Algebraic Iterative Schemes of Matrices in Models of Biological Systems – Problems and Applications (AlgebMIS)

Project no.: PDN1/18

Project description:

Iterative schemes are widely used and applied in dynamical systems. There are many scalar iterative maps that describe a single node. Nodes can be connected in a chain, cycle or grating and form a coupled map lattice. But instead of joining nodes into a lattice the replacement of scalar variable by second order square matrix in a single node model was proposed in 2011. The logistic map of matrices was introduced. Later, the whole class of iterative maps of square matrices of order 2 was presented. Iterative maps of matrices exhibited the effect of explosive divergence which is not common to any extension of iterative map or coupled map lattice. This effect served as a natural motivation to form promising areas of scientific research and applications.
The project can be divided into three main research objectives:
1. Coupled map lattices when a model of a single node is iterative map of matrices of order two. Coupling of iterative maps of matrices is completely new (has no analogous in literature) field of research.
2. Vagal nerve stimulation data analysis using matrix theory-based techniques. This objective is related to identification of differences in stimulation effects between healthy persons and in diabetics. This task is based on real world problem.

3. Investigation of synchronization between heart rate variability and other ECG parameters using delayed matrix technique.

Project funding:

KTU Business Support Fund


Project results:

The original 2-dimensional coupled map lattice of matrices was presented where each node is related to its four adjacent neighbors. Scalar nodal variables are replaced by a nilpotent matrix of order 2. Analytical expressions of simplified nilpotent model of the logistic coupled map lattice of matrices were derived. The digital image hiding scheme that does not require the difference image was proposed. Moreover, complex networks (regular, feed-forward, random, small-world) of coupled maps of matrices were investigated. It was demonstrated that chimera states of spatiotemporal divergence do exist in such networks. The visualization of chimera states of spatiotemporal divergence for networks of different structure in the plane of network coupling and network density parameters was provided. The technique based on delayed matrices was proposed to investigate the synchronization between two signals. Numerical experiments were performed using data from neuron models, Rössler equations as well as vagal nerve stimulation data.
Project results have been validated by publications in high-level scientific journals.

Period of project implementation: 2018-09-17 - 2019-09-16

Project coordinator: Kaunas University of Technology

Head:
Rasa Šmidtaitė

Duration:
2018 - 2019

Department:
Department of Applied Mathematics, Faculty of Mathematics and Natural Sciences

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