The existence of localised two-dimensional patterns has been observed and studied in numerous experiments and simulations: ranging from optical solitons, to patches of desert vegetation, to fluid convection. And yet, our mathematical understanding of these emerging structures remains extremely limited beyond one-dimensional examples.
In this talk I will discuss how adding a compact region of spatial heterogeneity to a PDE model can not only induce the emergence of fully localised 2D patterns, but also allows us to rigorously prove and characterise their bifurcation. The idea is inspired by experimental and numerical studies of magnetic fluids and tornados, where our compact heterogeneity corresponds to a local spike in the magnetic field and temperature gradient, respectively. In particular, we obtain local bifurcation results for fully localised patterns both with and without radial or dihedral symmetry, and rigorously continue these solutions to large amplitude. Notably, the initial bifurcating solution (which can be stable at bifurcation) varies between a radially-symmetric spot and a 'dipole' solution as the width of the spatial heterogeneity increases.
This work is in collaboration with David J.B. Lloyd and Matthew R. Turner (both University of Surrey).
Rhythmic oscillations in brain activity and their disturbances are key biomarkers of brain function in healthy and diseased states. Modulating neural activity and restoring normal dynamics therefore require a quantitative understanding of these oscillatory processes. Mathematical models of neural oscillators provide a powerful framework for explaining how neuronal populations respond to external stimulation and can predict stimulation paradigms that achieve desired modulation.
In this talk, I will begin with short-term effects, introducing phase-reduced models and illustrating their capability by comparing the Kuramoto model with experimental data from Parkinsonian rats. While oscillators with fixed limit cycles capture transient phase and amplitude modulation, long-term responses—driven by connectivity changes and synaptic plasticity—are accompanied by a deformation of the underlying limit cycle. In the second part, I will present a predictive plasticity framework that models such network adaptation under stimulation, using an extended Wilson–Cowan model in which evolving connectivity reshapes the system’s oscillations.
Together, these two studies highlight how dynamical models can bridge theory and experiment to optimize brain stimulation strategies and ultimately provide translational benefits for treating neurological disorders.
Collective behaviour in social systems is shaped not only by who interacts with whom, but by the valence of these interactions: trust, alliance, tension, and conflict. I will discuss recent advances on modelling and measuring these signed structures. Using a multiscale measure of structural balance based on semi-walks, we show how patterns of affective alignment and frustration can be detected across scales and related to emerging group organisation (Talaga et al., Commun. Phys., 2023). Building on this, we analyse a multi-year longitudinal real signed network of high-schoolers and show that, despite continuous turnover in ties, the system exhibits equilibrium-like stability in macroscopic distributions (González-Casado, Teixeira & Sánchez, Commun. Phys., 2025).
I will end by outlining a new direction on modelling human–LLM dynamics and how there are two sides to study empathy and the possible individual-level (micro) spillover effects on cognitive–emotional networks, and their potential macroscopic consequences for social stability and cohesion.
Collective locomotion of flying animals is fascinating in terms of individual-level fluid mechanics and group-level structure and dynamics. In this talk, I will introduce a model of formation flight that views the collective as a material whose properties arise from the flow-mediated interactions among its members. It builds on an aerodynamic model that describes how flapping flyers produce vortex wakes and how they are influenced by the wakes of others. Long in-line arrays show that the group behaves as a soft “crystal” with regularly ordered member “atoms” whose positioning is susceptible to deformations and dynamical instabilities. Perturbing a member produces longitudinal waves that pass down the group while growing in amplitude; with these amplifications even causing collisions. The model explains the aerodynamic origin of the spacing between the flyers, the springiness of the interactions, and the tendency for disturbances to resonantly amplify. Our findings suggest analogies with material systems that could be generally useful in the analysis of animal groups.
Joint work with Daniel Baldwin (Leeds) and Alan McKane (Manchester emeritus)
Traditionally, stochastic processes are modelled using either a Fokker-Planck PDE approach, or a Langevin SDE approach. There is also a third way: the functional or path integral. Originally developed by Wiener in the 1920s to model Brownian motion, path integrals were famously applied to quantum mechanics by Feynman in the 1950s. However, they also offer much to classical stochastic processes. In this talk I will introduce the formalism, focussing on the one-dimensional case when the noise is weak. There exists a remarkable correspondence between the most-probable stochastic paths and Hamiltonian mechanics in an effective potential [1,2,3]. It turns out that in some cases, the paths that dominate the integral, and hence quantities like the potential barrier crossing (Kramers) rate, depart from the real line. This is in some sense analogous to the way the residues at complex poles control ordinary integrals along the real line.
