I am associate professor at the Department of Applied Mathematics of the University of Granada. I work on mathematical models in biology and physics, mainly based on partial differential equations. This includes kinetic equations, coagulation and fragmentation models, and nonlocal PDE in several contexts. I am interested in analytic properties of these models, their asymptotic behaviour, and related mathematical techniques.
I am a member of the Institute of Mathematics of the University of Granada (IMAG).
Here's a summary of things on this site:
- Research. Some updates on my research, a full list of preprints and publications, some informal notes on various topics, and other things!
- Teaching / docencia. List of courses I have taught and some publicly accessible lecture notes and material (most of them in Spanish).
- Other things. Some LaTeX templates for papers and posters, technical details on LaTeX things, and anything else which doesn't fit elsewhere.
- Contact information. My email is canizo@ugr.es and my phone (+34) 958 24 08 27. See here for mailing address and other details.
Preprints
- María J. Cáceres, José A Cañizo and Nicolas Torres.
On the local stability of the elapsed-time model in terms of the transmission delay and interconnection strength.
(2025)
The elapsed-time model describes the behavior of interconnected neurons through the time since their last spike. It is an age-structured non-linear equation in which age corresponds to the elapsed time since the last discharge, and models many interesting dynamics depending on the type of interactions between neurons. We investigate the linearized stability of this equation by considering a discrete delay, which accounts for the possibility of a synaptic delay due to the time needed to transmit a nerve impulse from one neuron to the rest of the ensemble. We state a stability criterion that allows to determine if a steady state is linearly stable or unstable depending on the delay and the interaction between neurons. Our approach relies on the study of the asymptotic behavior of related Volterra-type integral equations in terms of theirs Laplace transforms. The analysis is complemented with numerical simulations illustrating the change of stability of a steady state in terms of the delay and the intensity of interconnections.
- José A. Cañizo, Alejandro Gárriz and Fernando Quirós.
Large-time estimates for the Dirichlet heat equation in exterior domains.
(2025)
We give large-time asymptotic estimates, both in uniform and $L^1$ norms, for solutions of the Dirichlet heat equation in the complement of a bounded open set of $\R^d$ satisfying certain technical assumptions. We always assume that the initial datum has suitable finite moments (often, finite first moment). All estimates include an explicit rate of approach to the asymptotic profiles at the different scales natural to the problem, in analogy with the Gaussian behaviour of the heat equation in the full space. As a consequence we obtain by an approximation procedure the asymptotic profile, with rates, for the Dirichlet heat kernel in these exterior domains. The estimates on the rates are new even when the domain is the complement of the unit ball in $\R^d$, except for previous results by Uchiyama in dimension 2, which we are able to improve in some scales. We obtain that the heat kernel behaves asymptotically as the heat kernel in the full space, with a factor that takes into account the shape of the domain through a harmonic profile, and a second factor which accounts for the loss of mass through the boundary. The main ideas we use come from entropy methods in PDE and probability, whose application seems to be new in the context of diffusion problems in exterior domains.
- María J. Cáceres, José A. Cañizo and Nicolas Torres.
Comparison principles and asymptotic behavior of delayed age-structured neuron models.
(2025)
In the context of neuroscience the elapsed-time model is an age-structured equation that describes the behavior of interconnected spiking neurons through the time since the last discharge, with many interesting dynamics depending on the type of interactions between neurons. We investigate the asymptotic behavior of this equation in the case of both discrete and distributed delays that account for the time needed to transmit a nerve impulse from one neuron to the rest the ensemble. To prove the convergence to the equilibrium, we follow an approach based on comparison principles for Volterra equations involving the total activity, which provides a simpler and more straightforward alternative technique than those in the existing literature on the elapsed-time model.
Some recent publications
- José A. Cañizo and Nicccolò Tassi.
A uniform-in-time nonlocal approximation of the standard Fokker-Planck equation.
Discrete & Continuous Dynamical Systems 46:348-386 (2026)
We study a nonlocal approximation of the Fokker-Planck equation in which we can estimate the speed of convergence to equilibrium in a way which respects the local limit of the equation. This uniform estimate cannot be easily obtained with standard inequalities or entropy methods, but can be obtained through the use of Harris's theorem, finding interesting links to quantitative versions of the central limit theorem in probability. As a consequence one can prove that solutions of this nonlocal approximation converge to solutions of the usual Fokker-Planck equation uniformly in time—hence we show the approximation is asymptotic-preserving in this sense. The associated equilibrium has some interesting tail and regularity properties, which we also study.
- Ricardo Alonso, Véronique Bagland, José A. Cañizo, Bertrand Lods and Sebastian Throm.
One-dimensional inelastic Boltzmann equation: Stability and uniqueness of self-similar $L^1$-profiles for moderately hard potentials.
