# MENS, an Info-Computational Model for (Neuro-)cognitive Systems Capable of Creativity

## Abstract

**:**

## 1. Introduction

## 2. Why Category Theory in Memory Evolutive Systems?

_{i}with some distinguished links f: P

_{i}→ P

_{j}. A collective link from P to an object N is a family (s

_{i}) of links s

_{i}from the different P

_{i}to N, such that fs

_{j}= s

_{i}for each distinguished link f: P

_{i}→ P

_{j}of P. The pattern admits a colimit (or inductive limit [5]) M if there is a collective link (l

_{i}) from P to M which factorizes any other collective link, so that the collective links (s

_{i}) from P to any N are in 1-1 correspondence with the links s: M → N binding them; formally, for each i, we have the equation s

_{i}= l

_{i}s. (cf. Figure 1)

**Remark.**Collective links and, a fortiori, colimits make an essential use of the composition of the category (via the above equations, as seen on the figure) and could not be defined in simple graphs. Category theory extensively uses diagrams, in particular commutative diagrams (in which two paths with the same extremities have the same composite). Proofs are often made more intuitive by reasoning on figures rather than writing the corresponding long sequences of equations; it is what is called the “diagram chasing” process, ubiquitous in articles using categories.

- (i)
- The structure of the system is changing, its components and their links varying over time. The few models of complex systems using category theory (e.g., inspired by [8]) only consider one category representing the invariant structure of system. On the contrary, in MES the system is not represented by a unique category but by what we call [9] an Evolutive System; it consists of a family of categories K
_{t}, representing the successive configurations of the system at each time t, and partial transition functors from K_{t}to K_{t'}accounting for the change from t to t' (a functor [4] is a map between categories which preserves their composition and identities.) - (ii)
- The system is hierarchical, with a tangled hierarchy of components varying over time. A component C of a certain level ‘binds’ at least one pattern P of interacting components of lower levels so that C, and P acting collectively, have the same functional role. Modelling this hierarchy raises the Binding Problem: how do simple objects bind together to form “a whole that is greater than the sum of its parts” [10] and how can such “wholes” interact? In the categorical setting, the ‘whole’ C is represented by the colimit of the pattern P of interacting simple objects; and the interactions between wholes are described in terms of simple links and complex links (cf. [3] and Section 4).
- (iii)
- There is emergence of complex multiform components, with development of a flexible central memory. Whence the Emergence Problem: how to measure the ‘real’ complexity of an object and what is the condition making possible the emergence over time of increasingly complex structures and processes? We characterize this condition as the Multiplicity Principle [3], a kind of ‘flexible redundancy’ which ensures the existence of components which binds various lower level patterns which are not isomorphic nor even well connected; such components are said to be multiform. And we prove that the Multiplicity Principle is necessary for the emergence of increasingly complex objects and processes with multiform presentations, constructed by iterated complexification processes. The complexification of a category K describes the ‘universally constructed’ category K' deduced from K after changes of the following kinds: addition of given new objects, formation (or preservation, if it exists) of a new object which becomes the colimit of some given pattern, suppression or decomposition of some objects. K' has been explicitly constructed in [9] and [3] (its links are both simple and complex links, cf. Section 4), and the construction is amenable to usual computations.
- (iv)
- The system has a multi-agent self-organization. Its global dynamic is modulated by the cooperation/competition of a network of internal functional subsystems, called the co-regulators, acting as agents with the help of a long-term memory. Each co-regulator operates locally with its own rhythm, procedures and complexity, but the commands of the different co-regulators can be conflicting and must be harmonized. While the local dynamics are amenable to conventional computations, the problem is different for the global one.

