Neural modelling of the
McCollough Eﬀect in color vision
Giacomo Spigler (s1360784)
Master of Science
Cognitive Science (Neural Computation & Neuroinformatics)
School of Informatics
University of Edinburgh
The present work investigates the neural mechanisms underlying the McCollough Eﬀect
through the simulation of three diﬀerent models of color visual systems: a dichromatic
and a trichromatic one that are inspired to the anatomy of the primates’ visual system
and an idealised trichromatic model designed to aid the self-organization of a simulated
Primary Visual Cortex to a physiologically-plausible level.
The McCollough Eﬀect allows for a direct investigation of cortical plasticity and home-
ostatic adaptation, and can provide a reference for works in the cortical modelling of color
perception. Within this dissertation we compare our results to previous psychological
data on humans and to the relevant work in modelling the eﬀect, eﬀectively reproducing
its the main properties.
I would like to thank all my friends, with whom I shared so many wonderful experi-
ences, with a special thanks to Daniele and Simone, for the amazing time spent together,
and to Valerio, one of my oldest friends, for the biggest laughs in the last decade.
A big thanks goes to The University of Edinburgh, for the great environment it
provided, and to my amazing supervisor James Bednar for the great advices that he
could always provide. A thanks also goes to Chris Ball, for the time he spent discussing
and explaining his model and code.
The biggest thanks however is for my parents, who have always supported me and
gave me countless opportunities to travel, study and learn, and they taught me the value
of work and education.
Edinburgh, August 13th
I declare that this thesis was composed by myself, that the work contained herein is my
own except where explicitly stated otherwise in the text, and that this work has not been
submitted for any other degree or professional qualiﬁcation except as speciﬁed.
(Giacomo Spigler (s1360784))
Table of Contents
1 INTRODUCTION 1
1.1 Organization of the work . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 BACKGROUND 5
2.1 ColorVision.................................. 5
2.1.1 Retina................................. 6
2.1.2 Lateral Geniculate Nucleus . . . . . . . . . . . . . . . . . . . . . . 6
2.1.3 Cortical Processing . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 McColloughEﬀect .............................. 9
2.3 Models of the McCollough Eﬀect . . . . . . . . . . . . . . . . . . . . . . 11
2.4 RelatedWork ................................. 12
3 MATERIALS AND METHODS 15
3.1 RetinaandLGN ............................... 15
3.2 GCAL - Model of Cortical Processing . . . . . . . . . . . . . . . . . . . . 16
3.3 Modelsofcolorvision............................. 18
3.3.1 Dichromatic Vision . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3.2 Trichromatic Vision . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3.3 Idealised Trichromatic Vision . . . . . . . . . . . . . . . . . . . . 22
3.3.4 Training ................................ 22
3.4 McColloughEﬀect .............................. 24
3.4.1 Induction and Testing . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4.2 Decoding the Perceived color . . . . . . . . . . . . . . . . . . . . 26
4 EXPERIMENTS AND RESULTS 29
4.1 TheMcColloughEﬀect............................ 29
4.2 Orientation dependency of the McCollough Eﬀect . . . . . . . . . . . . . 32
4.3 Diﬀerent Orientation of the Inducing Patterns . . . . . . . . . . . . . . . 33
4.4 Role of Homeostasis and Lateral Inhibitory Connections . . . . . . . . . 35
5 DISCUSSION AND CONCLUSIONS 41
5.1 Summaryoftheresults............................ 41
5.1.1 Neural mechanisms underlying the McCollough Eﬀect . . . . . . . 42
5.2 Futurework.................................. 44
5.2.1 Relevance to Neuroscience . . . . . . . . . . . . . . . . . . . . . . 44
The McCollough Eﬀect is a special color aftereﬀect  that makes white, oriented grat-
ings appear to be colored [31, 16, 42, 40, 8]. The eﬀect is shown in Fig. 1.1 : before
induction, two orthogonal white gratings appear to be colorless; induction is then per-
formed by showing colored gratings in alternation (usually red and green, which are known
to produce the strongest eﬀect) for a total time of 5 to 15 minutes. After the adaptation
to the patterns, the white gratings appear to possess a slight shade of color, the com-
plementary to the one of the induction grating with the same orientation. Notably, the
eﬀect has been observed to last for days and even months, which greatly constrains the
possible underlying mechanisms [27, 38]. Another interesting aspect of the eﬀect is its
contingency to both orientation and color of the patterns. Diﬀerently oriented gratings
are still perceived as colored, but the eﬀect fades approximately linearly with the dif-
ference in the orientation of the patterns (with respect to the induction gratings), and
vanishes at around 45o.
The McCollough Eﬀect was ﬁrst studied in depth by Celeste McCollough, who used
special devices to allow subjects to cancel the perceived color by compensating with its
opponent. The amount of ”correction” was used to measure the (perceived) colorimetric
purity of the stimulus, which in turn was found to be a good measure for the strength of
the ME (McCollough Eﬀect) .
The special importance of the McCollough Eﬀect is due to two main reasons. First,
it requires the interaction of color and orientation processing mechanisms at some point
within the visual hierarchy, which means that we can learn how diﬀerent features interact
in visual cortices and how they are combined . Second, even though many simple color
aftereﬀects are known (e.g., adaptation to colored patches), they don’t last a long time.
Understanding the McCollough Eﬀect will thus probably uncover mechanisms that aﬀect
Figure 1.1: Example McCollough Eﬀect. colorless grids (left) are perceived as slightly colored
(right) after 5−15 minutes of adaptation with similar, colored patterns. The perceived color
is complementary to the one of the induction grid with the same orientation.
the long-term properties and behavior of the visual system.
However, many problems still persist. There is still no strong agreement on where
is the neural locus which gives rise to the eﬀect, and it is still not clear what are its
exact mechanisms. Within the scope of our work, we will investigate the diﬀerent roles
of homeostatic adaptation and cortical plasticity in order to understand the dynamic
adaptation of the Primary Visual Cortex to varying input statistics [45, 46]. To test
our work, we will examine several adaptations of previously proposed models, which are
characterized by diﬀerent degrees of anatomical and functional biological plausibility.
While we are grounding our research on previous, similar, work , we can now
make use of advanced tools that make the understanding of the system easier and more
1.1 Organization of the work
Within this dissertation we are ﬁrst going to discuss the context of the problem with a
broad review of the relevant background (Chapter 2). Then, we will present our models
and the experiments we performed, with special focus in particular on three diﬀerent
models of color vision, namely a dichromatic and a trichromatic vision models, which
are easy to handle and are anatomically inspired to the biological system, albeit with
many limitations, and a model of primate color vision which is designed to self-organize
in a manner that reproduces physiological data, although employing speciﬁc techniques
to ensure statistical balance in the feature dimensions (Chapters 3 and 4).
In the end, we are going to discuss our results, comparing them to available psycho-
logical data collected in the past decades (Chapter 5).
The relevant background to the present work can be broadly divided into three main
parts. The ﬁrst one deals with color vision in primates and mammals, the second tackles
our current understandying of the McCollough Eﬀect together with a complete summary
of the most popular models and theoretical explanations of the phenomenon, and the last
one discusses the relationship between our dissertation and previous, similar, work.
2.1 Color Vision
Most animals possess the capacity to perceive the spectral composition of light, thus
experiencing color in some way. Usually, the light that is emitted by sources or is reﬂected
by objects is composed by diﬀerent wavelengths, and its speciﬁc spectrum determines the
Diﬀerent animals have diﬀerent color vision capacities, with some species being able
to perceive broader spectra (e.g., in the ultraviolet band) or narrower ones. Within the
scope of this work we are speciﬁcally interested in mammal vision and in particular in
primates’. Most mammals are generally dichromats (i.e., their retinas are sensitive to
one of two wavelenth bands only, and can thus only perceive a limited color space) or
trichromats, like humans (i.e., the retina is sensitive to three wavelength bands, and
allows for perception of the full range of colors we are used to) [10, 26]. It is also known
that some individuals (more generally women, as the genes involved are located on the
Xchromosome) possess tetrachromatic vision .
Visual processing in primates starts with light ﬂowing into the eyes and onto the Retina
(see Fig. 2.1). There, it is detected by the photo-receptors, special sensory cells ﬁlled
with molecules that undergo a speciﬁc chain of chemical reactions when hit by photons.
