MA 568 – Statistical Analysis of Point Process Data

MA 568留學生做業代寫、代作Matlab課程設計做業、代寫Gaussian filter做業、Matlab實驗做業代作
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 () ()http://www.6daixie.com/contents/12/2387.html
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?

 

由於專業，因此值得信賴。若有須要，請加QQ：99515681 或郵箱：99515681@qq.com 

微信：codinghelp