diff --git a/projects/project_lif/lif.tex b/projects/project_lif/lif.tex index dc24b05..b42042d 100644 --- a/projects/project_lif/lif.tex +++ b/projects/project_lif/lif.tex @@ -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. diff --git a/projects/project_noiseficurves/noiseficurves.tex b/projects/project_noiseficurves/noiseficurves.tex index a2df3df..279a0fc 100644 --- a/projects/project_noiseficurves/noiseficurves.tex +++ b/projects/project_noiseficurves/noiseficurves.tex @@ -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? diff --git a/projects/project_populationvector/populationvector.tex b/projects/project_populationvector/populationvector.tex index 4f98614..15c3398 100644 --- a/projects/project_populationvector/populationvector.tex +++ b/projects/project_populationvector/populationvector.tex @@ -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