Building a mathematical model and an algorithm for training a neural network with sparse dipole synaptic connections for image recognition




mathematical model, neural network, sparse dipole synaptic connections, image recognition


Large enough structured neural networks are used for solving the tasks to recognize distorted images involving computer systems. One such neural network that can completely restore a distorted image is a fully connected pseudospin (dipole) neural network that possesses associative memory. When submitting some image to its input, it automatically selects and outputs the image that is closest to the input one. This image is stored in the neural network memory within the Hopfield paradigm. Within this paradigm, it is possible to memorize and reproduce arrays of information that have their own internal structure.

In order to reduce learning time, the size of the neural network is minimized by simplifying its structure based on one of the approaches: underlying the first is «regularization» while the second is based on the removal of synaptic connections from the neural network. In this work, the simplification of the structure of a fully connected dipole neural network is based on the dipole-dipole interaction between the nearest adjacent neurons of the network.

It is proposed to minimize the size of a neural network through dipole-dipole synaptic connections between the nearest neurons, which reduces the time of the computational resource in the recognition of distorted images. The ratio for weight coefficients of synaptic connections between neurons in dipole approximation has been derived. A training algorithm has been built for a dipole neural network with sparse synaptic connections, which is based on the dipole-dipole interaction between the nearest neurons. A computer experiment was conducted that showed that the neural network with sparse dipole connections recognizes distorted images 3 times faster (numbers from 0 to 9, which are shown at 25 pixels), compared to a fully connected neural network

Author Biographies

Vasyl Lytvyn, Lviv Polytechnic National University

Doctor of Technical Sciences, Professor

Department of Information Systems and Networks

Roman Peleshchak, Lviv Polytechnic National University

Doctor of Physical and Mathematical Sciences, Professor

Department of Information Systems and Networks

Ivan Peleshchak, Lviv Polytechnic National University

Postgraduate Student

Department of Information Systems and Networks

Oksana Cherniak, Drohobych Ivan Franko State Pedagogical University

Postgraduate Student

Department of Mathematics

Lyubomyr Demkiv, Lviv Polytechnic National University

Doctor of Technical Sciences, Professor

Department of Information Systems and Networks


Peleshchak, I., Peleshchak, R., Lytvyn, V., Kopka, J., Wrzesien, M., Korniak, J. et. al. (2020). Spectral Image Recognition Using Artificial Dynamic Neural Network in Information Resonance Mode. Artificial Intelligence and Industrial Applications, 313–322. doi:

Lytvyn, V., Peleshchak, I., Peleshchak, R., Holoshchuk, R. (2018). Detection of multispectral input images using nonlinear artificial neural networks. 2018 14th International Conference on Advanced Trends in Radioelecrtronics, Telecommunications and Computer Engineering (TCSET). doi:

Greenberg, S., Guterman, H. (1996). Neural-network classifiers for automatic real-world aerial image recognition. Applied Optics, 35 (23), 4598. doi:

Andriyanov, N. A., Dementiev, V. E., Kargashin, Y. D. (2021). Analysis of the impact of visual attacks on the characteristics of neural networks in image recognition. Procedia Computer Science, 186, 495–502. doi:

Simard, P. Y., Steinkraus, D., Platt, J. C. (2003). Best practices for convolutional neural networks applied to visual document analysis. Seventh International Conference on Document Analysis and Recognition, 2003. Proceedings. doi:

Zhou, Y., Song, S., Cheung, N.-M. (2017). On classification of distorted images with deep convolutional neural networks. 2017 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP). doi:

Ha, M., Byun, Y., Kim, J., Lee, J., Lee, Y., Lee, S. (2019). Selective Deep Convolutional Neural Network for Low Cost Distorted Image Classification. IEEE Access, 7, 133030–133042. doi:

Li, B., Tian, M., Zhang, W., Yao, H., Wang, X. (2021). Learning to predict the quality of distorted-then-compressed images via a deep neural network. Journal of Visual Communication and Image Representation, 76, 103004. doi:

Guan, X., Li, F., He, L. (2020). Quality Assessment on Authentically Distorted Images by Expanding Proxy Labels. Electronics, 9 (2), 252. doi:

Peleshchak, R., Lytvyn, V., Peleshchak, I., Vysotska, V. (2021). Stochastic Pseudo-Spin Neural Network with Tridiagonal Synaptic Connections. 2021 IEEE International Conference on Smart Information Systems and Technologies (SIST). doi:

Slyadnikov, E. E. (2007). Fizicheskaya model' i associativnaya pamyat' dipol'noy sistemy mikrotrubochki citoskeleta. Zhurnal tehnicheskoy fiziki, 77 (7), 77–86. Availale at:

Slyadnikov, E. E. (2011). Fizicheskie osnovy, modeli predstavleniya i raspoznavaniya obrazov v mikrotrubochke citoskeleta neyrona. Zhurnal tehnicheskoy fiziki, 81 (12). Availale at:

Penrouz, R. (2005). Teni razuma: v poiskah nauki o soznanii. Moscow-Izhevsk: IKI, 688. Availale at:

Hameroff, S. R. (1994). Quantum coherence in microtubules: A neural basis for emergent consciousness? Journal of Consciousness Studies, 1 (1), 91–118. Availale at:

Brown, J. A., Tuszynski, J. A. (1999). A review of the ferroelectric model of microtubules. Ferroelectrics, 220 (1), 141–155. doi:

Tuszyński, J. A., Hameroff, S., Satarić, M. V., Trpisová, B., Nip, M. L. A. (1995). Ferroelectric behavior in microtubule dipole lattices: Implications for information processing, signaling and assembly/disassembly. Journal of Theoretical Biology, 174 (4), 371–380. doi:

Stebbings, H. (1995). Microtubule-based intracellular transport of organelles. The Cytoskeleton: A Multi-Volume Treatise, 113–140. doi:

Hopfield, J. J. (1982). Neural networks and physical systems with emergent collective computational abilities. Proceedings of the National Academy of Sciences, 79 (8), 2554–2558. doi:

Yurkovych, N. V., Herasimov, O. V., Yurkovych, V. M., Mar’yan, M. I. (2014). Composition of neural networks by hebb algorithm and direct spreading in characters encoding systems. Uzhhorod University Scientific Herald. Series Physics, 36, 161–167. Availale at:

Chernіak, O., Peleshchak, R., Doroshenko, M. (2020). Reduction of display time of input images by pseudo-spin neural network due to rarefaction of synaptic connections. Modern problems in science. Abstracts of VIII International Scientific and Practical Conference. Prague, 680–686. Availale at:



How to Cite

Lytvyn, V., Peleshchak, R., Peleshchak, I., Cherniak, O., & Demkiv, L. (2021). Building a mathematical model and an algorithm for training a neural network with sparse dipole synaptic connections for image recognition. Eastern-European Journal of Enterprise Technologies, 6(4 (114), 21–27.



Mathematics and Cybernetics - applied aspects