Photo-receptors are divided into two types, rods and cones. Rods are broadly selective
to light of any wavelength in the visible spectrum, while cones have a narrow preference
for determined ones. Most mammals have either dichromatic or trichromatic vision.
In the case of trichromatic vision the wavelength selectivity of each cone is shaped as
shown in Fig. 2.2, with peak preference around 420 −440nm for the S”blue” cone
(Short wavelength), 534 −555nm for the M”green” one (Medium) and 564 −580nm
for the L”red” one (Long). Notably, despite the common naming of the three cones as
Red/Green/Blue, the Long and Medium wavelength cones share a strongly correlated,
The information that is passed through the optic nerve is the further processed re-
sult of computation by Retinal Ganglion Cells (RGCs). Such cells pool diﬀerent photo-
receptors and ﬁlter them through receptive ﬁelds that are either ON-Center or OFF-
Center (see Fig. 2.3): ON-Center cells are active when bright stimuli surrounded by
darker backgrounds are seen within their receptive ﬁeld, while OFF-Center cells are more
active when the surround of their receptive ﬁeld is brighter than the center. A similar
type of ON/OFF cells are found that encode single color-opponent information: instead
of being selective to contrast in luminance, they are selective to constrast in color (Long-
Medium, Medium-Long, Short-(Medium+Long) [23, 9, 26].
2.1.2 Lateral Geniculate Nucleus
The Optic Nerves cross at the Optic Chiasm and ﬁnally reach corresponding structures in
the opposite hemispheres, the Lateral Geniculate Nuclei (LGN). The response properties
of LGN neurons are similar to those of the corresponding RGCs, with the addition of
temporal lags and reﬁnement of the receptive ﬁelds (possibly to reduce redundancy) .
LGN neurons project to a number of other structures. Most of their connections target
the Primary Visual Cortex (V1) at diﬀerent layers, keeping separate pathways segrated
(magnocellular, parvocellular and koniocellular). Some of the LGN outputs reach higher
cortices directly, too, bypassing V1 (e.g., V2 and V3), as found with studies related to
The LGN of humans is composed of six layers, the ﬁrst two of which are part of the
Figure 2.1: On the left is a schematic of a primate’s eye: light comes in through the
cornea and the lens, and arrives onto the retina, which is shown on the right. Light is
detected by the photo-receptors (rods and cones) and information is transmitted over the op-
tic nerve. Image adapted from Scholarpedia (http://www.scholarpedia.org/article/Retina)
Figure 2.2: Normalized response of retinal cone receptors in primates to light at diﬀerent
wavelengths (x-axis): Red corresponds to Long-wavelength cones, Green to Medium and Blue
to Short. Image adapted from Wikipedia [http://en.wikipedia.org/wiki/Cone cell].
Magnocellular system (one for each eye) which is involved in the perception of movement
and depth, while the other four are part of the Parvocellular system (two for each eye)
Figure 2.3: ON-Center and OFF-Center (left) Retinal Ganglion Cells’ and LGN neurons’ re-
ceptive ﬁelds. Single color-opponent RGC’s and LGN’s receptive ﬁelds (center and right): ON-
and OFF- Center cells selective to Long vs Medium (Red-Green, and vice versa) opponency
and Short (Blue) vs both the other receptors.
which is tuned for color and ﬁne form perception. In between the M and P layers are
Koniocellular cells, which are part of the system that processes information from the
Short-wavelength ”blue” cones.
2.1.3 Cortical Processing
Within the scope of our work, we will only model color processing at the level of the
Primary Visual Cortex (V1). Further stages of processing, in particular V2 and V4,
which evidence suggests are involved in the perception of hue  will not be discussed.
We decide to follow this approach for three main reasons. First, we are interested in
investigating the role of early cortical processing in color vision and in producing the
McCollough Eﬀect; second, simpler models can provide more insight into the basics of
color processing in the brain, whereas it can be more diﬃcult to get clear answers out of
more complex and detailed ones; last, though the neural locus of the eﬀect is still debated,
there is good evidence that suggests that V1 could account for most of the eﬀect.
It is known that some neurons in V1 are selective to color, and evidence was found
to support the idea that color selective neurons closely match the organization of CO
(Cytochrome Oxidase) blobs in the Primary Visual Cortex , though the response
properties of such neurons are still not fully understood. Experiments found that most of
the non-opponent color neurons (i.e., neurons that were found to respond to stimulation
of a single type of cones, and not to the others) in V1 were also tuned to orientation
(around 60% of the total color selective neurons), while single- and double-opponent
neurons (i.e., neurons that receive inputs from two or more types of cones, and in the
case of double-opponent, that are also sensitive to spatial contrast) were generally not .
Other studies however found a stricter independence between such neurons’ responses to
color and orientation .
2.2 McCollough Eﬀect
The McCollough Eﬀect is a visual aftereﬀect that links the perception of color and form
(orientation) (See Fig. 1.1).
The eﬀect was ﬁrst studied by Celeste McCollough in 1965  but, despite the
numerous studies on the matter, we still don’t understand the phenomenon fully, nor we
know where is the exact neural locus where the eﬀect is triggered.
Many studies have been done exploring the McCollough Eﬀect and its properties. The
ﬁrst thing we should acknowledge is that it is long-lasting: induction times of 5 to 15
minutes can lead to a persistent eﬀect (which continuously dacays during normal vision,
but not while sleeping ) that can last from hours to months [27, 38]. Experiments
have shown that the eﬀect can be produced using any pair of colored gratings, but it
is stronger when the colors are complementary (red and green, in particular, have been
found to induce the strongest eﬀect). Induction is possible with a single colored grating,
too, in which case a form of indirect aftereﬀect can be observed in gratings that are
orthogonal to the induction pattern. The eﬀect is perceived the best when the spatial
frequency of the gratings is similar between the induction and test stimuli. Moreover,
induction patterns need not to be orthogonal, though diﬀerent angles reduce the strength
of the eﬀect, up to a complete cancellation with patterns rotated less than around 11o
. Also, diﬀerent kinds of McCollough Eﬀects have been found that link diﬀerent types
of visual modalities, like motion and color, spatial frequency and color , and others.
Another peculiar property of the eﬀect is its dependency on two separate properties
of visual perception, the orientation of lines and their color. Ellis  in particular inves-
tigated this aspect of the phenomenon, measuring the colorimetric purity (the amount
of complementary colored light to be added to a stimulus to make it be perceived as
white) of gratings at diﬀerent orientations before and after induction with ME patterns :
the results found that the strength of the eﬀect, measured as the diﬀerence in colorimet-
ric purity before and after induction, decreases linearly with the angle the gratings are
rotated by, and is null at around 45o. (see Fig. 2.4 for the plot).
Last, it is still highly debated where is the actual neural locus that is responsible for
the eﬀect. The most agreed candidate is the Primary Visual Cortex, for a number of
reasons. First, the eﬀect depends on the orientation of gratings in retinal coordinates, as
Figure 2.4: Orientation dependency of the McCollough Eﬀect: plot of the diﬀerence in
colorimetric purity of test gratings before and after induction with colored stimuli, computed
using test gratings at diﬀerent orientations. Figure adapted from Ellis, 1977 .
tilting a subject’s head results in a weaker perception of the eﬀect (after induction). Also,
V1 is the ﬁrst stage of visual processing in primates where orientation selective neurons
are found, which makes it the candidate lowest in the hierarchy that could account for
the eﬀect. Moreover, no intra-ocular transfer of the eﬀect was found [31, 18], while areas
higher than V1 are know to have a high degree of binocularity.
Recent arguments, however, have been casted in favour of higher areas, ﬁrst of all
V2 and V4 , and even suggested the involvement of feedback connections. It is also
been claimed that V1 is not directly linked to conscious perception , and could not
thus be the locus of ”read-out” of the eﬀect, but critics say that conscious perception
is not required at all, as it has been shown that the ME can be induced using patterns
alternating at up to 50Hz, too quick for their color diﬀerences to be perceived consciously
. Other evidence in support of V1 is that the McCollough Eﬀect has been shown to
respond to the wavelength of the inducing patterns rather than perceived color [42, 47],
which is characteristic of V1 rather than V2, where most color-selective neurons respond
to hue. Also, higher cortical lesions were found not to aﬀect the ME .
