相关论文: Geometroneurodynamics
Common measures of neural representational (dis)similarity are designed to be insensitive to rotations and reflections of the neural activation space. Motivated by the premise that the tuning of individual units may be important, there has…
We consider a system with a discrete configuration space. We show that the geometrical structures associated with such a system provide the tools necessary for a reconstruction of discrete quantum mechanics once dynamics is brought into the…
Studies have shown that the Hilbert spaces of non-Hermitian systems require nontrivial metrics. Here, we demonstrate how evolution dimensions, in addition to time, can emerge naturally from a geometric formalism. Specifically, in this…
Physics relies on mathematical spaces carefully matched to the phenomena under study. Phase space in classical mechanics, Hilbert space in quantum theory, configuration spaces in field theory all provide representations in which physical…
The question of controllability of natural and man-made network systems has recently received considerable attention. In the context of the human brain, the study of controllability may not only shed light into the organization and function…
A growing literature in computational neuroscience leverages gradient descent and learning algorithms that approximate it to study synaptic plasticity in the brain. However, the vast majority of this work ignores a critical underlying…
The aim of the present article is to give an introduction to the concept of quasi-unitary equivalence and to define several (pseudo-)metrics on the space of self-adjoint operators acting possibly in different Hilbert spaces. As some of the…
Conventional neuroimaging analyses have revealed the computational specificity of localized brain regions, exploiting the power of the subtraction technique in fMRI and event-related potential analyses in EEG. Moving beyond this convention,…
Taking several statistical examples, in particular one involving a choice of experiment, as points of departure, and making symmetry assumptions, the link towards quantum theory developed in Helland (2005a,b) is surveyed and clarified. The…
Advances in experimental neuroscience have transformed our ability to explore the structure and function of neural circuits. At the same time, advances in machine learning have unleashed the remarkable computational power of artificial…
Geometric graphs are a special kind of graph with geometric features, which are vital to model many scientific problems. Unlike generic graphs, geometric graphs often exhibit physical symmetries of translations, rotations, and reflections,…
Uniquely among primates, humans possess a remarkable capacity to recognize and manipulate abstract structure in the service of task goals across a broad range of behaviors. One illustration of this is in the visual perception of geometric…
Useful relations describing arbitrary parameters of given quantum systems can be derived from simple physical constraints imposed on the vectors in the corresponding Hilbert space. This is well known and it usually proceeds by partitioning…
This paper gives an introduction to \textit{Cognidynamics}, that is to the dynamics of cognitive systems driven by optimal objectives imposed over time when they interact either with a defined virtual or with a real-world environment. The…
Starting with a similarity function between objects, it is possible to define a distance metric on pairs of objects, and more generally on probability distributions over them. These distance metrics have a deep basis in functional analysis,…
The extraordinary computational power of the brain may be related in part to the fact that each of the smaller neural networks that compose it can behave transiently in many different ways, depending on its inputs. Mathematically, input…
Isomorphisms of separable Hilbert spaces are analogous to isomorphisms of n-dimensional vector spaces. However, while n-dimensional spaces in applications are always realized as the Euclidean space R^n, Hilbert spaces admit various useful…
Quantifying similarity between neural representations -- e.g. hidden layer activation vectors -- is a perennial problem in deep learning and neuroscience research. Existing methods compare deterministic responses (e.g. artificial networks…
Experimental sciences have come to depend heavily on our ability to organize and interpret high-dimensional datasets. Natural laws, conservation principles, and inter-dependencies among observed variables yield geometric structure, with…
Comprehending the interplay between spatial and temporal characteristics of neural dynamics can contribute to our understanding of information processing in the human brain. Graph neural networks (GNNs) provide a new possibility to…