Leeds Applied Nonlinear Dynamics (LAND)

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.

Abstract: This talk will be a mix of "communicating" and "doing" science in different ways, aided by modern conceptual and computational tools. I will demonstrate the value of communication across disciplinary boundaries using real-time interactive simulations through VisualPDE.com. I will describe how past and ongoing work with developmental biologists, geographers, ecologists, and microbiologists can be enriched with these tools, and in particular how deep insights can be rapidly communicated without the need for vast disciplinary expertise. Among other examples, I will discuss stochasticity in 2D Rayleigh-Bénard convection, as well as the role of subcritical diffusion-driven instabilities in oncological and embryological settings, demonstrating the ease with which nuanced and technical theoretical ideas can be illustrated and explored in real-time. I will end by discussing fundamental limitations of phenomenological dynamical systems modelling in general, giving rise to the need for better frameworks to do science.

ABSTRACT

We study both a variational and a non-variational (complex) version of the cubic-quintic Swift–Hohenberg model (SH35) in a finite disk with Neumann boundary conditions. These prototype models are relevant in contexts such as fluid dynamics, combustion experiments, and nonlinear optics.

Using a combination of numerical methods (direct numerical simulations and continuation techniques via pde2path) and analytical approaches (linear and weakly nonlinear analysis), we identify and study branches of extended and localised patterns. In the variational case, we find three qualitatively distinct types of branches termed: wall modes, axisymmetric modes, and extended states. For each branch type, secondary bifurcations give rise to localised states organised in an extension of the so-called snaking scenario.

In the non-variational case, many of the observed branches correspond to dynamical counterparts of the variational branches. The system supports travelling, modulated travelling, standing, and localised standing waves. Due to numerical constraints on the disk radius, we also analyse a one-dimensional version of the non-variational problem with periodic boundary conditions. This reduced problem provides a good description of the organisation of wall-mode solutions, enabling analytical tractability via weakly nonlinear methods.

Hil Meijer (University of Twente, NL),
Title: Synchrony across the brain; a harmonic balance approach to delay-coupled oscillators
Abstract:  Delays are a natural component of computational models of large-scale brain dynamics. Delays combined with local synaptic activity typically lead to oscillations, but the question is whether synchrony or some out-of-phase solution is stable. Here we present a machinery using harmonic balance and accounting for symmetries to look for instabilities of the synchronous solution. We first analyse a simple model on a ring where ``travelling waves'' with activity jumping to the nearest or next-nearest neighbour appear. Employing numerical continuation, we also track which pattern exists as we change the delay. For stability of the asynchronous solutions, we rely on simulations. We then move on to the Wilson-Cowan model with similar results, and highlight some of the additional numerical challenges.


Davide Liessi (University of Udine, IT).
Title: Stability of periodic orbits of delay equations
Abstract: The local stability properties of periodic orbits of a delay differential equation or of a renewal equation can be studied by computing the Floquet multipliers, i.e the eigenvalues of the monodromy operators of the linearized equation. These operators can be approximated via pseudospectral collocation, resulting in a matrix whose eigenvalues can be computed with standard methods. In this seminar I will recall the key ideas of the Floquet theory for delay equations, based on the sun-star perturbation theory, and I will present the pseudospectral approximation method, along with some examples showing its effectiveness.

Non-recurrent dynamics - processes in which a system goes through a sequence of states while never returning to a previous state - encompasses a range of important problems, perhaps most notably the spreading of epidemics. We consider a system in which individuals go through a set of states as a consequence of interactions with their neighbours on an underlying network substrate. Predicting the evolution of such processes on a general network is an exponentially difficult problem. On certain classes of networks, however, very precise approximations can be given in the form of message-passing equations, which make use of the fact that non-recurrent dynamics can, in principle, be solved exactly on trees. Here we offer an intuitive interpretation of the message-passing approximation, discuss its connection to the non-backtracking matrix of the given network substrate and explore when (and how) this approximation is expected to fail.

The Antarctic Ice Sheet can undergo non-linear dynamics due to the Marine Ice Sheet Instability (MISI). Observations of ocean-driven grounding line retreat in the Amundsen Sea Embayment in Antarctica raise the question of an imminent collapse of the West Antarctic Ice Sheet due to MISI. This would raise global sea levels by more than three metres, impacting coastal regions and communities worldwide. A collapse would be caused by irreversible retreat of the ice sheet’s grounding lines – the positions where the formerly grounded ice starts to float. Here we analyse whether Antarctic grounding lines are undergoing a Marine Ice Sheet Instability in their current position. Furthermore, we investigate the committed evolution of Antarctic grounding lines under present-day ocean and atmospheric conditions and put this into past context, in order to understand the stability of the (West) Antarctic Ice Sheet.