[1] Ge, Hao, and Hong Qian. Int. J. Mod. Phys. B 26.24 1230012 (2012)
[2] SPF et al. J. Chem. Phys. 158.12 (2023)
[3] Honour, Tom and SPF. J. Phys. A 57 175002 (2024)
Abstract: At their simplest, heteroclinic cycles are unions of a finite number of equilibrium points along with heteroclinic trajectories connecting these equilibria in a cyclic fashion. They can arise in modelling the dynamics of evolving populations when there are several species competing with each other, and the game of ``Rock--Paper--Scissors'' is an archetypal example. Heteroclinic cycles can be robust when the connecting trajectories lie within subspaces where one or more species is extinct. Heteroclinic networks arise when trajectories have a choice of more than one route when leaving an equilibrium point. I will discuss recent work with Sofia Castro (Porto) and Claire Postlethwaite (Auckland) on heteroclinic cycles in pluridimensions and depth-two heteroclinic networks.
Understanding and predicting critical transitions in complex systems—ranging from neural circuits to climate subsystems—requires the recovery of both their underlying dynamics and network structure directly from noisy and limited observations. In this talk, I will present a data-driven framework for reconstructing the governing equations and interaction topology of weakly coupled chaotic networks, combining ideas from model reduction and system identification. A key insight is to leverage stochastic fluctuations, typically considered as noise, as informative signals encoding the hidden network structure. These signatures allow us to infer effective models that combine local dynamics with statistically estimated coupling rules, enabling the prediction of critical regime shifts [1]. Under suitable assumptions, we further refine the approach to recover exact dynamics, and validate it using both synthetic data inspired by cortical circuits and experimental recordings from the mouse neocortex [2]. The method is robust to short time series and sparse sampling, making it applicable in practical settings where full observability is rarely achievable. I will conclude with a discussion of open challenges in the reconstruction of dynamical networks from partial data, and outline how incorporating concepts from normal form theory and synchronization dynamics helps overcome current limitations [3].
References:
[1] D. Eroglu, M. Tanzi, S. van Strien, T. Pereira, Phys. Rev. X 10, 021047 (2020).
[2] I. Topal, D. Eroglu, Phys. Rev. Lett. 130, 117401 (2023).
[3] E. Nijholt, J.L. Ocampo-Espindola, D. Eroglu, I.Z. Kiss, T. Pereira, Nat. Commun. 13, 4849 (2022).
Three-wave interactions (3WIs) arise from a set of three wavevectors where the sum of first two gives the third one. We consider problems with two critical wavenumbers, where resonant triads form between two waves of a larger wavenumber and a third wave of a smaller wavenumber. 3WIs can be used to explain pattern-forming behaviour in the Faraday wave experiment close to onset. The experiment involves periodically forcing a container of fluid up and down and observing the patterns formed on the surface. When the forcing exceeds some threshold, the flat state becomes unstable, which can lead to a variety of patterns.
In this talk we will discuss the types of patterns formed by 3WIs with two wavenumbers and how these are different from patterns comprising a single wavelength. The dynamics exhibited by 3WIs can be represented as a system of complex ODEs, one equation for each amplitude of the wavevectors. We will consider how well this ODE system can predict the dynamics of a model PDE, an adaptation of the Lifschitz—Petrich equation.
Title:
Oscillatory neural dynamics in a phase-amplitude framework
Abstract
Model reduction techniques can provide useful insight into the dynamics behaviour of high dimensional oscillatory systems such as networks of neurons or neural field models. In this talk we will discuss the recently introduced phase-isostable framework which extends the classical technique of phase reduction to include a notion of a distance from limit cycle. This allows for representation of off cycle trajectories, the description of a greater variety of dynamics and greater accuracy in capturing the behaviour of the full model. We will highlight how this framework can be utilised to reveal bifurcations of phase-locked states in discrete networks and how this can be extended to networks with conduction delays and networks where the node oscillations are induced by delays. If time allows, we will also see how the framework can be applied to continuum neural field models to investigate instabilities of oscillatory phase waves to more exotic patterned states.
References:
[1] R Nicks, R Allen and S Coombes 2024 Insights into oscillator network dynamics using a phase-isostable framework, Chaos, Vol 34, 013141
[2] R Nicks, R Allen and S Coombes 2024 Phase and amplitude responses for delay equations using harmonic balance, Physical Review E, Vol 110, L012202
Abstract: How do collectives solve problems that individuals cannot? Our analysis shows how collective computational resources give rise to traditional forms of collective intelligence (the wisdom of the crowd, collective sensing, specialisation, cultural learning), and highlights underexplored research areas (collective reasoning, deliberation, collective adaptation). We explore this framework through case studies of specific collective behaviours in vigilance for predators, cooperative hunting, gradient navigation, and cumulative culture.
