The color phi phenomenon: Not so special, after all?

Abstract

We show how anomalous time reversal of stimuli and their associated responses can exist in very small connectionist models. These networks are built from dynamical toy model neurons which adhere to a minimal set of biologically plausible properties. The appearance of a “ghost” response, temporally and spatially located in between responses caused by actual stimuli, as in the phi phenomenon, is demonstrated in a similar small network, where it is caused by priming and long-distance feedforward paths. We then demonstrate that the color phi phenomenon can be present in an echo state network, a recurrent neural network, without explicitly training for the presence of the effect, such that it emerges as an artifact of the dynamical processing. Our results suggest that the color phi phenomenon might simply be a feature of the inherent dynamical and nonlinear sensory processing in the brain and in and of itself is not related to consciousness.

Full text

RESEARCH ARTICLE
The color phi phenomenon: Not so special,
after all?
Lars KeuninckxID
☯¤
*, Axel Cleeremans
☯
Consciousness, Cognition & Computation Group (CO3), Center for Research in Cognition & Neurosciences
(CRCN), Universite
´Libre de Bruxelles, Bruxelles, Belgium
☯These authors contributed equally to this work.
¤Current address: Interuniversity Microelectronics Centre (IMEC), Leuven, Belgium
*lars.keuninckx@gmail.com
Abstract
We show how anomalous time reversal of stimuli and their associated responses can exist
in very small connectionist models. These networks are built from dynamical toy model neu-
rons which adhere to a minimal set of biologically plausible properties. The appearance of a
“ghost” response, temporally and spatially located in between responses caused by actual
stimuli, as in the phi phenomenon, is demonstrated in a similar small network, where it is
caused by priming and long-distance feedforward paths. We then demonstrate that the
color phi phenomenon can be present in an echo state network, a recurrent neural network,
without explicitly training for the presence of the effect, such that it emerges as an artifact of
the dynamical processing. Our results suggest that the color phi phenomenon might simply
be a feature of the inherent dynamical and nonlinear sensory processing in the brain and in
and of itself is not related to consciousness.
Author summary
Well known visual illusions are used to investigate how objective physical stimuli translate
to subjective conscious experiences and are therefore believed to offer clues at how
human consciousness works. One such illusion, the color phi phenomenon is marked by
an anomalous time reversal of what is shown to test subjects and what they report to have
seen. The color phi phenomenon has been a prime example to illustrate that conscious
experience, instead of being merely a recording of the facts as they happen, involves both
pre- and post-constructive processes resulting in a coherent narrative. We provide an
alternative interpretation by showing that all aspects of this phenomenon can be ade-
quately modeled in small neural networks. These networks are composed of simple leaky
integrator neurons, emphasizing the importance of taking into account dynamical behav-
ior. We illustrate how anomalous temporally-reversed perception of stimuli can appear
based exclusively on bottom-up dynamics. Since the color phi phenomenon emerges as a
feature of dynamical processing in our model, we conclude that it may explain less about
human consciousness than previously thought and that it could be nothing more than a
feature of dynamical processing in the brain.
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OPEN ACCESS
Citation: Keuninckx L, Cleeremans A (2021) The
color phi phenomenon: Not so special, after all?
PLoS Comput Biol 17(9): e1009344. https://doi.
org/10.1371/journal.pcbi.1009344
Editor: Daniele Marinazzo, Ghent University,
BELGIUM
Received: February 10, 2021
Accepted: August 12, 2021
Published: September 3, 2021
Peer Review History: PLOS recognizes the
benefits of transparency in the peer review
process; therefore, we enable the publication of
all of the content of peer review and author
responses alongside final, published articles. The
editorial history of this article is available here:
https://doi.org/10.1371/journal.pcbi.1009344
Copyright: ©2021 Keuninckx, Cleeremans. This is
an open access article distributed under the terms
of the Creative Commons Attribution License,
which permits unrestricted use, distribution, and
reproduction in any medium, provided the original
author and source are credited.
Data Availability Statement: All data underlying
our results stem from model simulations in the
form of Python scripts and are included in the
Supporting Material.
Funding: AC and LK were supported by the
European Research Council under the Advanced

Introduction
The color phi phenomenon (CPP), Fig 1, is perhaps one of the most baffling visual illusions
known [1]. In the experiment, two different colored dots are shown in rapid succession and at
some distance from one another. For certain intervals and spacings, which depend on the
viewer, the perception is that of the first dot moving to the position of the second dot and
changing to the color of the second dot somewhere along the perceived path. The perception
of a mix of the colors of the dots is also possible. An interactive demonstration of the effect can
be found online [2]. Clearly, the CPP is comprised of two separate effects. First, there is the phi
phenomenon of perceived movement, which has been described since at least 1912 [3]. A
swiftly moving “ghost” presence is perceived, located in between the positions of the discretely
presented dots. The phi phenomenon has been the basis for moving pictures. The second part
and the more striking anomaly of the CPP is that the color change is perceived before the sec-
ond dot at the final position is. In the words of Herzog et al. [4]: “The conscious percept cannot
have formed in a time-ordered fashion, but must have been constructed retrospectively”. The
importance of the CPP is that it shows us that our conscious perception does not necessarily
preserve the chronological order of the stimuli, nor does it need to correspond to actual
applied stimuli.
The study of visual illusions such as Hermann grids and Mach bands which are direct illus-
trations of the existence of receptive fields, can offer important insights into our neural archi-
tecture [5]. Visual phenomena that are time-involved rather than static can hint at what kind
of causal connections, feedforward, horizontal or feedback, exist between different brain areas
[6]. In Ref. [7], it is argued that the receptive field of neurons in the visual cortex is mainly
determined by feedforward processing, while feedback connections can modulate responses in
V1 corresponding to perceptual grouping. In this way, the activity of feedback connections
may correlate to perceptual awareness. However, as Bachmann [8] argues, it is difficult to dis-
tinguish experimentally whether top-down signaling to lower earlier activated feature-encod-
ing neurons is due to higher level activation that is specific to target features or originates from
non-specific activation linked to general arousal or expectation modulation that was initiated
by feedforward non-feature-encoding reticulo-thalamic feedforward connections. In the
majority of cases, top-down processing may be influenced by both specific and non-specific
feedback.
Many illusions similar to the CPP exist that demonstrate that our perception is flawed
when time is involved. For example, in the flash-lag illusion, the perception of the position of a
Fig 1. The color phi phenomenon. Two dots of different color are shown in rapid succession. For certain spatial and
temporal distances, the perception is that of the first dot moving to the position of the second dot while abruptly
changing color somewhere along the path.
https://doi.org/10.1371/journal.pcbi.1009344.g001
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The color phi phenomenon: Not so special, after all?
