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Simulated Electrocortical Activity at Microscopic, Mesoscopic,
and Global Scales
JJ Wright*
,1,2,3,6
, CJ Rennie
1,4,5
, GJ Lees
2
, PA Robinson
1,4
, PD Bourke
3
, CL Chapman
3
, E Gordon
1,7
and DL
Rowe
1,4
1
Brain Dynamics Centre, Westmead Hospital and University of Sydney, Westmead, Australia;
2
Department of Psychiatry, University of Auckland
School of Medicine, Auckland, New Zealand;
3
Mental Health Research Institute, Parkville, Victoria, Australia;
4
Theoretical Physics Group, School of
Physics, University of Sydney, Australia;
5
Department of Medical Physics, Westmead Hospital, Westmead, Australia;
6
Liggins Institute, University
of Auckland, Auckland, New Zealand;
7
Department of Psychological Medicine, Westmead Hospital and University of Sydney, Westmead,
Australia
Simulation of electrocortical activity requires (a) determination of the most crucial features to be modelled, (b) specification of state
equations with parameters that can be determined against independent measurements, and (c) explanation of electrical events in the
brain at several scales. We report our attempts to address these problems, and show that mutually consistent explanations, and
simulation of experimental data can be achieved for cortical gamma activity, synchronous oscillation, and the main features of the EEG
power spectrum including the cerebral rhythms and evoked potentials. These simulations include consideration of dendritic and synaptic
dynamics, AMPA, NMDA, and GABA receptors, and intracortical and cortical/subcortical interactions. We speculate on the way in which
Hebbian learning and intrinsic reinforcement processes might complement the brain dynamics thus explained, to produce elementary
cognitive operations.
Neuropsychopharmacology (2003) 28, S80–S93. doi:10.1038/sj.npp.1300138
Keywords: gamma rhythm; cortical synchrony; evoked potentials; NMDA; AMPA; GABA
INTRODUCTION
Neuropsychopharmacology is a discipline attempting the
unification of events at different scales in the brain. This is
part of the larger task of relating brain events to conscious
experience and learning processes of normal people and
people suffering from mental disorders. The problems
involved in this unification are legion, as is well known to
readers of this journal. It is commonly found that different
types of events in the brain are difficult to relate to each
other, except as purely empirical findings. For example, it is
seldom clear how findings made by PET and fMRI scanning
are functionally related to most EEG measures. Nor how
EEG measures might reflect specific changes in synaptic
physiology. Nor how neurochemical effects can be func-
tionally related to psychological events. These are all
difficult questions, despite the huge growth in detailed
knowledge about specific processes and components
in the brain. No simple one-to-one ways exist, to relate
these very complex phenomena, one to the others. Yet
unifying principles must be sought, if the discipline is to
advance in a coherent fashion. One way of moving towards
this unification is through the discipline of Brain Dynamics.
The aim of the subject of Brain Dynamics is to give a
simplified, but sufficient, mathematical description of the
operation of the brain, in terms of the brain’s observable
electrical activity (Freeman, 1975; Basar, 1976; Nunez, 1981,
1995).
Obtaining a ‘simplified, but sufficient, mathematical
description’ can be broken down into a number of subtasks.
These are:
(a) Abstraction from physiological data of the most
important properties of the neurones of the brain, to
be used in the mathematical model.
(b) Determination of the form of the state equations, and
the values of parameters.
(c) Comparison of the equations’ properties to observable
aspects of brain function, at as many temporal and
spatial scales as possible.
The observable aspects of brain function to be accounted
for include the global EEG, the electrocorticogram and local
field potentials, and the statistics of action potentials. If a
wide match to experimental data can be achieved, then the
model that has emerged should exhibit dynamic properties
akin to a real brain, and it might then be asked what
consequences these dynamic properties have for adaptive
Received 01 September 2002; revised 03 November 2002; accepted
03 December 2002
*Correspondence: Dr JJ Wright, Brain Dynamics Centre, Westmead
Hospital, University of Sydney, Westmead, NSW 2145, Australia, Fax:
+61 3 9387 5061, E-mail: jjw@mhri.edu.au
Neuropsychopharmacology (2003) 28, S80–S93
&
2003 Nature Publishing Group All rights reserved 0893-133X/03
$
25.00
www.neuropsychopharmacology.org
learning. Does the introduction of plausible learning rules
then lead to an explanation of adaptive behavior?
Attempts to approach a theory of the brain along these
lines stem from the work of McCulloch and Pitts (1943), and
have led to extensive work on the properties of artificial
neural networks (eg Amit, 1989). Lines of theoretical
development more explicitly concerned with physiological
dynamics, and especially the dynamics underlying EEG,
flow from Wilson and Cowan (1973), Freeman (1975),
Nunez (1981), Lopes da Silva (van Rotterdam et al, 1982),
Haken (Haken et al, 1985), and others (eg Arbib et al, 1998).
Work by Freeman (1975 and subsequently), and of Singer
and Gray and others (eg Gray and Singer, 1989; Gray et al,
1989; Eckhorn et al, 1988) has been of particular importance
in revealing the dynamics of the brain from an experimental
perspective.
The work to be described from our own group has
evolved from early attempts to define circumstances in
which linear methods of analysis could be applied (Wright,
1990). Simple numerical simulations followed (Wright and
Liley, 1996; Wright, 1997, 1999), leading to more advanced
methods, including the development of wave equations
(Robinson et al, 1997, 1998a, b, 2001). We have progres-
sively introduced more refined physiological parameters,
and descriptions of anatomical organization (Liley and
Wright, 1994; Rennie et al, 1999, 2000, 2002), with the object
of developing a single model to account for events in the
brain at a number of different scales.
OVERALL BRAIN ORGANIZATION
Figure 1 highlights elementary features of gross brain
organization. The cortical mantle is the terminating area for
major sensory pathways, and the source of much of the
signals organizing motor activity. The cortex never acts
alone, but via continuous interaction with subcortical
systems, notably the thalamus, the limbic system including
hippocampus, the basal ganglia, etc (Alexander et al, 1990;
Posner and Petersen, 1990). Any model complete to first
approximation must consider interactions within the
cortex, and also between cortex and subcortical structures,
including events at microscopic, mesoscopic, and macro-
scopic (global or whole-brain) scales.
