2004 Special Issue

Using afterimages to test neural mechanisms for perceptual filling-in

Gregory Francis*, Justin Ericson

Department of Psychological Sciences, Purdue University, 703 Third Street, West Lafayette, IN 47907-2004, USA

Received 30 January 2004; accepted 30 January 2004

Abstract

Many theories of visual perception propose that brightness information spreads from edges to define the perceived intensity of the interior

of visual surfaces. Several theories of visual perception have hypothesized that this filling-in process is similar to a diffusion of information

where the signals coding brightness spread to nearest neighbors. This paper shows that diffusive mechanisms fail to account for the

characteristics of certain afterimage percepts that seem to be dependent on the filling-in process. A psychophysical experiment tests a key

property of diffusion-based filling-in mechanisms and finds data that rejects this class of models. A non-diffusive based filling-in mechanism

is proposed and is shown to act much like the diffusive based mechanism in many instances, but also produces afterimage percepts that match

the experimental data.

q 2004 Elsevier Ltd. All rights reserved.

Keywords: Afterimage; Brightness perception; Filling-in

1. Introduction

The front cover of this journal includes alternating pink

and blue crosses in the intersections of a grid of hypothetical

neural crossings. This image produces a visual effect known

as neon color spreading where there appears to be a fuzzy

pink disc around each of the pink crosses (da Pos & Bressan,

2003; Van Tuijl, 1975). This visual appearance is entirely

illusory, as all the pink on the printed page is actually within

the lines of the crosses and the surrounding areas are printed

in white. Under the proper viewing conditions the effect can

be very striking so that an illusory pink disc hides a physical

pink cross. Sometimes the discs in different intersections

merge together to form diagonal bands across the grid.

Neon color spreading and related phenomena have been

taken as evidence that visual perception involves a filling-in

process that computes information about perceived colors

and brightness across surfaces (Gerrits & Vendrik, 1970;

Pessoa, Thompson, & Noe, 1998). In one of the most

detailed descriptions of this process (Grossberg & Mingolla,

1985a), a feature contour system (FCS) utilizes a filling-in

process that computes and distributes brightness and color

information across a region, but the filling-in is restricted by

signals from a boundary contour system (BCS) that block

the filling-in process from spreading into adjacent regions.

In the particular case visible on the cover of this journal, the

pink color of a cross leaks out of its normal position and

fills-in an illusory contour that is created by the darker lines

that surround the cross (see Gove, Grossberg, and Mingolla

(1995) for simulations of these processes).

Filling-in processes have also been hypothesized to play

a critical role in accounting for visual perception in general

(both veridical and illusory percepts). Such applications

include brightness perception (Grossberg & Todorovic,

1988; Todorovic, 1987), properties of McCollough after-

images (Broerse, Vladusich, & O’Shea, 1999; Grossberg,

Hwang, & Mingolla, 2002), properties of color complement

afterimages (Shimojo, Kamitani, & Nishida, 2001), figure-

ground segmentation (Grossberg & Wyse, 1991), and some

aspects of 3D perception (Grossberg, 1997).

This paper investigates and theorizes about the compu-

tational properties of the filling-in process. Historically,

filling-in has been described as an isotropic diffusion of

information from edges to interiors of regions (Gerrits &

Vendrik, 1970). Much psychophysical data is consistent

with this idea (Pessoa et al., 1998; Rudd & Arrington, 2001).

For example, Paradiso and Nakayama (1991) demonstrated

that a circular mask appeared to block the diffusive spread

of brightness information (see also Arrington (1994) and

Stoper and Mansfield (1978)).

0893-6080/$ - see front matter q 2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.neunet.2004.01.007

Neural Networks 17 (2004) 737–752

www.elsevier.com/locate/neunet

* Corresponding author. Tel.: þ1-765-494-6934.

E-mail address: gfrancis@psych.purdue.edu (G. Francis).

On the other hand, neurophysiological evidence on

filling-in mechanisms has been less clear. Early reports on

the representation of edge and surface information in area

V1 of monkeys suggested that while edges were coded by

orientation-sensitive neurons, color-sensitive neurons were

not orientationally tuned (e.g. Livingstone & Hubel, 1984).

This has been taken as evidence for an anatomical

segregation of form (edges) and surface (color) information

in visual cortex. Also in agreement with the existence of a

filling-in process, Komatsu, Kinoshita, and Murakami

(2000) measured activity from cells responding to a

homogeneous pattern that covered the blind spot. Since

the blind spot receives no signals from the eye, this activity

implies the presence of a filling-in mechanism. While these

findings suggest the existence of some type of filling-in

mechanism, they do not help identify the properties of the

mechanism. Contrary to the idea of filling-in as an isotropic

diffusion of information, Friedman, Zhou, and von der

Heydt (2003) reported that many color-sensitive cells are

also highly orientation-selective. Friedman et al. argued that

current neurophysiological evidence no longer supports the

hypothesized anatomical separation of form and color

information. If the filling-in mechanism involves these

neurons, then it seems that the mechanism has some

orientation specificity and does not involve isotropic

diffusion.

In this paper we investigate computational mechanisms

for filling-in. We show how the properties of a recently

discovered afterimage (Francis & Rothmayer, 2003;

Vidyasagar, Buzas, Kisyarday, & Eysel, 1999) can be

used to test models of filling-in. Francis and Rothmayer

(2003) found that viewing two orthogonal bar gratings,

presented one after the other, produced an afterimage

similar to the first of the gratings. The orthogonal orientation

of the second bar grating relative to the first seemed to

be critical for the appearance of the afterimage. Francis

and Rothmayer (2003) also reported simulations

of the FACADE model proposed by Grossberg and

colleagues (Cohen & Grossberg, 1984; Grossberg &

Mingolla, 1985a,b; Grossberg, 1987, 1994, 1997) which

includes a stage of filling-in. Francis and Rothmayer (2003)

argued that the filling-in stage played a critical role in the

appearance of the afterimage. We now continue this line of

analysis to argue that the filling-in mechanism cannot be

based on isotropic diffusion. We report experimental results

that contradict a key prediction of such a mechanism and

suggest an alternative mechanism that better agrees with the

experimental data.