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Grant titled "Consciousness: The Radical Plasticity
Thesis", SH4, ERC-2013-ADG. The funders had no
role in study design, data collection and analysis,
decision to publish, or preparation of the
manuscript.
Competing interests: The authors have declared
that no competing interests exist.

moving object at an instant in time that another object is briefly shown, is systematically
advanced relative to the true position [9]. The cutaneous rabbit illusion [10] is a tactile version
of the phi phenomenon: taps or electrical pulses near the wrist and elbow in rapid succession
create the feeling of a sequence moving up the arm, without there being any stimulus applied
in between. The cutaneous rabbit illusion is accurately modeled by assuming perception fol-
lows a Bayesian inference process, based on a prior expectation of the stimulus having a low
speed and uncertainty about its position [11,12]. This results in an apparent distance contrac-
tion between the stimuli, perceptually leading to the rabbit hops. Given their success, these
statistically based models add strength to the view of the brain as a (Bayesian) predictive infer-
ence machine [13], however they do not incorporate or highlight the underlying physiological
connectionist basis.
In this article, we follow an approach that is more in line with O’Brien [14], in that percep-
tual awareness is seen as coinciding with at least temporally stable states or activation patterns
existing in a connectionist system. Such an approach requires that phenomena such as the
CPP are demonstrated and explained in a connectionist fashion, that is as existing in a neural
network built up from elementary (artificial) neural units which have properties that are
understood or accepted as being of similar nature as those of actual neurons in the brain. Most
importantly, all these units operate continuously in parallel, hence the naming parallel distrib-
uted models (PDPs). Cohen et al. [15] presented a PDP account of the Stroop effect in a strictly
feedforward dynamical model. Interestingly, their model had “attention” modulating inputs to
steer the network’s task to respond to either the word or the color it was presented. Such atten-
tion-modulating input would in reality clearly be a part of a top-down feedback path, even if
it is not explicitly derived from output activations present in this network. Since this was a
feedforward network, it does not have memory obtained from attractor states. In contrast,
Dehaene et al. [16] present a dynamical model that relies heavily on both short and long dis-
tance feedback between individual simulated thalamo-cortical columns. Interaction between
these columns, which consist of integrate-and-fire units, allows them to simulate the atten-
tional blink phenomenon. Conscious reportability is seen as caused by the synchronization or
locking of several simulated columns in a global activity pattern. The apparent similarity across
modalities between the cutaneous rabbit illusion and the phi phenomenon suggests a common
underlying mechanism which, in a connectionist view, could be supported by a common neu-
ronal microcircuit. This motivates why the study of archetypal (dynamical) microcircuits that
demonstrate comparable abstractions of these effects is important.
In this article, we offer fully detailed and testable connectionist models which demonstrate
stimulus-response order-of-arrival reversals, as well as “ghost” perceptions. We do so by
employing a rather crude neuron model, that nevertheless has all the necessary properties an
artificial neuron requires to be able to perform general computation. By choosing to position
ourselves at a well-chosen distance from more biologically informed models, we are able to
illustrate general principles.
With subjective experience changing from moment to moment, it is clear that conscious-
ness itself takes time. Therefore, dynamical systems are an appropriate basis for modeling that
offers insight in how things might be the way they are as, argued by Cleeremans and French
[17], that is all modeling does. Differential equations (DEs) and their discrete-time counter-
parts difference equations form the natural language of dynamical systems [18]. Coupling of
the same type of DE that describes an aspect of neural activity is how dynamical connectionist
networks are built.
The CPP received some fame for being the centerpiece in an argument formulated by Den-
net against the existence of a “Cartesian theater”, an informational finish line in the brain
where the order-of-arrival coincides exactly with the order of conscious perception [19].
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However, we argue, no matter in which manner the anomalous perception such as that of the
CPP comes about, at some point, somewhere in the brain, there must exist an activation that
reflects this subjective perception, if one is to accept that perception itself is physical. Therefore
there must exist some neuronal circuit that transforms the applied stimuli in objective activa-
tions that in turn either are or cause the anomalous perception. Our goal here is to demon-
strate that these phenomena can be modeled in the most simple way as generic dynamical
effects. Note that we choose in this article to focus on mechanisms. Therefore we specifically
do not correlate our models with results of behavioral experiments yet. Also, we remain agnos-
tic as to the existence of feature vectors that include time labels as proposed in Herzog et al.
[4]; our models do not require them as far as the CPP goes, but could be made compatible
with that idea.
The remainder of this article is organized as follows. In the next section we introduce the
dynamical neuron model we employ. Thereafter we describe a four-neuron network that illus-
trates the stimulus-response arrival order reversal. We introduce a model for the phi phenom-
enon, the operation of which depends on long distance feedforward “primer” connections. We
then model the complete CPP using an echo state network (ESN), a recurrent dynamical neu-
ral network built from a large number of randomly connected toy model neurons. Finally, we
summarize our results and propose ways in which we see this work continue.
Toy model neuron
Dynamical neuron models fall into two broad categories: spiking [20,21] and non-spiking.
Although spiking neuron models, which are inherently time-dependent, come closer to bio-
logical reality, they also form an extra hurdle since neural coding, apart from rate coding, is
not completely understood and still actively researched. We choose here to go with a dynam-
ical version of the simplest of all non-spiking neuron models, the perceptron [22]. Specifically,
we aim to develop our arguments starting from three well accepted basic prepositions:
• The computational power of a neuronal systems (brains) stems from the connection of
many, nearly identical and relatively simple units, called neurons.
• Neuronal signals are limited in amplitude and rate. Neurons and neuronal systems can satu-
rate and thus are inherently nonlinear. If stimuli aand belicit responses f(a) and f(b) respec-
tively, then the simultaneous application of aand bdoes in general not result in response
f(a+b).
• Neurons and neuronal systems have a certain slowness to them. They cannot respond imme-
diately to a change in input stimulus.
