相关论文: Convexity of Neural Codes with Four Maximal Codewo…
Functional connectivity is a fundamental property of neural networks that quantifies the segregation and integration of information between cortical areas. Due to mathematical complexity, a theory that could explain how the parameters of…
The visual information in V1 is processed by an array of modules called orientation preference columns. In some species including humans, orientation columns are radially arranged around singular points like the spokes of a wheel, that are…
Every ordered collection of sets in Euclidean space can be associated to a combinatorial code, which records the regions cut out by the sets in space. Given two ordered collections of sets, one can form a third collection in which the…
We consider the problem of constructing binary codes for correcting deletions that are localized within certain parts of the codeword that are unknown a priori. The model that we study is when $\delta \leq w$ deletions are localized in a…
Human brains are arguably the most complex entities known. Composed of billions of neurons, connected via a highly detailed structure where the underlying method by which functionality occurs is still debated. Here we consider one theory…
The local structure theory for cellular automata (CA) can be viewed as an finite-dimensional approximation of infinitely-dimensional system. While it is well known that this approximation works surprisingly well for some cellular automata,…
The presence of non-convexity in smooth optimization problems arising from deep learning have sparked new smoothness conditions in the literature and corresponding convergence analyses. We discuss these smoothness conditions, order them,…
The cooperative behavior of neurons and neuronal areas associated with the synchronization behavior proves to be a fundamental neural mechanism. In addition, abnormal levels of synchronization have been related to unhealthy neural…
Sparse coding algorithms trained on natural images can accurately predict the features that excite visual cortical neurons, but it is not known whether such codes can be learned using biologically realistic plasticity rules. We have…
A covering code is a set of codewords with the property that the union of balls, suitably defined, around these codewords covers an entire space. Generally, the goal is to find the covering code with the minimum size codebook. While most…
Life relies on the efficient performance of molecular codes, which relate symbols and meanings via error-prone molecular recognition. We describe how optimizing a code to withstand the impact of molecular recognition noise may be…
Convolutional neural networks use pooling and other downscaling operations to maintain translational invariance for detection of features, but in their architecture they do not explicitly maintain a representation of the locations of the…
This paper investigates the conformal isometry hypothesis as a potential explanation for the hexagonal periodic patterns in grid cell response maps. We posit that grid cell activities form a high-dimensional vector in neural space, encoding…
Let $C$ be a convex subset of a locally convex space. We provide optimal approximate fixed point results for sequentially continuous maps $f\colon C\to\bar{C}$. First we prove that if $f(C)$ is totally bounded, then it has an approximate…
For 20 years the beautiful structure in the grid cell code has presented an attractive puzzle: what computation do these representations subserve, and why does it manifest so curiously in neurons. The first question quickly attracted an…
We study a model of contraction-based cell motility inside a microchannel to investigate the regulation of cell polarization and motion by the mechanical resistance of the environment. A positive feedback between the asymmetry of the…
In this paper, we examine an important problem of learning neural networks that certifiably meet certain specifications on input-output behaviors. Our strategy is to find an inner approximation of the set of admissible policy parameters,…
A convex network can be defined as a network such that every connected induced subgraph includes all the shortest paths between its nodes. Fully convex network would therefore be a collection of cliques stitched together in a tree. In this…
We study a simple map as a minimal model of excitable cells. The map has two fast variables which mimic the behavior of class I neurons, undergoing a sub-critical Hopf bifurcation. Adding a third slow variable allows the system to present…
Connected decision boundaries are useful in several tasks like image segmentation, clustering, alpha-shape or defining a region in nD-space. However, the machine learning literature lacks methods for generating connected decision boundaries…