Trends in Differential Equations, Mathematical Neuroscience and other biological topics

Mathematics can contribute to the understanding of brain function by providing models and analytical tools to describe and analyze the mechanisms underlying neuronal activity. Over the years, we have developed approaches based on dynamical systems and computational modeling to explore how neuronal dynamics shape computation and behavior.
In this talk, we will review several concepts and modeling frameworks that have emerged from this work with Prof. Guillamon. These include the study of isochrons and phase descriptions of oscillatory systems to characterize neuronal rhythms and their response to perturbations; mathematical techniques to address inverse problems, aimed at inferring experimentally inaccessible quantities such as synaptic conductances; and dynamical models of perceptual multistability, which provide insight into the competition between alternative neural representations in perception.
Together, these contributions illustrate how mathematical modeling can shed light on the mechanisms driving neuronal dynamics and their functional consequences for neural computation.

The origin of life in a prebiotic scenario and the storage of large amounts of information share the so-called error catastrophe or critical mutation rate. This abrupt change is due to the expected large accumulation of errors during replication (Eigen 1971; Eigen and Schuster 1979; Swetina and Schuster 1982; Küppers 1985; Maynard-Smith and Szathmary 1995) and appears to be intrinsically related to the information code length (Eigen’s paradox).
To solve this paradox, Eigen and Schuster (1979) proposed a plausible biochemical structure which they called the “hypercycle.” Hypercycles are catalytic networks of smaller replicators (units that self-copy and share information) arranged in a cyclic architecture. This spatial configuration avoids the error catastrophe and exhibits scenarios of stability; however, its robustness in the presence of parasites is not always clear. Its richness from the point of view of Dynamical Systems Theory is significant and displays many interesting features, as various papers by Solé, Sardanyés, Fontich, Guillamon, and others have shown.
In this talk we will provide a brief overview of some of these results.

Differential systems are often studied with the aid of symmetries, either to put them into a more convenient form, or to reduce their study to a proper portion of the space. In some cases finding a symmetry can even lead to discover dynamic properties of the system otherwise difficult to prove. This is the case of mirror symmetries, that can show the existence of periodic solutions in presence of rotating orbits. For planar systems this is the easiest way to prove the existence of cycles in absence of known first integrals. The argument showing that every rotating orbit of a planar system that meets the symmetry axis is a cycle applies without changes to mirror-like nonlinear symmetries. This provides a more general tool for proving local integrability. Such symmetries σ, usually called reversibilities, are characterized by the following flow property,

The existence of such a reversibility can be proved without knowing the flow by showing that

where Jσ is the Jacobian matrix of σ. A different flow property,

characterizes another class of symmetries, whose existence can be proved without knowing the flow by showing that

For the sake of simplicity we call “symmetries” only the second type trans‑ formations, using the term “reversibilities” for the first type.

We show that measure-preserving symmetries of an n-dimensional dif‑ ferential system preserve its divergence and the divergence derivatives along the solutions. Also, we prove that measure-preserving reversibilities pre‑ serve odd-order divergence derivatives along the solutions, and that even‑ order derivatives are multiplied by −1. We apply such results to find all the area-preserving symmetries and reversibilities of planar Lotka-Volterra and Liènard systems.

Ghosts have long fascinated our imagination. They inhabit myths and religions, appear in literature, inspire songs and films, and populate entire genres of storytelling. They are described as presences that linger after something has vanished — traces without substance, influence without embodiment. One does not see them directly; one perceives their effects. Whether or not such entities exist outside folklore, mathematics provides a precise setting in which ghosts undeniably appear.
Ghosts arise when invariant structures disappear but their dynamical influence persists. After a bifurcation, equilibria or invariant manifolds may collide, lose stability, or cease to  exist in the real phase space. Yet trajectories still slow down, linger near the vanished set, and exhibit robust scaling laws in their transient times. What disappears geometrically continues to organize the flow.
In this talk, I present a unified view of ghosts as remnants of normally hyperbolic invariant objects. Classical saddle–node bifurcations provide the simplest example: the annihilation of equilibria leaves behind bottleneck delays. But this mechanism extends far beyond local bifurcations. When curves of quasi-neutral equilibria break down through global bifurcations, their remnants generate qualitatively different slowing-down phenomena, reflecting the higher-dimensional nature of the underlying invariant structure.
Intrinsic noise does not destroy these ghosts. On the contrary, stochastic dynamics reveal their geometry even more clearly, as scaling laws emerge from the Hamiltonian structure underlying large deviations. Across deterministic and stochastic systems, local and global bifurcations, a common principle emerges: the character of the transient remembers the dimension of the object that has vanished.
A central insight is that ghosts cannot be fully understood by restricting attention to the real phase portrait alone. The invariant structures responsible for delayed transitions do not simply vanish; they persist in an extended geometric setting where their influence remains dynamically meaningful. To see a ghost, one must look in a space rich enough to contain its continuation.
In this sense, ghosts are not anomalies but universal transient generators, revealing that bifurcations reorganize geometry before they reorganize dynamics. I will finish the talk by showing evidence of a real ghost.

