## 2 Mathematical Framework
### Broadband Power Spectral Density  
In this subsection, the Markov matrix approach is extended to an arbitrary number \(N_{i}\) of synchronized closed loops of firing neurons. Under the assumption of fluctuations in \(N_{i}\), the expression of the power spectrum containing a broadband component is derived. The syncronized firing of neurons belonging to the same state \(\alpha\) will be described by the superposition of individual firing functions in terms of \(F_{\alpha}(t)=\sum_{i=1}^{N_{i}}F_{i,\alpha}(t)=N_{i}\langle F_{i,\alpha}(t)\rangle\), where the brackets \(\langle\ \rangle\) refer to the average over the ensemble of \(N_{i}\) loops. Owing to the large number of loops \(N_{i}\) and to the imperfect synchronization of the elementary impulsesbelonging to the same \(\alpha\) state, the firing events are described as Gaussian functions; hence, \(\langle F_{i,\alpha}(t)\rangle\) can be written as:  
\[\langle F_{i,\alpha}(t)\rangle=\frac{\langle A_{\alpha}\rangle}{\sigma_{\alpha} \sqrt{2\pi}}\exp\biggl{(}-\frac{t^{2}}{2\sigma_{\alpha}^{2}}\biggr{)}\quad, \tag{13}\]  
with variance \(\sigma_{\alpha}\), amplitude \(\langle A_{\alpha}\rangle=\int_{-\infty}^{\infty}\langle F_{i,\alpha}(t) \rangle dt\) and the time origin of the impulses taken at the maximum of the Gaussian function relative to every state \(\alpha\). The Fourier transform of the function \(F_{\alpha}(t)\) can be written as \(S_{\alpha}(\omega)=N_{i}\langle S_{i,\alpha}(\omega)\rangle\), where:  
\[\langle S_{i,\alpha}(\omega)\rangle=\langle A_{\alpha}\rangle\exp\biggl{(}- \frac{\omega^{2}\sigma_{\alpha}^{2}}{2}\biggr{)}\quad. \tag{14}\]  
When the fluctuations in the number \(N_{i}\) of synchronized loops are taken into account in the general expression of the power spectal density, the quantity \(S_{\alpha}(\omega)^{*}S_{\alpha}(\omega)\) is written as \((\overline{N_{i}})^{2}\overline{(<|S_{i,\alpha}(\omega)|>)^{2}}\). The first term in the same equation is \((\overline{N_{i}^{2}})\)\((<|S_{i,\alpha}(\omega)|>)^{2}\). Hence, the power spectrum is:  
\[\Phi(\omega)=\
u\Bigl{(}\overline{N_{i}^{2}}-\overline{N_{i}^{2}}\Bigr{)} \overline{\langle|S_{i,\alpha}(\omega)|\rangle^{2}}\quad. \tag{15}\]  
The quantity \(\overline{N_{i}^{2}}-\overline{N_{i}^{2}}\), yielding the fluctuations in \(N_{i}\) around its average value \(\overline{N}_{i}\), can be estimated by assuming that \(N_{i}\) is a stationary random variable described by a normalized Gaussian probability function \(P(N_{i})\):  
\[P(N_{i})=\frac{1}{\sigma_{N_{i}}\sqrt{2\pi}}\cdot\exp\frac{-(N_{i}-\overline{ N_{i}})^{2}}{2\sigma_{N_{i}}^{2}}\quad, \tag{16}\]  
with \((\overline{N_{i}})^{2}=\Bigl{(}\int_{-\infty}^{\infty}(N_{i}\cdot P(N_{i}))dN _{i}\Bigr{)}^{2}\) and \(\overline{(N_{i}^{2})}=\int_{-\infty}^{\infty}\bigl{(}N_{i}^{2}\cdot P(N_{i}) \bigr{)}dN_{i}=(\overline{N_{i}})^{2}+\sigma_{N_{i}}^{2}\) and the variance is \(\sigma_{N_{i}}\). Hence, Equation (15) is:  
\[\Phi(\omega)=\
u\sigma_{N_{i}}^{2}\overline{\langle|S_{i,\alpha}(\omega)| \rangle^{2}}\quad, \tag{17}\]  
which yields the broadband component of the noise power spectrum, whose intensity has been estimated for normally distributed fluctuations in the number of syncronized loops. The amplitude of the broadband component depends on the fluctuations in \(N_{i}\) through the variance \(\sigma_{N_{i}}\). The width of the Gaussian \(\sigma_{\alpha}\) for each state \(\alpha\) in Equation (14) is related to the cut-off frequency of the power spectrum and, thus, to the cut-off of the highest harmonic frequencies of the periodic components. If the pulses are Gaussian with Fourier transform given by Equation (14), the cut-off frequency may be estimated as \(\omega_{c}=2\pi f_{c}=1/\overline{\sigma}_{\alpha}\), corresponding to a reduction factor of \(1/e\approx 0.36\).