## 3. Properties of the Neural System

- (i)
- The graph of neurons at an instant t: Its objects represent the states N
_{t}of the neurons N existing at t (measured by their activity around t), its links from N_{t}to N'_{t}represent the states of the synapses from N to N', weighted by their propagation delay around t and by their strength (to transmit an activation of N to N'). A synapse can be active or passive at t. The activity of N at t is a sum of the activities of the neurons connected to N by an active link, pondered by the strength of this link. The graph changes over time: some neurons ‘die’, new neurons are formed, and the same for synapses; the activity of a neuron varies, delays and strengths of synapses may also slowly change. - (ii)
- The structural core. The graph of neurons has a central sub-graph, called its structural core, discovered by Hagmann et al. [11] in 2008: “Our data provide evidence for the existence of a structural core in human cerebral cortex. This complex of densely connected regions in posterior medial cortex is both spatially and topologically central within the brain <….> the core may be an important structural basis for shaping large-scale brain dynamics <...> linked to self-referential processing and consciousness.” Recently (2011) it has been found that this core is a sub-graph with several hubs (e.g., the precuneus and the cingulated cortex) forming a “rich club” [12].
- (iii)
- Synchronous assemblies of neurons. Already in the forties Hebb [13] has noted the formation, persistence and intertwining of more or less complex and distributed assemblies of neurons whose synchronous activation is associated to specific mental processes: “Any frequently repeated, particular stimulation will lead to the slow development of a ‘cell-assembly’ as a close system”. And he gives the Hebb rule for synaptic plasticity: “When an axon of cell A is near enough to excite B and repeatedly or persistently takes part in firing it <…> A’s efficiency, as one of the cells firing B, is increased.” This rule has been experimentally verified in different brain areas.
- (iv)
- Degeneracy property of the neural code. Emphasized by Edelman, it says that: “more than one combination of neuronal groups can yield a particular output, and a given single group can participate in more than one kind of signaling function.” [14]. Thus the mental representation of a stimulus should be the common ‘binding’ of the more or less different neural patterns which it can synchronously activate in different contexts or at different times.
- (v)
- Modular organization. The brain has a modular organization, with a variety of ‘modules’ or areas of the brain with a specific function, from small specialized parts (the “treatment units” of Crick [15]) such as visual centres processing colour, to large areas such as the visual or motor areas, or nuclei of the emotive brain (brain stem and limbic system) or the associative cortex. These modules interact to direct the self-organized dynamic of the system.

_{t}, which is the category of paths of the graph of neurons at t: its objects model the states of the neurons N existing at t, the links model the synaptic paths between them, labelled by their propagation delay and strength (defined as the sum of those of their factors).

_{t}at t of a neuron N its new state N

_{t'}at t' provided that N still exists at t', and similarly for the links. The transitions describe what has changed, but they do not indicate the kind of computation (as processing of information) which is internally responsible for the change. A component of NEUR models a neuron through the sequence of its successive states; it is still called neuron, or cat-neuron of level 0. NEUR constitutes the level 0 of MENS, from which higher levels are constructed by iterated complexification processes.

## 4. Category-Neurons and Their Links

_{i}) from P to N, allowing that all the s

_{i}transmit an activation of P

_{i}to N at the same time; in particular this imposes that all the zigzags of links between P

_{i}and P

_{j}have the same propagation delay. A pattern with this property is said to be polychromous.

_{t}models the present state of the neural, mental and cognitive system; its objects are the cat-neurons of any level (from the level 0 of neurons up) existing at t with their activity, and their links with their propagation delay and strength; a link is active or not at t.

_{t'}at t' is obtained as the complexification of MENS

_{t}[cf. Section 2(iii)] with respect to a procedure Pr having objectives of the preceding kinds (cf. Figure 2). Such a complexification is solution of the "universal problem" of constructing a category in which the objectives of Pr are satisfied in the ‘best’ way. We have given an explicit construction of the complexification, in particular of the links between cat-neurons; and we have shown in [18] how, using its universal property, the propagation delays and strengths of synaptic paths (at the level 0) can be extended to the links of any level, as well as the Hebb rule.