CORTICAL DYNAMICS AT MICROSCOPIC SCALE
Figure 2 shows the two elementary components of the
cortexFexcitatory (pyramidal) cells which make up about
90% of cortical cells, and which send axons to remote
cortical locations, as well as interacting with near neighbors
via intracortical axonsFand inhibitory cells, which give
only local intracortical axons. The excitatory cells utilize
glutamate as a neurotransmitter, and act principally on fast-
acting AMPA receptors, and slower-acting NMDA recep-
tors. The NMDA receptors are voltage dependentFthat is,
they influence the postsynaptic dendrites when the receiv-
ing cell is depolarized, and thereby emitting action
potentials. The inhibitory cells use GABA as their neuro-
transmitter. These are the most important fast forms of
neuronal interaction.
A mathematical account of interactions in the cortex of
these restricted types is given in the appendix, and the
values we have obtained for all the parameters required in
this mathematical model are given in Table 1. The values of
the parameters are taken largely from the findings of
Braitenberg and Schuz (1991), Thomson et al (1996), and
Thomson (1997), either directly, or from further
calculations based on these (Liley and Wright, 1994; Rennie
et al, 2000). Recent additions regarding synaptic physiology
Figure 1 Overall brain organization and the EEG. The electroencepha-
logram arises from the cortical mantle. Cortical neurone cells interact locally
with each other, and with many subcortical systems, each with complex
delays and spatial dispersions.
Figure 2 Neurones involved in intracortical interactions. Excitatory
glutaminergic neurones (grey) interact locally with inhibitory gabaminergic
neurones (black). Corticocortical connections arise from the excitatory
cells and couple cortical locales together over long range.
Simulation of electrocortical activity
JJ Wright et al
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Neuropsychopharmacology
are derived from the findings of Tones and Westbrook
(1996), Lester and Jahr (1992), Dominguez-Perrot et al
(1996), Hausser and Roth (1997), and Partin et al (1996).
Our account treats the cortex as a continuum, rather than
as a set of separate cells, although the properties of the
continuum are closely related to the physiology of
individual neurones. It is thus an account of population
dynamics, rather than of cell-by-cell interactions, and is
suitable for numerical solution of the events over a large
extent of the cortical surface, as well as in small domains of
Table 1 State Variables and Parameters
State
variable
Description Range (units)
V
e
Local space-average dendritic potential, excitatory neurones 0V
i
rev
(V)
V
i
Local space-average dendritic potential, inhibitory neurones 0V
i
rev
(V)
j
e
Afferent excitatory pulse density 0Q
e
max
(s
1
)
j
i
Afferent inhibitory pulse density 0Q
i
max
(s
1
)
Q
e
Efferent excitatory pulse density 0Q
e
max
(s
1
)
Q
i
Efferent inhibitory pulse density 0Q
i
max
(s
1
)
Parameter Description Value (units)
N
ee,cc
Excitatory to excitatory corticocortical synapses/cell 3710 (dimensionless)
N
ie,cc
Excitatory to inhibitory corticocortical synapses/cell 3710 (dimensionless)
N
ee,ic
Excitatory to excitatory intracortical synapses/cell 410 (dimensionless)
N
ei,ic
Inhibitory to excitatory intracortical synapses/cell 800 (dimensionless)
N
ie,ic
Excitatory to inhibitory intracortical synapses/cell 410 (dimensionless)
N
ii,ic
Inhibitory to inhibitory intracortical synapses/cell 800 (dimensionless)
g
e
[0]
Excitatory gain per synapse at rest potential 2.4 10
6
(V s)
g
i
[0]
Inhibitory gain per synapse at rest potential 5.9 10
6
(V s)
Q
e
max
Maximum firing rate of excitatory cells 100 (s
1
)
Q
i
max
Maximum firing rate of inhibitory cells 200 (s
1
)
y
q
Mean dendritic potential when 50% of neurones firing 0.045 (V)
s
q
Standard deviation of neurone firing probability, versus mean
dendritic potential
0.005 (V)
V
e
rev
Excitatory reversal potential 0 (V)
V
i
rev
Inhibitory reversal potential 0.070 (V)
V
q,p
[0]
Resting membrane potential 0.064 (V)
a
ee
EPSP decay time constant in excitatory cells 68 (s
1
)
b
qp
EPSP and IPSP rise time constants 500 (s
1
)
a
ei
IPSP decay time constant in excitatory cells 47 (s
1
)
a
ie
EPSP decay time constant in inhibitory cells 176 (s
1
)
a
ii
IPSP decay time constant in inhibitory cells 82 (s
1
)
n Axonal conduction velocity Maximum 9 m s
1
(for largest scale simulations)
a Dendritic history regression slope coefficient 86 364 (V
1
s
1
)
b Maximum dendritic history time constant 1000 (s
1
)
k[R
c
, R
V
] Fraction of excitatory or inhibitory receptors nonvoltage-,
and voltage-dependent
k[AMPA] ¼0.5
k[NMDA] ¼0.5
k[GABA
a
] ¼1.0 (dimensionless)
l[R
c
, R
V
] Receptor adaptation pulse-efficacy decay constants [AMPA] ¼0.012
[NMDA] ¼0.037
[GABAa] 0.005 (s)
B
n
[R]
Receptor onset coefficients [AMPA]
1
¼1.0
[NMDA]
1
¼1.0
[GABAa]
1
¼1.0 (dimensionless)
A
n
[R]
Receptor offset coefficients [AMPA]
1
¼0.0004
[AMPA]
2
¼0.6339
[AMPA]
3
¼0.3657
[NMDA]
1
¼0.298
[NMDA]
2
¼0.702
[GABA]
1
¼0.0060
[GABA]
2
¼0.9936 (dimensionless)
b
n
[R]
Receptor onset time constants [AMPA]
1
¼760.0
[NMDA]
1
¼50.5
[GABA]
1
¼178.0 (s
1
)
a
n
[R]
Receptor offset time constants [AMPA]
1
¼21.8
[AMPA]
2
¼60.3
[AMPA]
#
¼684.0
[NMDA]
1
¼0.608
[NMDA]
2
¼3.3
[GABA]
1
¼11.2
[GABA]
2
¼127 (s
1
)
Simulation of electrocortical activity
JJ Wright et al
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Neuropsychopharmacology
macrocolumnar, or lesser, dimension. In addition to
numerical solutions, the equations can be solved for some
purposes in linear approximation, to yield wave equations
and dispersion relations, etc (Robinson et al, 1997, 1998a, b;
Rennie et al, 2000). Comparable treatments have been given
in Jirsa and Haken (1996), Liley et al (1999, 2002), and Jirsa
et al (2001).
