thalamic tumor
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Journal of Theoretical Biology 233 (2005) 271–286
Analysis of the electroencephalographic activity associated with
thalamic tumors
S.C. O’Connora,b,, P.A. Robinsona,b
aSchool of Physics, University of Sydney, Broadway, Sydney, NSW 2006, AustraliabBrain Dynamics Center, Westmead Hospital, Westmead, NSW 2145, Australia
Received 14 June 2004; accepted 7 October 2004
Available online 30 November 2004
Abstract
A physiologically based model of corticothalamic dynamics is used to investigate the electroencephalographic (EEG) activity
associated with tumors of the thalamus. Tumor activity is modeled by introducing localized two-dimensional spatial non-
uniformities into the model parameters, and calculating the resulting activity via the coupling of spatial eigenmodes. The model is
able to reproduce various qualitative features typical of waking eyes-closed EEGs in the presence of a thalamic tumor, such as the
appearance of abnormal peaks at theta (E3Hz) and spindle (E12Hz) frequencies, the attenuation of normal eyes-closed
background rhythms, and the onset of epileptic activity, as well as the relatively normal EEGs often observed. The results indicate
that the abnormal activity at theta and spindle frequencies arises when a small portion of the brain is forced into an over-inhibited
state due to the tumor, in which there is an increase in the firing of (inhibitory) thalamic reticular neurons. The effect is heightened
when there is a concurrent decrease in the firing of (excitatory) thalamic relay neurons, which are in any case inhibited by the
reticular ones. This is likely due to a decrease in the responsiveness of the peritumoral region to cholinergic inputs from the
brainstem, and a corresponding depolarization of thalamic reticular neurons, and hyperpolarization of thalamic relay neurons,
similar to the mechanism active during slow-wave sleep. The results indicate that disruption of normal thalamic activity is essential
to generate these spectral peaks. Furthermore, the present work indicates that high-voltage and epileptiform EEGs are caused by a
tumor-induced local over-excitation of the thalamus, which propagates to the cortex. Experimental findings relating to local over-
inhibition and over-excitation are discussed. It is also confirmed that increasing the size of the tumor leads to greater abnormalities
in the observable EEG. The usefulness of EEG for localizing the tumor is investigated.
r 2004 Elsevier Ltd. All rights reserved.
Keywords: Thalamic tumor; EEG; Pathological theta or delta; Continuum model
1. Introduction
The disruption to normal brain function induced by a
tumor of the thalamus can be detected by electroence-phalograms (EEGs), which measure electrical activity
via electrodes on the head. A typical feature of waking
EEG in the presence of thalamic tumors, is unusually
large activity near 3 Hz (Gibbs and Gibbs, 1964; Hirose
et al., 1975; Janati and Hester, 1986; Newmark et al.,
1983). This frequency is variously called ‘delta’ or ‘theta’
in thalamic literature; in the present work we refer to it
as the theta frequency. Other EEG correlates of thalamic tumor include the focal attenuation of normal
background rhythms (Janati and Hester, 1986; New-
mark et al., 1983), epileptiform discharges (Cheek and
Taveras, 1966; Janati and Hester, 1986; Hirose et al.,
1975; O’Brien et al., 1997), non-reactive alpha (Janati
and Hester, 1986), and diffuse (Hirose et al., 1975;
Newmark et al., 1983) or focal (Scarff and Rahm, 1941)
background slowing. There have also been reports of
12–14 Hz ‘spindles’ during waking EEG (Hirose et al.,
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0022-5193/$- see front matter r 2004 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jtbi.2004.10.009
Corresponding author. School of Physics, CUDOS, University of
Sydney, Sydney NSW 2006, Australia. Tel.: +612 93515635; fax:
+61 2 9351 7726.
E-mail address: suzie@physics.usyd.edu.au (S.C. O’Connor).
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1975; Janati and Hester, 1986), whereas spindling in
healthy brains is only seen during sleep (Steriade, 2000).
Furthermore, there has been a report of dissociation
between the cerebral hemispheres during sleep (Kanno
et al., 1977), whereby one hemisphere lags behind the
other in cycling through the normal sleep stages.
Thus, there is a wide range of reported abnormalEEG phenomena associated with thalamic tumors.
However, around 10–30% (Cheek and Taveras, 1966;
Hirose et al., 1975) of subjects exhibit normal EEGs in
the presence of such tumors. Hence, a model of thalamic
tumors would need to account for a large variety of
observed phenomena, ranging from normal EEGs,
through to rarely observed, highly abnormal features
such as EEG spindling during waking.
A model of corticothalamic dynamics has recently
been generalized to include spatial non-uniformities in
the parameters via the coupling of spatial eigenmodes
(O’Connor and Robinson, 2004c; Robinson et al.,
2003). This model has been successful in predicting the
form of individual EEG spectra, including the spectral
peaks such as the alpha rhythm (Robinson et al., 2001a).
Indeed, a key advantage of the model is its ability to unify
large-scale cortical activity of many different types into a
single framework. For example, the model has predicted
trends seen in various states of arousal (Robinson et al.,
2001a), certain seizure onsets and dynamics (Robinson
et al., 2002), and evoked response potentials (Rennie et al.,
2002). In the spatial domain, it has successfully addressed
coherence and correlations (Robinson, 2003), as well as
unifying on-going and evoked activity recorded from the
scalp and cortex (O’Connor et al., 2002; O’Connor andRobinson, 2003, 2004b).
Thalamic tumors represent relatively small-scale non-
uniformities in brain activity, typically projecting to a
cortical area less than 5 cm in width. We have shown
previously (O’Connor and Robinson, 2004c) that such
non-uniformities cannot be adequately addressed using
a uniform model. Thus, in the present work, we use the
spatially non-uniform analysis to investigate thalamic
tumors. The aim of the present work is to provide
insight into the physiological and anatomical mechan-
isms responsible for the range of observed EEG features
associated with thalamic tumors.Tumor tissue is electrically inactive regardless of the
type of tumor (Scarff and Rahm, 1941), which implies
that the morphological details of the tumor are likely to
be unimportant in determining its influence on EEG
recordings. Furthermore, a study on gliomas reported
that EEG findings were unrelated to the malignancy of
the glioma (Newmark et al., 1983), or to the histology of
the tumor (Newmark et al., 1983). Thus, in this study we
do not attempt to model the morphology of the tumor,
nor to investigate tumors of different types separately,
since these factors do not influence the electrical activity
of the brain.
In Section 2, we provide the necessary theoretical
background by outlining the non-uniform version of the
model in which the parameters can vary across the
brain, and the ensuing spectrum is calculated from the
resulting coupled spatial eigenmodes; thus, a localized
parameter variation can affect the spectrum at distant
sites on the cortex. We also review a reduced three-dimensional parameter space in which activity and
stability in the model can be easily visualized. In Section
3, we examine the pathophysiology of tumors and their
influence on surrounding tissue to determine an appro-
priate method of modeling them. In Section 4, we model
two-dimensional (2D) tumor-like non-uniformities, and
investigate the effects of varying the tumor size. In
Section 5, we study thalamic tumors superimposed on
realistic front-to-back parameter non-uniformities, as
determined by previous work fitting the non-uniform
model to the EEG spectra of 98 normal subjects
(O’Connor and Robinson, 2004c).