The first two properties give traditional artificial feedforward neural networks their general
computational abilities, under certain mathematical conditions. The third property brings
time and dynamical systems into the discussion. We start from the well-known perceptron
neuron:
y¼fX
N
i¼1
wixiþb
!;ð1Þ
in which the synaptic weights w
i
determine the input strength of input x
i
and fis a nonlinear
activation function. The weights w
i
can be positive (excitatory) or negative (inhibitory). In gen-
eral, a constant bias bis also included. A straightforward dynamical extension is given by the
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following ordinary differential equation:
tdy
dt ¼  yþfX
N
i¼1
wixiþb
!;ð2Þ
which is also known as the leaky-integrator neuron [23], not to be confused with the spiking
leaky integrate-and-fire neuron. Here, we will just call it a toy model neuron, since that is what
it is: an abstract entity with minimal neuron-like properties. The derivative dy/dt is the velocity
at which the output ychanges. Here τis a timescale, with higher τ’s meaning slower neurons.
Note that for τ!0 the response y(t) of Eq (2) equals that of Eq (1). The same is true for t!
1, with a constant input x, which can be seen from setting dy/dt to zero. Thus Eq (2) repre-
sents a “slow” perceptron. The nonlinear activation function is usually a hyperbolic tangent,
with output values ranging from −1 to 1 or a sigmoid f(x) = 1/(1 + exp
−x
), having a range of
0. . .1. Both are monotonic functions: increasing the stimulus strength leads to an increased
response, up to saturation. Eq (2) can be interpreted as the average activity or firing rate of a
single neuron as well as the activity of a larger neuronal (sub)structure. Our argument is that
any phenomenon that can exist in a small network of these crude dynamical neurons, can also
be present in the vastly more complex brain. Models built using this neuron then show which
computational pathways might exist. Note that in Eq (2) all variables, including time t, are con-
sidered de-dimensionalized, as this simplifies notation [18].
Example: Backward temporal masking
Merely as a starting point and to illustrate what is possible with this model, consider the sin-
gle-neuron circuit shown in Fig 2A. We will show it can serve as an elegant model for back-
ward temporal sensory masking. The reader who is familiar with nonlinear dynamics can skip
the remainder of this section. The circuit in Fig 2 is represented by:
tdy
dt ¼  yþf w1yþw2xð Þ ð3Þ
in which the parameters are chosen as: τ= 1 and w
1
=w
2
= 1.5. Here, for simplicity, the activa-
tion function is chosen to be a piece-wise linear function, of which the shape is reminiscent of
a hyperbolic tangent:
fðxÞ ¼
1if x<1;
xif 1x1;
1if x>1:
8
>
>
>
<
>
>
>
:ð4Þ
Plotting the right hand side of Eq (3),Fig 2B (top trace), for input x= 0, shows three points
where dy/dt is zero. In other words, this system has three equilibrium points. A small positive
push from the origin y= 0, will result in dy/dt >0, thus the system evolves to more positive y
values, until it reaches the right-hand side equilibrium y= 1. Here it remains, because a small
positive nudge towards values y>1 results in a negative change dy/dt <0. The treatment for
the left hand equilibrium at y=−1 is similar, showing it is also stable. Thus the system has a
single unstable equilibrium in the center y= 0 (indicated by an empty dot) and two stable
equilibria at y=±1 (filled dots). The latter are the results of the above-unity positive feedback
weight w
1
.
As discussed in the Introduction, stable attractor states in the brain, at least temporary
ones, have been hypothesized as the basis for conscious percepts [14,24]. We emphasize
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that although the example given in this section is original work that to the best of our knowl-
edge has not been shown earlier, it is hardly original in spirit and only serves as an introduc-
tory example. Note that technically, besides stable states, attractors can also include stable
oscillations known as limit cycles and chaotic oscillations, which take in extended regions
Fig 2. Backward temporal masking with a single dynamical neuron. (A) Neuron circuit with excitatory self-feedback, Eq (3), illustrates backward
temporal sensory masking. w
1
=w
2
= 1.5, τ= 1. (B) Positive feedback creates two stable equilibria at y=±1 (top trace). These form the minima of a
gradient-descent function V(y) (bottom trace). (C) Depending on the interval between a priming and a countering stimulus of opposite polarity,
masking can occur. (D) The effectiveness of the masking stimulus also depends on its amplitude.
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rather than isolated fixed points, in the phase space of a dynamical system. Mathis and
Mozer [24] were able to correlate several priming experiments to a discrete-time model.
Their model is based on a radial basis function (RBF) attractor network, instead of a specific
type of (networked) artificial neuron, and falls into the category of gradient descent systems.
Formally, in a (multivariable) gradient descent system, the change of the system state
~
yðtÞ ¼ ðy1ðtÞ;...;yKðtÞÞ can be written as:
d~
y
dt ¼  ~
rVð~
yÞ þ~
xðtÞ;ð5Þ
where ~
ris the Nabla operator: ~
rVð~
yÞ ¼ @V
@y1;...;@V
@yK
 . When undriven by a stimulus ~
xðtÞ,
the state evolves downward towards a minimum of a scalar function Vð~
yÞ:RK!R, much
like a golf ball rolling downward to the lowest point of a valley it happens to find itself in.
The influence of the input stimulus is such that it can drive the ball, representing the state of
the system at every point in time t, closer or farther away from specific local minima of the
Vð~
yÞ, corresponding to influencing both which perception the stimulus elicits and the time
it takes to do so. Eq (3) is actually also a gradient system, albeit a one-dimensional one. This
is shown in Fig 2B (lower trace). The piece-wise linear activation function Eq (4) allows inte-
gration of the top trace, so the undriven system can be written as dy/dt =−dV(y)/dy. This
leads to the piece-wise parabolic gradient function V(y), shown in Fig 2B (bottom trace). The
local minima of V(y) are at the stable equilibria y=±1 of the system. The maximum at y=
0 is an unstable equilibrium. An animation in the supporting information, S1 Animation,
shows how the stable “pockets” or valleys are created as the positive feedback w
1
increases.
In above setting, conscious perception coincides with a state sufficiently close to an equilib-
rium of a, not necessarily explicitly described, gradient system. Thus this exemplary system is
capable of two percepts y= +1 and y=−1, when started from y= 0 at t= 0. The outcome
depends on the stimulus x(t) during the time in which the dynamical state y(t) changes. Note
that short-lived or transient states can also represent distinguishable conscious percepts, if
they can cause distinguishable stable states in subsequent processing units exhibiting longer,
possibly infinite, memory.
