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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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3 changed files with 67 additions and 57 deletions
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41
projects/project_lif/lif.tex
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@ -2,19 +2,21 @@
\newcommand{\ptitle}{Integrate-and-fire neuron} \newcommand{\ptitle}{Integrate-and-fire neuron}
\input{../header.tex} \input{../header.tex}
\firstpagefooter{Supervisor: Jan Benda}{phone: 29 74573}% \firstpagefooter{Supervisor: Jan Benda}{}%
{email: jan.benda@uni-tuebingen.de} {email: jan.benda@uni-tuebingen.de}
\begin{document} \begin{document}
\input{../instructions.tex} \input{../instructions.tex}
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.
%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%% The temporal evolution of the membrane voltage $V(t)$ of a passive
neuron is described by the membrane equation
\begin{questions}
\question The temporal evolution of the membrane voltage $V(t)$ of a
passive neuron is described by the membrane equation
\begin{equation} \begin{equation}
\label{passivemembrane} \label{passivemembrane}
\tau \frac{dV}{dt} = -V + E \tau \frac{dV}{dt} = -V + E
@ -22,23 +24,25 @@
where $\tau=10$\,ms is the membrane time constant and $E(t)$ is the where $\tau=10$\,ms is the membrane time constant and $E(t)$ is the
reversal potential that also depends on time $t$. reversal potential that also depends on time $t$.
Such a differential equation can be numerically solved with the Euler method. Such a differential equation can be numerically solved with the Euler
For this the time is discretized by a time step $\Delta t=0.1$\,ms. method. For this the time is discretized by a time step $\Delta
The $i$-th time point is then at time $t_i = i \cdot \Delta t$. t=0.1$\,ms. The $i$-th time point is then at time $t_i = i \cdot
In matlab we get the time points $t_i$ simply by \Delta t$. In matlab we get the time points $t_i$ simply by
\begin{lstlisting} \begin{lstlisting}
dt = 0.1; dt = 0.1;
tmax = 100.0; tmax = 100.0;
time = [0.0:dt:tmax]; % t_i time = [0.0:dt:tmax]; % t_i
\end{lstlisting} \end{lstlisting}
When the membrane potential at time $t_0 = 0$ is $V_0$, the so When the membrane potential at time $t_0 = 0$ is $V_0$, the so called
called ``initial condition'', then we can iteratively compute the ``initial condition'', then we can iteratively compute the membrane
membrane potentials $V_i$ for successive time points $t_i$ according to potentials $V_i$ for successive time points $t_i$ according to
\begin{equation} \begin{equation}
\label{euler} \label{euler}
V_{i+1} = V_i + (-V_i + E_i) \frac{\Delta t}{\tau} V_{i+1} = V_i + (-V_i + E_i) \frac{\Delta t}{\tau}
\end{equation} \end{equation}
\begin{questions}
\question Passive membrane
\begin{parts} \begin{parts}
\part Write a function that computes the time course of the \part Write a function that computes the time course of the
membrane potential of the passive membrane. The function gets as membrane potential of the passive membrane. The function gets as
@ -80,8 +84,9 @@ time = [0.0:dt:tmax]; % t_i
neuron. neuron.
How does the filter function depend on the membrane time constant? 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 The passive neuron can be turned into a spiking neuron by
introducing a fixed voltage threshold. Whenever the computed introducing a fixed voltage threshold. Whenever the computed
@ -89,10 +94,12 @@ time = [0.0:dt:tmax]; % t_i
threshold a spike is generated and the membrane voltage is set to 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 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 spike'' only means that we note down the time of the threshold
crossing as a time where an action potential occurred. The crossing as a time where an action potential occurred. The waveform
waveform of the action potential is not modeled. Here we use a of the action potential is not modeled. Here we use a voltage
voltage threshold of 1\,mV. threshold of 1\,mV.
\begin{parts}
\part
Write a function that implements this leaky integrate-and-fire Write a function that implements this leaky integrate-and-fire
neuron by expanding the function for the passive neuron neuron by expanding the function for the passive neuron
appropriately. The function returns a vector of spike times. appropriately. The function returns a vector of spike times.

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

31
projects/project_populationvector/populationvector.tex
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@ -2,23 +2,23 @@
\newcommand{\ptitle}{Orientation tuning} \newcommand{\ptitle}{Orientation tuning}
\input{../header.tex} \input{../header.tex}
\firstpagefooter{Supervisor: Jan Benda}{phone: 29 74573}% \firstpagefooter{Supervisor: Jan Benda}{}%
{email: jan.benda@uni-tuebingen.de} {email: jan.benda@uni-tuebingen.de}
\begin{document} \begin{document}
\input{../instructions.tex} \input{../instructions.tex}
%%%%%%%%%%%%%% Questions %%%%%%%%%%%%%%%%%%%%%%%%% In the visual cortex V1 orientation sensitive neurons respond to bars
\begin{questions} 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 How is the orientation of a bar encoded by the activity of a
population of orientation sensitive neurons? population of orientation sensitive neurons?
\begin{questions}
\question Orientation tuning of the neurons
In an electrophysiological experiment, 6 neurons have been recorded In an electrophysiological experiment, 6 neurons have been recorded
simultaneously. First, the tuning of these neurons was characterized simultaneously. First, the tuning of these neurons was characterized
by presenting them bars in a range of 12 orientation angles. Each by presenting them bars in a range of 12 orientation angles. Each
@ -30,13 +30,6 @@
times are given in seconds. The stimulation with the bar starts a times are given in seconds. The stimulation with the bar starts a
time $t_0=0$ and ends at time $t_1=200$\,ms. 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} \begin{parts}
\part Illustrate the spiking activity of the V1 cells in response \part Illustrate the spiking activity of the V1 cells in response
to different orientation angles of the bars by means of spike 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. modulation depth of the firing rate, and $a$ is an offset.
Why is there a factor two in the argument of the cosine? 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 \part How can the orientation angle of the presented bar be read
out from one trial of the population activity of the 6 neurons? out from one trial of the population activity of the 6 neurons?
One possible method is the so called ``population vector'' where One possible method is the so called ``population vector'' where