title: Modelling the anaestheto-dynamic phase transition of the cerebral cortex

authors: Steyn-Ross, D. Alistair

abstract: This thesis examines a stochastic model for the electrical behaviour of the cerebral cortex under the influence of a general anaesthetic agent. The modelling element is the macrocolumn, an organized assembly of ∼10⁵ cooperating neurons (85% excitatory, 15% inhibitory) within a small cylindrical volume (∼1 mm³) of the cortex. The state variables are hₑ and hᵢ, the mean-field average soma voltages for the populations of excitatory (e) and inhibitory (i) neurons comprising the macrocolumn. The random fluctuations of hₑ about its steady-state value are taken as the source of the scalp-measured EEG signal. The randomness enters by way of four independent white-noise inputs representing fluctuations in the four types (e-e, i-e, e-i, i-i) of subcortic alactivity.

Our model is a spatial and temporal simplification of the original set of eight coupled partial differential equations (PDEs) due to Liley et al. [Neurocomputing 26-27, 795 (1999)] describing the electrical rhythms of the cortex. We assume (i) spatial homogeneity (i.e., the entire cortex can be represented by a single macrocolumn), and (ii) a separation of temporal scales in which all inputs to the soma “capacitor” are treated as fast variables that settle to steady state very much more rapidly than do the soma voltages themselves: this is the “adiabatic approximation.” These simplifications permit the eight-equation Liley set to be collapsed to a single pair of first-order PDEs in hₑ and hᵢ. We incorporate the effect of general anaesthetic as a lengthening of the duration of the inhibitory post-synaptic potential (PSP) (i.e., we are modelling the GABAergic class of anaesthetics), thus the effectiveness of the inhibitory firings increases monotonically with anaesthetic concentration.

These simplified equations of motion for hₑ,ᵢ are transformed into Langevin (stochastic) equations by adding small white-noise fluctuations to each of the four subcortical spike-rate averages. In order to anchor the analysis, I first identify the t → ∞ steady-state values for the soma voltages. This is done by turning off all noise sources and setting the dhₑ/dt and dhᵢ/dt time derivatives to zero, then numerically locating the steady-state coordinates as a function of anaesthetic effect λ, the scale-factor for the lengthening of the inhibitory PSP. We find that, when plotted as a function of λ, the steady-state soma voltages map out a reverse-S trajectory consisting of a pair of stable branches—the upper (active, high-firing) branch, and the lower (quiescent, low-firing) branch—joined by an unstable mid-branch. Because the two stable phases are not contiguous, the model predicts that a transit from one phase to the other must be first-order discontinuous in soma voltage, and that the downward (induction) jump from active-awareness to unconscious-quiescence will be hysteretically separated from (i.e., will occur at a larger concentration of anaesthetic than) the upward (emergence) jump for the return of consciousness.

By reenabling the noise terms, then linearizing the Langevin equations about one of the stable steady states, we obtain a two-dimensional Ornstein-Uhlenbeck (Brownian motion) system which can be analyzed using standard results from stochastic calculus. Accordingly, we calculate the covariance, time-correlation, and spectral matrices, and find the interesting predictions of vastly increased EEG fluctuation power, attended by simultaneous redistribution of spectral energy towards low frequencies with divergent increases in fluctuation correlation times (i.e., critical slowing down), as the macrocolumn transition points are approached. These predictions are qualitatively confirmed by clinical measurements reported by Kuizenga et al. [British Journal of Anaesthesia 80, 725 (1998)] of the so-called EEG biphasic effect. He used a slew-rate technique known as aperiodic analysis, and I demonstrate that this is approximately equivalent to a frequency-scaling of the power spectral density.

Changes in the frequency distribution of spectral energy can be quantified using the notion of spectral entropy, a modern measure of spectral “whiteness.” We compare the spectral entropy predicted by the model against the clinical values reported recently by Viertiӧ-Oja et al. [Journal of Clinical Monitoring 16, 60 (2000)], and find excellent qualitative agreement for the induction of anaesthesia.

To the best of my knowledge, the link between spectral entropy and correlation time has not previously been reported. For the special case of Lorentzian spectrum (arising from a 1-D OU process), I prove that spectral entropy is proportional to the negative logarithm of the correlation time, and uncover the formula which relates the discrete H₁ Shannon information to the continuous H₂ “histogram entropy,” giving an unbiased estimate of the underlying continuous spectral entropy Hω. The inverse entropy-correlation relationship suggests that, to the extent that anaesthetic induction can be modelled as a 1-D OU process, cortical state can be assessed either in the time domain via correlation time or, equivalently, in the frequency domain via spectral entropy.

