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# Dr Jen Creaser

## MRC Skills Development Fellowship (LSI)

** **

## Research

## Publications

Key publications | Publications by category | Publications by year

### Publications by category

### Journal articles

**Creaser J, Ashwin P, Postlethwaite C, Britz J**(In Press). Noisy network attractor models for transitions between EEG microstates.

## Abstract:

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.

Abstract. Author URL. 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.

**Ashwin P, Creaser J, Tsaneva-Atanasova K**(In Press). Sequential escapes: onset of slow domino regime via a saddle connection.

## Abstract:

Sequential escapes: onset of slow domino regime via a saddle connection

We explore sequential escape behaviour of coupled bistable systems under the

influence of stochastic perturbations. We consider transient escapes from a

marginally stable "quiescent" equilibrium to a more stable "active"

equilibrium. The presence of coupling introduces dependence between the escape

processes: for diffusive coupling there is a strongly coupled limit (fast

domino regime) where the escapes are strongly synchronised while for

intermediate coupling (slow domino regime) without partially escaped stable

states, there is still a delayed effect. These regimes can be associated with

bifurcations of equilibria in the low-noise limit. In this paper we consider a

localized form of non-diffusive (i.e pulse-like) coupling and find similar

changes in the distribution of escape times with coupling strength. However we

find transition to a slow domino regime that is not associated with any

bifurcations of equilibria. We show that this transition can be understood as a

codimension-one saddle connection bifurcation for the low-noise limit. At

transition, the most likely escape path from one attractor hits the escape

saddle from the basin of another partially escaped attractor. After this

bifurcation we find increasing coefficient of variation of the subsequent

escape times.

Abstract. Author URL. Full text.influence of stochastic perturbations. We consider transient escapes from a

marginally stable "quiescent" equilibrium to a more stable "active"

equilibrium. The presence of coupling introduces dependence between the escape

processes: for diffusive coupling there is a strongly coupled limit (fast

domino regime) where the escapes are strongly synchronised while for

intermediate coupling (slow domino regime) without partially escaped stable

states, there is still a delayed effect. These regimes can be associated with

bifurcations of equilibria in the low-noise limit. In this paper we consider a

localized form of non-diffusive (i.e pulse-like) coupling and find similar

changes in the distribution of escape times with coupling strength. However we

find transition to a slow domino regime that is not associated with any

bifurcations of equilibria. We show that this transition can be understood as a

codimension-one saddle connection bifurcation for the low-noise limit. At

transition, the most likely escape path from one attractor hits the escape

saddle from the basin of another partially escaped attractor. After this

bifurcation we find increasing coefficient of variation of the subsequent

escape times.

**Creaser J, Tsaneva-Atanasova, Ashwin P**(2018). Sequential noise-induced escapes for oscillatory network dynamics.

*SIAM Journal on Applied Dynamical Systems*,

*17(1)*, 500-525.

## Abstract:

Sequential noise-induced escapes for oscillatory network dynamics

It is well known that the addition of noise in a multistable system can

induce random transitions between stable states. The rate of transition can be

characterised in terms of the noise-free system's dynamics and the added noise:

for potential systems in the presence of asymptotically low noise the

well-known Kramers' escape time gives an expression for the mean escape time.

This paper examines some general properties and examples of transitions between

local steady and oscillatory attractors within networks: the transition rates

at each node may be affected by the dynamics at other nodes. We use first

passage time theory to explain some properties of scalings noted in the

literature for an idealised model of initiation of epileptic seizures in small

systems of coupled bistable systems with both steady and oscillatory

attractors. We focus on the case of sequential escapes where a steady attractor

is only marginally stable but all nodes start in this state. As the nodes

escape to the oscillatory regime, we assume that the transitions back are very

infrequent in comparison. We quantify and characterise the resulting sequences

of noise-induced escapes. For weak enough coupling we show that a master

equation approach gives a good quantitative understanding of sequential

escapes, but for strong coupling this description breaks down.

