Empirical models of spiking in neural populations

Jakob H. Macke, Lars Büsing,

John P. Cunningham, Byron M. Yu,

Krishna V. Shenoy and Maneesh Sahani

NIPS 2011December 15, 2011

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Characterizing neural functionI Single neuron recordings

I Simultaneous recordings from populations of neurons

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→ need for appropriate statistical models

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Characterizing neural functionI Single neuron recordings

I Simultaneous recordings from populations of neurons

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→ need for appropriate statistical models() 2 / 19

Models of population activity: direct couplings or latent variables?

model class I

model class II

generalized linear models (GLM)

latent dynamical system (DS)

externalinput

directcouplings

observedneurons

externalinput

dynamical system

observedneurons

e.g. [Chornoboy et al., Biol Cybern, 1988]

[Gat et al., 97, Network], [Yu et al., NIPS, 2006]

[Paninski et al., Neuro Comp, 2004]

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Models of population activity: direct couplings or latent variables?

model class I model class II

generalized linear models (GLM) latent dynamical system (DS)

externalinput

directcouplings

observedneurons

externalinput

dynamical system

observedneurons

e.g. [Chornoboy et al., Biol Cybern, 1988] [Gat et al., 97, Network], [Yu et al., NIPS, 2006][Paninski et al., Neuro Comp, 2004]

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Statistical models offer insights into neural computation

model class I

I GLMs improve decoding of visualstimuli from retina recordings

[Pillow et al., Nature, 2008]

model class II

I DS facilitate single trial analysis ofcortical multi-electrode recordings

[Chruchland et al, Nature Neurosci., 2010]

I Brain-computer interfaces[Wu et al., IEEE Tans. Neural Systems, 2000]

→What are good models for cortical multi-electrode recordings?

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Statistical models offer insights into neural computation

model class I

I GLMs improve decoding of visualstimuli from retina recordings

[Pillow et al., Nature, 2008]

model class II

I DS facilitate single trial analysis ofcortical multi-electrode recordings

[Chruchland et al, Nature Neurosci., 2010]

I Brain-computer interfaces[Wu et al., IEEE Tans. Neural Systems, 2000]

→What are good models for cortical multi-electrode recordings?

() 4 / 19

Statistical models offer insights into neural computation

model class I

I GLMs improve decoding of visualstimuli from retina recordings

[Pillow et al., Nature, 2008]

model class II

I DS facilitate single trial analysis ofcortical multi-electrode recordings

[Chruchland et al, Nature Neurosci., 2010]

I Brain-computer interfaces[Wu et al., IEEE Tans. Neural Systems, 2000]

→What are good models for cortical multi-electrode recordings?

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Data: Multi-electrode recordings from primate cortex

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ple

raste

r (s

ingle

trial)

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euro

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Target on Go cue

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30Data used in analyses

Popula

tion P

ST

H

R

ate

(hz)

I Data from Shenoy’s lab

I Array with 96 electrodes in motor / premotor areas

I 92 units = dimensions, 120 s of recordings , 10ms binning

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Model class I: Generalized linear models (GLMs)

I Instantaneous firing rate λi(t) of neuron i

λi(t) = exp([

det. innovations︷︸︸︷b(t) + D︸︷︷︸

couplings

·

spike history︷︸︸︷s(t) ]i)

I Poisson observations

yi(t) | s(t) = Poisson(λi(t))

I We used up to five different decayingexponentials as basis functions for the spikehistory:time constants 0.1 - 80ms

b(t)

I We used L1-regularization of the coupling matrix D to avoid over-fitting

e.g. [Chornoboy et al., Biol Cybern, 1988], [Paninski et al., Neuro Comp, 2004]

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Model class I: Generalized linear models (GLMs)

I Instantaneous firing rate λi(t) of neuron i

λi(t) = exp([

det. innovations︷︸︸︷b(t) + D︸︷︷︸

couplings

·

spike history︷︸︸︷s(t) ]i)

I Poisson observations

yi(t) | s(t) = Poisson(λi(t))

I We used up to five different decayingexponentials as basis functions for the spikehistory:time constants 0.1 - 80ms

b(t)

I We used L1-regularization of the coupling matrix D to avoid over-fitting

e.g. [Chornoboy et al., Biol Cybern, 1988], [Paninski et al., Neuro Comp, 2004]

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Model class I: Generalized linear models (GLMs)

I Instantaneous firing rate λi(t) of neuron i

λi(t) = exp([

det. innovations︷︸︸︷b(t) + D︸︷︷︸

couplings

·

spike history︷︸︸︷s(t) ]i)

I Poisson observations

yi(t) | s(t) = Poisson(λi(t))

I We used up to five different decayingexponentials as basis functions for the spikehistory:time constants 0.1 - 80ms b(t)

