Decoding: Summary of previous slides
"
"
2. Decoding (continued)
Decoding: for neuro-prostheses and/or for understanding the relationship
between the brain’s activity and perception or action
AA- 'Unaware'
'Unaware'
Different strategies are possible: optimal decoders (e.g. ML, MAP) vs
simple decoders (e.g.
winner take all, population vector), depending on
Adaptation!
Adaptation!
State
what we know about State
the encoding model, and constraints.
Population !
Population !
Response
Response
ss
!!
!!
B- 'Aware'
B- 'Aware'
25
0
!180
!90
0
!test
90
Fixed !
Fixed
!
!
!
Decoder
Decoder
ŝŝ?
B ! Population Response
spike
Responses
A ! Tuning Curves
50
rr
Encoder
Encoder
r
50
25
Adaptation!
0
Adaptation!
State
!180
State
180
!90
0
!test
90
180
preferred
orientation
Population !
Encoder
s
A- 'Unaware'
Encoder
s
A- 'Unaware'
!!
!!
Adaptation!
Encoder
r
Fixed !
Decoder
!
Decoder
ŝŝ1
ŝ2
..
ŝN
! E.g. stimulus = oriented Gabor patch;
! Encoder = set Adaptation!
of noisy tuning curves, e.g. in V1;
! r =population response;
State
! Decoder = model we choose for this (e.g optimal);
!
! ŝ = perceived orientation of Population
Gabor patch;
s
•
!!
Encoder
r
Fixed !
Decoder
!
Decoder
B- 'Aware'
Bias. b(s) = E(ŝ|s) − s
ŝŝ1
ŝ2
..
ŝN
If E(ŝ|s) = s the estimator is said to be unbiased.
Adaptation!
State
Response
! How can we relate this model
of perception and the statistics of
Adaptive !
!
!
the estimates with
psychophysical performance?
Encoder
!
Decoder
r
ŝ
ŝ
Population !
Response
B- 'Aware'
s
Adaptive !
!
Adaptive
!
Decoder
!
Decoder
is our decoder ?
a bit of estimation theory ...
Population !
Response
!!
r
r
Adaptation!
How
good
State
From Population State
Codes to Psychophysical Performances
s
Population
Response !
Response
ŝ
Population !
Response
s
!!
Encoder
r
Adaptive !
!
Decoder
ŝ
A- 'Unaware'
A- 'Unaware'
Adaptation!
How
good
State
Adaptation!
is our decoder ?
a bit of estimation theory ...
From Population State
Codes to Psychophysical Performances
Population !
Response
s
•
•
!!
Encoder
Population !
Response
r
Fixed !
Decoder
!
Decoder
B- 'Aware'
Bias. b(s) = E(ŝ|s) − s
ŝŝ1
ŝ2
..
ŝN
If E(ŝ|s) = s the estimator is said to be unbiased.
Adaptation!
Variance var(ŝ)
State
Estimation theory tells us that, knowing the encoder model P[r|s], there is a
Population !
lower bound on the variance that can be achieved by any decoder. This
quantity is known as the Cramer-RaoResponse
Bound. The denominator is known as
Adaptive !
Fisher Information.
!!
Encoder
(1 + b!Decoder
(s))! 2
var(ŝ) ≥
r
s
ŝ
s
!!
Encoder
r
B- 'Aware'
Response
! How can we relate this model
of perception and the statistics of
Adaptive !
!!
the estimates
with
psychophysical performance?
Encoder
!
r
s
Decoder
ŝ
b) Discrimination Tasks
! The measured quantity is the difference between the perceived
orientation and the real orientation:
ŝŝ1
ŝ2
..
ŝN
! E.g. stimulus = oriented Gabor patch;
! Encoder = set Adaptation!
of noisy tuning curves, e.g. in V1;
! r =population response;
State
! Decoder = model we choose for this (e.g optimal);
!
! ŝ = perceived orientation of Population
Gabor patch;
IF (s)
a) Estimation tasks
Fixed !
Decoder
!
Decoder
! Is the second grating of the same
< ŝ > −s
orientation as the first grating, or a different
orientation?
! The measured quantity is the Discrimination
Threshold a.k.a Just Noticeable difference (JND)
Stare at this
for 20 sec
Then look at
that
Tilt after-effet
Poggendorf
Illusion
Zollner Illusion
Fraser Illusion
Discrimination threshold depends on the overlap
between the internal ‘representation’ of the 2
stimuli: p[s1|r] and p[s2|r]:
• The bias of the internal representation
(expansion/contraction of the ‘distance’ between
the stimuli)
• How noisy the internal representation is (the
variance of the estimates)
Adaptation!
Adaptation!
State
State
ss
Population !
Population
Response !
Response
rr
Encoder
Encoder
!!
!!
Responses
A ! Tuning Curves
Fixed !
!
Fixed
!
Decoder
!
Decoder
ŝŝŝ
1
ŝ2
B ! Population Response
..
50
50
B- 'Aware'
B- 'Aware'
0
!180
!90
Fisher information: gives the discrimination threshold that would be obtained
(asymptotically) by an optimal decoder, for eg. ML (units of var ^-1)
!
ŝN
25
25
Fisher information: the best possible discrimination
performance for a given encoder model
0
0
!test
90
180
Adaptation!
Linking
Adaptation!
State
State
!180
!90
0
!test
90
180
b(ŝ) =< ŝ > −s
Population !
Population !
Response
Response
ssvar(ŝ)Encoder
Encoder
!!
!!
rr
IF (s) = − <
perceptual bias
Adaptive !
Adaptive
!
!
!