Abstract. Full text.induce random transitions between stable states. The rate of transition can be

characterised in terms of the noise-free system's dynamics and the added noise:

for potential systems in the presence of asymptotically low noise the

well-known Kramers' escape time gives an expression for the mean escape time.

This paper examines some general properties and examples of transitions between

local steady and oscillatory attractors within networks: the transition rates

at each node may be affected by the dynamics at other nodes. We use first

passage time theory to explain some properties of scalings noted in the

literature for an idealised model of initiation of epileptic seizures in small

systems of coupled bistable systems with both steady and oscillatory

attractors. We focus on the case of sequential escapes where a steady attractor

is only marginally stable but all nodes start in this state. As the nodes

escape to the oscillatory regime, we assume that the transitions back are very

infrequent in comparison. We quantify and characterise the resulting sequences

of noise-induced escapes. For weak enough coupling we show that a master

equation approach gives a good quantitative understanding of sequential

escapes, but for strong coupling this description breaks down.

**Ashwin P, Creaser J, Tsaneva-Atanasova K**(2017). Fast and slow domino regimes in transient network dynamics.

*Physical Review E*,

*96*(5).

## Abstract:

Fast and slow domino regimes in transient network dynamics

© 2017 American Physical Society. It is well known that the addition of noise to a multistable dynamical system can induce random transitions from one stable state to another. For low noise, the times between transitions have an exponential tail and Kramers' formula gives an expression for the mean escape time in the asymptotic limit. If a number of multistable systems are coupled into a network structure, a transition at one site may change the transition properties at other sites. We study the case of escape from a "quiescent" attractor to an "active" attractor in which transitions back can be ignored. There are qualitatively different regimes of transition, depending on coupling strength. For small coupling strengths, the transition rates are simply modified but the transitions remain stochastic. For large coupling strengths, transitions happen approximately in synchrony - we call this a "fast domino" regime. There is also an intermediate coupling regime where some transitions happen inexorably but with a delay that may be arbitrarily long - we call this a "slow domino" regime. We characterize these regimes in the low noise limit in terms of bifurcations of the potential landscape of a coupled system. We demonstrate the effect of the coupling on the distribution of timings and (in general) the sequences of escapes of the system.

Abstract. Full text.**Creaser JL, Krauskopf B, Osinga HM**(2017). Finding first foliation tangencies in the Lorenz system.

*SIAM Journal on Applied Dynamical Systems*,

*16*(4), 2127-2164.

## Abstract:

Finding first foliation tangencies in the Lorenz system

Classical studies of chaos in the well-known Lorenz system are based on reduction to the one-dimensional Lorenz map, which captures the full behavior of the dynamics of the chaotic Lorenz attractor. This reduction requires that the stable and unstable foliations in a particular Poincaré section are transverse locally near the chaotic Lorenz attractor. We study when this so-called foliation condition fails for the first time and the classic Lorenz attractor becomes a quasi-attractor. This transition is characterized by the creation of tangencies between the stable and unstable foliations and the appearance of hooked horseshoes in the Poincaré return map. We consider how the three-dimensional phase space is organized by the global invariant manifolds of saddle equilibria and saddle periodic orbits—before and after the loss of the foliation condition. We compute these global objects as families of orbit segments, which are found by setting up a suitable two-point boundary value problem (BVP). We then formulate a multi-segment BVP to find the first tangency between the stable foliation and the intersection curves in the Poincaré section of the two-dimensional unstable manifold of a periodic orbit. It is a distinct advantage of our BVP setup that we are able to detect and readily continue the locus of first foliation tangency in any plane of two parameters as part of the overall bifurcation diagram. Our computations show that the region of existence of the classic Lorenz attractor is bounded in each parameter plane. It forms a slanted (unbounded) cone in the three-parameter space with a curve of terminal-point, or T-point, bifurcations on the locus of first foliation tangency; we identify the tip of this cone as a codimension-three T-point-Hopf bifurcation point, where the curve of T-point bifurcations meets a surface of Hopf bifurcation. Moreover, we are able to find other first foliation tangencies for larger values of the parameters that are associated with additional T-point bifurcations: each tangency adds an extra twist to the central region of the quasi-attractor.

