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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): Functional analysis Category theory with Charles Ehresmann Transdisciplinary research: natural complex systems (biological, social, cognitive) </td> </tr>
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