Microsoft Word – Problem Set #2.docx
MA 568 – Statistical Analysis of Point Process Data
Problem Set #2
Due October 15, 2018
In Problem Set #1, we modeled baseline neural spiking as a simple Poisson process,
based on the assumption that the data was stationary and independent of past history.
More typically, we want to characterize how a point process varies as a function of other
time-varying signals. Inhomogeneous Poisson process models provide one approach to
relating point process data to these other observed signals.
In this problem we construct, fit, and evaluate a model of neural spiking activity in primate
primary motor cortex. Neurons in this region of the brain have been shown to relate to
limb movement kinematics. Recently there has been a lot of research into characterizing
the coding properties of these neurons for use in driving brain computer interfaces, for
paralyzed and movement-disabled individuals.
1. Please download the M1_spikes.mat file from the course website. The data
consists of simulated spiking activity as a primate performs an 8 second hand movement
in two dimensions. Loading this dataset in MATLAB creates the following variables:
T – Time index in ms.
X,Y – (x,y) coordinates of hand during a two-dimensional
movement task
V – Speed of hand movement
phi – Direction of hand movement in radians (0 =
movement to the right)
spiketimes – Time index of spikes (in ms.)
spikes – Time series of spikes binned at 1 ms. resolution
Plot the spiking activity as a time series. Plot the hand position trajectory (X vs. Y) with
the hand position at the spike times overlaid. Plot the velocity and direction of movement
(V vs. phi) at the spike times as well.
2. An occupancy-normalized histogram is a tool to visualize the firing rate as a function of
another covariate. Construct an occupancy-normalized histogram of the spiking activity
as a function of movement direction by first dividing the domain of possible directions into
discrete bins. For each bin, divide the total number of spikes observed during movement
in that range of directions by the total amount of time spent moving in those directions.
From your histogram, estimate the direction of movement in which this neuron is most
likely to fire.
The spiking properties of these neurons have previously been described by the following
cosine tuning model:
preferred
max
( )
( ( ), ( )) cos( ( ) )= + −
v t
v t t t
v
λ φ α β φ φ .
Under this model, for a particular hand velocity, the firing rate is maximal when the hand
is moving in the preferred direction, preferredφ , and drops off according to a cosine function
in other directions.
page 2: MA 568 – Problem Set 2
3. Assume that 30=α , 30=β , and max 16.1 cm/sec=v are known. Compute the
likelihood of the data as a function of preferred direction, preferred( )L φ , for a range of values
for preferredφ . Find the maximum likelihood estimate preferred,MLφ̂ . Compute the observed
Fisher information and construct a 95% confidence interval for this parameter.
4. Compute ML
ˆ ( )tλ , the time series of the estimated firing rate using the maximum
likelihood estimate for the preferred direction. Plot ML
ˆ ( )tλ as a function of time along with
the spike times.
Goodness-of-Fit:
5. Calculate a set of rescaled waiting times according to:
1
ML
ˆ ( )
−
= ∫
i
i
S
i S
Z t dtλ ,
where 0 0=S and iS is the i
th spike time. Plot histograms of the original interspike intervals
(ISIs), and the rescaled intervals. Construct a KS plot with 95% confidence bounds for
this data. Does the fit model pass the KS test?
6. Compute and plot the autocorrelation function of the rescaled intervals. What does this
suggest about the independence assumption for the inhomogeneous Poisson model?
7. Bin the rescaled event times into time bins of size 1. Compute the Fano Factor for this
rescaled and binned data, and 95% confidence bounds for this statistic under the
assumption that the data comes from an inhomogeneous Poisson (the sample Fano factor
is distributed according to a Gamma(n/2,2/n)). What, if anything, can you conclude about
the data from this result?
8. What conclusions can you draw about the firing properties of this neuron from the data?