2. FACADE model and afterimages

FACADE is an acronym for Form And Color And

DEpth. It is an extension of a model proposed by Grossberg

and Mingolla (1985a,b), who suggested that computational

processing in the visual system is divided into a BCS that

processes edge, or boundary, information and a FCS that

retains information about surface colors and brightness and

also provides stages for filling-in of that information to

identify the color and brightness of surfaces. The BCS is

concerned with identifying the location and orientation of

edge-like information. The filling-in stage in the FCS uses

the layout of BCS boundary information to define the spread

of surface information. The FCS signals diffusively spread

information only within a set of connected BCS signals.

Separate closed regions correspond to surfaces with

different perceived brightness or color.

Fig. 1 schematizes the key components of FACADE that

will be used in the following discussion. Not all connections

and interactions are drawn in this schematic representation,

and for simplicity the discussion is restricted to achromatic

colors. The input image projects to a pixel representation of

the black and white components of the image. At each pixel,

the opposite color representations compete in a gated dipole

circuit (Grossberg, 1972), which creates after responses at

offset of a stimulus. A gated dipole circuit includes parallel

channels that compete with each other as signals pass from

lower to higher levels of the circuit. Feeding this

Fig. 1. A schematic of the main components of FACADE theory. The input

image feeds into a retinotopic representation of black and white, which

compete in a gated dipole circuit. The gated dipole circuit produces

complementary after responses. The black and white information then feeds

into edge detection in the BCS, which also contains a gated dipole circuit

whose after responses code orthogonal orientations. The edges in the BCS

guide the spread of black and white information in the filling-in stage to

limit the spread of color and brightness information.

G. Francis, J. Ericson / Neural Networks 17 (2004) 737–752738

competition are inputs gated by habituative transmitter gates

(schematized as boxes). At the offset of stimulation, a gated

dipole circuit produces a reduction in cross-channel

inhibition from the stimulated channel to the unstimulated

channel. This leads to a transient rebound of activity in the

unstimulated pathway. Thus, for a color gated dipole circuit,

offset of input to the white channel leads to a brief after

response in the black channel, and vice-versa.

The color signal at the output of the gated dipole then

projects to two different systems: the BCS and the FCS.

Cells in the BCS are sensitive to oriented patterns of

intensity and correspond to the simple and complex cells of

areas V1 and V2 (see Grossberg, Mingolla, and Ross (1997)

and Raizada and Grossberg (2003) for neurophysiological

interpretations of the BCS). Within the BCS is another gated

dipole circuit that codes orthogonal orientations. Thus,

offset of input driving a horizontally tuned cell will lead to

an after response in a vertically tuned cell that codes the

same retinal position, and vice-versa. Other computations,

such as excitatory feedback that groups together common

orientations, also take place in the BCS to insure that

boundaries define and segment appropriate regions of an

image (Grossberg and Mingolla, 1985a,b). The orientation

gated dipole helps to control the duration of persisting

responses that are generated by the excitatory feedback

loops (Francis, Grossberg, & Mingolla, 1994).

Activities from the top level of the color gated dipole and

the BCS both feed into the FCS filling-in stage. The filling-

in stage contains separate systems for the representation of

black and white. The cells in these systems are closely

connected so that activity quickly spreads to neighboring

cells. As a result, representations of black and white quickly

diffuse across cells to fill-in different regions. The diffusion

of neural activity is blocked by the presence of oriented

BCS signals. In this way, the BCS orientation signals

separate different regions, which can then fill-in with

different brightness intensities.

The filling-in process can be understood by observing the

model’s response to a classic brightness illusion. Fig. 2a

demonstrates a simulation where the input image is similar

to one that produces a brightness illusion called the Craik–

O’Brien–Cornsweet effect (Cornsweet, 1970; Todorovic,

1987). The image (lower left) is a gray box that is divided

into two halves. At the boundary between the halves, a

luminance cusp is either lighter (left) or darker (right) than

the gray of the box. Moving away from the center edge

causes the lighter or darker intensity to fade into the gray of

the box. At the far left and far right side of the gray box, the

intensity is identical. For a properly constructed image of

this type (the image used in the simulation was deliberately

chosen to show the basic structure of the image rather than

to produce the visible COC effect) an observer will report

that the left half is brighter than the right half. Significantly,

the percept is of the entire half being brighter or darker, not

only the parts next to the central cusp. Thus, it appears that

brightness information at the central edge has spread to, or

filled-in, an entire region. This effect has been taken as

evidence of some kind of filling-in mechanism (Todorovic,

1987).

Fig. 2a shows simulation results using a diffusive filling-

in mechanism as in Grossberg and Todorovic (1988).

Details of the simulation are given in the appendix. Each

image plots values across all the pixels from one level of the

model. Each image combines information from two sources

(e.g. black and white colors or horizontal and vertical

orientations) by subtracting the simulation value of one

component from the other at each pixel. The resulting

Fig. 2. Simulation results of the Craik–O’Brien–Cornsweet effect. For both

simulations the input image (lower left), color gated dipole responses (lower

right), and oriented boundary signals (upper left) are the same. The filled-in

activity is quite similar for both filling-in mechanisms. (a) Simulation results

using a diffusive based filling-in mechanism. (b) Simulation results using a

non-diffusive based filling-in mechanism.

G. Francis, J. Ericson / Neural Networks 17 (2004) 737–752 739

difference of values is plotted with an intensity that codes

the relative value of activity. This is done by always setting

the value zero as middle gray. Positive values are then

plotted as lighter grays and negative values are plotted as

darker grays. For each plot, the simulated values are

normalized and then mapped to gray levels. Thus, the

largest absolute value of each set of difference values will be

set to black (if negative) or white (if positive). Above or

below each image is a symbol or phrase to indicate what

model-level activity is being plotted. The lower left image is

the input image. The lower right image corresponds to the

output of the color-gated dipole. The upper left image

corresponds to the representation of vertical and horizontal

orientations after the orientation gated dipole and the

feedback grouping process. The upper right image corre-

sponds to the black and white filling-in stages. The

distribution of activities across the filling-in stages corre-

sponds to the visual percept. These levels are discussed in

detail below.

The image on the bottom right of Fig. 2a shows the

activation of the cells tuned to the presence of white and

black in the input image. At each pixel location, middle gray

indicates that neither a white nor black cell is responding.

