The commutative diagram that I’ve chosen as a logo should be interpreted as a dynamical system. It “wants” to become commutative. If I am not mistaken and my wild shot in the dark hits the mark, it will capture the essence of cortical column computation.
The interpretation below is not the most general one. Y need not be a discrete space and the smoothing operator μ may involve averaging over memory content instead of majority decision. (CMAC). The situation is not entirely clear to me, but there seems to be some kind of duality between cognition and action, between glueing and cleaving apart.
X is a large topological space, an approximation of a true continuum. Y is a small set of symbols, with discrete topology. A continuous function f : X ⟶ Y on an open subset is locally constant. The h i are such functions, locally constant maps from large subsets of X into small spaces Xi. They are address spaces of bulk digital memories. Evaluation of f i is just memory lookup.
Now we have two maps into Yn : Δ ∘ f and (f i) ∘ (h i)
As x ∈ X varies, these maps may or may not agree.
On the one hand, this is just a complicated way of storing a given classifier f : X ⟶ Y by exploiting the vernier principle ( see also ). On the other hand, we may turn it into a dynamical system by the addition of a simple feedback rule: The map μ: Yn ⟶ Y is a majority decision whose results are, under certain conditions, fed back into the memory cells addressed by the h i.
A brain is made up of myriads of such interlocking cognitive wheels.
Next: Cortical Column Computation