Attendance at this conference
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Trends in Differential Equations, Mathematical Neuroscience and other biological topics
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Conference
From April 16, 2026
to April 18, 2026
to April 18, 2026
Venue: Hotel Riu Fluvià d'Olot (Girona)
Registration deadline 27 / 02 / 2026
SCHEDULE
REGISTRATION
VENUE
The event will take place at the Hotel Riu Fluvià d’Olot (Girona), where participants will be accommodated in shared rooms.
Introduction
This conference will be focused on three areas:
1) The qualitative theory of ordinary differential equations in the plane (including Hilbert’s 16th problem, period functions, etc.), with a special emphasis on predator-prey models.
2) Neuroscience, approached from the perspective of dynamical systems.
3) Mathematical modelling in biology.
The conference will have a twofold objective: on the one hand, to feature invited talks by national and international experts in these fields; on the other, to promote scientific discussion between established and early-career researchers.
speakers
Short-term plasticity in strongly coupled neural networks: a relevant model in neuropsychiatry
Albert Compte
Institut d’Investigacions Biomèdiques August Pi i Sunyer
Abstract
Dynamics of a discrete time hypercycle
Ernest Fontich
Universitat de Barcelona
Abstract
Modeling experience encoding in a minimal animal
Jordi Garcia-Ojalvo
Universitat Pompeu Fabra
Abstract
Persistent homology meets neuronal networks
Esther Ibáñez
Universitat Oberta de Catalunya
Abstract
Multiple timescale dynamics in neural population models
Elif Köksal
Institut National de Recherche en Informatique et en Automatique
Abstract
Kahan-Hirota-Kimura maps associated with isochronous centers
Víctor Mañosa
Universitat Politècnica de Catalunya
Abstract
Demystifying Arnold’s tongues in reversible planar periodic linear systems
Enrique Ponce
Universidad de Sevilla
Abstract
Interaction of segregated resonant mechanisms along the dendritic axis in CA1 pyramidal cells: Interplay of cellular biophysics and spatial structure
Horacio Rotstein
New Jersey Institute of Technology
Abstract
Marco Sabatini
Università degli Studi di Trento
Breaking Network Synchrony with Sine Waves: Arnold Tongue Structure in Network Desynchronization
María Victoria Sánchez-Vives
Institut d’Investigacions Biomèdiques August Pi i Sunyer – Institut Català de Recerca i Estudis Avançats
Abstract
Josep Sardanyés
Centre de Recerca Matemàtica
Louis Tao
Peking University
Monotonicity criteria for the period function
Jordi Villadelprat
Universitat Autònoma de Barcelona
Abstract
schedule
| Thursday April 16th |
Friday April 17th |
Saturday April 18th |
|
|---|---|---|---|
| 9.30–10.10 | Arrival in Olot | Tomás Lázaro Universitat Politècnica de Catalunya – CRM |
Excursion & departure |
| 10.10–10.50 |
Dynamics of a discrete time hypercycle Ernest Fontich |
||
| 10.50–11.20 | Coffee break | ||
| 11.20–12.00 | Registration & Welcome | Josep Sardanyés Centre de Recerca Matemàtica |
|
| 12.00–12.40 |
Gemma Huguet
Cati Vich |
Modeling experience encoding in a minimal animal Jordi García-Ojalvo |
|
| 12.40–13.20 |
Interaction of segregated resonant mechanisms along the dendritic axis in CA1 Horacio Rotstein |
Persistent homology meets neuronal networks Esther Ibáñez |
|
| Lunch | |||
| 15.00–15.40 |
Multiple timescale dynamics in neural population models Elif Köksal |
Armengol Gasull Universitat Autònoma de Barcelona |
|
| 15.40–16.20 | TBP | Marco Sabatini Università degli Studi di Trento |
|
| 16.20–16.50 | Coffee break | Coffee Break | |
| 16.50–17.30 |
Synchrony with Sine Waves: Arnold Tongue Structure in Network Desynchronization Mavi Sánchez-Vives |
Kahan-Hirota-Kimura maps associated with isochronous centers Víctor Mañosa |
|
| 17.30–18.10 |
Short-term plasticity in strongly coupled neural networks: a relevant model in neuropsychiatry Albert Compte |
Demystifying Arnold’s tongues in reversible planar periodic linear systems Enrique Ponce |
|
| 18.10–18.50 | Louis Tao Peking University |
Monotonicity criteria for the period function Jordi Villadelprat |
|
| 20.00 | Dinner | ||
ORGANISING committee
Armengol Gasull | Universitat Autònoma de Barcelona
Gemma Huguet | Universitat Politècnica de Catalunya – Centre de Recerca Matemàtica
J. Tomás Lázaro | Universitat Politècnica de Catalunya – Centre de Recerca Matemàtica
Catalina Vich | Universitat de les Illes Balears
registration
Registration deadline 27 february 2026
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For inquiries about this event please contact the Scientific Events Coordinator Ms. Núria Hernández at nhernandez@crm.cat
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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.
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.
) 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.
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.
In the late 1990s and early 2000s there was a renewed interest in the study of isochronous centers. The interest was twofold: on the one hand, the characterization of isochronicity in families of centers was a problem analogous to the classical center-focus problem and the bifurcation of critical periods was analogous to the bifurcation of small-amplitude limit cycles. On the other hand, it was shown that isochronous centers exhibit very interesting geometric properties such as the existence of commuting vector fields, linearizations and explicit first integrals. See, for instance, [1, 2, 3, 5, 9, 10, 12, 13, 15].
