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: Group model building to understand ‘wicked’ problems in New Zealand freshwater management

authors: Connolly, Justin Dennis

subjects: group model building

abstract: Freshwater resources are coming under increasing pressure across New Zealand. High levels of water use and contaminant loss from both point and non-point sources are contributing to growing concern around both the quality of freshwater and its availability. National government regulation has provided greater guidance on the freshwater standards required in water ways, while increasingly collaborative processes are being applied to develop directions for both national and regional policy. A shift towards the inclusion of stakeholders in policy development follows global trends; however, its application in reality is difficult, due to scarce resources (especially data and time) and the high biophysical and social complexity of the issues being addressed.
This research investigates whether the use of causal-loop diagrams (CLDs), applied in a group model building (GMB) process, increases the shared understanding of a freshwater issue by a group. CLDs are a qualitative system dynamics (SD) modelling tool, which seeks to identify the interdependency between elements of a system and the direction of these relationships. They are straightforward to generate and can provide insight into the underlying causes of suboptimal behaviour in complex, dynamic environments. 
The GMB process applied in this research involves the extension of an existing script-driven process, after extensive piloting. It is applied with only two supporting personnel within a limited timeframe, to explore how useful this process is within an environment involving scarce resources. The developed process is applied to two case studies, both involving freshwater management by New Zealand communities. The effectiveness of the GMB process was explored using a mixed methods approach. A post-workshop survey gathered self-reported quantitative and qualitative findings, while a qualitative analysis of data from semi-structured follow-up interviews was undertaken using thematic analysis. 
The self-reported data indicated not only very high levels of individual satisfaction with the process, but also that the process had helped the group’s shared understanding of the issues and that the output (the CLD) would be useful to the group in the future. The thematic analysis identified four broad themes: that there was a desire for a different approach to decision-making; that the GMB process was a refreshing change; that the GMB process built both a group AND a group model; and that, while it was useful, it also did not provide enough detail for decision-making on its own.
The desire for a different approach was underpinned by the complexity of the decision-making process; as well as frustration from some of those involved that they either did not feel that their views were being (or had been) appreciated or from a direct sense of frustration from dealing with council (currently or in the past). It was felt that the GMB approach was a refreshing change as it was a simple and focused process, involving a small group of relevant stakeholders. More importantly, the fact that the group members were actively involved in the process, it was participant led and independently facilitated were all very positive aspects identified.
However, the process for GMB that was applied was found not to be a panacea for solving problems, with more detailed, numerical modelling required to provide insight into the relative importance of key factors.

text: Group model building to understand ‘wicked’ problems in New Zealand freshwater management Freshwater resources are coming under increasing pressure across New Zealand. High levels of water use and contaminant loss from both point and non-point sources are contributing to growing concern around both the quality of freshwater and its availability. National government regulation has provided greater guidance on the freshwater standards required in water ways, while increasingly collaborative processes are being applied to develop directions for both national and regional policy. A shift towards the inclusion of stakeholders in policy development follows global trends; however, its application in reality is difficult, due to scarce resources (especially data and time) and the high biophysical and social complexity of the issues being addressed.
This research investigates whether the use of causal-loop diagrams (CLDs), applied in a group model building (GMB) process, increases the shared understanding of a freshwater issue by a group. CLDs are a qualitative system dynamics (SD) modelling tool, which seeks to identify the interdependency between elements of a system and the direction of these relationships. They are straightforward to generate and can provide insight into the underlying causes of suboptimal behaviour in complex, dynamic environments. 
The GMB process applied in this research involves the extension of an existing script-driven process, after extensive piloting. It is applied with only two supporting personnel within a limited timeframe, to explore how useful this process is within an environment involving scarce resources. The developed process is applied to two case studies, both involving freshwater management by New Zealand communities. The effectiveness of the GMB process was explored using a mixed methods approach. A post-workshop survey gathered self-reported quantitative and qualitative findings, while a qualitative analysis of data from semi-structured follow-up interviews was undertaken using thematic analysis. 
The self-reported data indicated not only very high levels of individual satisfaction with the process, but also that the process had helped the group’s shared understanding of the issues and that the output (the CLD) would be useful to the group in the future. The thematic analysis identified four broad themes: that there was a desire for a different approach to decision-making; that the GMB process was a refreshing change; that the GMB process built both a group AND a group model; and that, while it was useful, it also did not provide enough detail for decision-making on its own.
The desire for a different approach was underpinned by the complexity of the decision-making process; as well as frustration from some of those involved that they either did not feel that their views were being (or had been) appreciated or from a direct sense of frustration from dealing with council (currently or in the past). It was felt that the GMB approach was a refreshing change as it was a simple and focused process, involving a small group of relevant stakeholders. More importantly, the fact that the group members were actively involved in the process, it was participant led and independently facilitated were all very positive aspects identified.
However, the process for GMB that was applied was found not to be a panacea for solving problems, with more detailed, numerical modelling required to provide insight into the relative importance of key factors.

