Detection by neuron populations

European Mathematical Psychology Group,

Graz, September 9 – 11, 2008

Uwe MortensenUniversity 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

Probability summation

No probability summation

Aim of detection experiments

Models/ Noise

correlated white

deterministic stochastic

Extreme values theory

Quick‘s model Nonlinear pooling

Network/popu-lation models

Max mean detection

Test

Identification of neural mechanisms

Inconsistency!

descriptive!

Can be fitted to most data – meaningful?

Theoretical status unclear!

1

| |

( ) 1 2

nbi

i

ch

c

Quick (1974)

Equivalent to Weibull-function

with log 2 absorbed into

( ) 1 bcc e

| |bii

h

channels spatial positions

| )|( ) 1 e

cht dt

W c

| |1bich

ip e Temporal PS

Watson (1979)(pooling, Minkowski-Metric)

Canonical models for PS in visual psychophysics

unit respones

c contrast

Proof/derivation?Tinkering with maxima of Gaussian variables

Goal directed ad hoc mathematics

Quotes indicating use of Quick‘s approach:

Similar statements by Meese \& Williams (2000), Tversky, Geisler \& Perry (2004) on contour grouping, Monnier (2006), Meese \& Summers (2007),

… probability summation […] requires that the noises associated with different stimulations be uncorrelated.'' Gorea, Caetta, Sagi (2005, p.2531)

Watson \& Ahumada (2005) take Quick/Minkowski as a basis for a general model of contrast detection – everything is explained (?) Justification: as usual.

''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.)

Assumptions:

2 2

0

1( 1 exp[ exp[ ( ( ; )) ]]

2 2)

Tc S g t c dt

2 ''(0)R where the second spectral moment.

2

2

small noise fluctuates slowly

large noise fluctuates fast.

Detection and temporal probability summation – correlated noise

20

2

1

12 ( ) 1 ''(0) ( )

2R R o

( ) Noise is Gaussian and stationary

( ) Autocorrelation satisfies

22 ''(0) [( '( )) ] 0R E t

Illustration of second spectral moment:

0 0Power spectrum: ( ) , S k

0 0

0

sin( )Autocorrelation: ( )

2

kR

30

2 3

k

0 2Autocorrelations for different and :

2 /22( ) ''(0)R e R

( ( ) ( ) ) ( white noise)R

Example 1:

Example 2:

Roufs & Blommaert (1981): Determination of the impulse and step response by means of the perturbation technique

Data: Prediction

( , ) e

.079, 1/12.67, 3

p tg t c c t

b p

Temporal probability summation or maximum mean detection?

2For all values

of !

Stimulus

Pre-filter (lens, retina, LGB)

Hebb‘s rule implies adaptation of neuron – local matched filter

Detection by a population of matched neurons

Defined by a DOG-function

Test of matched neuron model: no probability summation of any sort!

StimulusResponse of pre-filter

Response of matched neurons

Data and predictions

To be estimated: four free parameters of the pre-filter!

„Channels“ and neuron populations: a stochastic model

Channel = Population of N neurons

( , ) [ , ]an t t t t t t number of active neurons within .

( , )an t t t

N

proportion of active neurons

0

( , )1( ) lim a

t

n t t tA t t

t N

activity of population at time

The meaning of activity

(based on a model of Gerstner (2000))

( )

1

( ) ( ) ( )

i

Nf ext

i ij jj

I i

I t w t t I t

Input current for the -th neuron

ijw i j synaptic coupling of -th neuron with th neuron ,

( )

( )

( fj

fj

t t

t

) time course of postsynaptic current

generated by spike at time

( )extI t is mean response of sensoric neurons activating

observed population

0 /ijw k N homogeneous case: all-to-all coupling

0 0 excitatoryk

0 0 inhibitoryk

0 0 independencek

( ), 1,...,im i i

duu RI t i N

dt

Integrate-and-fire neurons:

m RC time constant of cell membrane

r iu u

Activity of an individual neuron

Threshold (spike generation)

Resting potential

Membrane potential of i-th neuron

0

0

0 0( , )( , ) ,

u u

N u

n u u up u t du

N

lim

Membrane potential density

0 0( , )n u u u

N

0 0u u u

Proportion neurons with membrane potential between

and

Membrane potential density

( , ) ( , )p u t p u tTaylor-expansion of Fokker-Planck-equation for

Stochastic differential equation for individual trajectory of u:

( , )specify activity specify p u t

20 0( ) [ ( ) ( ) ( )] ( ) ( ),

( )

ext

r

du t a u t I t t dt t dW t

u u t

Activity.from stimulus

Activity from environment

Derivative of Brownian motion = white noise

( ) is restricted to this

interval!

u t

Drift Diffusion

0 0 0( ) [ ( ) ( ) ( ))] ( ) ( ),extdu t a u t I t t dt k 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.

20 0( ) [ ( ) ( ) ( )] ( ) ( ),

( )

ext

r

du t a u t I t t dt t dW t

u u t

Response to short pulses and step functions

( ) ( ) exp( ) ( Roufs & Blommaert, 1981)ext pI t c at at

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!

(Response to a 2 ms pulse!)

0g xS

max

Detection model:

Ground activity: determines probability of false alarm

Maximum of mean activity

Noise (= activity) from environment ( > 0)

Threshold value

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…

The probability of detection depends on how pronounced the (mean) activity generated by the stimulus is with respect to the overall activity.

Operationalised:

Summary:

1. Quick‘s (1974) model (white noise) may lead to arbitrary interpretations of data

2. Correlated activity is the norm, not the exception3. 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!

0

1( ) 1 exp[ exp[ ( ( , ))] ]Gauss:

Tc S g t c dt

T

0

1( ) 1 exp[ ( ( , ) ) ], 0Weibull: with

Tc g t c S dt S

T

Probability summation over time - the white noise case:

Application of extreme value statistics for independent variables

0

1(0) 1 exp[ ( ) ] 1 exp[ ( ) ], 0

T

S dt S ST

0

1(0) 1 exp[ exp( ) ] 1 exp[ ( )]

T

S dt exp ST

(0) 0 independent of T

0

0

1 exp[ ( ) ] (0) 1 exp[ ] 0, T

ot dt T But:

Hazard function

Detection by TPS, Gaussian coloured noise

The form of the psychometric function and its approximation by a Weibull function; different stimulus durations.

Mean response g(t) Psychometric function:2 2

0

1( 1 exp[ exp[ ( ( ; )) ]]

2 2)

Tc S g t c dt

Does not approach the expression for white noise if lambda-2 approaches infinity!

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-of-mean 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

peak detection?

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.

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?

Disinhibition?

Flanking line about 75% of test line!

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)

Probability summation does not predict disinhibition, - rather, inhibition!

No pseudo-disinhibitionfor „white noise“!

Pseudo-inhibition for higher flanking contrasts, no pseudo-disinhibition!

LSF-prediction by Quick‘s rule; stimulus configuration A

Explore mechanisms

Prob. Summation. No Prob. Summation

Correlated noise

Whitenoise

Deterministic models

Stochasticmodels

Quick (1974)

1

| |

( ) 1 2

nbi

i

ch

c

Pooling

with log 2 absorbed into

( ) 1 bcc e

Max Mean response

Test

(Minkowski-Metric)

Implies inconsistency

Equivalent to Weibull-function

1/

1/ | |b

b bi

i

h

(Canonical model in visual psychophysics)