Comparison Of Characteristic Features By Implementing The Hindmarsh-Rose (Hr) Neuron Model With Google Colab, Op-Amp And Arm Controllers

Abstract

Today, artificial intelligence technologies are advancing day by day, and neurons, which are the building blocks of artificial neural networks, one of the artificial intelligence technologies, are supporting this advancement. By studying the biological structures of neurons found in the human brain, dynamic models are created using mathematical differential equations. These models are implemented using analogue circuit elements or digital controllers such as computers to investigate their dynamic properties. Within the scope of this study, time series, phase portraits, bifurcation diagrams, and Lyapunov exponent spectrum analyses were performed using Google Colab, a new generation numerical analysis development environment, to reveal the dynamic properties of the Hindmarsh-Rose (HR) neuron model, which is effectively used in neuron modelling. In Google Colab, solutions were obtained using the Euler, Heun, RK4, solve_ivp, Adams-Bashforth/Moulton, and Z-transform methods in the numerical analysis solutions of the HR model; it was determined that the RK4 method, which has sufficient speed and high accuracy, is more suitable for microcontroller applications. Subsequently, an HR analogue circuit model was designed using Op-Amp operational elements in the OrCAD PSpice program, and simulations were performed to validate the Colab dynamic numerical analysis results. Finally, simulation results were obtained using a microcontroller with a Cortex-A72 (ARM v8) processor, which confirmed the Colab dynamic RK4 numerical analysis results of the HR neuron model. The applicability of membrane potential, fast recovery, and slow adaptation conditions to mini-systems for data processing was demonstrated.

Keywords

• Hindmarsh-Rose model
• Numerical analysis
• Analog circuit
• Simulation
• Microcontroller

Abstract

Today, artificial intelligence technologies are advancing day by day, and neurons, which are the building blocks of artificial neural networks, one of the artificial intelligence technologies, are supporting this advancement. By studying the biological structures of neurons found in the human brain, dynamic models are created using mathematical differential equations. These models are implemented using analogue circuit elements or digital controllers such as computers to investigate their dynamic properties. Within the scope of this study, time series, phase portraits, bifurcation diagrams, and Lyapunov exponent spectrum analyses were performed using Google Colab, a new generation numerical analysis development environment, to reveal the dynamic properties of the Hindmarsh-Rose (HR) neuron model, which is effectively used in neuron modelling. In Google Colab, solutions were obtained using the Euler, Heun, RK4, solve_ivp, Adams-Bashforth/Moulton, and Z-transform methods in the numerical analysis solutions of the HR model; it was determined that the RK4 method, which has sufficient speed and high accuracy, is more suitable for microcontroller applications. Subsequently, an HR analogue circuit model was designed using Op-Amp operational elements in the OrCAD PSpice program, and simulations were performed to validate the Colab dynamic numerical analysis results. Finally, simulation results were obtained using a microcontroller with a Cortex-A72 (ARM v8) processor, which confirmed the Colab dynamic RK4 numerical analysis results of the HR neuron model. The applicability of membrane potential, fast recovery, and slow adaptation conditions to mini-systems for data processing was demonstrated.

