Peter Ashwin

Bubbling of attractors and synchronisation of chaotic oscillators

We present a system of two coupled identical chaotic electronic circuits that exhibit a blowout bifurcation resulting in loss of stability of the synchronised state. We introduce the concept of bubbling of an attractor, a new type of intermittency that is triggered by low levels of noise, and demonstrate numerical and experimental examples of this behaviour. In particular we observe bubbling near the synchronised state of two coupled chaotic oscillators. We give a theoretical description of the behaviour associated with locally riddled basins, emphasising the role of invariant measures. In general these are non-unique for a given chaotic attractor, which gives rise to a spectrum of Lyapunov exponents. The behaviour of the attractor depends on the whole spectrum. In particular, bubbling is associated with the loss of stability of an attractor in a dynamically invariant subspace, and is typical in such systems.

The dynamics ofn weakly coupled identical oscillators

SummaryWe present a framework for analysing arbitrary networks of identical dissipative oscillators assuming weak coupling. Using the symmetry of the network, we find dynamically invariant regions in the phase space existing purely by virtue of their spatio-temporal symmetry (the temporal symmetry corresponds to phase shifts). We focus on arrays which are symmetric under all permutations of the oscillators (this arises with global coupling) and also on rings of oscillators with both directed and bidirectional coupling. For these examples, we classify all spatio-temporal symmetries, including limit cycle solutions such as in-phase oscillation and those involving phase shifts. We also show the existence of “submaximal” limit cycle solutions under generic conditions. The canonical invariant region of the phase space is defined and used to investigate the dynamics. We discuss how the limit cycles lose and gain stability, and how symmetry can give rise to structurally stable heteroclinic cycles, a phenomenon not generically found in systems without symmetry. We also investigate how certain types of coupling (including linear coupling between oscillators with symmetric waveforms) can give rise to degenerate behaviour, where the oscillators decouple into smaller groups.

Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system

Tipping points associated with bifurcations (B-tipping) or induced by noise (N-tipping) are recognized mechanisms that may potentially lead to sudden climate change. We focus here on a novel class of tipping points, where a sufficiently rapid change to an input or parameter of a system may cause the system to ‘tip’ or move away from a branch of attractors. Such rate-dependent tipping, or R-tipping, need not be associated with either bifurcations or noise. We present an example of all three types of tipping in a simple global energy balance model of the climate system, illustrating the possibility of dangerous rates of change even in the absence of noise and of bifurcations in the underlying quasi-static system.

Controlled and stochastic retention concentrates dynein at microtubule ends to keep endosomes on track

Bidirectional transport of early endosomes (EEs) involves microtubules (MTs) and associated motors. In fungi, the dynein/dynactin motor complex concentrates in a comet‐like accumulation at MT plus‐ends to receive kinesin‐3‐delivered EEs for retrograde transport. Here, we analyse the loading of endosomes onto dynein by combining live imaging of photoactivated endosomes and fluorescent dynein with mathematical modelling. Using nuclear pores as an internal calibration standard, we show that the dynein comet consists of ∼55 dynein motors. About half of the motors are slowly turned over (T1/2: ∼98 s) and they are kept at the plus‐ends by an active retention mechanism involving an interaction between dynactin and EB1. The other half is more dynamic (T1/2: ∼10 s) and mathematical modelling suggests that they concentrate at MT ends because of stochastic motor behaviour. When the active retention is impaired by inhibitory peptides, dynein numbers in the comet are reduced to half and ∼10% of the EEs fall off the MT plus‐ends. Thus, a combination of stochastic accumulation and active retention forms the dynein comet to ensure capturing of arriving organelles by retrograde motors.

Calculation of the periodic spectral components in a chaotic DC-DC converter

A simple mapping is derived, which describes the behavior of a peak current-mode controlled boost converter operating chaotically. The invariant density of this mapping is calculated iteratively and, from this, the power density spectrum of the input current at the clock frequency and its harmonics are deduced. The calculation is presented, along with experimental verification. The possibility of a novel application of chaos-amelioration of power supply interference-is discussed,.

Early endosome motility spatially organizes polysome distribution

To distribute the protein translation machinery throughout the cytoplasm, polysomes in the fungus Ustilago maydis associate with mobile early endosomes, resulting in long-range motility along microtubules.

Nonlinear dynamics: When instability makes sense

Mathematical models that use instabilities to describe changes of weather patterns or spacecraft trajectories are well established. Could such principles apply to the sense of smell, and to other aspects of neural computation?

Three identical oscillators with symmetric coupling

This paper is theoretical and experimental investigation of three identical oscillators with weak symmetric coupling. The authors find that the dynamics is governed by an ODE on the 2-torus with S3 symmetry. A new type of codimension-one global bifurcation in such systems is predicted, and found experimentally in a network of three coupled van der Pol oscillators.

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 where 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.

Dynamics on Networks of Cluster States for Globally Coupled Phase Oscillators

Systems of globally coupled phase oscillators can have robust attractors that are heteroclinic networks. We investigate such a heteroclinic network between partially synchronized states where the phases cluster into three groups. For the coupling considered there exist 30 different three-cluster states in the case of five oscillators. We study the structure of the heteroclinic network and demonstrate that it is possible to navigate around the network by applying small impulsive inputs to the oscillator phases. This paper shows that such navigation may be done reliably even in the presence of noise and frequency detuning, as long as the input amplitude dominates the noise strength and the detuning magnitude, and the time between the applied pulses is in a suitable range. Furthermore, we show that, by exploiting the heteroclinic dynamics, frequency detuning can be encoded as a spatiotemporal code. By changing a coupling parameter we can stabilize the three-cluster states and replace the heteroclinic network...