siberean

VGavrilov
August 25, 2011

/Draft/

Abstract:
High-level cognitive functions can be decomposed into 3 primitive math operations: associative memory storing, partitioning and disjoint (in sets terminology), or surjection and injection and so it is likely that during the evolution the abilities to do those primitive functions by a brain have been evolved in different times: disjoint first (visual cortex), then storing (by means of hippocampus), and partitioning - the last. If future fMRI research will show localization of the partitioning function in a separate from the visual cortex region, for example in neocortex - it will be in favour of the model.


Associative memory (as opposed to address or location-based memory) is the very basic memory while associative learning is the very basic type of brain operations in the human brain (and not only human). The area of associative processing is the same in monkeys and in humans *) and it is done in lower, 'sensory' areas rather than in the ITC (inferior temporal cortex), how it was thought in the past *). ITC is currently considered to be responsible for more complex processing such as visual processing and complex object features and shapes.
As for the associative memory apparatus - one of the most critical regions is hippocampus, one of the so-called "middle brain" regions, i.e. pretty 'low-level' layer of the brain comparing to the neocortex, which had been developed significantly only in our closest ancestors. Hippocampus is not only critical for memory formation but it is also responsible for spatial abilities and navigation in mammals, and evolutionary it is an ancient organ.
So associative memory by itself (as a storage and basic routine) is rather not unique to human beings and appeared long time ago during the evolution for processing of basic routines and is not different from mammals or at least monkeys (which is shown by fMRI).
However, the processes - how stored associations are used subsequently in the information processing pipeline - are not well understood by neurobiologists yet.
Understanding the complex parallel processing which may involve multiple brain areas may take time, and for achieving that - some models of cognitive functions should be built.

We take a different, interdisciplinary approach to come to the models. Let's take math trying to find the most basic processing primitives, necessary for the cognitive function. By optimality of nature and evolution and Occam's razor - the real mechanisms should not be much more complex than that (a sequence of primitive functions). At least - the real mechanisms should be able to be gained by an adding of elementary useful functions as the result of collected useful mutations, not simultaneous appearance of multiple necessary functions at once, because despite of usefulness of such complex function - the probability of simultaneous appearance of multiple features would be small.
The model should be close to the real brain not only functionally but ideally it should also reflect partitioning of the real functional parts, which appeared evolutionary in different times, into separate units, as they appeared in the brain (example: memory, vision). That will allow not only modelling brain functionally but also being able to find a similar mechanism in the brain when it is not found yet, but been demanded by the model. So, when a feature, which is following from the model, will be later found in the brain - this may give a clue about adequacy of the model or vice-versa.

What is necessary for any complex processing? Work of the brain is a function. Mathematically it may be represented as a function of inputs from the environment and the internal states (brain's memory). The brain creates a "virtual world" in itself (in consciousness loop) which models the real world, based on experience (memory, been collected during the lifetime and the state of the brain been inherited). That experience is been collected as associations (the very basic and simple form of memory), which are formed in hippocampus and later stored in long-term memory. So, the associative memory is the 1st necessary function (storing function). However, hippocampus is more high-level organ than visual cortex, and it is visual cortex which makes that most basic operations (processing) function - even before memory capacity is necessary. Let's move to the math and will come to this from a different side.





