Controlling a neuron by stimulating a coupled neuron. (English) Zbl 1416.92040
Summary: Despite the intensive studies on neurons, the control mechanism in real interactions of neurons is still unclear. This paper presents an understanding of this kind of control mechanism, controlling a neuron by stimulating another coupled neuron, with the uncertainties taken into consideration for both neurons. Two observers and a differentiator, which comprise the first-order low-pass filters, are first designed for estimating the uncertainties. Then, with the estimated values combined, a robust nonlinear controller with a saturation function is presented to track the desired membrane potential. Finally, two typical bursters of neurons with the desired membrane potentials are proposed in the simulation, and the numerical results show that they are tracked very well by the proposed controller.
MSC:
| 92C20 | Neural biology |
| 34H05 | Control problems involving ordinary differential equations |
| 93C10 | Nonlinear systems in control theory |
References:
| [1] | HODGKIN, A. L., HUXLEY, A. F., and KATZ, B. Measurement of current-voltage relations in the membrane of the giant axon of Loligo. Journal of Physiology, 116(4), 424-448 (1952) |
| [2] | MORRIS, C. and LECAR, H. Voltage oscillations in the barnacle giant muscle fiber. Biophysical Journal, 35(1), 193-213 (1981) |
| [3] | FITZHUGH, R. Impulses and physiological states in theoretical models of nerve membrane. Biophysical Journal, 1(6), 445-466 (1961) |
| [4] | BERTRAM, R., BUTTE, M. J., KIEMEL, T., and SHERMAN, A. Topological and phenomenological classification of bursting oscillations. Bulletin of Mathematical Biology, 57(3), 413-439 (1995) · Zbl 0813.92010 |
| [5] | IZHIKEVICH, E. M. Neural excitability, spiking and bursting. International Journal of Bifurcation and Chaos, 10(6), 1171-1266 (2000) · Zbl 1090.92505 |
| [6] | CRUNELLI, V., KELLY, J. S., LERESCHE, N., and PIRCHIO, M. The ventral and dorsal lateral geniculate nucleus of the rat: intracellular recordings in vitro. The Journal of Physiology, 384(1), 587-601 (1987) |
| [7] | JOHNSON, S. W., SEUTIN, V., and NORTH, R. A. Burst firing in dopamine neurons induced by N-methyl-D-aspartate: role of electrogenic sodium pump. Science, 258(5082), 665-667 (1992) |
| [8] | Rinzel, J., Bursting oscillation in an excitable membrane model, 304-316 (1985), New York · Zbl 0584.92003 |
| [9] | DE VRIES, G. Multiple bifurcations in a polynomial model of bursting oscillations. Journal of Nonlinear Science, 8(3), 281-316 (1998) · Zbl 0901.34045 |
| [10] | RUSH, M. E. and RINZEL, J. Analysis of bursting in a thalamic neuron model. Biological Cybernetics, 71(4), 281-291 (1994) · Zbl 0804.92008 |
| [11] | ZEITlER, M., DAFFERTSHOFER, A., and GIELEN, C. C. Asymmetry in pulse-coupled oscillators with delay. Physical Review E, 79(6), 065203 (2009) |
| [12] | IZHIKEVICH, E. M. Subcritical elliptic bursting of Bautin type. SIAM Journal on Applied Mathematics, 60(2), 503-535 (2000) · Zbl 1026.92009 |
| [13] | REN, G., XU, Y., and WANG, C. Synchronization behavior of coupled neuron circuits composed of memristors. Nonlinear Dynamics, 88(2), 893-901 (2017) |
| [14] | FERRARI, F. A. S., VIANA, R. L., LOPES, S. R., and STOOP, R. Phase synchronization of coupled bursting neurons and the generalized Kuramoto model. Neural Networks, 66, 107-118 (2015) · Zbl 1396.92014 |
| [15] | IZHIKEVICH, E. M. Class 1 neural excitability, conventional synapses, weakly connected networks, and mathematical foundations of pulse-coupled models. IEEE Transactions on Neural Networks, 10(3), 499-507 (1999) |
| [16] | ERMENTROUT, B. Type I membranes, phase resetting curves, and synchrony. Neural Computation, 8(5), 979-1001 (1996) |
| [17] | CHAPIN, J. K., MOXON, K. A., MARKOWITZ, R. S., and NICOLELIS, M. A. L. Real-time control of a robot arm using simultaneously recorded neurons in the motor cortex. Nature Neuro-science, 2(7), 664-670 (1999) |
| [18] | HAO, Y. Y., ZHANG, Q. S., ZHANG, S. M., ZHAO, T., WANG, Y. W., CHEN, W. D., and ZHENG, X. X. Decoding grasp movement from monkey premotor cortex for real-time prosthetic hand control. Chinese Science Bulletin, 58(20), 2512-2520 (2013) |
| [19] | HAN, J. S. Acupuncture: neuropeptide release produced by electrical stimulation of different frequencies. Trends in Neurosciences, 26(1), 17-22 (2003) |
| [20] | XIE, Y., AIHARA, K., and KANG, Y. M. Change in types of neuronal excitability via bifurcation control. Physical Review E, 77(2), 021917 (2008) |
| [21] | HONG, K. S. Synchronization of coupled chaotic FitzHugh-Nagumo neurons via Lyapunov functions. Mathematics and Computers in Simulation, 82(4), 590-603 (2011) · Zbl 1241.92014 |
| [22] | CHEN, T., LIU, X., and LU, W. Pinning complex networks by a single controller. IEEE Transactions on Circuits and Systems I: Regular Papers, 54(6), 1317-1326 (2007) · Zbl 1374.93297 |
| [23] | SHI, X. and DAI, S. Intracellular solitary pulse calcium waves in frog sympathetic neurons. Applied Mathematics and Mechanics (English Edition), 26(2), 150-159 (2005) https://doi.org/10.1007/BF02438236 · Zbl 1144.92305 |
| [24] | CHEN, W. H. and GUO, L. Control of nonlinear systems with unknown actuator nonlinearities. IFAC Proceedings Volumes, 37(13), 1347-1352 (2004) |
| [25] | ZHOU, Y. and WANG, Z. H. A robust optimal trajectory tracking control for systems with an input delay. Journal of the Franklin Institute, 353(12), 2627-2649 (2016) · Zbl 1347.93114 |
| [26] | SHI, M. and WANG, Z. H. Abundant bursting patterns of a fractional-order Morris-Lecar neuron model. Communications in Nonlinear Science and Numerical Simulation, 19(6), 1956-1969 (2014) · Zbl 1457.92032 |
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