In Fig 2C (top trace), we show how this simple one-neuron system is useful in a computa-
tional version of a stimulus onset asynchrony experiment. Here, a priming stimulus of unit
strength and duration 0.5 starts the system on a path to experience the positive percept (bot-
tom trace). Some time later, during the transient, a single negative masking stimulus of ampli-
tude −1.5 is administered. Depending on the time elapsed between the priming and masking
stimulus, the system ends up in either one or the other equilibrium. The self-reinforcing feed-
back can be interpreted as the system making a top-down prediction: in the absence of new
information, it will move towards the state that coincides with the direction that the initial
input pushed it towards. Contrasting information, in the form of input steering the state in the
opposite direction can be taken into account, however it must overcome the already accumu-
lated “proof” which, especially in later stages, becomes more difficult to do. That is why in this
example, even though the masking stimuli are stronger than the prime, only the one closest in
succession to the primary stimulus is able to mask it successfully. In Fig 2D we repeat this sim-
ulation for different amplitudes of the masking stimulus but for a fixed time post-stimulus. As
expected, the strength of the mask influences its effectiveness. Note that even if a masking
stimulus is not effective in blocking the prime, it can still delay the response. Thus this simple
model predicts that effective backward sensory masking depends on both amplitude and
elapsed time since the primary stimulus. We have included an animation in the supporting
information, S2 Animation, showing how the state moves towards a minimum of the gradient
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function. This model, given merely to exemplify the merits of dynamical systems, is of course
too simple to catch all scenarios. For example blanking, leading to no perception, is nearly
impossible here since the mask must cancel the primary stimulus exactly. This is due to the
unstable equilibrium in the origin. Adding noise to Eq (3), such as is present in a drift model,
randomizes the outcome and leads to a probability distribution that can be compared against
results from actual experiments. Note that a non-dynamical sum-saturate neuron can be used
to show (discrete-time) dynamical effects such as memory when it is used in the context of a
recurrent network. An example is given by the simple recurrent network [25], where due to
discrete state updates previous inputs play a role in determining the current output.
In the next section, we show a small network of toy model neurons that illustrates the first
part of the CPP, namely how perceptual states and their respective causal stimuli can appear to
be chronologically reversed under certain circumstances.
A dynamical model for stimulus-response order of arrival reversals
In Fig 3A, we show a neuronal circuit with four neurons of the type described by Eq (2). Two
inputs x
a
and x
b
, feed into two chains each having two neurons, top: a
1
,a
2
, bottom: b
1
,b
2
. The
respective outputs of the chains, y
a2
, and y
b2
, are taken to coincide with the perception of the
corresponding stimulus, when above a chosen threshold value. Inhibitory (negative) “long dis-
tance” connections cross couple the inputs to the opposite chain. Formally, the circuit is repre-
sented by:
ta1
dya1
dt ¼  ya1þfðwxa;a1xaþwa1;a1ya1þba1Þ;
ta2
dya2
dt ¼  ya2þfðwa1;a2xaþwa2;a2ya2þwxb;a2þba2Þ;
tb1
dyb1
dt ¼  yb1þfðwxb;b1xaþwb1;b1yb1þbb1Þ;
tb2
dyb2
dt ¼  yb2þfðwb1;b2xaþwb2;b2yb2þwxa;b2þbb2Þ;
ð6Þ
where we chose the parameters as: τ
a1
=τ
b1
= 2, τ
a2
=τ
b2
= 10, w
xa,a1
=w
xb,b1
= 1, w
a1,a2
=w
b1,
b2
= 1, w
xa,b2
=w
xb,a2
=−2, b
a1
=b
a2
=b
b1
=b
b2
=−0.5. As activation function we choose:
fðxÞ ¼
0if x<0;
xif 0x1;
1if x>1:
8
>
>
>
<
>
>
>
:ð7Þ
The threshold of perception was chosen as 0.5, equal to midway of the range of the nonlinear
function f. Conceptually, the existence of this arbitrarily chosen threshold does not have to be
problematic, as subsequent structures that take the outputs of the circuit may be tuned to have
an all-or-nothing response based on this threshold. In Fig 3B, we show how differently spaced
input stimuli (of width δ= 10), cause different response times RT
a
(red) and RT
b
(blue). For
both large positive and negative input intervals, Δt
in
=t
in,b
−t
in,a
, the response arrival order is
unaffected, i.e. first in first out. However, for Δt
in
−12 to 12, we see Δt
out
=t
out,b
−t
out,a
(black) has the opposite sign of Δt
in
, i.e. last in first out. Without the reversal effect, the green
dashed line would be followed. Note that if both inputs are activated simultaneously, then both
RTs are increased, however the interval between the responses remains zero.
Fig 3C shows how the model behaves if we vary both the amplitudes of the input stimuli
and the interval between them. Here, the sum of the amplitudes is kept at 1.5 and their ratios
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are varied. In Fig 3C, the lighter colored central part marks output intervals that are temporally
reversed relative to the input interval. There is a tendency, within certain limits, that the more
stronger a later arriving stimulus is, the easier it can overtake an earlier one. Thus the model
demonstrates how both temporal and amplitude-related characteristics of complex stimuli
influence their temporal perception in a non trivial way.
Fig 3D and 3E illustrate this process in detail. The input stimuli trigger a latching value at
the output of the first neurons, y
a1
and y
b1
, in both chains. These in turn drive the subsequent
stages. This however takes some time due to the slower timescale of neurons a
2
and b
2
. If the
Fig 3. A four-neuron model for demonstrating stimulus-response arrival order reversals. (A) The model. See below Eq (6) for parameter values. (B) A Scan over
stimuli intervals Δt
in
=t
in,b
−t
in,a
showing the influence on response times RT
a
=t
out,a
−t
in,a
(red), RT
b
=t
out,b
−t
in,b
(blue) and the response time interval Δt
out
=t
out,b
−t
out,a
(black). For a certain central region of the input interval, there exists a reversal of the order in which responses happen, compared to the order in which the
stimuli were applied. (C) Scan over stimuli intervals and amplitude ratios, illustrating how temporal and amplitude-related stimuli characteristics influence their
temporal perception. (D) Detailed signals for t
in,b
t
in,a
such that the output order corresponds to the input order. (E) Detailed signals in the input-output reversal
regime, where Δt
in
>0 and Δt
out
<0.
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input x
b
is activated while the output value y
a2
is still accumulating, Fig 3E, then the inhibitory
synapse slows down the a-branch response. This leads to y
b2
crossing the threshold before
y
a2
, even if stimulus x
a
was present before x
b
was. Since the circuit is symmetrical along the
branches, swapping the input stimuli leads to swapped output signals.
The circuit of Fig 3A clearly illustrates how dynamics analogous to anomalous order-
reversed perception can be found in small dynamical networks. The key ingredient is the set of
cross coupled inhibitory connections which jump forward in the chains, such that information
that arrives at a later time can intercept partially processed information at the opposing chain
as it flows from input to output. Loosely speaking, we make an exchange of time for space.
