413 citations · 434 across the 13 of their papers we have counts for
15 papers
Dowker's theorem for higher-order relations
Vin de Silva, Chad Giusti, Vladimir Itskov +5
Given a relation between two sets, Dowker's Theorem (1952) states that the homology groups of two associated simplicial complexes, now known as Dowker comp…
On the spectrum of the Hodge Laplacian on sequences
Hannah Santa Cruz Baur, Vladimir Itskov
Hodge Laplacians have been previously proposed as a natural tool for understanding higher-order interactions in networks and directed graphs. Here we introduce a Hodge-theoretic ap…
The combinatorial code and the graph rules of Dale networks
Nikola Milićević, Vladimir Itskov
We describe the combinatorics of equilibria and steady states of neurons in threshold-linear networks that satisfy Dale's law. The combinatorial code of a Dale network is character…
A Topological Approach to Inferring the Intrinsic Dimension of Convex Sensing Data
Min-Chun Wu, Vladimir Itskov
We consider a common measurement paradigm, where an unknown subset of an affine space is measured by unknown continuous quasi-convex functions. Given the measurement data, can one…
Hyperplane Neural Codes and the Polar Complex
Vladimir Itskov, Alex Kunin, Zvi Rosen
Hyperplane codes are a class of convex codes that arise as the output of a one layer feed-forward neural network. Here we establish several natural properties of stable hyperplane…
On open and closed convex codes
Joshua Cruz, Chad Giusti, Vladimir Itskov +1
Neural codes serve as a language for neurons in the brain. Convex codes, which arise from the pattern of intersections of convex sets in Euclidean space, are of particular relevanc…
Sobolev subspaces of nowhere bounded functions
Pier Domenico Lamberti, Giorgio Stefani
We prove that in any Sobolev space which is subcritical with respect to the Sobolev Embedding Theorem there exists a closed infinite dimensional linear subspace whose non zero elem…
Clique topology of real symmetric matrices
Carina Curto, Chad Giusti, Vladimir Itskov
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…
Clique topology reveals intrinsic geometric structure in neural correlations
Chad Giusti, Eva Pastalkova, Carina Curto +1
Detecting meaningful structure in neural activity and connectivity data is challenging in the presence of hidden nonlinearities, where traditional eigenvalue-based methods may be m…
A no-go theorem for one-layer feedforward networks
Chad Giusti, Vladimir Itskov
It is often hypothesized that a crucial role for recurrent connections in the brain is to constrain the set of possible response patterns, thereby shaping the neural code. This imp…
Combinatorial neural codes from a mathematical coding theory perspective
Carina Curto, Vladimir Itskov, Katherine Morrison +2
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 t…
The neural ring: an algebraic tool for analyzing the intrinsic structure of neural codes
Carina Curto, Vladimir Itskov, Alan Veliz-Cuba +1
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 fie…
Encoding binary neural codes in networks of threshold-linear neurons
Carina Curto, Anda Degeratu, Vladimir Itskov
Networks of neurons in the brain encode preferred patterns of neural activity via their synaptic connections. Despite receiving considerable attention, the precise relationship bet…
Flexible Memory Networks
Carina Curto, Anda Degeratu, Vladimir Itskov
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 flexibl…
Orbit reduction of exterior differential systems, and group-invariant variational problems
Vladimir Itskov
For For a given PDE system, or an exterior differential system possessing a Lie group of internal symmetries the orbit reduction procedure is introduced. It is proved that the solu…