131 lines
4.9 KiB
TeX
131 lines
4.9 KiB
TeX
\documentclass[a4paper,12pt,pdftex]{exam}
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\newcommand{\ptitle}{Integrate-and-fire neuron}
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\input{../header.tex}
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\firstpagefooter{Supervisor: Jan Benda}{}{email: jan.benda@uni-tuebingen.de}
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\begin{document}
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\input{../instructions.tex}
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The leaky integrate-and-fire model is a simple but powerful model of
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spiking neurons. It adds to the dynamics of a passive membrane a
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voltage threshold to simulate the generation of action potentials. In
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this project you learn how to simulate such a model and explore some
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of its stimulus encoding properties.
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The temporal evolution of the membrane voltage $V(t)$ of a passive
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neuron is described by the membrane equation
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\begin{equation}
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\label{passivemembrane}
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\tau \frac{dV}{dt} = -V + E
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\end{equation}
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where $\tau=10$\,ms is the membrane time constant and $E(t)$ is the
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reversal potential that also depends on time $t$.
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Such a differential equation can be numerically solved with the Euler
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method. For this the time is discretized by a time step $\Delta
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t=0.1$\,ms. The $i$-th time point is then at time $t_i = i \cdot
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\Delta t$. In matlab we get the time points $t_i$ simply by
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\begin{lstlisting}
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dt = 0.1;
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tmax = 100.0;
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time = [0.0:dt:tmax]; % t_i
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\end{lstlisting}
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When the membrane potential at time $t_0 = 0$ is $V_0$, the so called
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``initial condition'', then we can iteratively compute the membrane
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potentials $V_i$ for successive time points $t_i$ according to
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\begin{equation}
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\label{euler}
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V_{i+1} = V_i + (-V_i + E_i) \frac{\Delta t}{\tau}
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\end{equation}
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\begin{questions}
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\question Passive membrane
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\begin{parts}
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\part Write a function that computes the time course of the
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membrane potential of the passive membrane. The function gets as
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input arguments the initial condition $V_0$, the vector with the
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time course of $E(t)$, the value of the membrane time-constant
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$\tau$, and the time step $\Delta t$.
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\part In order to test your function set $V_0=1$\,mV and $E(t)=0$
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and compute $V(t)$ for $t_{max}=50$\,ms. Plot $V(t)$ and compare it to
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the expected result of $V(t) = \exp(-t/\tau)$.
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Vary the time step $\Delta t$ by factors of 10 and discuss the
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accuracy of the numerical solutions. What is a good time step?
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Why is $V=0$ the resting potential of this neuron?
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\part Response of the passive membrane to a step input.
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Set $V_0=0$. Construct a vector for the input $E(t)$ such that
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$E(t)=0$ for $t\le 20$\,ms or $t\ge 70$\,ms, and $E(t)=10$\,mV for
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$20$\,ms $<t<70$\,ms. Plot $E(t)$ and the resulting $V(t)$ for
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$t_{max}=120$\,ms.
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\part Response to sine waves.
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As an input we now use $E(t)=\sin(2\pi f t)$. Compute the time
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course of the membrane potential in response to this input
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($t_{max}=1$\,s). Vary the frequency $f$ between 1 and 100\,Hz.
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Be careful with the units within the sine function --- $f \cdot t$
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must be unitless.
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What do you observe?
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\part Filter function of the passive neuron.
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Measure the amplitude of the voltage responses evoked by the
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sinusoidal inputs as the maximum of the last 900\,ms of the
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responses. Plot the amplitude of the response as a function of
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input frequency. This is the filter function of the passive
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neuron.
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How does the filter function depend on the membrane time constant?
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\end{parts}
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\continue
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\question Leaky integrate-and-fire neuron
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The passive neuron can be turned into a spiking neuron by
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introducing a fixed voltage threshold. Whenever the computed
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membrane potential of the passive neuron crosses the voltage
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threshold a spike is generated and the membrane voltage is set to
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the reset potential $V_R$ that we here set to zero. ``Generating a
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spike'' only means that we note down the time of the threshold
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crossing as a time where an action potential occurred. The waveform
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of the action potential is not modeled. Here we use a voltage
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threshold of 1\,mV.
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\begin{parts}
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\part
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Write a function that implements this leaky integrate-and-fire
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neuron by expanding the function for the passive neuron
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appropriately. The function returns a vector of spike times.
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Illustrate how this model works by appropriate plot(s) and
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input(s) $E(t)$, e.g. constant inputs lower and higher than the
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voltage threshold.
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\part Show the response of the leaky integrate-and-fire neuron to
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a sine wave $E(t)=A\sin(2\pi ft)$ with $A=2$\,mV and frequency
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$f=10$, 20, and 30\,Hz.
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\part Compute the firing rate as a function of the frequency of
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the stimulating sine wave ($A=2$\,mV and frequencies between 5 and
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30\,Hz). For a spike train with $n$ spikes at times $t_k$ ($k=1,
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2, \ldots n$) the firing rate is
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\begin{equation}
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\label{firingrate}
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r = \frac{n-1}{t_n - t_1}
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\end{equation}
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What do you observe? Does the firing rate encode the frequency of
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the stimulus? Look at the spike trains in response to the sine
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waves to figure out what is going on.
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\end{parts}
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\end{questions}
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\end{document}
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