arXiv subjects

Dario Prandi

Publications and source records attributed to Dario Prandi.

At least 19 recordsLinked to original sources

arXiv2026

Modeling sequential cognitive states via population level cortical dynamics

In this work, we present a mathematical model for cyclic and sequential patterns of brain activity, combining heteroclinic dynamics with discrete neural-field models. We first show that spatial-discrete neural-field equations with biologically realistic equilibria cannot support heteroclinic cycles. On the other hand, heterocline dynamics often arise in Lotka-Volterra-type systems, but these equations do not directly correspond to neuronal processes. To address this, we use a version of the Universal Approximation Theorem to approximate any target dynamics by a neural network interpretable as a high-dimensional Amari-type neural-field system. When the target dynamics contains a heteroclinic cycle, the approximating vector field generates a periodic trajectory that closely follows the heteroclinic connection. As a case study, we consider the cognitive processes underlying focused-attention meditation. We show how the model reproduces sequential transitions among cognitive states and we conclude providing a neural interpretation of the approximating dynamics.

math.DS↗

arXiv2026

Unweighted Hardy Inequalities on the Heisenberg Group and in Step-Two Carnot Groups

We establish unweighted Hardy-type inequalities on step-two Carnot groups with one-dimensional vertical layer, with explicit lower bounds for the optimal Hardy constant. The approach is based on a quantitative integration-by-parts mechanism that replaces the non-horizontal Euler vector field by a suitably constructed horizontal vector field with controlled norm. As applications, we obtain fully explicit bounds in the Heisenberg group for both the Kor{\`a}nyi gauge and the Carnot--Carath{\'e}odory distance, and we extend the results to non-isotropic step-two structures through a generalized Kor{\`a}nyi-type homogeneous norm.

math.AP↗

arXiv2025

Spectral properties of magnetic fields on sub-Riemannian contact manifolds

Motivated by some recent studies of the magnetic Laplacian on Riemannian manifolds, we focus on the first eigenvalue of the magnetic horizontal Laplacian on contact manifolds. We characterize conditions for positive spectral shift, and provide some sharp upper bounds. In the Riemannian setting, a genus 1 assumption is known to force the underlying metric to be flat when equality holds in the sharp upper bounds. Interestingly, we show that the equivalent topological condition in the three--dimensional contact setting consists of having first Betti number equal to 2. In this case, equality in our upper bounds implies that the structure is that of a Heisenberg left--invariant nilmanifold. We conclude by showing that, in some specific three--dimensional contact settings, the knowledge of the first eigenvalue of the magnetic Laplacian uniquely determines the manifold Chern class, fully determining the topology of the underlying manifold.

math.DG↗

arXiv2025

Asymptotics of motion planning complexity for control-affine systems

In this paper, we study the complexity of the approximation of nonadmissible curves for nonlinear control-affine systems satisfying the strong H{\"o}rmander condition. Focusing on tubular approximation complexities, we provide asymptotic equivalences, with explicit constants, for all generic situations where the distribution, i.e., the linear part of the control system, is of co-rank one. Namely, we consider curves in step 2 distributions and any dimension. In the 3 dimensional case, we also consider the case of distributions with Martinet-type singularities that are crossed by the curve at isolated points.

math.DS↗

arXiv2025

A solution to the mystery of the sub-harmonic series via a linear model of the cochlea

In this paper, we study a simple linear model of the cochlea as a set of vibrating strings. We make hypothesis that the information sent to the auditory cortex is the energy stored in the strings and consider all oscillation modes of the strings. We show the emergence of the sub-harmonic series whose existence was hypothesized in the XVI century to explain the consonance of the minor chord. We additionally show how the nonlinearity of the energy can be used to study the emergence of the combination tone (Tartini's third sound) shedding new light on this long debated subject.