In conclusion, while everybody agrees that V1 is necessary to the eﬀect, not all agree
on it being suﬃcient. Within our and previous work  we found that models of V1
can indeed reproduce the eﬀect, though it is still possible that higher cortical areas might
have some eﬀect under diﬀerent conditions, and further investigation is necessary.
2.3 Models of the McCollough Eﬀect
Since the ﬁrst studies on the McCollough Eﬀect  many diﬀerent theoretical expla-
nations have been proposed. The ﬁrst idea, by Celeste McCollough herself, postulated
the existence of color sensitive edge neurons [31, 32] that would adapt to the inducing
patterns by means of homeostatic mechanisms. Any such theory that is based on home-
ostais requires the presence of units tuned to both orientation and color, which have been
found to exist, although in a relatively small number of color selective neurons. This
explanation was further suggested not to be suﬃcient to fully explain the phenomenon,
mostly because of its extremely long lasting behavior , which cannot be explained by
the short-term dynamics of homeostasis. It is noteworthy to say, however, that recent
studies found evidence that the eﬀect might be operating at two separate time-scales ,
and that thus the homeostatic mechanisms could account for part of the eﬀect.
Another popular theory takes into account a purely functional description of the phe-
nomenon, seeing it as the eﬀect of an underlying error-correcting system that keeps a
representation of the natural world and its long-term statistical correlations [14, 8]. Ori-
entation and color are generally uncorrelated in the world, but during induction of the
McCollough Eﬀect they become correlated, and the error-correcting mechanisms would
”correct” them. The main limitation of the model is that no neural details have been
included in the model, making it somehow impractical to evaluate experimentally.
The last big theory of the McCollough Eﬀect is based on the idea of classical condi-
tioning and Hebbian learning (associative learning) [1, 21]. The theory states that the
neocortex learns joint statistics of orientation and color, which can then be subjected to
homeostatic mechanisms or stronger mutual inhibition. The main problem of the model
is that the McCollough Eﬀect can only be generated using speciﬁc pairs of features, like
color and orientation or motion , and thus an approach based only on pure associative
learning would not ﬁt the eﬀect correctly, as it would require a suitable neural represen-
tation of the features before being able to learn associations between them.
To correct the major limitations of the associative model Barlow proposed direct
neural mechanisms that would implement such a behavior [3, 4, 8]. Within his model,
mutual inhibitory connections are learnt by anti-Hebbian learning to decorrelate visual
features, like color and orientation. During induction with colored patterns, then, the
mutual inhibition between, say, vertical and red, would increase, which in turn would
inhibit the population of red selective neurons when viewing colorless vertical grids, mak-
ing them appear more green. Notably a similar theory was tested successfully on the
Tilt-Aftereﬀect  and the McCollough Eﬀect itself . Barlow’s model is particularly
interesting as it could also explain the long time-course of the eﬀect and it doesn’t require
the existence of double-duty detectors.
Last, a number of neural network models were used to investigate the eﬀect, with
the most famous being the FACADE (Form-And-color-And-DEpth) general model of vi-
sion  and models that make use of ICA . However, FACADE explains the eﬀect
as part of a complex set of mechanisms for high-level visual processing, and within the
present work we are mostly interested in modelling low-level mechanisms. ICA-based
models, instead, are ruled out despite being good at modelling how color and form are
related in images because of their abstract relation to neural circuits, whose mechanistic
explanation is the focus of this thesis.
2.4 Related Work
Apart from the models and theories we discussed in section 2.3, the most relevant previous
work is the MSc thesis of Julien Ciroux .
Our work is extensively based on his, with a number of diﬀerences. First, we use
newer and state-of-the-art models of cortical function (GCAL/AL , as opposed to
the old one the author used (LISSOM ). Moreover, we work with models of color
vision that are more biologically realistic in either, depending on the speciﬁc model, the
functional anatomy of the system or its ﬁnal cortical organization (after learning). The
model used in that dissertation was related to previous works on color perception [6, 36]
which inspired the present work, too, but it only modelled dichromatic vision, as it is
particularly suitabile for modelling the McCollough Eﬀect, which is (generally) induced
using color pairs.
Within our simulations we used a diﬀerent dataset (Barcelona Natural Images )
that is more balanced than the Botanical dataset used in the previous work. For mea-
suring the strength of the McCollough Eﬀect we used a more advanced and complete
metric, too, which is necessary in order to measure this quantity in a trichromatic color
space (i.e., dichromatic color spaces are one-dimensional, and measuring the colorfulness
of a stimulus corresponds to computing a value on this line; trichromatic spaces, on the
contrary, are more complex and diﬀerent schema have to be designed in order to compute
a single value for the strength.)
Finally, within our work we expand the original proposition and scientiﬁc question
about the involvement of lateral inhibitory connections in the phenomenon to a more
complete evaluation including homeostatic mechanisms, which where not included in
the previous cortical model LISSOM . As we discussed, the McCollough Eﬀect has
been found to operate on two separate timescales , and is thus likely that many
mechanisms are involved. We investigate the role of homeostatic mechanisms in the
short-term dynamics of the eﬀect and the role of synaptic learning in lateral inhibitory
connections for the longer-term ones.
In any case, we will always point to the important similarities and diﬀerences between
this and any previous work within the next chapters.
MATERIALS AND METHODS
The core of our project is based on the simulation of three models of color-sensitive visual
systems. The underlying implementation makes use of state-of-the-art models of corti-
cal organization. The speciﬁc models have been implemented using the Tropographica
software  and Python scripts.
3.1 Retina and LGN
In order to describe the dynamics of Retinal Ganglion Cells (RGCs) and LGN neurons
we ﬁrst take input images from a color images dataset. Unless otherwise speciﬁed, the
dataset we used for our simulations is the Barcelona Natural Images dataset . Images
were either converted to the LMS format for greater match with the biological cone-
receptors or, for most of our simulations, they were kept in RGB format. Despite being
less biological realistic, RGB cones have the advantage of being more decorrelated than
LMS (where indeed the Long and Medium receptors have almost identical responses).
This is motivated by the fact that strong decorrelation mechanisms exist at subcortical
levels of color processing that further diﬀerentiate such cones’ responses, and thus using
RGB we can avoid modelling such speciﬁc mechanisms, still providing a good approximate
behavior as it would look like at the Primary Visual Cortex.
We further model the LGN as a set of populations of neurons, according to their tuning
preferences (e.g., ON- and OFF- Center color opponent units). To compute the output of
each unit, each aﬀerent (speciﬁc to the type of unit) is convolved with a Gaussian kernel,
and the weighted sum over all the aﬀerents is computed (the standard deviation of the
”surround” receptive ﬁeld was set to be four times the size of the ”center” one). In practice
the model LGN computes a Diﬀerence-Of-Gaussians response over diﬀerent inputs, e.g.,
Red ON-Center and Green OFF-Surround, or Blue ON-Center and (Red+Green) OFF-
That is, the output of the LGN unit j,ηLGN
j(t+δt) = f
where f(x) is a half-rectifying function ensuring that activations are positive, Fj,p is the
connection ﬁeld of LGN unit jon aﬀerent sheet p(i.e., R, G and B inputs), Ψi,p(t) is the
activation of the aﬀerent unit iin such ﬁeld and wij,p is the connection strength between
the aﬀerent and the LGN unit, which follows a Gaussian proﬁle speciﬁc to the projection.
Each projection is identiﬁed by the pair of the aﬀerent sheet and the LGN sheet it connects
to. For instance, a Red-Green LGN ON sheet would model the ON-Center population
of LGN neurons responding to Red-Green contrast, that is RedOn-Center, GreenOﬀ-
Surround, and would require wij,Red to be a narrow Gaussian (Center ﬁlter), and wij,Green
to be a broader (negated) Gaussian (Surround ﬁlter).