In the next section, we illustrate how the second aspect of the CPP, an apparent movement
or “ghost” perception, can be found in a similar small network motif.
A dynamical model for the phi phenomenon
As mentioned, the color phi phenomenon has two aspects. In this section we explain how the
appearance of a ghost percept, spatially and temporally placed in between percepts that corre-
spond to actual stimuli, might be understood as a dynamical priming effect. In Fig 4A, we
show a toy model neuron circuit consisting of 15 units, divided in three chains a,band c. It
represents a very crude one-dimensional visual system having three pixels. Stimuli on the
respective inputs x
a
,x
b
and x
c
,Fig 4B (top), lead after some propagation time to perceptions as
indicated by the outputs y
a
,y
b
and y
c
(bottom) being above a fixed threshold value. Each chain
couples to the adjacent chain(s) via weak long distance feedforward synapses, shown as dotted
connections in Fig 4A. These are driven by primer neurons p
a
,p
b
and p
c
,Fig 4B (middle), that
have a fast activation and a slow decay, the latter due to the strong self-feedback and lack of
negative bias. In Fig 4B (middle), the dotted line shows the combined input of the primers into
the final neuron of the b-chain. As long as the interval between the stimuli on the aand cchain
is sufficiently large, their combined priming signal does not exceed the activation threshold of
the b-chain, Fig 4C. However, when the stimulus on cswiftly follows the one on a, as in Fig
4D, priming exceeds the threshold and elicits a response at y
b
. As in the preceding section, we
find a timing anomaly: bis initially “perceived” after aand before c, here close to t= 115. An
animation showing the situation for intermediate a-cstimulus intervals can be found in the
supporting information, S3 Animation. Since the network is symmetric, the same effect exists
when the stimuli are applied in reverse.
One may wonder how this simple model can be interpreted within a biological context. If a
moving object, e.g. a prey animal, is seen at some position, say b, it is natural to assume that it
will be moving to an adjacent position, here either aor c, next. The priming signals thus per-
form the function of expectation. Priming for the adjacent positions helps reduce response
times, and increases the chance of catching that prey. Likewise, if an object is perceived as
quickly changing its location from ato c,Fig 4D, then by assuming movement and thereby
inferring that a path exists between the positions, the organism is potentially given useful
information about the environment. If the objects are perceived with a greater temporal inter-
val at positions that are not adjacent, as in Fig 4C, they are likely not the same and the existence
of a pathway cannot be inferred. Thus this simple model demonstrates that perception, even
when caused mainly by feedforward neuronal pathways, can in part be prediction [13]. Note
that these interpretations are meant to place the abstract model of Fig 4 in a more understand-
able context, and by no means do we want to state physiological facts.
Clearly, this simple hand assembled model fails in two ways. First, in Fig 4D, there is over-
lap between the band cpercepts. Second, if x
a
and x
c
are stimulated at the same time, y
b
will
also be seen, while in an actual experiment only the two outer position dots would be perceived
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simultaneously. These shortcomings could perhaps be alleviated at the expense of increased
model complexity, by using inhibitory connections from the inputs to the outputs. Similar to
the case of the order reversal model, as shown in Fig 3C, the stimulus characteristics influence
whether or not a phi phenomenon happens. Our code S1 Data contains a script that plots the
duration of the ghosts as a function of stimulus interval and amplitude ratio.
In the next section, we show a different approach to finding such a model and demonstrate
that the CPP could be understood as nothing more than a side effect of dynamical processing.
The color phi phenomenon in an echo state network
The models presented in the previous sections illustrate both aspects of the CPP as discussed.
Unfortunately, there seems to be no straightforward way to combine their structure into a sin-
gle CPP model. Therefore, we turned to neural network paradigms to find such a model.
Fig 4. An abstract one-dimensional three-pixel visual system, illustrating the phi phenomenon. (A) The circuit.
Outputs y
a
,y
b
and y
c
separately respond to inputs x
a
,x
b
and x
c
respectively, corresponding to perception when above a
certain threshold. Weights and τ-values of the aand c-chains are identical as those indicated in the b-chain. (B) Long
distance weakly coupled feedforward priming units p
a
,p
b
and p
c
(middle graph) with fast activation and slow decay
enable sensitivity to recent inputs to be carried to adjacent output units. (C) If the input pulses on x
a
and x
c
are
temporally sufficiently separated, the decayed priming signals p
a
and p
c
are unable to evoke a response from the middle
output y
b
. (D) When inputs x
a
and x
c
are stimulated adequately fast after one another, the priming signals combine
and a ghost percept on y
b
appears. This ghost happens after y
a
and before y
c
cross the threshold in opposite directions.
The ghost appears without the presence of an input stimulus on x
b
.
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Specifically, we employ an echo state network (ESN) [26]. For completeness, we briefly
explain the philosophy behind the ESN, after which we formally introduce our model. ESNs
where envisioned to circumvent the problem of slow backpropagation learning found in tradi-
tional multilayer neural networks [27,28] and to be able to exploit the computational abilities
of high-dimensional nonlinear physical substrates in which the constituents have a fixed and
often unknown interconnectivity. More in-depth discussions of ESNs, part of the larger group
of reservoir computing methods, can be found in Refs [29–31].
Instead of adjusting individual weights between neurons, such that the network itself per-
forms a certain prescribed function, in an ESN, Fig 5, one starts from a large recurrent net-
work, also called a reservoir, in which the weights between the network’s neurons or nodes are
chosen at random, within certain constraints, and kept fixed throughout the experiment. The
presence of recurrent connections implies the existence of time scales or memory on top of
those offered by the neurons’ individual dynamical behavior. An input signal is connected,
usually also via randomly chosen weights, and excites the network. For our purposes, the ESN
will have 3 ×2 inputs corresponding to left, middle and right positions in two colors, blue and
red, similar to the abstract one-dimensional visual system described in the previous section.
The complicated time-dependent response or “echo” in the network caused by the time-vary-
ing input is then recorded. Mathematically, the input signal is non-linearly projected into the
high-dimensional phase space of the dynamical network. The training phase of the ESN con-
sists of constructing the desired output response from a linear combination of the response
signals of the individual neurons. The weight values or coefficients of this linear combination
are collectively known as the readout layer. Mathematically, the readout layer forms a down-
projection from the phase space of the network onto a lower dimensional output space. The
main idea behind ESNs is that as long as the network shows what is loosely referred to as “rich
dynamical behavior” in response to the input, chances are good that a projection exists that
maps the dynamical response of the network sufficiently close to the desired output. The calcu-
lation of the readout layer is a simple linear regression, which is computationally relatively
cheap and fast. In this case, we have five outputs: three for the position and two for color. As
such, the quite easy task of the ESN is to classify a single active input. Furthermore, when sev-
eral inputs are active at the same time, we demand that all outputs must remain inactive. Since
the ESN is a dynamical system, it will show transition effects between different input stimuli.