eess.SP↗

arXiv2025

Schr\"odinger evolution on surfaces in 3D contact sub-Riemannian manifolds

Let $M$ be a 3-dimensional contact sub-Riemannian manifold and $S$ a surface embedded in $M$. Such a surface inherits a field of directions that becomes singular at characteristic points. The integral curves of such field define a characteristic foliation $\mathscr{F}$. In this paper we study the Schr\"odinger evolution of a particle constrained on $\mathscr{F}$. In particular, we relate the self-adjointness of the Schr\"odinger operator with a geometric invariant of the foliation. We then classify a special family of its self-adjoint extensions: those that yield disjoint dynamics.

math.SP↗

arXiv2024

Neural field equations with time-periodic external inputs and some applications to visual processing

The aim of this work is to present a mathematical framework for the study of flickering inputs in visual processing tasks. When combined with geometric patterns, these inputs influence and induce interesting psychophysical phenomena, such as the MacKay and the Billock-Tsou effects, where the subjects perceive specific afterimages typically modulated by the flickering frequency. Due to the symmetry-breaking structure of the inputs, classical bifurcation theory and multi-scale analysis techniques are not very effective in our context. We thus take an approach based on the input-output framework of control theory for Amari-type neural fields. This allows us to prove that, when driven by periodic inputs, the dynamics converge to a periodic state. Moreover, we study under which assumptions these nonlinear dynamics can be effectively linearised, and in this case we present a precise approximation of the integral kernel for short-range excitatory and long-range inhibitory neuronal interactions. Finally, for inputs concentrated at the center of the visual field with a flickering background, we directly relate the width of the illusory contours appearing in the afterimage with both the flickering frequency and the strength of the inhibition.

q-bio.NC↗

arXiv2024

Activity estimation via distributed measurements in an orientation sensitive neural fields model of the visual cortex

This paper investigates the online estimation of neural activity within the primary visual cortex (V1) in the framework of observability theory. We focus on a low-dimensional neural fields modeling hypercolumnar activity to describe activity in V1. We utilize the average cortical activity over V1 as measurement. Our contributions include detailing the model's observability singularities and developing a hybrid high-gain observer that achieves, under specific excitation conditions, practical convergence while maintaining asymptotic convergence in cases of biological relevance. The study emphasizes the intrinsic link between the model's non-linear nature and its observability. We also present numerical experiments highlighting the different properties of the observer.

math.OC↗

arXiv2024

Reproducibility via neural fields of visual illusions induced by localized stimuli

This paper focuses on the modeling of experiments conducted by Billock and Tsou [V. A. Billock and B. H. Tsou, Proc. Natl. Acad. Sci. USA, 104 (2007), pp. 8490--8495] using an Amari-type neural field that models the average membrane potential of neuronal activity in the primary visual cortex (V1). The study specifically focuses on a regular funnel pattern localized in the fovea or the peripheral visual field. It aims to comprehend and model the visual phenomena induced by this pattern, emphasizing their nonlinear nature. The research involves designing sensory inputs that mimic the visual stimuli from Billock and Tsou's experiments. The cortical outputs induced by these sensory inputs are then theoretically and numerically studied to assess their ability to model the experimentally observed visual effects at the V1 level. A crucial aspect of this study is the exploration of the effects induced by the nonlinear nature of neural responses. By highlighting the significance of excitatory and inhibitory neurons in the emergence of these visual phenomena, the research suggests that an interplay of both types of neuronal activities plays a crucial role in visual processes, challenging the assumption that the latter is primarily driven by excitatory activities alone.