Communications in Partial Differential Equations 50:931-983 (2025)
We prove the stability of $L^1$ self-similar profiles under the hard-to-Maxwell potential limit for the one-dimensional inelastic Boltzmann equation with moderately hard potentials, which, in turn, leads to the uniqueness of such profiles for hard potential collision kernels of the form $|\cdot|^\gamma$ with $\gamma > 0$ sufficiently small (explicitly quantified). Our result provides the first uniqueness statement for self-similar profiles of inelastic Boltzmann models allowing for strong inelasticity besides the explicitly solvable case of Maxwell interactions (corresponding to $\gamma = 0$). Our approach relies on a perturbation argument from the corresponding Maxwell model and a careful study of the associated linearized operator recently derived in this companion paper. The results can be seen as a first step towards a complete proof, in the one-dimensional setting, of a conjecture in a paper by Ernst and Brito regarding the determination of the long-time behavior of solutions to the inelastic Boltzmann equation, at least in a regime of moderately hard potentials.
- María J. Cáceres, José A. Cañizo and Alejandro Ramos-Lora.
On the asymptotic behavior of the NNLIF neuron model for general connectivity strength.
Communications in Mathematical Physics 406 (2025)
We prove new results on the asymptotic behavior of the nonlinear integrate-and-fire neuron model. Among them, we give a criterion for the linearized stability or instability of equilibria, without restriction on the connectivity parameter, which provides a proof of stability or instability in some cases. In all cases, this criterion can be checked numerically, allowing us to give a full picture of the stable and unstable equilibria depending on the connectivity parameter and transmission delay. We also give further spectral results on the associated linear equation, and use them to give improved results on the nonlinear stability of equilibria for weak connectivity, and on the link between linearized and nonlinear stability.
- María J. Cáceres, José A. Cañizo and Alejandro Ramos-Lora.
Sequence of pseudo-equilibria describes the long-time behaviour of the NNLIF model with large delay.
Physical Review E 110:064308 (2024)
There is a wide range of mathematical models that describe populations of large numbers of neurons. In this article, we focus on nonlinear noisy leaky integrate and fire (NNLIF) models that describe neuronal activity at the level of the membrane potential of neurons. We introduce a set of novel states, which we call "pseudo-equilibria", and give evidence of their defining role in the behaviour of the NNLIF system when a significant synaptic delay is considered. The advantage is that these states are determined solely by the system's parameters and are derived from a sequence of firing rates that result from solving a recurrence equation. We propose a new strategy to show convergence to an equilibrium for a weakly connected system with large transmission delay, based on following the sequence of pseudo-equilibria. Unlike with the direct entropy dissipation method, this technique allows us to see how a large delay favours convergence. We also present a detailed numerical study to support our results. This study explores the overall behaviour of the NNLIF system and helps us understand, among other phenomena, periodic solutions in strongly inhibitory networks.
- José A. Cañizo and Alejandro Ramos-Lora.
Discrete minimizers of the interaction energy in collective behavior: a brief numerical and analytic review.
Active Particles, Volume 4 (2024)
We consider minimizers of the N-particle interaction potential energy and briefly review numerical methods used to calculate them. We consider simple pair potentials which are repulsive at short distances and attractive at long distances, focusing on examples which are sums of two powers. The range of powers we look at includes the well-known case of the Lennard-Jones potential, but we are also interested in less singular potentials which are relevant in collective behavior models. We report on results using the software GMIN developed by Wales and collaborators for problems in chemistry. For all cases, this algorithm gives good candidates for the minimizers for relatively low values of the particle number $N$. This is well-known for potentials similar to Lennard-Jones, but not for the range which is of interest in collective behavior. Standard minimization procedures have been used in the literature in this range, but they are likely to yield stationary states which are not minimizers. We illustrate numerically some properties of the minimizers in 2D, such as lattice structure, Wulff shapes, and the continuous large-$N$ limit for locally integrable (that is, less singular) potentials.
- Ricardo Alonso, Véronique Bagland, José A. Cañizo, Bertrand Lods and Sebastian Throm.
Relaxation in Sobolev spaces and $L^1$ spectral gap of the 1D dissipative Boltzmann equation with Maxwell interactions.
Discrete & Continuous Dynamical Systems 45:4572-4605 (2024)
We study the dynamic relaxation to equilibrium of the 1D dissipative Boltzmann equation with Maxwell interactions in classical $H^s$ Sobolev spaces. In addition, we present a spectral shrinkage analysis and spectral gap estimates for the linearised 1D dissipative Boltzmann operator with such interactions. Based on this study, we explore the convergence in $H^s$ and $L^1$ spaces for the linear and nonlinear models. This study extends classical results found in the literature given for spaces with weak topologies.