Fig 5. Illustration of the color phi phenomenon via an echo state network. A large randomly connected network of neurons, also
called a reservoir, interacts in a complicated and unknown way in response to an input stimulus. Theconnections between the
neurons in the network remain fixed for the duration of the experiment. The desired output response is constructed from a linear
combination of the responses of the individual neurons that form the network. After training, the ESN can generalize to novel input
data.
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After training, the ESN will be tested for the CPP as further outlined below. It is crucial to
remark here that the network is not trained to show the CPP explicitly. As explained in detail
in the next section, the ESN consists of a discrete-time version of the toy model neurons,
Eq (2).
Interpreted as generating perceptions, the readout layer is the place where “it all comes
together”, i. e. the Cartesian theater. Note that it is not correct to interpret the readout layer
as simply being a reporting tool for the network, a voice box so to speak, since without it, the
computations that happen in the network do not get assembled into an active representation
and remain inert. Alternatively, the outputs of the readout layer could be seen as a behavioral
response. The reservoir itself could be extended to have specialized readout neurons, the out-
puts of which perform exactly as the readout layer.
ESN outline
In this section we formally describe the ESN we use to demonstrate the CPP. Starting from a
discretization with timestep T
s
, setting t=nT
s
, we replace differentials with differences as:
dy
dt yðnTsÞ  yðnTsTsÞ
Ts
;ð8Þ
such that the toy model neuron Eq (2) takes on the form of a recurrence equation:
ynþ1¼ ð1aÞynþafX
N
i
wixiþb
!;ð9Þ
in which nis the discrete timestep and the neuron timescale is given by α= (1 + τ/T
s
)
−1
. Com-
pared to the artificial neurons used in Cohen [15], in Eq (9) the differentiating action and the
saturating function are swapped, however their steady-state output remains the same. The
ESN state update equation is functionally equivalent to what is found in Ref [23]:
~
xnþ1¼ ðIAÞ~
xnþAf Wres~
xnþWin~
xin;nþ~
b
 ;ð10Þ
in which~
xn¼ ðx1;n;. . . ;xN;nÞis the network state vector at timestep n. In Eq (10), vectors are to
be interpreted as column matrices. Furthermore, W
res
is the N×Nnetwork connectivity matrix,
W
in
is the N×6 input weights matrix and ~
b¼ ðb1;. . . ;bNÞis a constant random bias vector.
Vector ~
xin represents the six inputs as shown in Fig 5. In our experiments, we set the network
size to N= 200 neurons. Network hyperparameters were chosen as suggested in Ref. [32]. The
network connectivity matrix is constructed by drawing random numbers from a normal distri-
bution Nðm¼0;s¼1Þin which 80% of the entries is set to zero (sparsity 0.8). Then the
matrix is scaled to obtain a spectral radius of 0.9. Similarly, the input matrix W
in
is constructed
from a normal distribution Nðm¼0;s¼0:5Þwith sparsity set to 0.8. The timescales of the
individual neurons are drawn from a uniform distribution aiUðlow ¼0:1;high ¼0:3Þ, and
collected in a diagonal matrix A
i,i
=α
i
. Here, fis the sigmoidal activation function. Both during
training and testing of the ESN, the network states in response to the input signal are collected
in a state matrix Xof dimensions K×(N+ 1). Each row of Xconsists of the network state ~
xnat
time n= 1, . . .,K. The last column of Xis constant. The readout layer consists of a (N+ 1) ×5
matrix W
out
. The output of the ESN is calculated as:
Y¼XWout;ð11Þ
where Yis a K×5 matrix, of which each column corresponds to an output signal as shown in
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Fig 5. The n-th row of Ycorresponds to the output vector at timestep n,~
yn¼ ðy1;n;. . . y5;nÞ.
The readout layer W
out
is determined using a linear regression, such that the output Y
train
in
response to a training input that causes network states X
train
comes close, in a least-squares
sense, to a predefined desired output Y
desired
:
Wout ¼ ðXT
trainXtrainÞ1XT
trainYdesired ð12Þ
A well-trained ESN can generalize to previously unseen input data. We refer the reader to
Refs [29–31], as further details about the ESN are not important for the remainder of this
article.
ESN experimental results
In this section we show how the CPP surfaces in the ESN we described above. As shown in Fig
6, the training input (top graph) consists of valid one-position/one-color stimuli, for which the
desired output (third graph) is the corresponding position and color. From n= 13000 onward,
the training stimulus consists of invalid input conditions in which multiple color and/or posi-
tion inputs are active simultaneously. For these confusing stimuli, we require the network to
not respond (keep all outputs at zero). A shift of 20 timesteps between input and desired out-
put was allowed to give the network time to respond. After training, i.e. calculating the readout
Fig 6. ESN during training. (top) Training input, (second graph) states of the first ten neurons, (third graph) desired output used to calculate the
readout layer, (bottom graph) actual output calculated by the readout layer from the neuron states after training.
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layer W
out
, we can show the resulting actual output in response to the training input (bottom
graph of Fig 6). The output is assembled from the complex network dynamics (first ten neu-
rons shown in the second graph of Fig 6). Ideally, it should be identical to the desired output.
An initial transient and some noise-like deviation is seen, however qualitatively the output
comes close to what is desired.
It is vital for this discussion to emphasize that we do not train with input conditions that
could lead to the CPP: transitions between the conditions left-to-right or right-to-left of oppo-
site color without passing via an activated middle input are explicitly excluded from the valid
training examples. Furthermore, input conditions are sufficiently spaced as to allow the net-
work’s internal dynamics to settle to a stable output. We are now ready to demonstrate that
this neural network can undergo a color phi illusion. In Fig 7A (top), we apply several CPP
input conditions, i.e. a stimulus that jumps from left-red to right-blue, with decreasing inter-
val. Note that the left-red and right-blue input pulses never overlap. Ideally, the middle output
would remain low (middle graph), corresponding to having not perceived anything anoma-
lous. However, as the left-red to right-blue jumps get closer to one another, at n3490, as
shown in the zoomed graph of Fig 7B (bottom graph), the middle output is briefly active above
the threshold of 0.5 (dashed horizontal line), while the right output is still below this threshold.
