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Detection by neuron populations
European Mathematical Psychology Group,
Graz, September 9 – 11, 2008
Uwe Mortensen
University of Münster, Germany
Detection
Probability summation
No probability summation
Models of neural mechanisms
The notion of probability summation:
Among channels/mechanisms.
Detection occurs if the activity in at least one of a number of channels exceeds
threshold.
Temporal:
Detection occurs if the activity at at least one point of time (within some
inerval J = [0, T]) exceeds threshold.
Spatial:
Detection occurs if the activity at at least one point in space (retinal coordinate)
exceeds threshold.
Usually just one type of PS is assumed in a given experiment
Aim of detection experiments
Probability
summation
Identification of
neural mechanisms
No probability
summation
Models/
Noise
deterministic
correlated
white
Inconsistency!
Network/population models
Extreme values theory
Quick‘s model
stochastic
Nonlinear
pooling
Can be fitted to most data –
meaningful?
Test
descriptive!
Theoretical
status unclear!
Max mean
detection
Quick (1974)
 (c )  1  2

n
|chi |b
i 1
Canonical models for
PS in visual
psychophysics

|cht )|


 W (c )  1  e

unit respones
Equivalent to
Weibull-function
channels
c contrast
 (c )  1  e
 c b
spatial
positions
   | hi |b
Temporal PS
1  pi  e|chi |
b
i
with log 2 absorbed into 
Tinkering with maxima of
Gaussian variables
(pooling, MinkowskiMetric)
Proof/derivation?
Watson (1979)
Goal directed ad hoc
mathematics

dt
Quotes indicating use of Quick‘s approach:
''To allow for the statistical nature of the detection process, the effects of
probability summation must be incorporated. … A convenient way to compute
the effects of spatial probability summation is based on Quick's (1974)
parameterization of the psychometric function. (Wilson 1978, p. 973; similarly
Wilson, Philips et al 1979, p. 594, Graham 1989, and many others.)
… probability summation […] requires that the noises associated with different
stimulations be uncorrelated.'' Gorea, Caetta, Sagi (2005, p.2531)
Similar statements by Meese \& Williams (2000), Tversky, Geisler \& Perry (2004) on
contour grouping, Monnier (2006), Meese \& Summers (2007),
Watson \& Ahumada (2005) take Quick/Minkowski as a basis for a general model of
contrast detection – everything is explained (?) Justification: as usual.
Detection and temporal probability summation – correlated noise
Assumptions:
( )1Noise is Gaussian and stationary
( )2Autocorrelation satisfies

 (c )  1  exp[ 2
2
1
R( )  1  R ''(0) 2  o( )
2

T
0
1
exp[ ( S  g (t; c))2 dt ]]
2
where 2  R ''(0) the second spectral moment.
0  2  
2 small  noise fluctuates slowly
2 large  noise fluctuates fast.
2   R ''(0)  E[( '(t )) 2 ]  0
Illustration of second spectral moment:
Example 1:
R( )  e
 2 /2
 R ''(0)    2  
(    R( )   ( ) (white noise) )
Example 2:
Power spectrum: S ( )  k ,  0    0
k0 sin(0 )
Autocorrelation: R ( ) 
2 0
k03
 2 
3
Autocorrelations for different 0 and 2 :
Temporal probability summation or maximum mean detection?
Roufs & Blommaert1981)
( : Determination of the impulse and step
response by means of the perturbation technique
Prediction
Data:
g (t , c)  c t p e   t
  .079, b  1/12.67, p  3
For all values
of 2 !
Detection by a population of matched neurons
Hebb‘s rule
implies adaptation
of neuron – local
matched filter
Pre-filter (lens, retina, LGB)
Stimulus
Defined by a
DOG-function
Test of matched neuron model: no probability summation of any sort!
Stimulus
Response of
pre-filter
Response of
matched neurons
To be estimated: four free parameters of the pre-filter!
Data and
predictions
„Channels“ and neuron populations: a stochastic model
(based on a model of Gerstner (2000))
Channel = Population of N neurons
The meaning of activity
na (t, t  t )
number of active neurons within [t, t  t ].
na (t , t  t )
N
proportion of active neurons
1 na (t , t  t )
activity of population at time t
t 0 t
N
A(t )  lim
Input current I i for the i-th neuron
N
I i (t )   wij  (t  t (j f ) )  I ext (t )
j 1
I ext (t ) is mean response of sensoric neurons activating
observed population
 (t  t (j f ) ) time course of postsynaptic current
generated by spike at time t (j f )
wij synaptic coupling of i-th neuron with j  th neuron ,
wij  k0 / N homogeneous case: all-to-all coupling
k0  0 excitatory
k0  0 inhibitory
k0  0 independence
Activity of an individual neuron
Integrate-and-fire neurons:
m
dui
 ui  RI i (t ), i  1,..., N
dt
Membrane potential of
i-th neuron
ur  ui  
Threshold (spike generation)
Resting potential
 m  RC time constant of cell membrane
Membrane potential density
n(u0 , u0  u )
N
Proportion neurons with membrane potential between
u0 and u0  u
limN 
u0 u
n(u0 , u0  u )