The way the mathematics are related to the cortical
anatomy and physiology is contained in three main
equations, next described, with explanation of the minor
terms and parameters left to the appendix and Table 1.
In the following formulae, subscript p represents associa-
tion with a presynaptic neurone, and subscript q association
with a postsynaptic neurone. Thus, p or q may each be
replaced by either e (excitatory) or i (inhibitory), and
subscript qp indicates a property arising from the action of
p upon q.
The principal observable property, to which the EEG can
be directly related, is the average dendritic membrane
potential, or local field potential, arising from the pyramidal
cells of the cerebral cortex. This is designated V
e
, in accord
with the above convention, when q ¼e.
The first equation describes the conversion of average
dendritic membrane potentials within a small locale at a
position r, into the density of action potentials within the
locale, as a sigmoidal function. Thus, V
q
(r, t), the average
membrane potential of excitatory or inhibitory cells
becomes converted into Q
q
(r, t), the instantaneous average
firing rate of cells of type q, within the locale
Q
q
¼ Q
max
q
=1 þ e
pðV
q
y
q
Þ=
ffiffi
3
p
s
q
:
With the introduction of p and
ffiffiffi
3
p
, the parameters y and
s correspond, respectively, to the average membrane
potential at which pulse density reaches 50% of maximum,
and the standard deviation of pulse density as a function of
average membrane potential. The probability that a
randomly chosen cell within the continuum is emitting an
action potential at a randomly chosen time, t, is given by
Q
q
=Q
max
q
.
Within the continuum formulation, Q
q
(r, t) can be further
treated as stochastic variables, by assuming that the
Poisson distribution of action potentials emitted from
individual neurones reflects random processes superim-
posed on the mass-action influences within the continuum.
This random perturbation of instantaneous pulse density
thus contributes one type of driving to the net cortical
activity.
The second equation describes the way that action
potentials reach the synapses at r from all the neurones
elsewhere in the cortex. This synaptic flux, j
p
, arises from
pulses, Q
p
which arose at many positions, r
0
, and were
conveyed at the velocity of axonal conduction, n
p
, to arrive
later at r:
j
p
¼
Z
f ðjr r
0
j; r
p
ÞQ
p
ðr
0
; t jr r
0
j=n
p
Þd
3
r
0
:
The final equation is more complicated, and describes the
dynamics of the synapses, and the dendritic membranes, in
converting the synaptic flux into the average dendritic
potential, which then gives rise to further action potentials.
This is
V
q
¼ V
½0
q
þ
X
p
N
qp
H
½D
qp
ðs
qp
G
p
j
p
Þ
including the densities of synapses of different types (N
qp
),
the time characteristics of EPSP and IPSP, (H
qp
[D]
), the
effects of reversal potentials and backpropagation of action
potentials into the dendritic tree (s
qp
), and the modulation
of AMPA, GABA, and NMDA receptor actions (G
p
). The last
term, G
p
, has parameters derived from recent models of the
kinetics of channel opening and closing in receptors,
consequent to varying equilibria among various tertiary
molecular configurations of the receptor-channel complex
(Lester and Jahr, 1992; Dominguez-Perrot et al, 1996;
Hausser and Roth, 1997; Partin et al, 1996). The derivation
of these receptor parameters will be reported elsewhere.
When compared to related and ancestral continuum
models, the novel features of the present treatment are
contained in the terms G
p
and s
qp
.
The features describing the adaptations of AMPA, NMDA,
and GABA receptors confer two properties. Firstly, because
of the relatively rapid decrease in efficacy of the excitatory
receptors compared to that of the inhibitory receptors as
both types of synaptic flux increase, there is a tendency to
stabilize locally the rate of generation of action potentials.
Secondly, the voltage-dependent operation of NMDA
receptors confers a capacity for the transient amplification
of fields of synchronous oscillation, as will be shown below.
The features encapsulated in s
qp
have consequences for
the frequency spectrum of local cortical activity. They
confer a switch-like property, such that for relatively low
pulse densities the 1/f background spectrum typical of EEG
is the predominant feature. At a critical point as firing rate
increases, a switch to oscillation in the gamma band occurs,
of greater local amplitude than the 1/f background. This
switch depends crucially on a convolution, H
½M
q
V
q
,
contained in s
qp
(see appendix). Physically, H
q
[M]
acts as a
sharply adjusted filtering process conferring the switching
effect, which is attributed to the impact of backpropagation
of action potentials into the proximal dendritic tree (Stuart
and Sakmann, 1994).
The combination of these properties enables simulation
of a two-dimensional wave medium, which can be
compared to experimental data, as is shown in Figures 3
and 4.
(a) At low levels of nonspecific cortical activation, when
stimulated by weak afferent volleys (approximated as a
white noise) to any part of the cortical surface, or in
response to the intrinsic perturbations of Q
q
(r, t) alone,
wave motion propagates outward from the site of stimula-
tion. The wave motion propagates at realistic speed for
electrocortical activity and has a ‘1/f ’ frequency content,
identical to that seen in background EEG activity. At very
low levels of cortical activation, a small peak of activity is
also seen in the theta range, but this cannot account fully for
the power seen in the theta band in matched experimental
data (Figure 3). The failure to account fully for the theta
activity is one example of why the cortical simulation needs
to be supplemented with consideration of cortical/subcor-
tical interactionsFsee further below.