Abstract. Full text.**Creaser JL, Krauskopf B, Osinga HM**(2015). α-flips and T-points in the Lorenz system.

*Nonlinearity*,

*28*(3), R39-R65.

## Abstract:

α-flips and T-points in the Lorenz system

© 2015 IOP Publishing Ltd. &. London Mathematical Society. We consider the Lorenz system near the classic parameter regime and study the phenomenon we call an α-flip. An α-flip is a transition where the one-dimensional stable manifolds Ws(p±) of two secondary equilibria p± undergo a sudden transition in terms of the direction from which they approach p±. This is a bifurcation at infinity and does not involve an invariant object in phase space. This fact was discovered by Sparrow in the 1980s but the stages of the transition could not be calculated and the phenomenon was not well understood (Sparrow 1982 the Lorenz equations (New York: Springer)). Here we employ a boundary value problem set-up and use pseudo-arclength continuation in AUTO to follow this sudden transition of Ws(p±) as a continuous family of orbit segments. In this way, we geometrically characterize and determine the moment of the actual α-flip. We also investigate how the α-flip takes place relative to the two-dimensional stable manifold of the origin, which shows no apparent topological change before or after the α-flip. Our approach allows for easy detection and subsequent two-parameter continuation of the first and further α-flips. We illustrate this for the first 25 α-flips and find that they end at terminal points, or T-points, where there is a heteroclinic connection from the secondary equilibria to the origin. It turns out that α-flips must occur naturally near T-points. We find scaling relations for the α-flips and T-points that allow us to predict further such bifurcations and to improve the efficiency of our computations.

Abstract. ### Publications by year

### In Press

**Creaser J, Ashwin P, Postlethwaite C, Britz J**(In Press). Noisy network attractor models for transitions between EEG microstates.

## Abstract:

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.

Abstract. Author URL. 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.

**Ashwin P, Creaser J, Tsaneva-Atanasova K**(In Press). Sequential escapes: onset of slow domino regime via a saddle connection.

## Abstract:

Sequential escapes: onset of slow domino regime via a saddle connection

We explore sequential escape behaviour of coupled bistable systems under the

influence of stochastic perturbations. We consider transient escapes from a

marginally stable "quiescent" equilibrium to a more stable "active"

equilibrium. The presence of coupling introduces dependence between the escape

processes: for diffusive coupling there is a strongly coupled limit (fast

domino regime) where the escapes are strongly synchronised while for

intermediate coupling (slow domino regime) without partially escaped stable

states, there is still a delayed effect. These regimes can be associated with

bifurcations of equilibria in the low-noise limit. In this paper we consider a

localized form of non-diffusive (i.e pulse-like) coupling and find similar

changes in the distribution of escape times with coupling strength. However we

find transition to a slow domino regime that is not associated with any

bifurcations of equilibria. We show that this transition can be understood as a

codimension-one saddle connection bifurcation for the low-noise limit. At

transition, the most likely escape path from one attractor hits the escape

saddle from the basin of another partially escaped attractor. After this

bifurcation we find increasing coefficient of variation of the subsequent

escape times.

