In
the file rodcurrents.mat you will find data on the current produced by toad
rods in response to dim flashes of light.
These data are from experiments done by FM Rieke and colleagues at the
University of Washington, and I am grateful to Fred for providing these data in
raw form. To download the file you
should click here, and use the �save downloaded file as �� option
in your browser. Put the file someplace
where you can load it into MATLAB; you can do this using load rodcurrents.mat.
The data consist of 350 examples of a 7.95 second
recording, sampled in 10 msec bins.
A light flash of fixed intensity (but of course with a variable number
of photons!) is delivered in the 101st bin, i.e. one second after the start of
the recording. Current is measured
in picoAmps.
1.
Just to be sure you
have the data in the right form, plot the average current as a function of
time. Be sure to give a real time
axis, in seconds. You should find
that the mean current peaks roughly two seconds after the flash, and this peak
is ~0.65 pA in amplitude.
2.
Start by focusing
on the �background noise� that occurs before the light flash. Convince yourself that the mean current
in this one second window really is close to zero. How big is the standard deviation of the current? Do these random currents come from a
Gaussian distribution? In order to
answer this last question, you need to collect all of the samples (you have
350x100 of them) and use these samples to estimate the distribution. The MATLAB command hist is useful here, but try to get the distribution
into the right units, with the correct normalization (the integral under the
curve should be one; explain what you can do to check that this is true). Compare your results to a Gaussian
distribution with the same mean and variance.
3.
You can do a little
better in problem 2 if you can put error bars on the distribution. To do this, choose half of the 350
trials at random, and make an estimate of the distribution from this half of
the data. Do it again, and again
many times. Now you have many �experiments,�
each with 175 trials, that generate slightly different distributions. You can form an error bar by computing
the standard deviation across these different experiments. Make a plot comparing the observed
distribution, with error bars, to a Gaussian. It might be best to use a semilog plot, since probability
distributions have a broad dynamic range.
How close is the real distribution to being Gaussian? Explain why you might expect the
Gaussian to be a good model for this background noise.
4.
Suppose that you
look at the current on each trial at one moment in time, say the moment when
the mean current hits its peak. Can
you look at the probability distribution of the current and �see� the different
peaks corresponding to zero, one, two, � photons? Maybe looking at the raw data is too hard; suppose that you
smooth the data by averaging together all the currents within 100 msec of the
peak. Does this smoothed data show the different quanta more clearly?
5.
Looking at the
distribution of (smoothed) peak currents that you found in [4], how big is the
response to one photon? Use the
distribution as a guide to cut the trials into groups and count how many times
you see zero, one, two, � photons.
What is the mean number of photons that are counted in these
trials? Does the distribution of
counts (as well as you can measure it) agree with the expected Poisson
distribution?
6.
Looking in more
detail, does the piece of the distribution of peak currents that corresponds to
counting zero photons agree with your measurements on the distribution of
background noise? When you cut the
data into trials with zero, one, two � photons, you made a decision. Estimate the probability that (in the
context of this experiment) you will make a mistake and say that there is one
photon when in fact there were zero.