Simulation of electrocortical activity
JJ Wright et al
S83
Neuropsychopharmacology
(b) At slightly higher levels of nonspecific cortical
activation, broadband activity in the gamma range appears
against the 1/f background, and can be matched to
experiments (Figure 3).
(c) At still higher levels of nonspecific cortical activation,
with or without introduction of specific sensory input,
oscillation appears in the simulated cortical field, and this
has close similarity to high levels of gamma or ‘40 Hz’
observed in many cortical experiments. Distant from the
site of this autonomous oscillation, it is found that
the gamma activity has acted as a driving source for the
background 1/f activity, by contributing a further increase
in the amplitude of random fluctuations of pulse density.
(Figure 4). At the site of the autonomous gamma oscillation,
the signal amplitude is very high compared to the
background 1/f activity at remote sites, and is closely
correlated with local pulse activity. Pulse densities are
comparable to those observed physiologically (Steriade et
al, 2001). At shorter distances of transmission, or with high
coupling strength between locales, the direct transmission
of activity in the gamma band is apparent.
Thus, depending upon local input conditions, transitions
can take place between a condition of autonomous
oscillation in the gamma band, and a dissipative, point-
attractor condition, in which the cortex is a passive medium
of wave transmission. Transition between these states
occurs sharply, depending on a very small change in local
cell firing rate, and near this transition, both the 1/f
background activity and the gamma activity have a stability
factor close to zero. This capacity for sharp transition of
state, analogous to a thermodynamic change of phase, bears
comparison to the ‘edge of chaos’ type of dynamics
described by Langton (1986, 1990), without necessarily
implying that the cellular dynamics are chaotic in either
state. The occurrence of fluctuating changes of state at a
critical level of activation accords with physiological
observations (Freeman and Rogers, 2002; Phillips and
Pflieger, 1999). There are, in turn, possible implications
for information processing, storage and recall from memory
(Kay and Phillips, 1997; Phillips and Singer, 1997; Wright,
1997b; Arhem and Liljenstrom, 2001; Liljenstrom, 2002).
Also, patches of active cortex can enter into synchronous
oscillation, as is next described.
CORTICAL INTERACTIONS AT MESOSCOPIC SCALE
At a scale from fractions of a millimeter to many
centimeters of cortex, patches of active cells have been
experimentally observed to enter into synchronous oscilla-
tionFthat is, cross-correlations of pulse density, or of
mean local field potential at the separated loci are maximal
at zero lag. Synchronous oscillation has been widely,
although controversially, considered to act as a substrate
for association processes in the cortex. (eg Eckhorn et al,
1988; Singer, 1994; Singer and Gray, 1995; Stryker, 1989;
Bressler et al, 1993; Livingstone, 1996; Miltner et al, 1999;
Neuenschwander and Singer, 1996; Palm and Wennekers,
1997; Steriade et al, 1996; Gray and Singer, 1989; Gray et al,
1989).
Results very similar to classic findings of synchronous
oscillation readily appear when suitable stimuli are
introduced into the simulated cortex, as is shown in Figure
5. This property was first observed in numerical simulations
(Traub et al, 1996; Wright, 1997a; Wright et al, 2000), and
then explained analytically (Robinson et al, 1998a; Chap-
2.0
1.5
1.0
0.5
0.0
2.0
1.5
1.0
0.5
0.0
4
3
2
1
0
4
3
2
1
0
5
4
3
2
1
0
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0 1 2
0 1 2
0 1 2
0 1 2
0 1 2 0 1 2
log 10 (freq(Hz))
log 10 (reference power)
Qns = 0.29
Qns = 0
Qns = -1
Figure 3 Log/log power spectra from real and simulated electrocortico-
gram. Experimental data (left-hand graphs) from Professor Walter
Freeman’s laboratory show typical samples of spectral content from
neocortical sites in a variety of states of cortical activation. Simulated data
(right-hand graphs) show, from top to bottom, performance of the cortical
simulation at high, medium, and low levels of cortical activation. (Qns is a
measure of cortical activation.) Note the model failure at the lowest level of
cortical activation (see text for discussion).
60000
0
0.00011
0.0012
0.0000
0.00000
power (mV
**
2
*
1e9)
0 50 100
0 50 100
0 50 100
0 50 100
0.09
0.00
Origin
5 mm
14 mm
10 mm
frequency (Hz)
Figure 4 Spontaneous gamma activity interacting with the surrounding
cortical field. Top left graph shows the power spectrum of electrocortical
activity associated with gamma-band oscillation at a site on the cortical
surface stimulated only by a DC input. Subsequent graphs show how the
spectrum of activity is transformed during the propagation of waves
through the quiescent surrounding cortex, to give rise to the 1/f cortical
background activity.
Simulation of electrocortical activity
JJ Wright et al
S84
Neuropsychopharmacology
man et al, 2002). Cross-correlation at zero lag can be shown
to occur at all frequencies, but, as is the case in
physiological experiments, large oscillations in the gamma
band, at about 40 Hz, are associated with the largest
amplitudes.
If two points on the simulated cortical surface are driven
by separate, uncorrelated, inputs (or in response to
uncorrelated autonomous local cortical activity at two
separated points) then, within a few milliseconds cross-
correlated activity, maximal at zero lag, appears in the
neighborhood of both active sites. For this effect to occur
axonal delay must be small compared to the rise and fall
time of the dendritic response, and the speed of onset of
synchrony depends mainly on the axonal delay.
Figure 6 sums up the mechanism. The results shown in
the top two rows of the figure are derived from a simulated
field of electrocortical activity, which is not shown
explicitly. The relevant state-variable in the simulated field
was V
e
(r, t), the mean dendritic potential of excitatory cells,
and the field activity was generated by delivering two
independent and uncorrelated time series of white noise to
two points on the simulated cortical field. Wave activity
radiated outward from both sites of input. The relevance for
the generation of synchronous oscillation in the brain lies in
the way these two independent fields of travelling waves can
be shown to interact.