2. Non-uniform model
In this section, we summarize the corticothalamic
model, which has been recently generalized to incorpo-
rate spatial non-uniformities via coupling of eigen-
modes. The full details and justification of the model can
be found elsewhere (Robinson et al., 2003). The cortex is
modeled as a continuous, bounded sheet. Precise
boundary conditions prove not to be very important in
determining activity in this model (O’Connor and
Robinson, 2004a; Robinson et al., 2001b; Robinson,2003), and the detailed geometry of cortical convolu-
tions is ignored. The corticothalamic connectivity
assumed in the model is shown in Fig. 1, and involves
three components: the specific nuclei, labeled s, which
relay subthalamic input fn to the cortex, and feed
cortical signals back to the cortex; the thalamic reticular
nucleus, labeled r, which inhibits the relay nuclei; and
the cortex, which contains both excitatory (e) and
inhibitory (i ) neurons, receives projections from the
relay nuclei, projects to both the reticular and relay
nuclei, and is densely connected to itself. Note the four
feedback loops: the intracortical loop, involving ex-citatory and inhibitory cortical neurons; the ‘direct’
corticothalamic loop which involves the cortex and the
relay nuclei; the ‘indirect’ corticothalamic loop which
involves the cortex, reticular nucleus, and relay nuclei;
and the intrathalamic loop which contains the thalamic
reticular nucleus and the relay nuclei.
Measured large-scale cortical potentials are propor-
tional to the mean cellular membrane currents, which
are in turn proportional to the firing rates fe;i : Cortical
excitatory neurons generate most of the measurable
potential on the scalp, because they are larger than
inhibitory neurons and better aligned to generate
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observable signals (Nunez, 1995; O’Connor and Robin-
son, 2003). Thus, in the absence of skull volume
conduction, the power spectrum on the head is wellapproximated by the squared modulus of the signal fe;to within a constant of proportionality. The effects on
the spectrum of conduction through the cerebrospinal
fluid, skull, and scalp, have been studied in the context
of this model (O’Connor et al., 2002; Robinson et al.,
2001a); they filter out high-wavenumber activity, k
\15 m1, (O’Connor et al., 2002; Robinson et al.,
2001a) and hence high-frequency activity (via the
dispersion relation). Thus, at the frequencies of interest
in the present work,o20 Hz, the effects are minimal and
are not considered further.
In our continuum treatment of the cortex the firingrate of signals emitted by excitatory or inhibitory
neurons, which depend on individual cell body poten-
tials, are averaged to give mean values of the outgoing
pulse field faðr; tÞ; where a ¼ e; i : The mean rate of
generation of neuronal pulse density depends on the
mean local cell-body potential via a smooth sigmoidal
function that increases from 0 to its maximum value as
the potential increases from 1 to 1: We approximate
the sigmoidal function here by a linear function on the
assumption that deviations from the steady state are
small in normal, non-seizure states. This approximation
has been found to yield excellent agreement with
observed frequency spectra and other phenomena
(O’Connor et al., 2002; O’Connor and Robinson,
2003, 2004b, c; Rennie et al., 2002; Robinson et al.,
2001a, 2002, 2003; Robinson, 2003).
The local mean cell body potential of neurons of type
a in the cortex is a function of inputs from other cortical
neurons, and from excitatory subcortical neurons.Incoming activity is received in the dendritic tree and
filtered as it spreads along the dendrites to the cell body.
The quantity Lab is a dendritic low-pass filter function
which accounts for the temporal delay and smearing of
an incoming signal from a neuron of type b ¼ e; i ; s as it
travels along the dendritic tree to the cell body of a
neuron of type a ¼ e; i : It can be written (Rennie et al.,
1999)
Labðr;oÞ ¼ 1
½1 io=aabðrÞ½1 io=babðrÞ; (1)
where bab and aab are the inverse rise and decay times of the dendritic potential, respectively.
Outgoing pulses from each neuron propagate along
its axonal tree at a velocity v(r)E1 0 m s1. This
propagation can be described by damped wave equa-
tions for the fields fa (Rennie et al., 1999). After Fourier
transforming in time, one finds,
Daðr;oÞfaðr;oÞ ¼X
b
J abðr;oÞfbðr;oÞ; (2)
where
Daðr;oÞ ¼ ½1 io=gaðrÞ2 r2ar 2; (3)
J abðr;oÞ ¼ Labðr;oÞG abðrÞeiotabðrÞ; (4)
gaðrÞ ¼ vðrÞ=ra is a measure of the damping, ra is the
mean range of axons a, the gain G ab represents the
scaled response strength in neurons a due to a unit signal
incident from neurons of type b, and tab represents a
pure delay—as in signal transmission between the cortex
and the thalamus, for example—and appears as an
exponent due to the temporal Fourier transform. Note
that (2) makes explicit the approximately one-to-one
mapping between locations in the cortex and thalamus,
particularly the specific nuclei of the thalamus.
Using (2) and the connectivities shown in Fig. 1, thewave equation for excitatory cortical neurons follows as
Deðr;oÞfeðr;oÞ ¼ J eeðr;oÞfeðr;oÞ þ J ei ðr;oÞfi ðr;oÞ
þ J esðr;oÞfsðr;oÞ: ð5Þ
The analogous equations for cortical inhibitory, specific
relay, and reticular neurons can also be deduced from
(2), and the quantities fi ; fs; and fr can be eliminated to
give the transfer function of a stimulus fn to fe
(Robinson et al., 2003). This is of the form
Aðr;oÞfeðr;oÞ ¼ B ðr;oÞfnðr;oÞ; (6)
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φe, φ
i
φs
φn
φr
φe
reticularnucleus
cortex
relaynuclei
ee,ei
es
rs re
srse
sn
Fig. 1. Schematic of corticothalamic connections showing the cortex,
reticular nucleus, and relay nuclei. The cortex is extensively connected
to itself, and also projects to and receives projections from the
thalamus. There are two loops through the thalamus: a direct loop
passing only through the relay nuclei, and an indirect loop which also
passes through the reticular nucleus. There is also an intrathalamic
loop. Locations at which gains G ab act are indicated on the diagram.
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where
Aðr;oÞ ¼ ð1 io=geÞ2
r2e
r 2
J eeð1 J srJ rsÞðJ se þ J srJ reÞ
r2e ð1 J ei Þð1 J srJ rsÞ
; ð7Þ
B ðr;oÞ ¼ J esJ sn
r2e ð1 J ei Þð1 J srJ rsÞ
: (8)
The uniform case previously studied is a special case.
Activity in this model is relatively insensitive to precise
boundary conditions (O’Connor and Robinson, 2004a;
Robinson et al., 2001b; Robinson, 2003), so we
investigate a simple cortical geometry in the first
instance. Taking the Fourier transform in space, and
applying periodic boundary conditions on a rectangular
cortex of size l x l y, the expression (6) becomes
XK
Aðk K;oÞfeðK;oÞ ¼X
KB ðk K;oÞfnðK;oÞ;
(9)
where k and K range over the values kmj ; Kmj ¼
ð2pm=l x; 2p j =l yÞ; and m and j are integers. Only a finite
number of modes can be included in the calculation, and
most of the activity can actually be captured using a
relatively small number of modes (Robinson et al.,
2003); thus we choose M max such that jmj; j j jpM max:Eq. (9) can be written as a matrix equation AUe ¼ BUn;or Ue ¼ A1BUn ¼ MUn; where the size of each matrix
depends on the number of modes M max retained after
truncation [for the 2D cortex studied here, A and B are(2M max+1)2 (2M max+1)2 matrices and Ue is a
(2M max+1)2 1 column matrix] (Robinson et al.,
2003). We have shown previously (Robinson et al.,
2003) that the power spectrum at a given r is given by
P ðr;oÞ ¼ jfnðoÞ2jXm;n
exp½iðkm knÞ rðMMyÞmn (10)
for spatially white noise, where m and n label matrix
elements. By averaging (10) over position, the mean
power can be written
P ðoÞ ¼ jfnðoÞ2jTrðMMyÞ; (11)
where fn is independent of o if the noise is also
temporally white.
