TRACKING OF ARBITRARY REGIMES FOR SPIKING AND BURSTING IN THE HODGKIN-HUXLEY NEURON
DOI:
https://doi.org/10.20319/mijst.2017.32.560576Keywords:
Hodgkin-Huxley Neuron, Neuron Spiking and Bursting, Feedback, Tracking Control, Speed Gradient, Target Attractor FeedbackAbstract
We propose here efficient mathematical tracking control algorithms to design the spiking or bursting behavior in the four dimensional dynamical system modeling biological neurons represented by the Hodgkin-Huxley (HH) differential equations. The stimulating external electrical current serves as a control signal, while the membrane action potential is the target output. We use two alternative feedback algorithms, Fradkov’s speed gradient and Kolesnikov’s ‘synergetic’ target attractor control, to produce arbitrary spiking or bursting regimes in the model and to track the action potential of the system. Both algorithms demonstrate high efficiency and robustness for the controlled HH dynamics. For virtually any initial condition we are able to form a single spike at the chosen moment of time, the train with any number of spikes, the arbitrary-shaped burst, and also to switch between regular and chaotic regimes of bursting. Two approaches developed here could be easily adopted for the networks of neural clusters and used effectively for the purposes of neuro-informatics and for modeling neural dysfunctions like epileptiform or other abnormal behavior in Hodgkin-Huxley neuron clusters. This work has been supported by the TÜBİTAK project 116F049 “Controlling Spiking and Bursting Dynamics in Hodgkin-Huxley Neurons”.
References
Ahmadian, Y., Packer, A. M., Yuste, R., Paninski, L. (2011). Designing optimal stimuli to control neuronal spike timing. Journal of Neurophysiology, 106(2), 1038-1053. https://doi.org/10.1152/jn.00427.2010
Awadalla M. H. A., Sadek, M. A. (2012). Spiking neural network-based control chart pattern recognition. Alexandria Engineering Journal, 51, 27-35. https://doi.org/10.1016/j.aej.2012.07.004
Bandopadhayay, P., Stiles, C. D. (2017). Population control: Cortical interneurons modulate oligodendrogenesis. Neuron, 94(3), 415-417. https://dx.doi.org/10.1016/j.neuron.2017.04.032
Bower, J. M. (Ed.). (2013). 20 Years of Computational Neuroscience. New York: Springer Sieser in Computational Neuroscience. https://doi.org/10.1007/978-1-4614-1424-7
Brody, C. D., Hopfield, J. J. (2003). Simple networks for spike-timing-based computation, with application to olfactory processing. Neuron, 37, 843-852. https://doi.org/10.1016/S0896-6273(03)00120-X
Cymbalyuka, G. S., Calabrese, R. L., Shilnikov, A. L. (2005). How a neuron model can demonstrate co-existence of tonic spiking and bursting. Neurocomputing, 6566, 869875. https://doi.org/10.1016/j.neucom.2004.10.107
Danzl, P., Moehlis, J. (2008). Spike timing control of oscillatory neuron models using impulsive and quasi-impulsive charge-balanced inputs. 2008 American Control Conference, Seattle, 171-176. https://doi.org/10.1109/ACC.2008.4586486
DiLorenzo, P. M., Victor, J. D. (Eds.). (2013). Spike Timing: Mechanisms and Function. Boca Raton: CRC Press. https://doi.org/10.1201/b14859
Ding, L., Hou, C. (2010).Stabilizing control of Hopf bifurcation in the Hodgkin-Huxley model via washout filter with linear control term. Nonlinear Dynamics, 60, 131-139. https://doi.org/10.1007/s11071-009-9585-x
Fourcaud-Trocme, N., Hansel, D., van Vreeswijk, C., Brunel, N. (2003). How spike generation mechanisms determine the neuronal response to fluctuating inputs. The Journal of Neuroscience, 23(37), 11628-11640.
Fradkov, A. L., Pogromsky, A. Yu. (1998). Introduction to Control of Oscillations and Chaos. Singapore: World Scientific. https://doi.org/10.1142/3412
Fradkov A. L. (2007). Cybernetical Physics: From Control of Chaos to Quantum Control. Berlin, Heidelberg: Springer.
Guckenheimer, J., Oliva, R. A. (2002). Chaos in the Hodgkin-Huxley model. The SIAM Journal on Applied Dynamical Systems, 1(1), 105-114. https://doi.org/10.1137/S1111111101394040
Haddad, W. M., Hui, Q., Bailey, J. M. (2014). Human brain networks: Spiking neuron models, multistability, synchronization, thermodynamics, maximum entropy production, and anesthetic cascade mechanisms. Entropy, 16, 3939-4003. https://doi.org/10.3390/e16073939
Hodgkin A. L., Huxley A. А. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiology, 117, 500-544.
Hoppensteadt, F. (2013). Heuristics for the Hodgkin-Huxley system. Mathematical Biosciences, 245(1), 56-60. https://doi.org/10.1016/j.mbs.2012.11.006
Horng, T.-L., Huang, M.-W.(2006). Spontaneous oscillations in Hodgkin-Huxley model. Journal of Medical and Biological Engineering, 26(4): 161-168. https://doi.org/10.1140/epjst/e2010-01282-3
Izhikevich, E. M. (2000). Neural excitability, spiking and bursting. International Journal of Bifurcation and Chaos, 10(6), 1171-1266. https://doi.org/10.1142/S0218127400000840
Jolivet, R., Lewis, T. J., Gerstner, W. (2003). The Spike Response Model: A Framework to Predict Neuronal Spike Trains. ICANN/ICONIP 2003, LNCS 2714, 846-853. https://doi.org/10.1007/3-540-44989-2_101
Kolesnikov, A. (2012). Synergetic Control Methods for Complex Systems. Moscow: URSS Publ.