The top frames of Figure 6 show how a large field of zero-
lag synchrony surrounds both sites of uncorrelated input.
The middle frames show eigenmode decomposition of the
same field of wave activity, which breaks the field of activity
into components of spatial activity, each independent of the
others. The first (dominant) eigenmode defines the field of
synchrony. The bottom frames convey the essence of the
physical process, which can be explained as follows. The
uncorrelated inputs can be decomposed into their even and
odd componentsF roughly, the parts of each driving signal
which are coincidentally in phase with the other, and those
parts of coincidentally reversed phaseFthe coincidences
0.0001
-0.0001
5E-05
-5E-05
0
6E-05
3E-05
-3E-05
-6E-05
0
6E-05
3E-05
-3E-05
-6E-05
0
4E-11
3E-11
2E-11
1E-05
0
1.5E-11
1E-11
5E-12
0
1.5E-11
1E-11
5E-12
0
0 100 200 300 400 500 600 700 800
0 100 200 300 400 500 600 700 800
0 100 200 300 400 500 600 700 800
ms
ms
ms
0 10 20 30 40 50 60 70 80
Hz
0 10 20 30 40 50 60 70 80
Hz
0 10 20 30 40 50 60 70 80
Hz
1
-1
0.8
-0.8
0.6
-0.6
0.4
-0.4
0.2
-0.2
0
-100 -80 -60 -40 -20 0 20 40 60 80 100
Delay (ms)
-100 -80 -60 -40 -20 0 20 40 60 80 100
Delay (ms)
-100 -80 -60 -40 -20 0 20 40 60 80 100
Delay (ms)
-0.5
0.4
0.5
-0.4
0.3
-0.3
0.2
-0.2
0.1
-0.1
0
-0.5
0.4
0.5
-0.4
0.3
-0.3
0.2
-0.2
0.1
-0.1
0
Local field potential
Power
Pulse cross correlation
Figure 5 Simulation of synchronous oscillation induced by moving bars. Local field potential time series, power spectral content, and cross-correlations of
two sites in the cortical receptive field, stimulated by combinations of moving bars in the visual field. Delivery of the visual stimuli to the visual cortex was
simulated by the delivery of moving bars of zero-mean white noise to the simulated cortical surface. Input signals within a bar are spatially coherent, while the
signals input to separate bars are uncorrelated. (In these simulations, the stimulus threshold for conversion to spontaneous oscillation in the gamma range
was not exceeded.).
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JJ Wright et al
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Neuropsychopharmacology
occurring randomly over time. Wave activity radiates from
both sites as travelling waves, but since dendrites act as
summing junctions, they dissipate even and odd compo-
nents in their inputs selectively. Even components arriving
at each junction reinforce one another, while odd compo-
nents tend to cancel about the signal mean. Dendrites also
perform a running time average of coinciding signals, over a
period longer than the axonal conduction lags. Thus, the
activity induced by the even components in the driving
signals at the two sites dominates throughout the field, and
emerges as the first eigenmode of the wave activity.
It may be noted in passing that in this simulation the
cortical field remained below transition into the active
gamma range, and the boundary conditions of the cortical
field were toroidal. Since spatial damping is high, the
boundary conditions are not important. The impact upon
synchrony of concurrent transition into autonomous
gamma activity at separate cortical locations is currently
under investigation.
It can be shown that the magnitude of the zero-lag
correlated field is sensitive to dendritic delay time, axonal
conduction lag between sites, and the relative strengths of
couplings in the field. Figure 7 shows the importance of the
strength of coupling between the two active sites. Notably,
when two sites on the cortex are concurrently active, the
strength of synchronous oscillation between the sites will be
a function of both their pre-existing degree of anatomical
connectivity, and of the activation of voltage-dependent
excitatory receptors. Thus, in these simulations, NMDA acts
as a dynamic coupling, enhancing synchrony, and can
contribute to the generation of fluctuating fields of
synchrony, in appropriate conditions of cortical activation.
INTERACTIONS WITH SUBCORTICAL SYSTEMS
Figure 8 expands the notions of cortical and subcortical
interaction sketched in Figure 1, with emphasis now on the
first-order loops of interaction of cortical and thalamic
structures. This architecture has been introduced (Robinson
et al, 2001) to a mathematical description of the cortex
similar to that described above, but omitting the properties
encapsulated in G
p
and s
qp
, which confer the capacity for
autonomous gamma-band oscillation. Since corticothalamic
interactions take place at large scale, between generally
quiescent masses of neurones, it has been considered
sufficient at present to treat the cortex at large scale as if
driven by spatio-temporal white noise, of extrinsic or
intrinsic origin. In future, the detailed interaction with
locally active cortex must be considered. With that proviso,
introduction of coresonance with the thalamus can account
for all the additional frequency bands of activity regularly
seen in the EEG, in sleep and waking (Walter et al, 1967;
Steriade et al, 1990).
Emphasis has been placed on the contribution made to
global resonance by the shortest paths of interaction of
cortex and subcortexFthat is, in loops which are a few
neurones long, modulated by interactions between the
cortex and the excitatory and inhibitory components within
the thalamus, or related systems. Obviously, longer path-
ways of interaction are of great functional importance, but
within the parameters applied in our models, longer
pathways cannot generate such relatively ‘sharp’ reso-
nances, as dendritic processes act as a low-pass filter,
progressively blurring the response in loops of many
neurones. Multineurone loops appear functionally suited
to adiabatic regulation of cortical tone, and thus may be
reflected in slow cortical potentials, rather than the major
EEG rhythms. Again, this adiabatic regulation is a feature
yet to be introduced into any of our models.
