A bit about reverse correlation
stimulus design and analysis
Yonatan Aljadeff, University of Chicago
aljadeff@uchicago.edu
1/27/2016, UCSD
Outline:
I
Reverse correlation experiments and linear non-linear models
I
Why use white noise?
I
Why not use white noise?
Analysis of responses to non-white noise stimuli
I
I
I
synthetic model
rat thalamus
Reverse correlation experiments:
The model we will construct:
response(t)
= f [stimulus(t0 < t), response(t0 < t)]
The model we will construct:
response(t)
= f [stimulus(t0 < t), response(t0 < t)]
= f [stimulus(t0 < t)]
The model we will construct:
response(t)
= f [stimulus(t0 < t), response(t0 < t)]
= f [stimulus(t0 < t)]
If by knowing f we can predict responses to stimuli that weren’t used
to fit f we have hope to understand the computation the cell is
performing.
Important model assumptions: no explicit time dependence (no
adaptation), fixed stimulus statistics
Two steps of constructing model
1. dimensionality reduction
response(t)
= f [stimulus(t0 < t)]
Two steps of constructing model
1. dimensionality reduction
response(t)
= f [stimulus(t0 < t)]
=
f [small number of stimulus components(t0 < t)]
Two steps of constructing model
1. dimensionality reduction
response(t)
= f [stimulus(t0 < t)]
=
f [small number of stimulus components(t0 < t)]
We want to focus on stimulus statistics, so small number = 1.
The first component is the Spike
Triggered Average:


X
1
ϕSTA = 
s(t) − hsi
Nspike t=t
spike
Two steps of constructing model
1. dimensionality reduction
response(t)
= f [stimulus(t0 < t)]
=
f [small number of stimulus components(t0 < t)]
We want to focus on stimulus statistics, so small number = 1.
spike
r(t) =
f [ϕSTA · s]
4
3
2
1
s2
The first component is the Spike
Triggered Average:


X
1
ϕSTA = 
s(t) − hsi
Nspike t=t
0
-1
-2
-3
-4
-4
-2
0
s1
2
4
Two steps of constructing model
2. fitting nonlinearity
We use Bayes’ rule:
r(t)
=
p(spike|s)
Two steps of constructing model
2. fitting nonlinearity
We use Bayes’ rule:
r(t)
=
p(spike|s)
p(s|spike)p(spike)
p(s)
=
1
0.9
0.8
probability
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-3
-2
-1
0
φSTA · s
1
2
3
Why use white noise stimuli?
Why use white noise stimuli?
Avoid putting the answer in
Why use white noise stimuli?
Avoid putting the answer in
Makes analysis easy
= f [ϕST A · s]
p(s|spike)p(spike)
p(spike|s) =
p(s)
r(t)
Why not use white noise stimuli?
Why not use white noise stimuli?
active sensing
(self generated stimulus)
Why not use white noise stimuli?
active sensing
(self generated stimulus)
some neurons will never spike
Why not use white noise stimuli?
active sensing
(self generated stimulus)
some neurons will never spike
ask questions about
adaptation, optimal processing ...
What is the big deal?
4
3
2
s2
1
0
-1
-2
-3
-4
-4
-2
0
2
4
s1
1
0.9
0.8
probability
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-3
-2
-1
0
φSTA · s
1
2
3
What is the big deal?
4
3
3
2
2
2
1
1
1
s2
s2
0
-1
s2
4
3
4
0
0
-1
-1
-2
-2
-2
-3
-3
-3
-4
-4
-2
0
2
4
-4
s1
-4
-4
-2
0
2
4
-4
-2
s1
1
0
2
4
2
4
s1
4
4
3
3
2
2
1
1
0.9
0.6
s2
probability
0.7
s2
0.8
0
0
0.5
-1
-1
-2
-2
0.4
0.3
0.2
-3
-3
0.1
-4
0
-3
-2
-1
0
φSTA · s
1
2
3
-4
-4
-2
0
s1
2
4
-4
-2
0
s1
A simple exercise, model
p(spike|s) =
1
1 + exp [−A(φ · s − B)]
φ=
1
probability of a spike
A = slope
0
0
B
φ·s
A simple exercise, stimulus
white noise
10×10 patches from
strongly correlated
Gaussians
Spike Triggered Average
φ
STA computed from
responses to white
noise
STA computed from
responses to
correlated noise
Decorrelation
Analyze responses with decorrelated stimulus s̃
C
s̃(t)
=
cov(S) =
1
= C − 2 s(t)
T
1X
s(t)s(t)>
T t=1
Decorrelation
Analyze responses with decorrelated stimulus s̃
C
s̃(t)
=
cov(S) =
T
1X
s(t)s(t)>
T t=1
1
= C − 2 s(t)
cov(S̃) =
T
1X
s̃(t)s̃(t)>
T t=1
Decorrelation
Analyze responses with decorrelated stimulus s̃
C
s̃(t)
=
cov(S) =
T
1X
s(t)s(t)>
T t=1
1
= C − 2 s(t)
cov(S̃) =
T
1X
s̃(t)s̃(t)>
T t=1
= C
− 12
T
1X
s(t)s(t)>
T t=1
!
