arXiv subjects

Federico W. Pasini

Publications and source records attributed to Federico W. Pasini.

14 recordsLinked to original sources

arXiv2026

Ramification Subgroups of Knot Groups and their Profinite and Cohomological Structure

We formalize a ramification theory for finite covers of knot exteriors. Given a knot group $G_K$ and a finite-index subgroup $U\le G_K$, we define meridional inertia subgroups $U\cap g\langle m\rangle g^{-1}$ and the global ramification subgroup $M_U\triangleleft U$ as their normal closure. We then analyze $M_U$ from three complementary viewpoints: (1) finite quotients, where $U/M_U$ is shown to be the universal ``maximal meridionally unramified'' quotient of $U$; (2) profinite completions, where we identify the closed ramification subgroup $\widehat M_{\widehat U}$ as the closed normal subgroup generated by closed inertia and prove that meridian-preserving isomorphisms of profinite completions preserve inertia and ramification; (3) cohomology, where ``unramified'' $H^1$-classes (discrete and profinite) are characterized as those vanishing on all inertia subgroups, in direct analogy with number-theoretic inertia conditions in Galois cohomology.

math.GT↗

arXiv2025

A mathematical language for linking fine-scale structure in spikes from hundreds to thousands of neurons with behaviour

Recent advances in neural recording technology allow simultaneously recording action potentials from hundreds to thousands of neurons in awake, behaving animals. However, characterizing spike patterns in the resulting data, and linking these patterns to behaviour, remains a challenging task. The lack of a rigorous mathematical language for variable numbers of events (spikes) emitted by multiple agents (neurons) is an important limiting factor. We introduce a new mathematical operation to decompose complex spike patterns into a set of simple, structured elements. This creates a mathematical language that allows comparing spike patterns across trials, detecting sub-patterns, and making links to behaviour via a clear distance measure. We apply the method to dual Utah array recordings from macaque prefrontal cortex, where this technique reveals previously unseen structure that can predict both memory-guided decisions and errors in a virtual-reality working memory task. These results demonstrate that this technique provides a powerful new approach to understand structure in the spike times of neural populations, at a scale that will continue to grow more and more rapidly in upcoming years.

q-bio.QM↗

arXiv2024

Joins of normal matrices, their spectrum, and applications

Motivated by studies of oscillator networks, we study the spectrum of the join of several normal matrices with constant row sums. We apply our results to compute the characteristic polynomial of the join of several regular graphs. We then use this theorem to study several problems in spectral graph theory. In particular, we provide some simple constructions of Ramanujan graphs and give new proofs for some theorems in the classical book of Cvetković, Rowlinson, and Slobodan.

math.CO↗

arXiv2023

An exact mathematical description of computation with transient spatiotemporal dynamics in a complex-valued neural network

We study a complex-valued neural network (cv-NN) with linear, time-delayed interactions. We report the cv-NN displays sophisticated spatiotemporal dynamics, including partially synchronized ``chimera'' states. We then use these spatiotemporal dynamics, in combination with a nonlinear readout, for computation. The cv-NN can instantiate dynamics-based logic gates, encode short-term memories, and mediate secure message passing through a combination of interactions and time delays. The computations in this system can be fully described in an exact, closed-form mathematical expression. Finally, using direct intracellular recordings of neurons in slices from neocortex, we demonstrate that computations in the cv-NN are decodable by living biological neurons. These results demonstrate that complex-valued linear systems can perform sophisticated computations, while also being exactly solvable. Taken together, these results open future avenues for design of highly adaptable, bio-hybrid computing systems that can interface seamlessly with other neural networks.

cs.NE↗

arXiv2023

Spectral perturbation by rank-$m$ matrices

Let $A$ and $B$ designate $n\times n$ matrices with coefficients in a field $F$. In this paper, we completely answer the following question: For $A$ fixed, what are the possible characteristic polynomials of $A+B$, where $B$ ranges over matrices of rank $\le m$?

math.SP↗

arXiv2023

On the joins of group rings

Given a collection $\{ G_i\}_{i=1}^d$ of finite groups and a ring $R$, we define a subring of the ring $M_n(R)$ ($n = \sum_{i=1}^d|G_i|)$ that encompasses all the individual group rings $R[G_i]$ along the diagonal blocks as $G_i$-circulant matrices. The precise definition of this ring was inspired by a construction in graph theory known as the joined union of graphs. We call this ring the join of group rings and denote it by $\mathcal{J}_{G_1,\dots, G_d}(R)$. In this paper, we present a systematic study of the algebraic structure of $\mathcal{J}_{G_1,\dots, G_d}(R)$. We show that it has a ring structure and characterize its center, group of units, and Jacobson radical. When $R=k$ is an algebraically closed field, we derive a formula for the number of irreducible modules over $\mathcal{J}_{G_1,\dots, G_d}(k)$. We also show how a blockwise extension of the Fourier transform provides both a generalization of the Circulant Diagonalization Theorem to joins of circulant matrices and an explicit isomorphism between the join algebra and its Wedderburn components.

