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activation fields
e.g., space, movement parameters, feature dimensions, viewing
parameters, ...
dimension
activationfield
metric contents
information, probability, certaintydimension
activationfield
specified value
dimension
activationfield
no value specified
evolution of activation fields in time: neuronal dynamics
movement
parameter
time
activation
preshapedfield
specific inputarrives
the dynamics such activation fields is structured so that
localized peaks emerge as attractor
solutions
movement
parameter
time
activation
preshapedfield
specific inputarrives
dimension, x
local excitation: stabilizespeaks against decay
global inhibition: stabilizes peaks against diffusion
input
activation field u(x)
(u)
u
mathematical formalizationAmari equation
⌧ u̇(x, t) = �u(x, t) + h + S(x, t) +
Zw(x� x
0)�(u(x
0, t)) dx
0
where
• time scale is ⌧
• resting level is h < 0
• input is S(x, t)
• interaction kernel is
w(x� x
0) = wi + we exp
"
�(x� x
0)
2
2�
2i
#
• sigmoidal nonlinearity is
�(u) =
1
1 + exp[��(u� u0)]
1
Interaction: convolution
116
! ∗ ! ! !! = ! !! − !! ! ! !!!!!!!
!!!!!!!!!!!!!!!(B2.2)
where ! = (! − 1)/2 is the half-width of the kernel. The sum extends to indices outside the
original range of the field (e.g., for m=0 at ! = −!). But that doesn’t cause problems because we
extended the range of the field as shown in Figure 2.18.
Note again that to determine the interaction effects for the whole field, this computation
has to be repeated for each point !!. In COSIVINA all these problems have been solved for you,
so you don’t need to worry about figuring out the indices in Equations like B2.2 ever again!
[End Box 2.1]
Figure 2.18 Top: The supra-threshold activation, !(!(!!)), of a field is shown over a finite range (from 0 to 180 deg). Second from top: The field is expanded to twice that range by attaching the left half of the field on the right and the right half on the left, imposing periodic boundary conditions. Third from top: The kernel has the same size as the original field and is plotted here centered on one particular field location, ! = !" deg. Bottom: The matching portions of supra-threshold field (red line) and kernel (blue line) are plotted on top of each other. Multiplying the values of these two functions at every location returns the black line. The integral over the finite range of the function shown in black is the value of the convolution at the location ! = !".
30° 60° 90° 120° 150°0°
1
0180° space
1
0
0.4
-0.4
0
0.8
-30° -60° -90°0°30°90° 60°
x=50°
0.4
-0.4
0
0.8
-30° -60° -90°0°30°90° 60°
1.2
30° 60° 90° 120° 150°0°-30° 210° 240°180°-90° -60° 270°
extended spaceinteractionkernel
result
supra-tresholdactivation
supra-tresholdactivation
Relationship to the dynamics of discrete activation variables
self-excitation
mutualinhibition
s(x)u(x)
u1 u2
x
s1s2
self-excitation
=> simulations
solutions and instabilities
input driven solution (sub-threshold) vs. self-stabilized solution (peak, supra-threshold)
detection instability
reverse detection instability
selection
selection instability
memory instability
detection instability from boost
Detection instability
h
dimension0
h
dimension0
h
dimension0
h
dimension0
input
self-excited peak
sub-threshold hill
sub-threshold hill
self-excited peaksub-threshold hill
self-excited peak
the detection instability helps stabilize decisions
threshold piercing detection instability
?4
?2
0
2
4
6
activ
atio
n
0 200 400 600 800 1000 1200
?4
0
4
8
activ
atio
n
time0 200 400 600 800 1000 1200
time
threshold
stable state
the detection instability helps stabilize decisions
self-stabilized peaks are macroscopic neuronal states, capable of impacting on down-stream neuronal systems
(unlike the microscopic neuronal activation that just exceeds a threshold)
emergence of time-discrete events
the detection instability also explains how a time-continuous neuronal dynamics may create macroscopic, time-discrete events
behavioral signatures of detection decisions
detection in psychophysical paradigms is rife with hysteresis
but: minimize response bias
Detection instability
in the detection of Generalized Apparent Motion
Generalized Apparent Motion
(Johansson, 1950)
t
Left
Position
Right
Position
Lum
inance (
cd/m
2) 1
Lb
Lm
L1 2
L2 2 1
Detection instability
varying BRLC
Detection instability
hysteresis of motion detection as BRLC is varied
(while response bias is minimized)
184 H. S. Hock, G. Schöner / Seeing and Perceiving 23 (2010) 173–195
Figure 5. Hysteresis effect observed by gradually increasing or gradually decreasing the backgroundrelative luminance contrast (BRLC) for a participant in Hock et al.’s (1997) third experiment. Theproportion of trials with switches from the perception of motion to the perception of nonmotion, andvice versa, are graphed as a function of the BRLC value at which each ascending or descendingsequence of BRLC values ends. (Note the inversion of the axis on the right.)
