the Memory Evolutive Systems (MES). This model, based on Category Theory, has been developed by the authors in a series of papers spanning the last 15 years, summarized in their Internet site.

Here it is applied to the development of memory, semantics and consciousness by a higher animal. The main idea is that consciousness relies on the formation of a personal 'affective' memory of the animal, his body, his experiences and interactions with his environment, called the Archetypal Core, at the basis of the notion of self.

KEYWORDS. Complex system. Cognitive system; Consciousness. Mathematical model. Category Theory.


1. Introduction

The analysis of a natural complex system in its context, say a biological or a social system, raises the following problems: The system is organized into various complexity levels, with their own temporality, yet its interactions with its environment are coherent, and the components and the organization can be partially modified while the identity of the whole system is maintained. It is autonomous with a capability to learn to recognize features of the context and to develop adapted strategies in answer; how can it memorize a hierarchy of representations which are sufficiently stable though adaptable to the context? The formation of higher organizational levels introduces new properties (example: development of higher cognitive processes up to consciousness); how do they emerge from lower levels without being directly reducible to them?

These questions are studied in the frame of Memory Evolutive Systems, a mathematical model for natural complex systems, based on Category Theory, that the authors have developed in a series of papers since 1987 (e.g. [Ehresmann & Vanbremeersch (or EV) (1987; 1991; 1997; 2002)]). The model allows to describe the emergence of a hierarchy of complex objects, whose flexibility comes from their capacity of switching between various internal organizations, while keeping their identity in time.

Here the model is applied to cognitive systems, corresponding to the development and functioning of the nervous system of a higher animal. up to the formation of a semantic classification [EV (1992)], and still higher of a global invariant, the Archetypal Core, which gives a support to conscious processes and to the notion of 'Self'.


2. Development of a memory

To extract features of his environment, later recognize them and react with suitable behaviors, the animal develops a dynamic though sufficiently stable archive of his various experiences. This 'memory' is formed by records of sensory and motor inputs, internal states, behaviors and strategies.

2.1. Physiologically the response of the neural system to a simple stimulus consists in the activation of a specialized neuron; for instance, in the visual areas, there exist 'simple cells' representing segments of a given direction, and 'complex cells' representing a particular angle. But more complex stimuli, except for some exceptions, do not have their own 'grand-mother neuron', as neuronal imaging confirms. Following [Hebb (1947)] most authors agree that the response to a complex stimulus consists in the activation of a synchronous assembly of interacting neurons (called neuronal groups by [Edelman (1989)] .

2.2. However the memory must be dynamic, and not rigid as that of a computer. Indeed, the same object should be recognized under different aspects (a circle may appear as an ellipse), a behavior must interact with the context, e.g. the motor neurons activated for taking an object depend on its size. Thus an object is encoded as a multifold record which does not represent just one synchronous assembly of neurons but a whole class of such assemblies with possible switches between them; and its later recall consists in the activation of anyone of these assemblies.

2.3. The memory does not store only records but also the manner in which they can dynamically interact. In this way, patterns of interacting former records can be bound together to represent more complex objects, events or behaviors. For instance different features of an object (color, shape,…) are processed by lower level modules (specialized brain areas) and only bound together at a higher level; a complex behavior requires the successive activation of a large number of simpler ones. The ’binding problem’ (e.g. [von der Malsburg (1995)]) examines how patterns of records, generated by widely separated areas, are integrated, allowing the progressive formation of a hierarchy of records, corresponding to more and more widely distributed information. Such a complex record has several ramifications representing assemblies of assemblies of… neurons and can be recalled through anyone of them, possibly later switching to another one.

2.4. The memory is not fixed once for all; it can be modified to take into account changes in the environment. A record is a dynamic unit, gradually modified through feedback from the context; for instance, the animal will learn to recognize more subtle aspects of a prey and to respond with more adapted behaviors.


3. Development of a Semantics

The development of the memory allows for adapted responses to specific stimuli. However to give more independence from the context it is important that the animal can classify the objects in his memory, having for instance a general notion of preys in spite of their individual differences. This relies on the formation of a semantic memory Sem. The literature on semantics is particularly large. We adopt a perspective in which a 'concept' can be seen as an internal representation of a class of items with a "family likeness" in the sense of [Wittgenstein (1953)]. More precisely we model the development of Sem as a 3 step process [EV (1992)]:

3.1. A pragmatic classification with respect to a particular attribute is effected by lower modules; e.g. a visual area dealing with color will respond in the same way to all blue objects.