2.1. Reduced parameter space
In this section, we give a brief overview of stability in
the model as it pertains to the present work. In previous
work using the uniform model (i.e. with spatially
uniform parameters) to analyse normal arousal
states and epileptic seizures, we found that stability
boundaries in parameter space occur approximately
where (Robinson et al., 2002)
0 ¼ ð1 io=geÞ2 x yð1 G srsÞ
1 G srsL2 eiot0 ; (12)
x ¼ G ee=ð1 G ei Þ; (13)
y ¼ G ese þ G esre
ð1 G srsÞð1 G ei Þ; (14)
is satisfied, where for brevity we have written G srs ¼
G srG rs; G ese ¼ G esG se; and G esre ¼ G esG srG re; and t0 ¼
2tes ¼ 2tse ¼ 2tre is the corticothalamic loop propaga-
tion delay. The quantities x and y relate to cortical and
corticothalamic activity, respectively, and
z ¼ G srsab=ða þ bÞ2; (15)
parametrizes intrathalamic activity; thus, the system can
be approximately parametrized in a reduced three-
dimensional (xyz) space. The stability zone in xyz spacedefined by Eq. (12) is shown in Fig. 2, for the eyes-closed
parameters used throughout this work. The alpha
stability boundary is indicated on the upper right of
the figure. Proximity to this boundary manifests itself in
the spectrum as increased activity at the alpha
frequency, since cortical activity approaches instability
and hence the maximum firing rates typical of seizures.
Indeed, if the boundary is crossed, the brain goes into a
limit cycle near 10 Hz, which is plausibly related to
seizure activity, via an instability of the alpha peak
(Robinson et al., 2002). Also shown is the slow-wave
boundary (the front unshaded surface), through which
the brain passes into slow-wave (o1 Hz) instability. This
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EO EC
S2
S4
z
1.0
y1.0
-1.0
x
1.0
theta
spindle
alpha
delta
Fig. 2. Stability zone in xyz space (Robinson et al., 2002) for the
parameters in Table 1. The surface is shaded according to instability
type: the transparent front right face corresponds to a zero-frequency
instability; the top light-shaded right face corresponds to alpha-
frequency (E10 Hz) instability; the top central surface corresponds to
a spindle-frequency (E13 Hz) instability; and the top light-shaded left
face corresponds to a theta-frequency (E3 Hz) instability. Approx-
imate locations are shown of eyes-open (EO), eyes-closed (EC), and
normal sleep stages two (S2) and four (S4), with each state located at
the top of its bar.
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boundary follows the plane x+ y=1 (Robinson et al.,
2002). The spindle and theta boundaries are also indicated
in Fig. 2, through which the brain passes into spindle
(E12–14 Hz), and E3 Hz spike-and-wave theta instabil-
ities, respectively (Robinson et al., 2002). More generally,
proximity to any stability boundary manifests itself as
increased activity at the corresponding frequency.The normal states of arousal lie within the stability
zone in Fig. 2; the approximate locations of waking eyes
closed (EC), waking eyes open (EO), and normal sleep
stages two (S2) and four (S4) are indicated in the zone.
These approximate locations have been inferred by
examination of typical spectra, extensive investigation of
the model, and comparisons with data from a wide
variety of experiments. For a thorough discussion of the
model parameters see Robinson et al. (2004).
Consider the waking eyes-closed (EC) and eyes-open
(EO) states. We see from Fig. 2 that the eyes-closed state
lies closer to both the slow-wave and alpha stability
boundaries than does the eyes-open state. In accordance
with the above discussion, the eyes-closed state hence
has larger peaks at low and alpha frequencies than the
eyes-open state; i.e. it has more salient features than the
eyes-open spectrum because of its relatively marginal
stability. We have shown previously (O’Connor and
Robinson, 2004c) that the exact location in the stability
zone of the eyes-closed state varies across the head; the
back of the head lies at smaller x and larger y than the
front of the head, and is nearer the alpha boundary;
thus, the back of the head exhibits greater alpha activity
than the front (O’Connor and Robinson, 2004c).
Consider now the sleeping states shown in Fig. 2.Sleep stage two (S2) lies near the spindle stability
boundary; spindling is the typical EEG feature of this
state. Sleep stage four (S4), on the other hand, lies near
the E3 Hz theta stability boundary; theta activity is the
typical EEG signature of this state. In general, the sleep
states lie in the portion of the zone where yt0; and the
waking states lie in the portion y\0: This partition can
be understood by examining Eq. (14). When the direct
corticothalamic loop gain G ese is stronger than the
indirect one G esre, the state parameter y is also positive.
When the indirect loop gain is stronger, y becomes
negative. This is physiologically reasonable since sleep isassociated with increased activity in the thalamic
reticular nucleus, which acts to suppress relay nuclei
and to attenuate positive feedback to the cortex
(Coenen, 1995).
3. Modeling a tumor
We must determine an appropriate method of
modeling tumors. In order to do so, we examine the
physiological impact of tumors on the surrounding
tissue. Tumors themselves are electrically inactive
(Scarff and Rahm, 1941), and exert their influence on
observable electrical fields via disruption of an annulus
immediately surrounding the tumor (Hess, 1975; Scarff
and Rahm, 1941). This peritumoral tissue consists of
neurons which are intact, but characterized by a number
of concomitant pathophysiological derangements (Hess,
1975) as follows: (i) Alterations in regional cerebralblood flow (rCBF) due to damaged auto-regulation; the
damage can lead to either an excess of blood (hyper-
emia), or an under-supply of blood (ischemia), (ii)
disruption of extracellular metabolic balance, (iii)
disruption of thalamocortical neuronal circuits, and
(iv) changes in the balance of neurotransmitter levels,
synaptic receptors, or ion channels (O’Brien et al., 1997),
and the number of cells in specific neuronal populations
(Haglund et al., 1992).
An extensive literature review (Jueptner and Weiller,
1995) indicates that rCBF reflects local energy con-
sumption, which in turn reflects local neuronal activity.
Most glucose consumed by a neuron is used to maintain
membrane potentials and restore ion gradients, but
changes in rCBF can be used to monitor changes in
synaptic activity in a population of cells (Jueptner and
Weiller, 1995). Similarly, changes in metabolic balance
are associated with changes in neuronal activity, for
example during slow-wave sleep (Hofle et al., 1997).
Changes in neurotransmitter concentrations, or num-
bers of synapses and neurons, affect the gain parameters
in our model (Robinson et al., 2004), which would
therefore also change near the tumor. Thus, the
peritumoral tissue is active, but damaged. The degree
of damage to this tissue decreases with distance from thetumor (Hess, 1975).
A schematic of the damaged connectivity in the
presence of a thalamic tumor is shown in Fig. 3(a),
which shows a cross-section through one dimension of
the tumor. Note first that because the tumor itself is
electrically silent, and has no thalamocortical axons, it
does not project to the cortex (Scarff and Rahm, 1941).
