Neural Encoding Models · 2011. 11. 25. · STA. Non-linearity. STA unbiased for spherical...
Transcript of Neural Encoding Models · 2011. 11. 25. · STA. Non-linearity. STA unbiased for spherical...
Neural Encoding Models
Maneesh [email protected]
Gatsby Computational Neuroscience UnitUniversity College London
Term 1, Autumn 2011
Studying sensory systems
x(t) y(t)
Decoding: x(t)= G[y(t)] (reconstruction)Encoding: y(t)= F [x(t)] (systems identification)
General approach
Goal: Estimate p(spike|x,H) [or λ(t|x[0, t), H(t))] from data.
• Naive approach: measure p(spike, H|x) directly for every setting of x.
– too hard: too little data and too many potential inputs.
• Estimate some functional f (p) instead (e.g. mutual information)
• Select stimuli efficiently
• Fit models with smaller numbers of parameters
Spikes, or rate?
Most neurons communicate using action potentials — statistically de-scribed by a point process:
P(spike ∈ [t, t + dt)
)= λ(t|H(t), stimulus, network activity)dt
To fully model the response we need to identify λ. In general this de-pends on spike history H(t) and network activity. Three options:
• Ignore the history dependence, take network activity as source of“noise” (i.e. assume firing is inhomogeneous Poisson or Cox process,conditioned on the stimulus).
• Average multiple trials to estimate
λ(t, stimulus) = limN→∞
1
N
∑n
λ(t|Hn(t), stimulus, networkn)
the mean intensity (or PSTH), and try to fit this.
• Attempt to capture history and network effects in simple models.
Spike-triggered average
Decoding: mean of P (x | y = 1)
Encoding: predictive filter
Linear regression
y(t) =
∫ T
0
x(t− τ )w(τ )dτ
W (ω) =X(ω)∗Y (ω)
|X(ω)|2
x1 x2 x3 . . . xT xT+1 . . .x1 x2 x3 . . . xT︸ ︷︷ ︸x1 x2 x3 . . . xT xT︸ ︷︷ ︸
x1 x2 x3 . . . xT+1
x2 x3 x4 . . . xT+1
...×
wt...w3
w2
w1
=
yTyT+1
...
XW = Y
W = (XTX)︸ ︷︷ ︸ΣSS
−1 (XTY )︸ ︷︷ ︸STA
Linear models
So the (whitened) spike-triggered average gives the minimum-squared-error linear model.
Issues:
• overfitting and regularisation
– standard methods for regression
• negative predicted rates
– can model deviations from background
• real neurons aren’t linear
– models are still used extensively– interpretable suggestions of underlying sensitivity– may provide unbiased estimates of cascade filters (see later)
How good are linear predictions?
We would like an absolute measure of model performance.
Measured responses can never be predicted perfectly:
• The measurements themselves are noisy.
Models may fail to predict because:
• They are the wrong model.
• Their parameters are mis-estimated due to noise.
Estimating predictable power
Psignal
Pnoise
response︸ ︷︷ ︸r(n)
= signal + noise
P(r(n)) = Psignal + Pnoise
P(r(n)) = Psignal +1
NPnoise
⇒Psignal =
1
N − 1
(NP(r(n))− P(r(n))
)Pnoise = P(r(n))− Psignal
Signal power in A1 responses
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0
0.1
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0.4
Noise power (spikes2/bin)
Sig
nal p
ower
(sp
ikes
2 /bin
)
0 50 100 150Number of recordings
Testing a model
For a perfect prediction⟨P(trial)− P(residual)
⟩= P(signal)
Thus, we can judge the performance of a model by the normalized pre-dictive power
P(trial)− P(residual)
P(signal)
Similar to coefficient of determination (r2), but the denominator is thepredictable variance.
