### The role of brain networks in PTSD

Post-Traumatic Stress Disorder (PTSD) is a neurological condition that occurs in some individuals after experiencing a traumatic event, such as a terrorist attack, car crash or live combat. Symptoms include distressing recollections or dreams about the event, hypervigilance and irritability. PTSD is associated with alterations in the electrical activity of the brain that can be captured using a device called an electroencephalogram (EEG). There is a clear gap in our understanding how these alterations change with CBT and relate to symptom reduction. The aim of this research is to identify dynamic features of altered brain electrical activity associated with PTSD and CBT response and use mathematical modelling and analysis to identify features that will enable patient-specific prediction of treatment response.

### Domino-effect like brain dynamics

Analysis of transitions between different neurological states is crucial to understanding and effective treatment of neurological disorders. Research into these transitions, also called transient dynamics, is an emerging area of interdisciplinary research that presents many challenges to study. 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.

### Microstates sequences track consiousness

EEG recordings can be converted into sequences of activity maps where the topography of the EEG activity is fixed for a short time. These activity maps are called microstates, which have been described as "atoms of thought". We analyse the statistical propoerties of the sequences of microstates including the transition probabilities and distributino of time spent in each states. We use an excitable network model to track how these properties change under the influence of anesthesia.

### 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.

### Publications

- 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. - Creaser, J., Ashwin, P. and Tsaneva-Atanasova, K., 2020. 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. - 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. - 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. - 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. - P Ashwin, J Creaser and K Tsaneva-Atanasova, 2017, Fast and slow domino effects in transient network dynamics,
*Physical Review E*,**96(5):**052309. - 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