Fixes to resulsts of tuning curve

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@@ -4,4 +4,5 @@
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+*swp #sic!

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@@ -158,10 +158,10 @@ Two very important variables are the mean strength of the signal, equivalent to
  \includegraphics[width=0.45\linewidth]{{img/basic/basic_15.0_1.0_200_detail_with_max}.pdf}
  \includegraphics[width=0.45\linewidth]{{img/basic/n_basic_weak_15.0_1.0_200_detail}.pdf}
 \includegraphics[width=0.45\linewidth]{img/basic/n_basic_compare_50_detail.pdf}
-\label{cf_limit}
 \caption{A: Coding fraction as a function of noise for different population sizes. Green dots mark the peak of the coding fraction curve. Increasing population size leads to a higher peak and moves the peak to stronger noise.  
 B: Coding fraction as a function of population size. Each curve shows coding fraction for a different noise strength. 
 C: Peak coding fraction as a function of population size for different input parameters. \notedh{ needs information about noise}}
+\label{cf_limit}
 \end{figure}
 
  
@@ -228,11 +228,11 @@ For faster signals, the past of the signal plays a role: after a spike there is
  \includegraphics[width=0.4\linewidth]{{img/temp/best_approximation_spikes_50hz_0.01noi500s_10.5_1_1.dat_256_with_input}.pdf}
 \caption{Two ways to arrive at coherence and coding fraction. Left: The input signal (top, center) is received by LIF-neurons. The spiking of the neurons is then binned and coherence and coding fraction are calculated between the result and the input signal. 
 Right: Input signal (top, center) is transformed by the tuning curve (top right). The tuning curve corresponds to a function $g(V)$, which takes a voltage as input and yields a firing rate. Output is a modulated signal. We calculate coherence and coding fraction between input voltage and output firing rate. If the mean of the input is close to the threshold, as is the case here, inputs below the threshold all get projected to 0. This can be seen here at the beginning of the transformed curve.
-Bottom left: Tuning curves for different noise levels. x-Axis shows the stimulus in mV, the y-axis shows the corresponding firing rate. For low noise levels there is a strong non-linearity at the threshold. For increasing noise, firing rate increases particularly}
+Bottom left: Tuning curves for different noise levels. x-Axis shows the stimulus strength in mV, the y-axis shows the corresponding firing rate. For low noise levels there is a strong non-linearity at the threshold. For increasing noise, firing rate becomes larger than 0 for progressively weaker signals. For strong stimuli (roughly 13mV and more) there is little different in the firing rate depending on the noise.}
 \label{non-lin}
 \end{figure}
 
-The noise influences the shape of the tuning curve, with stronger noise linearizing the curve. The linearity of the curve is important, because coding fraction is a linear measure.  For strong input signals (around 15mV) the curve is almost linear, resulting in coding fractions close to 1. 
+The noise influences the shape of the tuning curve, with stronger noise linearizing the curve. The linearity of the curve is important, because coding fraction is a linear measure.  For strong input signals (around 13mV) the curve is almost linear, resulting in coding fractions close to 1. For signal amplitudes in this range firing rate is almost independent of noise strength. This tells us that the increase in coding fraction that follows a change in noise strength we saw in previous chapters is not simply due to the neurons spiking more frequently. 
 For slow signals (1Hz  cutoff frequency, up to 10Hz) the results from the tuning curve and the simulation for large populations of neurons match very well (figure \ref{accuracy}) over a range of signal strengths, base inputs to the neurons and noise strength.
 This means that the LIF-neuron tuning curve gives us a very good approximation for the limit of encoded information that can be achieved by summing over independent, identical LIF-neurons with intrinsic noise.
 For faster signals, the coding fraction calculated through the tuning curve stays constant, as the tuning curve only deforms the signal. As shown in figure \ref{cf_for_frequencies} e) and f), the coding fraction of the LIF-neuron ensemble drops with increasing frequency. Hence for high frequency signals the tuning curve ceases to be a good predictor of the encoding quality of the ensemble.