The simpler non-autonomous periodic differential equations differential equations that can have limit cycles are Riccati and Abel differential equations.
From many points of view, Riccati differential equations are well understood. For instance, their time-T flow is a M¨obius map, which implies that they can have at most two isolated periodic solutions (limit cycles). The
first aim of this talk is to show that these equations still present interesting open questions and, perhaps more importantly, that they arise naturally in the study of the exact number of limit cycles for several families of planar differential equations.
As an illustration, we describe some results on certain families of piecewise-rigid systems [4]. We also present new results concerning the exact number of limit cycles for several families of Riccati equations [3].
On the other hand, Abel equations are much more complicated and criteria to have upper bounds for some subfamilies of them are not easy to be found. We illustrate these facts with some results of [1,2]
The speaker is partially supported by the Spanish State Research Agency, through the project PID2022-136613NB-I00 grant.
References
[1] A. Gasull. Some open problems in low dimensional dynamical systems. SeMA J., 78, 233–269. 2021.
[2] A. Gasull, A. Guillamon. Limit cycles for generalized Abel equations. Internat. J. Bifur. Chaos Appl. Sci. Engrg., 16(2), 3737–3745. 2006.
[3] A. Gasull, D. D. Novaes, J. Torregrosa. Weak-Coppel problem for a class of Riccati differential equations. Preprint 2025.
[4] A. Gasull, J. Torregrosa. Limit cycles for piecewise rigid systems with homogeneous non-linearities. In preparation.

Understanding how brain stimulation interacts with intrinsic network dynamics is key for designing effective neuromodulation protocols. We studied the effects of alternating current (AC) fields with different amplitudes and frequencies on cortical slices showing spontaneous slow oscillations. Entrainment emerged within an Arnold tongue-like region around the endogenous frequency. In contrast, slightly detuned higher-frequency stimulation induced a desynchronized regime, suggesting a new strategy to disrupt pathological synchrony.
Adding a direct current (DC) offset broadened the modulatory range, enabling either entrainment or desynchronization depending on DC polarity. These results were quantitatively reproduced by a spiking-neuron model, supporting a nonlinear oscillator framework for predicting cortical responses to AC fields. The findings provide both mechanistic insight and a robust stimulation protocol with potential clinical relevance.

Strongly coupled neural networks are considered a plausible dynamical model of the cerebral cortex, and can explain features of multiple brain functions, such as short-term memory or neural oscillations. I will introduce the fundamental dynamical system motif that underlies our thinking of cortical network function, based on strong positive feedback and slower negative feedback. Based on this general principle, I will show how a biological network model based on strong coupling between excitatory and inhibitory neurons can be built to simulate closely two very different brain functions: spatial working memory, and slow-wave sleep. Finally, I will present our latest research on patients with anti-NMDAR encephalitis and persons with schizophrenia, where we have used this computational network model to interpret their working memory performance and their EEG during sleep. I will show how adding short-term synaptic plasticity to the model allows to interpret all these empirical results as a reduction in cortical short-term synaptic potentiation in these neuropsychiatric disorders.

Neuronal frequency filters play a crucial role in cognition, motor behavior, and the dynamics of information processing in neural networks in both health and disease. The filtering properties of neurons are shaped by several factors, including the intrinsic properties of neurons, their dendritic geometry, and the heterogeneous distribution of ionic currents. In CA1 pyramidal neurons, experimental results show the existence of two distinct theta (∼ 4 – 10 Hz) resonant mechanisms: (i) a perisomatic resonance mediated by an M-type potassium current ( ) and amplified by a persistent sodium current ( ), and (ii) a dendritic resonance mediated by a hyperpolarization-activated current ().
While the propagation of (amplitude) resonances along dendritic trees has been investigated before, it is unclear how the two experimentally observed biophysically different and spatially segregated types of resonance interact in the presence of a heterogeneous distribution of ionic currents and membrane potential variations. It is also unknown what are the interaction and propagation properties of the associated phasonances (phase-resonances) generated by similar, segregated mechanisms. In this work, we address these issues using CA1 pyramidal neurons as a case study. We use a multicompartmental model based on the Hodgkin-Huxley formalism. The model includes , and , spatially and heterogeneously distributed  along the dendrites, and oscillatory inputs at different locations. Our results reveal that voltage variations along the dendritic cable differentially activate ionic channels, creating a diverse range of resonant and phasonant responses. The spatial structure of the dendrites provides the neuron with remarkable flexibility to process these inputs and support a variety of scenarios of resonance interaction.

