finished fano tim eproject

projects/project_fano_time/fano_time.tex

@@ -51,83 +51,88 @@
 %%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
 
 \begin{questions}
-  \question You are recording the activity of a neuron in response to
+  \question An important property of sensory systems is their ability
-  two different stimuli $I_1$ and $I_2$ (think of them, for example,
+  to discriminate similar stimuli. For example, to discriminate two
-  of two light intensities with different intensities $I_1$ and $I_2$
+  colors, light intensities, pitch of two tones, sound intensity, etc.
-  and the activity of a ganglion cell in the retina). Within an
+  Here we look at the level of a single neuron. What does it mean that
-  observation time of duration $W$ the neuron responds stochastically
+  two similar stimuli can be discriminated given the spike train
-  with $n$ spikes.
+  responses that have been evoked by the two stimuli?
+
+  You are recording the activity of a neuron in response to two
+  different stimuli $I_1$ and $I_2$ (think of them, for example, of
+  two light intensities with different intensities $I_1$ and $I_2$ and
+  the activity of a ganglion cell in the retina). The neuron responds
+  to a stimulus with a number of spikes. You (an upstream neuron) can
+  count the number of spikes of this response within an observation
+  time of duration $T$. For perfect discrimination, the number of
+  spikes evoked by the stronger stimulus within $T$ is larger than for
+  the smaller stimulus. The situation is more complicated, because the
+  number of spikes evoked by one stimulus is not fixed but varies.
 
   How well can an upstream neuron discriminate the two
   stimuli based on the spike counts $n$? How does this depend on the
-  duration $W$ of the observation time? How is this related to the fano factor
+  duration $T$ of the observation time?
-  (the ratio between the variance and the mean of the spike counts)?
 
-  The neuron is implemented in the file \texttt{lifadaptspikes.m}.
+  The neuron is implemented in the file \texttt{lifspikes.m}.
-  Call it with the following parameters:
+  Call it like this:
     \begin{lstlisting}
 trials = 10;
 tmax = 50.0;
-input = 65.0; 
+input = 15.0; 
-Dnoise = 0.1;
+spikes = lifspikes(trials, input, tmax);
-adapttau = 0.2;
-adaptincr = 0.5;
-
-spikes = lifadaptspikes( trials, input, tmax, Dnoise, adapttau, adaptincr );
     \end{lstlisting}
     The returned \texttt{spikes} is a cell array with \texttt{trials}
     elements, each being a vector of spike times (in seconds) computed
-    for a duration of \texttt{tmax} seconds.
+    for a duration of \texttt{tmax} seconds. The intensity of the stimulus
+    is given by \texttt{input}.
 
-    Think of calling the \texttt{lifadaptspikes()} function as a
+    Think of calling the \texttt{lifspikes()} function as a
     simple way of doing an electrophysiological experiment. You are
-    presenting a stimulus with a constant intensity $I$ that you set. The
+    presenting a stimulus with an intensity $I$ that you set. The
     neuron responds to this stimulus, and you record this
     response. After detecting the time points of the spikes in your
-    recordings you get what the \texttt{lifadaptspikes()} function
+    recordings you get what the \texttt{lifspikes()} function
-    returns. The advantage over real data is, that you have the
+    returns.
-    possibility to simply modify the properties of the neuron via the
-    \texttt{Dnoise}, \texttt{adapttau}, and
-    \texttt{adaptincr} parameter.
 
     For the two inputs $I_1$ and $I_2$ use
     \begin{lstlisting}
-input = 65.0; % I_1
+input = 14.0; % I_1
-input = 75.0; % I_2
+input = 15.0; % I_2
     \end{lstlisting}
 
   \begin{parts}
     \part 
-    Show two raster plots for the responses to the two different stimuli.
+    Show two raster plots for the responses to the two different
+    stimuli.  Find an appropriate time window and an appropriate
+    number of trials for the spike raster.
+
+    Just by looking at the raster plots, can you discriminate the two
+    stimuli? That is, do you see differences between the two
+    responses?
+
+    \part Generate properly normalized histograms of the spike counts
+    within $T$ (use $T=100$\,ms) of the responses to the two different
+    stimuli. Do the two histograms overlap? What does this mean for
+    the discriminability of the two stimuli?
 
-    \part Generate histograms of the spike counts within $W$ of the
+    How do the histograms depend on the observation time $T$ (use
-    responses to the two different stimuli. How do they depend on the
+    values of 10\,ms, 100\,ms, 300\,ms and 1\,s)?
-    observation time $W$ (use values between 1\,ms and 1\,s)?
 
-    \part Think about a measure based on the spike count histograms
+    \part Think about a measure based on the spike-count histograms
     that quantifies how well the two stimuli can be distinguished
     based on the spike counts. Plot the dependence of this measure as
-    a function of the observation time $W$.
+    a function of the observation time $T$.
 
