We use a population neuron model using the Leaky-Integrate-And-Fire (LIF) neuron, described by the equation

\begin{equation}V_{t}^j = V_{t-1}^j + \frac{\Delta t}{\tau_v} ((\mu-V_{t-1}^j) + \sigma I_{t} + \sqrt{2D/\Delta t}\xi_{t}^j),\quad j \in [1,N]\end{equation}
with  $\tau_v = 10 ms$ the membrane time constant, $\mu = 15.0 mV$ or $\mu = 10.5 mV$ as offset. $\sigma$ is a factor which scales the standard deviation of the input, ranging from 0.1 to 1 and I the previously generated stimulus. $\xi_{t}$ are independent Gaussian distributed random variables with mean 0 and variance 1. The Noise D was varied between $1*10^{-7} mV^2/Hz$ and $3 mV^2/Hz$. Whenever $V_{t}$ was greater than the voltage threshold (10mV) a "spike" was recorded and the voltage has been reset to 0mV. $V_{0}$ was initialized to a random value uniformly distributed between 0mV and 10mV.
Simulations of up to 8192 neurons were done using an Euler method with a step size of $\Delta\, t = 0.01$ms. Typical firing rates were around 90Hz for an offset of 15.0mV and 35Hz for an offset of 10.5mV. Firing rates were larger for high noise levels than for low noise levels.
We simulated large populations (up to 2048) of LIF-neurons.

As stimulus we used Gaussian white noise signal with different frequency cutoff on both ends of the spectrum. By construction, the input power spectrum is flat between 0 and $\pm f_{c}$:

\begin{equation}
 S_{ss}(f) = \frac{\sigma^2}{2 \left| f_{c} \right|} \Theta\left(f_{c} - |f|\right).\label{S_ss}
\end{equation}
A Fast Fourier Transform (FFT) was applied to the signal so it can serve as input stimulus to the simulated cells. The signal was normalized so that the variance of the signal was 1mV and the length of the signal was 500s  with a resolution of 0.01ms.


\begin{figure}
\includegraphics[scale=0.5]{img/intro_raster/example_noise_resonance.pdf}
\caption{Snapshots of 200ms length from three example simulations with different noise, but all other parameters held constant.  Black: Spikes of 32 simulated neurons. The green curve beneath the spikes is the signal that was fed into the network. The blue curve is the best linear reconstruction possible from the spikes. The input signal has a cutoff frequency of 50Hz.
If noise is weak, the neurons behave regularly and similar to each other (A). For optimal noise strength, the neuronal population follows the signal best (B). If the noise is too strong, the information about the signal gets drowned out (C). D: Example coding fraction curve over the strength of the noise. Marked in red are the noise strengths from which the examples were taken.}
\label{example_spiketrains}
\end{figure}




\subsection*{Analysis}
For each combination of parameters, a histogram of the output spikes from all neurons or a subset of the neurons was created.
The coherence $C(f)$ was calculated \citep{lindner2016mechanisms} in frequency space as the fraction between the squared cross-spectral density $|S_{sx}^2|$ of input signal $s(t) = \sigma I_{t}$ and output spikes x(t), $S_{sx}(f) = \mathcal{F}\{ s(t)*x(t) \}(f) $, divided by the product of the power spectral densities of input ($S_{ss}(f) = |\mathcal{F}\{s(t)\}(f)|^2 $) and output ($S_{xx}(f) = |\mathcal{F}\{x(t)\}(f)|^2$), where $\mathcal{F}\{ g(t) \}(f)$ is the Fourier transform of g(t).
\begin{equation}C(f) = \frac{|S_{sx}(f)|^2}{S_{ss}(f) S_{xx}(f)}\label{coherence}\end{equation} 

 The coding fraction $\gamma$ \citep{gabbiani1996codingLIF, krahe2002stimulus} quantifies how much of the input signal can be reconstructed by an optimal linear decoder. It is 0 in case the input can't be reconstructed at all and 1 if the signal can be perfectly reconstructed\citep{gabbiani1996stimulus}. 
 It is defined by the reconstruction error $\epsilon^2$ and the variance of the input $\sigma^2$:

\begin{equation}\gamma = 1-\sqrt{\frac{\epsilon^2}{\sigma^2}}.\label{coding_fraction}\end{equation}


The variance is 
\begin{equation}\sigma^2  = \langle \left(s(t)-\langle s(t)\rangle\right)^2\rangle =  \int_{f_{low}}^{f_{high}} S_{ss}(f) df .\end{equation}

The reconstruction error is defined as

\begin{equation}\epsilon^2 = \langle \left(s(t) - s_{est}(t)\right)^2\rangle = \int_{f_{low}}^{f_{high}} S_{ss} - \frac{|S_{sx}|^2}{S_{xx}} = \int_{f_{low}}^{f_{high}} S_{ss}(f) (1-C(f)) df\end{equation} 
with the estimate $s_{est}(t) = h*x(t)$. $h$ is the optimal linear filter which has Fourier Transform $H = \frac{S_{sx}}{S_{xx}}$\citep{gabbiani1996coding}. 

We then analyzed coding fraction as a function of these cutoff frequencies for different parameters (noise strength, signal amplitude, signal mean/firing rate) in the limit of large populations.
The limit was considered reached if the increase in coding fraction gained by doubling the population size is small (4\%)(??). 
For the weak signals ($\sigma = 0.1mV$) combined with the strongest noise ($D = 10^{-3} \frac{mV^2}{Hz}$), convergence was not reached for a population size of 2048 neurons. The same is true for the combination of the weak signal, close to the threshold ($\mu = 10.5mV$) and high frequencies (200Hz).