The surrounding annulus of damaged tissue in the
thalamus projects to a circular region in the cortex, via
the connectivity shown in Fig. 1. The degree of damage
to this tissue decreases with distance from the tumor
(Hess, 1975); damage is indicated in Fig. 3(a) by brokenlines. We therefore model the cortical projections of the
damaged thalamic tissue as localized non-uniformities in
the normal parameters. We use 2D non-uniformities
with a Gaussian profile, which decreases with distance
from the center, as required by physiology. For a generic
model parameter, labeled P here, this gives in coordinate
space
P ðX ; Y Þ ¼ P 0 þ P 1 exp ðX X 1Þ2 ðY Y 1Þ2
2s2
;
(16)
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where P (X , Y ) takes a value P 0+P 1 at the tumor
centroid or ‘focus’ (X 1, Y 1), and the Gaussian has a
characteristic width of s: In the present work, we take
the background, or nominal value P 0 of each parameter
to be that for a waking, eyes-closed state; the parameters
have been extensively studied using the spatially uniform
model, and are consistent with anatomical and physio-
logical data (Robinson et al., 2004). Position in the two
dimensions is labeled with X and Y to avoid confusion
with the state parameters x, y, and z.In the following, we take as the linear dimension of
the brain l x=l y=0.8 m, the approximate circumference
of the brain, which was determined by scaling the head
circumference to account for cortical convolutions
(Nunez, 1981). We take the front of the head to
correspond to Y ¼ 0 m; and the back to Y ¼ 0:4 m: If
we consider a thalamic tumor that affects an area of
cortex with width s; the fraction of the cortex affected is
of order pð2s=l xÞ2; where we assume most of the affected
area lies within 2s of the tumor center. For s ¼ 0:03 m;the tumor affects less than 2% of the cortex, and for
s ¼ 0:05 m; the tumor affects around 5% of the cortex.
Thalamic tumors are particularly well suited to being
modeled in this way, since the tumor itself merely
perturbs cortical activity through distorted thalamocor-
tical projections, as seen in Fig. 3(a). Figs. 3(c) and (d)
represent schematic top views of the cortex, for thalamic
and cortical tumors, respectively. Panel (c) shows that
corticocortical fibers remain intact, albeit damaged, in
the presence of a thalamic tumor; the tumor itself does
not project to the cortex and we need therefore model
only the surrounding, damaged tissue. Cortical tumors,on the other hand, represent a region of complete
electrical silence in the cortex itself, as shown by the
black region in Fig. 3(d). The long-range corticocortical
fibers are disrupted, making the modeling of cortical
tumors significantly more complicated than modeling
subcortical ones. We therefore restrict our analysis to
thalamic tumors in the present work. The mappings
shown in Fig. 3 are deduced from experimental findings:
in tumors which invade the cortex, a zone of absolute
electrical silence in the region invaded by the tumor has
been observed (Hess, 1975; Hirsch et al., 1966). This
region is surrounded by an annulus in which normal
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side view top view
cortex
thalamus
damaged
tissue
damaged
tissue
tumor
projection of
damaged tissue
cortex
cortex
thalamus
tumor
cortex
tumor
damaged tissue
(a) (c)
(b) (d)
Fig. 3. Schematic diagram of the corticothalamic and corticocortical connections in the region near a tumor. Cross-sections through the 2D tumor
are shown from a side view (first column) and a top view (second column). Damaged connections are indicated by broken lines, or by a shaded
region; undamaged connections are indicated by unbroken lines, or by a white region. Panel (a) represents a thalamic tumor, in which the tumor itself
does not project to the cortex; intracortical connections remain intact. Panel (b) represents a cortical tumor; intracortical connections are damaged or
absent, and the tumor is more difficult to model. Panel (c) represents a top view of the cortex in the case of a thalamic tumor; a circle of cortical tissue
is damaged. Panel (d) represents a top view of the cortex in the case of a cortical tumor; corticocortical connections are absent at the tumor (black
region) and damaged around it (shaded region).
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electrical activity is disrupted (Hess, 1975; Hirsch et al.,
1966), due to damaged brain tissue adjacent to the
tumor (Scarff and Rahm, 1941). These features are seen
in Fig. 3(d). Significantly, the inner zone of absolute
electrical silence is not observed in subcortical tumors
(Hess, 1975; Hirsch et al., 1966), in accord with
the schematic in Fig. 3(c) and the mapping shown inFig. 3(a).
4. Involvement of the cortex and thalamus
In this section, we show how the pathological
observations typical of EEGs in the presence of a
tumor, outlined in Section 1, can be reproduced by our
model, as can normal EEGs in the presence of a tumor.
We concentrate on waking EEGs, for which the most
amount of data are available. The nominal model
parameters are shown in Table 1, and were found byfitting the model to data recorded from 98 normal
subjects in the waking, eyes-closed state (O’Connor and
Robinson, 2004c). We model a tumor by introducing
small parameter non-uniformities with Gaussian pro-
files, of the form (16). Initially, we study the reduced
three-dimensional xyz parameter space, which provides
a means of easily visualizing the range of effects. We
also study the effects of varying the tumor size.
4.1. Purely cortical effects
Fig. 4 shows the effect on the spectrum of introducingGaussian non-uniformities into the state parameters x,
y, and z, given by Eqs. (13)–(15), respectively. Fig. 4(a)
shows the variation in x across the head, representing
purely cortical non-uniformities. Note that we are not
modeling cortical tumors, for reasons discussed in
Section 3, but rather cortical effects of thalamic tumors;
for example possible changes in the local base cortical
activity and hence gains, due to damaged thalamocor-
tical afferents.
The tumor is two-dimensional, and a cross-section
through its center along the line Y =0.2 m is shown in
Fig. 4. The background level of x 0:75 is typical of the
waking, eyes-closed state (O’Connor and Robinson,
2004c; Robinson et al., 2004). The Gaussian patch non-uniformity in x is obtained here by introducing a patch
non-uniformity into the cortical gain parameter G ee; it is
centered at (X , Y )=(0.2, 0.2) m and has a width of s ¼
0:03 m: Recall from Section 3 that such a tumor affects
less than 2% of the cortical area. The region of the brain
surrounding the tumor is shown, from X =0–0.4 m,
representing half of the brain, which extends to
X =0.8 m, as discussed in Section 3. The state parameter
x is always positive, from (13), so it falls to its minimum
value at (0.2, 0.2) m in Fig. 4(a). All parameters other
than G ee are held uniform at their nominal eyes-closed
value, shown in Table 1.
Fig. 4(b) shows two spectra predicted from the model
for the non-uniformity in panel (a): the solid line
represents the local spectrum at the tumor focus, from
(10), and the dotted line represents the mean spectrum
across the head, from (11). We see that the patch
reduction in x gives a slight focal reduction in power at
all frequencies. The reduction is not evident at most
other sites, as seen by inspecting Fig. 4(c), which shows
the cortical power as a function of both frequency and
position. Again, for ease of illustration, only a 1D slice
through the 2D spectrum is shown, through the tumor
center. The dashed lines are at the tumor focus plus and
minus one characteristic width s of the Gaussian.Note that introducing a patch non-uniformity with
positive amplitude into x, scales the whole spectrum to
higher power near the patch focus, opposite to the
behavior in Fig. 4(a). However, thalamic tumors have
been associated with a hypometabolic cortical region
adjacent to the tumor, even if the tumor did not directly
involve the cortex (Newmark et al., 1983). Hence, an
increase in the state parameter x is an unlikely correlate
of thalamic tumors, and is not investigated in full here.
