Neural Network Maps as a Method for Constructing Mathematical Models

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Scientists from HSE University–Nizhny Novgorod and the Institute of Physics Belgrade, Serbia, are jointly exploring the application of machine learning techniques and neural networks to the study of nonlinear dynamics. Natalya Stankevich, Leading Research Fellow at the Laboratory of Topological Methods in Dynamics of the Faculty of Informatics, Mathematics, and Computer Science at HSE University–Nizhny Novgorod, spoke to the HSE News Service about this international project.

Marina Litvintseva

Director for Advanced Research at HSE University

'The project “Machine Learning and Nonlinear Dynamics: Intersection, Interplay, and Synthesis” was selected as one of the winners of the second HSE University International Academic Cooperation competition in 2025. The project is a collaboration between the Laboratory of Topological Methods in Dynamics at HSE University–Nizhny Novgorod and the Scientific Computing Laboratory at the Institute of Physics Belgrade.’

The theories developed by Russian and Serbian mathematicians and physicists make it possible to identify the key properties of dynamical systems and develop models that support the advancement of regenerative technologies for the treatment of complex cardiovascular diseases.

Natalya Stankevich

— Tell us how the project originated and evolved.

— In April 2025, the laboratory, together with the Institute of Physics Belgrade, which is part of the University of Belgrade and Serbia's National Research Institute, won in the HSE International Academic Cooperation competition with the project ‘Machine Learning and Nonlinear Dynamics: Intersection, Interplay, and Synthesis.’ Work on the project began in the summer of the same year.

— Why did you decide to join forces with your Serbian colleagues?

— We have built up extensive experience collaborating with colleagues from Belgrade, dating back to 2015, when Vladimir Klinshov, a researcher at our laboratory, met Prof. Igor Franović. They immediately generated numerous ideas for collaboration and have continued to develop these ideas ever since, resulting in more than ten joint publications. At the Scientific Computing Laboratory, Igor and his team conduct cutting-edge research on the effects of noise on dynamical systems, and their expertise in this area is extremely valuable to us. So, we decided to join forces.

— What are the key areas of your research?

— Our research focuses on machine learning methods, dynamical systems of various types, and complex phenomena within them, which we study using tools from dynamical systems theory as well as machine learning methods.

In 2020, together with Pavel Kuptsov, we began developing a research focus on the use of machine learning methods to construct mathematical models in the form of dynamical systems. We refer to such systems as neural network maps. One of the central goals of the project is to investigate the properties of these neural network maps.

A neural network map is a dynamical system represented by a network of artificial neurons with a relatively simple architecture, trained on data generated by a system of ordinary differential equations. This approach was first proposed by Pavel Kuptsov, Anna Kuptsova, and Nataliya Stankevich in the paper ‘Artificial Neural Network as a Universal Model of Nonlinear Dynamical Systems’ (Russian Journal of Nonlinear Dynamics, 17(1), 2021, pp. 5–21). The authors of the study implemented a simple neural network architecture consisting of two perceptrons and demonstrated that this approach can produce a neural network map that successfully reproduces the dynamics of benchmark nonlinear systems such as the Lorenz system, the Rössler system, and the Hindmarsh–Rose system. In the paper ‘Discovering Dynamical Features of a Hodgkin–Huxley-Type Model of a Physiological Neuron Using an Artificial Neural Network’ (Chaos, Solitons & Fractals 167 (2023), 113027) by Pavel Kuptsov, Nataliya Stankevich, and Elmira Bagautdinova, this method was extended to a model of a physiological neuron based on the Hodgkin–Huxley formalism. Neuron models are so-called fast–slow systems, meaning that they involve two widely separated time scales. In this case, a simple neural network architecture does not provide a sufficiently accurate machine learning model. However, a slight modification of the architecture resolves this issue.

— Is your work more focused on basic science or applied research?

— Within this project, we focus on basic science. Before a research team can move on to applied problems, it first needs well-established and validated methods. We therefore deliberately set out to study the instrumental properties of the neural network models we have developed, to explore the limits of their applicability, and to understand how effectively neural network maps 'absorb' the nonlinear properties of dynamical systems during training, as well as whether they are capable of reproducing nonlinear effects.