References

1. Pan, I., and Das, S., “Evolving chaos: identifying new attractors of the generalized Lorenz family”, Applied Mathematical Modelling, 57, 391-405. (2018). DOI: https://doi.org/10.1016/j.apm.2018.01.015
2. Wang, J., and Sun, S., “Comment on the paper ‘Identifying critical nodes in complex networks based on distance Laplacian energy’”, Chaos, Solitons and Fractals, 187, 115343, (2024). DOI: https://doi.org/10.1016/j.chaos.2024.115343
3. Demir M., and Köker R., “Development of Brain Computer Interface Software Based on Detection of EEG Signals with New Generation Sensors”, 17th International Conference on Engineering and Natural Sciences, /Prague, Czechia, May 03 -07,2025, S.207-215, (2025). DOI: https://doi.org/10.5281/zenodo.15673348
4. Demir M., “Control of Unmanned Aerial Vehicles Using EEG Signals with Artificial Neural Networks”, Master's Thesis, Sakarya University, Faculty of Technology, Department of Biomedical Engineering, Sakarya, 93s, (2019).
5. Hindmarsh J.L., and Rose R. M., “A model of neuronal bursting using three couple first order differential equations”, Proc. R. Soc. Lond. Biol. Sci., 22: 87-102, (1984). DOI: https://doi.org/10.1098/rspb.1984.0024
6. Storace, M., Linaro, D., and de Lange, E., “The Hindmarsh-Rose neuron model: Bifurcation analysis and piecewise-linear approximations”, Chaos, 18(3), 033128, (2008). DOI: https://doi.org/10.1063/1.2975967
7. Li Y., and Cao H., “Bifurcation and Comparison of a Discrete-Time Hindmarsh-Rose Model”, Journal of Applied Analysis and Computation, 13(1): 34-56, (2023). DOI: https://www.jaac-online.com/article/doi/10.11948/20210204
8. Pehlivan, İ., Özdemir, A., Güleryüz, E., Kalaycı, O., and Akgül, A., “Fundamental dynamic analyses and analog circuit simulations of the Hindmarsh-Rose biological neuron model”, 5th International Symposium on Innovative Technologies in Engineering and Science (ISITES2017), Baku, Azerbaijan, (2017).