3.2 GCAL - Model of Cortical Processing
In order to simulate the development and dynamics of the Primary Visual Cortex in our
models we use the GCAL (Gain-Controlled, Adaptive, Laterally connected) model ,
as opposed to previous work that was based on an older version of the model, LISSOM
. In practice we will use the AL variant of GCAL, as we avoid using gain control for
modelling the parvocellular pathway in a more biologically realistic fashion . We will
nonetheless refer to the model as GCAL and AL interchangeably in the text, since AL can
be obtained using suitable parameter settings for GCAL. All the simulations described
in the present work have been implemented and run using the Topographica simulator
The model belongs to the family of Self-Organizing Maps, but is endowed with lateral
connectivity (short-range excitatory and long-range inhibitory) and homeostatic adapta-
tion. The model simulates the activity of a square sheet of cortical units (each about the
level of cortical columns), each of which is connected to other units either by Aﬀerent
projections (inputs from lower stages, here only the diﬀerent LGN sheets) and Excitatory
and Inhibitory lateral connections (inputs from units belonging to the same simulated
sheet). Each projection deﬁnes connections between input and target units, each char-
acterized by a synaptic weight. During learning the synaptic weights of the aﬀerent and
lateral inhibitory connections are updated, while the lateral excitatory ones are kept con-
stant at a narrow Gaussian proﬁle. Example connectivity can be seen in Fig. 3.1, which
shows one of the color models, and features the artiﬁcial Retina, LGN, V1 (the lateral
connections are drawn in yellow).
The AL model is simulated in two steps. First the activation of the units is computed,
and then the plastic synapses are updated using Hebbian Learning.
The activation of V1 units is computed as
j(t) = f X
where pare the incoming projections (i.e., the LGN sheets and the lateral connections),
γpis their strength scaling factor and Cjp (t) is the contribution of sheet pto the output
of unit j, which is computed as
Cjp (t+δt) = X
ηi,p is the activation of unit ifrom the connection ﬁeld of V1 unit jin the input sheet
pand wij,p is the strength of the synapse linking units i(on p) and j(on V1).
f(x) is a half-wave rectifying function with a variable threshold θ, updated according
to the units’ outputs in order to model homeostatic adaptation.
The ﬁnal activation of each unit is taken after a ﬁxed number of settling steps
(tsettle = 16) during which the activity of V1 units is updated.
After settling, the threshold and all the synaptic weights are updated. As for the
threshold, an exponential running average of the activity of each unit is updated ﬁrst
¯ηj(t) = (1 −β)ηj(t) + β¯ηj(t−1)
where β= 0.991 is the smoothing factor. ¯ηj(0) = µ= 0.024, where µis V1 units’ target
activity average. The threshold is ﬁnally modiﬁed as
θ(t) = θ(t−1) + λ(¯ηj(t)−µ)
λ= 0.01 being the learning rate.
Finally, the synaptic weights of all the aﬀerent connections, together with the lateral
inhibitory ones, are updated according to Hebbian Learning. Each connection ﬁeld is
initialized at random and enveloped in a Gaussian. Each connection is updated as
ωij,p(t) = ωij,p(t−1) + αpηjηi
Pkωkj,p(t−1) + αpηjηk
where αpis the learning rate associated to the speciﬁc input connection.
3.3 Models of color vision
Within our experiments, we used three diﬀerent models of primates’ color visual systems.
The ﬁrst two are heavily inspired by previous work [6, 36], which however was based
on older models of cortical processing  than those used in the present work .
Such two models diﬀer in that they are designed to model Dichromatic and Trichromatic
vision separately. The third model we use for our investigation is based on contemporary
(unpublished) work by Chris Ball, at The University of Edinburgh, and contrary to
the other two models it was not modelled after the known anatomy of primates’ visual
systems, but was rather designed to develop physiologically plausible V1 Orientation and
Hue maps. These models allow for investigation of the relative importance of various
features of the anatomy and the functional organization of V1 for the McCollough Eﬀect.
All the three systems use the same model of cortical response, AL , and they all
share a similar structure. Every system is composed by the same model Retina (Section
3.1) and V1 (3.2), and has an intermediate layer composed by LGN sheets.
Each system has a number of LGN sheets implementing speciﬁc populations of op-
ponent neurons, and each type of opponency is present both as ON- and OFF-Center.
All LGN sheets are color-related but two Luminosity sheets (ON- and OFF-), and each
projection from LGN sheets to the model V1 is scaled by a factor which is diﬀerent for
color and Luminance sheets, thus allowing for tuning of the balance of the color / form
representation on the cortical surface (we will refer to this balance as the color strength).
It is possible to change the relative strengths of the retinal sheets (i.e., cone popula-
tions) to compensate for biases in the color distributions in the datasets, but in most of
our simulations (unless otherwise speciﬁed) we will work around the problem by adding a
uniformly random jitter in the input images’ hue  (i.e., we perform uniform hue rota-
tion on the images). We know that there exist neurons selective to any hue in the Primary
Visual Cortex , despite the natural world being very unbalanced in the distribution
of color . It is likely that intrisic mechanisms exists that help rebalancing the distri-
bution of color representations in the cortex, and while we are not going to investigate or
model such mechanisms in the present work, it makes it reasonable to use hue rotation.
In order to further improve the balance between the color channels, the average of the
mean channel values is enforced to be ﬁxed at 0.44 (the value was determined empirically
Each sheet is characterized by density and area parameters. The density is the number
of simulated units (RGCs, neurons, etc..) within a unit of simulated surface, while the
area parameter sets the size of the sheet (e.g., an area of 2.0 means that the simulated
surface is 2 ×2 bigger than the unit square, and thus has four times as many units as
deﬁned by the density parameter). Within our simulations, unless otherwise speciﬁed, the
density of the Retina, LGN and V1 sheets are ﬁxed to 24, 24 and 48, respectively, with
the only diﬀerence of the idealised model that features LGN sheets of density 48. The
default area parameter for V1 is set to 1.3 for the dichromatic and trichromatic models
(that is, the ﬁnal size is 62 ×62 units), and 1.0 for the idealised one. This was done
empirically to optimize the cortical representation of orientation and color. The ﬁnal size
of the Retina and LGN sheets are computed in a similar way, with the addition of extra
units on the sides to prevent border-eﬀects (i.e., every sheets’ units has a Receptive Field
that lies completely within its aﬀerent sheet) .
All three systems share the same parameters for the projections’ scaling factors and
learning rates in the model V1: the aﬀerent strength (i.e., connections from the LGN) is
set to be 1.0 in the dichromatic and trichromatic models, and 1.5 in the idealised one;
excitatory and inhibitory connections’ strengths are set to 1.7 and 1.4, respectively, and
the learning rates of the aﬀerent and inhibitory connections are set to 0.1 and 0.3. Lateral
excitatory connections are not updated. Learning rates are divided equally among the
units of a projection, and thus the eﬀective learning rate of a single unit in a projection
that has Nunits is
3.3.1 Dichromatic Vision
Our model of dichromatic vision is inspired to the anatomical organization of color vision
in primates, but it is limited to two cone populations. Such model has many advantages.
It is simple to analyse and suitable for exploring the McCollough Eﬀect, which by itself
is induced using a pair of colors, and it provides a good model of foveal vision, as the
fovea doesn’t have any Short-wavelength receptors (and is thus L-M dichromatic.)
Within our model the two cones are mostly uncorrelated in that we use Red and
Green activations computed from RGB images rather than LMS, which however can be
reasonable, as we discussed in Section 3.1.
The LGN sheets in this simpliﬁed model correspond to the ON- and OFF-Center
populations of Red-Green, Green-Red and Luminosity (Red+Green) opponent neurons.
The color units are double opponent, and for instance a Red-Green LGN ON neuron
would correspond to a biological R+G- center and R-G+ surround. The model is shown
in Fig. 3.1.
Fig. 3.2 shows Hue and Orientation maps from an example model trained with default
parameters (as we discussed in the previous section). The only modiﬁcation is a lower
color strength such that the total strength of the four color LGN sheets is 2
3of that of the
two Luminosity sheets (0.6 strength for each color channel and 1.8 for each luminance,
for total strengths of 2.4 and 3.6 respectively).
Figure 3.1: Schematics of the dichromatic (Red-Green, Long-Medium) model of color vision.
Each LGN stack represents ON- and OFF-Center populations of neurons with similar tuning
(e.g., Red-Green opponency).
3.3.2 Trichromatic Vision
The basic trichromatic vision model is similar to the dichromatic one discussed in the
previous section. However, the trichromatic model further includes an anatomically in-
spired Short wavelength (Blue) cone receptor, together with the corresponding opponent
neurons in the LGN (Blue-Center / [Red+Green]-Surround). The model’s schematics is
shown in Fig. 3.3, while example preference maps are shown in Fig. 3.4 (the simulation
was run using the default parameters as discussed in the previous sections, and in partic-
ular the color strength was set to 0.5, thus making the whole contribution of color-related
LGN sheets equal to that of the Luminance-only, colorless, channels). The dataset used
for training the example maps of Fig. 3.4 is the Botanical dataset.
