title: Analysis of pattern dynamics for a nonlinear model of the human cortex via bifurcation theories

authors: Wang, Kaier

subjects: bifurcation

abstract: This thesis examines the bifurcations, i.e., the emergent behaviours, for the Waikato cortical model under the influence of the gap-junction inhibitory diffusion D₂ (identified as the Turing bifurcation parameter) and the time-to-peak for hyperpolarising GABA response γi (i.e., inhibitory rate-constant, identified as the Hopf bifurcation parameter). The cortical model simplifies the entire cortex to a cylindrical macrocolumn (∼ 1 mm³) containing ∼ 10⁵ neurons (85% excitatory, 15% inhibitory) communicating via both chemical and electrical (gap-junction) synapses. The linear stability analysis of the model equations predict the emergence of a Turing instability (in which separated areas of the cortex become activated) when gap-junction diffusivity is increased above a critical level. In addition, a Hopf bifurcation (oscillation) occurs when the inhibitory rate-constant is sufficiently small. Nonlinear interaction between these instabilities leads to spontaneous cortical patterns of neuronal activities evolving in space and time. Such model dynamics of delicately balanced interplay between Turing and Hopf instabilities may be of direct relevance to clinically observed brain dynamics such as epileptic seizure EEG spikes, deep-sleep slow-wave oscillations and cognitive gamma-waves.
The relationship between the modelled brain patterns and model equations can normally be inferred from the eigenvalue dispersion curve, i.e., linear stability analysis. Sometimes we experienced mismatches between the linear stability analysis and the formed cortical patterns, which hampers us in identifying the type of instability corresponding to the emergent patterns. In this thesis, I investigate the pattern-forming mechanism of the Waikato cortical model to better understand the model nonlinearities. I first study the pattern dynamics via analysis of a simple pattern-forming system, the Brusselator model, which has a similar model structure and bifurcation phenomena as the cortical model. I apply both linear and nonlinear perturbation methods to analyse the near-bifurcation behaviour of the Brusselator in order to precisely capture the dominant mode that contributes the most to the final formed-patterns. My nonlinear analysis of the Brusselator model yields Ginzburg-Landau type amplitude equations that describe the dynamics of the most unstable mode, i.e., the dominant mode, in the vicinity of a bifurcation point. The amplitude equations at a Turing point unfold three characteristic spatial structures: honeycomb Hπ, stripes, and reentrant honeycomb H₀. A codimension-2 Turing–Hopf point (CTHP) predicts three mixed instabilities: stable Turing–Hopf (TH), chaotic TH, and bistable TH. The amplitude equations precisely determine the bifurcation conditions for these instabilities and explain the pattern-competition mechanism once the bifurcation parameters cross the thresholds, whilst driving the system into a nonlinear region where the linear stability analysis may not be applicable.
Then, I apply the bifurcation theories to the cortical model for its pattern predictions. Analogous to the Brusselator model, I find cortical Turing pattens in Hπ, stripes and H₀ spatial structures. Moreover, I develop the amplitude equations for the cortical model, with which I derive the envelope frequency for the beating-waves of a stable TH mode; and propose ideas regarding emergence of the cortical chaotic mode. Apart from these pattern dynamics that the cortical model shares with the Brusselator system, the cortical model also exhibits “eye-blinking” TH patterns latticed in hexagons with localised oscillations. Although we have not found biological significance of these model pattens, the developed bifurcation theories and investigated pattern-forming mechanism may enrich our modelling strategies and help us to further improve model performance.
In the last chapter of this thesis, I introduce a Turing–Hopf mechanism for the anaesthetic slow-waves, and predict a coherence drop of such slow-waves with the induction of propofol anaesthesia. To test this hypothesis, I developed an EEG coherence analysing algorithm, EEG coherence, to automatically examine the clinical EEG recordings across multiple subjects. The result shows significantly decreased coherence along the fronto-occipital axis, and increased coherence along the left- and right-temporal axis. As the Waikato cortical model is spatially homogenous, i.e., there are no explicit front-to-back or right-to-left directions, it is unable to produce different coherence changes for different regions. It appears that the Waikato cortical model best represents the cortical dynamics in the frontal region. The theory of pattern dynamics suggests that a mode transition from wave–Turing–wave to Turing–wave–Turing introduces pattern coherence changes in both positive and negative directions. Thus, a further modelling improvement may be the introduction of a cortical bistable mode where Turing and wave coexist.