q-bio.NC↗

arXiv2023

A mathematical model of the visual MacKay effect

This paper investigates the intricate connection between visual perception and the mathematical modeling of neural activity in the primary visual cortex (V1). The focus is on modeling the visual MacKay effect [D. M. MacKay, Nature, 180 (1957), pp. 849--850]. While bifurcation theory has been a prominent mathematical approach for addressing issues in neuroscience, especially in describing spontaneous pattern formations in V1 due to parameter changes, it faces challenges in scenarios with localized sensory inputs. This is evident, for instance, in MacKay's psychophysical experiments, where the redundancy of visual stimuli information results in irregular shapes, making bifurcation theory and multiscale analysis less effective. To address this, we follow a mathematical viewpoint based on the input-output controllability of an Amari-type neural fields model. In this framework, we consider sensory input as a control function, a cortical representation via the retino-cortical map of the visual stimulus that captures its distinct features. This includes highly localized information in the center of MacKay's funnel pattern "MacKay rays". From a control theory point of view, the Amari-type equation's exact controllability property is discussed for linear and nonlinear response functions. For the visual MacKay effect modeling, we adjust the parameter representing intra-neuron connectivity to ensure that cortical activity exponentially stabilizes to the stationary state in the absence of sensory input. Then, we perform quantitative and qualitative studies to demonstrate that they capture all the essential features of the induced after-image reported by MacKay.

math.OC↗

arXiv2023

Weyl's law for singular Riemannian manifolds

We study the asymptotic growth of the eigenvalues of the Laplace-Beltrami operator on singular Riemannian manifolds, where all geometrical invariants appearing in classical spectral asymptotics are unbounded, and the total volume can be infinite. Under suitable assumptions on the curvature blow-up, we show how the singularity influences the Weyl's asymptotics. Our main motivation comes from the construction of singular Riemannian metrics with prescribed non-classical Weyl's law. Namely, for any non-decreasing slowly varying function $\upsilon$ we construct a singular Riemannian structure whose spectrum is discrete and satisfies \[ N(λ) \sim \frac{ω_n}{(2π)^n} λ^{n/2} \upsilon(λ). \] Examples of slowly varying functions are $\logλ$, its iterations $\log_k λ= \log_{k-1}\logλ$, any rational function with positive coefficients of $\log_k λ$, and functions with non-lo\-ga\-rithmic growth such as $\exp\left((\log λ)^{α_1} \dots (\log_k λ)^{α_k} \right)$ for $α_i \in (0,1)$. A key tool in our arguments is a new quantitative estimate for the remainder of the heat trace and the Weyl's function on Riemannian manifolds, which is of independent interest.

math.DG↗

arXiv2023

Beyond $\ell_1$ sparse coding in V1

Growing evidence indicates that only a sparse subset from a pool of sensory neurons is active for the encoding of visual stimuli at any instant in time. Traditionally, to replicate such biological sparsity, generative models have been using the $\ell_1$ norm as a penalty due to its convexity, which makes it amenable to fast and simple algorithmic solvers. In this work, we use biological vision as a test-bed and show that the soft thresholding operation associated to the use of the $\ell_1$ norm is highly suboptimal compared to other functions suited to approximating $\ell_q$ with $0 \leq q < 1 $ (including recently proposed Continuous Exact relaxations), both in terms of performance and in the production of features that are akin to signatures of the primary visual cortex. We show that $\ell_1$ sparsity produces a denser code or employs a pool with more neurons, i.e. has a higher degree of overcompleteness, in order to maintain the same reconstruction error as the other methods considered. For all the penalty functions tested, a subset of the neurons develop orientation selectivity similarly to V1 neurons. When their code is sparse enough, the methods also develop receptive fields with varying functionalities, another signature of V1. Compared to other methods, soft thresholding achieves this level of sparsity at the expense of much degraded reconstruction performance, that more likely than not is not acceptable in biological vision. Our results indicate that V1 uses a sparsity inducing regularization that is closer to the $\ell_0$ pseudo-norm rather than to the $\ell_1$ norm.

q-bio.NC↗

arXiv2022

Cortical origins of MacKay-type visual illusions: A case for the non-linearity

To study the interaction between retinal stimulation by redundant geometrical patterns and the cortical response in the primary visual cortex (V1), we focus on the MacKay effect (Nature, 1957) and Billock and Tsou's experiments (PNAS, 2007). We use a controllability approach to describe these phenomena starting from a classical biological model of neuronal field equations with a non-linear response function. The external input containing a localised control function is interpreted as a cortical representation of the static visual stimuli used in these experiments. We prove that while the MacKay effect is essentially a linear phenomenon (i.e., the nonlinear nature of the activation does not play any role in its reproduction), the phenomena reported by Billock and Tsou are wholly nonlinear and depend strongly on the shape of the nonlinearity used to model the response function.