9. McCulloch, W.S., and Pitts, W., “A logical calculus of ideas immanent in nervous activity”, Bulletin of Mathematical Biophysics 5, 115-133, (1943). DOI: https://doi.org/10.1007/BF02478259
10. Hodgkin A. L., Huxley F. A. “A quantitative description of membrane current and its application to conduction and excitation in nerve”, Journal of Physiology, 117(4): 500-544, (1952). DOI: https://doi.org/10.1113/jphysiol.1952.sp004764
11. Fitzhugh R., “Mathematical models for excitation and propagation in nerve”, Biological Engineering, pp. 1-85, (1969).
12. Nagumo, J., and Sato, S., “On a response characteristic of a mathematical neuron model”, Kybernetik, 10, 155–164, (1972), DOI: https://doi.org/10.1007/BF00290514
13. Izhikevich E.M., “Simple model of spiking neurons”, IEEE Transactions on Neural Networks. 14(6): 1569-1572, (2003). DOI: https://doi.org/10.1109/TNN.2003.820440
14. Zhang, S., Yao, W., Xiong, L., Wang, Y., Tang, L., Zhang, X., and Yu, F., “A Hindmarsh-Rose neuron model with electromagnetic radiation control for the mechanical optimization design”, Chaos, Solitons and Fractals, 187, 115408, (2024). DOI https://doi.org/10.1016/j.chaos.2024.115408
15. Ponomarenko, V. I., and Prokhorov M. D., “Experimental realization of neuron-like activity generators”, 2024 8th Scientific School Dynamics of Complex Networks and their Applications (DCNA), (2024). DOI: https://doi.org/10.1109/DCNA63495.2024.10718569
16. Coşkun, S., Pehlivan, İ., Akgül, A., and Gürevin, B., “A new computer-controlled platform for ADC-based true random number generator and its applications”, Turkish Journal of Electrical Engineering and Computer Sciences, 27(2): 847–860, (2019). DOI: https://doi.org/10.3906/elk-1806-167
17. Xie, J., and Grancharova, A., “Nonlinear model predictive control based on deep learning for Hindmarsh-Rose neuron model”, Proceedings of 2024 IEEE 12th International Conference on Intelligent Systems (IS), (2024). DOI: https://doi.org/10.1109/IS61756.2024.10705274
18. Cai, J., Bao, H., Chen, M., Xu, Q., and Bao, B., “Analog/Digital multiplierless implementations for nullcline-characteristics-based piecewise linear Hindmarsh-Rose neuron model”, IEEE Transactions on Circuits and Systems-I: Regular Papers, 69(7): 2916–2927, (2022). DOI: https://doi.org/10.1109/TCSI.2022.3164068
19. Koparal, F. D., Aslan, N., Saltan, M., Özdemir, Y., and Demir, B., “Geometrical construction of a chaotic horseshoe-like map on Cantor dust C×C×C”, Chaos, Solitons and Fractals, 187, 115382, (2024). DOI: https://doi.org/10.1016/j.chaos.2024.115382
20. Sprott, J. C.,“Some simple chaotic flows”, Physical Review E, 50(2):647–650, (1994). DOI: https://doi.org/10.1103/PhysRevE.50.R647
21. Jenderny, S., Ochs, K., Gibson, M., and Hövel, P., “A simplified Hindmarsh-Rose model based on power-flow analysis”, 21st IEEE Interregional NEWCAS Conference (NEWCAS), (2023). DOI: https://doi.org/10.1109/NEWCAS57931.2023.10198053
22. Lu, J., Kim, Y.-B., and Ayers, J., “A low power 65nm CMOS electronic neuron and synapse design for a biomimetic micro-robot”, 2011 IEEE 54th International Midwest Symposium on Circuits and Systems (MWSCAS), (2011). DOI: https://doi.org/10.1109/MWSCAS.2011.6026468
23. Bao, H., Hu, A., Liu, W., and Bao, B., “Hidden bursting firings and bifurcation mechanisms in memristive neuron model with threshold electromagnetic induction”, IEEE Transactions on Neural Networks and Learning Systems, 31(2): 502–516, (2020). DOI: https://doi.org/10.1109/TNNLS.2019.2905137
24. Çam, İ., and Tek, F. B., “Focused neuron”, 2017 25th Signal Processing and Communications Applications Conference (SIU), (2017). DOI: https://doi.org/10.1109/SIU.2017.7960632
25. Hruboš, Z., Götthans, T., and Petržela, J.,“Circuit realization of the inertia neuron”, Proceedings of the 21st International Conference Radıoelektronıka, (2011). DOI: https://doi.org/10.1109/RADIOELEK.2011.5936435
26. Tuna, M., Alçın, M., Koyuncu, İ., Fidan, C. B., and Pehlivan, İ., “High speed FPGA-based chaotic oscillator design”, Microprocessors and Microsystems, 66: 72–80, (2019). https://doi.org/10.1016/j.micpro.2019.02.012
27. Tlelo-Cuautle, E., Díaz-Muñoz, J. D., González-Zapata, A. M., Li, R., León-Salas, W. D., Fernández, F. V., Guillén-Fernández, O., and Cruz-Vega, I., “Chaotic Image Encryption Using Hopfield and Hindmarsh-Rose Neurons Implemented on FPGA”, Sensors, 20(5), (2020). DOI: https://doi.org/10.3390/s20051326
28. Pano-Azucena, A. D., Ovilla-Martinez, B., Tlelo-Cuautle, E., Muñoz-Pacheco, J. M., and de la Fraga, L. G., “FPGA-based implementation of different families of fractional-order chaotic oscillators applying Grünwald-Letnikov method”, Communications in Nonlinear Science and Numerical Simulation, 72(11): 516-527, (2019). DOI: https://doi.org/10.1016/j.cnsns.2019.01.014
29. Köse, E., and Mühürcü, A. “Realization of a digital chaotic oscillator by using a low cost microcontroller”. Engineering Review, 37(3): 341–348. (2017).
30. Dayan, P., and Abbott, L. F., “Theoretical neuroscience: Computational and mathematical modeling of neural systems”, MIT press, (2001)
31. Fitzhugh, R.,“Impulses and physiological states in theoretical models of nerve membrane”, Biophysical Journal,1(6): 445–466,(1961). DOI: https://doi.org/10.1016/S0006-3495(61)86902-6
32. Kıran, H. E., “A novel chaos-based encryption technique with parallel processing using CUDA for mobile powerful GPU control center”, Chaos and Fractals, 1(1): 6–18, (2024). DOI: https://doi.org/10.69882/adba.chf.2024072
33. Emin, B., and Yaz, M.,“Digital implementation of chaotic systems using Nvidia Jetson AGX Orin and custom DAC converter”, Chaos and Fractals, 1(1): 38–41, (2024).DOI: https://doi.org/10.69882/adba.chf.2024075
34. Makouo, L., Metsebo, J., Ainamon, C., Abba, O. A., and Chamgoué, A. C.,“Microcontroller execution, random number generation and chaos annihilation in piecewise quadratic map”, Computer Science, 3(1): 13–17, (2026). DOI: https://doi.org/10.69882/adba.cs.2026013