3.2. This classification takes a 'meaning' only if it is internally detected at a higher level which reflects it by the formation of an abstract representation of an invariance class (we call it a concept). The later activation of a concept can recall the most adapted instance of it in the present context, or lead to balances between several of its instances.

3.3. More complex concepts are formed by binding together simpler ones. Let us note that no language is supposed here. The development of Sem gives a double degree of freedom to modulate the interactions with the context, the recall of a concept being done through the selection of anyone of its instances, and then of anyone of the ramifications of this instance.


4. The Archetypal Core

Now we define two important sub-systems of the memory.

4.1. The existence of a semantic memory allows for the development of a personal 'affective' memory of the animal, his body, his experiences and interactions with his environment, which we call the Archetypal Core (AC). It is formed by records activated more often and during a longer period, from birth on (for instance stable aspects of the environment in contrast to more variable ones, deep feelings,…), their links and their concepts. It develops to integrate the main sensorial, proprioceptive, motor experiences, …, with their emotional overtones and the basic strategies associated to them, and to connect them into patterns with links strengthened with time. It may be autonomously activated, so that a whole sub-system of the AC is activated as soon as a small part of it is stimulated. For instance a record in it (say the blue sky) is archetypally linked to other records not only of perceptions or of motor processes but also of internal states and emotions (sun, heat, well-being, swimming,…).

4.2. The self-activation of the AC is directed and maintained for long periods by specific bundles of strong, quickly activated links connecting each archetypal record to some other objects of the AC. These bundles, which we call fans (to be compared to the chreods of [Waddington (1940)]), act as channels through which the activation of the record resonates as an echo, and is propagated first to the target concepts, then oscillates through a sequence of loops based on balances between various instances of the concepts and switches among their ramifications (physiologically, it relies on the thalamo-cortical loops). The fans are gradually strengthened, leading to more and more integration of the whole AC.

The AC, as a permanent representation of the animal, his phenomenal experiences, his acquired knowledge, be it pragmatic, social or conceptual and the basic strategies associated to them could be the basis of the notion of self.

4.3. Some records and concepts represent experiences which are sufficiently significant to have strong links toward the AC, possibly without belonging to it; with their links they form the experiential memory Exp (to be related to the "value-category memory" of [Edelman (1989)]). Each record in Exp recalls the 'archetypal' experience most closely linked to it, whose activation spreads through the fans into loops which remain self-generated for a long time and diffuse to close domains. Each record in the memory is linked by a cluster to a pattern of related experiences in Exp. If the links from a record to Exp become strong enough, the record will become integrated in Exp, and possibly later on in the AC.

5. The consciousness process

Recently consciousness has become a much studied topic, with often diverging viewpoints (cf. papers in the Journal of Consciousness Studies, the Tucson Conferences, and numerous books, e.g. [Baars (1997); Chalmers (1996); Changeux (1983); Crick (1993); Damasio (1999); Dennett (1991);…]).

We assume that the development of the AC is at the basis of the existence of consciousness, and we have proposed [EV (1992); (2002)] to characterize consciousness as a 3 parts process which integrates the structural and temporal dimensions and leads to more adapted strategies.

5.1. Conscious processes are aroused by a relevant event without an automatic response, caused either by the context (unknown or threatening stimuli) or internally (related to feelings, activities or strategies in progress, e.g. 'fracture' at a higher level). It instantly determines an increase of awareness (activation of the reticular formation) and starts a semiotic search in the memory to try to recognize it and respond adequately. The search is done through modules of various modalities and levels, up to the semantic memory, and heavily relies on the archetypal core which, always in the background, acts as a referent and as a filter, propagating the information through its multifold records and their echo. The information so gathered form a 'holist' extended landscape, whose objects are the temporal perspectives of the event and of its internal reflections (to be compared with the "Global workspace model" of [Baars (1997)]). Physiologically its construction relies on the existence of what [Edelman (1989)] calls the "consciousness loop".

5.2. A retrospection process, also based on a series of balances and switches, will be operated on this extended landscape; it uncovers lower levels of the near past to find, by abduction, the possible causes of the present event.

5.3. Then a prospection process will select long term strategies for several steps ahead, to respond in the most adapted way. This is accomplished through the formation of virtual landscapes inside the extended landscape, with the help of the whole memory and more specially the archetypal core, in which successive strategies (selected as concepts) can be 'tried' without material cost for the system.


6. A mathematical model: the MES of neurons

Here we give a brief outline of a mathematical model representing the various processes described above. It is based on Category Theory for which we refer to [Mac Lane (1971)].