Light gray and white indicates activity from a cell tuned to

the presence of white in the image. Dark gray and black

indicates activity from a cell tuned to the presence of black

in the image. The net result is that this image looks much

like the input image. The image on the top left represents

activations of BCS boundary signals. The activity of

different orientations is coded by color. A pixel colored

middle gray indicates that neither a horizontally tuned nor a

vertically tuned cell is responding at that pixel location. A

white pixel indicates activity from a vertically tuned cell. A

black pixel indicates activity from a horizontally tuned cell.

The boundary representation of the BCS to the horizontal

grating includes horizontal and vertical responses where

there is contrast against the black background. Additionally,

the luminance cusp in the middle of the image produces a

vertical contour. Significantly, only a single vertical contour

is produced here. The blending of the brightest and darkest

peaks of the cusp into the gray background is too gradual to

produce boundary responses above threshold.

The image on the top right in Fig. 2a describes the

activities of cells in the filling-in stages. White and light

gray indicates responses among cells tuned to white. Black

and dark gray indicates responses among cells tuned to

black. Middle gray indicates no response from either the

white or black tuned cells. Each pixel in the representation

of color information (right image on bottom row) feeds in to

the pixel at the same position in the filling-in stage. An

active cell in the filling-in stage then shares its activation

with its nearest neighbors, unless a BCS boundary signal is

present. For example, at a pixel just right of the central cusp,

the filling-in stage receives a signal from the lower level that

there is ‘black’ at this location. That signal spreads (along

with many others) right, up, and down in the filling-in stage;

but the signal cannot spread left across the vertical

boundary. Likewise, the horizontal and vertical boundaries

at the edge of the gray square keep the signal in the gray

square. The net result is that the dark signals spread across

the entire region to the right of the cusp. A similar situation

occurs for pixels to the left of the central cusp, so that the

left side of the gray box appears brighter than the right side.

This simulation used a filling-in mechanism used by

Grossberg and Todorovic (1988), but there are several

alternative filling-in mechanisms that behave essentially the

same for this image (and other static images). However,

simulations discussed below demonstrate that different

filling-in mechanisms do not behave the same in response

to a sequence of images. In particular, the choice of filling-in

mechanism determines the appearance of visual afterimages

in the model. The kind of afterimage we use is based on the

findings of Francis and Rothmayer (2003) and Vidyasagar

et al. (1999). Fig. 3a schematizes the types of images used

by Francis and Rothmayer (2003) in a psychophysical

experiment. (An online demonstration can be found at

http://www.psych.purdue.edu/~gfrancis/Experiments/

MCAIs.) The first image was a horizontal bar grating that

appeared for 1 s. The second image was a flickering vertical

bar grating that also appeared for 1 s. The third image was a

blank screen. Fig. 3b schematizes the percepts reported by

the observers. The percepts of the first two images were

essentially veridical reports of the bar gratings, but the

percept of the third image does not appear blank. Instead,

observers reported seeing a horizontal bar grating of

alternating shades of gray. The experimental data suggested

that the presence of the vertical grating after the horizontal

grating was critical; if the second image was a horizontal

grating or blank, reports of a horizontal grating as an

afterimage were much reduced.

In the following sections we consider the behavior of

three different filling-in mechanisms: isotropic diffusion

with unoriented boundaries, isotropic diffusion with

oriented boundaries, and a non-diffusive mechanism. We

show how the properties of the afterimages reported by

Francis and Rothmayer (2003) provide tests of the filling-in

mechanisms.

3. Isotropic diffusion with unoriented boundaries

With this filling-in mechanism, a cell spreads brightness

information through connections with its nearest neighbors.

Since the brightness information can spread in any

direction, it is similar to isotropic diffusion. In addition,

the spread of brightness information can be blocked by the

presence of any boundary signal, regardless of the

boundary’s orientation. This is the filling-in mechanism

used by Grossberg and Todorovic (1988) and Gove et al.

(1995) for simulations of various brightness percepts and for

the simulation results in Fig. 2a. The same mechanism was

used by Grossberg and Wyse (1991) for segmentation of

G. Francis, J. Ericson / Neural Networks 17 (2004) 737–752740

figure and ground. Fig. 4 shows the model’s behavior to the

sequence of images schematized in Fig. 3.

The sequence starts with the presentation of a horizontal

black and white grating (Fig. 4a). The output of the color-

gated dipole reflects the input from the horizontal grating.

The boundary signals are primarily horizontal. The filling-in

stage shows a horizontal grating, and thus a veridical

percept.

After 1 s, the horizontal grating is replaced by a vertical

grating. As in the experiments of Francis and Rothmayer

(2003), this vertical grating flickered with its color

complement and what is shown in Fig. 4b is the behavior

of the model at the end of the last vertical grating. The output

of the color gated dipole shows vertically arranged black

and white color signals. However, faintly superimposed on

the vertical pattern are black and white horizontal bars. (The

faint horizontal stripes may not be visible in the reproduc-

tion of the image.) These horizontal stripes are color after

responses produced by the offset of the horizontal grating.

The orientation signals are predominately vertical (white)

because of two effects. First, the presentation of the vertical

image produces strong responses among vertically tuned

cells at the appropriate positions on the edges of the bars.

Second, the offset of the horizontal grating has caused

rebounds in the orientation gated dipole that produce strong

vertical boundary responses. After excitatory feedback

Fig. 3. A schematic of the stimuli and percepts from the experiment of Francis and Rothmayer (2003). (a) The stimulus consisted of two orthogonal gratings

followed by a blank image. (b) The percepts of the gratings were essentially veridical, but an afterimage of a horizontal grating was visible during the blank.

Fig. 4. Simulation results using isotropic diffusion with unoriented boundaries for a sequence of images that produces a horizontal grating afterimage. (a)

Model behavior during presentation of a horizontal grating. (b) Model behavior during the last of the flickering vertical bar gratings. (c) Model behavior 1 s

after offset of the vertical bar grating. The filling-in stage shows a checkerboard type pattern, which is inconsistent with observer’s reports.

G. Francis, J. Ericson / Neural Networks 17 (2004) 737–752 741

groups together cells with similar orientations, there is a

dense block of active vertically tuned cells across the entire

grating. The filling-in stage shows a vertical grating, which

corresponds to a veridical percept. (There is a faint

horizontal pattern that corresponds to the after responses

generated by the offset of the horizontal grating, but it is

quite weak relative to the vertical pattern).

