### Domino effects and tipping cascades

In natural systems sometimes a small change can induce large and abrupt change. For example, ice sheet collapses or forest dieback. These abrupt transitions can have a potentially large impact on environment, economy and society. The critical threshold of such transitions is often referred to as tipping point and is closely related to bifurcations in nonlinear dynamical systems. In the Earth’s climate system, many large-scale components have been identified as tipping elements, i.e., sub-systems that may undergo abrupt transitions, with a potentially large impact on environment, economy and society. Tipping elements are dynamically coupled within the climate system, potentially leading to global tipping cascades. We develop new mathematical theory for domino effects and tipping cascades on networks.

- P Ashwin, J Creaser and K Tsaneva-Atanasova, 2018, Sequential escapes: onset of slow domino caused by saddle connection,
*The European Physical Journal Special Topics*,**227:**pp.1091–1100. - P Ashwin, J Creaser and K Tsaneva-Atanasova, 2017, Fast and slow domino effects in transient network dynamics,
*Physical Review E*,**96(5):**052309.

**Seizure Onset in Epilepsy**

We construct networks of excitable nodes representing populations of neurones based on phenomemonlogical model of seizure onset. Each node transitions between a steady (non-seizure) state and an oscillatory (seizure-like) state under the influence of noisy external input. We characterise how changes to the noisy input and the strength of coupling between nodes affects the transition to a seizure-like state. In particular we find that cahnges coupling strength lead to changes in the speed of propogation of activity leading to a slow domino effect. We apply the model to seizure-like recordings and explore how the interplay between coupling strength and node excitability (how easily a node can transition to the seizure state) effect seizure onset patterns.

- Creaser, J., Ashwin, P. and Tsaneva-Atanasova, K., 2021. Sequential Escapes and Synchrony Breaking for Networks of Bistable Oscillatory Nodes,
*SIAM Journal on Applied Dynamical Systems*,**19(4):**pp.2829-2846. - J Creaser, C Lin, T Ridler JT Brown, W D’Souza, U Seneviratne, M Cook, JR Terry, and K Tsaneva-Atanasova, 2020, Domino-like transient dynamics at seizure onset in epilepsy,
*PLoS computational biology*,**16(9):**e1008206. - J Creaser, K Tsaneva-Atanasova and P Ashwin, 2018, Sequential noise-induced escapes for oscillatory network dynamics,
*SIAM Journal on Applied Dynamical Systems*,**17(1):**pp. 500–525.
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### Microstate brain dynamics

Brain recordings, such as scalp level electroencephalogram (EEG) recordings, can be converted into sequences of activity maps. It is observed that the maps of the EEG activity do not switch continuously but actually stay fixed for a short time before switching. These activity maps are called **microstates**, and they have been described as "atoms of thought". The properties of sequences of microstates - how the microstate maps switch over time - can give information about someones brain state

**Modelling healthy brain dynamics**

We created a new excitable network model for microstate sequences observed in healthy resting brain dynamics. The model captures the the statistical properties of the sequences of microstates including the transition probabilities and distribution of time spent in each state.

- J Creaser, P Ashwin, C Postlethwaite, and J Britz, 2021. Noisy network attractor models for transitions between EEG microstates,
*The Journal of Mathematical Neuroscience***11(1):**pp.1-25.

**Post-Traumatic Stress Disorder**

Together with collaborators in Psychology we used microstate analysis to investigate the brain dynamics of people with Post-Traumatic Stress Disorder (PTSD). PTSD is a neurological condition that occurs in some individuals after experiencing a traumatic event, such as a terrorist attack, car crash or global pandemic. Self compassion can help reduce the symptoms of PTSD and improve treatment response, but the neural mechanisms are not well known and it is hard to predict who will response to treatment. We use microstate analysis to identify altered brain activity response of people with PTSD to a self compassion exercise. The aim of this work is to identify features that will enable patient-specific prediction of treatment response.

- J Creaser, J Store, and A Karl, 2022. Brain Responses to a Self-Compassion Induction in Trauma Survivors With and Without PTSD,
*Preprint*.

### The Lorenz system near the loss of the foliation condition

The well known Lorenz system is a simplified model of thermal atmospheric convection introduced by meteorologist Edward Lorenz in 1963. It is a parameter-dependent system of three ordinary differential equations. Warwick Tucker proved in 1999 that chaos exists in the system for the classic parameter values used by Lorenz. The analytic techniques used by Tucker and others to study the system rely on the condition that there is a stable foliation of the return map. This, in turn, means that the system can be reduced to a one-dimensional map, called the Lorenz map. We study a parameter regime where the foliation condition fails and much remains unknown about the system, including the parameter value at which the foliation condition itself is lost. Specifically, at how two-dimensional surfaces and one-dimensional curves, called invariant manifolds, organise the three-dimensional phase space of the Lorenz system and give structure to the chaos.

- J Creaser, B Krauskopf and H M Osinga, 2017, Finding first foliation tangencies in the Lorenz system,
*SIAM Journal on Applied Dynamical Systems*,**16(4):**pp. 2127-2164. - J Creaser, B Krauskopf and H M Osinga, 2015, α-flips and T-points in the Lorenz system,
*Nonlinearity*,**28(3):**R39–R65.

### Posters

### On the Brink of Chaos

Sep 2014

First prize in the University of Auckland Faculty of Science poster competition and the Department of Mathematics poster competition, Auckland, New Zealand