arXiv subjects

Carina Curto

Publications and source records attributed to Carina Curto.

30 records · Page 2Linked to original sources

arXiv2016

What makes a neural code convex?

Neural codes allow the brain to represent, process, and store information about the world. Combinatorial codes, comprised of binary patterns of neural activity, encode information via the collective behavior of populations of neurons. A code is called convex if its codewords correspond to regions defined by an arrangement of convex open sets in Euclidean space. Convex codes have been observed experimentally in many brain areas, including sensory cortices and the hippocampus, where neurons exhibit convex receptive fields. What makes a neural code convex? That is, how can we tell from the intrinsic structure of a code if there exists a corresponding arrangement of convex open sets? In this work, we provide a complete characterization of local obstructions to convexity. This motivates us to define max intersection-complete codes, a family guaranteed to have no local obstructions. We then show how our characterization enables one to use free resolutions of Stanley-Reisner ideals in order to detect violations of convexity. Taken together, these results provide a significant advance in understanding the intrinsic combinatorial properties of convex codes.

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arXiv2016

Pattern completion in symmetric threshold-linear networks

Threshold-linear networks are a common class of firing rate models that describe recurrent interactions among neurons. Unlike their linear counterparts, these networks generically possess multiple stable fixed points (steady states), making them viable candidates for memory encoding and retrieval. In this work, we characterize stable fixed points of general threshold-linear networks with constant external drive, and discover constraints on the co-existence of fixed points involving different subsets of active neurons. In the case of symmetric networks, we prove the following antichain property: if a set of neurons $τ$ is the support of a stable fixed point, then no proper subset or superset of $τ$ can support a stable fixed point. Symmetric threshold-linear networks thus appear to be well suited for pattern completion, since the dynamics are guaranteed not to get "stuck" in a subset or superset of a stored pattern. We also show that for any graph G, we can construct a network whose stable fixed points correspond precisely to the maximal cliques of G. As an application, we design network decoders for place field codes, and demonstrate their efficacy for error correction and pattern completion. The proofs of our main results build on the theory of permitted sets in threshold-linear networks, including recently-developed connections to classical distance geometry.

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arXiv2016

What can topology tell us about the neural code?

Neuroscience is undergoing a period of rapid experimental progress and expansion. New mathematical tools, previously unknown in the neuroscience community, are now being used to tackle fundamental questions and analyze emerging data sets. Consistent with this trend, the last decade has seen an uptick in the use of topological ideas and methods in neuroscience. In this talk I will survey recent applications of topology in neuroscience, and explain why topology is an especially natural tool for understanding neural codes. Note: This is a write-up of my talk for the Current Events Bulletin, held at the 2016 Joint Math Meetings in Seattle, WA.

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arXiv2015

Clique topology of real symmetric matrices

This is the Supplementary Text for "Clique topology reveals intrinsic geometric structure in neural correlations," by Chad Giusti, Eva Pastalkova, Carina Curto, and Vladimir Itskov.

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arXiv2015

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.

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arXiv2013

The neural ring: an algebraic tool for analyzing the intrinsic structure of neural codes

Neurons in the brain represent external stimuli via neural codes. These codes often arise from stereotyped stimulus-response maps, associating to each neuron a convex receptive field. An important problem confronted by the brain is to infer properties of a represented stimulus space without knowledge of the receptive fields, using only the intrinsic structure of the neural code. How does the brain do this? To address this question, it is important to determine what stimulus space features can - in principle - be extracted from neural codes. This motivates us to define the neural ring and a related neural ideal, algebraic objects that encode the full combinatorial data of a neural code. Our main finding is that these objects can be expressed in a "canonical form" that directly translates to a minimal description of the receptive field structure intrinsic to the code. We also find connections to Stanley-Reisner rings, and use ideas similar to those in the theory of monomial ideals to obtain an algorithm for computing the primary decomposition of pseudo-monomial ideals. This allows us to algorithmically extract the canonical form associated to any neural code, providing the groundwork for inferring stimulus space features from neural activity alone.