Figure 3.2: Example cortical properties of the V1 in the dichromatic model. Left are the
preference maps (top is Hue, bottom is Orientation), while Right is the polar plot of the
Hue preferences, where the magnitude of the vectors represent the selectivity of each unit to
the speciﬁc hue (vectors are plotted taking a reference neuron out of 15, to avoid crowding
the plot.) The preference maps represent each unit’s preference (color coded) and selectivity
(color value; white pixels representing unselective neurons.)
The model relies heavily on previous work, with crucial modiﬁcations (we use of GCAL
/ AL instead of LISSOM, hence including homeostatic mechanisms) [6, 36].
Figure 3.3: Schematics of the trichromatic (Red-Green-Blue, Long-Medium-Short) model of
color vision. Each LGN stack represents ON- and OFF-Center populations of neurons with
similar tuning (e.g., Red-Green opponency).
3.3.3 Idealised Trichromatic Vision
The ﬁnal model is based on a still unpublished model by Chris Ball, developed during
his PhD at The University of Edinburgh. The schematics of the model is shown in Fig.
3.5, while example preference maps are reported in Fig. 3.6.
The model is named Idealised because its objective is to provide a (not necessarily
biological) subcortical system such that the resulting cortical maps in V1 would be as
close as possible to the physiological data available. As can be seen in Fig. 3.5, the LGN
is simulated to have populations of every color opponency (e.g., Red-Blue, which is not
found in any trichromatic animal), achieving a perfect symmetry between the diﬀerent
cones. Moreover, the total activity of the LGN sheets is re-normalized such that at any
iteration the mean activity of the sheets is set to a ﬁxed value, here 0.05.
Training was performed by extracting random patches from the Barcelona natural images
dataset  and presenting them to the models.
Figure 3.4: Example cortical properties of the V1 in the trichromatic model. Left are the
preference maps (top is Hue, bottom is Orientation), while Right is the polar plot of the Hue
preferences, where the magnitude of the vectors represent the selectivity of each unit to the
speciﬁc hue (all units are plotted.)
Figure 3.5: Schematics of the idealised model of trichromatic vision. Each LGN stack
represents ON- and OFF-Center populations of neurons with similar tuning (e.g., Red-Green
It is known that cortical learning modelled using LISSOM / GCAL results in cortical
maps that do not depend on the initial weights’ values, but rather only depend on the
input stream, that is the exact sequence of images used during training . We eﬀectively
use this fact to simulate diﬀerent subjects for higher statistical relevance, using diﬀerent
values for the random generators’ seeds.
The training of the model was performed by presenting input patches and simulating
the cortical response, and then updating the cortical model as described in Section 3.2.
Training lasted 10000 iterations for most of our simulations, which is around the number
of iterations generally performed with the model we use. Training for a longer time didn’t
result in signiﬁcantly better results on testing the McCollough Eﬀect, and thus the ﬁnal
number of iterations was decided from a computational eﬃciency point of view.
3.4 McCollough Eﬀect
The speciﬁc modelling of the McCollough Eﬀect was divided into three phases: testing
before induction, induction and testing after induction. The ﬁnal results compare the
decoded colorfulness of stimuli before and after adaptation to the colored gratings.
Figure 3.6: Example cortical properties of the V1 in the idealised trichromatic model. Left
are the preference maps (top is Hue, bottom is Orientation), while Right is the polar plot of
the Hue preferences, where the magnitude of the vectors represent the selectivity of each unit
to the speciﬁc hue (all units are plotted.)
3.4.1 Induction and Testing
The induction of the McCollough Eﬀect was based on previous work . The patterns
used in the present work are colored sine gratings, which are set by drawing gratings of
diﬀerent amplitudes on the diﬀerent retinal sheets, thus simulating the activation of the
diﬀerent cones. During stimulation with the induction patterns the aﬀerent connections
(LGN to V1) were set to have a null learning rate, thus only allowing lateral inhibitory
connections and homeostatic mechanisms to adjust themselves. In previous work, afteref-
fects were found to be dominated by the plasticity of lateral connections , and here we
can test such hypothesis on the McCollough Eﬀect by separately enabling and disabling
cortical plasticity and homeostatic mechanisms.
Each of the inducting pattern was presented in alternation for 5 iterations each time,
repeating the sequence until a total of 300 presentation steps were simulated for each
pattern. The simulated length of the induction was found to be in line with psychological
experiments on human subjects, as previous work used a correspondence of 90 iterations
for three minutes of induction (Tilt Aftereﬀect, ), thus making 600 iterations suitable
for modelling a 15 −20 minutes McCollough Eﬀect test.
The phase of the induction gratings used in each 5-iterations block was sampled from
a normal distribution with a variance of 1
8of the gratings’ period, eﬀectively modelling a
ﬁxed gaze experiment.
3.4.2 Decoding the Perceived color
The problem of decoding the color that is perceived by a cortical model at a given time
is diﬃcult for many reasons. We know that speciﬁc visual properties can be decoded
statistically using population codes in V1 , but problems might rise if the cortical
model is not well balanced (i.e., some features are over-represented). Also, color is not
unidimensional: in the case of dichromatic simulations any perceived color can lie only
on a line, that goes from ”more color A” to ”more color B”. In a general color system
however this is not true anymore.
In order to tackle this problem, we decided to take advantage of the simpliﬁed context
of dichromatic simulations, which are a good ﬁt for studying the McCollough Eﬀect, in
that color pairs are used. We can thus compute a measure of the colorfulness of a pattern
along an axis where the two opposite directions encode for the perception of two diﬀerent
colors. In practice, we assign a color vector to each V1 unit, whose argument is the
neuron’s preferred hue and its magnitude is be the product of the neuron’s selectivity to
that hue and its activation. We proceed by computing a unit vector ~v whose argument
is that of the average vector over the population,
∠~v = arctan Piφiηisin(hi)
where ηiis the activation of unit i,hiis its hue preference ofand φiits selectivity.
Then, we compute the dot product between the vector ~v and the unit vectors rep-
resenting the two colors we want to compare (e.g., Red and Green), and compute the
diﬀerence (the sign is arbitrary, depending on which color we take as positive), but within
this work we’ll take Red to be positive. See Fig. 3.7.
The result is a single number which is close to zero (provided the neural population has
balanced hue preferences) when colorless stimuli are presented as input, and increasingly
positive (negative) for stimuli with the ﬁrst color of the pair (the second). In the case
of Red-Green comparison, within the scope of the McCollough Eﬀect, stimuli that are
perceived as ”Red” will have a high positive value, while stimuli perceived as ”Green”
will have a high negative value. It is also worth noting that any baseline dependent on the
balance of neurons’ hue preferences can be evened out by computing diﬀerences between
such measures of colorfulness (e.g., before and after induction of the McCollough Eﬀect).
Figure 3.7: Example decoding of an average population vector scaled by hue preferences and
selectivities of V1 neurons (black). Red and green are reference vectors onto which the test
vector is projected (dark red and dark green).
EXPERIMENTS AND RESULTS
Within this chapter we are going to present diﬀerent aspects of the McCollough Eﬀect for
which physiological data is available for comparison. We will try to compare our diﬀerent
models wherever possible and we will prepare the basis for further discussion about the
models and the dynamics of the McCollough Eﬀect, which we will explore further in
4.1 The McCollough Eﬀect
We start with the most basic experiment to discuss the ﬁrst properties of the phenomenon.
In this setting, we present two colorless gratings with orthogonal orientation and same
spatial frequency (in particular, one vertical and one horizontal) before and after induc-
tion with colored gratings, and we compute the diﬀerence in colorfulness before and after
adaptation. We will present the exact values of the colorfulness of the white gratings
before and after induction, too. However, as there are biases dependent on the unbalance
of hue-preferences in the model V1, their diﬀerence will be the most meaningful measure.
