arXiv subjects

Daniele Avitabile

Publications and source records attributed to Daniele Avitabile.

At least 19 recordsLinked to original sources

arXiv2026

A functional analytic theory for differential equations on Banach spaces with slowly evolving parameters

This paper provides a functional analytic approach to differential equations on Banach spaces with slowly evolving parameters. We develop a Fenichel-like theory for attracting subsets of critical manifolds via a Lyapunov-Perron method. This functional analytic approach to invariant manifold theory for fast-slow systems of differential equations has not been fully developed before, especially for the case that the fast subsystem lives on an infinite-dimensional Banach space. We provide rigorous functional analytic proofs for both the persistence of attracting critical manifolds as smooth slow manifolds, as well as the validity of slow manifold reduction near slow manifolds. Several aspects of our proofs are new in the literature even for the finite-dimensional case. For instance, we require less smoothness assumptions on systems than comparable texts based on Lyapunov-Perron approaches. The theory as developed here provides a rigorous framework that allows one (for example) to derive formal statements on the dynamics of biologically meaningful spatially extended models with slowly varying parameters.

math.DS↗

arXiv2026

McKean-Vlasov Equations for Large Networks of Neurons with Adaptive Asymmetric Edges

Classical Mckean-Vlasov theory concerns high-dimensional particle systems for which the effect of one particle on another is instantaneous. However in many applications, particularly neuroscience, the effect of one particle on another is not instantaneous but delayed. For biophysically-accurate neuroscientific models, for each j to k one can introduce an additional stochastic process that describes the propagation of the signal from j to k. In this paper we determine a self-consistent autonomous SDE that describe the high-dimensional limit of such neural networks. We apply this system to a model of the visual cortex. Performing a bifurcation analysis, we determine conditions under which hopf bifurcations occur in the network.

math.PR↗

arXiv2026

State and Parameter Estimation for a Neural Model of Local Field Potentials

The study of cortical dynamics during different states such as decision making, sleep and movement, is an important topic in Neuroscience. Modelling efforts aim to relate the neural rhythms present in cortical recordings to the underlying dynamics responsible for their emergence. We present an effort to characterize the neural activity from the cortex of a mouse during natural sleep, captured through local field potential measurements. Our approach relies on using a discretized Wilson--Cowan Amari neural field model for neural activity, along with a data assimilation method that allows the Bayesian joint estimation of the state and parameters. We demonstrate the feasibility of our approach on synthetic measurements before applying it to a dataset available in literature. Our findings suggest the potential of our approach to characterize the stimulus received by the cortex from other brain regions, while simultaneously inferring a state that aligns with the observed signal.

q-bio.NC↗

arXiv2026

Exploring Clustering Capability of Inpainting Model Embeddings for Pattern-based Individual Identification

In this paper, we explore deep learning techniques for individual identification of animals based on their skin patterns. Individual identification is crucial in biodiversity monitoring, since it enables analysis of decline or growth of populations, or intra-species interactions within populations. Models trained for the task of individual identification often do not focus on the skin pattern of animals, but on background details or body shape details. These characteristics are not individually specific, or can change drastically through time. We focus on techniques that will make machine learning models more responsive to skin pattern structure when extracting individual visual embeddings from images. For this, we explore image inpainting of task-specific masks as an auxiliary task to enhance ML-based individual identification from animal skin patterns. We propose a comparative analysis among four models as an encoder backbone for the individual identification task. We focus on the case study of zebrafish, which is a widely recognized biological model organism, and which exhibits individually identifying skin patterns. To evaluate encoder backbone performance, we present standard metrics for classification accuracy, embedding clustering metrics, and GradCAM visualizations.

cs.CV↗

arXiv2026

Astrocytic resource diffusion stabilizes persistent activity in neural fields

Persistent neural activity underlying working memory requires sustained synaptic transmission, yet the metabolic and neurotransmitter support provided by astrocyte networks is largely absent from spatially extended neural circuit models. We introduce a coupled astrocyte-neural field model in which synaptic efficacy is regulated by depletion and recovery of a conserved resource pool recycled and spatially redistributed through diffusively coupled astrocytes. We obtain explicit stationary bump profiles and self-consistency conditions for bump width and amplitude on a canonical ring architecture. Linearizing about these solutions while carefully accounting for perturbations at bump boundaries, we analyze the resulting spectral problem governing stability. Our analysis, supported by numerical simulations and low-dimensional Fourier truncations, reveals a two-stage stabilization mechanism: astrocytic diffusion smooths resource asymmetries created by small bump displacements, and synaptic replenishment transfers this smoothing back to the synaptic pool. Together, sufficiently strong diffusion and replenishment suppress drift instabilities and enlarge the parameter regime in which stationary bumps persist.

