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140 Appendices
App 1.8.3 Races between Antagonistic Pairs
of LATER Units
A common situation is one of two competing LATER units, (μ
What is the proportion of wins by the
faster one as a function of the parameters?
In simpler cases, this can be determined
analytically, and we repeat here a simplified version of the analysis set out in
(Noorani and Carpenter 2016).
If we have (μ
), plotting the promptness of one
(μ
2,σ2
against the other over a number of trials
will generate a bivariate distribution
(Figure App 1.23(b)), with unit 1 winning
when the corresponding point lies to the
left of the line (λ
1
resultant ellipse to make it circular (Figure
App 1.23(c)).
, σ1) and (μ2, σ2).
1
) racing against
1,σ1
= λ2). We can scale the
Now unit 1 wins when the point lies to
the left of (σ
y = σ2x). The distance of this
1
line from the centre of the distribution is
Δμ
ffiffiffiffiffiffiffiffiffiffi
p
, where Δμ = μ
2
2
σ
þσ
1
2
Therefore, the required probability
Δμ
ffiffiffiffiffiffiffiffiffiffi
p
p = P
p ¼ P
2
2
σ
þσ
1
2
For the special case of σ
Δμ
ffiffi
p
; we can relate this to the
σ
2
μ
1
2
:
= σ2,
1
reciprobit plot by drawing the asymptotic
lines separately (Figure App 1.24): if we
put ξ =(Δμ / σ), then the vertical distance
between them is.
So, to find the hor izontal asymptote,
representing the required proportion of
successes for unit 1 in the presence of unit
2, draw a vertical line from μ
to intersect
2
the distribution for unit 1, and scale by a
Figure App 1.23 (a) Two competing units with different values of μ and σ. (b) Together, they generate an
elliptical bivariate distribution. (c) With suitable scaling, the distribution will be circular rather than elliptic. The
dashed line represents the division between trials where μ
Figure App 1.24 (a) To find the proportion of trials in which μ1wins, find the point (μ2, ξ): then the required
proportion is ξ/√2. Similarly (b), to find the proportion of trials in which μ
again be the required proportion.
wins and where μ2wins.
1
wins, find the point (μ2, ξ); ξ/√2 will
2
Appendices 141
factor 1/√2. Conversely, to find the hor izontal asymptote for unit 2 in the presence
of unit 1, draw a vertical line from μ
to
1
intersect the distribution for unit 2, and
scale by 1/√2.
App 1.8.4 Bayesian Races
The problem may also be approached
from a Bayesian perspective. Suppose we
have two hypotheses H
need not be mutually exclusive, and some
evidence E; let the probability of getting
result E on hypothesis H be p(E|H). Then
the likelihood of H given E is L(H|E), = kp
(E|H) where k is an arbitrary constant
(Fisher 1921), which then cancels out. So
the likelihood ratio is then L(H
E); the log of this quantity has been called
the weight of the evidence afforded by E
(W (H
pared with H
: E)) in favour of H1as com-
1/H2
(Good 1950, Good 1968,
2
Good 1975):
WðH
: EÞ¼log LðH1,H2jEÞ
1=H2
¼ log pð E jH
and H2, which
1
|E)/L(H2|
1
Þlog pðEjH2Þ,
1
(App 1.18)
with the advantage that the combined
weight of two pieces of evidence is simply
the sum of their individual weights.
App 1.8.5 Races between Many LATER Units
If we have a large set of possible hypotheses (for instance, the possible existence of
several stimulus objects), we do not want
to have to compare each with every other
one in pairs. Rise-to-threshold provides a
means of simultaneously comparing all
hypotheses and selecting the most likely
one, if each decision signal is taken as a
measure of the belief B
hypothesis H
|T)), and K is an arbitrary constant,
(L(H
j
, where Bjis equal to K + log
j
so that the log-odds log(p(H
any pair of hypotheses is given by (B
B
). Figure App 1.25 represents a scheme
j
of this kind; it has a set of hypotheses H
and a selection E composed of a selection
from m possible stimulus elements S
related by a matrix L
). Thus the log-likelihood for H
L(H
j|Si
given E is ΣI(log Lij), and the updated
posterior belief is B
in an associated
j
)/(HLj)) of
j
of the likelihoods
ij
0
= Bj+ Σi(log Lij)
j
j
–
j
,
i
j
Figure App 1.25 Schematic representation of parallel multiple implementation of the rise-to-threshold model.
