arXiv subjects

Rachel Nicks

Publications and source records attributed to Rachel Nicks.

9 recordsLinked to original sources

arXiv2026

Phase oscillator networks with multiple and state-dependent delays: A framework for exploring white matter plasticity in neurodynamics

Network science is increasingly focused on how node dynamics influence emergent phenomena such as oscillations, waves, chimeras, and turbulence. In oscillatory systems modeled by networks of coupled ordinary differential equations (ODEs), a common approach is to reduce the system to phase variables. When understanding how network delays shape emergent properties, the delays are often absorbed in the reduced description as a phase shift. The reduced system is a set of ODEs that loses some information about the full delay differential equation (DDE) system. We adopt a less restrictive approach and consider limit cycle oscillator networks with multiple delays that can be reduced to a DDE system. This captures the effects of delayed couplings, including coexistence and multistability of states. We analyze relative equilibria in the form of phase-locked states with tools previously developed for more general DDE settings. This allows us to explore patterning in networks with space-dependent delays, including in neuroscience, using symmetric bifurcation theory, linear stability analysis, and numerical simulations and continuation. In brain dynamics, time delays are determined by the speed of a communicating signal (action potential) along a fiber (axon). Importantly, these are now known to be state-dependent since the myelin (white matter) that insulates axons is plastic and can change in response to neuronal activity. A simple phenomenological model of this process (in the phase reduced description) is introduced and analyzed by extending techniques developed for fixed delays. Our analysis suggests that white matter plasticity can drive networks to more coherent behavior.

math.DS↗

arXiv2026

Adaptive conduction delays and phase locking in spiking Haken Lighthouse networks

We develop a theory of phase-locked activity in delayed spiking networks using the Haken Lighthouse model as an analytically tractable event-based description of neural dynamics. For networks with fixed delays, we derive self-consistency conditions for phase-locked states and an associated linear stability theory formulated directly in terms of spike-time perturbations. The framework is illustrated for a delayed autapse, a reciprocally coupled two-cell network, and spatially structured rings with distance-dependent coupling and conduction delays, where circulant symmetry allows stability to be decomposed into Fourier modes. We then introduce an activity-dependent white matter plasticity rule in which myelination modulates axonal conduction speed and hence communication delay. This leads naturally to a slow--fast system with state-dependent delays, in which frozen phase-locked branches organise the adaptive dynamics. The plasticity rule selects commensurate delay--period relationships, providing a mechanism for the emergence of synchrony, other frequency-locked states, slow switching between competing phase-locked patterns, and the organisation of heterogeneous delays into discrete delay--period classes. Direct simulations of the event-driven network support the analytical predictions and illustrate how adaptive conduction can reshape the attractor structure of a delayed spiking network and generate long-timescale transitions. These results provide a tractable mathematical framework for studying how activity-dependent myelination may regulate temporal coordination, synchrony, and communication through coherence in spiking neural systems.

q-bio.NC↗

arXiv2026

Network Attractors driven by Time-Delay Plasticity

We develop a framework for collective frequency selection and attractor formation by means of delay plasticity. Specifically, we consider adaptive axonal delays (AADs), motivated by activity-dependent myelination in the brain which regulates signal propagation speeds and thus communication delays. We demonstrate frequency selection and explosive network relaxation oscillations in systems of delay-coupled phase oscillators with AADs on brain connectivity data and fully coupled ring networks.

nlin.AO↗

arXiv2024

Phase and amplitude responses for delay equations using harmonic balance

Robust delay induced oscillations, common in nature, are often modeled by delay-differential equations (DDEs). Motivated by the success of phase-amplitude reductions for ordinary differential equations with limit cycle oscillations, there is now a growing interest in the development of analogous approaches for DDEs to understand their response to external forcing. When combined with Floquet theory, the fundamental quantities for this reduction are phase and amplitude response functions. Here, we develop a framework for their construction that utilises the method of harmonic balance.

math.DS↗

arXiv2023

Insights into oscillator network dynamics using a phase-isostable framework

Networks of coupled nonlinear oscillators can display a wide range of emergent behaviours under variation of the strength of the coupling. Network equations for pairs of coupled oscillators where the dynamics of each node is described by the evolution of its phase and slowest decaying isostable coordinate have previously been shown to capture bifurcations and dynamics of the network which cannot be explained through standard phase reduction. An alternative framework using isostable coordinates to obtain higher-order phase reductions has also demonstrated a similar descriptive ability for two oscillators. In this work we consider the phase-isostable network equations for an arbitrary but finite number of identical coupled oscillators, obtaining conditions required for stability of phase-locked states including synchrony. For the mean-field complex Ginzburg-Landau equation where the solutions of the full system are known, we compare the accuracy of the phase-isostable network equations and higher-order phase reductions in capturing bifurcations of phase-locked states. We find the former to be the more accurate and therefore employ this to investigate the dynamics of globally linearly coupled networks of Morris-Lecar neuron models (both two and many nodes). We observe qualitative correspondence between results from numerical simulations of the full system and the phase-isostable description demonstrating that in both small and large networks the phase-isostable framework is able to capture dynamics that the first-order phase description cannot.

