Because of their nature, biomarkers for neuropsychiatric diseases were out of the reach of medical diagnostic technology until the past few decades. In recent years, the confluence of greater, affordable computer power with the need for more efficient diagnoses and treatments has increased interest in and the possibility of their discovery. This review will focus on the progress made over the past ten years regarding the search for electroencephalographic biomarkers for neuropsychiatric diseases. This includes algorithms and methods of analysis, machine learning, and quantitative electroencephalography as applied to neurodegenerative and neurodevelopmental diseases as well as traumatic brain injury and COVID-19. Our findings suggest that there is a need for consensus among quantitative electroencephalography researchers on the classification of biomarkers that most suit this field; that there is a slight disconnection between the development of increasingly sophisticated methods of analysis and what they will actually be of use for in the clinical setting; and finally, that diagnostic biomarkers are the most favored type in the field with a few caveats. The main goal of this state-of-the-art review is to provide the reader with a general panorama of the state of the art in this field.

This article presents a modified Duffing system based on Lienards Theorem and the integral of Melnikov, the first is used to propose the interpolation Sinc as a non-linear damping function and the second is used to assure an asymptotically stable limit cycle. The Sin-Duffing system is driven into chaos by using its corresponding bifurcation diagram, Lyapunov exponents, and the Theory of Melnikov. Furthermore, the system is placed in a critical state which produced chaotic and periodic sequences, driving it into a regimen of intermittence between chaos and the self-sustained oscillations near the stable limit cycle. Intermittence is achieved by searching and tuning all involved parameters when a very systematic procedure is used. Also, such a regimen is presented here as a useful mechanism to estimate the frequency of a very low weak signal for detection applications. The latest is made possible because the system capabilities to distinguish the intermittent periods were strengthened by a new method based on Melnikovs function that only depends on the most influential parameter in the type-Lienard system. The complete system formed by the new Sinc-Duffing oscillator showed higher sensitivity compere to other chaotic systems such as the traditional Duffing or the Van der Pol-Duffing for weak signal detection with a signal-to-noise ratio down to -70 dB.

A non-linear Liènard-type system based on trigono-metric functions is designed and used as a highly sensitive signal detector. Based on the intermittence mechanism presented in some chaotic systems, an adaptive detection array formed by five chaotic oscillators is designed, tuned, and tested. In the array, each individual oscillator identifies changes in frequency for non-stationary weak signals and, with the array working in an adaptative mode, gives an estimate of the timing at which intermittence occurred allowing the representation of a time-frequency spectrogram even when high-level noise is present. This work also shows that the trigonometric based system has an asymptotically stable limit cycle centered at the origin through two methods: first under the fixed-point analysis and later by the Melnikov method that uses a family of periodic orbits. Once the self-sustained oscillations are produced, the system is configured to drive itself into a saddle-node bifurcation through an iterative method for searching and tuning parameters which, in turn, assures the route to chaotic intermittence. Through this mechanism, the new system can be used as a non-stationary signal detector that also provides a high resolution time-frequency representation. The computer experiments showed that the adaptive array is perfectly compatible with other second-order non-linear oscillators configured in the chaotic intermittence regime such as Duffing, Van der Pol, and Van der Pol - Duffing, achieving high detection rates under noisy conditions. Moreover, when the new Liènard-type oscillators are incorporated into the adaptive array, the detection sensitivity overcomes the SNR threshold present in all oscillators mentioned above and allows for reaching and maintaining a relative error around 1% under severe noise conditions down to about -60 dB SNR.