Ch 3. Likelihood Based Approach to Modeling the Neural Code Bayesian Brain: Probabilistic Approaches...

Ch 3. Likelihood Based Ch 3. Likelihood Based Approach to Modeling the Approach to Modeling the Neural CodeNeural Code

Bayesian Brain: Probabilistic Approaches to Neural Codingeds. K Doya, S Ishii, A Pouget, and R RaoDec. 18th, 2008Summarized by Seok Ho-Sik

© 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 1

Task, Conclusion, ContributionTask, Conclusion, Contribution Task: design a model depicting the probabilistic relationship

between stimuli and neural response. Conclusion

ML method is useful. ML

Problem: in a high-dimensional space, containing tens to hundreds of parameters(describing a neuron’s receptive field and spike-generation properties)

Model

Contributions Introducing neural coding models, their validating

methods(likelihood-based cross-validation, time-rescaling, model-based decoding).

For us, introducing what should be considered to model neural responses (nonlinearity, probabilistic relation etc.).


( )

LNP GLM

Generalized Integrate-and-Fire Model

3.1 The Neural Coding Problem(1/2)3.1 The Neural Coding Problem(1/2)


3.1 The Neural Coding Problem(2/2)3.1 The Neural Coding Problem(2/2) Probabilistic relations: relationship between stimuli and neural

response is probabilistic.

Difficulties for obtaining full response distributions The high dimensionality of stimulus space and the finite duration of

neurophysiology experiments. A classical approach

Assumption: neurons are sensitive to a restricted set of stimulus features.

Statistical approach: assumes a probabilistic model of neural response and attempts to fit the model parameter Goal: to find a simplified and computationally tractable description of

p(y|x).


3.2 Model Fitting with Maximum Likelihood

Data:

Given a particular model, parameterized by the vector ,

one can apply a maximum likelihood method to obtain an

asymptotically optimal estimate of

Difficulties For many models of neural response (e.g. detailed biophysical

models) it is very difficult to compute likelihood.

In most cases, lives in a high-dimensional space.

© 2008, SNU Biointelligence Lab, http://bi.snu.ac.kr/

3.2.1 The LNP Model (1/5)

The LNP (linear-nonlinear Poisson) model. The cascading of a linear filter(k), a point nonlinearity (f), and Posisson

spike generator. k: represents the neuron’s space-time receptive field, which describes how the

stimulus is converted to intracellular voltage. f: the conversion of voltage to an instantaneous spike rate. Instantaneous rate is converted to a spike train via an Poisson process.

Model parameters: : the parameters governing f.



Because the bins of the responses are conditionally independent of one another given the stimulus,

The second term on the right hand of the () converges to a vector proportional to k if the stimulus distribution p(x) is spherically symmetric.

The first term is proportional to the STA (spike-triggered average) if f`/f is constant (when f is exponential).

yi: the spike count in the ith time bin, xi : the stimulus vector associated with this bin.

Rate parameter,

Dot product,Δ: the width of the time bin

…()


Differentiating with respect to K


A comparison between the STA and the ML estimates of the linear filter k on spike trains simulated using three different nonlinearities.

Result The ML estimate outperforms the STA except when f is exponential. If f differs significantly from exponential, the traditional STA is suboptimal.



A comparing ML to an estimator derived from STC (spike-triggered covariance), which uses the principal eigenvector of the STC matrix to estimate k.

Result: the ML estimate outperforms the STC and STA except when f is exponential.


Limits The LNP model is not biophysically realistic (especially the assumption

of Poisson spiking). Computationally intensive. Can not be guaranteed to converge to global maximum.

Usefulness It provides a compact and reasonably accurate description of average

response in many early sensory areas. LNP model can be generalized to include multiple linear filters and a

multidimensional nonlinearity.

LNP models could use information-theoretic estimators for finding “maximally informative dimensions” or features of stimulus space.

Because its sensitivity to higher-order statistics of the spike-triggered ensemble, it is more powerful and more general than STA or STC.

3.2.2 Generalized Linear Model (1/2)

Incorporating feedback from the spiking process, allowing the model to account for history-dependent properties of neural spike trains.

If we let denote the instantaneous spike rate as following, then

3.2.2 Generalized Linear Model (2/2)

How to discover the global maximum? To constrain the model so that one can guarantee that the likelihood function is free

from local maxima. If one can show that the negative log-likelihood is convex, then the only maxima is

the global maxima. The problem of computing the ML estimate is reduced to a convex optimization

problem, for which there are tractable algorithms. A number of suitable functions seems like reasonable choices for describing the

conversion of intracellular voltage to instantaneous spike rate (e. g.).

The GLM framework is quite general, and can easily be expanded to include additional linear filters that capture dependence on spiking activity in nearby neurons, behavior of the organism, or additional external covariates of spiking activity.

3.2.3 Generalized Integrate-and-Fire Model (1/3)

Recent work: the leaky integrate-and-fire (IF) model, a canonical but simplified description of intracellular spiking dynamics, can reproduce the spiking statistics of real neurons and can mimic important dynamical behaviors of more complicated models.

The injected current: a linear function of the stimulus, the spike-train history, plus a Gaussian noise current that introduces a probability distribution over voltage trajectories.


The model of dynamics

The dependency structure: the probability of an entire interspikeinterval is depending on a relevant portionof the stimulus and spike-train history.


How to compute the likelihood fora single ISI under the generalized GIFmodel using Monte Carlo sampling Given a setting of the model parameters,

one can sample voltage trajectories fromthe model, drawing independent noisesamples for each trajectory, and following each trajectory until it hits threshold.

The probability of a spike occurring at the ith bin is simply the fraction of voltage paths crossing threshold at this bin.

Likelihood function

Sample pathsSample paths

Voltage path obtained in the absence of noise

3.3 Model Validation (1/2)

Validating the quality of the model fit. Likelihood-Based Cross-validation

Time Rescaling To use the model to convert spike times into a series of i.i.d random

variables. Given the cumulative density function for a random variable, a general

result from probability theory holds that it provides a remapping of that variable to the one randomly distributed unit interval[0,1].

Any correlation (or some other form of dependence) between successive pairs of remapped spike times, indicates a failure of the model.

3.3 Model Validation (2/2)

Model-Based decoding To perform stimulus decoding using the model-based likelihood function. One can obtain the most likely stimulus to have generated the response y

by maximizing the posterior for x, which gives the maximum a posteriori estimate of the stimulus

With mean-squared error, this estimator is given by Decoding allows to measure how well a particular model preserves the

stimulus-related information in the neural response. Even though a model fails to reproduce certain statistical features of the

response, it provides a valuable tool for assessing what information the spike train carries about the stimulus and gives a perhaps more valuable description of the neural code.

Ch 3. Likelihood Based Approach to Modeling the Neural Code Bayesian Brain: Probabilistic Approaches...

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