q-bio.NC↗

arXiv2025

Neural Field Equations with random data

We study neural field equations, which are prototypical models of large-scale cortical activity, subject to random data. We view this spatially-extended, nonlocal evolution equation as a Cauchy problem on abstract Banach spaces, with randomness in the synaptic kernel, firing rate function, external stimuli, and initial conditions. We determine conditions on the random data that guarantee existence, uniqueness, and measurability of the solution in an appropriate Banach space, and examine the regularity of the solution in relation to the regularity of the inputs. We present results for linear and nonlinear neural fields, and for the two most common functional setups in the numerical analysis of this problem. In addition to the continuous problem, we analyse in abstract form neural fields that have been spatially discretised, setting the foundations for analysing uncertainty quantification (UQ) schemes.

math.NA↗

arXiv2025

Radial Basis Function Techniques for Neural Field Models on Surfaces

We present a numerical framework for solving neural field equations on surfaces using Radial Basis Function (RBF) interpolation and quadrature. Neural field models describe the evolution of macroscopic brain activity, but modeling studies often overlook the complex geometry of curved cortical domains. Traditional numerical methods, such as finite element or spectral methods, can be computationally expensive and challenging to implement on irregular domains. In contrast, RBF-based methods provide a flexible alternative by offering interpolation and quadrature schemes that efficiently handle arbitrary geometries with high-order accuracy. We first develop an RBF-based interpolatory projection framework for neural field models on general surfaces. Quadrature for both flat and curved domains are derived in detail, ensuring high-order accuracy and stability as they depend on RBF hyperparameters (basis functions, augmenting polynomials, and stencil size). Through numerical experiments, we demonstrate the convergence of our method, highlighting its advantages over traditional approaches in terms of flexibility and accuracy. We conclude with an exposition of numerical simulations of spatiotemporal activity on complex surfaces, illustrating the method's ability to capture complex wave propagation patterns.

math.NA↗

arXiv2025

Guided filtering and smoothing for infinite-dimensional diffusions

We consider the filtering and smoothing problems for an infinite-dimensional diffusion process X, observed through a finite-dimensional representation at discrete points in time. At the heart of our proposed methodology lies the construction of a path measure, termed the guided distribution of X, that is absolutely continuous with respect to the law of X, conditioned on the observations. We show that this distribution can be incorporated as a potent proposal measure for both sequential Monte Carlo as well as Markov Chain Monte Carlo schemes to tackle the filtering and smoothing problems respectively. In the offline setting, we extend our approach to incorporate parameter estimation of unknown model parameters. The proposed methodology is numerically illustrated in a case study for the stochastic Amari equation.

math.PR↗

arXiv2025

Neural Fields and Noise-Induced Patterns in Neurons on Large Disordered Networks

We study pattern formation in class of a large-dimensional neural networks posed on random graphs and subject to spatio-temporal stochastic forcing. Under generic conditions on coupling and nodal dynamics, we prove that the network admits a rigorous mean-field limit, resembling a Wilson-Cowan neural field equation. The state variables of the limiting systems are the mean and variance of neuronal activity. We select networks whose mean-field equations are tractable and we perform a bifurcation analysis using as control parameter the diffusivity strength of the afferent white noise on each neuron. We find conditions for Turing-like bifurcations in a system where the cortex is modelled as a ring, and we produce numerical evidence of noise-induced spiral waves in models with a two-dimensional cortex. We provide numerical evidence that solutions of the finite-size network converge weakly to solutions of the mean-field model. Finally, we prove a Large Deviation Principle, which provides a means of assessing the likelihood of deviations from the mean-field equations induced by finite-size effects.

math.PR↗

arXiv2025

Stochastic collocation schemes for Neural Field Equations with random data

We develop and analyse numerical schemes for uncertainty quantification in neural field equations subject to random parametric data in the synaptic kernel, firing rate, external stimulus, and initial conditions. The schemes combine a generic projection method for spatial discretisation to a stochastic collocation scheme for the random variables. We study the problem in operator form, and derive estimates for the total error of the schemes, in terms of the spatial projector. We give conditions on the projected random data which guarantee analyticity of the semi-discrete solution as a Banach-valued function. We illustrate how to verify hypotheses starting from analytic random data and a choice of spatial projection. We provide evidence that the predicted convergence rates are found in various numerical experiments for linear and nonlinear neural field problems.