4.2. Purely thalamic effects
Let us turn our attention to variations in the purely
thalamic state parameter z, given by Eq. (15). Again, we
show our results over X =0–0.4 m, the part of the brain
near the tumor focus (X , Y )=(0.2, 0.2) m. In the present
work, the ratio b=a ¼ 4 is held constant, and z depends
only on the intrathalamic gain G srs. Non-uniformities in
z therefore correspond to non-uniformities in this gain.
A Gaussian patch non-uniformity was introduced into
G srs, and hence z, and is represented in Fig. 4(d); all
other parameters are at their nominal eyes-closed values,
shown in Table 1. The background value of z 0:1 is
typical of the waking, eyes-closed state (O’Connor and
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Table 1
Nominal model parameters for the waking eyes-closed state
Parameter Value Unit
G ee 7.5 — G ei 9.1 —
G ese 6.1 —
G esre 3.8 —
G srs 0.6 —
G sn 1.1 —
re 0.08 m
ge 180 s1
a 80 s1
t0 0.085 s
These mean values were obtained by fitting the model to data recorded
from 98 normal awake subjects with closed eyes (O’Connor and
Robinson, 2004c), and are consistent with physiology and anatomy
(Robinson et al., 2004).
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Robinson, 2004c; Robinson et al., 2004). Similarly to
panel (a), the Gaussian has a centroid at (X , Y )=(0.2,
0.2) m and a width of 0.03 m, and a one-dimensional
cross-section through the tumor center is shown. The
quantity z is always positive, from (15), and so the
minimum of z ¼ 0 at the tumor center is the minimum
possible value for z. A one-dimensional cross-section of
the local predicted spectrum at (0.2, 0.2) m is represented
by the solid line in Fig. 4(e), and the mean spectrum is
represented by the dotted line in the same panel. We see
that there is little effect on the spectrum as a result of the
patch reduction in z, which is already small for waking,
eyes-closed states. Similarly, the grayscale plot in Fig.
4(f) shows little effect.
If the sign of the Gaussian is reversed, and z reaches a
maximum at (X , Y )=(0.2, 0.2) m, then at the tumor
focus the alpha peak diminishes and the beta peak
grows, compared to the mean spectrum. This effect is
slight for small variations in z.
4.3. Corticothalamic effects
Consider now a non-uniformity in the corticothalamic
state parameter y, given by (14). Fig. 4(g) represents a
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(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
(j) (k) (l)
Fig. 4. Effects on the spectrum of localized Gaussian non-uniformities in the state parameters x, y, and z, centered at (X , Y )=(0.2, 0.2) m. The first
column shows a 1D cross-section at Y =0.2 m of the variation of the parameter across the head in the region X =0–0.4 m near the tumor; the second
column shows the spectrum; and the third column shows the power as a function of both frequency and distance X . The dotted line in each panel of
the second column is the mean spectrum, and the solid line is the local spectrum at the tumor center. The grayscale in the final column is logarithmic,
with contours separated by a factor of 1.5, and light shades corresponding to high power; the dashed lines represent one characteristic width s of the
Gaussian from the tumor focus. Panels (a)–(c) correspond to purely cortical non-uniformities in x; panels (d)–(f) correspond to purely thalamic non-
uniformities in z; panels (g)–(i) correspond to a non-uniformity in the corticothalamic state parameter y; panels (j)–(1) correspond to a non-
uniformity in y with larger width (s ¼ 0:05m).
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one-dimensional cross-section through Y =0.2 m of the
variation of y over the head where, as above, the patch is
a Gaussian localized at (X , Y )=(0.2, 0.2) m, with a
width of 0.03m, and the results are shown over
X =0–0.4 m. The background value of y 0:15 is typical
of the waking, eyes-closed state (O’Connor and Robin-
son, 2004c; Robinson et al., 2004). We reduce y to avalue of 1:1 at the patch center in Fig. 4(g), by
reducing G ese and increasing |G esre|; these parameters are
studied individually later in this subsection. Fig. 4(h)
shows the local spectrum at the tumor focus (solid line),
and the mean spectrum across the head (dotted line) for
the non-uniformity shown in panel (g). The local
spectrum differs in several important ways from the
mean spectrum: the background activity is attenuated at
low frequencies, and at alpha and beta frequencies; there
is an additional peak in the theta range near 3 Hz (often
called the delta range); and, the alpha peak is shifted to a
higher frequency. These effects are quite localized, as
seen from the grayscale plot in Fig. 4(i).
We now increase the size of the tumor, as seen in Fig.
4(j), where that the affected cortical area has a width of
0.05 m, which affects around 5% of the cortex. Fig. 4(h)
shows that there is a peak near 12 Hz, which lies in the
spindle frequency range. The amplitude of the theta
peak near 3 Hz also increases as the size of the tumor
increases. Figs. 4(k) and (l) show the large peak near
3 Hz, the attenuation of background frequencies, and
the peak in the spindle range, near 12 Hz. With reference
to the discussion in Section 2.1, we note that a localized
change in the sign of y would take that local portion of
the brain into a region of the stability zone in Fig. 2which is normally associated with sleep states; hence the
appearance of the typical sleep peaks near 3 and 12 Hz.
We explore the possibility that the 12Hz rhythm is
‘fast alpha’, rather than a spindle, in conjunction with
Fig. 6 below. This is also discussed thoroughly in
Section 6.
Consider first the effects on the spectrum of changing
the sign of the Gaussian patch, so that y takes amaximum at the patch focus. From Fig. 2, we see that
increasing y takes the brain nearer both the alpha and
slow-wave stability boundaries; indeed, in such a case we
see a corresponding increase in the power at alpha and
low frequencies. If the increase in y takes the brain
outside the stability boundaries in Fig. 2 then the brain
becomes unstable at those frequencies. This mechanism
could account for the epileptiform discharges sometimes
observed in the EEGs of subjects with a thalamic tumor
(Cheek and Taveras, 1966; Hirose et al., 1975; Janati
and Hester, 1986; Robinson et al., 2002), and is
discussed further in Section 6.
From (14) the sign of y depends only on the relative
magnitudes of G ese and G esre, since G srs and G ei are
always negative. We now examine the effects of G ese and
G esre independently. Figs. 5(a) and (c) show the local
(solid line) and mean (dotted line) spectra when
Gaussian non-uniformities are introduced into the gains
G ese and G esre, respectively, and the relevant gain falls to
zero at the tumor focus [(X , Y )=(0.2, 0.2) m, here].
Panel (e) shows the local and mean spectra when the
gain G esre is increased at the tumor center, reaching three
times its nominal value in this case. The bottom row of
Fig. 5 shows grayscale plots, where light shades
correspond to high power, representing a one-dimen-sional cross-section through the tumor center. Panels (a)
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(a)
(b)
(c) (e)
(d) (f)
Fig. 5. Effects on the spectrum of introducing a Gaussian non-uniformity into the corticothalamic gains. The dotted line in each of panels (a), (c),
and (e) represents the mean spectrum, and the solid line represents the local spectrum at the tumor focus [( X , Y )=(0.2, 0.2) m, here] in the region
X =0–0.4 m near the tumor. Panels (b), (d) and (f) are grayscale plots, with contours separated by a factor of 1.4, where light shades correspond to
high power; the dashed lines represent one characteristic width s from the tumor focus. The first column corresponds to a non-uniformity in which
the direct loop gain G ese falls to zero at the tumor center; the second column corresponds to a non-uniformity in which the indirect loop gain G esrefalls to zero; and the third column corresponds to a non-uniformity in which G esre reaches three times its nominal value at the tumor center.
