## 4 Concluding Remarks  
A mathematical framework, leading to the unified description of the discrete and continuous power spectral components of the EEG and MEG signals, was proposed. The model is based on the assumption that the complex network of interacting neurons spontaneously form closed-loops chains of \(N\) firing neurons, giving rise to the repetitive emission of electric and magnetic impulses. By describing the electric and magnetic impulses generated by the firing of a neuron as Gaussians of a given amplitude and width, the line power spectrum of this periodic function is obtained. Fundamental and harmonic components are shown to depend on the distribution of the amplitude and variance in the emitted impulses and on the distribution of the time intervals between their emission, which are expressions of the \(N\) states \(\alpha\) of the sequence. A general matrix equation gives the line power spectrum. Furthermore, by considering the fluctuations in the number \(N_{i}\) of synchronized firing neurons belonging to different closed loops, a broadband component emerges of the power spectrum. The syncronized pulses are represented as Gaussian time functions whose amplitude and variance depend on the degree of syncronization of the firing of the neurons belonging to different single loops but in the same state, \(\alpha\). As discussed in Section 3, the fluctuations in the number of synchronized loops \(N_{i}\) may change the amplitude of the signal and its waveform, which is determined by the harmonic components generated by the loops. It is worthy of note that, even if the fluctuations change the waveform of the signal, the frequency of the wave remains unchanged, as it depends only on the duration of the loop \(\sum_{i}u_{i}\) which is unchanged by the fluctuations in \(N_{i}\). An additional effect of the broadband component concerns the onset of a cut off in the line power spectrum, which is related to the width of the electric impulses represented by the variance in the Gaussian time functions. A larger variance implies a lower cut-off frequency and smaller number of harmonic components, thus filtering out a signal closer and becoming closer to a pure sinusoid. The main features explained by the proposed model are summarized here, below. The broad range of discrete frequencies observed in the EEG and MEG signals is obtained by considering different durations of closed-loops sequences of firing neurons. By assuming an average firing interval between successive neurons of \(5\cdot 10^{-3}\) s, sequences ranging from a few hundred neurons to only a few neurons cover the range from 0.5 Hz (the lowest frequency of _delta_ waves) to 50 Hz (the highest frequency of _gamma_ waves). The presence of several harmonic components of the fundamental sinusoidal wave in the graph of EEG is also accounted for, particularly at low frequency, as the _delta_ waves, where a pure sinusoid is never observed. High-frequency waves are less distorted since harmonics stand at frequencies multiple of the fundamental one, and the cut-off frequency of the detected power spectrum cuts the harmonics exceeding that frequency. In a few particular cases, all harmonics are cut off, or strongly reduced, giving a nearly pure sinusoidal wave. Part of the distortion can be due to the broadband noise created by the fluctuation in the number of \(N_{i}\). One of the effects discussed above, and observed in almost all EEG graphs, is the continuous change in the waveform, practically at every period, of the received signal produced by the loops. This change, which is associated to a heavy distortion of the fundamental wave, is due to the presence of several harmonic waves within the periodic signal. A change in the amplitude or the phase of one or a few of these harmonics during the fluctuation in \(N_{i}\) is enough to change the waveform over the period.