\section*{Discussion}

In this paper we have shown the effect of Suprathreshold Stochastic Resonance (SSR) in ensembles of neurons. We detailed how noise levels affect the impact of population size on the coding fraction. We looked at different frequency ranges and could show that the encoding of high-frequency signals profits particularly well from SSR. Using the tuningcurve we were able to provide a way to extrapolate the effects of SSR for very large populations. Because in general analysis of the impact of changing parameters is complex, we investigated limit cases, in particular the slow stimulus limit and the weak stimulus limit. For low-frequency signals, i.e. the slow stimulus limit, the tuningcurve also allows analyzing the impact of changing signal strength; in addition we were able to show the difference in sub-threshold SR and SSR for different noise levels. For the weak stimulus limit, where noise is relatively strong compared to the signal, we were able to provide an analytical solution for our observations.

\citep{hoch2003optimal} also shows that SSR effects hold for both LIF- and HH- Neurons. However, Hoch et al. have found that optimal noise level depends "close to logarithmatically" on the number of neurons in the population. They used a cutoff frequency of only 20Hz for their simulations. \notedh{Hier fehlt ein plot, der Population size und optimum noise in Verbindung setzt}


We investigated the impact of noise on homogeneous populations of neurons. Neurons being intrinsically noisy is a phenomenom that is well investigated (Grewe et al 2017, Padmanabhan and Urban 2010).
In natural systems however, neuronal populations are rarely homegeneous. Padmanabhan and Urban (2010) showed that heterogeneous populations of neurons carry more information that heterogenous populations.
%\notedh{Aber noisy! Zitieren: Neurone haben intrinsisches Rauschen (Einleitung?)} (Grewe, Lindner, Benda 2017 PNAS Synchronoy code) (Padmanabhan, Urban 2010 Nature Neurosci). 
Beiran et al. (2017) investigated SSR in heterogeneous populations of neurons. They made a point that heterogeneous populations are comparable to homogeneous populations where the neurons receive independent noise in addition to a deterministic signal. They make the point that in the case of weak signals, heterogeneous population can encode information better, as strong noise would overwhelm the signal. 
\notedh{Unterschiede herausstellen!} Similarly, Hunsberger et al. (2014) showed that both noise and heterogeneity linearize the tuning curve of LIF neurons. 
In summary, while noise and heterogeneity are not completely interchangeable. In the limit cases we see similar behaviour.

\citep{Sharafi2013} Sharafi et al. (2013) had already investigated SSR in a similar way. However, they only observed populations of up to three neurons and were focused on the synchronous output of cells. They took spike trains, convolved those with a gaussian and then multiplied the response of the different neurons. In our simulations we instead used the addition of spike trains to calculate the cohenrece between input and output. Instead of changing the noise parameter to find the optimum noise level, they changed the input signal frequency to find a resonating frequency, which was possible for suprathreshold stochastic resonance, but not for subthreshold stochastic resonance. For some combinations of parameters we also found that coding fraction does not decrease monotonically with increasing signal frequency (fig. \ref{cf_for_frequencies}). 
It is especially notable for signals that are far from the threshold (fig \ref{cf_for_frequencies} E,F (red markers)). 
That we don't see the effect that clearly matches Sharafi et al.'s observation that in the case of subthreshold stochastic resonance, coherence monotonically decreased with increasing frequency. Pakdaman et al. (2001) 
\notedh{Besser verkn\"upfen als das Folgende (vergleichen \"uber Gr\"o\ss{}enordnungen; vergleichen mit Abbildung 5\ref{}; mehr als Sharafi zitieren Stichwort ``Coherence Resonance''}  

Similar research to Sharafi et al. was done by (de la Rocha et al. 2007). They investigated the output correlation of populations of two neurons and found it increases with firing rate. We found something similar in this paper, where an increase in $\mu$ increases both the firing rate of the neurons and generally also the coding fraction \notedh{Verkn\"upfen mit output correlation}(fig. \ref{codingfraction_means_amplitudes}). Our explanation is that coding fraction and firing rate are linked via the tuningcurve. In addition to simulations of LIF neurons de la Rocha et al. also carried out \textit{in vitro} experiments where they confirmed their simulations.

\notedh{Konkreter machen: was machen die Anderen, das mit uns zu tun hat und was genau hat das mit uns zu tun?}
\notedh{Vielleicht nochmal Stocks, obwohl er schon in der Einleitung vorkommt? Heterogen/homogen}
\notedh{Dynamische stimuli! Bei Stocks z.B. nicht, nur z.B. bei Beiran. Wir haben den \"Ubergang.}

Examples for neuronal systems that feature noise are P-unit receptor cells of weakly electric fish (which paper?) and ...


In the case of low cutoff frequency and strong noise we were able to derive a formula that explains why in those cases coding fraction simply depends on the ratio between noise and population size, whereas generally the two variables have very different effects on the coding fraction.
%SNR has proven to be unsuitable as a measure of encoding an aperiodic signal \citep{bulsara1996threshold,Collins1995aperiodic, collins1995stochastic}. Bulsara and Zador tackled this question using only a single LIF neuron. One of their conclusions was that for suprathreshold signals, information rate increases monotonically with the SNR (same SNR as defined here), which does not hold for populations.
%Collins et al. investigated SR (subthreshold) with Gaussian correlated noise (20s correlation time) as input using the FHN model. They used a normalized power norm, similar to the coherence C we use in this paper to assess the coding fraction. Even though they used a large population of up to 1000 neurons, discussion in their paper has focused only on the sub-threshold properties of SR and considered noise as something which inhibits the ability of the network to display supra-threshold signals. They were unable to detect  an increase of input-output coherence, because they did use a very slow signal and we could show that wideband signals are necessary for SSR to manifest in large populations of neurons. \notedh{Sollte ein plot rein mit verschiedenen cutoff-Frequenzen?}
