Invited Talk

Location

Seminar, Mathematics Department, University of Oxford, Oxford, 2026.

Date

October 2026.

Invited Talk

Location

Barcelona Mathematical Days (BMD 2026), Institut d'Estudis Catalans, Barcelona, 2026.

Date

7–9 October 2026.

Links

Event

Avitabile D., Lord G. J., Meddouni K.

Abstract

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 capture certain features of the neural activity from the cortex of a mouse during natural sleep, obtained 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 parameters of the external forcing in the neural activity model, while simultaneously inferring a state that aligns with the observed signal.

Links‌

DOI PDF Arxiv

BibTeX‌

@article{avitabile2026state,
  author  = {Avitabile, Daniele and Lord, Gabriel J. and Meddouni, Khadija},
  title   = {State and Parameter Estimation for a Neural Model of Local Field Potentials},
  journal = {Journal of Mathematical Biology},
  year    = {2026},
  volume  = {93},
  number  = {4},
  pages   = {52},
  doi     = {10.1007/s00285-026-02451-w}
}

Doorakkers D., Avitabile D., van den Berg J. B.

Abstract

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.

Links‌

DOI PDF Arxiv

BibTeX‌

@article{doorakkers2026functional,
  author  = {Doorakkers, Dirk and Avitabile, Daniele and van den Berg, Jan Bouwe},
  title   = {A functional analytic theory for differential equations on Banach spaces with slowly evolving parameters},
  journal = {Nonlinearity},
  year    = {2026},
  volume  = {39},
  number  = {9},
  pages   = {095010},
  doi     = {10.1088/1361-6544/ae9818}
}

Invited Talk

Location

Workshop on data-driven modeling, simulation and inference for neurobiological problems, NITMB, Chicago, 2026.

Date

29 September 2026.

Organiser

Location

Workshop, National Institute for Theory and Mathematics in Biology (NITMB), Chicago, 2026. Co-organised with B. Doiron, D. Sanz-Alonso and M. Tamborrino.

Date

28 September - 2 October 2026.

Palmer N., Cihak H. L., Avitabile D., Kilpatrick Z. P.

Abstract

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.

Links

DOI PDF Arxiv

BibTeX

@article{palmer2026astrocytic,
  author  = {Palmer, Noah and Cihak, Heather L. and Avitabile, Daniele and Kilpatrick, Zachary P.},
  title   = {Astrocytic Resource Diffusion Stabilizes Persistent Activity in Neural Fields},
  journal = {SIAM Journal on Life Sciences},
  year    = {2026},
  volume  = {1},
  number  = {3},
  pages   = {485--508},
  doi     = {10.1137/26M1871238}
}

Invited Talk

Location

Mathematical Approaches to Spatial Biology, Durham University, Durham, 2026.

Date

10 September 2026.

Links

Event

Baspinar E., Avitabile D., Nouveau C., Desroches M., Campillo F., Mantegazza M.

Abstract

We present a novel astro-neural-field population model with application to migraine-related cortical spreading depolarization. The model is composed of four spatio-temporal state variables: excitatory and inhibitory membrane potentials, astrocytic potassium uptake recruitment, and extracellular potassium concentration. Extending a previous neural field model, we incorporate activity-dependent astrocytic potassium clearance via a nonlinear term coupled to astrocyte dynamics. The astrocyte transfer function, like its neural counterpart, exhibits three regimes governed by extracellular potassium, capturing its effect on clearance. This yields a more comprehensive framework, better fits experimental data, and provides new insights into the mechanisms of cortical spreading depolarization.

Links

DOI PDF

BibTeX

@article{baspinar2026astro,
  author  = {Baspinar, Emre and Avitabile, Daniele and Nouveau, Camilla and Desroches, Mathieu and Campillo, Fabien and Mantegazza, Massimiliano},
  title   = {An astro-neural-field model with application to cortical spreading depolarization},
  journal = {Journal of Computational Neuroscience},
  year    = {2026},
  volume  = {54},
  number  = {3},
  pages   = {559--576},
  doi     = {10.1007/s10827-026-00948-3}
}

Avitabile D., Cavallini F., Dubinkina S., Lord G. J.

Abstract

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.

Links‌

DOI PDF Arxiv

BibTeX‌

@article{avitabile2026neuralb,
  author  = {Avitabile, Daniele and Cavallini, Francesca and Dubinkina, Svetlana and Lord, Gabriel J.},
  title   = {Neural {{Field Equations}} with {{Random Data}}},
  journal = {SIAM/ASA Journal on Uncertainty Quantification},
  year    = {2026},
  volume  = {14},
  number  = {2},
  pages   = {534--567},
  doi     = {10.1137/25M1768126}
}