Figure 9 shows examples of the way the corticothalamic
model can be fitted to EEG power spectra over a wide range
of states of cortical activation, from sleep to waking. Here,
the process of curve fitting is the inverse of the way that the
simulation is used to generate simulated EEG data. In each
case, a theoretical curve has been fitted to the experimental
data by adjusting a number of free parameters, which are
Figure 6 Mechanism of synchrony within the simulated cortex. These
figures are derived from a simulated cortical field driven by two
uncorrelated white noise inputs, each delivered to one of two sites on
the cortical surface. Top: Sites of input shown by white squares. Cross-
correlations and delays, with respect to the reference site at the black
square. Middle: first and second eigenmodes of the travelling waves
radiating out from the sites of input. Bottom: A schematic representation of
the way in which the first and second eigenmodes arise from addition of
even components, and cancellation of odd components, in the extended
cortical field.
Simulation of electrocortical activity
JJ Wright et al
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Neuropsychopharmacology
mathematically related to groups of parameters used in the
cortical model. As noted above, it is assumed that the
system is driven by diffuse white noise, which may be
equated with specific and nonspecific inputs to cortex, or
with the effect of random perturbations of pulse density
partly associated with small patches of autonomous gamma
activity, or both. At optimal fit, the free parameters
obtained have values in the range expected from the
independently estimated parameters used in the cortical
model.
A further point of interest is that this account of the
origin of the gross EEG spectrum and the major cerebral
rhythms does not depend on specific details of the cerebral
boundary conditions, as has been proposed by Nunez
(1981). In our work it can be readily shown that the spatial
damping is very high, and thus global resonant modes play
no significant part in the generation of wave activity
(Wright, 2000; Robinson et al, 2001). Nunez’s (1995) more
recent work deals with the interaction of local and global
scales of cortical activity, and our present work is in closer
harmony with this modification to his earlier work.
Figure 10 extends the explanatory power of the model to
include auditory-evoked potentials (ERP). This figure shows
real auditory-evoked potentials, obtained by averaging over
time-locked epochs of EEG, obtained as the individual was
responding to ‘target’ tones, and ignoring ‘nontarget’ tones,
in a conventional ‘oddball’ experimental paradigm. Also
shown are simulated ERP produced in accord with methods
described in Rennie et al (2002). The prestimulus ‘resting’
EEG of the subject recorded in epochs just before the
delivery of each stimulus was characterized by fitting the
corticothalamic model in the same way as in Figure 9, to
obtain a set of descriptive parameters. These parameters
define the transfer function of the corticothalamic system,
assuming that the sources driving cortical activity over the
prestimulus periods are equivalent to white noise. The
Enhanced Coupling
61% 37%
19 mm
79% 20%
5 mm
Eigenvector 1 Eigenvector 2
Driving
Site
Separation
Gaussian Coupling
5 mm
19 mm
98% 1%
93% 5%
1
32
Figure 7 Effects of enhanced coupling on synchronous oscillation (compare to middle graphs in Figure 6). First and second eigenmodes of activity for two
sites driven by uncorrelated white noise inputs, separated by either 5 or 19 mm. The upper figures show relative proportions of synchronous and
antisynchronous (or even and odd) activity when the sites are coupled together only by excitatory surrounds, which decline in strength as a Gaussian
function of distance. The lower figures show the impact of enhanced direct coupling between the driven sites, such as would be produced by structured
anatomical connections, or by voltage-dependent receptor activation. Top bar shows relative amounts of covariance as shades of gray.
Simulation of electrocortical activity
JJ Wright et al
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Neuropsychopharmacology
transfer function thus obtained can then be used to predict
the average time-locked response, or impulse response, of
the cortex, to any additional, but time-locked, input. As
shown, this enables the ERP to be accounted for as time-
locked responses to short bursts of activity generated by
sensory input, and from within the brain. Unfortunately,
there is no means currently available to us to generate the
second input from within the simulation, as this must
depend upon a specific cognitive eventFthe subjects’
decision that the particular tone heard could be (correctly)
identified as a target, requiring a response. In principle at
least, we expect that this cognitive process might be reduced
to associative processes mediated by mechanisms as
described in the earlier sections on cortical gamma activity
and synchronous oscillation. This step toward closing the
paths of causality within brain modelling, is, of course, a
daunting taskFand one that might best be initially
addressed in approximately steady-state conditions, in
accord with the methods described by Jirsa et al (2001).
The corticothalamic model so far developed has some
interesting implications, which may influence future
modelling. Firstly, the role of the lower-frequency cerebral
rhythms on biasing transitions of state in the cortex is
largely unexplored. Secondly, the occurrence of long-range
inhibitory interactions in the thalamus, mediated by the
reticular nucleus, contrasts with the predominance of
excitation at long range in the cortex. This implies a
reversal of phase relations generated by synchronous
oscillations in thalamus, to those in cortex. Since thalamic
association nuclei and cortex interact with a high degree of
one-to-one mapping, the interactions in the entire system
might further modulate the onset and offset of fields of
synchrony in a dynamic way. Effects on alpha phase
suggestive of the operation of such a mechanism have been
observed, both experimentally, and in a related type of
thalamo-cortical simulation (Suffczynski et al, 2001).
Thirdly, the interactions of cortex and thalamus with other
subcortical systems over longer pathways may act to
modulate the fields of cortical activation, creating analogues
Figure 8 Schematic of interactions between the thalamus and the
cortex. The cerebral cortex gives and receives excitatory (red) connections
to and from the specific and secondary relay nuclei of the thalamus, and
gives connections also to the reticular nucleus of the thalamus. The reticular
nucleus gives inhibitory connections to the other thalamic nuclei, and
receives inputs from the specific and secondary nuclei. All components are
activated by the reticular activating system. Symbols f
ee
, f
es
, etc, represent
pulse fluxes, and G
rs
, G
sr
, etc, synaptic gains, in accord with a related
convention to that used in the purely cortical model.
Figure 9 The global EEG and states of cortical activation. Optimum fits of the thalamocortical model to log/log EEG power spectra from the same
individual, in alert and drowsy waking states, and the first two stages of sleep.