math.NA↗

arXiv2025

Poisson Hypothesis and large-population limit for networks of spiking neurons

We study mean-field descriptions for spatially-extended networks of linear (leaky) and quadratic integrate-and-fire neurons with stochastic spiking times. We consider large-population limits of continuous-time Galves-Löcherbach (GL) networks with linear and quadratic intrinsic dynamics. We prove that that the Poisson Hypothesis holds for the replica-mean-field limit of these networks, that is, in a suitably-defined limit, neurons are independent with interaction times replaced by independent time-inhomogeneous Poisson processes with intensities depending on the mean firing rates, extending known results to networks with quadratic intrinsic dynamics and resets. Proving that the Poisson Hypothesis holds opens up the possibility of studying the large-population limit in these networks. We prove this limit to be a well-posed neural field model, subject to stochastic resets.

math.PR↗

arXiv2024

Well-posedness and regularity of solutions to neural field problems with dendritic processing

We study solutions to a recently proposed neural field model in which dendrites are modelled as a continuum of vertical fibres stemming from a somatic layer. Since voltage propagates along the dendritic direction via a cable equation with nonlocal sources, the model features an anisotropic diffusion operator, as well as an integral term for synaptic coupling. The corresponding Cauchy problem is thus markedly different from classical neural field equations. We prove that the weak formulation of the problem admits a unique solution, with embedding estimates similar to the ones of nonlinear local reaction-diffusion equations. Our analysis relies on perturbing weak solutions to the diffusion-less problem, that is, a standard neural field, for which weak problems have not been studied to date. We find rigorous asymptotic estimates for the problem with and without diffusion, and prove that the solutions of the two models stay close, in a suitable norm, on finite time intervals. We provide numerical evidence of our perturbative results.

math.AP↗

arXiv2022

Projection methods for Neural Field equations

Neural field models are nonlinear integro-differential equations for the evolution of neuronal activity, and they are a prototypical large-scale, coarse-grained neuronal model in continuum cortices. Neural fields are often simulated heuristically and, in spite of their popularity in mathematical neuroscience, their numerical analysis is not yet fully established. We introduce generic projection methods for neural fields, and derive a-priori error bounds for these schemes. We extend an existing framework for stationary integral equations to the time-dependent case, which is relevant for neuroscience applications. We find that the convergence rate of a projection scheme for a neural field is determined to a great extent by the convergence rate of the projection operator. This abstract analysis, which unifies the treatment of collocation and Galerkin schemes, is carried out in operator form, without resorting to quadrature rules for the integral term, which are introduced only at a later stage, and whose choice is enslaved by the choice of the projector. Using an elementary timestepper as an example, we demonstrate that the error in a time stepper has two separate contributions: one from the projector, and one from the time discretisation. We give examples of concrete projection methods: two collocation schemes (piecewise-linear and spectral collocation) and two Galerkin schemes (finite elements and spectral Galerkin); for each of them we derive error bounds from the general theory, introduce several discrete variants, provide implementation details, and present reproducible convergence tests.

math.NA↗

arXiv2022

Cross-scale excitability in networks of synaptically-coupled quadratic integrate-and-fire neurons

From the action potentials of neurons and cardiac cells to the amplification of calcium signals in oocytes, excitability is a hallmark of many biological signalling processes. In recent years, excitability in single cells has been related to multiple-timescale dynamics through canards, special solutions which determine the effective thresholds of the all-or-none responses. However, the emergence of excitability in large populations remains an open problem. Here, we show that the mechanism of excitability in large networks and mean-field descriptions of coupled quadratic integrate-and-fire (QIF) cells mirrors that of the individual components. We initially exploit the Ott-Antonsen ansatz to derive low-dimensional dynamics for the coupled network and use it to describe the structure of canards via slow periodic forcing. We demonstrate that the thresholds for onset and offset of population firing can be found in the same way as those of the single cell. We combine theoretical analysis and numerical computations to develop a novel and comprehensive framework for excitability in large populations, applicable not only to models amenable to Ott-Antonsen reduction, but also to networks without a closed-form mean-field limit, in particular sparse networks.

math.DS↗

arXiv2021

Bursting in a next generation neural mass model with synaptic dynamics: a slow-fast approach