Induction is performed using similarly oriented gratings, under two diﬀerent exper-
iments for each model: Red-Vertical, Green-Horizontal for the ﬁrst test and Green-
Vertical, Red-Horizontal for the second one. In both tests we expect the perceived color of
each test grating to be shifted towards the color opposite to the corresponding induction
For the experiments with the dichromatic model we trained 10 models using diﬀerent
random seeds to train them using diﬀerent input streams, thus eﬀectively simulating
diﬀerent subjects’ visual experiences and achieving diﬀerent neural layouts. Each model
was trained for N= 10000 iterations, and slightly more strength was given to Luminance
LGN sheets compared to color-speciﬁc ones (color strength = 0.4) in order to improve
the orientation map in the model V1. Random hue-jitter was also applied to input images
in order to obtain a more balanced representation of colors in the inputs, independent
from the speciﬁc natural images statistics (e.g., abundance of green and blue, for woods
and the sky). We computed the required measures for each model and then their average
and standard deviation. The results are shown in Table 4.1. Fig. 4.1 shows example
population activities and average population vectors. As we can see, the perception of
colorless gratings is shifted towards the color of the other induction grating. Vertical
gratings will thus look greener” when Red-Vertical patterns are used for induction, and
”redder” when Green-Vertical patterns are used. We also observe a good balance between
the shifts in the two directions (Red and Green).
TEST 1 Before After Diﬀerence
VERTICAL -0.154779 (0.176994) -0.626384 (0.191359) -0.471605 (0.139545)
HORIZONTAL -0.042584 (0.174001) 0.438041 (0.150593) 0.480625 (0.137430)
TEST 2 Before After Diﬀerence
VERTICAL -0.154779 (0.176994) 0.396533 (0.191535) 0.551312 (0.145786)
HORIZONTAL -0.042584 (0.174001) -0.555774 (0.157620) -0.513190 (0.155681)
Table 4.1: Dichromatic system. Measured colorfulness of white gratings Before and After
induction with colored gratings. The values represented the perceived color, as Green (below
zero) and Red (above zero). The data shown is the average over 10 simulated subjects, and it’s
reported as mean values with their standard deviation (in parentheses). Top table: ﬁrst test,
induction with Red-Vertical and Green-Horizontal gratings. After induction colorless vertical
gratings appear greener (negative) and horizontal ones appear redder (positive). Bottom
table: second test, using Green-Vertical and Red-Horizontal; vertical gratings look redder
(positive) while horizontal ones look greener (negative).
For the trichromatic model we simulated 5 diﬀerent subjects in the same way as we
did in the previous experiment. The color and the luminance LGN sheets are given a
total equal weight (i.e., the two luminance sheets have a strength that is equal to that
of all the color sheets). We computed the strength of the McCollough Eﬀect for each
subject, and we extracted statistical information, which is reported in Table 4.2. As we
Figure 4.1: Example population activity before and after induction in the dichromatic
model. Left is the population activity drawn using hue-preferences as the vectors’ arguments
and their selectivity times activation as their magnitude. Blue vectors are the responses to
a white horizontal grating before induction, Green vectors are the same after induction with
a Green-Horizontal grating. Right picture shows the vector averages: light orange is the
average before induction, darker orange is the average after induction. Red and Green vectors
are shown as reference.
can see from the variance of the values, the system is more unstable than the previous
simulation, which was well balanced and consistent. This can be for many reasons, and
a better understanding of the diﬀerences between the two simulations will be interesting
future work. As far as we could analyze the models, we found that the ﬁrst one had
neurons that represented most orientations with good accuracy (consistent with color-
less previous simulations using GCAL, which had a mean decoding error of 5oto 10o,)
while the second was poorer in such a representation. The representation of color, on the
other hand, was very balanced in both models.
Finally, in the case of the Idealised Trichromatic model we simulated another
5 test subjects and performed induction with the appropriate gratings as described in
Section 3.4.1. The results are reported in Table 4.3, and similarly to the case of the
TEST 1 Before After Diﬀerence
VERTICAL 0.028121 (0.841249) -0.176233 (0.735258) -0.204353 (0.335443)
HORIZONTAL -0.230547 (0.901317) 0.084596 (1.033288) 0.315143 (0.210030)
TEST 2 Before After Diﬀerence
VERTICAL 0.028121 (0.841249) 0.767211 (0.880755) 0.739090 (0.563988)
HORIZONTAL -0.230547 (0.901317) -0.770934 (0.479388) -0.540387 (0.432896)
Table 4.2: Trichromatic system. Measured colorfulness of white gratings Before and After
induction with colored gratings. The values are Green (below zero) and Red (above zero), and
represent the average over 5simulated subjects, reported together with their standard devia-
tion (in parentheses). Top table: ﬁrst test, induction with Red-Vertical and Green-Horizontal
gratings. After induction vertical gratings appear greener (negative) and horizontal ones ap-
pear redder (positive). Bottom table: second test, using Green-Vertical and Red-Horizontal;
vertical gratings look redder (positive) while horizontal ones look greener (negative).
trichromatic model, the variance is very high, for similar reasons. Interestingly we found
that the simulations that had the strongest eﬀect had the lowest orientation decoding
error on average, while simulations that failed in achieving a good representation of the
orientation space could not reproduce the McCollough Eﬀect. This is reasonable because
the McCollough Eﬀect relies on the interactions between the perceived orientation of
patterns and their color, and a poor representation of either of these features is likely to
prevent the eﬀect to be perceived.
Moreover, and critical for the following sections, we could only achieve a reasonable
representation of both color and orientation in the dichromatic model, with the other two
achieving partial representations and thus partial results.
4.2 Orientation dependency of the McCollough Eﬀect
Ellis  studied the dependency of the strength of the McCollough Eﬀect on gratings
with diﬀerent orientations (i.e., not only vertical and horizontal). Fig. 2.4 shows her
data, which is characterized by two main features: gratings oriented around 45ooﬀ the
induction gratings appear to be colorless, and the decay is the strength of the eﬀect is
TEST 1 Before After Diﬀerence
VERTICAL 0.339216 (1.296216) -0.216267 (0.231275) -0.555484 (1.168237)
HORIZONTAL 0.033844 (1.482212) 0.483650 (0.172860) 0.449805 (1.390989)
TEST 2 Before After Diﬀerence
VERTICAL 0.339216 (1.296216) 0.467627 (0.128034) 0.128411 (1.207255)
HORIZONTAL 0.033844 (1.482212) -0.325781 (0.219666) -0.359625 (1.413137)
Table 4.3: Idealised Trichromatic system. Measured colorfulness of white gratings Before
and After induction with colored gratings. The values are Green (below zero) and Red (above
zero), and represent the average over 5simulated subjects, reported together with their
standard deviation (in parentheses). Top table: ﬁrst test, induction with Red-Vertical and
Green-Horizontal gratings. After induction vertical gratings appear greener (negative) and
horizontal ones appear redder (positive). Bottom table: second test, using Green-Vertical
and Red-Horizontal; vertical gratings look redder (positive) while horizontal ones look greener
linear to the angular divergence of the test patterns with comparison to the induction
In the case of our dichromatic model we replicated a similar ﬁgure by averaging over
10 simulated subjects, using the same parameters as in Section 4.1. The resulting plot is
shown in Fig. 4.2. As we can see, not only the plot resembles Ellis’ in great detail, but
it also perfectly matches the described features.
As we discussed in Section 4.1, the overall quality of the representation of orientations
in the other two models is lower, and this makes it diﬃcult to produce a clear plot like
4.3 Diﬀerent Orientation of the Inducing Patterns
The ﬁnal test we performed tried to replicate the psychological data collected by Fidell,
in 1970 , which was also studied in the previous thesis by Ciroux , thus allowing
for a direct comparison between our models and human data. For the scope of the present
simulations we only used the dichromatic model because of its good representation of
Figure 4.2: Orientation dependency of the ME in the dichromatic model, averaged over
10 simulated subjects. The strength shown is the diﬀerence in colorfulness of white gratings
before and after induction with Green-Horizontal (0o) and Red-Vertical (90o) gratings. Positive
values mean that the perception is shifted towards Red. Bars represent the standard deviation
of the measures.
orientations, which was required for the experiments.
Fidell used pairs of gratings at diﬀerent orientations during induction and testing
(induction and testing was always performed with the same gratings, except for their
color). In the ﬁrst experiment (Fidell, 1, cyan line, in Fig. 4.3), as a control, induction
was performed using a Red grating oriented at the angle shown on the x-axis, and a
Green grating orthogonal to it. The strength of the McCollough Eﬀect, measured using
test gratings with the same orientation as the induction ones, was found to be independent
from the exact orientation of the patterns, as long as they were orthogonal. Our results
for the same setting are shown in Fig. 4.3 (experiment 1, blue line), with the diﬀerence
that we could aﬀord computing more sample points than in the original experiment. Each
point corresponds to the average of 5 simulated subjects, with the bars representing the
standard deviations. Our simulations show the independence from the orientation of the
induction patterns (as long as orthogonal) with even more detail than the original data.