A stimulus event E consists of the detection of a set of stimulus elements S
set of potential hypotheses H
weight of the evidence from E, is calculated from the learnt associations p(S
hypotheses, and used to update the corresponding belief function B
reaches a criterion level, at which point the appropriate response is initiated and all the Bjare reset.
concerning the existence of particular targets, of which there are n, the sum of the
j
, of which there are m. For each of a
i
) between stimulus elements and
i|Hj
. This process continues until one of the B
j
j
142 Appendices
If we are not interested in the odds for
individual pairs of hypotheses, but only in
the best overall hypothesis, then all we need
do is run a race. B
starts at some level
j
representing the prior likelihood, and then,
if updating occurs at constant intervals of
time, E will rise at a rate proportional to the
log of the likelihood ratio, until it reaches a
criterion level which may be taken to represent a sufficient degree of belief to permit
action. The dynamic properties of the
model can be taken care of simply by arranging for feedback such that the B
0
1 is equal to B
at time t.
j
at timet +
j
App 1.9 Learning
In Section 4.1 in Chapter 4, we saw that a
change in prior prob ability in the middle
of a run causes the reaction time on each
side to alter in much the same way as a
‘static’ prior probability that is constant
throughout the run. The time-course of
this alteration appears to be roughly exponential, taking about 70 trials finally to
settle down at the new equilibrium. It
was pointed out there that the phenomenon is better considered as a process of
forgetting, in that the contribution of old
evidence gradually contributes less new,
being discounted by what we called a
Lethean factor λ, whose value is typically
around 0.05. This is not, of course, how a
strict Bayesian model should behave: all
evidence, whether recent or not, should
count equally. It is, however, relatively
easy to model at the synaptic level, using
a Hebbian-like mechanism.
An alternative formulation (Peirce
1878) just needs two opposed neurons
coding for I(H) and I(~H), an event
E increasing the firing of each in proportion to I(E|H) and I(E|~H) respectively.
Synapses must therefore strengthen in
proportion to Peircean probability of
E with respect to H and ~H.
The distinction between ‘probability’
(for hypotheses) and ‘chance’ (for events)
unnecessary: both are talking about the
same kind of neural activity: to a neuron,
so an event is a hypothesis. It is also clear
that probability and causation are the
same kind of phenomenon. And so is
perception itself: the reconstruction of
the ‘real’ world is as much guesswork as
the extrapolation into future possible
worlds, and involves precisely similar
neural processes. Though we tend to compare the uncertainty of the future with the
certainty of the past, as every historian
knows, we guess the past just as much as
the future.
Probability is represented in two quite
different ways: strengths of synapses represent conditional probabilities or likelihoods, while firing frequency represents
degree of belief in a hypothesis. The patterns of activity of afferent synapses represent patterns of circumstantial support,
while sub-threshold depolarisation represents prior probability represented. Then
the Lethean factor λ (Chapter 4, Section
4.1) represents the speed with which synaptic strength can change.
App1.9.1 HebbianSynapsesas
Bayesian Computers
We have seen that in an uncertain world
decisions are all about probability, and
showed how sensory inf ormation is used
as evidence and determines the strength of
one’s belief in hypotheses about the outside world, and how to quantify this
process.
The final stage of this whole process is
something called confirmation, which is at
the heart of how neurons learn. So far, the
various conditional probabilities p(E|H)
have been presented as if they were given,
but of course they have to be learnt
through experience, being themselves
updated when the truth about whether
the prediction was actually correct is
finally revealed. This is exactly equivalent
to the use of parametric feedback after the
Appendices 143
Figure App 1.26 Highly simplified representation of Pavlovian conditioning. In the untrained animal there is
an intrinsic, hard-wired link by which food (the unconditional stimulus, or UCS) causes salivation (a). After
sufficient of food with a conditional stimulus (CS) such as a bell, the CS will trigger salivation even in the absence
of food (b). The inescapable conclusion is that there are now two chains of neurons, from the CS and UCS, and
that there must be at least one neuron (X) that is common to both (c). If this neuron shows Hebbian learning, its
connection from the CS will be strengthened (d).
