arXiv subjects

Claire Postlethwaite

Publications and source records attributed to Claire Postlethwaite.

7 recordsLinked to original sources

arXiv2026

Local interaction of two systems with saddle-node bifurcations: mutualistic and mixed cases

The saddle-node bifurcation is the simplest example of a generic bifurcation in smooth ordinary differential equations, and is associated with the creation or destruction of a pair of equilibria. In this paper we examine the unfolding of the dynamics that occur when two generically coupled systems have simultaneous saddle-node bifurcations. We note that four parameters are required to generically unfold the interactions, and the dynamics are surprisingly complicated relative to the simplicity of a single saddle-node bifurcation. In the unfolding, in addition to saddle-node, Hopf and codimension-two local bifurcations, we also find a variety of global bifurcations, including homoclinic, SNIC, SNICeroclinic and non-central SNIC bifurcations. The latter two are codimension-two bifurcations that occur at the termination of a curve of SNIC bifurcations. A further contribution of this work is the development of numerical continuation techniques for the tracking of these codimension-two bifurcations through parameter space.

math.DS↗

arXiv2020

The Echo Index and multistability in input-driven recurrent neural networks

A recurrent neural network (RNN) possesses the echo state property (ESP) if, for a given input sequence, it ``forgets'' any internal states of the driven (nonautonomous) system and asymptotically follows a unique, possibly complex trajectory. The lack of ESP is conventionally understood as a lack of reliable behaviour in RNNs. Here, we show that RNNs can reliably perform computations under a more general principle that accounts only for their local behaviour in phase space. To this end, we formulate a generalisation of the ESP and introduce an echo index to characterise the number of simultaneously stable responses of a driven RNN. We show that it is possible for the echo index to change with inputs, highlighting a potential source of computational errors in RNNs due to characteristics of the inputs driving the dynamics.

math.DS↗

arXiv2019

Noisy network attractor models for transitions between EEG microstates

The brain is intrinsically organized into large-scale networks that constantly re-organize on multiple timescales, even when the brain is at rest. The timing of these dynamics is crucial for sensation, perception, cognition and ultimately consciousness, but the underlying dynamics governing the constant reorganization and switching between networks are not yet well understood. Functional magnetic resonance imaging (fMRI) and electroencephalography (EEG) provide anatomical and temporal information about the resting-state networks (RSNs), respectively. EEG microstates are brief periods of stable scalp topography, and four distinct configurations with characteristic switching patterns between them are reliably identified at rest. Microstates have been identified as the electrophysiological correlate of fMRI-defined RSNs, this link could be established because EEG microstate sequences are scale-free and have long-range temporal correlations. This property is crucial for any approach to model EEG microstates. This paper proposes a novel modeling approach for microstates: we consider nonlinear stochastic differential equations (SDEs) that exhibit a noisy network attractor between nodes that represent the microstates. Using a single layer network between four states, we can reproduce the transition probabilities between microstates but not the heavy tailed residence time distributions. Introducing a two layer network with a hidden layer gives the flexibility to capture these heavy tails and their long-range temporal correlations. We fit these models to capture the statistical properties of microstate sequences from EEG data recorded inside and outside the MRI scanner and show that the processing required to separate the EEG signal from the fMRI machine noise results in a loss of information which is reflected in differences in the long tail of the dwell-time distributions.

q-bio.NC↗

arXiv2016

Quantifying noisy attractors: from heteroclinic to excitable networks

Attractors of dynamical systems may be networks in phase space that can be heteroclinic (where there are dynamical connections between simple invariant sets) or excitable (where a perturbation threshold needs to be crossed to a dynamical connection between "nodes"). Such network attractors can display a high degree of sensitivity to noise both in terms of the regions of phase space visited and in terms of the sequence of transitions around the network. The two types of network are intimately related---one can directly bifurcate to the other. In this paper we attempt to quantify the effect of additive noise on such network attractors. Noise increases the average rate at which the networks are explored, and can result in "macroscopic" random motion around the network. We perform an asymptotic analysis of local behaviour of an escape model near heteroclinic/excitable nodes in the limit of noise $η\rightarrow 0^+$ as a model for the mean residence time $T$ near equilibria. We also explore transition probabilities between nodes of the network in the presence of anisotropic noise. For low levels of noise, numerical results suggest that a (heteroclinic or excitable) network can approximately realise any set of transition probabilities and any sufficiently large mean residence times at the given nodes. We show that this can be well modelled in our example network by multiple independent escape processes, where the direction of first escape determines the transition. This suggests that it is feasible to design noisy network attractors with arbitrary Markov transition probabilities and residence times.