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and (b) show that the reduction in G ese and hence y, and
consequently increased distance from the slow-wave and
alpha stability boundaries indicated in Fig. 2, produces
a focal attenuation of background activity, similar to
that in Figs. 4(h) and (k). In panels (c) and (d), we see
the opposite effect: the reduction in |G esre| and hence
increase in y, and consequently increased proximity tothe slow-wave and alpha stability boundaries indicated
in Fig. 2, produces a focal increase of background
rhythms. Furthermore, there is a slight downward shift
of the alpha peak frequency in panel (c), or ‘background
slowing’, due to disruption of the propagation delay in
the corticothalamic pathway. In panels (e) and (f), there
is a focal theta (E3 Hz) peak, and a focal shift in the
alpha peak towards the spindle frequency. We thus see
that reducing G ese attenuates the background peaks by
flattening the spectrum, and increasing |G esre| introduces
the abnormal theta peak. The latter effect is due to the
sign change of the state parameter y and consequent
localized over-inhibited state.
The spectral effects of non-uniformities in the
parameters G ei and G srs are more complicated, since
they affect more than one of the state parameters x, y,
and z. In general, decreasing |G ei | increases both x and y,
and hence gives higher power at spectral peaks.
Decreasing |G srs| increases y but decreases z, from Eqs.
(14) and (15), so its effect depends on the location in the
state space in Fig. 2. Fig. 6(a) shows one example of
varying G srs. It represents the spectral dependence on
position on the head, where at the tumor center [(X ,
Y )=(0.2, 0.2) m, here] G ese falls to zero, and |G esre| and
|G srs| increase to four times and two times their nominalvalue, respectively, and the Gaussian has s ¼ 0:04 m:Power is shown over a one-dimensional slice of the
cortex, in the region X =0–0.4 m near the tumor focus.
As observed clinically, we see focal attenuation of the
normal waking eyes-closed background rhythms: the
slow-wave rhythm, below 2 Hz, and the alpha rhythm,
at 9.5 Hz. These are most attenuated at the site of the
tumor. We also see the focal theta peak near 3 Hz, and
the focal peak at the spindle frequency of 11–12 Hz
which is enhanced by the increase in |G srs| and hence z.
Let us consider now whether the spindle-frequency
peak shown in Figs. 4 and 6(a) is a true spindle, or an
alpha variant. The difference between the two lies in
their genesis: alpha in our model is generated as a
resonance in the corticothalamic feedback loop, and its
frequency is primarily determined by the time taken to
complete one loop, with only a minor influence from
synaptodendritic kinetics. Sleep spindles, on the other
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(a)
(b)
Fig. 6. Contour plots of power as a function of (cross-sectional 1D) position in the region X =0–0.4 m near the tumor, and frequency. The gray scale
is logarithmic and common to both plots, with contours separated by a factor of 1.2. Light shades correspond to high power. The dashed horizontal
lines are at one characteristic width s from the tumor focus. Gaussian non-uniformities are introduced into the parameters: in panel (a), G ese is zero at
(0.2, 0.2) m, G esre and G srs are increased to four and two times their nominal value, respectively. In panel (b), G ese is zero at (0.2, 0.2) m, G esre and G srsare increased to four and eight times their nominal value, respectively, and a and b fall to 60% of their nominal value at (0.2, 0.2)m.
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hand, are generated in the intrathalamic loop shown in
Fig. 1, and their frequency is determined largely by
synaptodendritic kinetics, according to o ffiffiffiffiffiffiab
p (Ro-
binson et al., 2002). The values of b and a; the inverse
rise and decay times of the dendritic potential, were
uniform in all the figures shown thus far; we can
calculate a spindle frequency from their values in Table1, with b ¼ 4a; of f E25 Hz. This indicates that the peak
near 12 Hz in Fig. 6(a) is a fast alpha peak, rather than a
true spindle. A true spindle can, however, be generated
in a waking state if the intrathalamic loop gain G srs is
significantly increased, and a; and hence b; are reduced
at the site of the tumor. Such a reduction in a and b is
consistent with the dominance of GABAB during sleep,
which has slower kinetics than GABAA (Robinson et
al., 2004). Fig. 6(b) shows a contour plot with the same
non-uniformities as in panel (a), except a is reduced to
50 at the site of the tumor and G srs is increased to eight
times its nominal value, or 4.8. The increase in G srsincreases the state parameter z, from (15), and decreases
the magnitude of y, from (14), although y is still
negative. The decrease in | y| reduces the power at the
theta peak, but the greatly increased z takes the brain
near the spindle instability boundary. If G srs, and hence
z, is increased still further, the brain passes into an
instability at f ffiffiffiffiffiffiab
p =ð2pÞ 15Hz: For the case
shown, the brain is not yet at the instability boundary
and the frequency is slightly lower than 15 Hz.
We see therefore that both fast alpha and spindle
peaks can be generated in the spindle-frequency range,
both of which might be classed as spindles in an
experimental study. Furthermore, the theta peak (oftenknown as a delta peak) which typifies tumor EEGs, can
be reproduced using our model by changing the sign of
the corticothalamic state parameter y. These are
discussed further in Section 6.
5. Tumor localization
In this section we model a tumor in a brain in which
realistic underlying, normal front-to-back (anteriopos-
terior) non-uniformities are also present; these non-
uniformities were deduced by comparison of the modelwith data from 98 normal subjects in an earlier study
(O’Connor and Robinson, 2004c). The present work has
implications for the usefulness of EEG studies in
determining the location of a thalamic tumor.
A previous study fitted the spatially non-uniform
model to data from 98 normal awake subjects with
closed eyes (O’Connor and Robinson, 2004c). The
parameter non-uniformities deduced indicate that aver-
age cortical gains decrease, and average thalamic gains
increase, towards the back of the head. Sinusoidal
variations about the mean values in Table 1 were found
to fit the observations well. Furthermore, the cortical
state parameter x was found to decrease towards the
back of the head, whereas the state parameters y and z
both increased (O’Connor and Robinson, 2004c). These
deduced parameter non-uniformities are consistent with
physiological and anatomical data (Robinson et al.,
2004).
In the present work, these anterioposterior (front-to-back) parameter non-uniformities were introduced into
the two-dimensional model, while parameters in the
mediolateral direction (side-to-side across the head)
were kept constant. Figs. 7(a) and (b) show the power
across the head in the presence of these normal non-
uniformities, at 3 Hz (theta) and at the alpha frequency,
respectively; there is little spatial variation at 3 Hz
compared to at the alpha frequency. Power is shown
versus position in the anterioposterior direction Y , and
position in the mediolateral (left-to-right) direction X
where the ranges X , Y =0–0.4 m are plotted. Each point
in panel (a) represents the power at 3 Hz, with the
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(a) (c)
(b) (d)
Fig. 7. Contour plots of power as a function of distance in both the
anterioposterior direction Y (front-to-back) and the mediolateral
direction X (side-to-side) in the region X , Y =0–0.4 m near the tumor.
