Optimal entrainment of neural oscillator ensembles

In this paper, we derive the minimum-energy periodic control that entrains an ensemble of structurally similar neural oscillators to a desired frequency. The state-space representation of a nominal oscillator is reduced to a phase model by computing its limit cycle and phase response curve, from which the optimal control is derived by using formal averaging and the calculus of variations. We focus on the case of a 1:1 entrainment ratio and suggest a simple numerical method for approximating the optimal controls. The method is applied to asymptotically control the spiking frequency of neural oscillators modeled using the Hodgkin-Huxley equations. Simulations are used to illustrate the optimality of entrainment controls derived using phase models when applied to the original state-space system, which is crucial for using phase models in control synthesis for practical applications. This work addresses a fundamental problem in the field of neural dynamics and provides a theoretical contribution to the optimal frequency control of uncertain oscillating systems.

[1]  H. Eichenbaum,et al.  Oscillatory Entrainment of Striatal Neurons in Freely Moving Rats , 2004, Neuron.

[2]  Leonidas D. Iasemidis,et al.  Control of Synchronization of Brain Dynamics leads to Control of Epileptic Seizures in Rodents , 2009, Int. J. Neural Syst..

[3]  P. Olver.  Nonlinear Systems , 2013 .

[4]  L. Perko.  Differential Equations and Dynamical Systems , 1991 .

[5]  Ghanim Ullah,et al.  Tracking and control of neuronal Hodgkin-Huxley dynamics. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Ilʹi︠a︡ Izrailevich Blekhman,et al.  Synchronization in science and technology , 1988 .

[7]  A. Peressini,et al.  The Mathematics Of Nonlinear Programming , 1988 .

[8]  S. V. Fomin,et al.  Ergodic Theory , 1982 .

[9]  S. Brendle,et al.  Calculus of Variations , 1927, Nature.

[10]  Aronson,et al.  Entrainment regions for periodically forced oscillators. , 1986, Physical review. A, General physics.

[11]  S. Strogatz.  Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering , 1995 .

[12]  E. Izhikevich,et al.  Oscillatory Neurocomputers with Dynamic Connectivity , 1999 .

[13]  Jürgen Kurths,et al.  Synchronization in small-world networks. , 2008, Chaos.

[14]  Jr-Shin Li,et al.  Optimal Asymptotic Entrainment of Phase-Reduced Oscillators , 2011, ArXiv.

[15]  Jr-Shin Li,et al.  Optimal design of minimum-power stimuli for phase models of neuron oscillators. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Ingo Fischer,et al.  Synchronization of chaotic semiconductor laser dynamics on subnanosecond time scales and its potential for chaos communication , 2000 .

[17]  Bard Ermentrout,et al.  Type I Membranes, Phase Resetting Curves, and Synchrony , 1996, Neural Computation.

[18]  J. Moehlis,et al.  On the Response of Neurons to Sinusoidal Current Stimuli: Phase Response Curves and Phase-Locking , 2006, Proceedings of the 45th IEEE Conference on Decision and Control.

[19]  Warren M. Grill,et al.  Selection of stimulus parameters for deep brain stimulation , 2004, Clinical Neurophysiology.

[20]  Brian H. Houston,et al.  Frequency entrainment for micromechanical oscillator , 2003 .

[21]  H. Pinsker.  Aplysia bursting neurons as endogenous oscillators. I. Phase-response curves for pulsed inhibitory synaptic input. , 1977, Journal of neurophysiology.

[22]  Jürgen Kurths,et al.  Synchronization - A Universal Concept in Nonlinear Sciences , 2001, Cambridge Nonlinear Science Series.

[23]  Jr-Shin Li.  Control of Inhomogeneous Ensembles , 2006 .

[24]  Eric Shea-Brown,et al.  On the Phase Reduction and Response Dynamics of Neural Oscillator Populations , 2004, Neural Computation.

[25]  Satoshi Nakata,et al.  Arnold tongue of electrochemical nonlinear oscillators. , 2009, The journal of physical chemistry. A.

[26]  F. Hanson.  Comparative studies of firefly pacemakers. , 1978, Federation proceedings.

[27]  S. Strogatz,et al.  Synchronization of pulse-coupled biological oscillators , 1990 .

[28]  Allan Peterson,et al.  The Theory of Differential Equations: Classical and Qualitative , 2003 .

[29]  P. McClintock.  Synchronization:a universal concept in nonlinear science , 2003 .

[30]  Denis Efimov,et al.  Phase Resetting Control Based On Direct Phase Response Curve , 2010 .

[31]  Lai-Sang Young,et al.  Ergodic Theory of Differentiable Dynamical Systems , 1995 .

[32]  A. Hodgkin,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve , 1952, The Journal of physiology.

[33]  Ali Nabi,et al.  Charge-Balanced Optimal Inputs for Phase Models of Spiking Neurons , 2009 .

[34]  S. Strogatz.  From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators , 2000 .

[35]  Yoshiki Kuramoto,et al.  Chemical Oscillations, Waves, and Turbulence , 1984, Springer Series in Synergetics.

[36]  Peter A. Tass,et al.  A model of desynchronizing deep brain stimulation with a demand-controlled coordinated reset of neural subpopulations , 2003, Biological Cybernetics.

[37]  Hanson Fe.  Comparative studies of firefly pacemakers. , 1978 .

[38]  Takahiro Harada,et al.  Optimal waveform for the entrainment of a weakly forced oscillator. , 2010, Physical review letters.

[39]  Jürgen Kurths,et al.  Synchronization: Phase locking and frequency entrainment , 2001 .

[40]  Bard Ermentrout,et al.  Simulating, analyzing, and animating dynamical systems - a guide to XPPAUT for researchers and students , 2002, Software, environments, tools.

[41]  Monika Sharma,et al.  Chemical oscillations , 2006 .

[42]  J. D. Hunter,et al.  Amplitude and frequency dependence of spike timing: implications for dynamic regulation. , 2003, Journal of neurophysiology.

[43]  Kurths,et al.  Phase synchronization of chaotic oscillators. , 1996, Physical review letters.

[44]  E. Izhikevich.  Weakly Coupled Oscillators , 2006 .

[45]  G. BARD.  Beyond . a pacemaker ’ s entrainment limit : phase walk-through , .

[46]  Denis V. Efimov,et al.  Controlling the phase of an oscillator: A phase response curve approach , 2009, Proceedings of the 48h IEEE Conference on Decision and Control (CDC) held jointly with 2009 28th Chinese Control Conference.

[47]  E. Izhikevich,et al.  Weakly connected neural networks , 1997 .

[48]  T. Aprille,et al.  A computer algorithm to determine the steady-state response of nonlinear oscillators , 1972 .

[49]  G. M.,et al.  Theory of Differential Equations , 1902, Nature.

[50]  Sean M Montgomery,et al.  Entrainment of Neocortical Neurons and Gamma Oscillations by the Hippocampal Theta Rhythm , 2008, Neuron.

[51]  G. Ermentrout.  n:m Phase-locking of weakly coupled oscillators , 1981 .

[52]  X.L. Chen,et al.  Deep Brain Stimulation , 2013, Interventional Neurology.

[53]  W. Govaerts,et al.  Computation of the Phase Response Curve: A Direct Numerical Approach , 2006, Neural Computation.

[54]  John L Hudson,et al.  Emerging Coherence in a Population of Chemical Oscillators , 2002, Science.