The rotation-transition procedure of the Fitzhugh-Nagumo neuron model and its hardware verification

Yıl 2024, Cilt: 30 Sayı: 3, 316 - 323, 29.06.2024

Nimet Korkmaz

Öz

The biological neuron models, which have the biologically significant, describe the characteristics of neurons in the living body. These models can be defined similar to oscillators. A great of the theorems that describe the characteristics of oscillator structures, such as stability control and synchronization control, can also be used to examine the biological neuron models. Recently, the rotation-transition process has become a remarkable issue in the nonlinear dynamical system applications. After the rotation-transition process; the dynamical attractor of a nonlinear system can be directed to any desired direction by changing the rotation angle. One of the most known examples of the nonlinear dynamical systems is the chaotic oscillator structures. There are many studies on the dynamical attractor control of the chaotic oscillators by means of rotation-transition in the literature. However; although the rotation changes are observed in the dynamical characteristics of the real biological systems, there isn’t any study dealing with the rotation controls of the dynamical attractors of biological neuron models. Therefore, the rotation-transition procedure of the Fitzhugh-Nagumo (FHN) model has been handled in this study. The equilibrium points of the rotated FHN neuron model are calculated for getting its characteristic outputs. After the rotation-transition process, the changes on the rotation of the dynamic attractors of the FHN neuron have been observed by numerical simulation results. Finally, the rotated-controlled FHN neuron has also been realized with the ‘Field Programmable Gate Array- (FPGA)’, which is a programmable and reconfigurable device, in order to both support the functionality of the rotation transformation process and to obtain the real-time signals requiring for the bio-inspired systems. Thus, it has been shown that thanks to the proposed rotation-transition process, the phase adjustment of the system dynamics in neural systems can be intervened without requiring any coupling definition. Based on this view; the mathematical descriptions of the rotated-FHN neuron model has been pointed out, this model is promoted by the numerical simulations and confirmed by the hardware implementation studies.

Anahtar Kelimeler

Fitzhugh-Nagumo neuron model, Rotation-Transition method, Euler rotation theorem, Field programmable gate array (FPGA)

Fitzhugh-Nagumo nöron modelinin rotasyon dönüşüm prosedürü ve donanım doğrulaması

Öz

Biyolojik anlamlılığa sahip olan biyolojik nöron modelleri, canlı vücudundaki nöronların karakteristiklerini tanımlamaktadır. Bu modeller, osilatörlere benzer şekilde tanımlanabilmektedir. Osilatör yapılarının karakteristiklerini tanımlayan kararlılık kontrolü ve senkronizasyon kontrolü gibi teoremlerin pek çoğu, biyolojik nöron modellerinin incelenmesi için de kullanılabilmektedir. Son zamanlarda, doğrusal olmayan dinamik sistem uygulamalarında rotasyon dönüşüm işlemi de dikkat çeken bir konu haline gelmiştir. Rotasyon dönüşüm işlemi sonrasında; doğrusal olmayan bir sistemin dinamik çekeri, rotasyon açısının değişimi ile arzu edilen herhangi bir doğrultuya yönlendirilebilmektedir. Literatürde doğrusal olmayan dinamik sistemlerin en bilinen örneklerinden biri kaotik osilatör yapılarıdır. Kaotik osilatörlerin dinamik çekerinin rotasyon dönüşümü vasıtası ile kontrolü üzerine yapılan pek çok çalışma literatürde mevcuttur. Bununla birlikte; gerçek biyolojik sistemlerin dinamik karakteristiklerinde rotasyon değişimi gözlemlenmesine rağmen, biyolojik nöron modellerinin dinamik çekerinin rotasyon kontrolünü ele alan bir çalışmaya rastlanılmamıştır. Bu sebeple bu çalışmada, Fitzhugh-Nagumo (FHN) modelinin rotasyon dönüşüm işlemi ele alınmıştır. Rotasyonlu FHN nöron modelinin karakteristik çıktılarını elde etmek için denge noktaları hesaplanmıştır. Rotasyon dönüşüm işlemi sonrasında, FHN nöron modelinin dinamik çekerinin rotasyonundaki değişim nümerik simülasyon sonuçlarıyla gözlemlenmiştir. Son olarak hem rotasyon dönüşüm işleminin işlevselliğini desteklemek hem de biyolojiden esinlenerek geliştirilen sistemler için ihtiyaç duyulan gerçek zamanlı işaretleri elde etmek amacıyla; rotasyonlu FHN nöronu programlanabilir ve yeniden yapılandırılabilir bir eleman olan ‘Alan Programlanabilir Kapı Dizisi-(FPGA)’ ile de gerçeklenmiştir. Böylece, önerilen rotasyon dönüşüm işlemi sayesinde nöral sistemlerde herhangi bir kuplajlama tanımlamasına ihtiyaç duyulmadan sistem dinamiklerinin faz ayarlamasına müdahile edilebildiği gösterilmiştir. Bu görüşten yola çıkarak; rotasyonlu FHN nöron modeli; matematiksel olarak modellenmiş, nümerik simülasyonlarla desteklenmiş ve donanım gerçekleştirim çalışması ile de doğrulanmıştır.