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arXiv2013

Encoding binary neural codes in networks of threshold-linear neurons

Networks of neurons in the brain encode preferred patterns of neural activity via their synaptic connections. Despite receiving considerable attention, the precise relationship between network connectivity and encoded patterns is still poorly understood. Here we consider this problem for networks of threshold-linear neurons whose computational function is to learn and store a set of binary patterns (e.g., a neural code) as "permitted sets" of the network. We introduce a simple Encoding Rule that selectively turns "on" synapses between neurons that co-appear in one or more patterns. The rule uses synapses that are binary, in the sense of having only two states ("on" or "off"), but also heterogeneous, with weights drawn from an underlying synaptic strength matrix S. Our main results precisely describe the stored patterns that result from the Encoding Rule -- including unintended "spurious" states -- and give an explicit characterization of the dependence on S. In particular, we find that binary patterns are successfully stored in these networks when the excitatory connections between neurons are geometrically balanced -- i.e., they satisfy a set of geometric constraints. Furthermore, we find that certain types of neural codes are "natural" in the context of these networks, meaning that the full code can be accurately learned from a highly undersampled set of patterns. Interestingly, many commonly observed neural codes in cortical and hippocampal areas are natural in this sense. As an application, we construct networks that encode hippocampal place field codes nearly exactly, following presentation of only a small fraction of patterns. To obtain our results, we prove new theorems using classical ideas from convex and distance geometry, such as Cayley-Menger determinants, revealing a novel connection between these areas of mathematics and coding properties of neural networks.

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arXiv2012

Combinatorial neural codes from a mathematical coding theory perspective

Shannon's seminal 1948 work gave rise to two distinct areas of research: information theory and mathematical coding theory. While information theory has had a strong influence on theoretical neuroscience, ideas from mathematical coding theory have received considerably less attention. Here we take a new look at combinatorial neural codes from a mathematical coding theory perspective, examining the error correction capabilities of familiar receptive field codes (RF codes). We find, perhaps surprisingly, that the high levels of redundancy present in these codes does not support accurate error correction, although the error-correcting performance of RF codes "catches up" to that of random comparison codes when a small tolerance to error is introduced. On the other hand, RF codes are good at reflecting distances between represented stimuli, while the random comparison codes are not. We suggest that a compromise in error-correcting capability may be a necessary price to pay for a neural code whose structure serves not only error correction, but must also reflect relationships between stimuli.

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arXiv2011

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.

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arXiv2008

Threefold Flops via Matrix Factorization

The explicit McKay correspondence, as formulated by Gonzalez-Sprinberg and Verdier, associates to each exceptional divisor in the minimal resolution of a rational double point a matrix factorization of the equation of the rational double point. We study deformations of these matrix factorizations, and show that they exist over an appropriate "partially resolved" deformation space for rational double points of types A and D. As a consequence, all simple flops of lengths 1 and 2 can be described in terms of blowups defined from matrix factorizations. We also formulate conjectures which would extend these results to rational double points of type E and simple flops of length greater than 2.

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arXiv2008

Matrix model superpotentials and ADE singularities

We use F. Ferrari's methods relating matrix models to Calabi-Yau spaces in order to explain much of Intriligator and Wecht's ADE classification of $\N=1$ superconformal theories which arise as RG fixed points of $\N = 1$ SQCD theories with adjoints. We find that ADE superpotentials in the Intriligator-Wecht classification exactly match matrix model superpotentials obtained from Calabi-Yaus with corresponding ADE singularities. Moreover, in the additional $\Hat{O}, \Hat{A}, \Hat{D}$ and $\Hat{E}$ cases we find new singular geometries. These `hat' geometries are closely related to their ADE counterparts, but feature non-isolated singularities. As a byproduct, we give simple descriptions for small resolutions of Gorenstein threefold singularities in terms of transition functions between just two coordinate charts. To obtain these results, we develop an algorithm for blowing down exceptional $\PP^1$s, described in the appendix.

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arXiv2005

Matrix Model Superpotentials and Calabi-Yau Spaces: an ADE Classification

We use F. Ferrari's methods relating matrix models to Calabi-Yau spaces in order to explain Intriligator and Wecht's ADE classification of $\N=1$ superconformal theories which arise as RG fixed points of $\N = 1$ SQCD theories with adjoints. The connection between matrix models and $\N = 1$ gauge theories can be seen as evidence for the Dijkgraaf--Vafa conjecture. We find that ADE superpotentials in the Intriligator--Wecht classification exactly match matrix model superpotentials obtained from Calabi-Yau's with corresponding ADE singularities. Moreover, in the additional $\Hat{O}, \Hat{A}, \Hat{D}$ and $\Hat{E}$ cases we find new singular geometries. These `hat' geometries are closely related to their ADE counterparts, but feature non-isolated singularities. As a byproduct, we give simple descriptions for small resolutions of Gorenstein threefold singularities in terms of transition functions between just two coordinate charts. To obtain these results we develop techniques for performing small resolutions and small blow-downs, including an algorithm for blowing down exceptional $\PP^1$'s. In particular, we conjecture that small resolutions for isolated Gorenstein threefold singularities can be obtained by deforming matrix factorizations for simple surface singularities -- and prove this in the length 1 and length 2 cases.

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