Figure App 1.27 (a) Chains of neurons connecting CS and UCs to the response R. (b) Neuron X has a Hebbian
synapse ultimately driven by the CS. (c) If we identify U with a hypothesis H, and E with an observed event (the
conditional stimulus CS), then the strength of the synapse (p(E|U) represents likelihood and therefore embodies
Bayesian learning:
event in motor control and is quite an
enlightening way at looking at many kinds
of learning.
Take the simplest of all – Pavlovian
conditioning (Pavlov 1927), Figure
App 1.26. After the conditioning has been
learnt, the dog is in effect using the bell as
evidence E for the hypothesis U that food
is in the offing. And when it does finally
appear, the value of p(E|U), p(bell|food) is
increased. This updating or confirmation
is what psychologists mean by reinforcement, but it is obviously an example of a
Bayesian process as well.
Learning of this kind can be performed by Hebbian synapses (Hebb
1949), of which NMDA synapses are a
well-known example. When they get
stronger as a result of association between
pre-synaptic and post-synaptic activity,
this is equivalent to altering p(E|H), so
that next time the hypothesis is considered even more likely when the bell is
heard, Figure App 1.27. Things that tend
to happen together in the outside world
tend to get associated together in the
brain: ‘fire together, wire together’ We
are building in our brain a model of probability relationships in the outside world,
perpetually predicting what’s coming
next (Carpenter and Williams 1995,
Brodersen, Penny et al. 2008), Figure
App 1.28.
Can we work out what the rule for
Hebbian strengthening (and weakening)
must be if a synapse is to function as a
Bayesian element? If it acts linearly, its
strength S should be proportional to log
(p(U|C)). Its history consists of the
number of instances of U andC (n
UCS
occurring together, and the total number
of instances of C on its own (n
S = log(n
UC/nC
).
). Then
C
So how must S change in response to
C or UCS in order to generate this function? The answer is that after every occurrence of C the strength should decline by
log(1 + 1/n
), and if U occurs as well it
C
)
144 Appendices
Figure App 1.28 Thanks to Hebbian synapses,
the connections between neurons in the brain
come to correspond to neural connections
forming a model of the outside world.
should also increase by log (1 + 1/nUC).
Since log (1 + x)=x – x
2
/2 + x3/3, ... ,to
a first approximation the strength should
change by –1/n
and 1/nUC, respectively.
C
Unfortunately, this implies that the synapse needs to have a memory not only of
its own strength, but also of its tally of
UC and C events. This may sound unreasonable: what we want is an updating
rule that uses only the current S and the
fact of the event U&C or C. However, the
overall strength of the synapse might be
the result of two parameters representing
independently the history of C and the
history of UC. They could be the
numbers of two different kinds of membrane channel, for example, NMDA
versus AMPA: is it plausible that the total
excitation could be a log function of the
number of active channels? If S =log
) – log(nC), then the rule could be
(n
UCS
extremely simple: after U&C, the number
of each type of channel increases by one;
after C alone, n
increases by one. So,
C
one might predict two sets of channels,
one excitatory and increasing after conjunction, the other inhibitory and
increasing after presynaptic activity only,
though this is not in fact how AMPA and
NMDA receptors behave. Physiologists
will recogn ise that the formula for S is
in effect the same as for the Nernst
potential (V = k(log(C
), where C
1/C2
1
and C2are the numbers of each of the
ions on each side of the membrane.
App 1.10 Information and Probability
App 1.10.1 Uncertainty as Lack
of Information
Another conceivable scenario regarding
the input of information regarding a
hypothesis is that some information is
provided about this hypothesis, but then
a contradictory signal is given that cancels
that previous message. Such conflicting
input can be encoded as illustrated in
Figure App 1.29.