math.DS↗

arXiv2023

Oscillatory networks: Insights from piecewise-linear modeling

There is enormous interest -- both mathematically and in diverse applications -- in understanding the dynamics of coupled oscillator networks. The real-world motivation of such networks arises from studies of the brain, the heart, ecology, and more. It is common to describe the rich emergent behavior in these systems in terms of complex patterns of network activity that reflect both the connectivity and the nonlinear dynamics of the network components. Such behavior is often organized around phase-locked periodic states and their instabilities. However, the explicit calculation of periodic orbits in nonlinear systems (even in low dimensions) is notoriously hard, so network-level insights often require the numerical construction of some underlying periodic component. In this paper, we review powerful techniques for studying coupled oscillator networks. We discuss phase reductions, phase-amplitude reductions, and the master stability function for smooth dynamical systems. We then focus in particular on the augmentation of these methods to analyze piecewise-linear systems, for which one can readily construct periodic orbits. This yields useful insights into network behavior, but the cost is that one needs to study nonsmooth dynamical systems. The study of nonsmooth systems is well-developed when focusing on the interacting units (i.e., at the node level) of a system, and we give a detailed presentation of how to use \textit{saltation operators}, which can treat the propagation of perturbations through switching manifolds, to understand dynamics and bifurcations at the network level. We illustrate this merger of tools and techniques from network science and nonsmooth dynamical systems with applications to neural systems, cardiac systems, networks of electro-mechanical oscillators, and cooperation in cattle herds.

math.DS↗

arXiv2018

Clusters in nonsmooth oscillator networks

For coupled oscillator networks with Laplacian coupling the master stability function (MSF) has proven a particularly powerful tool for assessing the stability of the synchronous state. Using tools from group theory this approach has recently been extended to treat more general cluster states. However, the MSF and its generalisations require the determination of a set of Floquet multipliers from variational equations obtained by linearisation around a periodic orbit. Since closed form solutions for periodic orbits are invariably hard to come by the framework is often explored using numerical techniques. Here we show that further insight into network dynamics can be obtained by focusing on piecewise linear (PWL) oscillator models. Not only do these allow for the explicit construction of periodic orbits, their variational analysis can also be explicitly performed. The price for adopting such nonsmooth systems is that many of the notions from smooth dynamical systems, and in particular linear stability, need to be modified to take into account possible jumps in the components of Jacobians. This is naturally accommodated with the use of \textit{saltation} matrices. By augmenting the variational approach for studying smooth dynamical systems with such matrices we show that, for a wide variety of networks that have been used as models of biological systems, cluster states can be explicitly investigated. By way of illustration we analyse an integrate-and-fire network model with event-driven synaptic coupling as well as a diffusively coupled network built from planar PWL nodes, including a reduction of the popular Morris--Lecar neuron model. We use these examples to emphasise that the stability of network cluster states can depend as much on the choice of single node dynamics as it does on the form of network structural connectivity.

math.DS↗

arXiv2015

Mathematical frameworks for oscillatory network dynamics in neuroscience

The tools of weakly coupled phase oscillator theory have had a profound impact on the neuroscience community, providing insight into a variety of network behaviours ranging from central pattern generation to synchronisation, as well as predicting novel network states such as chimeras. However, there are many instances when this theory is expected to break down, say in the presence of strong coupling, or must be carefully interpreted, as in the presence of stochastic forcing. There are also surprises in the dynamical complexity of the attractors that can robustly appear - for example, heteroclinic network attractors. In this review we present a set of mathematical tools that are suitable for addressing the dynamics of oscillatory neural networks, broadening from a standard phase oscillator perspective to provide a practical framework for further successful applications of mathematics to understanding network dynamics in neuroscience.

nlin.AO↗

arXiv2013

A classification of the symmetries of uniform discrete defective crystals

Crystals which have a uniform distribution of defects are endowed with a Lie group description which allows one to construct an associated discrete structure. These structures are in fact the discrete subgroups of the ambient Lie group. The geometrical symmetries of these structures can be computed in terms of the changes of generators of the discrete subgroup which preserve the discrete set of points. Here a classification of the symmetries for the discrete subgroups of a particular class of three-dimensional solvable Lie group is presented. It is a fact that there are only three mathematically distinct types of Lie groups which model uniform defective crystals, and the calculations given here complete the discussion of the symmetries of the corresponding discrete structures. We show that those symmetries corresponding to automorphisms of the discrete subgroups extend uniquely to symmetries of the ambient Lie group and we regard these symmetries as (restrictions of) elastic deformations of the continuous defective crystal. Other symmetries of the discrete structures are classified as `inelastic' symmetries.

math-ph↗