The existence of a time-reversal symmetry in periodic linear systems imposes some structure in the monodromy matrix. For planar systems, we study the local geometry of their stability boundaries around some critical values of parameters. We explain how each curve of the Arnold’s tongues are associated to the vanishing of certain entry in the monodromy matrix. This is a joint work with Emilio Freire and Manuel Ordóñez.

Neural population models provide a low-dimensional framework for describing the collective dynamics of large neuronal ensembles. Despite their reduced dimensionality, these models can generate rich activity patterns spanning multiple temporal scales. Such multiscale dynamics naturally emerge from underlying mechanisms including synaptic interactions, recurrent connectivity, and activity-dependent plasticity. In this talk, I will present a general overview of multiple-timescale dynamics in neural mass models. I will illustrate how distinct temporal processes give rise to structured activity patterns such as oscillations, slow modulations, and transient dynamics. Furthermore, I will discuss how geometric singular perturbation theory provides a principled mathematical framework to dissect and interpret these multiscale phenomena. This approach allows us to separate fast and slow subsystems, uncover invariant manifolds, and systematically analyze the mechanisms underlying complex population activity. Together, these perspectives offer both conceptual and analytical tools for understanding how multiscale structure shapes neural population dynamics.

Eigen and Schuster introduced hypercycles as systems of coupled macromolecules such that each catalyzes the self-replication of the next in a closed-loop structure. They were introduced as possible models of prebiotic evolution from simple elements to more complex structures. Subsequently, hypercycles have been used to model many different phenomena. Here we consider a discrete-time version of a hypercycle introduced by Hofbauer and study some features of its dynamics. By moving a parameter, we can study the transition from global cooperation to degradation from one member to the next. This transition is reflected in a degenerate Neimark-Sacker transcritical bifurcation. This was a joint work with Toni Guillamón, Julia Perona and Josep Sardanyés.

The study of period function dates back at least to 1673, when Christiaan Huygens observed that the pendulum clock has a monotonic period function and therefore oscillates with a shorter period as its energy decreases—that is, as the clock’s spring unwinds. Motivated by the need for greater accuracy in marine navigation, he sought to construct a clock exhibiting truly isochronous oscillations. His solution, the cycloidal pendulum, is perhaps the earliest known example of a nonlinear isochronous system.

In the second half of the twentieth century, many researchers resumed the study of questions related to the behavior of the period function. Pioneering contributions in this  direction were made by Loud, Urabe, Opial, Chow, Obi, and Chicone, among others.

In this talk, we will focus on results concerning the monotonicity of the period function. Our starting point will be a beautiful theorem by Freire, Gasull, and Guillamon, which relates the derivative of the period function to the existence of a normalizer for the vector field X. This is one of the few results that applies to general vector fields. We will then specialize to the case in which X is a potential vector field and review the monotonicity criteria established by Schaaf, Chicone, Chow and Wang, and Rothe. If time permits, we will sketch the main ideas of their proofs, present a slight refinement of one of these criteria, and discuss some relevant applications.

When adapting to changing environments, living systems benefit from modulating their response to present conditions according to their past experiences. In this talk I will discuss a specific example of such an experience-dependent behavior in a minimal model organism, namely the nematode C. elegans. I will discuss how mathematical modeling can help us validate the molecular mechanisms used by this worm to encode both its present and its recent past. I will also show how this molecular level of description is combined with the cellular and organismal levels to provide an integrated understanding of experience encoding in a minimal model organism.

The Brunel network is a neuronal network model composed of excitatory and inhibitory leaky integrate-and-fire spiking neurons. This model is known to exhibit four distinct activity states, which can be primarily characterized by the synchronicity and regularity of neuronal firing.
Topological Data Analysis (TDA) is a field within algebraic topology and computational geometry that emerged in the early 2000s, aimed at uncovering the topological structure of complex datasets. Its central tool is persistent homology, an algebraic framework that analyzes how topological features—such as cavities or holes in different dimensions— appear and disappear across scales.
The goal of this talk is to show that persistent homology can be useful to identify the dynamics of neuronal networks. In this case, we apply persistent homology to distinguish the four activity states generated by the Brunel network. Additionally, we use these topological techniques to detect and characterize the dynamical changes induced by synaptic plasticity, specifically short-term depression, in the Brunel network.