-    For which observation times can the two stimuli perfectly discriminated?
+    For which observation times can the two stimuli perfectly
+    discriminated?
 
     \underline{Hint:} A possible readout is to set a threshold
     $n_{thresh}$ for the observed spike count.  Any response smaller
     than the threshold assumes that the stimulus was $I_1$, any
     response larger than the threshold assumes that the stimulus was
-    $I_2$. For a given $W$ find the threshold $n_{thresh}$ that
+    $I_2$. For a given $T$ find the threshold $n_{thresh}$ that
-    results in the best discrimination performance.
+    results in the best discrimination performance. How can you
-
+    quantify ``best discrimination'' performance?
-    \part Also plot the Fano factor as a function of $W$. How is it
-    related to the discriminability?
-
-    \uplevel{If you still have time you can continue with the
-      following question:}
-
-    \part You may change the two stimuli $I_1$ and $I_2$ and the
-    intrinsic noise of the neuron via \texttt{Dnoise} (change it in
-    factors of ten, higher values will make the responses more
-    variable) and repeat your analysis.
 
  \end{parts}
 

projects/project_fano_time/solution/fanotime.m

@@ -1,13 +1,12 @@
 %% general settings for the model neuron:
 trials = 10;
 tmax = 50.0;
-D = 0.01;
 
 %% generate and plot spiketrains for two inputs:
 I1 = 14.0;
 I2 = 15.0;
-spikes1 = lifspikes(trials, I1, tmax, D);
+spikes1 = lifspikes(trials, I1, tmax);
-spikes2 = lifspikes(trials, I2, tmax, D);
+spikes2 = lifspikes(trials, I2, tmax);
 subplot(1, 2, 1);
 tmin = 10.0;
 spikeraster(spikes1, tmin, tmin+2.0);
@@ -18,7 +17,7 @@ title(sprintf('I_2=%g', I2))
 %savefigpdf(gcf(), 'spikeraster.pdf')
 
 %% spike count histograms:
-Ts = [0.01 0.1 0.5 1.0];
+Ts = [0.01 0.1 0.3 1.0];
 cmax = 100;
 figure()
 for k = 1:length(Ts)
@@ -36,7 +35,7 @@ end
 
 %% discrimination measure:
 T = 0.1;
-cmax = 20;
+cmax = 15;
 [d, thresholds, true1s, false1s, true2s, false2s, pratio] = discriminability(spikes1, spikes2, tmax, T, cmax);
 figure()
 subplot(1, 3, 1);
@@ -45,6 +44,7 @@ hold on;
 plot(thresholds, true2s, 'b');
 plot(thresholds, false1s, 'r');
 plot(thresholds, false2s, 'r');
+xlim([0 cmax])
 hold off;
 % Ratio:
 subplot(1, 3, 2);
@@ -57,7 +57,7 @@ plot(false2s, true1s);
 %% discriminability:
 Ts = 0.01:0.01:1.0;
 cmax = 100;
-ds = zeros(length(Ts), 1)
+ds = zeros(length(Ts), 1);
 for k = 1:length(Ts)
     T = Ts(k);
     [c1, b1] = counthist(spikes1, 0.0, tmax, T, cmax);

projects/project_fano_time/solution/lifspikes.m

@@ -1,9 +1,8 @@
-function spikes = lifspikes(trials, input, tmaxdt, D)
+function spikes = lifspikes(trials, input, tmax)
 % Generate spike times of a leaky integrate-and-fire neuron.
 % trials: the number of trials to be generated
-% input: the stimulus either as a single value or as a vector
+% input: the stimulus intensity
-% tmaxdt: in case of a single value stimulus the duration of a trial
+% tmax: the duration of a trial
-%         in case of a vector as a stimulus the time step
 % D: the strength of additive white noise
     
     tau = 0.01;
@@ -12,21 +11,18 @@ function spikes = lifspikes(trials, input, tmaxdt, D)
     end
     vreset = 0.0;
     vthresh = 10.0;
+	D = 0.01;
     dt = 5e-5;
     
-    if max(size(input)) == 1
+    n = ceil(tmax/dt);
-        input = input * ones(ceil(tmaxdt/dt), 1);
-    else
-        dt = tmaxdt;
-    end
     spikes = cell(trials, 1);
     for k=1:trials
         times = [];
         j = 1;
         v = vreset + (vthresh-vreset)*rand(1);
-        noise = sqrt(2.0*D)*randn(length(input), 1)/sqrt(dt);
+        noise = sqrt(2.0*D)*randn(n, 1)/sqrt(dt);
         for i=1:length(noise)
-            v = v + (- v + noise(i) + input(i))*dt/tau;
+            v = v + (- v + noise(i) + input)*dt/tau;
             if v >= vthresh
                 v = vreset;
                 spiketime = i*dt;