Kolesnikov, A. (2014). Introduction of Synergetic Control.2014 American Control Conference, Portland, 3013-3016. https://doi.org/10.1109/ACC.2014.6859397
Lewis, J. E., Lindner, B., Laliberte, B., Groothuis, S. (2007). Control of neuronal firing by dynamic parallel fiber feedback: Implications for electrosensory reafference suppression. Journal of Experimental Biology, 210, 4437-4447. https://doi.org/10.1242/jeb.010322
Li, L., Brockmeier, A., Chen, B., Seth, S., Joseph T., Francis, J. T., Sanchez, J. C., Príncipe, J. C. (2013). Adaptive inverse control of neural spatiotemporal spike patterns with a Reproducing Kernel Hilbert Space (RKHS) framework. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 21(4), 532-543. https://doi.org/10.1109/TNSRE.2012.2200300
Lu, J., Tucciarone, J., Padilla-Coreano, N., He, M., Gordon, J. A., Huang, Z. J. (2017). Selective inhibitory control of pyramidal neuron ensembles and cortical subnetworks by chandelier cells. Nature Neuroscience, 20, 1377-1383. https://doi.org/10.1038/nn.4624
Lu, H., Balmer, T. S., Romero, G. E., Trussell, L. O. (2017). Slow AMPAR synaptic transmission is determined by stargazin and glutamate transporters. Neuron, 96(1), 73-80. http://dx.doi.org/10.1016/j.neuron.2017.08.043
Moss, F., Gielen, S. (2001). Neuro-Informatics and Neural Modelling, Amsterdam: Elsevier.
Nabi, A., Moehlis, J. (2012). Time optimal control of spiking neurons. Journal of Mathematical Biology, 64(6), 981-1004. http://dx.doi.org/10.1007/s00285-011-0441-5
Neiman, A. B., Dierkes, K., Lindner, B., Han, L., Shilnikov, A. L. (2011). Spontaneous voltage oscillations and response dynamics of a Hodgkin-Huxley type model of sensory hair cells. The Journal of Mathematical Neuroscience, 1, 1-24. https://doi.org/10.1186/2190-8567-1-11
Purali, N. (2002). Firing properties of the soma and axon of the abdominal stretch receptor neurons in the crayfish (Astacusleptodactylus). General Physiology and Biophysics, 21, 205-226.
Qiao, N., Mostafa, H., Corradi, F., Osswald, M., Stefanini, F., Sumislawska, D., Indiveri, G. (2015). A reconfigurable on-line learning spiking neuromorphic processor comprising 256 neurons and 128 K synapses. Frontiers in Neuroscience, 9, Article141. https://doi.org/10.3389/fnins.2015.00141
Qin, Y.-M., Wang, J., Men, C., Chan, W.-L., Wei, X.-L. Deng, B. (2013). Control of synchronization and spiking regularity by heterogenous aperiodic high-frequency signal in coupled excitable systems. Communications in Nonlinear Science and Numerical Simulation, 18(10), 2775-2782. https://doi.org/10.1016/j.cnsns.2013.02.010
Rabinovich, M. I., Abarbanel, H. D. I. (1998). The role of chaos in neural systems. Neuroscience, 87(1), 5-14. https://doi.org/10.1016/S0306-4522(98)00091-8
Rasmussen, R. G., Schwartz, A., Chase, S. M. (2017). Dynamic range adaptation in primary motor cortical populations. Computational and Systems Biology, Neuroscience, 6, e21409. https://doi.org/10.7554/eLife.21409.
Saha, A. K., Choudhury, S., Majumder, M. (2017). Performance efficiency analysis of water treatment plants by using MCDM and neural network model. MATTER: International Journal of Science and Technology, 3(1), 27 - 35. https://dx.doi.org/10.20319/Mijst.2017.31.2735
Schultheiss, N. W., Prinz, A. A., Butera, R. J. (2011). Phase Response Curves in Neuroscience: Theory, Experiment, and Analysis. New York: Springer Science & Business Media.
Strogatz, S. (1994). Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. Massachusetts: Perseus Books Publishing.
Subashini, P., Dhivyaprabha, T. T., Krishnaveni, M. (2017). Cellular organism based Particle Swarm Optimization Algorithm for complex non-linear problems. MATTER: International Journal of Science and Technology, 3(2), 209 - 229. https://dx.doi.org/10.20319/mijst.2017.32.209229
Tonnelier, A. (2005). Categorization of neural excitability using threshold models. Neural Computation, 17(7), 1447-1455. https://dx.doi.org/10.1162/0899766053723087
Wang, J., Chen, L., Fei, X. (2007). Analysis and control of the bifurcation of Hodgkin-Huxley model. Chaos, Solitons and Fractals, 31(1), 247-256. https://doi.org/10.1016/j.chaos.2005.09.060
Downloads
Published
How to Cite
Issue
Section
License
Copyright of Published Articles
Author(s) retain the article copyright and publishing rights without any restrictions.
All published work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
