33 research outputs found
Flexible Memory Networks
Networks of neurons in some brain areas are flexible enough to encode new
memories quickly. Using a standard firing rate model of recurrent networks, we
develop a theory of flexible memory networks. Our main results characterize
networks having the maximal number of flexible memory patterns, given a
constraint graph on the network's connectivity matrix. Modulo a mild
topological condition, we find a close connection between maximally flexible
networks and rank 1 matrices. The topological condition is H_1(X;Z)=0, where X
is the clique complex associated to the network's constraint graph; this
condition is generically satisfied for large random networks that are not
overly sparse. In order to prove our main results, we develop some
matrix-theoretic tools and present them in a self-contained section independent
of the neuroscience context.Comment: Accepted to Bulletin of Mathematical Biology, 11 July 201
Clique topology reveals intrinsic geometric structure in neural correlations
Detecting meaningful structure in neural activity and connectivity data is
challenging in the presence of hidden nonlinearities, where traditional
eigenvalue-based methods may be misleading. We introduce a novel approach to
matrix analysis, called clique topology, that extracts features of the data
invariant under nonlinear monotone transformations. These features can be used
to detect both random and geometric structure, and depend only on the relative
ordering of matrix entries. We then analyzed the activity of pyramidal neurons
in rat hippocampus, recorded while the animal was exploring a two-dimensional
environment, and confirmed that our method is able to detect geometric
organization using only the intrinsic pattern of neural correlations.
Remarkably, we found similar results during non-spatial behaviors such as wheel
running and REM sleep. This suggests that the geometric structure of
correlations is shaped by the underlying hippocampal circuits, and is not
merely a consequence of position coding. We propose that clique topology is a
powerful new tool for matrix analysis in biological settings, where the
relationship of observed quantities to more meaningful variables is often
nonlinear and unknown.Comment: 29 pages, 4 figures, 13 supplementary figures (last two authors
contributed equally
Diversity of emergent dynamics in competitive threshold-linear networks: a preliminary report
Threshold-linear networks consist of simple units interacting in the presence
of a threshold nonlinearity. Competitive threshold-linear networks have long
been known to exhibit multistability, where the activity of the network settles
into one of potentially many steady states. In this work, we find conditions
that guarantee the absence of steady states, while maintaining bounded
activity. These conditions lead us to define a combinatorial family of
competitive threshold-linear networks, parametrized by a simple directed graph.
By exploring this family, we discover that threshold-linear networks are
capable of displaying a surprisingly rich variety of nonlinear dynamics,
including limit cycles, quasiperiodic attractors, and chaos. In particular,
several types of nonlinear behaviors can co-exist in the same network. Our
mathematical results also enable us to engineer networks with multiple dynamic
patterns. Taken together, these theoretical and computational findings suggest
that threshold-linear networks may be a valuable tool for understanding the
relationship between network connectivity and emergent dynamics.Comment: 12 pages, 9 figures. Preliminary repor
The combinatorial code and the graph rules of Dale networks
We describe the combinatorics of equilibria and steady states of neurons in
threshold-linear networks that satisfy the Dale's law. The combinatorial code
of a Dale network is characterized in terms of two conditions: (i) a condition
on the network connectivity graph, and (ii) a spectral condition on the
synaptic matrix. We find that in the weak coupling regime the combinatorial
code depends only on the connectivity graph, and not on the particulars of the
synaptic strengths. Moreover, we prove that the combinatorial code of a weakly
coupled network is a sublattice, and we provide a learning rule for encoding a
sublattice in a weakly coupled excitatory network. In the strong coupling
regime we prove that the combinatorial code of a generic Dale network is
intersection-complete and is therefore a convex code, as is common in some
sensory systems in the brain.Comment: 22 pages, 4 figures, added discussion section, corrected typos,
expanded the background on convex code
Cell Groups Reveal Structure of Stimulus Space
An important task of the brain is to represent the outside world. It is unclear how the brain may do this, however, as it can only rely on neural responses and has no independent access to external stimuli in order to ‘‘decode’’ what those responses mean. We investigate what can be learned about a space of stimuli using only the action potentials (spikes) of cells with stereotyped—but unknown—receptive fields. Using hippocampal place cells as a model system, we show that one can (1) extract global features of the environment and (2) construct an accurate representation of space, up to an overall scale factor, that can be used to track the animal’s position. Unlike previous approaches to reconstructing position from place cell activity, this information is derived without knowing place fields or any other functions relating neural responses to position. We find that simply knowing which groups of cells fire together reveals a surprising amount of structure in the underlying stimulus space; this may enable the brain to construct its own internal representations
Cell Groups Reveal Structure of Stimulus Space
An important task of the brain is to represent the outside world. It is unclear how the brain may do this, however, as it can only rely on neural responses and has no independent access to external stimuli in order to “decode” what those responses mean. We investigate what can be learned about a space of stimuli using only the action potentials (spikes) of cells with stereotyped—but unknown—receptive fields. Using hippocampal place cells as a model system, we show that one can (1) extract global features of the environment and (2) construct an accurate representation of space, up to an overall scale factor, that can be used to track the animal's position. Unlike previous approaches to reconstructing position from place cell activity, this information is derived without knowing place fields or any other functions relating neural responses to position. We find that simply knowing which groups of cells fire together reveals a surprising amount of structure in the underlying stimulus space; this may enable the brain to construct its own internal representations
Understanding short-timescale neuronal firing sequences via bias matrices
The brain generates persistent neuronal firing sequences across varying timescales. The short-timescale (~100ms) sequences are believed to be crucial in the formation and transfer of memories. Large-amplitude local field potentials known as sharp-wave ripples (SWRs) occur irregularly in hippocampus when an animal has minimal interaction with its environment, such as during resting, immobility, or slow-wave sleep. SWRs have been long hypothesized to play a critical role in transferring memories from the hippocampus to the neocortex [1]. While sequential firing during SWRs is known to be biased by the previous experiences of the animal, the exact relationship of the short-timescale sequences during SWRs and longer-timescale sequences during spatial and nonspatial behaviors is still poorly understood. One hypothesis is that the sequences during SWRs are “replays” or “preplays” of “master sequences”, which are sequences that closely mimic the order of place fields on a linear track [2,3]. Rather than particular hard-coded “master” sequences, an alternative explanation of the observed correlations is that similar sequences arise naturally from the intrinsic biases of firing between pairs of cells. To distinguish these and other possibilities, one needs mathematical tools beyond the center-of-mass sequences and Spearman’s rank-correlation coefficient that are currently used