Finally, the vertical image was replaced by a blank gray

screen. Fig. 4c shows the model’s response 1 s after the

offset of the vertical grating. The responses of the color

gated dipoles produce alternating bright and dark rows/

columns of checks. This pattern is due to after responses

generated by both the vertical and horizontal bar gratings.

The orientation signals are primarily horizontal, because

offset of the vertical bar grating produces after responses

among horizontally tuned cells. The filling-in stage in Fig. 4c

is a replication of the activity at the color gated dipoles. This

occurs because the dense array of horizontal boundary

signals prevents spreading of brightness information in any

direction. Significantly, the predicted percept of a checks

pattern afterimage is contrary to what observers actually

report. Observers report seeing a horizontal bar grating. This

conflict suggests that the filling-in mechanism is not

isotropic diffusion with unoriented boundaries.

What is needed for the model to match the percept is for

brightness information to be able to spread in a direction

parallel to the orientation of boundary signals. This would

allow brightness information from the checks to spread

horizontally but not vertically and would thereby produce a

percept that matched the observers’ reports. This type of

filling-in is discussed in Section 4.

4. Isotropic diffusion with oriented boundaries

This mechanism differs from the isotropic diffusion

with unoriented boundaries version only in the interaction

between boundaries and the direction of diffusive flow. In

this new version a boundary signal at a particular location

will allow brightness information to flow to a neighboring

cell if the neighboring cell is in a direction parallel to the

orientation of the boundary signal. Thus, a horizontally

tuned boundary signal would allow brightness

information to flow left or right but would block

brightness information from flowing up or down. Fig. 5

shows the behavior of the model if oriented boundaries

are used in the filling-in mechanism. The behavior of the

color and orientation gated dipoles are identical to what is

shown in Fig. 4. The only differences will be at the

filling-in stages. See Appendix A for details of this filling-

in mechanism.

For the horizontal grating (Fig. 5a), the percept is

veridical, as the horizontal and vertical boundary signals

block the flow of brightness information from spreading

outside the bars. For the vertical grating (Fig. 5b), the

percept is again veridical. Here there are many vertical

boundary signals (due to the orientation after responses and

excitatory feedback grouping mechanisms). However, these

vertical signals allow brightness information to

spread vertically and thereby still produce a vertical grating.

(This vertical spreading of information also washes out the

faint horizontal pattern that was produced with the

unoriented boundary simulations in Fig. 4b.) Finally, offset

of the vertical grating (Fig. 5c) produces an after response of

a horizontal grating at the filling-in stages. This occurs

because the offset of the vertical grating produces a dense

array of horizontal boundary signals. These horizontal

boundary signals allow the checks of a given row to spread

left and right, which produce an entire row of similar

brightness. The horizontal boundaries prohibit the spread of

information up or down, so alternating rows maintain

different filled-in values. Thus, introducing oriented bound-

ary signals into the filling-in process allows the model to

produce after responses that match the afterimage percept.

Francis and Rothmayer (2003) used an idealized version of

this type of filling-in mechanism to simulate a variety of

these afterimage percepts. Section 5 identifies a key

characteristic of diffusive mechanisms and a psychophysical

experiment that tests the characteristic.

5. Spatial extent of diffusive spread: a prediction

Perceptual filling-in seems to be able to travel quite long

distances without diminishing the strength of the signal.

Holding the front cover of this journal at a distance of

approximately 25 cm from the eyes makes the extent of the

illusory pink circles of the neon color spreading effect cover

around 80 min of visual arc. If colored diagonal lines are

perceived, the spreading has traveled even further. In the

Craik–O’Brien–Cornsweet effect, brightness information

can spread several degrees of visual angle to influence

brightness percepts away from the center of the display (e.g.

Todorovic, 1987).

To account for such long-range spreading of brightness

information, filling-in mechanisms based on diffusion have

assumed that properties of the medium that might prohibit

spatial spread are negligible and that spatial spread is halted

only by the presence of appropriate boundary signals. As a

result, neural mechanisms based on diffusion have essen-

tially unlimited spatial extent. Although information

diffuses locally (to nearest neighbors) a chain of nearest

neighbors allows brightness information to spread long

distances until blocked by appropriate boundary signals.

This unlimited spatial extent is demonstrated in Fig. 6,

which shows three different simulation runs. The first image

in a sequence is a horizontal grating with a centered vertical

gap (the same color as the gray background) that separates

the left and right sides of the image. As Fig. 6a–c show, the

model with isotropic diffusion and oriented boundaries

predicts no effect of gap width on the shape of the

afterimage percept. In every case the model predicts that

G. Francis, J. Ericson / Neural Networks 17 (2004) 737–752742

the left side and right side of each horizontal row should

spread horizontally and fill-in the gap to produce an

afterimage percept of a complete horizontal grating. We

tested this prediction with a psychophysical experiment.

5.1. Methods and procedures

All stimuli were created and presented with MATLAB,

using the Psychophysics Toolbox extensions (Brainard,

1997; Pelli, 1997), on a PC running Windows 98 with a

monitor that refreshed at 75 Hz.

Fig. 7 schematizes one of the trials. The first stimulus was

always a gapped horizontal grating of 10 black (0.6 cd/m2)

and nine white (128 cd/m2) bars on a gray (24 cd/m2)

background. On a given trial, the gap width was one of 10

widths that varied between 15 and 45 min of visual arc. The

grating was shown for 1 s and then replaced by a vertical bar

grating of 20 alternating black and white bars (without a

gap). The bars of the vertical grating were in a window the

same size and position as the horizontal grating, with

the vertical bars translating to the left or right by shifting one

pixel with every refresh of the monitor. Bars that moved off

the window on one side reappeared on the opposite side.

The vertical grating was presented for a total of 3.2 s, which

meant that each bar shifted six positions by the end of the

movement. After offset of the vertical grating the screen

showed only the gray background for 1 s. This was then

replaced by a box of random dots, which was a cue for the

observer to report any afterimage that was seen just before

the random dots appeared. At the viewing distance of 60 cm,

the maximum extent of the gratings was approximately 108

visual angle in height and width. The box of random dots

had the same size. Observers were advised that if the

afterimage percept changed during the blank interval, they

were to report the shape of what was visible just before the

random dots appeared.

Observers were instructed to report on the appearance of

the afterimage by identifying whether they saw a horizontal

grating without a gap, a horizontal grating with a gap, or

other. Responses were entered by a keyboard and another

keypress started the next trial. To minimize interactions

across trials, a forced delay of 15 s was introduced before

the start of the next trial.

