Recalculation of Lost Information in Neuron with Quadratic Spline Interpolation

Abstract

The main function of neurons in a living creature is to transmit information. Neurons carry out information transmission without loss despite environmental and internal noise sources. However, sometimes there may be losses in the transmission of information. This results in diseases such as Alzheimer's, MS, and Epilepsy. In this study, the information lost in neurons is recalculated with the Quadratic Spline Interpolation method. In cases where it is difficult or impossible to calculate a function, the process of calculating the corresponding value of an unmeasured variable is called interpolation. In this study, first of all, three sample neuron behaviours are created with the Fitzhugh-Nagumo model, and the action potential and recovery parameter variables are obtained. Then, some data in the variables are deleted, resulting in unhealthy neuron behaviour. Then, these deleted data are recalculated using the Quadratic Spline Interpolation method. Various error values are obtained by comparing the actual and calculated data. The data lost in the action potential-recovery variable are detected with a very low error rate of 0.2630-0.0524%, 0.2885-0.0165% and 0.2543-0.0781% for the three sample neuron behaviours, respectively. With this study, it has been demonstrated that information lost or incorrectly coded in neurons for any reason can be corrected. It is also understood that this study can be used to prevent losses in real-time measurement results from biological neurons and to recalculate erroneous values.

Keywords

• Quadratic Spline Interpolation
• Cubic Spline Interpolation
• FitzHugh-Nagumo
• Hodgkin-Huxley
• Neuron Model

İkinci Dereceden İnterpolasyon ile Nöronda Kayıp Bilginin Yeniden Hesabı

Abstract

Canlılarda nöronların temel görevi bilgi iletimidir. Nöronlar çevresel ve içsel gürültü kaynaklarına rağmen bilgi iletimini kayıpsız olarak gerçekleştirirler. Fakat kimi zaman bilgi iletiminde kayıplar meydana gelebilir. Bu durum Alzheimer, MS, Epilepsi gibi hastalıklar ile sonuçlar. Bu çalışmada nöronlarda kaybolan bilgi İkinci Dereceden Şerit İnterpolasyon yöntemi ile yeniden hesaplanması sağlanmıştır. Bir fonksiyonun hesaplanmasının zor veya mümkün olmayan durumlarda, değeri ölçülmemiş bir değişkenine karşılık gelen değerinin hesaplanması işlemine interpolasyon adı verilir. Bu çalışamda öncelikle Fitzhugh-Nagumo model ile üç örnek nöron davranışı oluşturulmuş ve aksiyon potansiyeli ile toparlanma parametresi değişkenleri elde edilmiştir. Ardından değişkenlerdeki bazı veriler silinerek sağlıksız bir nöron davranışı sağlanmıştır. Daha sonra İkinci Dereceden Şerit İnterpolasyon yöntemi kullanılarak silinen bu veriler yeniden hesaplanmıştır. Gerçek ve hesaplanan veriler karşılaştırılarak çeşitli hata değerleri elde edilmiştir. Aksiyon potansiyeli- toparlanma parametresinde kaybolan veriler, üç örnek nöron davranışı için sırasıyla %0.2630-%0.0524, %0.2885-%0.0165 ve %0.254-%0.0781 gibi çok düşük bir hata oranıyla tespit edilir. Bu çalışma ile nöronlarda herhangi bir sebepten dolayı kaybolan veya yanlış kodlanan bilgi düzeltilebilir olduğu ortaya konmuştur. Ayrıca bu çalışmanın biyolojik nöronlardan gerçek zamanlı ölçüm sonuçlarındaki kayıpları önlemek ve hatalı değerleri yeniden hesaplamak için kullanılabileceği anlaşılmaktadır.

Keywords

• İkinci Dereceden İnterpolasyon
• Kübik İnterpolasyon
• FitzHugh-Nagumo
• Hodgkin-Huxley
• Nöron Model