which there were switches during trials with a particular end-point BRLC valuewas different, depending on whether that aspect ratio was preceded by an ascend-ing (vertical axis on the left side of the graph) or a descending sequence of BRLCvalues (the inverted vertical axis on the right side of the graph). For example, whenthe end-point BRLC value was 0.5, motion continued to be perceived without aswitch to non-motion for 90% of the descending trials, and non-motion continuedto be perceived without a switch to motion for 58% of the ascending trials. Percep-tion therefore was bistable for this BRLC value and other BRLC values near it; bothmotion and non-motion could be perceived for the same stimulus, the proportion ofeach depending on the direction of parameter change. It was thus confirmed thatthe hysteresis effect obtained for single-element apparent motion was indicative ofperceptual hysteresis, and was not an artifact of ‘inferences from trial duration’.
7. Near-Threshold Neural Dynamics
The perceptual hysteresis effect described above indicates that there are two stableactivation states possible for the motion detectors stimulated by generalized ap-parent motion stimuli, one suprathreshold (motion is perceived) and the other sub-threshold (motion is not perceived). Because of this stabilization of near-thresholdactivation, motion and non-motion percepts both can occur for the same stimu-lus (bistability), and both can resist random fluctuations and stimulus changes thatwould result in frequent switches between them.
7.1. Why Stabilization Is Necessary
Whether an individual detector is activated by a stimulus or not, a random per-turbation will with equal probability increase or decrease its activation. Assume it
selection instability
input
activationfield
input
dimension
activationfield
input
activationfield
dimension
dimension
stabilizing selection decisions
behavioral signatures of selection decisions
in most experimental situations, the correct selection decision is cued by an “imperative signal” leaving no actual freedom of “choice” to the participant (only the freedom of “error”)
reasons are experimental
when performance approaches chance level, then close to “free choice”
because task set plays a major role in such tasks, I will discuss these only a little later
one system of “free choice”
selecting a new saccadic location
Analysis of the eye movement trace may allow us to understand whychanges are so hard to detect and what is the origin of the difference betweenthe Central and Marginal Interest cases.
Eye Movement Measures
Figure 2 shows a typical eye movement scanning pattern for a picture. It is seenthat even though the observer was looking at the picture for 48 sec, and search-ing actively for possible changes that might occur anywhere in the picture, theeye continued to follow a surprisingly stereotyped, repetitive, scanpath inwhich large areas of the picture are never directly fixated. Similar observationswere made by Yarbus (1967) and other authors, who observed that many por-tions of a picture are never directly fixated, and that the particular scanpath thatis used depends on what the observer is looking for in the picture.
Could this be the reason why some changes are not noticed? Could it be thatthose cases when the change is missed correspond to cases where the scanpathhappens not to include the change location? This hypothesis might explain thedifference between the MI and CI changes: Thus, it might be that MI locations,being less “interesting” to observers, tend to be less likely to be included in thescanpath than CI locations.
198 O’REGAN ET AL.
FIG. 2. Typical scanpath while a subject searched for changes. The original picture was in colour. Thechange that occurred in this picture was a vertical displacement of the railing in the background to thelevel of the man’ s eyes. In this record, the change was detected at the moment that the observer blinkedfor the fourth time. The positions of the eye when the blinks occurred are shown as white circles. Thelast, “effective” blink, marked “E”, occurred when the eye was in the region of the bar.
[O’Reagan et al., 2000]
input
input
saccadic
end-point
targets
targets
saccadic
end-point
activation field
activation fieldactivation
field
[after Kopecz, Schöner: Biol Cybern 73:49 (95)]
bistable
initial
fixation
visual
targets
[after: Ottes et al., Vis. Res. 25:825 (85)]
saccade generation
2 layer Amari fields
to comply with Dale’s law
and account for difference in time course of excitation (early) and inhibition (late)
_ +
+
excitatorylayer
inhibitorylayer
inhibitorykernel
excitatorykernel
[figure: Wilimzig, Schneider, Schöner, Neural Networks, 2006]
2 layer Amari model
time course of selection
early: input driven
intermediate: dominated by excitatory interaction
late: inhibitory interaction drives selection
Wilimzig, Schneider, Schöner, Neural Networks, 2006
=> early fusion, late selection
-10
0
10 (A)
double target paradigm
100 200 300 400 500 600 700
-10
0
10 (B)
target distractor paradigm
dire
ctio
n of
sac
cade
[deg
]
latency
Figure 16 Wilimzig Schneider Schöner
Wilimzig, Schneider, Schöner, Neural Networks, 2006
studying selection decisions in the laboratory
using an imperative signal...