App 1.10.2 Information in
Extended Displays
In Section 4.6 in Chapter 4, we looked at
various types of tasks in which the subject
is required to make a judgement about
whether in a field filled randomly with a
mixture of two or more individual categories of discrete stimuli (for instance,
red and green dots), there are more of
one of the categories than the other. In
the Type 1 version of this task, a proportion a of the total of N items are the same,
while the remainder are random (for
example, in an RDK experiment a dots
may move consistently to the right, while
the others execute a random walk). In a
Type 2 version, there are a items of one
kind (for example, moving to the right),
and the remainder (N – a) are of the other
kind, so that discrimination gets more
difficult as a approaches (N/2). A simple
Appendices 145
Figure App 1.29 (a) Initial probability for some hypothesis H. (b) A message is received telling us the value B of
p(H). (c) A second message is received, saying that the previous message was untrue. P reverts to its original (red),
but the total path length is increased.
example of a Type 1 task was addressed in
Reddi and Carpenter (2003). One needs to
bear in mind that estimating the information content of such displays is not entirely
straightforward and depends on certain
assumptions, in particular on how local
the estimation of direction of movement
for any one of the detector units is. Here we
assume it to be very local, as also have
Weiss and Adelson (Weiss and Adelson
1998, Reddi, Asrress et al. 2003); the topic
has been thoughtfully discussed by Barlow
and Tripathy (Barlow and Tripathy 1997).
We start with the two opposed
hypotheses, that at a particular moment
the majority of dots are moving to the
right (H
) or to the left (HL). Using
R
Bayes, the observation E of one dot
moving to the right will increase the log
likelihood ratio for H
versus HLby log
R
(C), where C is the likelihood ratio, (p(E|
)/(pE|HL).
H
R
In a Type 1 experiment, with N items,
the probability of a particular item
moving rightwards if H
is tr ue is (aN +
R
(N – aN)/2)/N,or(1+a)/2. Similarly, the
probability of an item moving leftwards if
is true is (1 – a)/2; so the likelihood
H
L
ratio for any one item will be C =(1+a)/
(1 – a): in many ways, C can be thought of
as a kind of velocity contrast. The support
for H
against HLfor the entire display
R
will be N log C, so in terms of LATER the
median rate of rise of the decision signal
be proportional to log (C). Then the
median reaction time will be
þ k=ðN log CÞ,
T
0
where k is an arbitrary constant that is
likely to vary from person to person, and
is the constant delay encapsulating all
T
0
those factors, such as conduction time,
synaptic delay, and the time needed to
activate muscle, that can be regarded as
constant for any particular task. The
larger a is, the shorter the reaction time
will be.
In a Type 2 experiment, the difference
is simply that the probability of observing
a parti cular dot moving rightwards if H
is true is a, but (1 – a)ifHLis true. So, C
is now given by a/(1 – a).
R
Appendix 2 Clinical
That along with excitation of discharges of
nervous arrangements in the cerebrum,
mental states occur, I, of course, admit; but
how this is I do not enquire; indeed, so far
as clinical medicin e is concerned, I do not
care.
J. Hughlings Jackson, Selected Writings
(1932)
Clinical neurology is currently a numberfree zone (Antoniades and Carpenter 2012,
Carpenter 2012). Compared with, say,
haematology or cardiology we lack quantitative measures of impairment of response
to treatment. One of the attractions of saccadometry – the measurement of saccadic
latency distributions – is the wealth of very
detailed quantitative information that it
can provide about the very highest levels
of cerebral function. Even if we cannot
explain why the distributions are as they
are, they provide an objective way of
seeing whether behaviour has significantly
improved, deteriorated, or remained
essentially the same. They may also suggest
the nature of the underlying dysfunction.
However, a snag with saccadometry is
that it is idiosyncratic. As we saw in
Figure 2.9, Chapter 2, although two or
three parameters are all we need to summarise the performance of a single subject, and in the absence of disease these
parameters remain stable over time, they
vary greatly from one person to another.