From the point of view of discrete dynamics, there are two problems analogous to the isochronicity problem: the study and characterization of globally periodic maps, that is, maps for which all orbits are periodic with the same period; and the characterization of maps preserving foliations of curves on which a rotation number can be defined and it is constant. Both problems naturally arise in the context of the study of so-called integrable maps.
In this work, we consider integrable birational maps with rational first integrals, more specifically the so-called Kahan-Hirota-Kimura (KHK) maps, which arise as a discretization method of quadratic ODEs developed, independently, by Kahan (1993) and Hirota and Kimura (2000). These maps play an important role in the study of integrable systems, since they often admit first integrals when they are derived from systems having first integrals [4, 11].
The results presented in this talk are part of a work in which we present a methodology to characterize the dynamics of KHK maps that preserve fibrations, [8]. However, we will focus on KHK maps associated with quadratic vector fields with isochronous centers. These fields were characterized in [7] (see also [14]), obtaining four different cases, namely with , following the notation in [3].
fibrations, [8]. However, we will focus on KHK maps associated with quadratic vector fields with isochronous centers. These fields were characterized in [7] (see also [14]), obtaining four different cases, namely 
with 
, following the notation in [3].In particular, we focus on how to study the dynamics of a one-parameter family of KHK maps, which depend on the integration step size, associated with . The maps in the family preserve the first integral of the vector field and, moreover, the original vector field is a Lie symmetry of the map. Furthermore, on each level curve of the first integral, the dynamics is conjugate to a rotation, and for each fixed step size, the rotation number is constant. As a consequence, for a dense set of values of the step size parameter, the maps are globally periodic, realizing all possible periods except period 2.
. The maps in the family preserve the first integral of the vector field and, moreover, the original vector field is a Lie symmetry of the map. Furthermore, on each level curve of the first integral, the dynamics is conjugate to a rotation, and for each fixed step size, the rotation number is constant. As a consequence, for a dense set of values of the step size parameter, the maps are globally periodic, realizing all possible periods except period 2.For the remaining cases , , and , we have found numerical evidence suggesting the non-integrability of the associated KHK maps. For these cases, we also define a new class of maps, which we call pseudo-KHK maps. These maps preserve the first integrals, the associated vector fields are Lie symmetries, and we provide a methodology for the study of their dynamics, based on the results in [6]. Their dynamics turns out to be significantly richer than the one of the continuous flows that they approximate.
, 
, and 
, we have found numerical evidence suggesting the non-integrability of the associated KHK maps. For these cases, we also define a new class of maps, which we call pseudo-KHK maps. These maps preserve the first integrals, the associated vector fields are Lie symmetries, and we provide a methodology for the study of their dynamics, based on the results in [6]. Their dynamics turns out to be significantly richer than the one of the continuous flows that they approximate.The results presented have been obtained jointly with Chara Pantazi, [8].
References
[1] A. Algaba, M. Columé Reyes, A. Bravo. Geometry of the uniformly isochronous centers with polynomial commutator, Differential Equations Dynam. Systems 10: 257–275, 2002.
[2] J. Chavarriga, J. Giné, I. A. García. Isochronous centers of a linear center perturbed by homogeneous polynomials, in Proceedings of the 3rd Catalan Days on Applied Mathematics (Lleida, 1996), 65–80, Univ. Lleida, Lleida, 1996.
[3] J. Chavarriga, M. Sabatini. A survey of isochronous centers, Qual. Theory Dyn. Syst. 1: 1–70, 1999.
[4] E. Celledoni, R. I. McLachlan, B. Owren, G. R. W. Quispel. Geometric properties of Kahan’s method. J. Phys. A: Math. Theor., 46(2):025201, 12, 2013.
[5] E. Freire, A. Gasull, A. Guillamon. A characterization of isochronous centres in terms of symmetries, Rev. Mat. Iberoamericana 20: 205–222, 2004.
[6] M. Llorens, V. Mañosa. Lie symmetries of birational maps preserving genus 0 fibrations. J. Math. Anal. & Appl 432: 531–549, 2015.
[7] W. S. Loud. Behavior of the period of solutions of certain plane autonomous systems near centers, Contrib. Differential Equations 3: 21–36, 1964.
[8] V. Mañosa, Ch. Pantazi. Dynamics of Kahan-Hirota-Kimura maps with rational invariant fibrations. In preparation, 2026.
[9] P. Mardešić, C. Rousseau, B. Toni. Linearization of isochronous centers. J. Differential Equations 121: 67–108, 1995.
[10] L. Mazzi, M. Sabatini. Commutators and linearizations of isochronous centers. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 11: 81–98, 2000.
[11] M. Petrera, A. Pfadler, Y. B. Suris. On integrability of Hirota-Kimura type discretizations. Regul. Chaotic Dyn., 16: 245–289, 2011.
[12] M. Sabatini. Characterizing isochronous centres by Lie brackets. Differential Equations Dynam. Systems, 5: 91–99, 1997.
[13] M. Sabatini. Dynamics of commuting systems on two-dimensional manifolds. Ann. Mat. Pura Appl., 173: 213–232, 1997.
[14] M. Sabatini. Quadratic isochronous centers commute. Appl. Math. (Warsaw), 26: 357–362, 1999.
[15] M. Villarini. Regularity properties of the period function near a center of a planar vector field, Nonlinear Anal. 19: 787–803, 1992.