Related papers: A no-go theorem for one-layer feedforward networks
Associative memory has been a prominent candidate for the computation performed by the massively recurrent neocortical networks. Attractor networks implementing associative memory have offered mechanistic explanation for many cognitive…
Combinatorial neural codes are $0/1$ vectors that are used to model the co-firing patterns of a set of place cells in the brain. One wide-open problem in this area is to determine when a given code can be algorithmically drawn in the plane…
We introduce combinatorial interpretability, a methodology for understanding neural computation by analyzing the combinatorial structures in the sign-based categorization of a network's weights and biases. We demonstrate its power through…
This dissertation explores applications of discrete geometry in mathematical neuroscience. We begin with convex neural codes, which model the activity of hippocampal place cells and other neurons with convex receptive fields. In Chapter 4,…
In this work, we analyze the capabilities and practical limitations of neural networks (NNs) for sequence-based signal processing which can be seen as an omnipresent property in almost any modern communication systems. In particular, we…
Predictive coding is a unifying framework for understanding perception, action and neocortical organization. In predictive coding, different areas of the neocortex implement a hierarchical generative model of the world that is learned from…
It is widely believed that deep neural networks contain layer specialization, wherein neural networks extract hierarchical features representing edges and patterns in shallow layers and complete objects in deeper layers. Unlike common…
Neural codes are lists of subsets of neurons that fire together. Of particular interest are neurons called place cells, which fire when an animal is in specific, usually convex regions in space. A fundamental question, therefore, is to…
A long standing open problem in the theory of neural networks is the development of quantitative methods to estimate and compare the capabilities of different architectures. Here we define the capacity of an architecture by the binary…
A pressing scientific challenge is to understand how brains work. Of particular interest is the neocortex,the part of the brain that is especially large in humans, capable of handling a wide variety of tasks including visual, auditory,…
We propose a framework for constructing combinatorial complexes (CCs) from fMRI time series data that captures both pairwise and higher-order neural interactions through information-theoretic measures, bridging topological deep learning and…
We consider deep multi-layered generative models such as Boltzmann machines or Hopfield nets in which computation (which implements inference) is both recurrent and stochastic, but where the recurrence is not to model sequential structure,…
Neural codes are binary codes that are used for information processing and representation in the brain. In previous work, we have shown how an algebraic structure, called the {\it neural ring}, can be used to efficiently encode geometric…
Studies of the dynamics of nonlinear recurrent neural networks often assume independent and identically distributed couplings, but large-scale connectomics data indicate that biological neural circuits exhibit markedly different…
Polynomial functions have plenty of useful analytical properties, but they are rarely used as learning models because their function class is considered to be restricted. This work shows that when trained properly polynomial functions can…
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…
Convex neural codes are subsets of the Boolean lattice that record the intersection patterns of convex sets in Euclidean space. Much work in recent years has focused on finding combinatorial criteria on codes that can be used to classify…
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…
Recurrently coupled oscillators that are sufficiently heterogeneous and/or randomly coupled can show an asynchronous activity in which there are no significant correlations among the units of the network. The asynchronous state can…
Cortical neurons are complex, multi-timescale processors wired into recurrent circuits, shaped by long evolutionary pressure under stringent biological constraints. Mainstream machine learning, by contrast, predominantly builds models from…