— What have you been able to achieve?

— One of the objectives of the first year of our project was to study the simplest form of bistability in neural network maps. Bistability is a widespread phenomenon in dynamical systems, characterised by the coexistence of multiple attractors in phase space, where the system may converge to different states depending on the initial conditions. A familiar example of bistability can be found in optical illusions or ambiguous images, where different patterns are perceived depending on viewing angle, distance, or other factors.

As part of the project, we studied models of simple radiophysical oscillators, such as the van der Pol oscillator, in which stable self-sustained oscillations and a stable equilibrium state can coexist. We considered three types of oscillators: a classical one, which does not exhibit bistability but undergoes an Andronov–Hopf bifurcation, in which the equilibrium loses stability and periodic self-sustained oscillations emerge; and two modified versions in which a subcritical Andronov–Hopf bifurcation is possible, giving rise to a region in parameter space where stable equilibria coexist with stable self-sustained oscillations. For all three oscillators, we trained neural network maps that successfully reproduced the dynamical regimes observed in the models, including regions of bistability.

One of the central questions we sought to answer was whether neural network maps can reliably reproduce bistable states depending on the dataset used for training. If the neural network is exposed to all possible states of the system during training, it will, of course, be able to reproduce them with relative ease. We created several training datasets in which the neural network was not exposed to the system’s final states. We found that when the network was trained on short trajectory segments sampled across the entire phase space, the resulting neural network map was still able to reproduce the system’s full dynamics with high accuracy. We then considered more challenging scenarios by removing from the training data a region of phase space containing one of the coexisting attractors. In this case, the performance of the neural network map deteriorated. However, when the data exclusion was less severe, the dynamics were reproduced more accurately, and the machine learning model was able to identify the second attractor. This, in turn, made it possible to investigate codimension-two bifurcations, which the neural network model was also able to reproduce successfully.

Based on these results, we prepared a paper that was accepted for publication in Chaos, an A-ranked journal, on May 1. We are currently conducting a similar study on a mathematically analogous dynamical system based on a Hodgkin–Huxley-type neuron model.

— Are there any results from your work that have practical significance?

— The laboratory is involved in several other projects with more explicitly applied objectives. We aim to translate insights gained from our basic research into practical applications. We also maintain close collaboration with colleagues working on applied problems to better understand how our theoretical findings can be put to use.

For example, one of the objectives of our project on neural network maps is to investigate whether they can exhibit the phenomenon of synchronisation. In its classical form, synchronisation refers to the adjustment of the frequencies of interacting self-oscillatory systems. If two self-oscillating oscillators with slightly different frequencies are coupled, increasing the coupling strength can lead to the emergence of oscillations at a common frequency. As part of the project, we plan to explore whether neural network maps are capable of reproducing this type of behaviour.

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This objective appears promising for developing digital twins of the electrical activity of cells. It should be noted that synchronisation is a widespread phenomenon across many fields. Living organisms are full of oscillatory processes operating on multiple time scales. A classic example of a rhythm that we all experience is the heartbeat, which reflects the coordinated oscillatory activity of cardiac cells—cardiomyocytes—in heart tissue. In certain cardiovascular diseases, cardiomyocytes die and are replaced by non-conductive tissue, known as fibroblasts, which can lead to serious cardiac dysfunction. Regenerative technologies based on stem cells are currently being developed with the aim of replacing such non-conductive regions. A key challenge in advancing these approaches is ensuring that the newly generated cells can operate synchronously with the surrounding tissue. We believe that, in the future, neural network maps derived from experimental data could be used to simulate interactions between different cells and to investigate the possibility of their synchronous activity.

We are actively exploring potential applications where our methods could be used. One of our strategies for identifying such opportunities is to participate in conferences with a strong applied focus. This summer, a small group of project members plans to attend the Biomedical Data Science Conference in Budapest, where they will present the results of their work and also learn about cutting-edge developments in biological and medical research based on large-scale data analysis. We hope that the conference will help us identify promising applications and formulate new research objectives.