\subsection*{Refractory Periods}

We analyzed the effect of non-zero refractory periods on the previous results. We added a 1ms or a 5ms refractory period to each of the LIF-neurons. Then, we repeated the same simulations as before. Results are summarized in figure \ref{refractory_periods}.
Results change very little for a refractory period of 1ms, especially for large noise values. For a refractory period of 5ms resulting coding fraction is lower for almost all noise values. Paradoxically, for high frequencies in smallband signals and very small noise, coding fraction actually is larger for 5ms refractory period than for 1ms. In spite of this, coding fraction is still largest for the LIF-ensembles without refractory period.

We also find all other results replicated even with refractory periods of 1ms or 5ms: Figure (??) shows that the optimal noise stills grows with \(\sqrt{N}\) for both the 1ms and the 5ms refractory period. We see an increase in the value of the optimum noise with an increase of the refractory period. The achievable coding fraction is lower for the neurons with refractory periods, especially at the maximum. In the limit of large noise, the neurons with 1ms refractory period and the ones with no fractory period also result in similar coding fractions, over a wide range of population sizes. However, this is not true for the neurons with 5ms refractory period.


\begin{figure}
 \includegraphics[width=0.8\linewidth]{img/ordnung/refractory_periods_coding_fraction.pdf}
 \caption{Repeating the simulations adding a refractory period to the LIF-neurons shows no qualitative changes in the SSR behaviour of the neurons. Coding fraction is lower the longer the refractory period. The SSR peak moves to stronger noise; cells with larger refractory periods need stronger noise to work optimally.}
 \label{refractory_periods}
\end{figure}