text: Analysis of pattern dynamics for a nonlinear model of the human cortex via bifurcation theories This thesis examines the bifurcations, i.e., the emergent behaviours, for the Waikato cortical model under the influence of the gap-junction inhibitory diffusion D₂ (identified as the Turing bifurcation parameter) and the time-to-peak for hyperpolarising GABA response γi (i.e., inhibitory rate-constant, identified as the Hopf bifurcation parameter). The cortical model simplifies the entire cortex to a cylindrical macrocolumn (∼ 1 mm³) containing ∼ 10⁵ neurons (85% excitatory, 15% inhibitory) communicating via both chemical and electrical (gap-junction) synapses. The linear stability analysis of the model equations predict the emergence of a Turing instability (in which separated areas of the cortex become activated) when gap-junction diffusivity is increased above a critical level. In addition, a Hopf bifurcation (oscillation) occurs when the inhibitory rate-constant is sufficiently small. Nonlinear interaction between these instabilities leads to spontaneous cortical patterns of neuronal activities evolving in space and time. Such model dynamics of delicately balanced interplay between Turing and Hopf instabilities may be of direct relevance to clinically observed brain dynamics such as epileptic seizure EEG spikes, deep-sleep slow-wave oscillations and cognitive gamma-waves.
The relationship between the modelled brain patterns and model equations can normally be inferred from the eigenvalue dispersion curve, i.e., linear stability analysis. Sometimes we experienced mismatches between the linear stability analysis and the formed cortical patterns, which hampers us in identifying the type of instability corresponding to the emergent patterns. In this thesis, I investigate the pattern-forming mechanism of the Waikato cortical model to better understand the model nonlinearities. I first study the pattern dynamics via analysis of a simple pattern-forming system, the Brusselator model, which has a similar model structure and bifurcation phenomena as the cortical model. I apply both linear and nonlinear perturbation methods to analyse the near-bifurcation behaviour of the Brusselator in order to precisely capture the dominant mode that contributes the most to the final formed-patterns. My nonlinear analysis of the Brusselator model yields Ginzburg-Landau type amplitude equations that describe the dynamics of the most unstable mode, i.e., the dominant mode, in the vicinity of a bifurcation point. The amplitude equations at a Turing point unfold three characteristic spatial structures: honeycomb Hπ, stripes, and reentrant honeycomb H₀. A codimension-2 Turing–Hopf point (CTHP) predicts three mixed instabilities: stable Turing–Hopf (TH), chaotic TH, and bistable TH. The amplitude equations precisely determine the bifurcation conditions for these instabilities and explain the pattern-competition mechanism once the bifurcation parameters cross the thresholds, whilst driving the system into a nonlinear region where the linear stability analysis may not be applicable.
Then, I apply the bifurcation theories to the cortical model for its pattern predictions. Analogous to the Brusselator model, I find cortical Turing pattens in Hπ, stripes and H₀ spatial structures. Moreover, I develop the amplitude equations for the cortical model, with which I derive the envelope frequency for the beating-waves of a stable TH mode; and propose ideas regarding emergence of the cortical chaotic mode. Apart from these pattern dynamics that the cortical model shares with the Brusselator system, the cortical model also exhibits “eye-blinking” TH patterns latticed in hexagons with localised oscillations. Although we have not found biological significance of these model pattens, the developed bifurcation theories and investigated pattern-forming mechanism may enrich our modelling strategies and help us to further improve model performance.
In the last chapter of this thesis, I introduce a Turing–Hopf mechanism for the anaesthetic slow-waves, and predict a coherence drop of such slow-waves with the induction of propofol anaesthesia. To test this hypothesis, I developed an EEG coherence analysing algorithm, EEG coherence, to automatically examine the clinical EEG recordings across multiple subjects. The result shows significantly decreased coherence along the fronto-occipital axis, and increased coherence along the left- and right-temporal axis. As the Waikato cortical model is spatially homogenous, i.e., there are no explicit front-to-back or right-to-left directions, it is unable to produce different coherence changes for different regions. It appears that the Waikato cortical model best represents the cortical dynamics in the frontal region. The theory of pattern dynamics suggests that a mode transition from wave–Turing–wave to Turing–wave–Turing introduces pattern coherence changes in both positive and negative directions. Thus, a further modelling improvement may be the introduction of a cortical bistable mode where Turing and wave coexist.