This has two consequences. First, a single
measurement in one patient is essentially
meaningless, unless it is grossly different
from the norm (or if there is left/right
difference). What is essential is to use
longitudinal studies, with one distribution
measured before the injury or treatment
and the other after; and preferably, of
146
course, more measurements in the
follow-up period. The second consequence is that comparisons of populations
tend to be weak, since the idiosyncrasies
greatly increase the variance of measures
across the population. However, to compensate, the equipment and protocols
lend themselves to international standardisation, effectively increasing the size of
the populations being compared
(Antoniades, Ettinger et al. 2013).
In this Appendix we skim briefly over
some examples of using saccadometry
more or less successfully in providing
quantitative measures of higher cerebral
function. Much more thorough accounts,
with wider scope, can be found in, for
example, Leigh and Kennard (2004), and
Leigh and Zee (2015).
App 2.1 Degenerative Conditions
App 2.1.1 Parkinson’sDisease
Michell, Xu et al. (2006), Perneczky,
Ghosh et al. (2011) – L-dopa was found
to increase saccadic latency of Parkinson’s
disease (PD) patients, and this could be
modelled by an increased threshold in the
LATER model. Higher μ (shorter latency)
was positively correlated with grey matter
volume of the prefrontal cortex and cerebellar vermis in PD patients (Perneczky,
Ghosh et al. 2011).
Antoniades, Xu et al. (2013) – Saccadic
latency and manual (hand) response
latency are both similarly increased in
PD compared with normal control subjects, reflecting a similar increase in μ.
App 2.1.2 Deep Brain Stimulation
Temel, Visser-Vandewalle et al. (2008) –
In PD patients with deep brain
Appendices 147
stimulators (DBS) inserted in the subthalamic nuclei, bilateral electrical stimulation led to an increased saccadic latency
that corresponded to an increased μ in the
LATER model. This implies that DBS
enhances the gain of the descending
basal ganglia pathways that initiate
saccades. It has subseque ntly been shown
that this improvement in response
times also applies to manual responses
implying a more general effect not limited
to saccades (Antoniades, Carpenter
et al. 2012). Note that insertion of the
stimulators themselves gives a transient
increase of saccadic latency prior to this
improvement (Antoni ades, Buttery et al.
2012).
App 2.1.3 Huntington’sDisease
Ali, Michell et al. (2006) – The use of
saccadometry in patients with
Huntington’s disease (HD) revealed these
patients have increased latency and more
early saccades compared with normal
subjects, and parameterising these differences was sensitive enough to diagnose
HD patients (Antoniades, Altham et al.
2007). Monitoring of HD patients over
three years from prio r to disease manifestation to established disease revealed a
clear progression of saccadic abnormalities, suggesting that studying these eye
movements may help track disease progression (Robert, Nachev et al. 2009,
Antoniades, Zheyu et al. 2010). It is
thought that there are two parallel paths
descending from frontal cortex to substantia nigra pars reticulata, which then
inhibits the colliculus: an indirect tonically inhibitory one via GPe and the subthalamus, and an excitatory one going
directly from the caudate and putamen
to SNpr; in HD the indirect pathway is
impaired, leading to an in crease in spontaneous, unsuppressed, movement
(Peitsch, Hoffman et al. 2008) and giving
longer saccadic latencies.
App 2.1.4 Progressive Supranuclear Palsy
Antoniades, Bak et al. (2007) – Analysing
saccadic latency distributions revealed
that progressive supranuclear palsy (PSP)
patients can be discriminated from
patients with other parkinsonian condi-
tions based on parameters of LATER
modelling of these distributions, suggest-
ing saccadometry may be a useful diag-
nostic tool here.
Ghosh, Carpenter et al. (2013) –
Saccadic abnormalities in PSP patients
progress at different rates compared with
other motor and cognitive deficits, likely
reflecting differential deterioration in their
underlying cortical– subcortical circuits.
