讲解MA 568、Statistical Analysis辅导、讲解Matlab、Matlab语言程序讲解

日期：2018-12-14 09:45

MA 568 – Statistical Analysis of Point Process Data

Problem Set #5 (Extra Credit)

Due December 11, 2018

In class, we discussed the construction of state-space estimation algorithms based on point

process observations. In this problem, we will apply the approximate Gaussian filter to the

problem of reconstructing a wind stimulus from an ensemble of simulated neurons from

the cricket cercal system.

Crickets have pairs of specialized sensory structures on their lower abdomen called cerci

that are used to detect air currents such as those generated by rapidly approaching

predators. Each of these cerci is covered with up to several thousand hairs of various sizes

and shapes that can be deflected in one direction. Each hair grows from a single sensory

neuron and each neuron is tuned to respond to different air current dynamics. This system

is unique in that the sensory neuron closest to the periphery already fires spikes. This

makes it well-suited for studying how neural representations can arise.

It has been shown that for small magnitude wind stimuli, a good model for the firing

activity of the neurons closest to the periphery is a Poisson process with rate function

λ αβ ( ) exp{ ( )} t vt = + , where v t( ) is the wind speed in the direction to which the neuron

responds, and α and β are model parameters for that cell.

Please download the file CricketData.mat (.csv) from the course website. Loading

this file into MATLAB will produce the following data structures:

trainingStim - A 1 second wind stimulus signal for the training period.

trainingSpikes - The spiking activity for each of 10 neurons during the

training period

testStim - The wind stimulus during the test period to be decoded

testSpikes - The spiking activity during the test period.

Each signal is recorded at a sampling rate of 1 KHz.

1. Construct a linear Gaussian state model for the wind stimulus, k kk 1 1 v Av ε = + , with

2

1 ~ (0, ) ε σ k N , using the data in trainingStim. Begin by plotting the stimulus value

at each time against its value at the previous time step. Recall that the linear regression

estimate for the equation y Ax = +ε is given by ( ) 2 / A xy x =∑ ∑ ii i . Use linear

regression to find the optimal estimate of the parameter A . Plot a histogram of the

residuals 1 k k v Av , and estimate the state transition variance, 2 σ? .

2. Estimate the α and β parameters for each of the ten neurons from the trainingStim

data using the MATLAB glmfit function. Are all of these parameters significant?

Construct a KS Plot for one of these neurons. How well does this model fit the observed

spiking data?

page 2: MA 568 - Problem Set 4

3. Use the approximate Gaussian filter we developed in class to decode testStim from

a) only the spiking activity of the first neuron, b) only the activity from neurons 1 and 2,

and c) from the entire ensemble of 10 neurons. Begin by setting 0|0 v? = 0 and 0|0

W = 0.03.

Recursively estimate the posterior mean and variance at each time step. For each case, plot

the posterior variance as a function of time. Also plot the test stimulus and the decoded

estimate on the same graph. Compute the mean-squared error of the estimate of the test

stimulus in each case. How does this compare to the endpoint of the posterior variance

estimate? How well are you able to decode with a single neuron? How does the decoding

accuracy improve as a function of ensemble size?

4. Use the exact posterior density equation to numerically compute the posterior density of

the state given the full ensemble spiking activity for two time steps. That is compute the

distributions of { }

10 ( )

1 1 1 | c

c v N = Δ and { }

10 () ()

21 2 1 | , c c

c vNN = Δ Δ . Plot these distributions on the

same graph as the Gaussian estimates from the approximate Gaussian filter. Is the

Gaussian approximation appropriate for this problem?