## 2 Mathematical Framework
### Line Power Spectral Density  
In this subsection, we will show how to derive the power spectral density of a single closed loop of Markov-chain correlated pulses. When the Markov matrix \(m\) in Equation (1) describes random events organized in closed loop sequences, the correlation matrix \(M\) yields physically sound line spectra as those observed in brain waves. The lines in the power spectrum correspond to singularities of Equation (6) when \(\det(I-M(\omega))=0\). The singularities occur for \(q_{\alpha,\alpha^{\prime}}(u)=\delta(u-u_{\alpha,\alpha^{\prime}})\) where \(u_{\alpha,\alpha^{\prime}}\) is the time interval between consecutive events characterized by the states \(\alpha,\alpha^{{}^{\prime}}\). Then:  
\[Q_{\alpha,\alpha^{{}^{\prime}}}(\omega)=\exp\Bigl{(}i\omega u_{\alpha,\alpha^ {{}^{\prime}}}\Bigr{)} \tag{7}\]  
and Equation (3) takes the form:  
\[M_{\alpha,\alpha^{{}^{\prime}}}(\omega)=m_{\alpha,\alpha^{{}^{\prime}}}\exp \Bigl{(}i\omega u_{\alpha,\alpha^{{}^{\prime}}}\Bigr{)}\quad. \tag{8}\]  
A closed loop involving \(N\) states \(\alpha\) can be expressed by a \(N\times N\) matrix \(M(\omega)\) where all the rows are made up of zeroes except one entry equal to \(1\) (with \(m_{\alpha,\alpha^{{}^{\prime}}}=0\) except if \(\alpha_{i+1}=\alpha^{\prime}\) and \(\alpha_{i}=\alpha\ m_{\alpha,\alpha^{\prime}}=1\)). The condition \(m_{N-1,N}=m_{N,1}=1\) ensures the closeness of the loops. With the above choices, the relationship \(\det(I-M(\omega))=0\) becomes:  
\[1-\exp[i\omega(u_{1,2}+u_{2,3}+\ldots+u_{N,1})]=0 \tag{9}\]  
with solutions:  
\[\omega=\omega_{n}=n\omega_{o}=\frac{2\pi}{\sum_{i=1}^{N}u_{i,i+1}} \tag{10}\]  
Equation (10) yields the fundamental frequency \(\omega_{o}\) and the harmonics of the periodic component. \(n\) is an integer. \(\omega_{o}\) decreases as \(N\) and \(u_{i,i+1}\) increase. The amplitude of the spectral lines at \(\omega=\omega_{n}\) is given by:  
\[A_{n}=\frac{2\pi\
u^{2}}{N^{2}}\operatorname{Re}\sum_{\alpha\alpha^{\prime}} S_{\alpha}^{*}(\omega_{n})S_{\alpha^{\prime}}(\omega_{n})C_{\alpha\alpha^{ \prime}}(\omega_{n}) \tag{11}\]  
where \(\
u\) and \(S_{\alpha}^{*}(\omega)S_{\alpha^{\prime}}(\omega)\) have been defined after Equation (5) and \(C_{\alpha\alpha^{\prime}}\) are the entries of the adjoint matrix: \(C(\omega_{n})=\operatorname{adj}(I-M(\omega_{n}))\). The amplitude given by Equation (11) holds only at the singular frequency values \(\omega=\omega_{n}\), when the determinant is zero. At frequencies \(\omega\) different from the singularities \(\omega_{n}\), the amplitude is equal to zero as expected for strictly periodic functions. Thus, the power spectral density of the periodic components is:  
\[\Phi_{\ell}(\omega)=\sum_{n=-\infty}^{\infty}A_{n}\ \delta(\omega-\omega_{n}) \tag{12}\]  
The amplitude \(A_{n}\) has been estimated for different values of the parameters \(\
u\), \(N\) and Fourier transform of the single pulse \(S_{\alpha}\) in [23]. The power spectral density of an arbitrary Markov-correlated pulse sequence Equation (5) was worked out by averaging over time from \(-\infty\) to \(\infty\), by assuming stationarity. In the case of a single closed-loop sequence, the power spectrum of a perfectly periodic signal is obtained, i.e., a line power spectrum taking discrete positive values at \(n\ \omega_{0}\) and zero at any other frequency, and no broadband noise is generated. The expression within square brackets in Equation (5) corresponds to the real part of the sum of \(N\) elements of the principal diagonal of the matrix \(K(\omega)\) multiplied by \(S_{\alpha}(\omega)S_{\alpha}(\omega)^{*}\) and by \(p_{\alpha}\), while the off-diagonal elements do not contribute. The real parts of the diagonal elements are equal to \(-1/2\), multiplied by \(p_{\alpha}=1/N\) and by \(2\), and give \(-\overline{|S(\omega)|^{2}}\), which summed to the first term \(\overline{|S(\omega)|^{2}}\) of the same equation, cancel each other.