The first column shows the distribution of power due to the underlyingnormal uniformities, in the absence of a tumor; the model parameters
vary sinusoidally about their nominal values in Table 1, in a manner
deduced from fitting the model to data from 98 normal subjects
(O’Connor and Robinson, 2004c). The second column show the
distribution of power when both the normal underlying non-
uniformities and a tumor at (X , Y )=(0.2, 0.2) m are present. The
position (X , Y )=(0.2, 0.2) is indicated by a cross. Panels (a) and (c)
show the distribution of power at 3 Hz; panels (b) and (d) show the
distribution of power at the alpha frequency, which was calculated for
each point, and was found to vary from 9 to 10 Hz. The gray scale is
logarithmic, with light shades corresponding to high power. Contours
are separated by a factor of 1.03, which corresponds to overall factors
from lightest to darkest of E1.6 and E2.2 in panels (c) and (d), for
example.
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frequency fixed since there is no peak near this
frequency for the normal eyes-closed state. There is a
definite alpha peak, on the other hand, the frequency of
which varies across the head (Markand, 1990; O’Connor
and Robinson, 2004c). To avoid introducing artifact
related to the differing alpha frequency across the head,
we show the power at the alpha peak frequency at eachpoint in Fig. 7(b), which was found to vary from E9 Hz
at the front of the head to E10 Hz at the back of the
head, in agreement with previous studies (Markand,
1990; O’Connor and Robinson, 2004c; Robinson et al.,
2003). Note that the plots at a single frequency do not
differ significantly from those at the (variable) peak
frequency, provided the width in frequency of the
spectral peak is larger than the change in peak frequency
across the head. However, for very narrow peaks, the
two-dimensional spatial plot at a single frequency can be
quite misleading if the peak moves from above to below
the chosen frequency as the scalp is traversed; in this
case it is more informative to plot the power at the
(variable) peak frequency instead.
A two-dimensional thalamic tumor with s ¼ 0:03 m
whose center maps to (X , Y )=(0.2, 0.2) m, where local
yo0, was then superimposed on the normal background
non-uniformities. The results are shown in Figs. 7(c) and
(d), at 3 Hz, and the (differing) alpha frequency,
respectively. Fig. 7(c) shows that a theta peak is
introduced into the waking spectrum by the over-
inhibited state of the peritumoral region, as discussed
in Section 6. This theta peak is not centered above the
patch focus at (0.2, 0.2) m, indicated by the cross, but is
shifted slightly towards the back of the head by theunderlying ‘normal’ eyes-closed non-uniformities. Thus,
the EEG cannot necessarily be used directly to exactly
locate the tumor, where ‘direct’ localization assumes a
direct correspondence between the location of the peak
and the location of the underlying tumor. Fig. 7(d)
shows the spatial distribution of power at the alpha
frequency. There is a general increase towards the back
of the head, as for normal eyes-closed EEG; however,
the alpha peak is attenuated due to the tumor near the
center of the head. These competing effects mean that
near the tumor the peak alpha power is at the sides of
the head. Power at the alpha frequency would be of littledirect localizing value for the tumor, since the effects of
the tumor are hidden somewhat by the significant
underlying variation of power at the alpha frequency.
The EEG would best be used in conjunction with fitting
to the model predictions, thereby enabling the tumor to
be localized. Thus we see that, in agreement with clinical
observations (Hess, 1975), the EEG alone is often of
little localizing value, at least if used without the
deeper insights provided by modeling. Our results
indicate that activity at the theta frequency is of better
direct localizing value than activity at the alpha
frequency.
6. Discussion
We have successfully explained the range of EEG
features associated with thalamic tumors using a unified
framework. In a key advance, we have demonstrated via
modeling that the theta peak (sometimes known as a
delta peak) can be accounted for by an over-inhibitionof the peritumoral region of brain, due to a disruption of
signal propagation through the corticothalamic path-
ways. This finding not only explains theta-peak genera-
tion in the presence of thalamic tumors, but can likely be
generalized to explain theta generation in other brain
disorders too, since the present results are not specifi-
cally dependent on properties of the tumor, such as
pathophysiology or malignancy, but on non-uniformi-
ties in corticothalamic gains. Furthermore, we have used
the same model to explain other qualitative features of
thalamic tumor EEGs, including background peak
attenuation, background slowing of the alpha rhythm,
high-voltage bursts and the onset of unstable (probably
epileptiform) discharges, and spindle-frequency activity
during waking. The model is based closely on physiol-
ogy, and is thus able to provide insight into the
physiological mechanisms responsible for generating
these features. Note that each of the features can be
reproduced only by the parameter combinations dis-
cussed, enabling specific conclusions to be drawn from
these results. These conclusions and insights are
discussed in paragraphs (i)–(v) below. Much of the
attraction and power of the present approach lies in the
ability of the model to unify the above features into a
single framework. Furthermore, the same model hasbeen used successfully to explain a wide range of other
phenomena, and we can thus have confidence that the
model accurately represents activity in the brain.
(i) Background attenuation is the phenomenon where-
by at the focus of the tumor the usual waking eyes-
closed background peaks are diminished compared
to elsewhere on the cortex. These background peaks
are the slow-wave peak, which occurs below 1 Hz,
the alpha peak at around 7–12 Hz, and the beta
peak at around 15–25 Hz. In the context of the
model, these peaks are attenuated when either of the state parameters x or y is slightly reduced, since
this takes the part of the brain near the tumor
further from the slow-wave and alpha stability
boundaries. Hence, from the discussion in Section
2.1, there is less power at the slow-wave and alpha
frequencies in the region near the tumor. From Eqs.
(13) and (14), the parameters which most strongly
affect x and y are G ee or G ei , G ese, and G esre. The
corticothalamic gains G ese and G esre strongly affect
the amplitude of the background peaks, as seen in
Fig. 5, whereas the cortical gain has an effect which
is independent of frequency, as in Fig. 4(b). This
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finding is consistent with a previous result which
showed that focal attenuation of background
rhythms is correlated with involvement of the
thalamus (Newmark et al., 1983).
(ii) Background ‘slowing’ is the phenomenon whereby
the peaks of the background rhythms are shifted to
lower frequencies. Published tumor data rarelyindicate the degree of slowing, although the shift
is generally small (Scarff and Rahm, 1941) and
related studies indicate that it is of order 1 Hz
(Schaffler et al., 1982). An effect of this magnitude
can be seen in the case of a focal reduction in |G esre|,
as in Fig. 5(c). A similar effect can be found by
simply decreasing the corticothalamic loop propa-
gation time t0 by 10–20% at the tumor, since this
delay is the dominant factor in determining the
alpha frequency in the model. If the loop time is
decreased further, the background frequencies are
further reduced. Similarly, if there are tumor-
induced local reductions in a and b; which describe
the synaptodendritic dynamics, then the alpha peak
is similarly slightly slowed. Note that background
slowing is common in neurological diseases (Walc-
zak and Jayakar, 1997), and thus need not be due to
tumor-specific physiology, which is consistent with
the fact that it does not feature prominently in these
results, and is often not reported at all in published
work.
(iii) Epileptiform discharges and high-voltage bursts of
activity are reported in 10–30% of patients with a
thalamic tumor (Cheek and Taveras, 1966; Hirose
et al., 1975; Janati and Hester, 1986; O’Brien et al.,1997). As discussed in Section 2.1, high-voltage
activity occurs with proximity to the stability
boundaries indicated in Fig. 2. Similarly, epilepti-
form discharges likely occur when a stability
boundary is crossed, and the frequency of the
activity depends on the boundary. Thus, from Eqs.
