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## Mini.dvi

**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 smallcylindrical volume (

*∼*1 mm3) of the cortex. The state variables are

*he *and

*hi*, the mean-ﬁeldaverage soma voltages for the populations of excitatory (

*e*) and inhibitory (

*i*) neurons comprisingthe macrocolumn. The random ﬂuctuations of

*he *about its steady-state value are taken as thesource 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 subcorticalactivity.

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 theduration 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
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 forthe 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 afunction of anaesthetic eﬀect

*λ*, the scale-factor for the lengthening of the inhibitory PSP. Weﬁnd that, when plotted as a function of

*λ*, the steady-state soma voltages map out a reverse-Strajectory 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

By reenabling the noise terms, then linearizing the Langevin equations about one of the sta-
ble steady states, we obtain a two-dimensional Ornstein–Uhlenbeck (Brownian motion) system
which can be analyzed using standard results from stochastic calculus. Accordingly, we calcu-
late 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 Jour-
nal 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 Vierti¨
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

*H*1 Shannon information tothe continuous

*H*2 “histogram entropy,” giving an unbiased estimate of the underlying continuousspectral entropy

*Hω*. The inverse entropy–correlation relationship suggests that, to the extentthat 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
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 “pseu-
docolumns.” 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

*ex-*
*citatory *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 at-
tempt 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.

Source: http://www.eng.waikato.ac.nz/pdfs/cortical/thesis/ASR_full_abstract.pdf

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