6.1. Let us briefly recall that a MES consists of the following data (cf. [EV (1987); (1991)]):

- An evolutive system K, that is a timescale T (subset of the real numbers), a family of categories Kt representing the state of the system at the instant t, and a (partial) functor transition from Kt to Kt' for each t < t', representing the change of state. The system is hierarchical if the objects of Kt are divided into 'complexity levels', so that an object of level n+1 is the colimit (or inductive limit [Kan (1958)]) of at least one pattern of linked objects (or diagram) of level n.

- A hierarchical evolutive sub-system Mem of K, called the memory, whose number of levels may increase in time.

- A net of evolutive sub-systems of K, called Coregulators (or CR) which direct the dynamics of the MES. The CRs act in parallel, but each at its own discrete timescale (finite subset of T) and complexity level and with a differential access to the central memory Mem.

At each step of its timescale, say from t to t+1, a CR operates a 3 parts trial-and-error learning process [EV (1991)]: 1. As an 'observer' it integrates the information it can receive from the system and its environment into its actual landscape Lt, which is a category whose objects are specific clusters of morphisms from an object of Kt to the CR; 2. As an 'actor' it selects a strategy S on Lt taking into account the different constraints and the information in Mem from earlier similar experiences; the objectives of the strategy are to bind certain patterns (by adjunction of a colimit), and to suppress or decompose some objects; if the strategy succeeds, the next landscape L(t+1) should be the complexification [EV (1987)] L' of Lt with respect to S. 3. As an 'evaluator' at the next step it compares L' to L(t+1) by a 'comparison' functor.

At a given time, the strategies of the various CRs are reflected to the system where an equilibration process (the 'interplay among strategies') is operated; it can lead to fractures for the CRs whose strategies cannot be coherently integrated. Thus the dynamics of the MES is modulated by the dialectics between heterogeneous CRs.

6.2. The MES associated to a cognitive system is the MES of neurons of the animal, based on the 'category of neurons' Neur of the animal at t, defined as follows (cf. [EV (1991); (1999)]): take the directed (multi-)graph whose objects are the neurons and the arrows the synapses between them; this graph is labeled by the strength of the synapses related to the manner they propagate an influx; form the category of paths of this graph and define the strength of a synaptic path as the product of the strengths of its factors. Then Neur is the quotient category of this category of paths by the equivalence identifying 2 synaptic paths having the same source, target and strength.

An assembly of neurons is modeled by a pattern of linked objects (or diagram) in this category, and its binding into a record corresponds to the adjunction of a colimit of this pattern in a suitable complexification of the category [EV (1987); (1991)].

The state-categories of the MES of neurons are obtained by successive complexifications of Neur. The complexification process gives a solution to the "binding problem" because it also determines what are the 'good' links between newly formed complex records, thus allowing to iterate the process and form increasingly complex assemblies of interacting records.

6.3. The memory of the MES of neurons corresponds to the evolutive sub-system Mem of the MES formed by the records and their links. Its flexibility comes from the fact that a record can be the colimit of different patterns (we say that the MES satisfies the Multiplicity Principle [EV (1997)]), and its later retrieval can be done through anyone of them.

The CRs of the MES model particular modules or areas of the brain.

The semantic memory is constructed as an evolutive sub-system Sem of Mem. The pragmatic classification corresponds to the fact that the same pattern of agents of a lower level CR is activated, and the associated concept is represented by the adjunction of a (projective) limit ([Kan (1958)]) of this pattern, through the operation of a higher CR [EV (1992)].

The experiential memory and the archetypal core are modeled by evolutive sub-systems Exp and AC of Mem. The categorical characteristics of these sub-systems are as follows: Exp is a final sub-system of Mem, and it admits AC as a reflective sub-system [Mac Lane (1971)]. Moreover the distinguished fans in AC equip it with a supplementary structure, namely a Grothendieck (co)topology [Grothendieck and Verdier (1972)].



In this model, the evolution of the archetypal core and of consciousness takes into account the whole experience of the animal, be it perceptive, behavioral or (as emphasized by [Damasio (1999)]) emotional, with an integration of the temporal dimensions. The degree of consciousness will depend on the development of the archetypal core. Consciousness gives a selective advantage, since it allows to search for the causes of an event instead of reacting only to its effects. And by programming on the long term, more effective strategies may be devised, less constrained by the immediate concerns. It does not need language (higher animals may have such a consciousness) though language allows for more efficient processes. And for the "hard problem", the qualia could correspond to perspectives of records of the archetypal core in the extended landscape.



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