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: Testing the efficacy of a motor analogy designed to promote safe landing by older adults who fall accidentally: a study protocol for a randomised control study

authors: Oladi, Sana

abstract: Introduction: Falling is associated with adverse effects on the health of older people. The majority of research into falls among older people has focused on prevention, with less attention to ‘how to fall safely’. Previous research suggests that motor analogies can be used to promote safe landing by young adults; however, the efficacy of this technique for older people remains unknown. This study aims to determine whether a motor analogy is useful for promoting safe falling in the older adult population. Methods and analysis: The study adopts a randomised, controlled, single- blinded study design. People 65 years and older will be randomly allocated to a control condition or a motor analogy condition. They will receive a nudge in a forward, backward or sideways direction (randomised order), which will initiate a fall. The nudge will occur at variable (randomised) time points, so participants will not be aware of when they will fall. Participants in the motor analogy condition will be instructed to ‘land like a feather’, whereas participants in the control condition will be instructed to ‘land safely’. The primary outcome parameters are maximum impact force (normalised by mass) applied to different body segments during impact and fracture risk ratio of wrists and hips. A two- way multivariate analysis of variance will be conducted to examine differences between the motor analogy and control conditions as a function of the different variables.

text: Testing the efficacy of a motor analogy designed to promote safe landing by older adults who fall accidentally: a study protocol for a randomised control study Introduction: Falling is associated with adverse effects on the health of older people. The majority of research into falls among older people has focused on prevention, with less attention to ‘how to fall safely’. Previous research suggests that motor analogies can be used to promote safe landing by young adults; however, the efficacy of this technique for older people remains unknown. This study aims to determine whether a motor analogy is useful for promoting safe falling in the older adult population. Methods and analysis: The study adopts a randomised, controlled, single- blinded study design. People 65 years and older will be randomly allocated to a control condition or a motor analogy condition. They will receive a nudge in a forward, backward or sideways direction (randomised order), which will initiate a fall. The nudge will occur at variable (randomised) time points, so participants will not be aware of when they will fall. Participants in the motor analogy condition will be instructed to ‘land like a feather’, whereas participants in the control condition will be instructed to ‘land safely’. The primary outcome parameters are maximum impact force (normalised by mass) applied to different body segments during impact and fracture risk ratio of wrists and hips. A two- way multivariate analysis of variance will be conducted to examine differences between the motor analogy and control conditions as a function of the different variables.

year: 2022

title: Bite

authors: Mitchell, Tori

subjects: poetry

abstract: Bite is a collection of blunt confessional poetry draped in honesty, identity, and 21st century feminism, examined through lived experience with eating disorders, sexuality, trauma, disconnect from family and culture, love and loss, and what it means to be a girl growing up. The collection explores these raw concepts with vulnerability, through the eyes of ‘wolf girl’, in hopes that talking loudly about these topics will force the public to challenge the silence and stigma around them.  

This thesis was inspired by Maori Mythology, and all of the Polynesian poets who showed me how important it is for women of colour to share their stories and experiences. It was also influenced by feminist artists Jenny Holzer, Tracey Emin, and Barbara Kruger, whose art embodies the content of these poems in a literal way: freedom of expression, stripped bare and unflinching. The poetry in this collection is confronting, visceral, and filled with bite.

text: Bite Bite is a collection of blunt confessional poetry draped in honesty, identity, and 21st century feminism, examined through lived experience with eating disorders, sexuality, trauma, disconnect from family and culture, love and loss, and what it means to be a girl growing up. The collection explores these raw concepts with vulnerability, through the eyes of ‘wolf girl’, in hopes that talking loudly about these topics will force the public to challenge the silence and stigma around them.  

This thesis was inspired by Maori Mythology, and all of the Polynesian poets who showed me how important it is for women of colour to share their stories and experiences. It was also influenced by feminist artists Jenny Holzer, Tracey Emin, and Barbara Kruger, whose art embodies the content of these poems in a literal way: freedom of expression, stripped bare and unflinching. The poetry in this collection is confronting, visceral, and filled with bite.

year: 2020