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[projects] fixed noiseficurves

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Jan Benda 2021-01-31 21:29:46 +01:00
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81
projects/project_lif/lif.tex
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@ -2,43 +2,47 @@
\newcommand{\ptitle}{Integrate-and-fire neuron}
\input{../header.tex}
\firstpagefooter{Supervisor: Jan Benda}{phone: 29 74573}%
\firstpagefooter{Supervisor: Jan Benda}{}%
{email: jan.benda@uni-tuebingen.de}
\begin{document}
\input{../instructions.tex}
%%%%%%%%%%%%%% 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}
The leaky integrate-and-fire model is a simple but powerful model of
spiking neurons. It adds to the dynamics of a passive membrane a
voltage threshold to simulate the generation of action potentials. In
this project you learn how to simulate such a model and explore some
of its stimulus encoding properties.
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}
\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{questions}
\question Passive membrane
\begin{parts}
\part Write a function that computes the time course of the
membrane potential of the passive membrane. The function gets as
@ -80,19 +84,22 @@ time = [0.0:dt:tmax]; % t_i
neuron.
How does the filter function depend on the membrane time constant?
\end{parts}
\part Leaky integrate-and-fire neuron.
\question 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 1\,mV.
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 1\,mV.
\begin{parts}
\part
Write a function that implements this leaky integrate-and-fire
neuron by expanding the function for the passive neuron
appropriately. The function returns a vector of spike times.

10
projects/project_noiseficurves/noiseficurves.tex
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@ -2,7 +2,7 @@
\newcommand{\ptitle}{Neural tuning and noise}
\input{../header.tex}
\firstpagefooter{Supervisor: Jan Benda}{phone: 29 74573}%
\firstpagefooter{Supervisor: Jan Benda}{}%
{email: jan.benda@uni-tuebingen.de}
\begin{document}
@ -15,7 +15,7 @@ $I$ (think of that, for example, as a current $I$ injected via a
patch-electrode into the neuron).
We first characterize the neurons by their tuning curves (also called
intensity-response curve). That is, what is the mean firing rate of
intensity-response curves). That is, what is the mean firing rate of
the neuron's response as a function of the constant input current $I$?
In the second part we demonstrate how intrinsic noise can be useful
@ -83,9 +83,9 @@ from different neurons with different noise properties by setting the
\question Subthreshold stochastic resonance
Let's now use as an input to the neuron a 1\,s long sine wave $I(t)
= I_0 + A \sin(2\pi f t)$ with offset current $I_0$, amplitude $A$,
and frequency $f$. Set $I_0=5$, $A=4$, and $f=5$\,Hz.
Let's now use a 1\,s long sine wave $I(t) = I_0 + A \sin(2\pi f t)$
with offset current $I_0$, amplitude $A$, and frequency $f$. Set
$I_0=5$, $A=4$, and $f=5$\,Hz as an input to the neuron.
\begin{parts}
\part Do you get a response of the noiseless ($D_{noise}=0$) neuron?

33
projects/project_populationvector/populationvector.tex
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@ -2,22 +2,22 @@
\newcommand{\ptitle}{Orientation tuning}
\input{../header.tex}
\firstpagefooter{Supervisor: Jan Benda}{phone: 29 74573}%
\firstpagefooter{Supervisor: Jan Benda}{}%
{email: jan.benda@uni-tuebingen.de}
\begin{document}
\input{../instructions.tex}
%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%%
\begin{questions}
In the visual cortex V1 orientation sensitive neurons respond to bars
in dependence on their orientation. In this project we explore the
question:
\question In the visual cortex V1 orientation sensitive neurons
respond to bars in dependence on their orientation. In this project
we explore the question:
How is the orientation of a bar encoded by the activity of a
population of orientation sensitive neurons?
How is the orientation of a bar encoded by the activity of a
population of orientation sensitive neurons?
\begin{questions}
\question Orientation tuning of the neurons
In an electrophysiological experiment, 6 neurons have been recorded
simultaneously. First, the tuning of these neurons was characterized
@ -30,13 +30,6 @@
times are given in seconds. The stimulation with the bar starts a
time $t_0=0$ and ends at time $t_1=200$\,ms.
Then the population activity of the 6 neurons was measured in
response to arbitrarily oriented bars. The responses of the 6
neurons to 50 presentation of a bar are stored in the
\texttt{spikes} variables of the \texttt{population*.mat} files.
The \texttt{angle} variable holds the angle of the presented bar.
\continue
\begin{parts}
\part Illustrate the spiking activity of the V1 cells in response
to different orientation angles of the bars by means of spike
@ -56,7 +49,17 @@
modulation depth of the firing rate, and $a$ is an offset.
Why is there a factor two in the argument of the cosine?
\end{parts}
\question Inferring stimulus orientation from neural activity
In the second part of the experiment the population activity of the
6 neurons was measured in response to arbitrarily oriented bars. The
responses of the 6 neurons to 50 presentation of a bar are stored in
the \texttt{spikes} variables of the \texttt{population*.mat} files.
The \texttt{angle} variable holds the angle of the presented bar.
\begin{parts}
\part How can the orientation angle of the presented bar be read
out from one trial of the population activity of the 6 neurons?
One possible method is the so called ``population vector'' where