I We used L1-regularization of the coupling matrix D to avoid over-fitting

e.g. [Chornoboy et al., Biol Cybern, 1988], [Paninski et al., Neuro Comp, 2004]

() 6 / 19

Model class I: Generalized linear models (GLMs)

I Instantaneous firing rate λi(t) of neuron i

λi(t) = exp([

det. innovations︷︸︸︷b(t) + D︸︷︷︸

couplings

·

spike history︷︸︸︷s(t) ]i)

I Poisson observations

yi(t) | s(t) = Poisson(λi(t))

I We used up to five different decayingexponentials as basis functions for the spikehistory:time constants 0.1 - 80ms b(t)

I We used L1-regularization of the coupling matrix D to avoid over-fitting

e.g. [Chornoboy et al., Biol Cybern, 1988], [Paninski et al., Neuro Comp, 2004]

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Model class II: Latent dynamical system (DS) models

I Latent variable x(t) evolves linearly, withGaussian innovations

x(t + 1) = A x(t)︸ ︷︷ ︸dynamics

+

innovations︷ ︸︸ ︷b(t) + η(t)

I Gaussian linear dynamical system (GLDS):

y(t) | x(t) ∼ N (Cx(t) + d,R)

I Poisson linear dynamical system (PLDS):

yi(t) | x(t) ∼ Poisson(exp(Ci,:x(t) + di + Disi(t)))

I Parameters were learned by the EM algorithm

I We used a global Laplace approximation to the posterior for the PLDS

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Model class II: Latent dynamical system (DS) models

I Latent variable x(t) evolves linearly, withGaussian innovations

x(t + 1) = A x(t)︸ ︷︷ ︸dynamics

+

innovations︷ ︸︸ ︷b(t) + η(t)

I Gaussian linear dynamical system (GLDS):

y(t) | x(t) ∼ N (Cx(t) + d,R)

I Poisson linear dynamical system (PLDS):

yi(t) | x(t) ∼ Poisson(exp(Ci,:x(t) + di + Disi(t)))

I Parameters were learned by the EM algorithm

I We used a global Laplace approximation to the posterior for the PLDS

() 7 / 19

Model class II: Latent dynamical system (DS) models

I Latent variable x(t) evolves linearly, withGaussian innovations

x(t + 1) = A x(t)︸ ︷︷ ︸dynamics

+

innovations︷ ︸︸ ︷b(t) + η(t)

I Gaussian linear dynamical system (GLDS):

y(t) | x(t) ∼ N (Cx(t) + d,R)

I Poisson linear dynamical system (PLDS):

yi(t) | x(t) ∼ Poisson(exp(Ci,:x(t) + di + Disi(t)))

I Parameters were learned by the EM algorithm

I We used a global Laplace approximation to the posterior for the PLDS

() 7 / 19

Model class II: Latent dynamical system (DS) models

I Latent variable x(t) evolves linearly, withGaussian innovations

x(t + 1) = A x(t)︸ ︷︷ ︸dynamics

+

innovations︷ ︸︸ ︷b(t) + η(t)

I Gaussian linear dynamical system (GLDS):

y(t) | x(t) ∼ N (Cx(t) + d,R)

I Poisson linear dynamical system (PLDS):

yi(t) | x(t) ∼ Poisson(exp(Ci,:x(t) + di + Disi(t)))

I Parameters were learned by the EM algorithm

I We used a global Laplace approximation to the posterior for the PLDS

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Measuring performance: predicting neural firing from the population

I On test trials, for every neuron i :

I Compute cross-prediction E[yi,:|y\i,:] of neuron yi,: given all other neurons y\i,:

I Compute MSE between prediction and true trajectory

I We report improvement over a constant predictor (i.e. higher = better)

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Cross-prediction: Dynamical systems outperform GLMs

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Percentage of zero−entries in coupling matrix

Var

−M

SE

GLM 10msGLM 100msGLM2 100msGLM 150msPLDS dim=5

I GLM performance as a function of L1-regularization parameter:controls zero-entries in coupling matrix

I PLDS with 5-dimensional state space

I For our recordings PLDS outperforms GLMs with different history filters

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Cross-prediction: Poisson observations improve performance

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Dimensionality of latent space

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GLDSPLDSPLDS 100ms

I Poisson observation model improves performance over Gaussian models

I Single neuron history filters result in an slight increase in performance

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Matching statistics of the data I: temporal cross-correlations

latent dynamical systems (DS)

generalized linear models (GLM)

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GLDSPLDSPLDS 100msrecorded data

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GLM 10msGLM 100msGLM 150srecorded data

I Data: peaks at zero time lag and broad flanks

I DS: matches cross-correlations well

I GLM: difficulties to reproduce peak due to lack of instantaneous couplings

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Matching statistics of the data I: temporal cross-correlations

latent dynamical systems (DS) generalized linear models (GLM)

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Cor

rela

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GLDSPLDSPLDS 100msrecorded data

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Cor

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GLM 10msGLM 100msGLM 150srecorded data

I Data: peaks at zero time lag and broad flanks

I DS: matches cross-correlations well

I GLM: difficulties to reproduce peak due to lack of instantaneous couplings

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Matching statistics of the data II: population spike counts

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Number of spikes in 10ms bin

Fre

quen

cy

GLMGLM 2GLDSPLDSreal data

I Gaussian DS: under-estimates large population events

I Poisson DS: good match to the data

I GLM: no regularization; GLM2: optimal regularization

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How much variance of the data can we explain using cross-prediction?