This corresponds to a ghost or phi phenomenon as elaborated on in the previous sections.
More so, at the same time the blue output is also active, thus the output of the ESN corre-
sponds with the perception of this ghost in the middle position and in the final color before
correctly perceiving the second applied stimulus of that color.
Fig 7. Echo state network during testing for the color phi phenomenon. (A) (top) Stimuli jumping between left-red and right-blue with decreasing
intervals, (middle graph) ideal response with low middle output, (bottom graph) actual output showing CPP near n3490. (B) Zoom-in of (A). The
vertical dashed green line in the bottom graph indicates the time at which the CPP occurs.
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The dashed vertical line in Fig 7B (bottom graph) indicates the point in time where this
condition first occurs. Thus the system portrays all the characteristics of undergoing a color
phi illusion. The supporting information contains a short clip S4 Animation produced directly
from the timeseries of Fig 7A, showing how the ESN “perceives” the CPP. The perception of
the final transition from left red to right blue briefly shows a blue dot in the middle.
Analysis. Since the model is essentially a system of N= 200 coupled nonlinear difference
equations, we cannot explain in detail how this behavior arises. It is obvious, however, that
simple dynamical interactions give rise to the existence of the CPP here. Nevertheless, it is
important to ask if there are any discernible differences between ESNs that do and those that
do not show the CPP effect. In Fig 8 we present four network characteristics that we specifi-
cally checked to identify such differences. The plots are based on N= 100000 randomly
assembled ESNs in total. In general, only a low fraction of the ESNs show the CPP. In our
experiments this was 1.87%. Fig 8A shows the distribution of α-parameters of the neurons.
In Fig 8B we plot the magnitude of the eigenvalues of the reservoir matrix W
res
. Both of these
metrics govern the timescales at which the dynamics evolve. There is no clear difference
between networks with and without the CPP. In Fig 8C we plot the ratio of excitatory (posi-
tive) to inhibitory (negative) synapses. Finally, Fig 8D shows the number of positive long dis-
tance feedforward connections (LDFF) in the networks. Here, a connection from neuron ito
neuron jis counted as LDFF, if ipositively influences jvia both a direct and an indirect path.
This test was suggested by our phi model, where priming was caused by LDFF connections.
Again, no differences between networks that show the CPP and those that do not come for-
ward. These negative results suggest -but of course do not prove- that it truly is the detailed
internal structure that matters, and the ESNs are otherwise macroscopically indistinguish-
able. We suspect only a small portion of the neurons in the ESNs assists in generating the
CPP.
In Fig 9, we show a brief exploration of how robust the results are with respect to the
choices of network parameters. Here, we scan over reservoir and input sparsity, spectral radius
and input scaling, the latter being the σ-value with which the input matrix W
in
is constructed.
The non-varied parameters are as described in the previous section. Each point is the fraction
pout of n= 200 ENSs that showed the CPP. The error bars are estimated at the 95% confidence
interval using Wald’s method: pzffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pð1pÞ=n
p, with z= 1.96.
It must be remarked that invariably such parameters are highly interdependent, such that
for the purpose of optimization they must be probed together. We did not explore this any fur-
ther as this is was outside of the scope of this article.
Fig 8. Some characteristics of the ESNs that do (N
with
= 1867) and do not (N
without
= 98133) show the CPP. (A) Distribution of neuron α-
timescales, (B) magnitude of the reservoir matrix eigenvalues, (C) ratio of excitatory to inhibitory synapses, (D) number of positive long distance
feedforward connections.
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In the next and final section we summarize our results and look at how they may be inter-
preted in light of the importance of the CPP.
Discussion
Our goal in this article was to show that small connectionist networks built from crude toy
model neurons can capture the main aspects of the color phi phenomenon (CPP). The CPP
was described by Dennett [19] as a landmark example of the constructive character of con-
scious experience: rather than merely recording events as they unfold over time, subjective
experience involves both preconstructive and reconstructive mechanisms that constitute a
coherent narrative for the subject whose experiences it is. There is thus no central “theater”
in which unfolding events play out; rather, says Dennett, consciousness builds on “multiple
drafts” out of which one narrative will eventually win out.
With this in mind, we simply asked which kinds of neural architectures might naturally
lead to the striking reversals observed in the CPP. Our approach was based on the following
assumption: if consciousness is something physical (as it must be), then there must be mecha-
nisms through which objective stimuli are mapped into objectively measurable neuronal acti-
vations which either constitute or drive conscious perception in a way that is congruent with
subjective reports. The core aspect of what needs to be explained in the case of the CPP is the
chronological anomaly it presents us with: how can the color of the second stimulus be experi-
enced before its position is? Here, we set out to explore which kinds of neural network archi-
tecture might produce such effects without further assumptions, and we showed that the
dynamical properties of relatively simple elementary networks naturally produce CPP-like
effects. Our models are relatively simple circuits that are assembled from toy model neurons.
They nevertheless have three essential properties in common with real biological neural sys-
tems: modularity, nonlinearity and time dependence. Our work is connected to the ideas
recently developed by Herzog et al. [4], who proposed an explanation of visual perception that
operates in two stages. The first stage consists of a continuous and unconscious integration of
sensory information into features, best representing the stimulus. These features include time
labels. During processing, all features are continuously updated yet not reportable. In a second
stage, the features are rendered into consciousness all at once. We demonstrated stimulus-
response order reversal in a four-neuron network that uses local recurrence, showing that, at
least in this scenario, such time labels are unnecessary.
In line with Herzog et al., our simple model shows continuous feature integration and a by
definition discrete, threshold-like rendering of features into conscious percepts. Additionally,
Fig 9. Fraction of ESNs that show the CPP as a function of several network parameters. (A) Reservoir sparsity, (B) input sparsity, (C) spectral
radius, (D) input scaling. Each point represents n= 200 tested networks. The error bars are estimated at the 95% confidence interval. Non-varied
parameters kept as described in the text.
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timing labels may be constructed as neuronal activaty, the value of which is related to the time
since the onset of a (not yet reportable) stimulus. A thresholded signal derived from the stimu-
lus as a whole acts as a gatekeeper. Naturally, a neural timing signal would have to be relative
to the larger perceptual context or mental state, rather than present any objective wall clock
time. In summary, the experience of time is just like any other percept: a construction. Our
four-neuron time reversal model illustrates how little machinery is needed to falsify this expe-
rience. Thus we conclude that our models, if needed, could incorporate time label features as
proposed by Herzog et al. [4].