Five replications for each of the gap widths were

randomly mixed in an experimental session of 50 trials.

Twenty naive observers were recruited from the exper-

imental subject pool at Purdue University. The observers

received course credit for participation in the experiment.

Each observer was run separately in a room that used regular

overhead lighting.

5.2. Results

Fig. 8 shows the proportion of different responses for

each of the gap widths. The proportions for each gap width

are based on 100 experimental trials. Across all gap widths

almost all reports were of a horizontal grating either with or

without a gap. Reports of ‘other’ hovered around 10% and

did not vary systematically with gap width. On the other

hand, reports of seeing the gap did vary with gap width. At

the smaller gap widths most reports were of a horizontal

grating without a gap. As gap width increased, these reports

became less frequent and at a gap width of around 40 min

visual arc, reports of a horizontal grating with a gap become

more frequent than reports of a horizontal grating without

a gap.

5.3. Discussion

The experimental data do not agree with the prediction of

the diffusion-based filling-in mechanism. The model pre-

dicted that observers would not see an afterimage with a gap

Fig. 5. Simulation results using isotropic diffusion with oriented boundaries for a sequence of images that produces a horizontal grating afterimage. (a) Model

behavior during presentation of a horizontal grating. (b) Model behavior during the last of the flickering vertical bar gratings. (c) Model behavior 1 s after offset

of the vertical bar grating. The filling-in stage shows a horizontal pattern, which is consistent with observer’s reports.

G. Francis, J. Ericson / Neural Networks 17 (2004) 737–752 743

regardless of the gap width. Although observers do not

generally see an afterimage with a gap for small gap widths,

they do see a gap in the afterimage for larger gap widths.

This casts doubt on diffusive-based mechanisms for

filling-in. In the context of these afterimages, a diffusive

filling-in mechanism would spread brightness information

across the gap because there are no vertical boundary

signals to stop the spread.

Fig. 6. Simulation results using isotropic diffusion with oriented boundaries. The model predicts that regardless of the gap width in the horizontal grating, the

afterimage will appear as a continuous horizontal grating. (a)–(c) show results for increasing gap widths.

G. Francis, J. Ericson / Neural Networks 17 (2004) 737–752744

This finding seems to place diffusion-based filling-in

mechanisms in a dilemma. On the one hand, to account for

the existence of the horizontal grating afterimage in Fig. 3,

brightness information must be able to easily spread across a

boundary signal in a direction parallel to the boundary’s

orientation. On the other hand, if brightness information can

easily spread this way, then the model predicts that gaps

should not be visible in the afterimage with the stimuli used

in the experiment. There seems to be no way a diffusion-

based filling-in mechanism can resolve this dilemma.

6. Non-diffusive filling-in

FACADE’s explanation of the afterimage percept

requires the existence of a filling-in mechanism that

spreads brightness information in the direction of the

oriented boundary signals. However, the explanation does

not require that the mechanism involve a diffusive

spreading of information. Fig. 9 schematizes a non-

diffusive filling-in mechanism first proposed by Francis

and Schoonveld (2003). Each cell in a filling-in stage is

hypothesized to have a receptive field that samples

activities from other cells in the filling-in stage. Part of

this receptive field is schematized in Fig. 9a, which

schematizes the existence of independent subfields. Each

subfield samples other cells in the surface representation

along a particular line drawn outward from the location of

the sampling cell.

The input into one of these subfields is schematized in

Fig. 9b. The white circles on the bottom indicate a row of

cells in the white filling-in stage. The schematized

Fig. 7. The sequence of frames during a trial of the experiment. The

observer’s task was to report on any seen afterimages just before the box of

random dots appeared. The observer was to report if the afterimage was a

horizontal grating with a gap, horizontal grating without a gap, or other.

Fig. 8. Proportion of reports of different afterimage percepts as a function of

gap width. Reports of ‘other’ were rare and unaffected by gap width.

Reports of a horizontal grating without a gap were most frequent for the

smallest gap widths and decreased as gap width increased. Reports of a

horizontal grating with a gap increased with gap width.

Fig. 9. A non-diffusive filling-in mechanism. (a) Each cell has many

independent subfields. The cell receives feedback if at least two of the

subfields are activated. (b) Details of one subfield for a cell in the white

filling-in plane. The subfield stretches horizontally to the left of the center

of the receptive field. At each location it samples input from cells in the

white filling-in stage. BCS boundary signals that are different from

the orientation of the subfield can block sampling from locations beyond the

boundary.

G. Francis, J. Ericson / Neural Networks 17 (2004) 737–752 745

subfield extends leftward from the center of the receptive

field and samples activities from other cells in the filling-

in stage. The sampling of cells is blocked by the presence

of a BCS boundary signal of an orientation different from

the direction of sampling. The lines at different pixel

locations in the subfield schematize the boundary

orientations that would block sampling for this subfield.

For example, a vertical boundary at the third location

from the right would block sampling from that cell and

from cells that are in more distant locations in this

subfield.

Each subfield functions in a similar way; sampling

from cells in a straight line away from the receptive field

center. Each subfield is blocked from sampling cells that

lie beyond a BCS boundary signal that is not the same

orientation as the direction of sampling. In addition, the

center of the receptive field samples from itself, if it is

active. The center of the receptive field is an additional

subfield of the receptive field. If at least two subfields

are active, a signal is fed back to the sampling cell. A

subfield is active if it samples from at least one active

cell in the filling-in stage. This feedback process allows

for further changes in activity among other cells and

eventually reaches an equilibrium state. (See Appendix A

for a mathematical description of the mechanism and

details of the simulations.)

Although this non-diffusive mechanism is substantially

more complicated (in terms of the amount of computation

involved and in the neurophysiological details needed for

biological implementation), in many respects it behaves

much the same as a diffusive filling-in mechanism. In

particular, without blocking boundary signals, brightness

information can freely spread from edges to the interior of

surfaces. This can be seen in Fig. 2b, which shows a

simulation of the Craik – O’Brien – Cornsweet effect

with the non-diffusive filling-in mechanism. Similar to

the diffusive filling-in mechanisms, the non-diffusive

filling-in mechanism spreads brightness information from

the central cusp to the entire left and right sides of the

square.