% of explained variance

Comparison with PSTH predictor

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cent

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PLDS, d=1d=2d=5d=10d=20

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PLDS, d=1d=2d=5d=10d=20

I For small bins, more variance can be explained than would be possible if firingconditioned on latent state were Poisson

I PLDS models can explain up to twice as much variance as a model withindependent activity conditioned on PSTH

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How much variance of the data can we explain using cross-prediction?

% of explained variance

Comparison with PSTH predictor

0 50 100 150 200 250 300 350 4000

5

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Binsize (ms)

Per

cent

age

of v

aria

nce

expl

aine

d

d=1d=2d=5d=10d=20Poisson

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Binsize (ms)

Impr

ovem

ent o

ver

PS

TH

pre

dict

or

PLDS, d=1d=2d=5d=10d=20

I For small bins, more variance can be explained than would be possible if firingconditioned on latent state were Poisson

I PLDS models can explain up to twice as much variance as a model withindependent activity conditioned on PSTH

() 13 / 19

How much variance of the data can we explain using cross-prediction?

% of explained variance Comparison with PSTH predictor

0 50 100 150 200 250 300 350 4000

5

10

15

20

25

Binsize (ms)

Per

cent

age

of v

aria

nce

expl

aine

d

d=1d=2d=5d=10d=20Poisson

0 50 100 150 200 250 300 350 400

0.8

1

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1.8

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Binsize (ms)

Impr

ovem

ent o

ver

PS

TH

pre

dict

or

PLDS, d=1d=2d=5d=10d=20

I For small bins, more variance can be explained than would be possible if firingconditioned on latent state were Poisson

I PLDS models can explain up to twice as much variance as a model withindependent activity conditioned on PSTH

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Summary & conclusions

I Dynamical systems match the investigated cortical recordings better than GLMs

I Cross-prediction

I Various statistics, e.g. cross-correlation structure

I GLM modelling assumptions (direct couplings) might not be appropriate for thistype of data:

I Multi-electrode arrays sample a neural population very sparsely

I Most inputs presumably come from outside the sampled population

I Latent dynamical systems are arguably the more natural models formulti-electrode recording from cortex

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Summary & conclusions

I Dynamical systems match the investigated cortical recordings better than GLMs

I Cross-prediction

I Various statistics, e.g. cross-correlation structure

I GLM modelling assumptions (direct couplings) might not be appropriate for thistype of data:

I Multi-electrode arrays sample a neural population very sparsely

I Most inputs presumably come from outside the sampled population

I Latent dynamical systems are arguably the more natural models formulti-electrode recording from cortex

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Summary & conclusions

I Dynamical systems match the investigated cortical recordings better than GLMs

I Cross-prediction

I Various statistics, e.g. cross-correlation structure

I GLM modelling assumptions (direct couplings) might not be appropriate for thistype of data:

I Multi-electrode arrays sample a neural population very sparsely

I Most inputs presumably come from outside the sampled population

I Latent dynamical systems are arguably the more natural models formulti-electrode recording from cortex

() 14 / 19

Acknowledgements

I Jakob Macke (Gatsby Unit, UCL)

I Maneesh Sahani (Gatsby Unit, UCL)

I Krishna Shenoy & lab (Stanford)

I John Cunningham (Cambridge)

I Byron Yu (Carnegie Mellon)

Funding:

I Gatsby Charitable FoundationI EU-FP7 Marie Curie FellowshipI DARPA: REPAIR N66001-10-C-2010I EPSRC EP/H019472/1I NIH CRCNS R01-NS054283I NIH Pioneer 1DP1OD006409

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Additional material

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Cross-prediction: area under ROC

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0.6

0.62

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Percentage of zero−entries in coupling matrix

AU

C

GLM 10msGLM 100msGLM2 100msGLM 150msPLDS dim=5

5 10 15 200.64

0.65

0.66

0.67

0.68

Dimensionality of latent space

AU

C

GPFAGLDSPLDSPLDS 100ms

I Same qualitative results as measured by MSE

I Benefits of history filters for PLDS are more pronounced

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Correlations in data are well reflected by DS models

data bin at 10ms 5-dimensional PLDS

Neuron index

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