q-bio.NC↗

arXiv2022

Reproducing sensory induced hallucinations via neural fields

Understanding sensory-induced cortical patterns in the primary visual cortex V1 is an important challenge both for physiological motivations and for improving our understanding of human perception and visual organisation. In this work, we focus on pattern formation in the visual cortex when the cortical activity is driven by a geometric visual hallucination-like stimulus. In particular, we present a theoretical framework for sensory-induced hallucinations which allows one to reproduce novel psychophysical results such as the MacKay effect (Nature, 1957) and the Billock and Tsou experiences (PNAS, 2007).

q-bio.NC↗

arXiv2021

Horizontal magnetic fields and improved Hardy inequalities in the Heisenberg group

In this paper we introduce a notion of magnetic field in the Heisenberg group and we study its influence on spectral properties of the corresponding magnetic (sub-elliptic) Laplacian. We show that uniform magnetic fields uplift the bottom of the spectrum. For magnetic fields vanishing at infinity, including Aharonov--Bohm potentials, we derive magnetic improvements to a variety of Hardy-type inequalities for the Heisenberg sub-Laplacian. In particular, we establish a sub-Riemannian analogue of Laptev and Weidl sub-criticality result for magnetic Laplacians in the plane. Instrumental for our argument is the validity of a Hardy-type inequality for the Folland--Stein operator, that we prove in this paper and has an interest on its own.

math.SP↗

arXiv2021

An auditory cortex model for sound processing

The reconstruction mechanisms built by the human auditory system during sound reconstruction are still a matter of debate. The purpose of this study is to refine the auditory cortex model introduced in [9], and inspired by the geometrical modelling of vision. The algorithm transforms the degraded sound in an 'image' in the time-frequency domain via a short-time Fourier transform. Such an image is then lifted in the Heisenberg group and it is reconstructed via a Wilson-Cowan differo-integral equation. Numerical experiments on a library of speech recordings are provided, showing the good reconstruction properties of the algorithm.

math.AP↗

arXiv2021

A cortical-inspired sub-Riemannian model for Poggendorff-type visual illusions

We consider Wilson-Cowan-type models for the mathematical description of orientation-dependent Poggendorff-like illusions. Our modelling improves two previously proposed cortical-inspired approaches embedding the sub-Riemannian heat kernel into the neuronal interaction term, in agreement with the intrinsically anisotropic functional architecture of V1 based on both local and lateral connections. For the numerical realisation of both models, we consider standard gradient descent algorithms combined with Fourier-based approaches for the efficient computation of the sub-Laplacian evolution. Our numerical results show that the use of the sub-Riemannian kernel allows to reproduce numerically visual misperceptions and inpainting-type biases in a stronger way in comparison with the previous approaches.

cs.CV↗

arXiv2020

A bio-inspired geometric model for sound reconstruction

The reconstruction mechanisms built by the human auditory system during sound reconstruction are still a matter of debate. The purpose of this study is to propose a mathematical model of sound reconstruction based on the functional architecture of the auditory cortex (A1). The model is inspired by the geometrical modelling of vision, which has undergone a great development in the last ten years. There are however fundamental dissimilarities, due to the different role played by the time and the different group of symmetries. The algorithm transforms the degraded sound in an 'image' in the time-frequency domain via a short-time Fourier transform. Such an image is then lifted in the Heisenberg group and it is reconstructed via a Wilson-Cowan differo-integral equation. Preliminary numerical experiments are provided, showing the good reconstruction properties of the algorithm on synthetic sounds concentrated around two frequencies.

eess.AS↗