In the second experiment, Fidell ﬁxed the Green grating to always be vertical (90o),
while the orientation of the Red grating was set to the value on the x-axis. In this setting,
the induction gratings are increasingly closer together, and it is hence possible to compute
the strength of the McCollough Eﬀect that results from induction patterns oriented at
diﬀerent angles (i.e., not orthogonal). Fig. 4.3 shows a comparison between Fidell’s
data (Fidell, 2, magenta line) and ours (experiment 2, red line), which are remarkeably
similar, showing a signiﬁcant decay in the strength of the eﬀect when patterns are less
than 45oapart, rapidly falling and becoming almost non-existent when the diﬀerence in
the orientation of the gratings is around 10o. Again, the computational approach allows
for sampling more points of the curve.
Our results match the previous work by Ciroux, too (see Fig. 4.6 of [Ciroux, 2006]
(), but in the case of our experiments we used a more accurate method for measuring
the strength of the eﬀect. For every simulation (i.e., pair of gratings at speciﬁc orien-
tations) we presented both combinations of Red and Green with each orientation, and
computed the average of the four values (sign-corrected) of the measured strength (i.e.,
2 test values corresponding to each induction grating, for each association Red-ﬁrst grat-
ing, Green-second grating and Green-ﬁrst grating, Red-second grating). For example, in
the default case of horizontal and vertical gratings we’d compute the diﬀerences in the
perceived colorfulness of the test patterns in each of the two combinations (obtaining
two values for each combination, one positive, perceived as ”redder” and one negative,
perceived as ”greener”). The resulting strength is then the average of the absolute values
of the 4 measurements.
4.4 Role of Homeostasis and Lateral Inhibitory Connec-
One critical aspect we tried to investigate is about the speciﬁc contribution of homeostatic
mechanisms and cortical plasticity (of lateral inhibitory connections) to the production
of the eﬀect.
One advantage of running computer simulations is that, contrary to the biological sys-
tem, we can selectively disable either mechanism, and we can thus compare the diﬀerences
in the McCollough Eﬀect that each of them triggers separately.
In particular, we run two sets of experiments, both using our dichromatic system. In
the ﬁrst one, we compared three scenarios, namely one with both mechanisms enabled
(i.e., homeostatic adaptation and plastic lateral inhibitory connections), another with
only homeostasis enabled, and the last one with only plasticity, and we compared the
strength of the eﬀect as measured using colorless gratings with the same orientation as
the induction patterns (horizontal and vertical). The results are reported in Table 4.4,
and compare the three cases (”default” meaning that both mechanisms are enabled)
in the two experimental settings, Red-Vertical, Green-Horizontal (Test 1) and Green-
Vertical, Red-Horizontal (Test 2), averaging each value over 10 simulated subjects. As
we could expect, the eﬀect is stronger when both mechanisms are present: this can also
ﬁt the ﬁnding that the McCollough Eﬀect operates at two diﬀerent timescales, likely
resulting from two diﬀerent mechanisms active at the same time . In the case of
the experiments run enabling a single mechanisms, the results from enabling homeostasis
(and disabling cortical learning) are slightly stronger and more balanced than by only
enabling cortical plasticity. Even though this could hint at a stronger dependency of the
eﬀect on homeostasis, it is unlikely to explain its long-lasting dynamics. Moreover, the
test patterns in the experiments share the same orientation as the induction gratings,
which would be expected to work using homeostatic mechanisms in any simulation that
had double-duty neurons (that is, neurons selective for both orientation and color).
In order to further investigate the role of homeostatic adaptation and cortical learning
in the McCollough Eﬀect we run a set of simulations similar to the ﬁrst ones we described,
but instead we plotted the strength of the eﬀect when using test patterns at diﬀerent
orientations (like in Section 4.2). Comparing the eﬀect using data from all the orientations
is useful to explain the stronger eﬀect observed when enabling the only homeostatic
adaptation. Indeed, we would expect the eﬀect to be very strong near the orientations
used for induction, because of the adaptation to such stimuli, but not to be able to
account for the full eﬀect as was measured in Section 4.2.
We show the results in Fig. 4.4, where each of the two plots compares the strength
of the McCollough Eﬀect tested at diﬀerent orientations using the default model (both
mechanisms enabled) and each of the two models where only one is enabled at a time.
Notably, as we would expect, when only homeostasis is enabled the peak of the curve
matches the default simulation: orientations similar to the induction patterns are directly
aﬀected by adaptation. However, it is interesting that in a such setting the width of the
curve is much narrower than the curve computed using the default simulation, and in
TEST 1 Default Only Homeostasis Only Plasticity
VERTICAL -0.471605 (0.139545) -0.459404 (0.103371) -0.342562 (0.168272)
HORIZONTAL 0.480625 (0.137430) 0.454146 (0.139887) 0.394118 (0.152049)
TEST 2 Default Only Homeostasis Only Plasticity
VERTICAL 0.551312 (0.145786) 0.497714 (0.147248) 0.453383 (0.216519)
HORIZONTAL -0.513190 (0.155681) -0.517360 (0.157267) -0.390162 (0.198635)
Table 4.4: Dichromatic system. Measured colorfulness of white gratings before and after
induction with colored gratings, averaged over 10 simulated subjects. Positive values indicate
a shift in perception towards Red while negative values are associated to Green. The standard
deviation of each measure is shown in parentheses. The two tables show data from two tests,
which diﬀer in the induction patterns used: Test 1 uses Red-Vertical and Green-Horizontal
gratings, while Test 2 uses Green-Vertical and Red-Horizontal ones. After induction in Test
1 vertical gratings appear ”greener” (negative values) while horizontal ones appear ”redder”
(positive values), and the opposite is true for Test 2. Note that diﬀerently from the tables
in Section 4.1, here we only reported the diﬀerences in the perceived colorfulness of the test
stimuli, that is the measure of the strength of the eﬀect, and not the exact values before and
particular the eﬀect decays at around 30o, as opposed to the psychologically measured 45o
in humans (note that this is partially similar in the default simulation, for the gratings
around 90o, right part of the graph).
In contrast, when only cortical plasticity is enabled, and homeostasis is disabled, the
peaks match the default curve worse (they appear to be more ”ﬂat”), but the width of
the curve is a perfect ﬁt to human data, which is indeed matched better than in the
default simulation. We suggest that the closer match hints at a major involvement of
cortical plasticity in the McCollough Eﬀect, and that the discrepancy near the induction
orientations is actually due to the poor representation of orientation in the model, which
is on average oﬀ by 5oto 20o, thus introducing errors in the plot.
In conclusion, homeostasis seems to be describing the eﬀect around the orientation of
the induction gratings very well, though longer-term eﬀects would need to be considered
(i.e., at some point in time the adaptation would fade out, leaving only the eﬀect of the
updated synapses), while cortical plasticity probably accounts for the accurate shape of
the orientation-dependent curve as measured in humans (i.e., linear decay of the eﬀect
with the orientation of the test patterns and width of the curve).
Figure 4.3: McCollough Eﬀect : dependency on the angle between the induction gratings.
Cyan and Magenta are data from [Fidell, 1970] (), while Red and Blue are the experiments
from the present work. The top ﬁgure shows a comparison between our data and Fidell’s ﬁrst
experiment. It represents the average strength of the ME inducted using orthogonal gratings,
the ﬁrst of which being oriented as the angles shown on the x-axis (e.g., points at 30ostand
for inducing patterns oriented at 30oand 30o+ 90o= 120o). The strength of the eﬀect is
constant regardless of the gratings used, as long as they are orthogonal. The bottom ﬁgure
shows the strength of the ME obtained using induction gratings with diﬀerent orientations:
the ﬁrst grating is oriented at the angle on the x-axis while the second is always vertical
(90o). The strength of the eﬀect is maximal when the induction gratings are orthogonal, and
it decreases quickly for patterns that are less than 45oapart. Data from our experiments is
averaged over 5simulated subjects. 39
Figure 4.4: Comparison between the dependency on the orientation of the test patterns of
the McCollough Eﬀect, here obtained using Green-Horizontal and Red-Vertical gratings. The
top plot shows the strength of the McCollough Eﬀect for gratings at each orientation when
only homeostatic adaptation is enabled (and cortical plasticity is disabled). The bottom one
shows similar data, obtained using the reverse combination (cortical plasticity is enabled and
homeostasis is disabled). In both plots a reference curve is shown in cyan, and corresponds
to the strength measured when both mechanisms are enabled.