We report a detailed analysis on the emergence of bursting in a recently developed neural mass model that takes short-term synaptic plasticity into account. The one being used here is particularly important, as it represents an exact meanfield limit of synaptically coupled quadratic integrate & fire neurons, a canonical model for type I excitability. In absence of synaptic dynamics, a periodic external current with a slow frequency ε can lead to burst-like dynamics. The firing patterns can be understood using techniques of singular perturbation theory, specifically slow-fast dissection. In the model with synaptic dynamics the separation of timescales leads to a variety of slow-fast phenomena and their role for bursting is rendered inordinately more intricate. Canards are one of the main slow-fast elements on the route to bursting. They describe trajectories evolving nearby otherwise repelling locally invariant sets of the system and are found in the transition region from subthreshold dynamics to bursting. For values of the timescale separation nearby the singular limit ε = 0, we report peculiar jump-on canards, which block a continuous transition to bursting. In the biologically more plausible regime of ε this transition becomes continuous and bursts emerge via consecutive spike-adding transitions. The onset of bursting is of complex nature and involves mixed-type like torus canards, which form the very first spikes of the burst and revolve nearby fast-subsystem repelling limit cycles. We provide numerical evidence for the same mechanisms to be responsible for the emergence of bursting in the quadratic integrate & fire network with plastic synapses. The main conclusions apply for the network, owing to the exactness of the meanfield limit.

math.DS↗

arXiv2020

Canonical models for torus canards in elliptic bursters

We revisit elliptic bursting dynamics from the viewpoint of torus canard solutions. We show that at the transition to and from elliptic burstings, classical or mixed-type torus canards can appear, the difference between the two being the fast subsystem bifurcation that they approach, saddle-node of cycles for the former and subcritical Hopf for the latter. We first showcase such dynamics in a Wilson-Cowan type elliptic bursting model, then we consider minimal models for elliptic bursters in view of finding transitions to and from bursting solutions via both kinds of torus canards. We first consider the canonical model proposed by Izhikevich (ref. [22] in the manuscript) and adapted to elliptic bursting by Ju, Neiman, Shilnikov (ref. [24] in the manuscript), and we show that it does not produce mixed-type torus canards due to a nongeneric transition at one end of the bursting regime. We therefore introduce a perturbative term in the slow equation, which extends this canonical form to a new one that we call Leidenator and which supports the right transitions to and from elliptic bursting via classical and mixed-type torus canards, respectively. Throughout the study, we use singular flows ($\varepsilon=0$) to predict the full system's dynamics ($\varepsilon>0$ small enough). We consider three singular flows: slow, fast and average slow, so as to appropriately construct singular orbits corresponding to all relevant dynamics pertaining to elliptic bursting and torus canards.

math.DS↗

arXiv2020

Local theory for spatio-temporal canards and delayed bifurcations

We present a rigorous framework for the local analysis of canards and slow passages through bifurcations in a wide class of infinite-dimensional dynamical systems with time-scale separation. The framework is applicable to models where an infinite-dimensional dynamical system for the fast variables is coupled to a finite-dimensional dynamical system for slow variables. We prove the existence of centre-manifolds for generic models of this type, and study the reduced, finite-dimensional dynamics near bifurcations of (possibly) patterned steady states in the layer problem. Theoretical results are complemented with detailed examples and numerical simulations covering systems of local- and nonlocal-reaction diffusion equations, neural field models, and delay-differential equations. We provide analytical foundations for numerical observations recently reported in literature, such as spatio-temporal canards and slow-passages through Hopf bifurcations in spatially-extended systems subject to slow parameter variations. We also provide a theoretical analysis of slow passage through a Turing bifurcation in local and nonlocal models.

math.DS↗

arXiv2020

Numerical investigation of a neural field model including dendritic processing

We consider a simple neural field model in which the state variable is dendritic voltage, and in which somas form a continuous one-dimensional layer. This neural field model with dendritic processing is formulated as an integro-differential equation. We introduce a computational method for approximating solutions to this nonlocal model, and use it to perform numerical simulations for neuro-biologically realistic choices of anatomical connectivity and nonlinear firing rate function. For the time discretisation we adopt an Implicit-Explicit (IMEX) scheme; the space discretisation is based on a finite-difference scheme to approximate the diffusion term and uses the trapezoidal rule to approximate integrals describing the nonlocal interactions in the model. We prove that the scheme is of first-order in time and second order in space, and can be efficiently implemented if the factorisation of a small, banded matrix is precomputed. By way of validation we compare the outputs of a numerical realisation to theoretical predictions for the onset of a Turing pattern, and to the speed and shape of a travelling front for a specific choice of Heaviside firing rate. We find that theory and numerical simulations are in excellent agreement.

math.DS↗