year: 2017

title: Whakarongo mai!: Transformative teaching to support Māori identity and success in a mainstream school

authors: Gilgen, Renée

subjects: Māori and mainstream education

abstract: This thesis topic emerged following a year’s teaching experience in a mainstream classroom during 2009. My cultural assumptions of being a Māori teacher in a mainstream class were challenged predominantly by a group of Māori students in the classroom. The level of despondency I felt as a result of disruptive student behaviours forced me to reassess my own socio-cultural constructs which had influenced the formation of my identity as Māori and as a mainstream teacher. The purpose of this study was to understand and explore a transformational behavioural shift which occurred within a group of Māori students during 2009 and the level of support they felt they received as Māori within a mainstream classroom context. National and international literature demonstrates that transformative praxis is both reflective and active. Successful inclusive teaching requires a personal and professional commitment to firstly understand individual cultural constructs to then better understand how learners are perceived and positioned. Culturally responsive pedagogy ensures that all students’ cultural practices and values are acknowledged and included into classroom teaching and learning programmes. In a New Zealand mainstream context, the Treaty of Waitangi principles of ‘partnership’, ‘participation’ and ‘protection’ serve as a bi-cultural ‘power-sharing’ metaphor from which all mainstream teachers of Māori students can successfully encourage student voice and foster reciprocal teaching and learning relationships. My use of qualitative research methods and kaupapa Māori principles to guide the research process ensured that Māori cultural protocols and practices were honoured. My role as a Māori insider researcher meant that my relationship with the participants was already established. Our reflections as Māori students and a Māori teacher in a mainstream classroom context were based from shared teaching and learning experiences during 2009. Researcher reflexivity has ensured that the research integrity, validity and reliability have remained high. The research findings present the reflections of both the participants and researcher as Māori students and as a Māori teacher during 2009. Participants believed that teaching and learning programmes which incorporated te reo Māori served to honour their identity as Māori. However, their awareness of participating with tikanga Māori principles of whānaungatanga (relationships), manaakitanga (caring), ako (reciprocity) and aroha (respect) embedded within classroom activities that were exclusively in English, were not identified as being relevant to them as Māori. Similarly, the Treaty of Waitangi principles lacked transparency within my teaching pedagogy. Two key implications for teaching and learning in mainstream schools and classrooms were identified. Firstly, there is an urgency for mainstream school managers, leaders and teachers to develop an understanding of the Treaty of Waitangi and its significance for the 21st Century mainstream teacher and learner. A bi-cultural understanding of ourselves as treaty partners support Māori cultural values and practices within increasingly diverse mainstream school and classroom contexts. Māori students‟ success becomes the responsibility of all mainstream managers, leaders and teachers as treaty partners. Secondly, inclusive and culturally responsive pedagogy demonstrates that Māori and non-Māori mainstream teachers are capable of effectively engaging Māori students with learning while acknowledging and reaffirming their cultural identity and language as tangata whenua. Overall, my commitment for Māori students to be supported as Māori within mainstream school and classroom context has been enriched and strengthened as a result of this research study. The cultural and professional tensions I have experienced in mainstream contexts have challenged the socio-cultural constructs that I had accepted as being my cultural ‘norm.’

text: Whakarongo mai!: Transformative teaching to support Māori identity and success in a mainstream school This thesis topic emerged following a year’s teaching experience in a mainstream classroom during 2009. My cultural assumptions of being a Māori teacher in a mainstream class were challenged predominantly by a group of Māori students in the classroom. The level of despondency I felt as a result of disruptive student behaviours forced me to reassess my own socio-cultural constructs which had influenced the formation of my identity as Māori and as a mainstream teacher. The purpose of this study was to understand and explore a transformational behavioural shift which occurred within a group of Māori students during 2009 and the level of support they felt they received as Māori within a mainstream classroom context. National and international literature demonstrates that transformative praxis is both reflective and active. Successful inclusive teaching requires a personal and professional commitment to firstly understand individual cultural constructs to then better understand how learners are perceived and positioned. Culturally responsive pedagogy ensures that all students’ cultural practices and values are acknowledged and included into classroom teaching and learning programmes. In a New Zealand mainstream context, the Treaty of Waitangi principles of ‘partnership’, ‘participation’ and ‘protection’ serve as a bi-cultural ‘power-sharing’ metaphor from which all mainstream teachers of Māori students can successfully encourage student voice and foster reciprocal teaching and learning relationships. My use of qualitative research methods and kaupapa Māori principles to guide the research process ensured that Māori cultural protocols and practices were honoured. My role as a Māori insider researcher meant that my relationship with the participants was already established. Our reflections as Māori students and a Māori teacher in a mainstream classroom context were based from shared teaching and learning experiences during 2009. Researcher reflexivity has ensured that the research integrity, validity and reliability have remained high. The research findings present the reflections of both the participants and researcher as Māori students and as a Māori teacher during 2009. Participants believed that teaching and learning programmes which incorporated te reo Māori served to honour their identity as Māori. However, their awareness of participating with tikanga Māori principles of whānaungatanga (relationships), manaakitanga (caring), ako (reciprocity) and aroha (respect) embedded within classroom activities that were exclusively in English, were not identified as being relevant to them as Māori. Similarly, the Treaty of Waitangi principles lacked transparency within my teaching pedagogy. Two key implications for teaching and learning in mainstream schools and classrooms were identified. Firstly, there is an urgency for mainstream school managers, leaders and teachers to develop an understanding of the Treaty of Waitangi and its significance for the 21st Century mainstream teacher and learner. A bi-cultural understanding of ourselves as treaty partners support Māori cultural values and practices within increasingly diverse mainstream school and classroom contexts. Māori students‟ success becomes the responsibility of all mainstream managers, leaders and teachers as treaty partners. Secondly, inclusive and culturally responsive pedagogy demonstrates that Māori and non-Māori mainstream teachers are capable of effectively engaging Māori students with learning while acknowledging and reaffirming their cultural identity and language as tangata whenua. Overall, my commitment for Māori students to be supported as Māori within mainstream school and classroom context has been enriched and strengthened as a result of this research study. The cultural and professional tensions I have experienced in mainstream contexts have challenged the socio-cultural constructs that I had accepted as being my cultural ‘norm.’

year: 2010