- (i)
- Simple links. They ‘bind’ clusters of lower level links as follows. Let M and M' be 2 cat-neurons binding lower level patterns P and P' respectively. If we have a cluster G of links from P to P' well correlated by the distinguished links of P and P', this cluster binds into a link from M to M', called a (P, P')-simple link (or n-simple link if P and P' are of level ≤ n). Such a link just translates at the level n+1 the information that P can coherently activate components of P' through the links of G; and this information is computable at the lower levels. A composite of n-simple links binding adjacent clusters is n-simple.
- (ii)
- Complex links. They emerge at a higher level, as composites of n-simple links binding non-adjacent clusters. Their existence is possible because of the existence of cat-neurons M which are multiform. Figure 3 presents a complex link from N to M' composite of a (Q', Q)-simple link with a (P, P')-simple link, where P and Q are non-connected decompositions of the multiform cat-neuron M. Such a link represents information emerging at the level n+1 by integration of the global structure of the lower levels, and not locally computable through lower level decompositions of N and M'; indeed the fact that the cat-neuron M is multiform imposes global conditions, calling out all its lower decompositions and their collective links; could it be amenable to some kind of unconventional computation?

**Remark.**Here we only speak of cat-neurons constructed by colimits. In fact there are also cat-neurons obtained by projective limits [5], which arise for instance in the construction of a semantic memory. When the procedure asks also for the formation of such ‘classifying’ cat-neurons, we speak of a mixed complexification; its construction is more complicated [3].

**Emergence Theorem**

**.**Iterated complexifications preserve the Multiplicity Principle and lead to the emergence in MENS of cat-neurons of increasing complexity order, representing more and more complex mental objects or cognitive processes.

## 5. Local and Global Dynamic of MENS

- (i)
- Formation of the landscape at t. It is a category L
_{t}which models the partial information accessible to CR through active links: its objects are clusters G from a cat-neuron B to CR with at least one link activating a cat-neuron in CR around t. It plays the role of a working memory for CR during the step. - (ii)
- Selection of an admissible procedure Pr to respond to the situation with adequate structural changes. It is done through the landscape, using the access of CR to Mem to recall how the information has been processed in preceding analogue events. For instance in presence of an object S, a CR treating colours will retain only information on the colour of S, and the objective of Pr could be to bind the pattern P of neurons activated by the colour to memorize the colour or, if already known, recall it.
- (iii)
- Commands of the procedure are sent to its effectors in MENS. In the above example, the binding of P into a CR-record of S consists in strengthening the distinguished links of P using Hebb rule. The dynamic by which the effectors should realize the commands during the continuous time of the step is computable, using differential equations (for instance of the “Cohen-Grossberg-Hopfield with delays” type) in the phase space of the landscape whose coordinates are the activities of the cat-neurons and the strengths of the links between them; we refer to [18] for more details.
- (iv)
- Evaluation at the beginning of the next step, by comparison of the anticipated landscape (which should be the complexification of L
_{t}with respect to Pr) with the new landscape; then Pr and its result are recorded. If the commands of Pr have not entirely succeeded, we say that there is a fracture for CR.

## 6. AC and Higher Cognitive Processes

- (i)
- A retrospection process (toward the past) proceeds by abduction (in the sense of Pierce [22]) to recollect information back in time thanks to its reinforcement in GL. Processing this information allows for analyzing the event S which has triggered the formation of GL and finding its possible causes, thus “sensemaking” of the present.
- (ii)
- A prospection process (toward the future) is then developed in GL, to try and select long term strategies. It is done through the formation, inside GL, of local ‘virtual’ landscapes (representing “mental spaces”), where successive procedures can be tried by internally constructing the corresponding complexifications, with evaluation of their benefits and of the risk of dysfunction. A sequence of alternating retrospection and prospection processes thus leads to various ‘scenarios’. Once a scenario is selected, the retrospection process allows back-casting to find sequences of procedures (implicating co-regulators of various levels) able to realize this long term program.

## 7. Conclusions and Future Work

## Acknowledgments

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Ehresmann, A.C. MENS, an Info-Computational Model for (Neuro-)cognitive Systems Capable of Creativity. *Entropy* **2012**, *14*, 1703-1716.
https://doi.org/10.3390/e14091703

**AMA Style**

Ehresmann AC. MENS, an Info-Computational Model for (Neuro-)cognitive Systems Capable of Creativity. *Entropy*. 2012; 14(9):1703-1716.
https://doi.org/10.3390/e14091703

**Chicago/Turabian Style**

Ehresmann, Andrée C. 2012. "MENS, an Info-Computational Model for (Neuro-)cognitive Systems Capable of Creativity" *Entropy* 14, no. 9: 1703-1716.
https://doi.org/10.3390/e14091703