
\subsection*{Narrowband stimuli}
Using the \(f_{cutoff} = 200 \hertz\usk\) signal, we repeated the analysis for only a part of the spectrum. We did so for two "low frequency" (0--8Hz, 0--50Hz) and two "high frequency" (192--200Hz, 150--200Hz) intervals. We then compared the results to the results we get from narrowband stimuli, with power only in those frequency bands.
To keep the power of the signal inside the two intervals the same as in the broadband stimulus, amplitude of the narrowband signals was less than that of the broadband signal. For the 8Hz intervals, amplitude (i.e. standard deviation) of the signal was 0.2mV, or a fifth of the amplitude of the broadband signal. Because signal power is proportional to the square of the amplitude, this was appropriate for a stimulus with a spectrum 25 times smaller. Similarly, for the 50Hz intervals we used a 0.5mV amplitude, or half of that of the broadband stimulus.
As the square of the amplitude is equal to the integral over the frequency spectrum, for a signal with a quarter of the width we need to half the amplitude to have the same power in the interval defined by the narrowband signals.

\subsection*{Smaller frequency intervals in broadband signals }

\begin{figure}
\includegraphics[width=0.45\linewidth]{img/small_in_broad_spectrum}
\includegraphics[width=0.45\linewidth]{img/power_spectrum_0_50}
	\includegraphics[width=0.49\linewidth]{{img/broad_coherence_15.0_1.0}.pdf}
	\includegraphics[width=0.49\linewidth]{{img/coherence_15.0_0.5_narrow_both}.pdf}
	\includegraphics[width=0.49\linewidth]{{img/broad_coherence_10.5_1.0_200}.pdf}
	\includegraphics[width=0.49\linewidth]{{img/coherence_10.5_0.5_narrow_both}.pdf}
	\caption{Coherence for broad and narrow frequency range inputs. a) Broad spectrum. 
	At the frequency of the firing rate (91Hz, marked by the black bar) and its first 
	harmonic (182Hz) the coding fraction breaks down. For the weak noise level (blue), 
	population sizes n=4 and n=4096 show indistinguishable coding fraction. 
	In case of a small population size, coherence is higher for weak noise (blue) than 
	for strong noise (green) in  the frequency range up to about 50\hertz. For higher
	frequencies coherence is unchanged. For the case of the larger population size and the 
	greater noise strength there is a huge increase in the coherence for all frequencies. 
	b) Coherence for two narrowband inputs with different frequency ranges. 
	Low frequency range: coherence for 
	slow parts of the signal is close to 1 for weak noise. SSR works mostly on the 
	higher frequencies (here >40\hertz). High frequency range: At 182Hz (first harmonic 
	of the firing frequency) there is a very sharp decrease in coding fraction, 
	especially for the weak noise condition (blue). Increasing the noise makes the drop 
	less clear. For weak noise (blue) there is another break down at 182-(200-182)Hz. 
	Stronger noise seems to make this sharp drop disappear. Again, the effect of SSR 
	is most noticeable for the higher frequencies in the interval.}
	\label{fig:coherence_narrow_15.0}
\end{figure}

We want to know how  good encoding works for different frequency intervals in the signal.
When we take out a narrower frequency interval from a broadband signal, the other 
frequencies in the signal serve as common noise to the neurons encoding the signal.
In many cases we only care about a certain frequency band in a signal of much wider bandwidth.
In figure \ref{fig:coherence_narrow_15.0} A we can see that SSR has very different
effects on certain frequencies inside the signal than on others. In blue we see the
case of very weak noise (\(10^{-6} \milli\volt\squared\per\hertz\)). Increasing the
population size from 4 neurons to 2048 neurons has practically no effect. Around
the average firing rate of the neurons, coherence becomes almost zero.  When
we keep population size at 4 neurons, but 
add more noise to the neurons (green, \(2\cdot10^{-3} \milli\volt\squared\per\hertz\)),
encoding of the low frequencies (up to about 50\hertz) becomes worse, while
encoding of the higher frequencies stays unchanged. When we increase the population
size to 2048 neurons we have almost perfect encoding for frequencies up to 50\hertz.
Coherence is still reduced around the average firing rate of the neurons, but at
a much higher level than before. For higher frequencies coherence becomes higher again.
In summary, the high frequency bands inside the broadband stimulus experience a 
much greater increase in encoding quality than the low frequency bands,
which were already encoded quite well. 
 