Decoder
Decoder discrimination threshold (76%
ŝŝ
std(ŝ)
threshold(ŝ) =
1 + b! (ŝ)
is expressed in terms of the encoding model P[r|s], i.e. in terms of the
tuning curves and the noise
!
the statistics of the model and psychophysics
correct)
just noticeable difference
From the Cramer Rao Bound, knowing the encoder model and independently of
the decoder, it is known that the threshold is bounded by the sqrt of Fisher
information: threshold(ŝ) ≥ ! 1
!
!
∂ 2 lnP [r|s]
>
∂s2
Interpreted as a measure of ‘information’ in the responses;
a useful tool to relate directly the properties of the neural responses with
discrimination performance.
! is related with Mutual information and Stimulus Specific Information
(Brunel and Nadal 1998, Challis and Series 2008).
IF (s)
A- 'Unaware'
From Population Responses to Psychophysics
From Population Responses to Psychophysics
Population !
Population
Population
Response !
Response
Response
ss
sensory
stimulus
!!
!!
rr
Encoder
Encoder
P [r|s]
A ! Tuning Curves
B- 'Aware'
B- 'Aware'
25
Responses
50
25
0
!180
!90
Fixed !
!
Fixed
!
Decoder
!
Decoder
B ! Population Response
50
Adaptation!
State
Population !
Response
s
ŝŝ
Perception
A ! Tuning Curves
0
90
180
test
Adaptation!
Adaptation!
State
State
!180
!90
!
0
90
50
50
25
25
0
!180
!90
Fixed !
Optimal
!
Decoder
var(ŝ) !
0
0
!test
90
ŝ
B ! Population Response
B- 'Aware'
180
!180
!90
0
!test
90
180
! Number of neurons?
?
!Tuning curves shapePopulation
!
! Noise correlations ? Response
180
test
s
!!
Encoder
r
Adaptive !
!
Decoder
1
IF (s)
1
thres(ŝ) ! !
IF (s)
Questions that weAdaptation!
can explore:
What changes in encoder
State would increase discrimination performances?
0
!
r
Encoder
!!
Responses
Two strategies:
! Assume the decoder is optimal: Compute Fisher information from P[r|s]. We
A- 'Unaware'
know thatAthis 'Unaware'
will give us the minimal possible variance of any unbiased decoder,
and the minimal threshold of any decoder (biased or unbiased).
! Construct explicitly Adaptation!
the decoder (e.g. population vector). Compute explicitly bias,
Adaptation!
variance, and threshold
of
estimates.
State
State
ŝ
Response
variance from
trial to trial
∆r
∆θ
-90
-60
-30
0
30
60
What are the factors that control performance?
Sharpening of tuning
curve
Responses (spk/s)
Responses (spk/s)
What are the factors that control performance?
90
∆r
∆θ
-90
-60
Orientation (!)
0
-30
0
30
60
Orientation (!)
Orientation
90
What are the factors that control performance?
Sharpening or
Gain
modulation
Response
variance from
trial to trial
Responses (spk/s)
Responses (spk/s)
Adaptation
Attention,
Perceptual learning
-60
60
Orientation
What are the factors that control performance?
-90
30
Orientation (!)
Orientation
Gain
modulation
-30
90
Aging? Disease ?
Drugs? Lack of sleep?
-90
-60
-30
0
30
60
Orientation (!)
Orientation
90
What are the factors that control performance?
What are the factors that control performance?
Fisher information formalizes intuition and provides a tool to explore
these questions precisely.
! For Poisson noise (under some assumptions on the tuning curves):
Fisher information formalizes intuition and provides a tool to explore
these questions precisely.
! For Poisson noise (under some assumptions on the tuning curves):
!
Ii (s) =
I(s) =
fi! (s)2
fi (s)
! f ! (s)2
i
i
fi (s)
!
Slope 2
Ii (s) =
variance
For independent neurons, FI of the
population is the sum of each
neurons’ FI
I(s) =
fi! (s)2
fi (s)
! f ! (s)2
i
i
!
fi (s)
Slope 2
variance
For independent neurons, FI of the
population is the sum of each
neurons’ FI
For Gaussian correlated noise:
1
IF (s) = f ! (s)Q−1 (s)f ! (s) + Trace[Q−1 (s)Q! (s)Q−1 (s)Q! (s)]
2
For correlated neurons, FI is modulated by
correlations.
Research questions (1)
What would be the ‘optimal’ shape for tuning curves?
adaptation, attention and learning a step towards more ‘optimal’ tuning
curves for the attended/trained stimulus ?
!
! Are
Research questions (2)
How many neurons participate in a psychophysical task ? (see also, lab 1)
1, 10, 100, 10000? How can we find out ?
! comparing performance (e.g. MT: Britten et al 1992). stimulating (MT:
Salzman, Britten, Newsome 1990).
!
Neurons in auditory midbrain of the guinea pig adjust their response to improve the
accuracy of the code close to the region of most commonly occurring sound levels.
[Dean, Harper & McAlpine, Nature Neuro, 2005]
Houweling & Brecht, Nature, 2008
Research questions (3)
Pooling from large populations of neurons thought to be a way to
average out the noise.
! Pairs of neurons show correlations in their variability: does pooling
more and more neurons increases (linearly) the accuracy of the
representation?
or Is information saturating over a certain number of neurons ?
[Zohary et al 1994]
!
Summary
"
"
The efficiency of Estimators / Decoders can be characterized by the bias
and the variance.
Fisher Information is related to the minimal variance of a unbiased
estimator.
In a model of a population of neurons, Fisher Information can be
expressed in terms of the tuning curves and the noise.
" Fisher information can be used to relate population responses and
discrimination performances. It gives a bound on the discrimination
threshold
" Fisher Information can be used to explore the factors that impact on the
precision of the code / behavioral performances.
"
Research questions (4)
Can the study of illusions inform us on the type of ‘decoder’
that is used in the brain?
!