math.RA↗

arXiv2023

Broadcasting solutions on networked systems of phase oscillators

Networked systems have been used to model and investigate the dynamical behavior of a variety of systems. For these systems, different levels of complexity can be considered in the modeling procedure. On one hand, this can offer a more realistic and rich modeling option. On the other hand, it can lead to intrinsic difficulty in analyzing the system. Here, we present an approach to investigate the dynamics of Kuramoto oscillators on networks with different levels of connections: a network of networks. To do so, we utilize a construction in network theory known as the join of networks, which represents ``intra-area" and ``inter-area" connections. This approach provides a reduced representation of the original, multi-level system, where both systems have equivalent dynamics. Then, we can find solutions for the reduced system and broadcast them to the original network of networks. Moreover, using the same idea we can investigate the stability of these states, where we can obtain information on the Jacobian of the multi-level system by analyzing the reduced one. This approach is general for arbitrary connection schemes between nodes within the same area. Finally, our work opens the possibility of studying the dynamics of networked systems using a simpler representation, thus leading to a better understanding of the dynamical behavior of these systems.

math.DS↗

arXiv2022

An algebraic approach to spike-time neural codes in the hippocampus

Although temporal coding through spike-time patterns has long been of interest in neuroscience, the specific structures that could be useful for spike-time codes remain highly unclear. Here, we introduce a new analytical approach, using techniques from discrete mathematics, to study spike-time codes. We focus on the phenomenon of ``phase precession'' in the rodent hippocampus. During navigation and learning on a physical track, specific cells in a rodent's brain form a highly structured pattern relative to the oscillation of local population activity. Studies of phase precession largely focus on its well established role in synaptic plasticity and memory formation. Comparatively less attention has been paid to the fact that phase precession represents one of the best candidates for a spike-time neural code. The precise nature of this code remains an open question. Here, we derive an analytical expression for an operator mapping points in physical space, through individual spike times, to complex numbers. The properties of this operator highlight a specific relationship between past and future in hippocampal spike patterns. Importantly, this approach generalizes beyond the specific phenomenon studied here, providing a new technique to study the neural codes within spike-time sequences found during sensory coding and motor behavior. We then introduce a novel spike-based decoding algorithm, based on this operator, that successfully decodes a simulated animal's trajectory using only the animal's initial position and a pattern of spike times. This decoder is robust to noise in spike times and works on a timescale almost an order of magnitude shorter than typically used with decoders that work on average firing rate. These results illustrate the utility of a discrete approach, based on the symmetries in spike patterns, to provide insight into the structure and function of neural systems.

q-bio.NC↗

arXiv2022

Equilibria in Kuramoto oscillator networks: An algebraic approach

Kuramoto networks constitute a paradigmatic model for the investigation of collective behavior in networked systems. Despite many advances in recent years, many open questions remain on the solutions for systems composed of coupled Kuramoto oscillators on complex networks. In this article, we describe an algebraic method to find equilibrium points for this kind of system without using standard approximations in the limit of infinite system size or the continuum limit. To do this, we use a recently introduced algebraic approach to the Kuramoto dynamics, which results in an explicitly solvable complex-valued equation that captures the dynamics of the original Kuramoto model. Using this new approach, we obtain equilibria for both the nonlinear original Kuramoto and complex-valued systems. We then completely classify all equilibria in the case of complete graphs originally studied by Kuramoto. Finally, we go on to study equilibria in networks of coupled oscillators with phase lag, in generalized circulant networks, multi-layer networks, and also random networks.

math.DS↗

arXiv2022

Joins of circulant matrices

We study the spectrum of the join of several circulant matrices. We apply our results to compute explicitly the spectrum of certain graphs obtained by joining several circulant graphs.

math.CO↗

arXiv2021

Spectral perturbation by rank one matrices

Let $A$ be a matrix of size $n \times n$ over an algebraically closed field $F$ and $q(t)$ a monic polynomial of degree $n$. In this article, we describe the necessary and sufficient conditions of $q(t)$ so that there exists a rank one matrix $B$ such that the characteristic polynomial of $A+B$ is $q(t)$.

math.SP↗

arXiv2020

Koszul algebras and quadratic duals in Galois cohomology

We investigate the Galois cohomology of finitely generated maximal pro-$p$ quotients of absolute Galois groups. Assuming the well-known conjectural description of these groups, we show that Galois cohomology has the PBW property. Hence in particular it is a Koszul algebra. This answers positively a conjecture by Positselski in this case. We also provide an analogous unconditional result about Pythagorean fields. Moreover, we establish some results that relate the quadratic dual of Galois cohomology with $p$-Zassenhaus filtration on the group. This paper also contains a survey of Koszul property in Galois cohomology and its relation with absolute Galois groups.

math.NT↗

arXiv2020

The ambient classifying space of a classical knot group

For a prime knot group, the classifying space for the family of the subgroups generated by the meridians can be seen as an abstract analogue of the ambient manifold in which the knot lives. An explicit model of this ambient classifying space is constructed as a branched covering space of the 3-sphere branched over the knot; more general branched covering spaces, obtained quotienting the ambient classifying space by finite-index normal subgroups, are studied. Various homological properties of said spaces are established, some of which have parallels in algebraic number theory. In particular, prime knot groups are shown to be Bieri-Eckmann, but not Poincaré, duality groups for Bredon cohomology with respect to the family of the meridians.

math.AT↗

arXiv2020

Enhanced Koszul properties in Galois cohomology

We prove that Galois cohomology satisfies several surprisingly strong versions of Koszul properties, under a well known conjecture, in the finitely generated case. In fact, these versions of Koszulity hold for all finitely generated maximal pro-$p$ quotients of absolute Galois groups which are currently understood. We point out several of these unconditional results which follow from our work. We show how these enhanced versions are preserved under certain natural operations on algebras, generalising several results that were previously established only in the commutative case.

math.RA↗