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# Memory Evolutive Neural Systems ## When Category Theory meets Neuroscience ##### Younesse Kaddar ##### Based on A. Ehresmann and J.-P. Vanbremeersch's work

### Introduction: what's wrong with biological systems? ____ ### I. Complex objects as colimits ### II. Emergence theorem ### III. Memory Evolutive Neural Systems

Introduction

Memory Evolutive Systems:

Categorical modelisation of hierarchical, evolving and self-regulating systems (≃ 30 years of research already!).

Authors

A. Ehresmann Mathematician (Université de Picardie):
  1. Functional analysis
  2. Category theory with Charles Ehresmann
  3. Transdisciplinary research: natural complex systems (biological, social, cognitive)
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J.-P. Vanbremeersch Physician (Université de Picardie): specialty in geriatry </tr> </table> # What's wrong with biology? ## 1. Emergence vs. Reductionism ## 2. Self-organisation - Flexibility - Adaptability
Dyslexia - Key parts of the brain not (→ but may become) developed for reading
# I. Make room for... Category theory! Structures preserved by morphisms$\qquad \overset{\text{+ associativity, identity}}{\rightsquigarrow} \qquad \underbrace{\text{Category Theory}}_{\text{focus on relations rather than objects}}$ ________________ *Examples*: - Objects preserved by morphisms: - Sets - Vector spaces, Topological spaces, Manifolds, Groups, Rings, ... - Logical formulas, Types (λ-calculus), ... - Structures forming *one* category: - Posets, Monoids, Groups, Groupoids, ... ## ... and even (small) categories themselves form a category!
- **Objects:** Categories - **Morphisms**: Functors

## Universal constructions **Limits**: : - Terminal objects - Products - Equalizers (Kernels), Pullbacks, ... **Colimits**: : - Initial objects - Coproducts - Coequalizers (Quotients), Pushouts, ... > ⟹ Colimits are **more** than sums ### Complex objects as colimits
Figure - Colimit of a pattern/diagram
>**Hierarchical category $H_t$**: layers of complexity: objects at level $n$ are colimits of patterns of level $<n$ # II. Hierarchical Evolutive Systems (HES)
Hierarchical Evolutive System (HES) $K$: : - a timescale $T ⊆ ℝ_+$ - $∀ t∈ℝ_+$, a hierarchical category $K_t$ (configuration at $t$) - $∀ t < t'$, a *transition* functor $k_{tt'}: K_{tt'} ⊆ K_{t} ⟶ K_{t'}$ - *Transitivity condition*: a *component* $C ∈ H$ is a maximal set of objects linked by transitions.
Figure - a Hierarchical Evolutive System (HES) $K$
# 3. Multiplicity Principle >**Multiplicity Principle:** for each time and each level, there is at least one multi-faceted component, *i.e.* two patterns with the same colimit that are not isomorphic in the category of patterns and clusters.
## Emergence of complex links
# Complexification "Standard changes" of living systems (cf. René Thom): *birth:* (addition of new components), *death* (addition of certain components), *collision* (formation of new pattern bindings), *scission* (destruction of certain bindings).
## Emergence Theorem > Complexification preserves the Multiplicity Principle. In a HES, there may be: - formation of multi-faceted components more and more complex - ⟶ emergence of complex links between them. ## III. Memory Evolutive Neural Systems ## Flexible Memory
### vs. Hopfield networks Hopfield rule: : to store patterns $\textbf{p}_1, ⋯, \textbf{p}_n$ in the Hopfield network, the weight matrix is set to: $$W = \frac 1 N \sum\limits_{ i } \textbf{p}_i \textbf{p}_i^\T$$ cf. Jupyter Notebook for simulations ________________ Hopfield networks: - the stored patterns are not apparent in the network - pattern capacity: $≃ 0.138 N$ (where $N$ is the number of neurons) - no "flexibility": if your mom's appearance changes when she ages (which is unfortunately likely to happen), you don't recognize her anymore! - **spurious patterns:** linear combinations of odd number of patterns also stored! - unoriented graph! # Conclusion