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\firstpageheader{Scientific Computing}{Project Assignment}{11/05/2014
  -- 11/06/2014}
%\runningheader{Homework 01}{Page \thepage\ of \numpages}{23. October 2014}
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\begin{document}
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\begin{center}
  \input{../disclaimer.tex}
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%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%

\begin{questions}
  \question The temporal evolution of the membrane voltage $V(t)$ of a
  passive neuron is described by the membrane equation
  \begin{equation}
    \label{passivemembrane}
    \tau \frac{dV}{dt} = -V + E
  \end{equation}
  where $\tau=10$\,ms is the membrane time constant and $E(t)$ is the
  reversal potential that also depends on time $t$.

  Such a differential equation can be numerically solved with the Euler method.
  For this the time is discretized by a time step $\Delta t=0.1$\,ms.
  The $i$-th time point is then at time $t_i = i \cdot \Delta t$.
  In matlab we get the time points $t_i$ simply by
  \begin{lstlisting}
dt = 0.1;
tmax = 100.0;
time = [0.0:dt:tmax]; % t_i
  \end{lstlisting}
  When the membrane potential at time $t_0 = 0$ is $V_0$, the so
  called ``initial condition'', then we can iteratively compute the 
  membrane potentials $V_i$ for successive time points $t_i$ according to
  \begin{equation}
    \label{euler}
    V_{i+1} = V_i + (-V_i + E_i) \frac{\Delta t}{\tau}
  \end{equation}

  \begin{parts}
    \part Write a function that computes the time course of the
    membrane potential of the passive membrane. The function gets as
    input arguments the initial condition $V_0$, the vector with the
    time course of $E(t)$, the value of the membrane time-constant
    $\tau$, and the time step $\Delta t$.

    \part In order to test your function set $V_0=1$\,mV and $E(t)=0$
    and compute $V(t)$ for $t_{max}=50$\,ms. Plot $V(t)$ and compare it to 
    the expected result of $V(t) = \exp(-t/\tau)$.

    Why is $V=0$ the resting potential of this neuron?

    \part Response of the passive membrane to a step input. 

    Set $V_0=0$. Construct a vector for the input $E(t)$ such that
    $E(t)=0$ for $t<20$\,ms and $t>70$\,ms and $E(t)=10$\,mV for
    $20$\,ms $<t<70$\,ms. Plot $E(t)$ and the resulting $V(t)$ for
    $t_{max}=120$\,ms.

    \part Response to sine waves.

    As an input we now use $E(t)=\sin(2\pi f t)$. Compute the time
    course of the membrane potential in response to this input
    ($t_{max}=1$\,s). Vary the frequency $f$ between 1 and 100\,Hz.  Be
    careful with the units within the sine function --- $ft$ must be
    unitless.

    What do you observe?

    \part Transfer function of the passive neuron.

    Measure the amplitude of the voltage responses evoked by the
    sinusoidal inputs as the maximum of the last 900\,ms of the
    responses.  Plot the amplitude of the response as a function of
    input frequency. This is the transfer function of the passive neuron.

    How does the transfer function depend on the membrane time
    constant?

    \part Leaky integrate-and-fire neuron.

    The passive neuron can be turned into a spiking neuron by
    introducing a fixed voltage threshold. Whenever the computed
    membrane potential of the passive neuron crosses the voltage
    threshold a spike is generated and the membrane voltage is set to
    the reset potential $V_R$ that we here set to zero. ``Generating a
    spike'' only means that we note down the time of the threshold
    crossing as a time where an action potential occurred. The
    waveform of the action potential is not modeled. Here we use a
    voltage threshold of one.

    Write a function that implements this leaky integrate-and-fire
    neuron by expanding the function for the passive neuron
    appropriate. The function returns a vector of spike times.

    Illustrate how this model works by appropriate plot(s) and
    input(s) $E(t)$, e.g. constant inputs lower and higher than the
    voltage threshold.

    \part Show the response of the leaky integrate-and-fire neuron to
    a sine wave $E(t)=A\sin(2\pi ft)$ with $A=2$\,mV and frequency
    $f=10$, 20, and 30\,Hz.

    \part Compute the firing rate as a function of the frequency of
    the stimulating sine wave ($A=2$\,mV and frequencies between 5 and
    30\,Hz). For a spike train with $n$ spikes at times $t_k$ ($k=1,
    2, \ldots n$) the firing rate is
    \begin{equation}
      \label{firingrate}
      r = \frac{n-1}{t_n - t_1}
    \end{equation}

    What do you observe? Does the firing rate encode the frequency of
    the stimulus?
  \end{parts}

\end{questions}

\end{document}