Fig. 10 shows that the non-diffusive filling-in mechanism

also behaves much like the diffusive mechanism with

oriented boundaries under the context of the afterimages

described above. The presentation of a horizontal grating

followed by a vertical grating produces a filled-in after-

image percept of a horizontal grating.

Additionally, the non-diffusive mechanism differs from

the diffusive mechanisms in that under the context of the

dense boundary signals present in the afterimages, the non-

diffusive mechanism has a limited spatial extent, which is

set by the size of the subfields. With a dense array of

horizontal boundary signals, the non-diffusive mechanism

requires that the left and right subfields be activated in order

to fill-in the gap (any subfields oriented up, down, or

diagonal will be blocked from sampling by the horizontal

boundary signals). If the gap is so large that the brightness

signals are beyond the reach of the subfields, then the gap

will not be filled in and will be visible in the afterimage.

Simulations of these effects are shown in Figs. 11a–c for

gaps of various sizes. For small gaps the afterimage has the

gap filled-in, but for larger gaps the afterimage includes a

visible gap. Thus, the non-diffusive mechanism better

matches the experimental data than the diffusive filling-in

mechanism.

Significantly, Hong and Grossberg (2003) also proposed

a filling-in mechanism utilizing long-range receptive fields.

Their motivation was that simulations of the diffusive

filling-in process was computationally slow (because

information travels only to nearest neighbors). They showed

that their alternative mechanism could be integrated into

Fig. 10. Simulation results using the non-diffusive filling-in mechanism for a sequence of images that produces a horizontal grating afterimage. (a) Model

behavior during presentation of a horizontal grating. (b) Model behavior during the last of the flickering vertical bar gratings. (c) Model behavior 1 s after offset

of the vertical bar grating. The filling-in stage shows a horizontal pattern, which is consistent with observer’s reports.

G. Francis, J. Ericson / Neural Networks 17 (2004) 737–752746

FACADE theory to account for many brightness phenom-

ena. Thus, there is both experimental and computational

support for a need to deviate from a diffusive filling-in

process.

Finally, the non-diffusive mechanism agrees with the

neurophysiological finding (Friedman et al., 2003) that

color-sensitive cells are also orientation-sensitive. The

separate subfields of the non-diffusive filling-in cells are

Fig. 11. Simulation results using the non-diffusive filling-in mechanism. The model’s after response is sensitive to the width of the gap in the horizontal grating.

For a small gap the model fills-in the gap in the afterimage, but for larger gaps the model does not fill-in the gap. (a)–(c) show results for increasing gap widths.

G. Francis, J. Ericson / Neural Networks 17 (2004) 737–752 747

both orientation and color selective, and the cells identified

by Friedman et al. (2003) may be neurophysiological

instantiations of this mechanism.

7. Conclusions

The afterimages identified by Francis and Rothmayer

(2003) and Vidyasagar et al. (1999) can be used to test

neural models of perceptual filling-in. The appearance of a

horizontal grating afterimage resulting from viewing a

horizontal then vertical grating conflicts with the predic-

tion of the neural mechanism used by Grossberg and

Todorovic (1988) to account for many brightness illusions

because that mechanism does not allow for spreading of

brightness signals parallel to the orientation of a boundary

signal.

A modified diffusion-based filling-in mechanism that

allows brightness information to spread to a neighbor that

lies in a direction parallel to the orientation of a boundary

signal can account for the afterimages identified by Francis

and Rothmayer (2003). However, additional analysis now

reveals that the diffusion-based filling-in mechanisms have

essentially unlimited spatial spread (except for the presence

of blocking boundary signals). This characteristic led to the

identification of an experimental test of diffusion-based

filling-in models. These models predict that if the horizontal

grating includes a central vertical gap, the afterimage should

not include the gap, regardless of the gap’s size. For these

diffusive-based filling-in mechanisms the brightness signals

should spread across the gap. The experimental results,

however, found that observers do see the gap for larger gap

sizes.

All of these results are consistent with a non-diffusive

based filling-in mechanism. This new mechanism behaves

much like a diffusive filling-in mechanism for static images,

and accounts for the existence of the afterimage reported by

Francis and Rothmayer (2003). This consistency with

earlier mechanisms is important because it implies that a

non-diffusive filling-in mechanism can be embedded in the

FACADE model and maintain the model’s explanations of

various visual percepts. The non-diffusive mechanism also

accounts for the appearance of the gap in the afterimage

percepts reported here and thereby extends the explanatory

power of filling-in models.

Thus, the cover of the journal Neural Networks is

particularly appropriate for this discussion because it

highlights how the study of computational neural models

and the study of visual percepts feed off each other. Having

precise computational neural models led to the generation of

psychophysical experiments that tested key properties of

those models. The results of the experiment, in turn, drove

the development of the models and helped to identify further

computational details of the models. These kinds of

interactions can lead to an upward spiral of model

development and testing that will ultimately merge with

neurophysiological studies to create detailed models of the

relationship between the brain, computational principles,

and visual perception.

Acknowledgements

GF was supported by a fellowship at the Hanse

Wissenschaftskolleg, Delmenhorst, Germany.

Appendix A. Simulations

A.1. Image plane: input image

Each pixel ði; jÞ had an input value Ii;j: All the images

used intensities between 21 (black) and þ1 (white), with 0

indicating middle gray. A 128 by 128 pixel plane was used

in all simulated images. Each image plane had its origin at

the upper left corner. Larger positive indices indicate

locations to the right or below the origin.

For the Craik–O’Brien–Cornsweet stimulus, the gray

box was 64 by 64 pixels. The intensity of the central cusp

started at 0.3 above or below the gray level at the center and

blended into the background according to the following

equation:

Ii;j ¼ ^0:3di;j 2 15

15

� �2

; ðA1Þ

where di;j is the distance of pixel ði; jÞ from the central cusp

and 15 sets the spatial extent of the cusp. Other intensity

values in the box were set to the gray background (zero).

The surrounding areas were black.

Each bar grating image was 96 by 96 pixels on a gray

background. The thickness of each black or white bar was

4 pixels. Each image was presented for 10 simulated time

units. The gap widths in Figs. 6 and 11 were 5, 15, and 25

pixels. The vertical grating flickered (black and white

values changed places) 10 times for one time unit duration

each.