Simulation of electrocortical activity
JJ Wright et al
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Neuropsychopharmacology
of attention and selective arousal. Such modulations would
be relatively slow compared to the cerebral rhythms, as
interactions between cortex and thalamus of order greater
than two neurones are generally smoothed by dendritic
filtering. Subcortical pathways might thus regulate ‘adia-
batic’ shifts in cortical activation, akin to the slow potentials
of the cerebral cortex. Fourthly, there is some evidence that
some theta activity might be related to pulsed interactions
between the cortex and the thalamus (Rennie et al, 2002).
Increased understanding of this, and of other sources for
theta rhythm may lead to explanation of the failure of our
cortical model to account for the theta content observed in
experiments, as noted in relation to Figure 3.
Finally, it may be noted that by negative feedback
mechanisms, corticothalamic and other cortical/subcortical
interactions may contribute to maintaining cortical activa-
tion near the level of transition between gamma and 1/f
conditions.
DISCUSSION
The results described above indicate that representation of
brain activity in mathematical terms is practicable, albeit in
a preliminary manner. Further work is needed to overcome
the numerous points upon which our treatment encounters
limitations, as pointed out in the preceding text. As such
problems are addressed, further physiological and anato-
mical detail might thus be introduced into an organized
framework, always subject to tests against experimentally
observed properties of the brain’s global properties.
The properties observed in our simulations suggest
approximation toward an account of the unified operation
of the brain may be possible. Within the simulations there
occur properties considered, in related contexts, to confer
universal computation, information storage, association
and recall, and self-organization (Freeman and Rogers,
2002; Arhem and Liljenstrom, 2001; Liljenstrom, 2002;
Langton, 1986, 1990; Singer, 1994; von Neumann, 1949;
Wolfram, 1984; Phillips and Singer, 1997; Kording and
Konig, 2000). Work showing that realistic anatomical
connectivity can appear on the basis of Hebbian learning,
in certain simplified neural networks (von der Marlsburg,
1973; Swindale, 1996; Alexander et al, 2000) is also of
interest, as this property appears likely to be transferable to
the more dynamically realistic continuum models described
here.
A further step would see, as well as the introduction of
Hebbian learning, the incorporation of innate positive and
negative reinforcement systems analogous to those known
to exist in the brain (Olds and Milner, 1954). It may be
presumed that such reinforcement systems act, via neuro-
Figure 10 Simulation of the ERP based upon impulse response of the thalamocortical model. Comparison of (a) modelled ERPs with (b) experimental
ERPs, in response to target and nontarget tones. To enable modelling of the experimental data, parameter values were obtained by fitting the
thalamocortical theoretical model (as in Figure 9) to the EEG power spectrum in the experimental prestimulus epochs. The modelled ERPs were then
generated as outputs utilizing the transfer function associated with the prestimulus state, while assuming an input sensory pulse of brief duration reached the
subcortex, and hence cortex at t ¼0.10 s from the time of stimulus delivery, for both target and nontarget tones. In the case of target tones, a second
impulse was assumed to be generated from autonomous activity within the cortex, a further 0.10 s later. Thus, the target response is the result of
superposition of two impulse responses, generated 0.1 s apart.
Simulation of electrocortical activity
JJ Wright et al
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Neuropsychopharmacology
modulation, to regulate synaptic consolidation. Some
survival behaviors related to activity in the reinforcement
pathways appear to be hard-wired, thanks to species
evolution. Such a priori reinforcement systems might also
act to supervise adaptive learning, if the reinforcement
system was initially activated by only a small subset of the
total environmental stimuli, but could, by associative
learning, come to be activated by more complex inputs,
and by internally generated brain states. Internally gener-
ated states that did not acquire association with basic
survival behaviors and the operation of the reinforcement
mechanisms would be extinguished if they did not activate
the survival behaviors and the synaptic consolidation
mechanisms over some critical time period. Thus, survi-
val-consistent behaviors would remain, and develop in
complexity.
Another goal is the fitting of models of this type to a
much larger range of data, particularly the large normative
and standardized data set under construction by Gordon
(2000, 2002), which includes EEG and fMRI data from both
normal people and people with a variety of psychopathol-
ogies. It is our hope that this will help us to contribute to the
unification of findings in neuropsychopharmacology, as
discussed at the beginning of this paper.
ACKNOWLEDGEMENTS
We thank Professor Walter Freeman for access to his library
of electrocortical recordings, Mr Nicholas Hawthorn for
technical assistance, and the University of Sydney, the
University of Auckland, the Swinburne University, Mel-
bourne, and the Mental Health Research Institute, Mel-
bourne, for the use of computer facilities. We thank also our
hosts at the Agora for BioSystems, Stockholm, and the
University of Potsdam, at whose conferences this paper was
delivered.
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APPENDIX: A MORE DETAILED DESCRIPTION OF
THE MATHEMATICS OF A CORTICAL MODEL AND
OF METHODS OF SIMULATION
In the following formulae, subscript p represents associa-
tion with a presynaptic neurone, and subscript q association
with a postsynaptic neurone. Thus p or q may each be
replaced by either e (excitatory) or i (inhibitory), and
subscript qp indicates a shared property. The extended
subscripts ic and cc indicate that either intracortical or
corticocortical synapses are specifically referred to.
The principal state variables are:
Q
q
(r, t), the point pulse densities of neurones of type q,
where t is the time, and r is the position on the cortical
surface (s
1
).
V
q
(r, t), the corresponding point local field potentials
(V).
j
p
(r, t), the synaptic flux densities (s
1
).
The rate at which pulses are generated is given by
Q
q
¼ Q
max
q
=1 þ e
pðV
q
y
q
Þ=
ffiffi
3
p
s
q
;
where Q
q
max
is the maximum pulse rate sustainable by
neurones of type q, and y
q
and s
q
are parameters defining
pulse density distribution as a function of V
q
. Perturbation
of the value of Q
q
(r, t) by random local factors in individual
neurones provides an internal source of white noise driving
to the system.