Predictive performance
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0.5
1
1.5
2
2.5
norm
alis
ed B
ayes
pre
dict
ive
pow
er
Training Error
−2 −1 0 1−2
−1.5
−1
−0.5
0
0.5
1Cross−Validation Error
normalised STA predictive power
Extrapolating the model performance
Jackknifed estimates
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0
0.5
1
1.5
2
2.5
3
Nor
mal
ized
line
arly
pre
dict
ive
pow
er
Normalized noise power
Extrapolated linearity
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0.5
1
1.5
2
2.5
−5 0 5 10 15 20 25 30
−0.2
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1
Normalized noise power
Nor
mal
ized
line
arly
pre
dict
ive
pow
er
[extrapolated range: (0.19,0.39); mean Jackknife estimate: 0.29]
Simulated (almost) linear data
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0.5
1
1.5
2
2.5
3
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1.1
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1.5
Normalized noise power
Nor
mal
ized
line
arly
pre
dict
ive
pow
er
[extrapolated range: (0.95,0.97); mean Jackknife estimate: 0.97]
Linear fits to non-linear functions
Linear fits to non-linear functions
(Stimulus dependence does not always signal response adaptation)
Approximations are stimulus dependent
(Stimulus dependence does not always signal response adaptation)
Consequences
Local fitting can have counterintuitive consequences on the interpreta-tion of a “receptive field”.
“Independently distributed” stimuli
Knowing stimulus power at any set of points in analysis space providesnoinformation about stimulus power at any other point.
DRC:
Space
Spectrotemporal
Ripple:
Independence is a property of stimulus and analysis space.
Christianson, Sahani, and Linden (2008)
Nonlinearity & non-independence distort RF estimates
Stimulus may have higher-order correlations in other analysis spaces— interaction with nonlinearities can produce misleading “receptive fields.”
Christianson, Sahani, and Linden (2008)
What about natural sounds?
Multiplicative RF
Time (ms)
Fre
q. (
kHz)
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1
2
3
4
5
6
7Multiplicative RF
Fre
q. (
kHz)
Time (ms)−30 −25 −20 −15 −10 −5
1
2
3
4
5
6
7
Finch Song
Fre
q. (
kHz)
Time (ms)−30 −25 −20 −15 −10 −5
1
2
3
4
5
6
7Finch Song
Fre
q. (
kHz)
Time (ms)−30 −25 −20 −15 −10 −5
1
2
3
4
5
6
7
Usually not independent in any space — so STRFs may not be conser-vative estimates of receptive fields.
Christianson, Sahani, and Linden (2008)
Beyond linearity
Beyond linearity
Linear models often fail to predict well. Alternatives?
•Wiener/Volterra functional expansions
– M-series– Linearised estimation– Kernel formulations
• LN (Wiener) cascades
– Spike-trigger covariance (STC) methods– “Maximimally informative” dimensions (MID) ⇔ ML nonparametric
LNP models– ML Parametric GLM models
• NL (Hammerstein) cascades
– Multilinear formulations
Non-linear models
The LNP (Wiener) cascade
k n
Rectification addresses negative firing rates. Possible biophysical justi-fication.
LNP estimation – the Spike-triggered ensemble
Single linear filter
STA.
Non-linearity.
STA unbiased for spherical (elliptical) data.
Bussgang.
Non-spherical inputs.
Biases.
Multiple filters
Distribution changes along relevant directions (and, usually, along alllinear combinations of relevant directions).
Proxies for distribution:
•mean: STA (can only reveal a single direction)
• variance: STC
• binned (or kernel) KL: MID “maximally informative directions” (equiv-alent to ML in LNP model with binned nonlinearity)
STC
Project out STA:
X = X − (Xksta)kTsta; Cprior =XTX
N;Cspike =
XTdiag(Y )X
Nspike
Choose directions with greatest change in variance:k- argmax‖v‖=1
vT(Cprior − Cspike)v
⇒ find eigenvectors of (Cprior − Cspike) with large (absolute) eigvals.
STC
Reconstruct nonlinearity (may assume separability)
Biases
STC (obviously) requires that the nonlinearity alter variance.