(13) and (14), activity of this type occurs when G eseor G ee is increased, or when |G esre| o r |G ei | is
decreased; that is, when the excitatory activity is
increased, and the inhibitory activity is decreased.
Thus, when a thalamic tumor leads to a local over-
excitation of the thalamus, cortical excitability issimilarly upregulated, and high-voltage or epilepti-
form activity can ensue. Such an increase in
thalamic activity in the presence of a tumor is likely
related to ‘luxury perfusion’, or hyperemia, the
increase in peritumoral cerebral blood flow some-
times observed in the presence of a tumor (Hess,
1975). Indeed, tumor-induced epileptiform activity
is primarily due to low-grade tumors (O’Brien et al.,
1997), which are rarely associated with pathological
evidence of ischemia (O’Brien et al., 1997).
Furthermore, several studies have found that
cerebral tumors may induce local changes in
neurotransmitter levels, and that tumor-induced
epilepsy is associated with increased local concen-
trations of excitatory neurotransmitters, such as
glutamine (the precursor to glutamate), and de-
creased concentrations of inhibitory ones such as
gamma-aminobutyric acid, GABA (Haglund
et al., 1992; O’Brien et al., 1997; Recht and Glantz,1997).
(iv) The appearance of a theta peak near 3 Hz is a highly
typical signature of thalamic tumor. From Figs. 2
and 4, theta activity occurs for negative values of
the corticothalamic state parameter y, which
effectively puts this portion of the brain into an
over-inhibited state, in which the inhibitory effect of
the thalamic reticular neurons is stronger than the
excitatory effect of the thalamic relay neurons.
There is an increase in power near 3 Hz when G ese is
reduced; however, for the parameters near our
(physiologically realistic) nominal ones, this forms a
peak only when |G esre| increases simultaneously. On
the other hand, an increase in |G esre| alone is
sufficient to generate a theta peak. Physiologically,
both effects are likely to occur together, since
increased firing of the reticular neurons will inhibit
the relay neurons; a similar effect is observed during
sleep (Braun et al., 1997; Hofle et al., 1997; Maquet
et al., 1997). Thus, the present work indicates that
involvement of the corticothalamic feedback path-
ways is essential to produce pathological theta. This
result confirms previous studies which found that
rhythmic theta activity was related to thalamic
dysfunction (Daly et al., 1975; Hirose et al., 1975),rather than an alternative proposal that irregular
theta activity might be related to deafferentation of
the cortex (Gloor et al., 1977). Note that patholo-
gical theta is observed in a number of brain
disorders, such as attention-deficit hyperactivity
disorder (ADHD) which could also be investigated
using this model (Rowe and Robinson, 2004; Rowe
et al., 2004), and is likely caused by a similar
mechanism; for example, the stimulants which are
often used to treat ADHD may act by bringing the
affected patch of the brain out of its over-inhibited
state.(v) Peaks in the spindle frequency range, between the
alpha and beta frequencies, have been reported in
the presence of thalamic tumors (Hirose et al., 1975;
Janati and Hester, 1986), although the phenomenon
is rare. These peaks are often called ‘sleep spindles’
and are a typical feature of normal sleep, as is a
theta (or ‘delta’) enhancement (Steriade, 2000).
Indeed, the present work indicates that both a theta
peak and spindle-frequency activity can arise when
the tumor shifts the nearby brain to an over-
inhibited state, as discussed in the previous para-
graph. As discussed in conjunction with Fig. 6, the
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of the tumor is not considered, since a study of gliomas
reported that EEG findings are unrelated to the
malignancy of the tumor (Newmark et al., 1983).
However, malignancy is associated with tumor growth
rate, and we have shown that size is related to EEG
abnormality. Thus, we confirm that an EEG recording
that becomes increasingly abnormal over time could beindicative of tumor malignancy (Hess, 1975).
In all cases shown in this work, the tumor was located
at (0.2, 0.2) m. We found that varying the position of the
tumor across the head does not affect the outcome; that
is, in the absence of underlying parameter non-
uniformities, the strongest effect is always at the tumor
center, with decreasing effect as distance from the center
increases. However, as shown in Fig. 7, underlying
normal non-uniformities can shift the activity. Fig. 7
shows an example in which the variation across the head
of activity at 3 Hz can help locate the thalamic tumor;
note, however, that the strongest activity at this
frequency does not occur directly over the tumor, but
is typically slightly shifted towards the back of the head
due to the underlying normal front-to-back parameter
non-uniformities. Activity at the alpha frequency,
however, from Fig. 7(d), is of little localizing value, if
we assume a direct correspondence between EEG peak
location and tumor location. This is consistent with
previous findings that the EEG is of localizing value in
40–50% of patients with a tumor of the thalamus
(Cheek and Taveras, 1966; Hirose et al., 1975), with
some EEGs providing false localizing signs. The total
contrast in Fig. 7 indicates that the highest power is
approximately twice that of the lowest power at thefrequencies investigated, so the topographical variations
should be reasonably easily detected. We have not yet
investigated side-to-side (mediolateral) non-uniformities
in normal brains; such non-uniformities might also shift
the peaks from the tumor focus, and thus further
contribute to the false localizations. However, using our
model, tumors could be better localized by improving
the inversion process to account for the underlying
background parameter non-uniformities, as well as
fitting the model to determine the parameter non-
uniformities due to the tumor.
Tumors are electrically inactive (Scarff and Rahm,1941), and so they can be modeled by investigating their
influence on signal transmission through various path-
ways in the peritumoral tissue; the morphology of the
thalamic tumors themselves is not important. The model
can reproduce a wide range of EEG features associated
with thalamic tumors, and the present work indicates
that different EEG features are due primarily to
variations in the pathophysiology of the tumor, in
particular whether it results in an excess or deficit of
blood supply, and its size; it was confirmed that
increasing the size of the tumor led to greater
abnormalities in the observable EEG.
The results of the present study indicate that the
abnormal theta and spindle-frequency peaks near 3 and
12 Hz, respectively, are due to a local over-inhibition
resulting from increased activity of the inhibitory
reticular thalamic neurons in the peritumoral tissue.
This is likely accompanied by a decrease in the
excitability of thalamic relay neurons, and an overallreduction in thalamic blood flow near the tumor. Brain
tumors have been shown to induce changes in local
neurotransmitter balance and synaptic receptors: we
postulate that these tumors induce a reduction in the
responsiveness of the peritumoral neurons to cholinergic
input from the brainstem, and a corresponding depolar-
ization of thalamic reticular neurons, and hyperpolar-
ization of thalamic relay neurons, similar to the
mechanism active during sleep. That is, a small portion
of the brain is forced into an over-inhibited sleep-like
state due to the tumor. Involvement of the thalamus is
essential to generate these abnormal spectral peaks. This
is the opposite mechanism to that which produces high-
voltage and epileptiform activity, which we propose are
associated with a local over-excitation due to increased
activity of thalamic relay neurons and decreased activity
of reticular ones. This over-excitation propagates to the
cortex, and is likely related to a luxury perfusion of
blood, rather than a deficit, and may be related to
increased sensitivity to cholinergic inputs. This conclu-
sion is consistent with findings that epileptiform activity
is uncorrelated with theta activity in the presence of
tumors (Newmark et al., 1983; O’Brien et al., 1997).
Acknowledgements
The authors thank C.R. Rennie for helpful comments.
This work was supported by the Australian Research
Council, an Australian Postgraduate Award, and a
Westmead Millenium Foundation Stipend Enhance-
ment Award.
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