A.2. Color gated dipole

The Ii;j value at each pixel fed directly into a color

gated dipole. Here black and white signals were sent

through opponent channels and habituation of those

channels occurred. The habituating gate for the white

channel at pixel ði; jÞ; gi;j; obeyed the differential

equation:

dgi;j

dt¼ ½A 2 Bgi;j 2 Cgi;jð½Ii;j�

þ þ JÞ�D: ðA2Þ

The term A 2 Bgi;j describes a process whereby the gate

amount increases toward the value A=B: The last

subtraction describes depletion of the gate by

G. Francis, J. Ericson / Neural Networks 17 (2004) 737–752748

the presence of a tonic signal J and by a white signal,

½Ii;j�þ; where the notation [ ]þ indicates that negative

values are set to zero. Parameter D controls the overall

rate of change of the equation.

The black opponent pathway had an identical type of

equation with only the term ½Ii;j�þ replaced by ½2Ii;j�

þ; so

that only black signals passed through the gate. In the

following equations the habituating gate for the black

pathway is labeled as Gi;j: Each simulation trial started with

an initial value of gates that corresponded to an equilibrium

value of the gates with no outside input:

gi;jð0Þ ¼ Gi;jð0Þ ¼A

B þ CJ: ðA3Þ

The parameters were set as A ¼ B ¼ C ¼ 1; D ¼ 0:1; and

J ¼ 0:5:

The white output of the color gated dipole was then

computed by weighting the net input into the white channel

by its gate and then subtracting the resulting value by the

weighted black input. The resulting difference was thre-

sholded and then scaled. The white output, wi;j was

computed as:

wi;j ¼ E½ðIi;j�þ þ JÞgi;j 2 ð½2Ii;j�

þ þ JÞGi;j 2 F�þ: ðA4Þ

Here, F ¼ 0:004 is a threshold, the term to its left is the

inhibitory input from the black channel, and the next term to

the left is the excitatory input from the lower levels of the

white channel. The difference is rectified so that negative

values are set to zero and then scaled by the multiplying

term E ¼ 100: The black output of the gated dipole, bi;j; had

a similar equation, with only the middle and left terms

trading excitatory and inhibitory roles. The value wi;j 2 bi;j

was plotted in the simulation figures (bottom right frame).

A.3. Boundary contour system

A.3.1. Edge detection and orientation gated dipole

The wi;j and bi;j values were sent to the BCS for edge

detection. At each pixel, detectors were sensitive for an

activation pattern that signaled either a vertical or horizontal

edge. For computational simplicity detectors were used that

just looked for a change in luminance intensity in a vertical

or horizontal direction. The absolute value of this change

was taken as the response of the detector. A boundary cell at

position ði; jÞ tuned to a vertical edge had an activity

yi;j ¼ ½lwi;j 2 wi21;jlþ lwi;j 2 wiþ1;jlþ lbi;j 2 bi21;jl

þ lbi;j 2 biþ1;jl2 K�þ: ðA5Þ

This receptive field looks to the left and right of the edge

location for any discontinuities among the white or black

representations of color. The term K ¼ 20 indicates a

threshold. Any values below K were set to zero. A similar

value, Yi;j; was computed for the horizontally tuned

boundary cells.

The equations for the BCS oriented gated dipole were

identical in form to those used for the color gated dipole.

The only differences from the equations of the color gated

dipole are that in the BCS version of Eqs. (A2) and (A4),

½Ii;j�þ is replaced by yi;j (vertical), ½2Ii;j�

þ is replaced by Yi;j

(horizontal), D ¼ 1:0; E ¼ 1:0; and F ¼ 0:0: The relevant

equations are:

dgi;j

dt¼ ½A 2 Bgi;j 2 Cgi;jðyi;j þ JÞ�D ðA6Þ

and

xi;j ¼ E½ðyi;j þ JÞgi;j 2 ðYi;j þ JÞGi;j 2 F�þ: ðA7Þ

Here, xi;j refers to the output of the orientation gated dipole

for a vertically tuned cell. Since the outputs of the color

gated dipole feed in to the orientation detectors, which in

turn feed into the habituative gates for the orientation gated

dipole, the differential equations for both sets of the

habituative gates were solved simultaneously.

A.3.2. Boundary grouping

Grouping of BCS signals is supported by model

bipole cells. Each vertically tuned bipole cell received

excitation from vertically tuned gated dipole output cells

and received inhibition from horizontally tuned gated

dipole output cells. A vertical bipole cell had two sides

(Up and Down) that separately combined excitatory and

inhibitory inputs from the lower level. Each side

combined information from a column of locations

above (Up) or below (Down) the location of the bipole

cell. For a vertical bipole cell at position ði; jÞ; define

intermediate terms:

Upi;j ¼X10

k¼2

ðxi;j2k 2 Xi;j2kÞ

" #þ

ðA8Þ

and

Downi;j ¼X10

k¼2

ðxi;jþk 2 Xi;jþkÞ

" #þ

ðA9Þ

where xi;j and Xi;j refer to the output of the orientation

gated dipole for vertically and horizontally tuned cells,

respectively. A bipole cell has a positive activity if both

of the intermediate terms are non-zero, or the bottom-up

edge information at the bipole cell’s pixel location is

non-zero and one of the intermediate terms is also non-

zero. Thus, the activity of the vertical bipole cell at pixel

ði; jÞ is:

Bi;j ¼ maxðUpi;jDowni;j; xi;jUpi;j; xi;jDowni;jÞ: ðA10Þ

This allows bipole cells to respond as if they interpolate,

but do not extrapolate, bottom-up contour information. A

vertical bipole cell will also respond if an oriented edge

is present at the location of the bipole cell and a vertical

oriented edge is above or below the location of the

bipole cell. Similar equations exist for each horizontally

G. Francis, J. Ericson / Neural Networks 17 (2004) 737–752 749

tuned bipole cell, which received excitatory input from

other horizontally tuned cells and inhibitory input from

vertically tuned cells.

To avoid proliferation of bipole cell responses, a second

competition between orientations at each pixel was

included. This competition was winner-take-all, so that the

orientation bipole cell with the largest value received a

value of one, while the other orientation bipole cell was

given a value of zero. The set of values of each orientation at

each pixel location was taken as the ‘output’ of the BCS. A

pixel was said to have a vertical boundary if the vertically

tuned cell won the competition here. A more elaborate

version of this type of competition can be found in

Grossberg et al. (1997). The outputs of the bipole boundary

signals are referred to as Vi;j and Hi;j for vertical and

horizontal signals, respectively.