reaction time (RT) paradigm
time
imperative signal=go signal
response
RT
task set
task set
that is the critical factor in most studies of selection!
for example, the classical Hick law, that the number of choices affects RT, is based on the task set specifying a number of choices
(although the form in which the imperative signal is given is varied as well... )
how do neuronal representations reflect the task set?
notion of preshape
movemen
t
param
eter
time
activation
1.0
taskinput
movement parameter
0.0preshapedfield
-0.4 0.0
preshapedfield
specific inputarrives
specific input
weak preshape in selection
specific (imperative) input dominates and drives detection instability
[Wilimzig, Schöner, 2006]
0
500
1000
1500
0
parameter, x
time,
t
activ
atio
n u(
x)
specific input + boostin different conditions
preshape
0
2
4
parameter, x
S(x)
-20
-10
0
10
u(x)
parameter, x
boost
using preshape to account for classical RT data
Hick’s law: RT increases with the number of choices
-4
-3
-4
-3
-4
-3
-4
-3
200 300 400 500 600
05
10
parameter, x
time, t
pres
hape
d u(
x)u m
ax(t)
1 2 4 8
[Erlhagen, Schöner, Psych Rev 2002]
metric effect
predict faster response times for metrically close than for metrically far choices
0 45 90 135 180
150 200 250 300 350 400 450-4
-2
0
2
4
time
pres
hape
d ne
ural
fiel
d
movement direction
peak
leve
l of a
ctiv
atio
n
wide
narrow
[from Schöner, Kopecz, Erlhagen, 1997]
faster
slower
experiment: metric effect
[McDowell, Jeka, Schöner ]
-4
-3
-2
0
2
4
-4
-3
250 350 450 550-2
0
2
4
time
-4
-3
0
2
4
250 350 450 550time
250 350 450 550time
pres
hape
d ac
tivat
ion
field
mai
xmal
act
ivat
ion
movement parameter
same metrics, different probability different metrics, same probability
highprobability
highprobability
highprobability
lowprobabilitylow
probability
lowprobability
movement parameter movement parameter
[from Erlhagen, Schöner: Psych. Rev. 2002]
WideFrequent
WideRare
NarrowFrequent
NarrowRare
320
300
280
260
240
220
200
Target
7
6
5
4
3
2
1
0Wide
FrequentWideRare
NarrowFrequent
NarrowRare
Target
[from McDowell, Jeka, Schöner, Hatfield, 2002]
wide narrow
Reaction Time P300 Amplitude Fz
Tim
e (
ms)
Am
plitu
de (
mic
roV
)
rarerare
frequent
frequent
boost-induced detection instability
dimensionactivation
preshape
dimensionactivation
dimensionactivation
self-excited activation peak
boost
boost
preshape
preshape
boost-driven detection instability
inhomogeneities in the field existing prior to a signal/stimulus that leads to a macroscopic response=“preshape”
the boost-driven detection instability amplifies preshape into macroscopic selection decisions
this supports categorical behavior
when preshape dominates
[Wilimzig, Schöner, 2006]
weak preshape in selection
specific (imperative) input dominates and drives detection instability
[Wilimzig, Schöner, 2006]
0
500
1000
1500
0
parameter, x
time,
t
activ
atio
n u(
x)
specific input + boostin different conditions
preshape
0
2
4
parameter, x
S(x)
-20
-10
0
10
u(x)
parameter, x
boost
distance effect
common in categorical tasks
e.g., decide which of two sticks is longer... RT is larger when sticks are more similar in length
interaction metrics-probability
Wilimzig, Schöner, 2006
opposite to that predicted for input-driven detection instabilities:
metrically close choices show larger effect of probability
0 100 200 300-20246
time, t
max
imal
activ
atio
n
0 100 200 300
parameter, x0
1
2
3
parameter, x
S(x)
time, ttime, t0 100 200 300
more probablechoice
less probablechoice
more probablechoice
less probablechoice
0
10
20
30
40
% e
rror
s
more probablechoice
less probablechoice
parameter, x
same metrics, different probability different metrics, same probability
Memory instability
h
dimension0
h
dimension0 input
self-excited
peak
input
self-excited
peak
h
dimension0
sub-threshold attractorh
dimension0
self-sustained
peak
“space ship” task probing spatial working memory
Metric�Working�Memory�Tasks10 sec delay2000 ms Ready, Set, Go!
