846 research outputs found
Robust spatial memory maps encoded in networks with transient connections
The spiking activity of principal cells in mammalian hippocampus encodes an
internalized neuronal representation of the ambient space---a cognitive map.
Once learned, such a map enables the animal to navigate a given environment for
a long period. However, the neuronal substrate that produces this map remains
transient: the synaptic connections in the hippocampus and in the downstream
neuronal networks never cease to form and to deteriorate at a rapid rate. How
can the brain maintain a robust, reliable representation of space using a
network that constantly changes its architecture? Here, we demonstrate, using
novel Algebraic Topology techniques, that cognitive map's stability is a
generic, emergent phenomenon. The model allows evaluating the effect produced
by specific physiological parameters, e.g., the distribution of connections'
decay times, on the properties of the cognitive map as a whole. It also points
out that spatial memory deterioration caused by weakening or excessive loss of
the synaptic connections may be compensated by simulating the neuronal
activity. Lastly, the model explicates functional importance of the
complementary learning systems for processing spatial information at different
levels of spatiotemporal granularity, by establishing three complementary
timescales at which spatial information unfolds. Thus, the model provides a
principal insight into how can the brain develop a reliable representation of
the world, learn and retain memories despite complex plasticity of the
underlying networks and allows studying how instabilities and memory
deterioration mechanisms may affect learning process.Comment: 24 pages, 10 figures, 4 supplementary figure
The Topology of Probability Distributions on Manifolds
Let be a set of random points in , generated from a probability
measure on a -dimensional manifold . In this paper we study
the homology of -- the union of -dimensional balls of radius
around , as , and . In addition we study the critical
points of -- the distance function from the set . These two objects
are known to be related via Morse theory. We present limit theorems for the
Betti numbers of , as well as for number of critical points of index
for . Depending on how fast decays to zero as grows, these two
objects exhibit different types of limiting behavior. In one particular case
(), we show that the Betti numbers of perfectly
recover the Betti numbers of the original manifold , a result which is of
significant interest in topological manifold learning
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