App 2.1.5 Amyotrophic Lateral Sclerosis
Burrell,Carpenter et al. (2013) – Measuring
saccadic latencies in amyotrophic lateral
sclerosis (ALS) patients revealed a higher
number of early saccades compared with
normal subjects, but all other saccadic parameters were normal.
App 2.1.6 Dementia
Burrell, Hornberger et al. (2012) – The
use of saccadometry in patients with frontotemporal dementia (FTD) demonstrated an increase in early saccades and
also the saccadic latency, corresponding
with reduced μ , compared with normal
subjects. These deficits were correlated
with atrophy of the left frontal eye field
in these patients, as determined by brain
imaging. Other work has similarly showed
that FTD patients have impaired ability
to withhold an antisaccade (Meyniel,
Rivaud-Péchoux et al. 2005).
App 2.2 General Neurology
App 2.2.1 Anaesthetics
Nouraei, de Pennington et al. ( 2003) – In
human subjects being administered the
general anaesthetic sevofluorane, increasing doses of drug led to increased saccadic
148 Appendices
Figure App 2.1 Effect of different levels of sevofluorane sedation. (a) Higher sedation levels are associated with
longer latencies. (b) Dose-response relationships, average of 5 subjects. 1 MAC represents the minimum average
anaesthetic dose (Nouraei, de Pennington et al. 2003).
Figure App 2.2 Saccadic latency distributions on the unoperated (a) and operated (b) side before and after a
stroke brought on by carotid endarterectomy. Note the very large increase in median latency on both sides, and
the larger number of early response on the operated side both before and after the operation (Nouraei, Roos
et al. 2010).
latency as well as stop signal reaction
times in a countermanding task, suggesting saccadic measurements could be used
to estimate the cortical effects of general
anaesthetics (Khan, Taylor et al. 1999),
Figure App 2.1.
2010). In general, deleterious effects are
more marked on the operated side
(Nouraei, Roos et al. 2010) , but so are
the benefits from improved cerebral perfusion (Figure App 2.2). This is a good
example of left–right differences in
latency providing more useful informa-
App 2.2.2 Endarterectomy
Endarterectomy and other vascular sur-
tion than can be generated, for example,
by paper-and-pencil ‘cognitive’ tests.
gery affecting the blood supply to the
brain can often be followed by cerebral
deterioration or even stroke because of
blocking of small blood vessels by dislodged debris (Walsh, Nouraei et al.
App 2.2.3 Migraine
Measurements of saccades in patients
with migraine (although not having
migraine episodes at the time) showed
Appendices 149
Figure App 2.3 Reversible effect of mild traumatic brain injury. Latencies for a saccadic step task just before a
boxing match (red), immediately after (blue), and at various times thereafter. The subject showed signs of mild
concussion during the bout, reflected in the increased latency after it; but over 12 days’ time, the distribution
gradually returned to normal (Pearson, Armitage et al. 2007).
these patients have reduced variability in
reaction times compared with normal
subjects, pointing towards a functional
deficit in noradrenergic systems influencing the cerebral cortex (Chandna,
Chandrasekharan et al. 2012). Moreover,
patients with more severe migr aines produced more anti-saccade errors, implying
a deficit in inhibitory control processes.
test can be greatly enhanced by making
baseline measurements in advance, so that
responses can be tracked in the same
person, rather than relying on group statistics. Preliminary work has been encouraging in suggesting that while blows to the
head may create quite substantial shifts of
the distribution to longer latencies, these
appear to revert to normal over a matter
of days (Figure App 2.3) (Pearson,
App 2.2.4 Traumatic Brain Injury
Armitage et al. 2007).
Another potential field of application is in
evaluating concussion (mild traumatic
brain injury) – an area that has recently
become the focus of much public concern,
particularly in contact sports such as football, in horse-riding, and in military personnel (Putukian, Echemendia et al.
2000). In all these cases, the value of the
App 2.2.5 Hepatic Encephalopathy
Patients with liver cirrhosis develop hepatic encephalopathy, which is typically
assessed only subjectively and therefore
is difficult to quantify (Krismer, Roos
et al. 2010). In patients with hepatic
encephalopathy due to liver cirrhosis,
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