To cognize is to categorize revisited: Category Theory is where Mathematics meets Biology

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1 To cognize is to categorize revisited: Category Theory is where Mathematics meets Biology Autonomous Systems Laboratory Technical University of Madrid

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3 Index 1. Category Theory 2. The structure-function trouble 3. The key is the structure 4. The NN approach in Cognitive Architecture 5. Modelling the Hippocampus 6. Algebra of concepts 7. Conclusions

4 Category theory Category theory was created in the 40 s by Eilenberg & McLane CT is the mathematical theory that studies the structure Think with arrows, express key concepts in terms of arrows rather than elements [Arbib & Manes Arrows, Structure and Functors ] ida:a A However counter-intuitive it may be, in category theory maps rather than objects have the primary role

5 Category theory A category K is composed of A collection of objects Obj(K) For each pair of objects A,B in Obj(K) a collection of morphisms K(A,B) i.e: f:a B Morphisms are composable f:a B g: B C, f o g : A C Every A of Obj(K) has an identitity 1A: A A such that 1A o f = f and g o 1A = g Composition is associative: if f:a B g: B C h: C D then h o( f o g) = (h o f) o g

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7 The structure-function trouble Cognitive and anatomical models of the brain are not valuable on their own but in terms of their mutual convergence In the neural correlate finding agenda, there is an implicit assumption: there are two well diferentiated domains that need to be causally connected: on the one hand the neural structures located in the brain and on the other the cognitive functions of the mind.

8 The structure-function trouble A global picture of the brain structure and function that is able to capture the principles of organisation in the brain, is still missing. We need a theory that dissolves the dualistic notion of structure-function mapping of the brain

9 The reverse inference trouble in brain studies Direct inference dictates which brain areas are active given a cognitive process. Inverse inference determines from the activation of a brain region, which particular cognitive process is engaged. [Gomez 08] direct inference can be defective in terms of precision, while reverse inference can also be a logical fallacy. Reverse Inference problem as it is formulated in neuroscience is ill-founded or at least mathematically outdated because considers the structure based upon naive set theory.

10 The key is the structure Need for a holistic, structural approach to brain function understanding. Formal sciences provide provable knowledge about the real world, thanks to the fact that FS principally deals with the structure and not the things themselves.[poincaré] The structure must reflect the invariance through transformations, which is embodied in the functor

11 The key is the structure

12 Functor m 2 m 2 Category C m 3 m 2 ο m 1 C Functor F F (B) Category D A model is a functor domain = classifying category codomain = any category with similar structure Functor is structure preserving mapping between categories F (m 1 ) F (A) F (m 3 ) F (m 2 ) ο F (m 1 ) F (m 2 ) F (C)

13 The key is the structure Exploit the notion of structure as a repeated pattern that can be mathematically formalised. The concept of structure needs of a rigorous formulation Disputes like whether the neural correlates of cognition are dispersed (the activity of a particular neuron is not representative) or sparse (the level of individual neurons is selective of a concrete feature) acquires a different insight

14 The key is the structure The unreasonable effectiveness of mathematics [Wigner] to explain the non mathematical realm lessen. Mathematics describes the structures and patterns, and the structures are present in the physical system itself. For example, a cognitive agent (the physical system), charts his way to the world through its own structures which are homomorphic to the structures instantiated in the objects of the world.

15 Symmetry & CT in Pattern formation Is symmetry an useful tool in neuroscience? Crystals, atom of hydrogen can be modeled by symmetry Using the groupoid formalism, all robust patterns of synchrony in a network can be worked out And CT build the colimit of the patterns

16 An example of symmetry in biology The leech heart consists of two lateral sinuses, each a tube-like series of chambers, located in the third to eighteenth segments of the leech s body [I. Stewart] Networking opportunity, Nature 2004

17 The NN approach in Cognitive Architecture Cognitive architectures conceived in the pre fmri era, like for example SOAR or ACT-R, reflected assumptions about human cognition which were based on facts derived from psychology experiments. ACT-R + adaptive linear algebra + fmri, conform a computational model not a mathematical model. Category thery could help with old linear algebra and naive mathematics in NN ACT-R circuits are really fibre bundles with lifting and projection maps from category theory and algebraic topology.

18 The NN approach in Cognitive Architecture A neuron is a neuron is a neuron. How of inspiration and How of realism is in the Cognitive Architectures based on NN

19 The NN approach in Cognitive Architecture Is NN a legitimate representation of neuropsychology? Any connetionist model that neglects the cytoarchitecture cannot be neurobiologically realistic NN postulates point-neurons whereas actual neurons are more, much more Indeed it is the neural arborisation (not just the preykarya or cell-body) that could cure Alzeheimer

20 Modelling the Hippocampus The Hippocampus as a laboratory for CT and Grupoids

21 Modelling the Hippocampus Place Cells The majority of principle cells in the hippocampus are show spatial correlated activity PC are not topographically organised, the randomly re-map for every new environment Grid Cells located in mec fire selectively when rat visits vertices in triangular grids mec shows a topographic organisation

22 Modelling the Hippocampus [The hippocampus and memory: insights from spatial processing Chris M. Bird & Neil Burgess Nature Reviews Neuroscience 9, (March 2008)]

23 Modelling the Hippocampus Maybe.the hippocampus does not store at all metrical relations? Some insist in extract and generate from the entorhinal cortical grid cell system There is more than topographical maps, euclidean space and general (point set topology) Differential topology: studies the structure of flows, attempt to inegrate local with global Algebraic topology: Algebraizes topological properties obtaining the equivalences

24 Algebra of concepts The animal has a hierarchy of mental objects, from the mental image of a simple object modeled by a single neuron, to mental objects activated by a synchronous assembly of neurons, up to more complex mental objects combining more elementary ones The neurons type A, place cells, are identified to cat-neurons of level 0. The cat-neurons of level1 bind patterns of neurons. To model more complex mental objects, we have to iterate the construction and form cat-neurons of increasing complexity levels, binding patterns of cat-neurons of lower complexity

25 Algebra of concepts [3]The iterative building of cat-neurons binding patterns of cat-neurons of lower levels, models the computation on mental objects, to form more complex ones. It gives explanatory power for the construction of an algebra of mental objects

26 Models with categorical representation [Edelman] Consciousness loops [Baars, Franklin] Global workspace [Grossberg] Adaptative Resonance Theory

27 Conclusions Science must proceed testing the theoretical framework against facts, empirically testable In the absence of a theory written in mathematical terms, the separation between the different disciplines within cognitive sciences will be progressively more acute and an understanding between them unattainable. We need a shift towards the formal sciences in cognitive sciences. Category theory is a powerful foundational framework for modeling in cognitive sciences worthy to be explored

28 THANKS! For more info, please visit: iceaproject.eu This work has been supported by ICEA (Integrating Cognition, Emotion and Autonomy) funded by IST Cognitive Systems Unit