## Research

### Transient Dynamics on Networks

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 investigate transient dynamics in a number of mathematical models:

*Domino-effect like brain dynamics*: We construct networks of excitable nodes representing populations of neurones. 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.*Microstate sequences*: Brain activity can be measured using a scalp electroencephalogram (EEG). EEG recordings can be split 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 use a network approach to identify how sequences of microstates 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

**In prep**

- P Ashwin, JC and K Tsaneva-Atanasova, Sequential escapes: onset of slow domino caused by saddle connection.
- JC, P Ashwin, J Britz and C Postlethwaite, Modelling brain-dynamics altered by anaesthesia using EEG-microstates.
- JC, C Lin, T Ridler, J Brown, W D’Souza, U Seneviratne, M Cook and K Tsaneva-Atanasova, The slow domino effect modelling framework for the onset of epileptiform activity.

**2018**

- JC, P Ashwin, and K Tsaneva-Atanasova, Sequential noise-induced escapes for oscillatory network dynamics,
*SIAM Journal on Applied Dynamical Systems*,**17(1):**pp. 500–525.

**2017**

- JC, B Krauskopf and H M Osinga, Finding first foliation tangencies in the Lorenz system,
*SIAM Journal on Applied Dynamical Systems*,**16(4):**pp. 2127-2164 - P Ashwin, JC and K Tsaneva-Atanasova, Fast and slow domino effects in transient network dynamics,
*Physical Review E*,**96(5):**052309.

**2015**

- JC, B Krauskopf and H M Osinga, α-flips and T-points in the Lorenz system,
*Nonlinearity*,**28(3):**R39–R65.