A.4. Filling-in with isotropic diffusion and unoriented

boundaries

Diffusive filling-in calculations were based on the

method used by Grossberg and Todorovic (1988). The

activity of a cell at position ði; jÞ in the white filling-in array

obeyed the differential equation:

dSij

dt¼ 2MSij þ

Xðp;qÞ

ðSpq 2 SijÞPpqij þ wij: ðA11Þ

Here parameter M ¼ 1:0 sets the rate of passive decay, the

term wij is input from the white pathway of the color gated

dipole, and the summation indicates a weighted sum of the

difference in activity between this cell and its nearest (four)

neighbors. The weighting term, Ppqij; is determined by the

presence of boundary signals. For the isotropic diffusion

with unoriented boundary signals (Figs. 2 and 4) the weights

are set by the following equation:

Ppqij ¼d

1 þ 1ðVij þ Hij þ Vpq þ HpqÞ: ðA12Þ

Here Vij and Hij refer to the vertical and horizontal outputs

of the oriented bipole cells at the same pixel location as the

cell and Vpq and Hpq refer to the vertical and horizontal

outputs of the oriented bipole cells at a neighboring pixel

location. Parameter d ¼ 600 sets the ease with which

brightness information spreads from one cell to a neighbor,

and parameter 1 ¼ 1000 sets the blocking effect that a

boundary has on the spread of brightness information. To

control for edge effects, each Ppqij weight was set equal to

zero at the edge of the pixel array.

Since the filling-in process is hypothesized to work

quickly, the reported values of the filling-in stage are

equilibrium values. The equilibrium values were found by

solving Eq. (A11) at equilibrium, so that

Sij ¼wij þ

Pðp;qÞSpqPpqij

M þP

ðp;qÞPpqij

: ðA13Þ

Equilibrium values were estimated by plugging in the

current values of Sij and solving for new values of Sij: This

process was repeated until the maximal change in value

from one iteration to the next across the entire array was less

than 0.0001. No oscillations were ever observed in the

calculations.

A separate filling-in stage computed darkness signals.

The equations were the same as above, except wij was

replaced by bij: The distribution of activities in the white and

black filling-in stages were displayed in the simulation

figures by subtracting the value in the black filling-in stage

from the value in the white filling-in stage at each pixel

location. The value zero was always assigned to middle

gray, the maximum positive (white) value was assigned the

brightest white, the most negative (black) value was

assigned the darkest black, and intermediate values were

scaled to appropriate gray levels.

A.5. Filling-in with isotropic diffusion and oriented

boundaries

The equations and parameters for the isotropic diffusion

with oriented boundaries (Figs. 5 and 6) were similar to the

equations for isotropic diffusion with unoriented bound-

aries. The only difference was the calculation of the

weighting term Ppqij: With oriented boundaries, the weight

was based on boundary signals only from the orientation

opposite the direction of the nearest neighbor. Thus, a

weight for a cell receiving brightness information from a

neighbor to its left would be:

Ppqij ¼d

1 þ 1ðVij þ VpqÞ; ðA14Þ

which does not include boundary signals of a horizontal

orientation. A similar equation defined the weights for other

neighbors.

A.5.1. Non-diffusive filling-in

Each cell in a filling-in stage had eight subfields that

stretched away from the center of the cell out 10 pixels

away. In the current simulations there were eight such

subfields (four along the main axes and four along the main

diagonals). The following notation indexes the properties of

a subfield based on pixel location ði; jÞ as fi;j;m and Fi;j;m;

m ¼ 0;…; 7:

For each subfield at position ði; jÞ; several terms were

computed. This discussion will describe the calculations for

the up subfield, which is indexed as m ¼ 0: The first term to

be computed was the extent of sampling, fi;j;0: This index

was defined by starting at pixel position ði; j 2 1Þ and

moving up along a vertical path ði; j 2 kÞ to find the first (if

any) pixel location that had a nonzero horizontal boundary

signal, Hi;j2k: Written mathematically:

fi;j;0 ¼ min½10; k such that Hi;j2k . 0�: ðA15Þ

G. Francis, J. Ericson / Neural Networks 17 (2004) 737–752750

Sampling of activities by this subfield was then the sum of

activities from other cells in the filling-in stage.

Fi;j;0 ¼Xfi;j;021

k¼1Wi;j2k: ðA16Þ

The definition of fi;j;0 restricts Fi;j;0 from summing terms that

are at or beyond a horizontal boundary. A third term

calculated the number of active inputs within each subfield:

Ci;j;0 ¼Xfi;j;021

k¼1TðWi;j2kÞ: ðA17Þ

where TðxÞ ¼ 1 if x . 0 and zero otherwise.

A similar calculation was made for every subfield.

Finally, if two subfields had sampled from at least one pixel

each, or if one subfield had sampled from at least one pixel

and Wi;j was greater than zero, then a new value of Si;j was

computed as the average sampled value across all subfields:

Si;j ¼Wi;j þ

Xm

Fi;j;m

1 þX

mCi;j;m

; ðA18Þ

The net result was that each pixel in the filling-in stage

computed the average activity among all sampled cells in

the filling-in stage. The sampling was restricted by the

presence of BCS boundary signals and by the physical

arrangement of the signals themselves.

The process involved feedback, so it was implemented

as an iterative computation. At initialization, Si;j was set

equal to the color signal feeding in from the color gated

dipole. (wi;j or bw;j). Separate calculations were performed

for a white and a black filling-in stage. Equilibrium values

were estimated by plugging in the current values of Si;j

and solving for new values of Si;j: This process was

repeated until the maximal change in value from one

iteration to the next across the entire array was less than

0.001. No oscillations were ever observed in the

calculations.

Finally, each cell activity was thresholded so that any

value less than 0.01 was set to zero. The values for black

were subtracted from the values for white and normalized

for plotting in the simulation figures (Figs. 2b, 10, and 11).

For Fig. 2b horizontal boundaries were artificially added to

the corners of the gray box. Without this addition, the non-

diffusive filling-in mechanism could spread brightness

information through the corners (because no horizontal or

vertical boundaries were naturally present at the corners). In

a more elaborate simulation with diagonally oriented

boundary signals, this addition would be unnecessary.

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