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Abstract:
The ability to make distinctions is one of the fundamental capacities underlying cognition, from perception through abstract (categorical) thought. The distinctions a cognitive system is capable of making, are manifested in its neural activity. Given a set of distinctions, the natural question that arises is whether this imposes constraints on the activity spaces which could embed such a set. We hypothesize that an activity space can embed a given set of distinctions only if its structure corresponds in some sense to the set of distinctions (that is it does not cause collapse of distinctions or undue elaborations within domains or clusters). Thus, we reason that the homology of an activity space approximates the rough structure of the underlying set of distinctions that is realized by the system's activity. Therefore, we refer to the structure of a given activity set as its representational capacity. We hypothesize that activity sets corresponding to different states of vigilance (for example wakefulness as compared to sleep) exhibit disparity in their representational capacity.
To render such general sentiments in tractable form we proceed as following : 1.) We assume that crude features of the signal which we refer to as the "structure" of activity suffice to determine it's membership in a class. 2.) Instances of activity are registered at different states of vigilance (anesthesia/quiet wakefulness/attention). We conjecture that what constitutes a state in terms of activity is similarity (invariance) in the structure of instances of activity. Thus, real (structure sensitive) functions can be utilized to classify activity according to state. 3.) The level sets of the typical value corresponding to a state can be calculated explicitly within a boundary of ε from the set of measurements 4.) Finally, the Betty numbers of such level sets, which give the rank of the corresponding homology groups, and the corresponding statistics (such as the dependency on ε), can be computed following (Munkres, 84, Robins, 2000, Kaczynski et al. 2004).
We present preliminary results obtained from analysis of voltage sensitive dye imaging (Grivald, 2004) data obtained from the primary visual cortex behaving primates.
