Abstract
Deep brain stimulation (DBS) is a common method of combating pathological conditions associated with Parkinson’s disease, Tourette syndrome, essential tremor, and other disorders, but whose mechanisms are not fully understood. One hypothesis, supported experimentally, is that some symptoms of these disorders are associated with pathological synchronization of neurons in the basal ganglia and thalamus. For this reason, there has been interest in recent years in finding efficient ways to desynchronize neurons that are both fast-acting and low-power. Recent results on coordinated reset and periodically forced oscillators suggest that forming distinct clusters of neurons may prove to be more effective than achieving complete desynchronization, in particular by promoting plasticity effects that might persist after stimulation is turned off. Current proposed methods for achieving clustering frequently require either multiple input sources or precomputing the control signal. We propose here a control strategy for clustering, based on an analysis of the reduced phase model for a set of identical neurons, that allows for real-time, single-input control of a population of neurons with low-amplitude, low total energy signals. After demonstrating its effectiveness on phase models, we apply it to full state models to demonstrate its validity. We also discuss the effects of coupling on the efficacy of the strategy proposed and demonstrate that the clustering can still be accomplished in the presence of weak to moderate electrotonic coupling.
Similar content being viewed by others
References
Adamchic, I., Hauptmann, C., Barnikol, U.B., Pawelczyk, N., Popovych, O., Barnikol, T.T., Silchenko, A., Volkmann, J., Deuschl, G., Meissner, W.G., Maarouf, M., Sturm, V., Freund, H.-J., Tass, P.A. (2014). Coordinated reset neuromodulation for Parkinson’s disease: proof-of-concept study. Movement Disorders, 29(13), 1679–1684.
Benabid, A.L., Benazzous, A., Pollak, P. (2002). Mechanisms of deep brain stimulation. Movement Disorders, 17(SUPPL. 3), 19–38.
Beric, A., Kelly, P.J., Rezai, A., Sterio, D., Mogilner, A., Zonenshayn, M., Kopell, B. (2002). Complications of deep brain stimulation surgery. Stereotactic and Functional Neurosurgery, 77(1–4), 73–78.
Brown, E., Moehlis, J., Holmes, P. (2004). On the phase reduction and response dynamics of neural oscillator populations. Neural Computation, 16(4), 673–715.
Cagnan, H., Brittain, J.S., Little, S., Foltynie, T., Limousin, P., Zrinzo, L., Hariz, M., Joint, C., Fitzgerald, J., Green, A.L., Aziz, T., Brown, P. (2013). Phase dependent modulation of tremor amplitude in essential tremor through thalamic stimulation. Brain: A Journal of Neurology, 136(10), 3062–3075.
Chen, C.C., Litvak, V., Gilbertson, T., Ku̇hn, A., Lu, C.S., Lee, S.T., Tsai, C.H., Tisch, S., Limousin, P., Hariz, M., Brown, P. (2007). Excessive synchronization of basal ganglia neurons at 20 Hz slows movement in Parkinson’s disease. Experimental Neurology, 205(1), 214–221.
Danzl, P., Hespanha, J., Moehlis, J. (2009). Event-based minimum-time control of oscillatory neuron models: Phase randomization, maximal spike rate increase, and desynchronization. Biological Cybernetics, 101(5-6), 387–399.
Ermentrout, B. (2002). Simulating, analyzing, and animating dynamical systems. Philadelphia: Society for Industrial and Applied Mathematics.
Ermentrout, G.B., & Terman, D.H. (2010). Mathematical foundations of neuroscience, volume 35 of interdisciplinary applied mathematics. New York: Springer.
Galan, R.F., Ermentrout, G.B., Urban, N.N. (2005). Efficient estimation of phase-resetting curves in real neurons and its significance for neural-network modeling. Physical Review Letters, 94(15), 1–4.
Golomb, D., & Hansel, D. (2000). The number of synaptic inputs and the synchrony of large, sparse neuronal networks. Neural Computation, 12(5), 1095–1139.
Hammond, C., Bergman, H., Brown, P. (2007). Pathological synchronization in Parkinson’s disease: networks, models and treatments. Trends in Neurosciences, 30(7), 357–364.
Hodgkin, A., & Huxley, A. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiology, 117, 500–544.
Hua, S.E., Lenz, F. a., Zirh, T. a., Reich, S.G., Dougherty, P.M. (1998). Thalamic neuronal activity correlated with essential tremor. Journal of Neurology, Neurosurgery, and Psychiatry, 64(2), 273–276.
Johnston, D., & Wu, S. M.-S. (1995). Foundations of cellular neurophysiology, 1st edn. Cambridge: MIT Press.
Keener, J., & Sneyd, J. (2009). Mathematical physiology. Interdisciplinary applied mathematics. New York: Springer.
Khalil, H. (2015). Nonlinear control, 1st edn. New York: Pearson.
Kuramoto, Y. (1984). Chemical oscillations, waves, and turbulence, volume 19 of springer series in synergetics. Springer: Berlin.
Levy, R., Hutchison, W., Lozano, A., Dostrovsky, J. (2000). High-frequency synchronization of neuronal activity in the subthalamic nucleus of Parkinsonian patients with limb tremor. The Journal of Neuroscience, 20(20), 7766–7775.
Li, J.-S., Dasanayake, I., Ruths, J. (2013). Control and synchronization of neuron ensembles. IEEE Transactions on Automatic Control, 58(8), 1919–1930.