+++
-40°
[Schutte, Spencer, JEP:HPP 2009]1977; Compte et al., 2000, for neural network models that usesimilar dynamics).
Considered together, the layers in Figure 3 capture the real-timeprocesses that underlie performance on a single spatial recall trial.At the start of the trial, the only activation in the perceptual fieldis at the location associated with the perceived reference axis (seehighlighted reference input in Figure 3a). This is a weak input andis not strong enough to generate a self-sustaining peak in theSWM field, though it does create an activation peak in theperceptual field (PFobj). Note that this input to the model isassumed to be generated by relatively low-level neural pro-cesses that extract symmetry using the visible edges of the taskspace (for evidence that symmetry axes are perceived as weaklines, see Li & Westheimer, 1997). We have not included thevisible edges in simulations of the model because they are quitefar from the target locations probed in our experiments. Giventhat neural interactions in the DFT depend on metric separation,these additional inputs far from the targets would have negli-gible consequences.
The next event in the simulation in Figure 3a is the targetpresentation. This event creates a strong peak in PFobj (see targetinput in Figure 3a) which drives up activation at associated sites inthe SWM field (SWMobj). When the target turns off, the targetactivation in PFobj dies out, but the target-related peak of activationremains active in SWMobj. In addition, activation from the refer-ence axis continues to influence PFobj because the reference axis issupported by readily available perceptual cues (see peak in PFobj
during the delay).Central to the DFT account of geometric biases is how the
reference-related perceptual input affects neurons in the workingmemory field during the delay. Figure 3c shows a time slice of theSWMobj field at the end of the delay. As can be seen in the figure,the working memory peak has slightly lower activation on the leftside. This lower activation is due to the strong inhibition aroundmidline created by the reference-related peak in PFobj (see high-
lighted reference input in Figures 3a & 3c). The greater inhibitionon the left side of the peak in SWM effectively “pushes” the peakaway from midline during the delay, that is, the maximal activityin SWM at the end of the trial is shifted to the right of the actualtarget location (for additional behavioral signatures of these inhib-itory interactions, see Simmering et al., 2006). Note that workingmemory peaks are not always dominated by inhibition as in Figure3c. For instance, if the working memory peak were positioned veryclose to or aligned with midline (location 0), it would be eitherattracted toward or stabilized by the excitatory reference input.This hints at how the DFT accounts for developmental changes ingeometric biases.
A simulation of the model with “child” parameters is shown inFigure 3b. This simulation is the same as the adult simulation inFigure 3a, except the interaction among neurons within each fieldand the projections between the fields have been scaled accordingto the spatial precision hypothesis: the neural interactions withinthe SWMobj and PFobj fields are weaker (relative to the adultparameters), the widths of the projections between the fields arebroader, and the excitatory and inhibitory projections areweaker (for a more detailed discussion see below). As can beseen in Figure 3b, these changes in interaction result in abroader peak in the SWMobj field. Additionally, the referenceinput is broader and weaker to reflect young children’s diffi-culty with reference frame calibration, that is, their ability tostably align and realign egocentric and allocentric referenceframes (see Spencer et al., 2007). The result of these changes isthat neural interactions in PFobj are not strong enough to builda reference-related peak during the delay. Consequently,SWMobj is only influenced by the broad excitatory input fromdetection of midline in the task space and the SWMobj peakdrifts toward the reference axis instead of away from the axis.
The simulations in Figure 3 demonstrate that the spatial preci-sion hypothesis and the DFT can capture the general pattern ofgeometric biases in early development and later development, but
Figure 4. Apparatus used for spaceship task. Inset shows sample target locations relative to the starting point.Targets are projected onto the table from beneath and responses are recorded using an Optotrak movementanalysis system. Note that the lights in the room are turned on for the photograph. During the experiment thelights were dimmed, and the table appeared black.
1702 SCHUTTE AND SPENCER
repulsion from midline/landmarks
0
4
8
cons
tant
err
or [
deg]
center20 deg40 deg 8
0
4
8
delay [s]
0 5 10 15 20
12
-4
-8
leftcenterright
delay [s]
0 5 10 15 20
[Schutte, Spencer, JEP:HPP 2009]
DFT account of repulsion: inhibitory interaction with peak representing landmark
DFT�$ Spatial�Recall
PF
SWM
Inhib
Act
ivat
ion
Location (°)
Time
midlinetarget
target
drift�away�from�midline
Simmering,�Schutte,�&�Spencer�(2007)�Brain�Research[Simmering, Schutte, Spencer: Brain Research, 2007]
Working memory as sustained peaks
implies metric drift of WM, which is a marginally stable state (one direction in which it is not asymptotically stable)
=> empirically real..