In order to investigate a thermodynamic analogy for the anaesthetic-driven (“anaestheto-dynamic”) phase transition of the cortex, we use the steady-state trajectories as an effective equation of state to uncouple the macrocolumn into a pair of (apparently) independent “pseudocolumns.” The stable steady states may now be pictured as local minima in a landscape of potential hills and valleys. After identifying a plausible temperature analogy, we compute the analogous entropy and predict discontinous entropy change—with attendant “heat capacity” anomalies—at transition. The Stullken dog experiments [Stullken et al., Anesthesiology 46, 28(1977)], measuring cerebral metabolic rate changes, seem to confirm these model predictions.

The penultimate chapter examines the impact of incorporating NMDA, an important excitatory neurotransmitter, in the adiabatic model. This work predicts the existence of a new stable state for the cortex, midway between normal activity and quiescence. An induction attempt using a pure anti-NMDA anaesthetic agent (e.g., xenon or nitrous oxide) will take the patient to this mid-state, but no further. I find that for an NMDA-enabled macrocolumn, a GABA induction can produce a second biphasic power event, depending on the brain state at commencement. The latest clinical report from Kuizenga et al. [British Journal of Anaesthesia 86, 354 (2001)] provides apparent confirmation.

text: Modelling the anaestheto-dynamic phase transition of the cerebral cortex This thesis examines a stochastic model for the electrical behaviour of the cerebral cortex under the influence of a general anaesthetic agent. The modelling element is the macrocolumn, an organized assembly of ∼10⁵ cooperating neurons (85% excitatory, 15% inhibitory) within a small cylindrical volume (∼1 mm³) of the cortex. The state variables are hₑ and hᵢ, the mean-field average soma voltages for the populations of excitatory (e) and inhibitory (i) neurons comprising the macrocolumn. The random fluctuations of hₑ about its steady-state value are taken as the source of the scalp-measured EEG signal. The randomness enters by way of four independent white-noise inputs representing fluctuations in the four types (e-e, i-e, e-i, i-i) of subcortic alactivity.

Our model is a spatial and temporal simplification of the original set of eight coupled partial differential equations (PDEs) due to Liley et al. [Neurocomputing 26-27, 795 (1999)] describing the electrical rhythms of the cortex. We assume (i) spatial homogeneity (i.e., the entire cortex can be represented by a single macrocolumn), and (ii) a separation of temporal scales in which all inputs to the soma “capacitor” are treated as fast variables that settle to steady state very much more rapidly than do the soma voltages themselves: this is the “adiabatic approximation.” These simplifications permit the eight-equation Liley set to be collapsed to a single pair of first-order PDEs in hₑ and hᵢ. We incorporate the effect of general anaesthetic as a lengthening of the duration of the inhibitory post-synaptic potential (PSP) (i.e., we are modelling the GABAergic class of anaesthetics), thus the effectiveness of the inhibitory firings increases monotonically with anaesthetic concentration.

These simplified equations of motion for hₑ,ᵢ are transformed into Langevin (stochastic) equations by adding small white-noise fluctuations to each of the four subcortical spike-rate averages. In order to anchor the analysis, I first identify the t → ∞ steady-state values for the soma voltages. This is done by turning off all noise sources and setting the dhₑ/dt and dhᵢ/dt time derivatives to zero, then numerically locating the steady-state coordinates as a function of anaesthetic effect λ, the scale-factor for the lengthening of the inhibitory PSP. We find that, when plotted as a function of λ, the steady-state soma voltages map out a reverse-S trajectory consisting of a pair of stable branches—the upper (active, high-firing) branch, and the lower (quiescent, low-firing) branch—joined by an unstable mid-branch. Because the two stable phases are not contiguous, the model predicts that a transit from one phase to the other must be first-order discontinuous in soma voltage, and that the downward (induction) jump from active-awareness to unconscious-quiescence will be hysteretically separated from (i.e., will occur at a larger concentration of anaesthetic than) the upward (emergence) jump for the return of consciousness.

By reenabling the noise terms, then linearizing the Langevin equations about one of the stable steady states, we obtain a two-dimensional Ornstein-Uhlenbeck (Brownian motion) system which can be analyzed using standard results from stochastic calculus. Accordingly, we calculate the covariance, time-correlation, and spectral matrices, and find the interesting predictions of vastly increased EEG fluctuation power, attended by simultaneous redistribution of spectral energy towards low frequencies with divergent increases in fluctuation correlation times (i.e., critical slowing down), as the macrocolumn transition points are approached. These predictions are qualitatively confirmed by clinical measurements reported by Kuizenga et al. [British Journal of Anaesthesia 80, 725 (1998)] of the so-called EEG biphasic effect. He used a slew-rate technique known as aperiodic analysis, and I demonstrate that this is approximately equivalent to a frequency-scaling of the power spectral density.