\begin{figure}
\includegraphics[width=0.45\linewidth]{img/broadband_optimum_newcolor.pdf}
\includegraphics[width=0.45\linewidth]{img/smallband_optimum_newcolor.pdf}
\centering
\includegraphics[width=0.9\linewidth]{img/max_cf_smallbroad.pdf}
\caption{
 A: Input signal spectrum of a broadband signal. The colored area marks the frequency ranges considered here. 
 B: Two narrowband signals (red and blue). The broadband signal from A (grey) is shown again for comparison.
 C and D: Best amount of noise for different number of neurons. The dashed lines show where coding fraction still is at least 95\% from the maximum. The width of the peaks is much larger for the narrowband signals which encompasses the entire width of the high-frequency interval peak. 
Optimum noise values for a fixed number of neurons are always higher for the broadband signal than for narrowband signals. 
In the broadband case, the optimum amount of noise is larger for the high-frequency interval than for the low-frequency interval and vice-versa for the narrowband case. %The optimal noise values have been fitted with a function of square root of the population size N, $f(N)=a+b\sqrt{N}$. We observe that the optimal noise value grows with the square root of population size.
E and F: Coding fraction as a function of noise for a fixed population size (N=512). Red dots show the maximum, the red line where coding fraction is at least 95\% of the maximum value.
G: An increase in population size leads to a higher coding fraction especially for broader bands and higher frequency intervals. Coding fraction is 
larger for the narrowband signal than in the equivalent broadband interval for all neural population sizes considered here. The coding fraction for the low frequency intervals is always larger than for the high frequency interval.
Signal mean $\mu=15.0\milli\volt$, signal amplitude $\sigma=1.0\milli\volt$ and $\sigma=0.5\milli\volt$ respectively.}
\label{smallbroad}
\end{figure}

\subsection*{Narrowband Signals vs Broadband Signals}




In nature, often an external stimulus covers a frequency range 
that starts at high frequencies, so that only using broadband white noise signals 
as input appears to be insufficient to describe realistic scenarios.
%, with bird songs\citep{nottebohm1972neural} and ???\footnote{chirps, in a way?}. 
%We see that in many animals receptor neurons have adapted to these signals. For example, it was found that electroreceptors in weakly electric fish have band-pass properties\citep{bastian1976frequency}. 
Therefore, we investigate the coding of narrowband signals in the ranges 
described earlier (0--50Hz, 150--200Hz). Comparing the results from coding of 
broadband and coding of narrowband signals, we see several differences. 

For both low and high frequency signals, the narrowband signal 
can be resolved better than the broadband signal for any amount of noise and (figure \ref{smallbroad}, bottom left).
That coding fractions are higher when we use narrowband signals can be
explained by the fact that the additional 
frequencies in the broadband signal are now absent. In the broadband signal
they are a form of "noise" that is common to all the input neurons.
Similar to what we saw for the broadband signal, 
the peak of the low frequency input is still much more broad than the peak of the high frequency input.
To encode low frequency signals the exact strength of the noise is not as 
important as it is for the high frequency signals which can be seen from the wider peaks.

\subsection{Discussion}
The usefulness of noise on information encoding of subthreshold signals by single neurons has been well investigated. However, the encoding of supra-threshold signals by populations of neurons has received comparatively little attention and different effects play a role for suprathreshold signals than for subthreshold signals \citep{Stocks2001}. This paper delivers an important contribution for the understanding of suprathreshold stochastic resonance (SSR). We simulate populations of leaky integrate-and-fire neurons to answer the question how population size influences the optimal noise strength  for linear encoding of suprathreshold signals. We are able to show that this optimal noise is well described as a function of the square root of population size. This relationship is independent of frequency properties of the input signal and holds true for narrowband and broadband signals.

In this paper, we show that SSR works in LIF-neurons for a variety of signals of different bandwidth and frequency intervals. We show that signal-to-noise ratio is for signals above a certain strength sufficient to describe the optimal noise strength in the population, but that the actual coding fraction depends on the absolute value of signal strength. %We furthermore show that increasing signal strength does not always increase the coding fraction.

We contrast how well the low and high frequency parts of a broadband signal can be encoded. We take an input signal with $f_{cutoff} = \unit{200}\hertz$ and analyse the coding fraction for the frequency ranges 0 to \unit{50}\hertz\usk and 150 to \unit{200}\hertz\usk separately. The maximum value of the coding fraction is lower for the high frequency interval compared to the low frequency interval. This means that inside broadband signals higher frequencies intervals appear more difficult to encode for each level of noise and population size. The low frequency interval has a wider peak (defined as 95\% coding fraction of its coding fraction maximum value), which means around the optimal amount of noise there is a large area where coding fraction is still good. The noise optimum for the low frequency parts of the input is lower than the optimum for the high frequency interval (Fig. \ref{highlowcoherence}). In both cases, the optimal noise value appears to grow with the square root of population size.

In general, narrowband signals can be encoded better than broadband signals.
narrowband vs broadband

Another main finding of this paper is the discovery of frequency dependence of SSR.
We can see from the shape of the coherence between the signal and the output of the simulated 
neurons, SSR works mostly for the higher frequencies in the signal. As the lower frequency
components are in many cases already encoded really well, the addition of noise
helps to flatten the shape of the coherence curve. In the case of weak noise, often there
are border effects which disappear with increasing strength of the noise.
In addition, for weak noise there are often visible effects from the firing rate of the neurons, in so far that the encoding
around those frequencies is worse than for the surrounding frequencies. Generally
this effect becomes less pronounced when we add more noise to the simulation, but
we found a very striking exception in the case of narrowband signals. 
Whereas for a firing rate of about 
91\hertz\usk  the coding fraction of the encoding of a signal in the 0-50\hertz\usk band is 
better than for the encoding of a signal in the 150-200\hertz\usk band. However, this is 
not the case if the neurons have a firing rate about 34\hertz. 
We were thus able to show that the firing rate on the neurons in the simulation is of 
critical importance to the encoding of the signal.

