  An assembly is a large population of neurons whose synchronous firing is
hypothesized to represent a memory, concept, word, and other cognitive
categories. Assemblies are believed to provide a bridge between high-level
cognitive phenomena and low-level neural activity. Recently, a computational
system called the Assembly Calculus (AC), with a repertoire of biologically
plausible operations on assemblies, has been shown capable of simulating
arbitrary space-bounded computation, but also of simulating complex cognitive
phenomena such as language, reasoning, and planning. However, the mechanism
whereby assemblies can mediate learning has not been known. Here we present
such a mechanism, and prove rigorously that, for simple classification problems
defined on distributions of labeled assemblies, a new assembly representing
each class can be reliably formed in response to a few stimuli from the class;
this assembly is henceforth reliably recalled in response to new stimuli from
the same class. Furthermore, such class assemblies will be distinguishable as
long as the respective classes are reasonably separated -- for example, when
they are clusters of similar assemblies. To prove these results, we draw on
random graph theory with dynamic edge weights to estimate sequences of
activated vertices, yielding strong generalizations of previous calculations
and theorems in this field over the past five years. These theorems are backed
up by experiments demonstrating the successful formation of assemblies which
represent concept classes on synthetic data drawn from such distributions, and
also on MNIST, which lends itself to classification through one assembly per
digit. Seen as a learning algorithm, this mechanism is entirely online,
generalizes from very few samples, and requires only mild supervision -- all
key attributes of learning in a model of the brain.