If so, subspace is unbiased if distribution
• radially (elliptically) symmetric
• AND independent
⇒ Gaussian.
May be possible to correct by transformation, subsampling or weighting(latter two at cost of variance).
More LNP methods
• Non-parametric non-linearities: “Maximally informative dimensions”(MID)⇔ “non-parametric” maximum likelihood.
– Intuitively, extends the variance difference idea to arbitrary differ-ences between marginal and spike-conditioned stimulus distribu-tions.
kMID = argmaxk
KL[P (k · x)‖P (k · x|spike)]
– Measuring KL requires binning or smoothing—turns out to be equiv-alent to fitting a non-parametric nonlinearity by binning or smooth-ing.
– Difficult to use for high-dimensional LNP models.
• Parametric non-linearities: the “generalised linear model” (GLM).
Generalised linear models
LN models with specified nonlinearities and exponential family noise.
In general (for monotonic g):
y ∼ ExpFamily[µ(x)]; g(µ) = βx
For our purposes easier to write
y ∼ ExpFamily[f (βx)]
(Continuous time) point process likelihood with GLM-like dependence ofλ on covariates is approached in limit of bins→ 0 by either Poisson orBernoulli GLM.
Mark Berman and T. Rolf Turner (1992) Approximating Point Process Likelihoods with GLIM
Journal of the Royal Statistical Society. Series C (Applied Statistics), 41(1):31-38.
Generalised linear models
Poisson distribution⇒ f = exp() is canonical (natural params = βx).
Canonical link functions give concave likelihoods⇒ unique maxima.
Generalises (for Poisson) to any f which is convex and log-concave:
log-likelihood = c− f (βx) + y log f (βx)
Includes:
• threshold-linear
• threshold-polynomial
• “soft-threshold” f (z) = α−1 log(1 + eαz).
Generalised linear models
ML parameters found by
• gradient ascent
• IRLS
Regularisation by L2 (quadratic) or L1 (absolute value – sparse) penal-ties (MAP with Gaussian/Laplacian priors) preserves concavity.
Linear-Nonlinear-Poisson (GLM)
stimulus filter pointnonlinearity
Poissonspiking
stimulus
k(t)
GLM with history-dependence
• rate is a product of stim- and spike-history dependent terms
• output no longer a Poisson process
• also known as “soft-threshold” Integrate-and-Fire model
exponentialnonlinearity
+post-spike filter
h
(t)
stimulus filter
(Truccolo et al 04)
k
Poissonspiking
conditional intensity(spike rate)
stimulus
filter output
traditional IF
filter output
“hard threshold”
“soft-threshold” IF
spik
e ra
te
GLM with history-dependence
exponentialnonlinearity
+post-spike filter
h
(t)
stimulus filter
k
Poissonspiking
• “soft-threshold” approximation to Integrate-and-Fire model
stimulus
GLM dynamic behaviors
time after spike time (ms)
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stimulus x(t)
post-spike waveform
0 0
stim-induced
spike-history
induced
regular spiking
GLM dynamic behaviors
stimulus x(t)
post-spike waveform
0 0
stim-filter output
spike-history
filter output
regular spiking
0 10 20
0
0 100 200 300 400 500
0
time after spike time (ms)
irregular spiking
GLM dynamic behaviors
stimulus x(t)
00
burstingpost-spike waveform
time after spike time (ms)
0 20 40
0
0 100 200 300 400 500-10
0
adaptation
Generalized Linear Model (GLM)
post-spike filter
exponentialnonlinearity
probabilisticspiking
stimulus
stimulus filter
+
multi-neuron GLM
exponentialnonlinearity
probabilisticspiking
stimulus
neuron 1
neuron 2
post-spike filter
stimulus filter
+
+
multi-neuron GLM
exponentialnonlinearity
probabilisticspiking
couplingfilters
stimulus
neuron 1
neuron 2
post-spike filter
stimulus filter
+
+
conditional intensity(spike rate)
...
time t
GLM equivalent diagram:
Multilinear models
Input nonlinearities (Hammerstein cascades) can be identified in a mul-tilinear (cartesian tensor) framework.
