\section*{Limit case of large populations}

\subsection*{For large population sizes and strong noise, coding fraction becomes a function of their quotient}

For the linear response regime of large noise, we can estimate the coding fraction. From Beiran et al. 2018 we know the coherence in linear response is given as

\eq{
C_N(\omega) = \frac{N|\chi(\omega)|^2 S_{ss}}{S_{x_ix_i}(\omega)+(N1)|\chi(\omega)|^2S_{ss}}
\label{eq:linear_response}
}

where \(C_1(\omega)\) is the coherence function for a single LIF neuron. Generally, the single-neuron coherence is given by \citep{??}

\eq{
C_1(\omega)=\frac{r_0}{D} \frac{\omega^2S_{ss}(\omega)}{1+\omega^2}\frac{\left|\mathcal{D}_{i\omega-1}\big(\frac{\mu-v_T}{\sqrt{D}}\big)-e^{\Delta}\mathcal{D}_{i\omega-1}\big(\frac{\mu-v_R}{\sqrt{D}}\big)\right|^2}{\left|{\cal D}_{i\omega}(\frac{\mu-v_T}{\sqrt{D}})\right|^2-e^{2\Delta}\left|{\cal D}_{i\omega}(\frac{\mu-v_R}{\sqrt{D}})\right|^2}
\label{eq:single_coherence}
}

where \(r_0\) is the firing rate of the neuron,
\[r_0 = \left(\tau_{ref} + \sqrt{\pi}\int_\frac{\mu-v_r}{\sqrt{2D}}^\frac{\mu-v_t}{\sqrt{2D}} dz e^{z^2} \erfc(z) \right)^{-1}\]. 
In the limit of large noise (calculation in the appendix) this equation evaluates to:

\eq{
C_1(\omega) = \sqrt{\pi}D^{-1}
\frac{S_{ss}(\omega)\omega^2/(1+\omega^2)}{2 \sinh\left(\frac{\omega\pi}{2}\right)\Im\left( \Gamma\left(1+\frac{i\omega}{2}\right)\Gamma\left(\frac12-\frac{i\omega}{2}\right)\right)}
\label{eq:simplified_single_coherence}
}

From eqs.\ref{eq:linear_response} and \ref{eq:simplified_single_coherence} it follows that in the case \(D \rightarrow \infty\) the coherence, and therefore the coding fraction, of the population of LIF neurons is a function of \(D^{-1}N\). We plot the approximation as a function of \(\omega\) (fig. \ref{d_n_ratio}). In the limit of small frequencies the approximation matches the exact equation very well, though not for higher frequencies. We can verify this in our simulations by plotting coding fraction as a function of \(\frac{D}{N}\). We see (fig. \ref{d_n_ratio}) that in the limit of large D, the curves actually lie on top of each other. This is however not the case (fig. \ref{d_n_ratio}) for stimuli with a large cutoff frequency \(f_c\), as expected by our evaluation of the approximation as a function of the frequency.


\begin{figure}
\centering 
 \includegraphics[width=0.32\linewidth]{{img/d_over_n/d_10.5_0.5_10_detail}.pdf}
 \includegraphics[width=0.32\linewidth]{{img/d_over_n/d_15.0_0.5_50_detail}.pdf}
 \includegraphics[width=0.32\linewidth]{{img/d_over_n/d_15.0_1.0_200_detail}.pdf}
 \includegraphics[width=0.32\linewidth]{{img/d_over_n/d_over_n_10.5_0.5_10_detail}.pdf}
 \includegraphics[width=0.32\linewidth]{{img/d_over_n/d_over_n_15.0_0.5_50_detail}.pdf}
 \includegraphics[width=0.32\linewidth]{{img/d_over_n/d_over_n_15.0_1.0_200_detail}.pdf}
 \label{d_n_ratio}
 \caption{Top row: Coding fraction as a function of noise.
 Bottom row: Coding fraction as a function of the ratio between noise strength and population size. For strong noise, coding fraction is a function of this ratio. 
 Left: signal mean 10.5mV, signal amplitude 0.5mV, $f_{c}$ 10Hz.
 Middle: signal mean 15.0mV, signal amplitude 0.5mV, $f_{c}$ 50Hz.
 Right: signal mean 15.0mV, signal amplitude 1.0mV, $f_{c}$ 200Hz.}
\end{figure}
