Neural Encoding Models

Maneesh Sahanimaneesh@gatsby.ucl.ac.uk

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

0 0.5 1 1.5 2 2.5 3 3.5 4

0

0.1

0.2

0.3

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

0 0.5 1 1.5 2 2.50

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

0 50 100 150−0.5

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

0 50 100 150−0.5

0

0.5

1

1.5

2

2.5

−5 0 5 10 15 20 25 30

−0.2

0

0.2

0.4

0.6

0.8

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

0 50 100 1500

0.5

1

1.5

2

2.5

3

−5 0 5 10 15 20 25 300.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

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)

−30 −25 −20 −15 −10 −5

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)

0 50 100 0 100 200 300 400 500

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

0.1

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