Latent models of stepping and ramping: an update on (the debate - - PowerPoint PPT Presentation

Latent models of stepping and ramping: 
 an update on (the debate over)
 single-trial dynamics in LIP

Jonathan Pillow

Princeton Neuroscience Institute, Princeton University

• Comp. Neurosci. Workshop: Computation, Cognition and the Brain

Rutgers University May 30, 2018

targets fixate

RF

saccade motion

“random dots” decision-making task

fixed duration or RT

targets fixate

RF

saccade motion

“random dots” decision-making task

fixed duration or RT

Q: what are the latent dynamics of spike trains in LIP during sensory evidence accumulation?

What do we mean by latent dynamics?

“direct” sensory 
 encoding model e.g., “LN” cascade:

stimulus filter nonlinearity

s

sensory stimulus spike response spike response

r

latent variable model

What do we mean by latent dynamics?

s r

sensory stimulus spike response spike response

x

latent variable noisy spike rate

latent variable model

What do we mean by latent dynamics?

s r

sensory stimulus spike response spike response

x

latent variable noisy spike rate

latent variable model

What do we mean by latent dynamics?

s r

sensory stimulus spike response spike response

x

latent variable noisy spike rate

latent dynamical model

xt+1

st+1 rt+1

xt

st rt

xt-1

st-1 rt-1

latent
 dynamics ... ... ... ... ...

What do we mean by latent dynamics?

sensory stimulus spike response spike response noisy trajectory

• bservations

dynamics

dx1 dx2 · · · dxT

probability of data:

[Gold & Shadlen 2007]

• neuron integrates motion “evidence”

time

Classic account: drift-diffusion model (DDM)

Accumulated evidence [logLR]

IN OUT time [sec] LIP response [spikes/sec] = Accumulated evidence [logLR]

... Roitman & Shadlen, 2002 Gold & Shadlen, 2002 Huk & Shadlen, 2005 Yang & Shadlen, 2007 Churchland & Shadlen, 2008 ...

• neuron integrates motion “evidence”

Classic account: drift-diffusion model (DDM)

normative modeling approach: what LIP neurons

• ught to do, based on a theory of “optimal” decision-making
• find most accurate statistical description of spike responses
• agnostic about function

descriptive modeling approach:

examples:

• log-probability (Shadlen & Newsome 1996)
• expected utility (Platt & Glimcher 1999)
• posterior probability (Beck et al 2008)
• change in the RL value function (Seo, Barraclough, & Lee 2009)

(“let the data speak for themselves”) example:

• variance of conditional expectation (varCE) - (Churchland et al 2011)

OUT motion

sp/s

classical analysis of LIP responses

[Shadlen & Newsome, 2001]

sp/s

IN motion

motion

• n

but what are the dynamics on single trials?

IN motion

motion

• n
• averaging obscures

single-trial dynamics spike train

sp/s

• ur goal: 


infer latent dynamics from spike trains discrete stepping noisy “ramping”

Formalizing the models

[Latimer et al 2015]

Chapter 1:

initial spike rate

ramping (“diffusion-to-bound”) model

bound height slope

200 400 600 time after motion onset (ms)

Poisson spikes: noise variance latent state: spiking:

ramping (“diffusion-to-bound”) model

200 400 600 time after motion onset (ms)

initial spike rate noise variance Poisson spikes: bound height latent state: spiking:

ramping (“diffusion-to-bound”) model

200 400 600 time after motion onset (ms)

initial spike rate noise variance Poisson spikes: bound height latent state: , spiking: ,

8 model parameters:

stepping (“discrete switching”) model

200 400 600 time after motion onset (ms)

step time distribution probability of “in” step spikes:

• semi-Markov model

step times latent spiking:

stepping (“discrete switching”) model

200 400 600 time after motion onset (ms)

probability of “in” step step time distribution

• semi-Markov model

spikes: step times latent spiking:

stepping (“discrete switching”) model

200 400 600 time after motion onset (ms)

probability of “in” step

14 parameters:

step time distribution

• semi-Markov model

spikes: step times latent spiking:

parameters

likelihood:

coherence

30 60

bound

spike rate (Hz)

requires summing over all possible latent paths

Fitting: difficult for both models

specified by model

• Bayesian inference: use MCMC to compute this integral

[Latimer et al 2015]

Which model is better?

naively, more parameters ⟹ more flexibility to fit the data Q: how to compare models with different numbers of parameters?