DISCUSSION AND CONCLUSIONS
In Chapter 4 we presented data from a number of experiments that we performed using
our models. In particular, we could reproduce the McCollough Eﬀect under a variety of
conditions, exploring most of its characteristics and comparing our results to available
In the speciﬁc case of dichromatic simulations we could reproduce the results from
the closest previous work  using the new models and GCAL, and we could compare
some of the similarities and diﬀerences between the models. The results are interesting
and accurate, though we found a major diﬀerence in the learnt models in the number of
”double-duty” units (units that are selective to both orientation and color), which in our
models were predominant. Interestingly, having more double-duty cells makes it easier for
homeostatic mechanisms alone to reproduce the eﬀect, even though in biological systems
their number is lower, and so is probably the contribution of the homeostasis. Further,
it is diﬃcult to compare our model with Ciroux’s  qualitatively as both works match
the available human data perfectly[16, 17].
5.1 Summary of the results
We ﬁrst reproduced the McCollough Eﬀect using our three models, and we found it easier
for the dichromatic system to produce a stronger, balanced eﬀect than using the others.
This could be due to many reasons, the most critical being that, even though all three
models have learnt a very balanced representation of the input color space (with great
help of the hue-rotation transformations), their representation of orientation was found
to be poor (as measured by decoding the orientation of white gratings and computing the
error). We tested the accuracy of the representation by presenting gratings at diﬀerent
orientations and decoding the perceived orientation (using a population average decoder),
and ﬁnally computing the error between the real and the decoded values. Between the
three models, the dichromatic system had the best accuracy (even though the error was
still considerable), while the other systems failed with even higher errors. As the eﬀect
relies on the precise interactions between the representation of color and orientation,
an inaccurate coding of orientation can eﬀectively prevent the eﬀect from matching the
We further tested this hypothesis while also validating our models presenting test pat-
terns (before and after induction) at diﬀerent orientations, with the scope of reproducing
the data collected by Ellis in 1977 , which shows the dependency of the strength of
the McCollough Eﬀect on the orientation of the test gratings. As we saw in Section 4.2,
the results from the dichromatic simulations matched closely the data, meaning that the
model is capable of reproducing the McCollough Eﬀect to a great level of realism. An-
other test we ran to determine the behavioral match between our dichromatic model and
the psychologically-measured eﬀect was based on the experiments by Fidell in 1970 ,
which tested the relationship between the mutual orientation of the induction gratings
and the strength of the resulting McCollough Eﬀect (e.g., for determining the minimum
diﬀerence in orientation capable of inducing the eﬀect). Again, our results match closely
the human data and are in agreement with previous results .
5.1.1 Neural mechanisms underlying the McCollough Eﬀect
Having determined the level of behavioral match between our model and human data
from previous works we investigated the possible neural mechanisms capable of producing
the McCollough Eﬀect. In particular, we know that the only changes in our models
during adaptation with the induction gratings are restricted to the plasticity of lateral
inhibitory connections and to homeostasis: in fact, learning in aﬀerent projections and
lateral excitatory connections was always disabled during our experiments, and every
other variable in the model is constant.
What was left to determine was the individual contributions of the two mechanisms
active during our experiments. It is interesting to note that using computer simulations
provides a great advantage over physiological experiments as we can turn oﬀ either mech-
anism without aﬀecting the rest of the system. This allows for predictions and a deeper
investigation of the phenomenon, which could then hopefully be tested on the biological
Preliminary results comparing the strength of the eﬀect computed for gratings oriented
like the induction patterns hinted that homeostasis could account for the phenomenon, as
it was found to induce strong and balanced eﬀects. However, computing the orientation-
dependent plot in each case (enabling only homeostatic adaptation or cortical plasticity)
resulted in more interesting data. In fact, plots produced by inducing the McCollough
Eﬀect using only homeostatic mechanisms resulted in curves that were signiﬁcantly nar-
rower than those measured in human experiments, even though they were extremely
accurate near the peaks (that is, at the orientations of the inducing patterns). On the
contrary, simulation that only enabled cortical plasticity were found to be a perfect ﬁt to
the main characteristics of the psychological data, accounting for the width of the curves
and the linear decay of the eﬀect, which faded completely at 45ooﬀ the orientation of
the induction patterns.
There are two main reasons that could explain the good ﬁt in the case of the homeostatic-
based simulations and for the higher error around the peaks in the case of the plasticity-
based ones. First, the low quality of the representation of the orientations in the model
introduces errors in the plot, possibly activating similar sets of neurons regardless of the
test patterns’ orientations (e.g., near the orientation of the induction patterns). Sec-
ond, our dichromatic model was characterized by a great number of double-duty cells
(actually, almost every unit in the model was double-duty, see Fig. 3.2), in contrast to
previous work . A Primary Visual Cortex with many double-duty cells selective to
many combinations of colors and orientations could give rise to a McCollough Eﬀect, but
its properties would likely be diﬀerent from those observed in humans, and it would be
unlikely to explain the long-term persistence of the eﬀect anyway. Even in the case of
a high number of double-duty cells, in fact, the width of the curve of the orientation-
dependent plot would mostly depend on the width of the orientation tuning curve of the
neurons and on the quality of the representation of the orientations.
It is worth noting that in the case of fewer double-duty neurons (as it is likely to be the
case in primates ) we would expect the curves to be narrower when only homeostasis
is enabled, and the overall eﬀect to be described with worse accuracy than in the present
simulations (i.e., the two plots, homeostasis-only and plasticity-only, Fig. 4.4, would be
signiﬁcantly diﬀerent from each other, with the homeostatic-based one showing strong,
narrow peaks at 0oand 90oand the plasticity-based one closely resembling the default
simulation). Also, having a model with fewer double-duty units would probably imply
a higher quality in the representation of orientations, as color and orientation would be
learnt in a decorrelated fashion.
5.2 Future work
Future work will be required to better determine the basic requirements for reproducing
the McCollough Eﬀect, with special attention to the individual contributions of home-
ostatic mechanisms and cortical plasticity, which can be best studied using computer
Our models need to be improved in their representation of orientations and it would be
useful to understand what does it drive the development and proportion of double-duty
units in the models, as having the possibility to change their proportion would allow for
further investigation on the role of homeostatic adaptation in the short-term dynamics
of the McCollough Eﬀect.
Also, physiological data suggests that the number of V1 neurons that are selective
to color is rather small compared to the number of orientation-selective neurons ,
and this should be taken into account as a target for better, more biologically realistic,
It would also be interesting to produce a model (even ”artiﬁcially”, that is, not learnt)
that had very narrow orientation tuning curves and a very accurate representation of
orientations. In this case the homeostatic mechanisms alone would likely account for
only a very little set of orientations close to the orientation of the induction gratings, and
wouldn’t probably produce a strong eﬀect when other test gratings are used.
At last, more physiological experiments are required to better understand the repre-
sentation of color in V1 and its relation to orientation, like determining the proportion and
distribution of double-duty neurons and the quality of the representation of orientations.
5.2.1 Relevance to Neuroscience
Interestingly, our computational investigations can have broader applications than just
the McCollough Eﬀect. The technical possibility of separately handling the various con-
tributions of adaptation and plasticity allows for a deep investigation of a range of cortical
phenomena that apply to all visual experience, and not only to this speciﬁc phenomenon.
As we brieﬂy mentioned, the selective disabling of either mechanism in the model
while avoiding the disruption of any other cortical function is a key advantage of running
computer simulations over physiological experiments. Both types of work are necessary,
and data from studies on the direct biological system is critical to the development of
accurate simulations. Having the possibility of running this type of simulations, however,
can in turn aid the understanding of speciﬁc phenomena and can also suggest more
experiments to be performed.
The speciﬁc dynamics of the McCollough Eﬀect that we simulated can be used not
only to explain other types of after-eﬀects, like it was done in previous studies, that
however lacked homeostatic mechanisms [7, 11], but also a range of phenomena related
to visual perception, like the learning of statistical properties of natural scenes  and
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