Abstract. Author URL. Full text.influence of stochastic perturbations. We consider transient escapes from a

marginally stable "quiescent" equilibrium to a more stable "active"

equilibrium. The presence of coupling introduces dependence between the escape

processes: for diffusive coupling there is a strongly coupled limit (fast

domino regime) where the escapes are strongly synchronised while for

intermediate coupling (slow domino regime) without partially escaped stable

states, there is still a delayed effect. These regimes can be associated with

bifurcations of equilibria in the low-noise limit. In this paper we consider a

localized form of non-diffusive (i.e pulse-like) coupling and find similar

changes in the distribution of escape times with coupling strength. However we

find transition to a slow domino regime that is not associated with any

bifurcations of equilibria. We show that this transition can be understood as a

codimension-one saddle connection bifurcation for the low-noise limit. At

transition, the most likely escape path from one attractor hits the escape

saddle from the basin of another partially escaped attractor. After this

bifurcation we find increasing coefficient of variation of the subsequent

escape times.

### 2018

**Creaser J, Tsaneva-Atanasova, Ashwin P**(2018). Sequential noise-induced escapes for oscillatory network dynamics.

*SIAM Journal on Applied Dynamical Systems*,

*17(1)*, 500-525.

## Abstract:

Sequential noise-induced escapes for oscillatory network dynamics

It is well known that the addition of noise in a multistable system can

induce random transitions between stable states. The rate of transition can be

characterised in terms of the noise-free system's dynamics and the added noise:

for potential systems in the presence of asymptotically low noise the

well-known Kramers' escape time gives an expression for the mean escape time.

This paper examines some general properties and examples of transitions between

local steady and oscillatory attractors within networks: the transition rates

at each node may be affected by the dynamics at other nodes. We use first

passage time theory to explain some properties of scalings noted in the

literature for an idealised model of initiation of epileptic seizures in small

systems of coupled bistable systems with both steady and oscillatory

attractors. We focus on the case of sequential escapes where a steady attractor

is only marginally stable but all nodes start in this state. As the nodes

escape to the oscillatory regime, we assume that the transitions back are very

infrequent in comparison. We quantify and characterise the resulting sequences

of noise-induced escapes. For weak enough coupling we show that a master

equation approach gives a good quantitative understanding of sequential

escapes, but for strong coupling this description breaks down.

Abstract. Full text.induce random transitions between stable states. The rate of transition can be

characterised in terms of the noise-free system's dynamics and the added noise:

for potential systems in the presence of asymptotically low noise the

well-known Kramers' escape time gives an expression for the mean escape time.

This paper examines some general properties and examples of transitions between

local steady and oscillatory attractors within networks: the transition rates

at each node may be affected by the dynamics at other nodes. We use first

passage time theory to explain some properties of scalings noted in the

literature for an idealised model of initiation of epileptic seizures in small

systems of coupled bistable systems with both steady and oscillatory

attractors. We focus on the case of sequential escapes where a steady attractor

is only marginally stable but all nodes start in this state. As the nodes

escape to the oscillatory regime, we assume that the transitions back are very

infrequent in comparison. We quantify and characterise the resulting sequences

of noise-induced escapes. For weak enough coupling we show that a master

equation approach gives a good quantitative understanding of sequential

escapes, but for strong coupling this description breaks down.

### 2017

**Ashwin P, Creaser J, Tsaneva-Atanasova K**(2017). Fast and slow domino regimes in transient network dynamics.

*Physical Review E*,

*96*(5).

## Abstract:

Fast and slow domino regimes in transient network dynamics

© 2017 American Physical Society. It is well known that the addition of noise to a multistable dynamical system can induce random transitions from one stable state to another. For low noise, the times between transitions have an exponential tail and Kramers' formula gives an expression for the mean escape time in the asymptotic limit. If a number of multistable systems are coupled into a network structure, a transition at one site may change the transition properties at other sites. We study the case of escape from a "quiescent" attractor to an "active" attractor in which transitions back can be ignored. There are qualitatively different regimes of transition, depending on coupling strength. For small coupling strengths, the transition rates are simply modified but the transitions remain stochastic. For large coupling strengths, transitions happen approximately in synchrony - we call this a "fast domino" regime. There is also an intermediate coupling regime where some transitions happen inexorably but with a delay that may be arbitrarily long - we call this a "slow domino" regime. We characterize these regimes in the low noise limit in terms of bifurcations of the potential landscape of a coupled system. We demonstrate the effect of the coupling on the distribution of timings and (in general) the sequences of escapes of the system.