nlin.AO↗

arXiv2015

Designing heteroclinic and excitable networks in phase space using two populations of coupled cells

We give a constructive method for realizing an arbitrary directed graph (with no one-cycles) as a heteroclinic or an excitable dynamic network in the phase space of a system of coupled cells of two types. In each case, the system is expressed as a system of first order differential equations. One of the cell types (the $p$-cells) interacts by mutual inhibition and classifies which vertex (state) we are currently close to, while the other cell type (the $y$-cells) excites the $p$-cells selectively and becomes active only when there is a transition between vertices. We exhibit open sets of parameter values such that these dynamical networks exist and demonstrate via numerical simulation that they can be attractors for suitably chosen parameters.

nlin.AO↗

arXiv2014

On designing heteroclinic networks from graphs

Robust heteroclinic networks are invariant sets that can appear as attractors in symmetrically coupled or otherwise constrained dynamical systems. These networks may have a very complicated structure that is poorly understood and determined to a large extent by the constraints and dimension of the system. As these networks are of great interest as dynamical models of biological and cognitive processes, it is useful to understand how particular graphs can be realised as robust heteroclinic networks that are attracting. This paper presents two methods of realizing arbitrarily complex directed graphs as robust heteroclinic networks for flows generated by ODEs---we say the ODEs {\em realise} the graphs as heteroclinic networks between equilibria that represent the vertices. Suppose we have a directed graph on $n_v$ vertices with $n_e$ edges. The "simplex realisation" embeds the graph as an invariant set of a flow on an $(n_v-1)$-simplex. This method realises the graph as long as it is one- and two-cycle free. The "cylinder realisation" embeds a graph as an invariant set of a flow on a $(n_e+1)$-dimensional space. This method realises the graph as long as it is one-cycle free. In both cases we find the graph as an invariant set within an attractor, and discuss some illustrative examples, including the influence of noise and parameters on the dynamics. In particular we show that the resulting heteroclinic network may or may not display "memory" of the vertices visited.

nlin.AO↗

arXiv2012

Resonance bifurcations of robust heteroclinic networks

Robust heteroclinic cycles are known to change stability in resonance bifurcations, which occur when an algebraic condition on the eigenvalues of the system is satisfied and which typically result in the creation or destruction of a long-period periodic orbit. Resonance bifurcations for heteroclinic networks are more complicated because different subcycles in the network can undergo resonance at different parameter values, but have, until now, not been systematically studied. In this article we present the first investigation of resonance bifurcations in heteroclinic networks. Specifically, we study two heteroclinic networks in $\R^4$ and consider the dynamics that occurs as various subcycles in each network change stability. The two cases are distinguished by whether or not one of the equilibria in the network has real or complex contracting eigenvalues. We construct two-dimensional Poincare return maps and use these to investigate the dynamics of trajectories near the network. At least one equilibrium solution in each network has a two-dimensional unstable manifold, and we use the technique developed in [18] to keep track of all trajectories within these manifolds. In the case with real eigenvalues, we show that the asymptotically stable network loses stability first when one of two distinguished cycles in the network goes through resonance and two or six periodic orbits appear. In the complex case, we show that an infinite number of stable and unstable periodic orbits are created at resonance, and these may coexist with a chaotic attractor. There is a further resonance, for which the eigenvalue combination is a property of the entire network, after which the periodic orbits which originated from the individual resonances may interact. We illustrate some of our results with a numerical example.

nlin.CD↗