Changes in the frequency distribution of spectral energy can be quantified using the notion of spectral entropy, a modern measure of spectral “whiteness.” We compare the spectral entropy predicted by the model against the clinical values reported recently by Viertiӧ-Oja et al. [Journal of Clinical Monitoring 16, 60 (2000)], and find excellent qualitative agreement for the induction of anaesthesia.

To the best of my knowledge, the link between spectral entropy and correlation time has not previously been reported. For the special case of Lorentzian spectrum (arising from a 1-D OU process), I prove that spectral entropy is proportional to the negative logarithm of the correlation time, and uncover the formula which relates the discrete H₁ Shannon information to the continuous H₂ “histogram entropy,” giving an unbiased estimate of the underlying continuous spectral entropy Hω. The inverse entropy-correlation relationship suggests that, to the extent that anaesthetic induction can be modelled as a 1-D OU process, cortical state can be assessed either in the time domain via correlation time or, equivalently, in the frequency domain via spectral entropy.

In order to investigate a thermodynamic analogy for the anaesthetic-driven (“anaestheto-dynamic”) phase transition of the cortex, we use the steady-state trajectories as an effective equation of state to uncouple the macrocolumn into a pair of (apparently) independent “pseudocolumns.” The stable steady states may now be pictured as local minima in a landscape of potential hills and valleys. After identifying a plausible temperature analogy, we compute the analogous entropy and predict discontinous entropy change—with attendant “heat capacity” anomalies—at transition. The Stullken dog experiments [Stullken et al., Anesthesiology 46, 28(1977)], measuring cerebral metabolic rate changes, seem to confirm these model predictions.

The penultimate chapter examines the impact of incorporating NMDA, an important excitatory neurotransmitter, in the adiabatic model. This work predicts the existence of a new stable state for the cortex, midway between normal activity and quiescence. An induction attempt using a pure anti-NMDA anaesthetic agent (e.g., xenon or nitrous oxide) will take the patient to this mid-state, but no further. I find that for an NMDA-enabled macrocolumn, a GABA induction can produce a second biphasic power event, depending on the brain state at commencement. The latest clinical report from Kuizenga et al. [British Journal of Anaesthesia 86, 354 (2001)] provides apparent confirmation.

year: 2002

title: Beyond altruistic and commercial contract motherhood: The professional model

authors: van Zyl, Liezl

subjects: contract motherhood

abstract: It has become common to distinguish between altruistic and commercial contract motherhood (or 'surrogacy'). Altruistic arrangements are based on the 'gift relationship': a woman is motivated by altruism to have a baby for an infertile couple, who are free to reciprocate as they see fit. By contrast, in commercial arrangements both parties are motivated by personal gain to enter a legally enforceable agreement, which stipulates that the contract mother or 'surrogate' is to bear a child for the intending parents in exchange for a fee. She is required to undergo medical examinations and to refrain from behaviour that could harm the foetus. The intending parents are the child's legal parents from the outset. The parties to the contract can, but are not expected to, maintain contact after the transaction is completed. We argue that contract motherhood should not be organized according to the norms of the gift relationship, and that contract mothers should be compensated for their labour. However, we accept that there are good reasons for rejecting the commercial model as a suitable framework for contract pregnancy, and argue, instead, in favour of viewing it as a profession.

text: Beyond altruistic and commercial contract motherhood: The professional model It has become common to distinguish between altruistic and commercial contract motherhood (or 'surrogacy'). Altruistic arrangements are based on the 'gift relationship': a woman is motivated by altruism to have a baby for an infertile couple, who are free to reciprocate as they see fit. By contrast, in commercial arrangements both parties are motivated by personal gain to enter a legally enforceable agreement, which stipulates that the contract mother or 'surrogate' is to bear a child for the intending parents in exchange for a fee. She is required to undergo medical examinations and to refrain from behaviour that could harm the foetus. The intending parents are the child's legal parents from the outset. The parties to the contract can, but are not expected to, maintain contact after the transaction is completed. We argue that contract motherhood should not be organized according to the norms of the gift relationship, and that contract mothers should be compensated for their labour. However, we accept that there are good reasons for rejecting the commercial model as a suitable framework for contract pregnancy, and argue, instead, in favour of viewing it as a profession.

year: 2012

title: Rata, the Effect of Māori Land Law on Ahikāroa

authors: Whaanga, Mere Joslyn

subjects: Ahikāroa

abstract: Rata, the effect of Māori land law on ahikāroa is an examination of the special relationship a whānau of Ngāi Tahu (ki Wairoa) has had since 1868 with their ancestral land Tutuotekaha. The effects of Māori land law on traditional concepts of land tenure are analysed from the advent of the Native Land Court in the 1860s through to a modern shareholding in the Anewa Trust. An analysis of Te Ture Whenua Māori Act 1993 looks at the basis for the current legislation and how the principles are applied in the case study central to this thesis, an application to have Rata farm returned to the Whaanga whānau.

