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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)
J.-P. Vanbremeersch Physician (Université de Picardie): specialty in geriatry

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.

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