• but with too many parameters, we will overfit (fit noise in data)!

• 1. Divide data into a training set and a test set
• 2. Fit two models to the training data:
• 3. Compare on the test data

Model comparison

• I. Cross validation

model 1 fit model 2 fit

vs Q: how to compare models with different numbers of parameters?

Model comparison

Q: how to compare models with different numbers of parameters?

• II. Penalized log-likelihood (AIC, BIC, etc).

Akaike’s information criterion:

• 2 × log-likelihood

penalty based on # parameters

Compute AIC for each model, take model with lowest value.

Estimate of the information lost when a given model used to represent the process that generated the data [Akaike 1974]

Model comparison

Q: how to compare models with different numbers of parameters?

• II. Penalized log-likelihood (AIC, BIC, etc).

Bayesian information criterion:

• 2 × log-likelihood

penalty based on # parameters

# samples

Comes from an approximation to the marginal likelihood:

[Schwarz, 1978]

Model comparison

Q: how to compare models with different numbers of parameters?

• II. Penalized log-likelihood (AIC, BIC, etc).

Deviance information criterion (DIC) samples from posterior

Spiegelhalter et al 2002 (9K cites)

penalty based on “effective” # parameters

main result: model comparison

stepping ramping

• 31 / 40 neurons

better fit by stepping model

• 25 vs. 6 with

“strong evidence”

DIC difference

• 250
• 50

number of cells 2 4 6

50 250

• 25

25

[Latimer et al 2015]

DIC difference

• 100

100 200 number of cells 2 4 6

ramping stepping

Reaction Time task: Roitman & Shadlen 2002

• 10 / 13 better

explained by stepping

[Latimer et al 2015]

stepping ramping

Validating consistency of DIC

250

• 800
• 400

400

• 400

400 800 1200 1600

4 8 12 16 number of cells DIC difference DIC difference

ramping simulations stepping simulations

• 200
• 100

100

"WAIC

2 4 6 8

stepping ramping

Using Watanabe-Akaike Information Criterion (WAIC)

• 26 / 40 neurons

better fit by stepping model

[Zoltowski et al, in prep]

• newer “IC” for Bayesian analyses, more robust to overfitting

[Watanabe 2010]

A simpler approach?

[Zhao & Körding 2018]

Chapter 2:

…

training set test set

ˆ θ = arg max

θ

p(rtrain|θ)

Fit parameters

p(rtest|ˆ θ)

evaluate test likelihood lower & upper spike rate (2 params)

Z p(rtrain|xstep, θ)P(xstep)dxstep

p(rtrain | {z }

x

“across-trial” cross-validation

…

training set test set

ˆ θ = arg max

θ

p(rtrain|θ)

Fit parameters

p(rtest|ˆ θ)

evaluate test likelihood lower & upper spike rate (2 params)

Z p(rtrain|xstep, θ)P(xstep)dxstep

p(rtrain | {z }

x

• vs. p(rtest|ˆ

θ2)

(eg “ramp” model)

“across-trial” cross-validation

…

training set test set Fit parameters & latents for each trial parameters + step time for each trial ˆ θ, ˆ xstep = arg max

θ,x p(rtrain|xstep, θ)

100 trials =

⇒

102 parameters!