The synaptic flux densities are given by equations of the
form
j
p
¼
Z
f ðjr r
0
j; r
p
ÞQ
p
ðr
0
; t jr r
0
j=n
p
Þd
3
r
0
;
where r
p
is the axonal range and n
p
is the average velocity of
axonal conduction. The axonal distribution functions
f(|rr
0
|, r
p
) may have any form, but are usually treated as
two-dimensional Gaussian distributions. Intracortical con-
nections are also usually approximated as having no
significant extension in space, unlike the corticocortical
connections, and thus
j
e
¼ j
e; ic
þ j
e; cc
;
j
e; ic
Q
e
;
j
i; ic
Q
i
;
j
s
and j
ns
are specific and nonspecific inputs to the cortex
from all sources, and are not explicitly described here, but
act as an external source of driving inputs to the cortical
system.
Local field potential is considered directly proportional to
the pyramidal cell point membrane potential, which is given
by
V
q
¼ V
½0
q
þ
X
p
N
qp
H
½D
qp
ðs
qp
G
p
j
p
Þ;
where V
q
[0]
is the resting membrane potential, N
qp
is the
number of synapses of type p per dendritic tree, and #
indicates convolution over time.
Normalized average EPSP/IPSP at the soma are given by
H
½D
qp
ðtÞ¼
a
qp
b
qp
b
qp
a
qp
ðe
a
qp
t
e
b
qp
t
Þ; tX0;
where 1/b
qp
,1/a
qp
are the rise and fall time constants.
The standardized synaptic strength is represented by g
p
[0]
.
It is the integral over time of the PSP, as recorded at the
soma, when the neurone is at its resting membrane
potential V
q
[0]
. The actual synaptic strength s
qp
is related
to this, through the ion channel reversal potentials, V
q,p
rev
,
and a measure of the membrane potential, thus:
s
qp
¼ g
½0
p
V
rev
p
H
½M
q
V
q
V
rev
p
V
½0
q
:
The convolution by H
q
[ M]
is introduced to account for an
effect of retrograde propagation of action potentials into
the dendritic tree. We assume that at low synaptic flux
and firing rates, cells are principally sensitive to synapses
proximate to the soma. During active firing, the site of
synaptic summation and further action potential generation
is shifted more distally, into the dendritic tree, and
now more remote synapses contribute greater weight to
determination of whether or not firing continues.
The contribution of remote synapses depends upon the
arrival of EPSP, IPSP generated at delay times in the distal
dendrites, and at earlier values of the membrane potential.
For the population of cells, the delay because of this effect
increases as average membrane potential decreases, so to
first approximation,
H
½M
q
¼ c
q
e
c
q
t
; tX0;
where c
q
is determined by a linear regression,
c
q
¼ aðV
q
V
½0
q
Þþb:
(The linear regression can be replaced by a more
physiologically realistic nonlinear function, which will be
reported in detail elsewhere.)
The impact of this membrane-voltage-dependent switch-
ing of a mean delay process in the dendrites is to regulate
partially the transition of local electrocortical activity into,
and out of, an active state in which gamma activity appears,
and generates strong fluctuations in the local pulse-density
and local field potential. This contributes a third type of
driving to wave propagation in the system, in addition to
that provided from external sources and stochastic pertur-
bations of pulse density.
Neurotransmitter receptor adaptation to continuing in-
put is included in G
p
, for which parameters
k
½R
p
; l½R; A
n
; a
n
; B
n
; b
n
have been obtained from model-
ling the behavior of receptors, from physiological measure-
ments of transformations among receptor tertiary
molecular configurations (references are given in main
text).
G
p
¼ H
½R
c
p
G
½R
c
p
þ H
½R
V
p
G
½R
V
p
:
The superscript ½R¼½R
c
; ½R
V
indicates whether or not
the receptor is voltage dependent. Thus if p ¼e, R
c
indicates
characteristics of an AMPA receptor, and R
V
an NMDA
receptor, while if p ¼i, then R
c
indicates a GABAa receptor,
Simulation of electrocortical activity
JJ Wright et al
S92
Neuropsychopharmacology
and no voltage-dependent channels are considered.
G
½R
c
p
¼ k
½R
c
p
e
l½R
c
f
p
;
G
½R
v
p
¼ k
½R
V
p
e
l½R
V
f
p
Q
q
=Q
max
q
:
The coefficients k
½R
p
describe the relative amplitude of the
two components of G
p
, and the multiplication by Q
q
/Q
q
max
is introduced in the second of the above equations because
voltage-dependent receptors are active only in that fraction
of neurones, which are currently firing.
The onset and offset of this modulation of synaptic gain
are described by normalized time functions, analogous to
H
½D
qp
:
H
½R
p
ðtÞ¼½B
½R
1
=b
½R
1
þ B
½R
2
=b
½R
2
þA
½R
1
=a
½R
1
A
½R
2
=a
½R
2
1
½B
½R
1
e
b
½R
t
1
þ B
½R
2
e
b
½R
t
2
A
½R
1
e
a
½R
t
1
A
½R
2
e
a
½R
t
2
:
These modulations of synaptic gain help to maintain
cortical stability close to the boundary between 1/f activity,
and active gamma, since AMPA activity is more strongly
downregulated with decreasing membrane voltage than is
GABA activity. The voltage dependence of NMDA activity
contributes a type of dynamic gain, enhancing synchronous
oscillation.
The simulation results shown in Figures 3 and 4 applied
random perturbations of the pulse densities Q
q
(r, t)in
accord with the assumed stochastic nature of individual
neurone firing. In Figures 5, 6, and 7, this perturbation was
not applied, although the moving bars or static inputs were
treated as zero-mean signals of white-noise type. The
theoretical functions applied to derive the results in Figures
9 and 10 assumed driving of the model system by diffuse
white noise. These differences represent, in part, stages in
the evolution of the family of models rather than differences
of principle. All relevant properties are retained in
simulations that include the more complex treatments of
the sources of driving.
The cortical simulations were applied with the cortical
surface ‘lumped’ into a 20 20 matrix of elements, with
toroidal boundary conditions. Variation of boundary
conditions is without significant effect.
Simulation of electrocortical activity
JJ Wright et al
S93
Neuropsychopharmacology