Following the breakdown of possible types of temporal consciousness in Herzog et al. [33],
we consider our models to come closest to the discrete retentional class, albeit with percepts of
finite non-zero duration which is at least in part governed by the stimuli. In our ESN model,
not all feature perceptions are updated simultaneously and this is where the CPP-like effect
originates from: the “color” feature is updated (meaning; crosses a perceptual threshold) asyn-
chronously from the“position” feature, see Fig 7B. Thus features can be updated all at once,
without noticeable timing differences, but this is not a given, as it generally depends on the
characteristics of the stimulus and probably also on the larger perceptual context. What our
ESN model indicates is that the CPP is the result of dispersion between features that stimulus-
wise belong to each other, but are processed by different pathways. Under the right conditions,
subtle timing differences caused by the different velocities with which the features traverse the
neural substrate cause a noticeable binding error.
Fekete et al. [34] have argued at length against a perception that is tied down in a “discrete
straightjacket”, on the basis of both empirical physiological data and philosophical arguments.
One strong argument against a fully discrete awareness is the asynchronicity of inter area com-
munications of the brain.
It is interesting to note that there are common elements in the structure of the order rever-
sal and the phi phenomenon networks we presented. Both have local recurrences and long-dis-
tance feedforward connections. Local recurrences evidently may induce long lasting states
without the need for large buffers and feedforward priming connections, as discussed in the
section on the phi phenomenon model, can be interpreted as a form of prediction. Sadly, these
simple models do not offer a clear path back into how they might be represented in the actual
neural architecture of the brain, although functionally similar circuits must exist at some level.
We see several directions in which we could expand this work. First, it is clear that the inter-
actions that take place in the networks we proposed are influenced by the amplitude, duration
and interval characteristics of the stimuli, something we did not fully explore, except briefly
for the introductory masking example and in Fig 3C. In the latter it was shown that the relative
strength of the stimuli influences the temporal perception of their intervals. Results from such
explorations could be compared to empirical data with the goal of finding similar scaling laws.
Second, we did not offer any explanation as to how the proposed neuronal structures could
have emerged in the first place. In other words: under which conditions might the relevant
weights be learned instead of merely being hand-tuned? It is straightforward to equip the leaky
integrator neurons with a learning rule that mimics spike-timing-dependent-plasticity, by set-
ting _
wij ¼Zxi_
yj, which can be interpreted as follows: when input x
i
is active while output y
j
increases, then this input predicts the increased output value, hence the associated weight
should increase. This learning rule was recently proposed by Bengio et al. [35,36]. It would be
interesting to investigate under which circumstances the CPP emerges in a plastic neural net-
work, as opposed to the ESN which has fixed randomly chosen internal weights.
In our models, we (naively?) interpreted the end of a percept as coinciding with crossing
the same threshold as its onset but in the opposite direction. The animation S4 Animation in
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the supplementary information of how the ESN “perceives” the CPP, based on the timeseries
of Fig 7, is constructed in this way. However, as many experiments have shown, there clearly is
an asymmetry in the sensitivity of when something emerges vs. when it disappears from our
perception, even in very simple visual scenarios. Such hysteresis effects have been hypothesized
by Poltoratski and Tong [37] to disambiguate visual perception during the processing of scene
transitions. In other words: the hysteretic thresholds are themselves a function of the scene in
its entirety. Since a continuous signal -the precursor of a conscious percept- cannot pass two
unequal values instantaneously, the existence of hysteresis naturally implies percepts of non-
zero duration. Hysteresis effects could be explored further using the modeling approach
shown in this article.
We stress again the crucial point that during the training phase of the ESN we explicitly
excluded input conditions that could invoke a CPP: the readout layer was constructed from
the network’s responses to input stimuli that did not contain direct left-to-right or right-to-left
jumps of opposite color. We found 1.87% of N= 100000 networks showed the CPP, with the
network parameters we chose. Kim and Senjowski [38] showed that for a spiking recurrent
neural network performing a working memory task, network characteristics that strongly
influence the emerging timescales, and which are thus good predictors of performance, can be
found. We similarly analyzed several characteristics of our ESNs, but failed to find markers
that predicted the presence or absence of the CPP.
Although the ESN, or more general reservoir computing, has been described as a “brain-
inspired” machine learning paradigm, one could argue that the readout layer, of which the
construction is fundamentally based on a matrix inversion, is all but biologically plausible.
Thus we can interpret the ESN simply as a function approximator which in itself has no rela-
tion to consciousness or the brain whatsoever. Stated otherwise: the ESN shown here exhibits
sensitivity to input conditions that invoke the CPP, but remains mute concerning conscious
perception. We conclude that the CPP, while an intriguing visual illusion in its own right,
might be nothing more than a consequence of the inherent dynamics that characterize infor-
mation processing in the brain.
Supporting information
S1 Animation. An animation showing the creation of stable states by feedback in a single
toy model neuron network used for demonstrating priming.
(WMV)
S2 Animation. An animation of a bistable system with positive input stimulus masked by
negative stimuli with decreasing interval.
(WMV)
S3 Animation. An animation of the phi phenomenon, the appearance of a “ghost” percept
in dynamical neuronal network representing an abstract three pixel visual system.
(WMV)
S4 Animation. An animation of the color phi phenomenon based on the timeseries found
in Fig 7A as “perceived” by the ESN during testing.
(WMV)
S1 Data. Contains the Python scripts that directly generate the graphs of this article.
(ZIP)
PLOS COMPUTATIONAL BIOLOGY
The color phi phenomenon: Not so special, after all?
PLOS Computational Biology | https://doi.org/10.1371/journal.pcbi.1009344 September 3, 2021 19 / 21

Acknowledgments
AC is a Research Director with the Fonds de la Recherche Scientifique F.R.S.-FNRS (Belgium).
Author Contributions
Conceptualization: Lars Keuninckx, Axel Cleeremans.
Formal analysis: Lars Keuninckx.
Funding acquisition: Axel Cleeremans.
Investigation: Lars Keuninckx, Axel Cleeremans.
Methodology: Lars Keuninckx, Axel Cleeremans.
Project administration: Lars Keuninckx, Axel Cleeremans.
Resources: Lars Keuninckx, Axel Cleeremans.
Software: Lars Keuninckx.
Supervision: Axel Cleeremans.
Validation: Axel Cleeremans.
Visualization: Lars Keuninckx.
Writing – original draft: Lars Keuninckx, Axel Cleeremans.
Writing – review & editing: Lars Keuninckx, Axel Cleeremans.
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