year: 2014

title: Analysing interaction in children's digital books

authors: Vanderschantz, Nicholas

subjects: children's books

abstract: Pedagogical practices in formal educational settings, together with the nature of communication technologies in a variety of digital media, mean that children will encounter screen-based learning opportunities in both formal educational settings and during their daily recreational pursuits. This research looks to address the lack of research informing best practices for the visual and interaction design of this material for children in a screen-based environment. This investigation provides a survey of interaction methods in on-screen reading material available to children in primary-school, middleschool and public libraries in New Zealand. From the data collected, a database chronicling variables relating to interaction in current digital books for children was created. Specifically, this database houses variables including navigation, orientation functions and in-line linking such as hypertext, as well as other factors that can impact a child's digital book use. Through analysis of the database this research provides insight into current trends in interaction techniques found in material designed for children's on-screen reading and provides recommendations for improvements that may be made to assist in the design of children's digital books.

text: Analysing interaction in children's digital books Pedagogical practices in formal educational settings, together with the nature of communication technologies in a variety of digital media, mean that children will encounter screen-based learning opportunities in both formal educational settings and during their daily recreational pursuits. This research looks to address the lack of research informing best practices for the visual and interaction design of this material for children in a screen-based environment. This investigation provides a survey of interaction methods in on-screen reading material available to children in primary-school, middleschool and public libraries in New Zealand. From the data collected, a database chronicling variables relating to interaction in current digital books for children was created. Specifically, this database houses variables including navigation, orientation functions and in-line linking such as hypertext, as well as other factors that can impact a child's digital book use. Through analysis of the database this research provides insight into current trends in interaction techniques found in material designed for children's on-screen reading and provides recommendations for improvements that may be made to assist in the design of children's digital books.

year: 2012

title: Whānau Māori explain how the Harti Hauora Tamariki tool assists better access to health services

authors: Masters-Awatere, Bridgette

abstract: In this paper whānau Māori highlight how a Kaupapa Māori centred intervention (the Harti Hauora Tamariki tool) has improved interactions with health services. The Harti tool is undergoing a Randomised Control Trial at Waikato Hospital in New Zealand. As part of the RCT, the authors engaged in a series of qualitative interviews with whānau members of tamariki Māori (children aged 0-5years) admitted to Waikato Hospital’s paediatric ward. Whānau who met at least one criteria for New Zealand’s domains of deprivation were included. Using a Kaupapa Māori approach to the study, participants shared their views on barriers and facilitators to accessing health resources and primary care services. Our interviews highlight how the Harti tool, when administered in a culturally appropriate and respectful manner that prioritised relationship-building, enabled better connection to healthcare services. Prevalent in our analysis were connections to wider determinants of health and ways to reduce existing health inequities. To conclude the paper, we highlight how the tool has enhanced feelings of being in control of health with the potential to reduce the likelihood of a hospital readmission.

text: Whānau Māori explain how the Harti Hauora Tamariki tool assists better access to health services In this paper whānau Māori highlight how a Kaupapa Māori centred intervention (the Harti Hauora Tamariki tool) has improved interactions with health services. The Harti tool is undergoing a Randomised Control Trial at Waikato Hospital in New Zealand. As part of the RCT, the authors engaged in a series of qualitative interviews with whānau members of tamariki Māori (children aged 0-5years) admitted to Waikato Hospital’s paediatric ward. Whānau who met at least one criteria for New Zealand’s domains of deprivation were included. Using a Kaupapa Māori approach to the study, participants shared their views on barriers and facilitators to accessing health resources and primary care services. Our interviews highlight how the Harti tool, when administered in a culturally appropriate and respectful manner that prioritised relationship-building, enabled better connection to healthcare services. Prevalent in our analysis were connections to wider determinants of health and ways to reduce existing health inequities. To conclude the paper, we highlight how the tool has enhanced feelings of being in control of health with the potential to reduce the likelihood of a hospital readmission.

year: 2019