A minimization problem for the sum of weighted convolutions’ difference and a novel approach to the inverse problem of ECG- and PPG-signals recovering

In this paper, we consider an unstudied problem of approximation of an observed pulse train by by a quasiperiodic signal generated by a pulse with a given pattern (reference) shape. The quasiperiodicity allows variation of time intervals between repetitions of the pattern pulse, as well as nonlinear expansions of the pattern in time. Such inverse problems are common in electrocardiogram (ECG) and photoplethysmogram (PPG) features extraction. The following two variants of the problem are considered. In the first variant, the number of the pulse repetitions is unknown, while in the second one, that number is given. The polynomial-time solvability of the both variants of the problem is constructively proved.

Weak Quasiperiodic Signal Propagation through Multilayer Feed-Forward Hodgkin–Huxley Neuronal Network

Quasiperiodic signal is ubiquitous and entrenched in neuronal networks, and thus taking it into consideration is necessary. The Wiener process with the intensity of σ is used here to model randomly fluctuated phase in external weak quasiperiodic signal. The departure from the normal periodicity can be governed by the parameter σ. Then, the effects of randomly fluctuated phase of signal and time-periodic coupling intensity of synaptic junctions between neurons on propagation of weak quasiperiodic signal through feed-forward Hodgkin–Huxley network are explored in detail. Increasing σ makes more neurons fire simultaneously, and better synchronous state is observed. Consequently, the external weak quasiperiodic signal introduced into all neurons in the first layer can be effectively transmitted through the whole feed-forward network via synchronization mechanism. In the case of time-periodic synaptic coupling intensity, when oscillatory frequency of synaptic coupling intensity is equal precisely to average frequency of external quasiperiodic signal, the propagation of weak quasiperiodic signal is optimal. Additionally, rapid oscillation of synaptic coupling intensity hinders or even kills the propagation of quasiperiodic signal to great depths of neuronal network, provided σ is not large enough.