Lu̇cken, L., Yanchuk, S., Popovych, O.V., Tass, P.A. (2013). Desynchronization boost by non-uniform coordinated reset stimulation in ensembles of pulse-coupled neurons. Frontiers in Computational Neuroscience, 7, 63.
Lysyansky, B., Popovych, O.V., Tass, P.A. (2011). Desynchronizing anti-resonance effect of m: n ON-OFF coordinated reset stimulation. Journal of Neural Engineering, 8(3), 036019.
Lysyansky, B., Popovych, O.V., Tass, P.A. (2013). Optimal number of stimulation contacts for coordinated reset neuromodulation. Frontiers in Neuroengineering, 6(July), 5.
Matchen, T., & Moehlis, J. (2017). Real-time stabilization of neurons into clusters. In American controls conference (pp. 2805–2810). Seattle.
Rodriguez-Oroz, M.C., Obeso, J.A., Lang, A.E., Houeto, J.L., Pollak, P., Rehncrona, S., Kulisevsky, J., Albanese, A., Volkmann, J., Hariz, M.I., Quinn, N.P., Speelman, J.D., Guridi, J., Zamarbide, I., Gironell, A., Molet, J., Pascual-Sedano, B., Pidoux, B., Bonnet, A.M., Agid, Y., Xie, J., Benabid, A.L., Lozano, A.M., Saint-Cyr, J., Romito, L., Contarino, M.F., Scerrati, M., Fraix, V., Van Blercom, N. (2005). Bilateral deep brain stimulation in Parkinson’s disease: a multicentre study with 4 years follow-up. Brain: A Journal of Neurology, 128(10), 2240–2249.
Rosenblum, M., & Pikovsky, A. (2004). Delayed feedback control of collective synchrony: an approach to suppression of pathological brain rhythms. Physical Review E, 70(4), 041904.
Rubin, J.E., & Terman, D. (2004). High frequency stimulation of the subthalamic nucleus eliminates pathological thalamic rhythmicity in a computational model. Journal of Computational Neuroscience, 16(3), 211–235.
Sacrė, P., & Sepulchre, R. (2014). Sensitivity analysis of oscillator models in the space of phase-response curves: oscillators as open systems. IEEE Control Systems, 34(2), 50–74.
Savica, R., Stead, M., Mack, K.J., Lee, K.H., Klassen, B.T. (2012). Deep brain stimulation in Tourette syndrome: a description of 3 patients with excellent outcome. Mayo Clinic Proceedings, 87(1), 59–62.
Schmidt, G.S., Wilson, D., Allgower, F., Moehlis, J. (2014). Selective averaging with application to phase reduction and neural controls. Nonlinear Theory and Its Application IEICE, 5(4), 424–435.
Schnitzler, A., & Gross, J. (2005). Normal and pathological oscillatory communication in the brain. Nature Reviews Neuroscience, 6(4), 285–96.
Tass, P.A. (2003a). A model of desynchronizing deep brain stimulation with a demand-controlled coordinated reset of neural subpopulations. Biological Cybernetics, 89(2), 81–88.
Tass, P.A. (2003b). Desynchronization by means of a coordinated reset of neural sub-populations - a novel technique for demand-controlled deep brain stimulation. Progress of Theoretical Physics Supplement, 150(150), 281–296.
The Deep-Brain Stimulation for Parkinson’s Disease Study Group. (2001). Deep-brain stimulation of the subthalamic nucleus or the pars interna of the globus pallidus in Parkinson’s disease. New England Journal of Medicine, 345(13), 956–963.
Uhlhaas, P.J., & Singer, W. (2006). Neural synchrony in brain disorders: relevance for cognitive dysfunctions and pathophysiology. Neuron, 52(1), 155–168.
Wilson, D., & Moehlis, J. (2014). Optimal chaotic desynchronization for neural populations. SIAM Journal on Applied Dynamical Systems, 13(1), 276–305.
Wilson, D., & Moehlis, J. (2015). Clustered desynchronization from high-frequency deep brain stimulation. PLoS Computational Biology, 11(12), 1–26.
Wilson, C.J., Beverlin, B., Netoff, T. (2011). Chaotic desynchronization as the therapeutic mechanism of deep brain stimulation. Frontiers in Systems Neuroscience, 5, 50.
Zhao, C., Wang, L., Netoff, T., Yuan, L.L. (2011). Dendritic mechanisms controlling the threshold and timing requirement of synaptic plasticity. Hippocampus, 21(3), 288–297.
Zlotnik, A., & Li, J.-S. (2014). Optimal subharmonic entrainment of weakly forced nonlinear oscillators. SIAM Journal on Applied Dynamical Systems, 13(4), 1654–1693.
Zlotnik, A., Nagao, R., Kiss, I.Z., Li, J.-S. (2016). Phase-selective entrainment of nonlinear oscillator ensembles. Nature Communications, 7, 1–7.
Acknowledgements
Support for this work by National Science Foundation Grants No. NSF-1264535/1631170 and NSF-1635542 is gratefully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interests
The authors declare that they have no conflict of interest.
Additional information
Action Editor: Steven J. Schiff
Rights and permissions
About this article
Cite this article
Matchen, T.D., Moehlis, J. Phase model-based neuron stabilization into arbitrary clusters. J Comput Neurosci 44, 363–378 (2018). https://doi.org/10.1007/s10827-018-0683-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10827-018-0683-y