Anahtar Kelimeler

Fitzhugh-Nagumo nöron modeli, Rotasyon dönüşüm yöntemi, Euler rotasyon teoremi, Alan programlanabilir kapı dizisi (FPGA)

Kaynakça

• [1] Hodgkin A, Huxley A. “A quantitative description of membrane current and its application to conduction and excitation in nerve”. The Journal of physiology, 117, 500–544, 1952.
• [2] Morris C, Lecar H. “Voltage oscillations in the barnacle giant muscle fiber”. Biophysical Journal, 35, 193–213, 1981.
• [3] Wilson HR, Cowan J.D. “Excitatory and inhibitory interactions in localized populations of model neurons”. Biophysical Journal, 12(1), 1-24, 1972.
• [4] FitzHugh R. Mathematical Models for Excitation and Propagation in Nerve. Editor: Schawn, H.P. Biological Engineering, 1-85, New York, USA, McGraw-Hill, 1969.
• [5] Hindmarsh JL, Rose RM. “A model of neural bursting using three couple first order differential equations”. Proceedings of the Royal Society of London. Series B, 221(1222), 87–102, 1984.
• [6] Izhikevich EM. “Simple model of spiking neurons”. IEEE Transactions on Neural Networks, 14(6), 1569-1572, 2003.
• [7] Korkmaz N, Öztürk İ, Kılıç R. “Multiple perspectives on the hardware implementations of biological neuron models and programmable design aspects”. Turkish Journal of Electrical Engineering and Computer Sciences, 24(3), 1729-1746, 2016.
• [8] Ciszak M, Euzzor S, Geltrude, Arecchi FT, Meucci R. “Noise and coupling induced synchronization in a network of chaotic neurons”. Communications in Nonlinear Science and Numerical Simulation, 18(4), 938-945, 2013.
• [9] Wei DQ, Luo XS, Zhang B, Qin YH. “Controlling chaos in space-clamped FitzHugh–Nagumo neuron by adaptive passive method”. Nonlinear Analysis: Real World Applications, 11(3), 1752-1759, 2010.
• [10] Doruk R, Ihnısh H. “Bifurcation control of Fitzhugh-Nagumo models”. Süleyman Demirel University Journal of Natural and Applied Sciences, 22, 375-391, 2018.
• [11] Andrievsky B, Fradkov AL, Liberzon D. “Robustness of Pecora–Carroll synchronization under communication constraints”. Systems & Control Letters, 111, 27-33, 2018.
• [12] Chen Q, Wang J, Yang S, Qin Y, Deng B, Wei X. “A real-time FPGA implementation of a biologically inspired central pattern generator network”. Neurocomputing, 244, 63-80, 2017.
• [13] Minati L, Frasca M, Yoshimura N, Koike Y. “Versatile locomotion control of a hexapod robot using a hierarchical network of nonlinear oscillator circuits”. IEEE Access, 6, 8042-8065, 2018.
• [14] Cristiano J, Puig D, Garcia MA. “Efficient locomotion control of biped robots on unknown sloped surfaces with central pattern generators”. Electronics Letters, 51(3), 220-222, 2015.
• [15] Tabor M. Chaos and Integrability in Nonlinear Dynamics: an Introduction. New York, USA, John Wiley & Sons, 1989.
• [16] Kim MY, Roy R, Aron JL, Carr TW, Schwartz IB. “Scaling behavior of laser population dynamics with time-delayed coupling: theory and experiment”. Physical Review letters, 94(8-088101), 1-4, 2005.