The research and writing is multi-disciplinary, combining history, Māori land law (ture) and Māori land lore (tikanga and traditional ecological knowledge). An auto-ethnographic approach is balanced by the commentary and judgements of the Māori Land Court and the Māori Appellate Court which are documented in the final section of the thesis.

This thesis questions the canons of Māori land ownership and management – consolidation, amalgamation and incorporation – and posits whether these forms of land holding have become yet another type of alienation.

The three main areas are
I) Ahikāroa
Chapters 2 and 3 examine traditional Māori land tenure; definitions of mana whenua, the obligations as well as the authority; and traditional ecological knowledge, agriculture and Māori land lore. The latter illustrates ways of knowing the land, how, why and when its resources were used. Members of the community of Iwitea were interviewed to provide accounts of usage of some of the traditional resources known to remain on the Tutuotekaha blocks.

II) From gardeners and gatherers to agricultural shareholders
Chapters 4, 5 and 6 examine the shift from traditional tenure and land usage to large-scale agricultural shareholdings. The changes as land passed through the Native Land Court from 1868 are contextualised within the upheaval and radical reconfiguring of Māori society in the Mahia and Wairoa areas as Pākehā settlement increased, and the region was engulfed by the Hauhau and Te Kooti conflicts. This section contains an overview of Māori land law from 1862 to 1987 and the ideological conflict between the colonisers and the colonised. As well, it looks at the influence of Sir James Carroll and Sir Āpirana Ngata, the reasons that Māori went into large-scale farming and how Ngata’s 1931 policies set the model for the incorporations and trusts that still hold most of the district’s Māori land today.

III) Te Ture Whenua Māori Act 1993, philosophy versus implementation
Chapters 7 through to 11 centre on Te Ture Whenua Māori Act 1993 and the application to the Māori Land Court to partition Rata farm from the Anewa Trust. The philosophy of the Act is examined, aided by Deputy Chief Judge Fox’s commentary. This section documents the processes of filing the application to partition, the discussions and outcomes of a judicial hearing, two Māori Land Court hearings and subsequent appeals to the Māori Appellate Court.

text: Rata, the Effect of Māori Land Law on Ahikāroa Rata, the effect of Māori land law on ahikāroa is an examination of the special relationship a whānau of Ngāi Tahu (ki Wairoa) has had since 1868 with their ancestral land Tutuotekaha. The effects of Māori land law on traditional concepts of land tenure are analysed from the advent of the Native Land Court in the 1860s through to a modern shareholding in the Anewa Trust. An analysis of Te Ture Whenua Māori Act 1993 looks at the basis for the current legislation and how the principles are applied in the case study central to this thesis, an application to have Rata farm returned to the Whaanga whānau.

The research and writing is multi-disciplinary, combining history, Māori land law (ture) and Māori land lore (tikanga and traditional ecological knowledge). An auto-ethnographic approach is balanced by the commentary and judgements of the Māori Land Court and the Māori Appellate Court which are documented in the final section of the thesis.

This thesis questions the canons of Māori land ownership and management – consolidation, amalgamation and incorporation – and posits whether these forms of land holding have become yet another type of alienation.

The three main areas are
I) Ahikāroa
Chapters 2 and 3 examine traditional Māori land tenure; definitions of mana whenua, the obligations as well as the authority; and traditional ecological knowledge, agriculture and Māori land lore. The latter illustrates ways of knowing the land, how, why and when its resources were used. Members of the community of Iwitea were interviewed to provide accounts of usage of some of the traditional resources known to remain on the Tutuotekaha blocks.

II) From gardeners and gatherers to agricultural shareholders
Chapters 4, 5 and 6 examine the shift from traditional tenure and land usage to large-scale agricultural shareholdings. The changes as land passed through the Native Land Court from 1868 are contextualised within the upheaval and radical reconfiguring of Māori society in the Mahia and Wairoa areas as Pākehā settlement increased, and the region was engulfed by the Hauhau and Te Kooti conflicts. This section contains an overview of Māori land law from 1862 to 1987 and the ideological conflict between the colonisers and the colonised. As well, it looks at the influence of Sir James Carroll and Sir Āpirana Ngata, the reasons that Māori went into large-scale farming and how Ngata’s 1931 policies set the model for the incorporations and trusts that still hold most of the district’s Māori land today.

III) Te Ture Whenua Māori Act 1993, philosophy versus implementation
Chapters 7 through to 11 centre on Te Ture Whenua Māori Act 1993 and the application to the Māori Land Court to partition Rata farm from the Anewa Trust. The philosophy of the Act is examined, aided by Deputy Chief Judge Fox’s commentary. This section documents the processes of filing the application to partition, the discussions and outcomes of a judicial hearing, two Māori Land Court hearings and subsequent appeals to the Māori Appellate Court.

year: 2012