Input nonlinearities
The basic linear model (for sounds):
r(i)︸︷︷︸predicted rate
=∑jk
wtfjk︸︷︷︸
STRF weights
s(i− j, k)︸ ︷︷ ︸stimulus power
,
How to measure s? (pressure, intensity, dB, thresholded, . . . )
We can learn an optimal representation g(.):
r(i) =∑jk
wtfjkg(s(i− j, k)).
Define: basis functions {gl} such that g(s) =∑lw
llgl(s)
and stimulus array Mijkl = gl(s(i− j, k)). Now the model is
r(i) =∑j
wtfjkw
llMijkl or r = (wtf ⊗ wl) •M.
Multilinear models
Multilinear forms are straightforward to optimise by alternating least squares.
Cost function:E =
∥∥∥r− (wtf ⊗ wl) •M∥∥∥2
Minimise iteratively, defining matrices
B = wl •M and A = wtf •M
and updating
wtf = (BTB)−1BTr and wl = (ATA)−1ATr.
Each linear regression step can be regularised by evidence optimisa-tion (suboptimal), with uncertainty propagated approximately using vari-ational methods.
Some input non-linearities
25 40 55 70
0
l (dB−SPL)
wl
Parameter grouping
Separable STRF: (time) ⊗ (frequency). The input nonlinearity model isseparable in another sense: (time, frequency) ⊗ (sound level).
intensitytime
freq
uenc
y
time
freq
uenc
y
intensityw
eigh
t
Other separations:
• (time, sound level) ⊗ (frequency): r = (wtl ⊗ wf) • M,
• (frequency, sound level) ⊗ (time): r = (wfl ⊗ wt) • M,
• (time) ⊗ (frequency) ⊗ (sound level): r = (wl ⊗ wf ⊗ wl) • M.
(time, frequency) ⊗ (sound level)
t (ms)
f (kH
z)
180 120 60 0
32
16
8
4
2
0
25 40 55 70
0
l (dB−SPL)w
l
(time, sound level) ⊗ (frequency)
t (ms)
l (dB
−S
PL)
180 120 60 0
70
55
40
25
0
25 50 100
0
f (kHz)
wf
(frequency, sound level) ⊗ (time)
f (kHz)
l (dB
−S
PL)
2 4 8 16 32
70
55
40
25
0
180 120 60 0
0
t (ms)
wt
Contextual influences
stimulus
wτφ
wtfPSTH
time
frequency
rate(i) = c +∑jkl
wtjw
fkw
ll gl(soundi−j,k)(
c2 +∑mnp
wτmwφn w
λp gp(soundi−j−m,k+n)
)
Contextual influences
Introduce extra dimensions:
• τ : time difference between contextual and primary tone,
•φ: frequency difference between contextual and primary tone,
•λ: sound level of the contextual tone.
Model the effective sound level of a primary tone by
Level(i, j)→ Level(i, j) · (const + Context(i, j)) .
and the context by
Context(i, j) =∑m,n,p
wτmwφn w
λp hp(s(i−m, j + n))
This leads to a multilinear model
r = (wt ⊗ wf ⊗ wl ⊗ wτ ⊗ wφ ⊗ wλ) •M.
Inseparable contexts
We can also allow inseparable contexts (and principal fields), droppingthe level-nonlinearity to reduce parameters.
r(i) = c +∑jk
wtfjk soundi−j,k
(1 +
∑mn
wτφmn soundi−j−m,k+n)
stimulus
wτφ
wtfPSTH
time
frequency
Performance
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0.5
0.9
Ext
rapo
late
d no
rmal
ised
pred
ictiv
e po
wer
STRF
Separable Context
Inseparable Context
test set
training set