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We investigate different dynamical regimes of a small neuronal circuit. This circuit includes two cells that are interconnected by means of dynamical synapses (excitatory and inhibitory). On the individual level, each neuron is modelled by FitzHugh–Nagumo equations. To analyze complex patterns and transitions between them in this small circuit, we apply wavelet analysis. We analyse the influence of synaptic kinetics on the synchronization in this circuit. We also show that the wavelet analysis could be applicable to systems that are difficult to investigate using methods of classical bifurcation analysis. [ABSTRACT FROM AUTHOR]

We propose an approach to the fast numerical evaluation of the continuous wavelet transform with the Morlet wavelet based on the replacement of the Gaussian factor in the latter by a hierarchical sequence of B-splines and their application to the image formed by the analyzed signal multiplied by a set of periodic functions. We show that the suggested approach results in a significant reduction of the resulting data storage (almost linear dependence on the sample length vs. the quadratic one required by conventional methods). In order to illustrate the workability of the proposed technique we applied it to the analysis of the data on the blood flow velocity data obtained in vivo from chorioallantoic membrane of chicken embryo, which exhibit non-stationarity, bi- or multirhythmicity. [ABSTRACT FROM AUTHOR]

We propose two approaches to the analysis of sparse stochastic data, which exhibit a power-law dependence between their first and second moments (Taylor's law), and determining the respective power index, when it has a value between 1 and 2. They are based on the analysis of components of the Haar wavelet expansion. The first method uses a dependence between the first iterations of averaging and wavelet coefficients with the subsequent studying the statistics of zero paddings on the line, which corresponds the linear dependence between two first moments of the analysed distributions. The second method refers to Taylor's plot formed by the full set of wavelet coefficients. It is discussed that such representations provide also an opportunity to check time–scale stability of analysed data and distinguish between particular cases of the Tweedie probabilistic distribution. Both simulated series and real marine species abundance data for five spatial regions of the Pacific are used as example illustrating an applicability of these approaches. • The Haar wavelet transform allow calculating the Tweedie power index. • Taylor(Tweedie)'s law can be distinguished from the Gamma-distributed data with zeros. • Real fish abundance data are analysed as an example. [ABSTRACT FROM AUTHOR]

We propose an approach to analysing single trajectories of a particle, which moves randomly on a landscape distinct parts of which result in sufficiently various diffusion coefficients. The method based on the mapping the cumulative sum of step-wise elementary displacement squared into a complex oscillating function with the subsequent continuous wavelet analysis of the later allows the localization of relatively homogeneous sub-regions and determining values of the diffusion coefficient within each such sub-region. This approach is applied to demonstrable test examples as well as to the reconstruction of the biological membrane's properties from the real data of single molecule walk tracing. • Revealing patches with different diffusivity from a single particle track. • An improved continuous wavelet transform for exploring random walks. • Processing a case study of real biological membrane visited by a walking molecule. [ABSTRACT FROM AUTHOR]

We investigated phases and phase differences between coupled mechanical and chemical self-oscillations running in two kinds of newly synthesised chemomechanical gels. Processing of experimental data was model-free and it was based on spectral and wavelet approaches of the theory of dynamical systems. It was applied to samples of different sizes and compositions (Ru- and Fe-based gels). It was revealed that the Ru-based gels demonstrate a faster mechanical response to the redox changes of the Ru-moiety coupled to the Belousov-Zhabotinsky chemical reaction than the Fe-based gels. This demonstrated a significant phase difference between the redox changes of the Fe-moiety and the mechanical oscillations. Special attention was paid to the phase-coupling features for averaged higher harmonics, which are significant due to the high degree of nonlinearity of the chemomechanical oscillatory process. • New chemomechanical gels with Ru- and Fe-based catalysts are investigated. • The phase difference between coupled chemical and volume oscillations is traced. • A method to address phase compounds of highly nonlinear processes in liquids and soft gel's components. [ABSTRACT FROM AUTHOR]