1
C− 2
Decorrelation
Analyze responses with decorrelated stimulus s̃
C
s̃(t)
=
cov(S) =
T
1X
s(t)s(t)>
T t=1
1
= C − 2 s(t)
cov(S̃) =
T
1X
s̃(t)s̃(t)>
T t=1
= C
= I
− 12
T
1X
s(t)s(t)>
T t=1
!
1
C− 2
For finite data, need regularization
eigenvalue of covariance matrix
10 2
10 0
10 -2
10 -4
0
50
eigenvalue number
100
For finite data, need regularization
model overlap
eigenvalue of covariance matrix
10 2
10 0
10 -2
computed from full inverse
10 -4
0
50
eigenvalue number
100
50
pseudoinverse order
100
0.6
0.4
0.2
0
0
computed from pseudoinverse
Rat thalamus data
B
Φ=0
Maximum protraction
Midpoint
Maximum
retraction
100
Spike rate (Hz)
A
Φ = ±π
0o
50
0
0
π
−π
Phase in whisk cycle
Vibrissa angle
C
80o
Measured
Reconstructed from Hilbert transform
100o
120o
140o
Spike train
20o
0o
-20o
Position
Relative power in eigenvalue
of stimulus covariance
1s
Time
1
E
10−1
F
1
2
3
10−2
4
5
6
10−3
10−4
7
8
-0.3 -0.2 -0.1
Time (s)
−5
10
−6
< 10
0
50
100
Rank of eigenvalue
0
Interval distribution, Hz
Angle
about midpoint
D
40 Inter-spike
interval
20
Inter-whisk
interval
NT
150
0
0
0.2
Time (s)
0.4
0.2
0.2
STA
0.1
mode 1
0
0
mode 3
-0.1
-0.1
-0.2
-0.3
-0.2
-0.1
0
-0.2
-0.3
0
-0.1
-0.2
Time (s)
Time (s)
B
Eigenvalue of
covariance matrix
Whitened
STC mode 1
STC
mode 2
Whitened
STA
0.1
0.2
mode 1
0.1
mode 2
0
mode 3
-0.1
Null hypothesis
-0.2
0
50
100
150
Rank of eigenvalue
C
Marginal density of
input projected on
whitened STA mode
Input probability density
projected on
whitened STC mode 1
0.6
Prior
distribution
100
80
60
0.4
40
0.2
20
0
4.0
00
-2 180
-3
120
60
-4 0
Predicted
spike rate (Hz)
0.8
Spikeconditional
distribution
Predicted marginal
spike rate (Hz)
1.0
Predicted marginal
spike rate (Hz)
40 60 80 100
20
2.0
0
-2.0
-4.0
-4.0
-2.0
0
2.0
4.0 0 0.2 0.4 0.6 0.8 1.0
Input probability density
Marginal density of
projected on
input projected on
whitened STA mode
whitened STC mode 1
Spike triggered covariance
A
Spike triggered average
Rat thalamus data
Rat thalamus data
Angle about
midpoint
A
Vibrissa C4
30o
0o
-30o
Spike train
B
Spike-triggered average
0.2
0.1
0
Whitened spike-triggered average
Predicted probability of spiking
0.2
0.1
0
0.2
0.1
0
0.2
0.1
0
Combined spike triggered covariance and spike triggered average
Combined, whitened spike triggered covariance and whitened spike triggered average
Maximum noise entropy
0.4
0.3
0.2
0.1
0
0.3
0.2
0.1
0
Generalized linear model
Phase tuning
D
Log(Power)
0
-0.30
Time (s)
-0.35
|Coherence|
E
-0.45
STA
whitened STA
STC+STA
whitened STC+STA
MNE
GLM
Tune
Null
-0.40
1s
0
Measured train
-5
0
Stimulus
-5
-10
π/2
1.0
0
-π/2
Phase
C
Log-likelihood (1/s)
0.2
0.1
0
0.5
0
0.95 confidence
0
5
10
Frequency (Hz)
15
20
Conclusion
Think about stimulus design and the appropriate analyses
methods the stimulus you choose implies.
Conclusion
Think about stimulus design and the appropriate analyses
methods the stimulus you choose implies.