• [17] Cruz JM., Escalona J, Parmananda P, Karnatak R, Prasad A, Ramaswamy R. “Phase-flip transition in coupled electrochemical cells”. Physical Review E, 81(4), 046213, 2010.
• [18] Adhikari BM, Prasad A, Dhamala M. “Time-delay-induced phase-transition to synchrony in coupled bursting neurons”. Chaos, 21(2-023116), 1-8, 2011.
• [19] Skiadas CH., Skiadas C. “Chaotic modeling and simulation in rotation–translation models”. International Journal of Bifurcation and Chaos, 21(10), 3023-3031, 2011.
• [20] Dai S, Sun K, He S, Ai W. “Complex chaotic attractor via fractal transformation”. Entropy, 21(11), 1115, 2019.
• [21] Wang M, Deng Y, Liao X, Li Z, Ma M, Zeng, Y. “Dynamics and circuit implementation of a four-wing memristive chaotic system with attractor rotation”. International Journal of Non-Linear Mechanics, 111, 149-159, 2019.
• [22] Prasad A, Dana SK, Karnatak R, Kurths J, Blasius B, Ramaswamy R. “Universal occurrence of the phase-flip bifurcation in time-delay coupled systems”. Chaos, 18(2-023111), 1-9, 2008.
• [23] Bhowmick SK., Ghosh D, Dana SK. “Synchronization in counter-rotating oscillators”. Chaos, 21(3), 1-9, 2011.
• [24] Prasad A. “Universal occurrence of mixed-synchronization in counter-rotating nonlinear coupled oscillators”. Chaos, Solitons & Fractals, 43(1-12), 42-46, 2010.
• [25] Sayed WS, Radwan AG, Elnawawy M, Orabi H, Sagahyroon A, Aloul F, El-Sedeek A. “Two-dimensional rotation of chaotic attractors: Demonstrative examples and FPGA realization”. Circuits, Systems, and Signal Processing, 38(10), 4890-4903, 2019.
• [26] Orabi H, Elnawawy M, Sagahyroon A, Aloul F, Elwakil AS, Radwan AG. “On the Implementation of a Rotated Chaotic Lorenz System on FPGA”. IEEE Asia Pacific Conference on Circuits and Systems (APCCAS), Bangkok, Thailand, 11-14 November 2019.
• [27] Takagi S, Ueda T. “Emergence and transitions of dynamic patterns of thickness oscillation of the plasmodium of the true slime mold Physarum polycephalum”. Physica D: Nonlinear Phenomena, 237(3), 420-427, 2008.
• [28] Hargreaves EL, Yoganarasimha D, Knierim, JJ. “Cohesiveness of spatial and directional representations recorded from neural ensembles in the anterior thalamus, parasubiculum, medial entorhinal cortex, and hippocampus”. Hippocampus, 17(9), 826-841, 2007.
• [29] Bhowmick SK, Bera BK, Ghosh D. “Generalized counter-rotating oscillators: Mixed synchronization”. Communications in Nonlinear Science and Numerical Simulation, 22(1-3), 692-701, 2015.
• [30] Binczaka S, Jacquira S, Bilbaulta JM, Kazantsevb VB, Nekorkinb VI. “Experimental study of electrical MFHN neurons”. Neural Networks, 19, 684-693, 2006.
• [31] Chen M, Qi J, Wu H, Xu Q, Bao B. “Bifurcation analyses and hardware experiments for bursting dynamics in non-autonomous memristive FitzHugh-Nagumo circuit”. Science China Technological Sciences, 63, 1035–1044, 2020.
• [32] Keener J, Sneyd J. Mathematical Physiology. New York, USA, Springer, 1998.
• [33] FitzHugh R. “Impulses and physiological states in theoretical models of nerve membrane”. Biophysical Journal, 1(6), 445-466, 1961.