References

1. Blu, Thierry, Philippe Thévenaz, and Michael Unser. 2004. “Linear Interpolation Revitalized.” IEEE Transactions on Image Processing 13(5): 710–19.
2. Casado, José Manuel. 2003. “Synchronization of Two Hodgkin-Huxley Neurons Due to Internal Noise.” Physics Letters, Section A: General, Atomic and Solid State Physics 310(5–6): 400–406.
3. Effenberger, Cedric, and Daniel Kressner. 2012. “Chebyshev Interpolation for Nonlinear Eigenvalue Problems.” BIT Numerical Mathematics 52(4): 933–51.
4. Faisal, A. Aldo, Luc P.J. Selen, and Daniel M. Wolpert. 2008. “Noise in the Nervous System.” Nature Reviews Neuroscience 9(4): 292–303.
5. FitzHugh, Richard. 1961. “Impulses and Physiological States in Theoretical Models of Nerve Membrane.” Biophysical Journal 1(6): 445–66.
6. Gardner, Floyd M. 1993. “Interpolation in Digital Modems—Part I: Fundamentals.” IEEE Transactions on Communications 41(3): 501–7.
7. Hindmarsh, J. L., and R. M. Rose. 1984. “A Model of Neuronal Bursting Using Three Coupled First Order Differential Equations.” Proceedings of the Royal Society of London. Series B, Containing papers of a Biological character. Royal Society (Great Britain) 221(1222): 87–102.
8. Hodgkin, A. L., and A. F. Huxley. 1952. “A Quantitative Description of Membrane Current and Its Application to Conduction and Excitation in Nerve.” The Journal of Physiology 117(4): 500–544.

9. Izhikevich, Eugene M. 2003. “Simple Model of Spiking Neurons.” IEEE Trans. Neural Netw. 14(6): 1569–72.
10. Kang, Yanmei et al. 2020. “Formation of Spiral Wave in Hodgkin-Huxley Neuron Networks with Gamma-Distributed Synaptic Input.” Communications in Nonlinear Science and Numerical Simulation 83: 105112.
11. Keys, Robert G. 1981. “Cubic Convolution Interpolation for Digital Image Processing.” IEEE Transactions on Acoustics, Speech, and Signal Processing 29(6): 1153–60.
12. Koziel, Slawomir, John W. Bandler, and Kaj Madsen. 2006. “Space-Mapping-Based Interpolation for Engineering Optimization.” IEEE Transactions on Microwave Theory and Techniques 54(6): 2410–21.
13. Li, Han et al. 2020. “Overview of Cannabidiol (CBD) and Its Analogues: Structures, Biological Activities, and Neuroprotective Mechanisms in Epilepsy and Alzheimer’s Disease.” European Journal of Medicinal Chemistry 192: 112163.
14. Lunardi, Alessandra, and Scuola normale superiore (Italy). 2009. Interpolation Theory. Edizioni Della Normale.
15. Morris, C., and H. Lecar. 1981. “Voltage Oscillations in the Barnacle Giant Muscle Fiber.” Biophysical Journal 35(1): 193–213. Nagumo, J., S. Arimoto, and S. Yoshizawa. 1962. “An Active Pulse Transmission Line Simulating Nerve Axon*.” Proceedings of the IRE 50(10): 2061–70.
16. Nakamura, Osamu, and Katsumi Tateno. 2019. “Random Pulse Induced Synchronization and Resonance in Uncoupled Non-Identical Neuron Models.” Cognitive Neurodynamics 13(3): 303–12.
17. Narang, Sunil K., Akshay Gadde, and Antonio Ortega. 2013. “Signal Processing Techniques for Interpolation in Graph Structured Data.” ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings: 5445–49.
18. Polytechnica, Periodica, and Ser Civ Eng. 1999. “GIS Functions - Interpolation.” Periodica Polytechnica Civil Engineering 43(1): 63–87.
19. Prof, Assoc et al. 2014. “Reservoir Engineer, Emerson Process Management Level 11, Menara Chan.” Applied Mathematical Sciences 8(102): 5083–98.
20. Purves, Dale et al. 2019. Neurosciences, 6th Edition Neurosciences, 6th Edition.
21. Sauer, Thomas, and Yuan Xu. 1995. “On Multivariate Lagrange Interpolation.” Mathematics of Computation 64(211): 1147–70.
22. Schafer, Ronald W., and Lawrence R. Rabiner. 1973. “A Digital Signal Processing Approach to Interpolation.” Proceedings of the IEEE 61(6): 692–702.
23. Scheuerer, Michael. 2009. “A Comparison of Models and Methods for Spatial Interpolation in Statistics and Numerical Analysis.”
24. Werner, Wilhelm. 1984. “Polynomial Interpolation: Lagrange versus Newton.” Mathematics of Computation 43(167): 205.