Abstract. Full text.**Creaser JL, Krauskopf B, Osinga HM**(2017). Finding first foliation tangencies in the Lorenz system.

*SIAM Journal on Applied Dynamical Systems*,

*16*(4), 2127-2164.

## Abstract:

Finding first foliation tangencies in the Lorenz system

Classical studies of chaos in the well-known Lorenz system are based on reduction to the one-dimensional Lorenz map, which captures the full behavior of the dynamics of the chaotic Lorenz attractor. This reduction requires that the stable and unstable foliations in a particular Poincaré section are transverse locally near the chaotic Lorenz attractor. We study when this so-called foliation condition fails for the first time and the classic Lorenz attractor becomes a quasi-attractor. This transition is characterized by the creation of tangencies between the stable and unstable foliations and the appearance of hooked horseshoes in the Poincaré return map. We consider how the three-dimensional phase space is organized by the global invariant manifolds of saddle equilibria and saddle periodic orbits—before and after the loss of the foliation condition. We compute these global objects as families of orbit segments, which are found by setting up a suitable two-point boundary value problem (BVP). We then formulate a multi-segment BVP to find the first tangency between the stable foliation and the intersection curves in the Poincaré section of the two-dimensional unstable manifold of a periodic orbit. It is a distinct advantage of our BVP setup that we are able to detect and readily continue the locus of first foliation tangency in any plane of two parameters as part of the overall bifurcation diagram. Our computations show that the region of existence of the classic Lorenz attractor is bounded in each parameter plane. It forms a slanted (unbounded) cone in the three-parameter space with a curve of terminal-point, or T-point, bifurcations on the locus of first foliation tangency; we identify the tip of this cone as a codimension-three T-point-Hopf bifurcation point, where the curve of T-point bifurcations meets a surface of Hopf bifurcation. Moreover, we are able to find other first foliation tangencies for larger values of the parameters that are associated with additional T-point bifurcations: each tangency adds an extra twist to the central region of the quasi-attractor.

Abstract. Full text.### 2015

**Creaser JL, Krauskopf B, Osinga HM**(2015). α-flips and T-points in the Lorenz system.

*Nonlinearity*,

*28*(3), R39-R65.

## Abstract:

α-flips and T-points in the Lorenz system

© 2015 IOP Publishing Ltd. &. London Mathematical Society. We consider the Lorenz system near the classic parameter regime and study the phenomenon we call an α-flip. An α-flip is a transition where the one-dimensional stable manifolds Ws(p±) of two secondary equilibria p± undergo a sudden transition in terms of the direction from which they approach p±. This is a bifurcation at infinity and does not involve an invariant object in phase space. This fact was discovered by Sparrow in the 1980s but the stages of the transition could not be calculated and the phenomenon was not well understood (Sparrow 1982 the Lorenz equations (New York: Springer)). Here we employ a boundary value problem set-up and use pseudo-arclength continuation in AUTO to follow this sudden transition of Ws(p±) as a continuous family of orbit segments. In this way, we geometrically characterize and determine the moment of the actual α-flip. We also investigate how the α-flip takes place relative to the two-dimensional stable manifold of the origin, which shows no apparent topological change before or after the α-flip. Our approach allows for easy detection and subsequent two-parameter continuation of the first and further α-flips. We illustrate this for the first 25 α-flips and find that they end at terminal points, or T-points, where there is a heteroclinic connection from the secondary equilibria to the origin. It turns out that α-flips must occur naturally near T-points. We find scaling relations for the α-flips and T-points that allow us to predict further such bifurcations and to improve the efficiency of our computations.

Abstract. Jen_Creaser Details from cache as at 2020-01-20 17:54:48