Neural encoding models & maximum likelihood Jonathan Pillow 1 - - PowerPoint PPT Presentation

Neural encoding models & maximum likelihood

Statistical modeling and analysis of neural data NEU 560, Spring 2018 Lecture 7 Jonathan Pillow

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true mean: [0 0.8] true cov: [1.0 -0.25

• 0.25 0.3]

sample mean: [-0.05 0.83] sample cov: [0.95 -0.23

• 0.23 0.29]

700 samples

sampling (measurement) Inference (“fitting”)

probability leftovers: sampling vs inference

Model Data

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Estimation

( 1

2

, , r r = r



neuron # spike count

parameter

(“stimulus”)

measured dataset


(“population response”)

An estimator is a function

model

• often we will write or just

ˆ θ(x)

p(x|θ) x = {r1, r2, . . . , rn}

f : x − → ˆ θ

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Properties of an estimator

bias:

• “unbiased” if bias=0

variance:

• “consistent” if bias and variance both go


to zero asymptotically

Q: what is the bias of the estimator 
 (i.e., estimate is 7 for all datasets x)

ˆ θ(x) = 7

“expected” value 
 (average over draws of x)

Q: what is the variance of that estimator?

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y x

stimuli spike trains

neural coding problem

Q: what is the probabilistic relationship between stimuli and spike trains?

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y x

stimuli spike trains

neural coding problem

Q: what is the probabilistic relationship between stimuli and spike trains?

“encoding model”

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y x

stimuli spike trains

today: single-neuron encoding

“encoding model”

P(yi|~ x, ✓)

Question: what criteria for picking a model?

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model desiderata

fittability / tractability richness / flexibility

(can be fit to data) (capture realistic neural properties)

linear, Gaussian multi- compartment Hodgkin-Huxley sweet spot

GLM

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Example 1: linear Poisson neuron

spike count spike rate encoding model: stimulus parameter

−3 −2 − 1 2 3 4 5 6 7 8 9 10

= mean = variance

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20 40 20 40 60

(contrast) (spike count)

20 40 60

conditional distribution

p(y|x)

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20 40 20 40 60

(contrast) (spike count)

20 40 60

conditional distribution

p(y|x)

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20 40 20 40 60

(contrast) (spike count)

20 40 60

conditional distribution

p(y|x)

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Maximum Likelihood Estimation:

• given observed data , find that maximizes

all spike counts all stimuli parameters

}

single-trial probability

Q: what assumption are we making about the responses? A: conditional independence across trials!

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Q: when do we call a likelihood? Maximum Likelihood Estimation:

• given observed data , find that maximizes

all spike counts all stimuli parameters

}

single-trial probability

Q: what assumption are we making about the responses? A: conditional independence across trials! A: when considering it as a function of !

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20 40 20 40 60

(contrast) (spike count)

Maximum Likelihood Estimation:

• given observed data , find that maximizes

p(y|x)

• could in theory do this by turning a knob

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20 40 20 40 60

(contrast) (spike count)

Maximum Likelihood Estimation:

• given observed data , find that maximizes

p(y|x)

• could in theory do this by turning a knob

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20 40 20 40 60

(contrast) (spike count)

Maximum Likelihood Estimation:

• given observed data , find that maximizes

p(y|x)

• could in theory do this by turning a knob

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likelihood

Likelihood function: as a function of

Because data are independent:

1 2

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1 2

log-likelihood log

Likelihood function: as a function of

Because data are independent:

1 2

likelihood

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1 2

log-likelihood

Do it: solve for

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• Closed-form solution when model in “exponential family”

1 2

log-likelihood

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Properties of the MLE (maximum likelihood estimator)

• consistent

(converges to true in limit of infinite data)

• efficient 


(converges as quickly as possible, 
 i.e., achieves minimum possible asymptotic error)

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Example 2: linear Gaussian neuron

spike count spike rate encoding model: stimulus parameter

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20 40 20 40 60

(contrast) (spike count)

20 40 60

All slices have same width encoding distribution

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Log-Likelihood Do it: differentiate, set to zero, and solve.

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Log-Likelihood Maximum-Likelihood Estimator:

(“Least squares regression” solution) (Recall that for Poisson, )

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Example 3: unknown neuron

• 25

25 25 50 75 100 (contrast) (spike count)

Be the computational neuroscientist: what model would you use?

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Example 3: unknown neuron

More general setup:

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25 25 50 75 100 (contrast) (spike count)

for some nonlinear function f

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Answer: stimulus likelihood function - useful for ML stimulus decoding!

Quick Quiz:

• 1. as a function of y?
• 2. as a function of ?

The distribution P(y|x, ) can be considered as a function of y, x, or .

spikes stimulus parameters

Answer: encoding distribution - probability distribution over spike counts Answer: likelihood function - the probability of the data given model params

• 3. as a function of x?

What is P(y|x, ) :

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20 40 20 40 60

(contrast) (spike count)

stimulus decoding likelihood

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