From math we know that any function can be decomposed into a surjection and an injection (we can decompose to a different combination, but the motivation of taking surjection will be clear soon). Surjective function - is a function on sets if every element in domain can be obtained by the function from some element in the range.
Injection function is one that preserves distinctness. In the terms of sets - surjection and injection are partitionining and disjoint, correspondingly.
So, any complex function collapses ("divide and coquer" approach) into two simpler sequential operations (and we know that the brain and evolution "likes" sequential operations: it is easier to do 2 simpler operations one after another rather than to "invent" a complex operation).
Why we took surjection and injection?
Let's consider the first one. Partitioning (or surjection) in the material (not mathematical) world of the brain is a process of taking all objects (from a domain) and finding an attribute or a property (i.e. a function) - to divide it into non-intersecting categories. Those categories from now on can be distinguished only by that single attribute (set of attributes always can be concatenated into one) but not by different "intersecting" attributes (not to compare apples and oranges). So, this basic operation, surjection, is "simplifying" the world, without which the complex world could not be understood. In other words - it is "encoding" of the world into a model (in case of a humans - into linguistic model). Simple example: all objects are divided into black, white and colored, three not intersecting sets. There are no objects which exist in co-domain but not in the image (range), i.e. each object is attributed by one of those 3 "encodings": "black", "white" or "colored". It will require a very short message - to transfer or store the information in memory using such efficient encoding. It may be seen that it is partitioning, i.e. a surjection.
We could do the following formal decomposing of the surjection into projection and bijection, but the latest in the "brain world" may be understood as a simple re-mapping or changing of the language, or attribute names, so we can ignore that treating surjection and projection as synonyms (in our "brain world"). So, let's leave surjection (partitioning) as a basic operation of the brain.
We should notice, that dichotomy, i.e. partitioning of a set into 2 disjoint sets - is a special case of more general partitioning, a "binary" version of it.
Starting from the most basic, association operation, multiple associations may lead to one "right-side" of the association, ending in the same category, becoming indistinguishable. With time - "neurons that fire together - wire together" and we have a compressed grammar (of the "left-sides") to manipulate the "right-sides". We should notice here that here we speak about the model of the world (in a brain) as a non-contradictory, deterministic function, ignoring changing of the brain state in time, random fluctuations (which are not detected) and voluntary agreed contradictions (politics) and considering only a consistent non-contradictive "virtual world". At least in a short time of brain activity this is a reasonable assumption.
Phenomenon of organizing associations into sets may be also named as cluster analysis. If something does not have an association - it must be included into some set, so the whole range is consisting of non-intersecting regions (dichotomies: "day" and "night", "bad" and "good", as well as non-dichotomies: "black", "white" and "color", 4 seasons, 12 months etc). However, "compression" must not be only by means of a human language, and in case of animals it may be a reflex.

The second necessary addendum to surjection (to be able to simulate any function) is injection. Injection is a one-to-one function which preserves distinctness and never maps two elements in the domain to one element in the co domain (which surjection do). It does not require image to "fill" the co-domain, so there may be elements in co-domain which are not in the image (which do not have associations yet), however if 2 "left-sides" are distinct - so are the "right-sides" (of the association or a function). This is a function of distinguishing 2 different objects (taken arbitrarily), finding their difference and putting into disjoint sets which are not intersecting. It is intuitively obvious that it is also a "generator" of new features or properties of objects, of new terms (necessarily "invented" as a consequence of the distinctness of the two objects), the source of increasing the vocabulary and new connections in the brain. This also does not require assigning human language labels, or verbal attributes to each category and role of "labels" or "language" may be played by reflexes.

Mathematicians know that any complex function may be decomposed into superposition of surjection and injection. So, it is possible that any arbitrarily complex function (and a cognitive function is not an exception) may be also decomposed into 3 sequential operations: storing associative memories, partitioning and disjoint (on sets of objects been referenced by the associative memory). If fMRI scanning in the future will confirm that partitioning takes place in a separate field from disjoint area (which is processed by the early stages of visual cortex *) - it might be a clue that the above mentioned model is adequate and those primitive cognitive functions were developed by evolution in separate known stages: first, after vision emerged - appeared an ability for disjoint function, then - hippocampus allowed to store more associative memories, then the partitioning was added to the set. On the latest stages - all those 3 functions allowed to model and process any arbitrarily complex function with subsequent encoding it using more advanced (linguistic) encoding but still using the same basic elementary mechanisms (including associations). With the most advanced cognitive abilities - we, humans, still use those primitive operations, inherited on different stages of evolution: disjoint (injection), partitioning (surjection) and association (for both encoding and storage).

Added (Dec 1, 2011):
One of candidates for the surjection region in the brain is right fusiform gyrus which is responsible for object agnosia, i.e. inability to distinguish and recognize abstract objects.

*)
http://www.sciencedirect.com/science/article/pii/S0896627311004818
http://www.sciencedaily.com/releases/2007/03/070314134812.htm
http://www.apa.org/science/about/psa/2005/02/suzuki.aspx
http://www.ncbi.nlm.nih.gov/pubmed/16343950
http://jn.physiology.org/content/93/1/603.full
http://www.cell.com/neuron/abstract/S0896-6273%2811%2900481-8