p(u, t )du,
u
0
N
Membrane potential density
specify activity  specify p(u, t )
Taylor-expansion of p(u, t )  Fokker-Planck-equation for p(u, t )
Stochastic differential equation for individual
trajectory of u:
Diffusion
Drift
du(t )  [a0u(t )  0 I ext (t )   (t )]dt   2 (t )dW (t ),
ur  u(t )  
Activity.from
stimulus
u (t ) is restricted to this
interval!
Activity from
environment
Derivative of
Brownian motion =
white noise
du(t )  [a0u(t )  0 I ext (t )   (t )]dt   2 (t )dW (t ),
ur  u(t )  
du(t )  [a0 u(t )  0 I ext (t )   (t ))]dt  k0 (t )dW (t ),
 varies slowly compared to stimulus driven activity
(Leopold, Murayama,Logothetis, 2003)
determines (i) the level of activation not due to stimulus,
and (ii) its variance,
  constant within trial, varies randomly between trials.
Response to short pulses and step functions
I ext (t )  c(at ) p exp(at ) (Roufs & Blommaert, 1981)
(Response to a 2 ms pulse!)
  15.5
  7.5
  2.5
1. The amplitude of mean response g is the same in all three cases – the
smaller eta, the more pronounced is g
2. The peaks of the activity (spike rate) are extremely short compared to the
mean response to the stimulus – prob. summation is unlikely!
Detection model:
The probability of detection depends on how pronounced the (mean) activity
generated by the stimulus is with respect to the overall activity.
Operationalised:
Maximum of mean activity
Ground activity: determines
probability of false alarm
g max  x0

S
Threshold value
Noise (= activity) from environment ( > 0)
Yeshurun & Carrasco, 1998, 1999; Treue, 2003; Martinez-Trujillo &
Treue, 2004: focussing attention on a position or feature will reduce
noise and enhance the response.
However: Reynolds & Desimone, 2003: attention increases
contrast gain in V4-neurons…
Summary:
1. Quick‘s (1974) model (white noise) may lead to arbitrary
interpretations of data
2. Correlated activity is the norm, not the exception
3. More realistic models (correlated noise) of probability summation
show that probability summation is not a general mode of detection
with max-mean or peak detection a special case
4. There may be adaptive processes – mechanisms are not
necessarily invariant with respect to stimulation
5. Construct dynamic network or population models, - not diffuse
„nonlinear summation“ models
Thank you for your attention!
Probability summation over time - the white noise case:
Application of extreme value statistics for independent
variables
Weibull:
 (c)  1  exp[
1
T
Gauss:
 (c)  1  exp[
1
T

T
0

T
0
( g (t , c)  S )  dt ], with S  0
exp[( S  g (t , c))]dt ]
T
1
 (0)  1  exp[  (  S )  dt ]  1  exp[(  S )  ], S  0
T0
T
1
 (0)  1  exp[  exp(  S )dt ]  1  exp[exp(  S )]
T0
 (0)  0 independent of T
T
But:
  1  exp[   (t )dt ]   (0)  1  exp[To ], 0  0
0
Hazard function
Detection by TPS, Gaussian coloured noise
Mean response g(t)
Psychometric function:
 (c )  1  exp[
2
2

T
0
1
exp[ ( S  g (t; c))2 dt ]]
2
Does not approach the
expression for white noise if
lambda-2 approaches infinity!
The form of the psychometric function and its
approximation by a Weibull function; different
stimulus durations.
Roufs & Blommaert (1981): Direct measurement of impulse and step
responses by means of a perturbation technique.
Impulse response, transient channel,
as determined by perturbation method
Impulse response, as derived from
MTF: true according to Watson (1981)
(although the additional assumption of
a minimum phase system has to be
made).
Roufs & Blommaert (1981): Direct measurement of impulse and step
responses by means of a perturbation technique (assuming maximum-ofmean detection).
Impulse response for transient channels: 3- or 2-phasic?
Watson (1981):
triphasic impulse response is an
artifact
Quick‘s model with exponent
between 2 and 7 yields 3-phasic
impulse repsonse. True response is
2-phasic, as derived from MTF.
Artifact?
Assumption of
probability
summation?
peak
detection?
Spatial probability summation:
Templates or matched filters for circular discs of different diameters,
superimposed on subthreshold Bessel-Jo-patterns for various spatial frequency
parameters: neither temporal nor spatial probability summation.
Data and predictions of template/MF-model, based on
temporal peak detection
There is no „nonlinear Minkowski-summation claimed by Watson &
Ahumada (2005) as a necessary element in the detection process!
Determination of line spread function – Hines (1976)
Rentschler & Hilz (1976) – Disinhibition in LSF-measurements?
Flanking line about 75% of test line!
Disinhibition?
Wilson, Philips et al (1979) – no disinhibition, but spatial
probability summation, as modelled by Quick‘s rule
LSF and LSF-estimates – probability summation, correlated noise
p(false alarm) = .1
LSF and LSF-estimates – probability summation, correlated noise
p(false alarm) = .01
Probability summation: correlated noise
(q: luminance proportion of flanking lines, P(fA) = .1)
No pseudo-disinhibition
for „white noise“!
Pseudo-inhibition for higher flanking
contrasts, no pseudo-disinhibition!
Probability summation does not predict disinhibition, - rather, inhibition!
LSF-prediction by Quick‘s rule; stimulus configuration A
Explore mechanisms
Prob. Summation.
Correlated
noise
No Prob. Summation
White
noise
Deterministic
models
Stochastic
models
Implies
Quick (1974)
inconsistency
 (c )  1  2
(Canonical
model in visual
psychophysics)

n
|chi |b
Max Mean
response
i 1
Pooling
Equivalent to
Weibull-function
1/ b
 ( c )  1  e  c
b
with log 2 absorbed into 
 1/ b


  | hi |b 
 i

(Minkowski-Metric)
Test