Cartesian Abstraction Can Yield ‘Cognitive Maps’

AbstractIt has been long debated how the so called cognitive map, the set of place cells, develops in rat hippocampus. The function of this organ is of high relevance, since the hippocampus is the key component of the medial temporal lobe memory system, responsible for forming episodic memory, declarative memory, the memory for facts and rules that serve cognition in humans. Here, a general mechanism is put forth: We introduce the novel concept of Cartesian factors. We show a non-linear projection of observations to a discretized representation of a Cartesian factor in the presence of a representation of a complementing one. The computational model is demonstrated for place cells that we produce from the egocentric observations and the head direction signals. Head direction signals make the observed factor and sparse allothetic signals make the complementing Cartesian one. We present numerical results, connect the model to the neural substrate, and elaborate on the differences between this model and other ones, including Slow Feature Analysis [17]

Singularities of orthogonal and symplectic determinantal varieties

Let either $GL(E)\times SO(F)$ or $GL(E)\times Sp(F)$ act naturally on the space of matrices $E\otimes F$. There are only finitely many orbits, and the orbit closures are orthogonal and symplectic generalizations of determinantal varieties, which can be described similarly using rank conditions. In this paper, we study the singularities of these varieties and describe their defining equations. We prove that in the symplectic case, the orbit closures are normal with good filtrations, and in characteristic $0$ have rational singularities. In the orthogonal case we show that most orbit closures will have the same properties, and determine precisely the exceptions to this