“within-trial” cross-validation

evaluate test likelihood

p(rtest|ˆ xstep, ˆ θ)

Problem: application to simulated data

simulated data: step model w/ rates 40Hz & 80Hz

firing rate (sp/s)

• 84% error rate

40 80

within-trial CV ramp better step better

test LLR

• Latimer 2018 [http://www.kennethlatimer.com/blog/simplicity-is-good-but-only-when-its-correct]

simulated data: step model w/ rates 40Hz & 80Hz

firing rate (sp/s)

• 0% error rate

40 80

across-trial CV ramp better step better

test LLR

• Latimer 2018 [http://www.kennethlatimer.com/blog/simplicity-is-good-but-only-when-its-correct]

Problem: application to simulated data

summary: Zhao & Kording 2018

• within-trial CV: fit models with >100 parameters 


(ramp or step for each trial)

• advantage: simple
• disadvantage: wrong 


(can’t identify true model in a lineup!)

• identifies “constant spike rate” model as best!

constant ramp step Can’t account for PSTH:

[Roitman & Shadlen 02]

summary: Zhao & Kording 2018

• within-trial CV: fit models with >100 parameters 


(ramp or step for each trial)

• advantage: simple
• disadvantage: wrong 


(can’t identify true model in a lineup!)

• identifies “constant spike rate” model as best!

constant ramp step

• take home: it is hard to get anything

useful out of models that are badly overfit

fortunately: across-trial CV (integrating over latent) can overcome these problems

• fit only ~10 parameters to entire dataset
• can identify true model in a lineup

David Zoltowski, unpublished

applied to real data (w/ original step & ramp models)

step vs. ramp

(Step 22/40)

step vs. constant

(Step 37/40) const better step better ramp better step better

across-trial CV and WAIC largely agree

ramp better ramp better step better step better cross-validation

David Zoltowski, unpublished

• Zylberberg & Shadlen 2016
• Latimer et al 2017
• Zoltowski et al, in preparation

Chapter 3: ramping and its discontents

ramping with vs. without lower bound

• Zylberberg & Shadlen 2016

true ramp (DDM) with lower bound fit w/ orig. ramp model fit w/ step model

simulation analysis

[Zylberberg & Shadlen 2016]

• stochastically remove spikes so baseline ≈ 0

step better ramp better

8 step : 8 ramp result: (vs. 13 : 3)

[Zoltowski et al, in prep] firing rate latent

• nonlinear relationship between

latent and FR

• GLM spike-history filter

extended ramping model (DDM)

[Zoltowski et al, in prep]

• baseline

Model comparison

[Zoltowski et al, in prep]

Step model + spike history ramp model + spike history + baseline + sqrt nonlinearity ramp model + spike history + baseline

• ≈1/2 better explained by each model: rare point of agreement!

Caveat:

[Zoltowski et al, in prep]

• single trial trajectories of fitted ramp model

may resemble steps! PSTH simulated single-trial rates

ramp model + spike history + baseline + sqrt nonlinearity

Future directions

• richer models (eg. “race” model w/ 2 latent trajectories, HMM)
• apply to multi-cell data (latent models of population dynamics)

Summary

• WAIC: improved model comparison tool over DIC
• cross-validation also good (but be sure to integrate

the latents; don’t fit them!)

• Zhao & Kording: suggest adding variable baseline
• new “ramp model with lower-bound”: 


≈1/2 LIP neurons better fit by each model

• Bayesian approach: powerful formalism for fitting and

comparing models; easily extended to richer model

Acknowledgments

Huk lab (UT Austin)

Alex Huk

Miriam Meister

(postdoc,


• U. Washington)

Jacob Yates

(postdoc,


• U. Rochester)

McKnight Scholar’s award
 Simons Global Brain Award NIH (CRCNS: R01-EY017366) NIH (R01-MH099611) NSF CAREER Award IIS-1150186

Funding Pillow lab

Kenneth Latimer

(postdoc, U. Chicago)

David Zoltowski

(Ph. D. Student)