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Existence and Stability Criteria for Phase-Locked Modes in Ring Networks Using Phase-Resetting Curves and Spike Time Resetting Curves

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Book cover Phase Response Curves in Neuroscience

Part of the book series: Springer Series in Computational Neuroscience ((NEUROSCI,volume 6))

Abstract

Both phase resetting curves (PRCs) and spike time resetting curves (STRCs) are tools used for predicting the phase locked modes in neural networks and investigating their stability. The PRC tabulates the relative change in the duration of the interspike intervals due to a presynaptic input. In networks of neurons that generate brief action potentials (spiking neurons) with pulsatile couplings, the first order PRC, which accounts only for the change in the duration of the interspike interval that includes the perturbation itself, accurately predicts the phase-locked modes. Although the STRC-based method is equivalent to the first order PRC, it has the advantages of unambiguously defining the temporal intervals in networks of dissimilar neurons and a straightforward geometric interpretation. When using the PRC to predict the phase-locked modes it is generally assumed that (1) the open loop stimulus is identical to the recursive inputs received by the same neuron in closed loop setup, (2) the isolated neural oscillator returns to its unperturbed limit cycle before receiving the next input, and (3) each neuron receives only one presynaptic input and sends its output to only one other neuron in the network. These assumptions limit the applicability of the PRC and the STRC methods to 1:1 phase-locked modes to ring networks. For spiking neurons, the PRC/STRC-based theoretical predictions regarding the phase-locked modes and their stability were in very good agreement with the closed loop numerical simulations. Since neurons that generate sustained and closely spaced trains of action potentials (bursts) maintain their postsynaptic coupling over a significant duration, their influence on the postsynaptic neurons extend beyond the first order PRC. For bursting neurons, the first two assumptions were successfully relaxed by considering higher order PRCs and the burst resetting curves (BRCs) in predicting the phase-locked modes in hybrid networks controlled via dynamic clamp.

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References

  • Aiken SP, Kuenzi FM, Dale N (2003) Xenopus embryonic spinal neurons recorded in situ with patch-clamp electrodes – conditional oscillators after all? European Journal of Neuroscience 18:333–343.

    Article  PubMed  Google Scholar 

  • Arshavsky Y, Deliagina TG, Orlovsky GN, Yu V. P (1989) Control of feeding movements in the pteropod mollusc, Clione limacina. Exp Brain Res 78:387–397.

    PubMed  Google Scholar 

  • Arshavsky YI, Deliagina TG, Orlovsky GN (1997) Pattern generation. Current Opinion in Neurobiology 7:781–789.

    Article  PubMed  CAS  Google Scholar 

  • Bal T, Nagy F, Moulins M (1988) The pyloric central pattern generator in Crustacea: a set of conditional neuronal oscillators. Journal of Comparative Physiology A: Neuroethology, Sensory, Neural, and Behavioral Physiology 163:715–727.

    Article  Google Scholar 

  • Bartos M, Manor Y, Nadim F, Marder E, M/P. N (1999) Coordination of fast and slow rhythmic neuronal circuits. J Neurosci 19:6650–6660.

    Google Scholar 

  • Brown TG (1914) On the nature of the fundamental activity of the nervous centres; together with an analysis of the conditioning of rhythmic activity in progression, and a theory of the evolution of function in the nervous system. J Physiol 48:18–46.

    PubMed  CAS  Google Scholar 

  • Buono P-L (2001) Models of central pattern generators for quadruped locomotion II. Secondary gaits. Journal of Mathematical Biology 42:327–346.

    Article  CAS  Google Scholar 

  • Buono PL, Golubitsky M, Palacios A (2000) Heteroclinic cycles in rings of coupled cells. Physica D 143:74–108.

    Article  Google Scholar 

  • Calabrese RL (1995) Oscillation in motor pattern-generating networks. Current Opinion in Neurobiology 5:816–823.

    Article  PubMed  CAS  Google Scholar 

  • Calabrese RL, Nadim F, Olsen OH (1995) Heartbeat control in the medicinal leech: a model system for understanding the origin, coordination, and modulation of rhythmic motor patterns. J Neurobiol 27:390–402.

    Article  PubMed  CAS  Google Scholar 

  • Canavier CC, Baxter DA, Clark JW, Byrne JH (1999) Control of multistability in ring circuits of oscillators. Biological Cybernetics 80:87–102.

    Article  PubMed  CAS  Google Scholar 

  • Canavier CC, Butera RJ, Dror RO, Baxter DA, Clark JW, Byrne JH (1997) Phase response characteristics of model neurons determine which patterns are expressed in a ring circuit model of gait generation. Biol Cybern 77:367–380.

    Article  PubMed  CAS  Google Scholar 

  • Cheron G, Cebolla AM, Saedeleer CD, Bengoetxea A, Leurs F, Leroy A, Dan B (2007) Pure phase-locking of beta/gamma oscillation contributes to the N30 frontal component of somatosensory evoked potentials. BMC Neuroscience 8.

    Google Scholar 

  • Cherry EM, Fenton FH (2007) A tale of two dogs: analyzing two models of canine ventricular electrophysiology. Am J Physiol 292:H43–H55.

    CAS  Google Scholar 

  • Dror RO, Canavier CC, Butera RJ, Clark JW, Byrne JH (1999) A mathematical criterion based on phase response curves for stability in a ring of coupled oscillators. Biological Cybernetics 80:11–23.

    Article  PubMed  CAS  Google Scholar 

  • Efimov DV (2011) Phase resetting control based on direct phase response curve. Journal of Mathematical Biology 63(5):855–879.

    Article  PubMed  CAS  Google Scholar 

  • Eldridge FL (1991) Phase resetting of respiratory rhythm – experiments in animals and models. In: Springer Series in Synergetics, Rhythms in Physiological Systems, pp 165–175. Berlin: Springer-Verlag.

    Chapter  Google Scholar 

  • Ermentrout B (1996) Type I Membranes, Phase Resetting Curves, and Synchrony. Neural Computation 8:979–1001.

    Article  PubMed  CAS  Google Scholar 

  • Galan RF, Ermentrout GB, Urban NN (2005) Efficient Estimation of Phase-Resetting Curves in Real Neurons and its Significance for Neural-Network Modeling. Physical Review Letters 94:158101.

    Article  PubMed  Google Scholar 

  • Golubitsky M, Stewart I, Buono P-L, Collins JJ (1998) A modular network for legged locomotion. Physica D: Nonlinear Phenomena 115:56–72.

    Article  Google Scholar 

  • Golubitsky M, Stewart I, Buono P-L, Collins JJ (1999) Symmetry in locomotor central pattern generators and animal gaits. Nature 401.

    Google Scholar 

  • Govaerts W, Sautois B (2006) Computation of the Phase Response Curve: A Direct Numerical Approach. Neural Computation 18:817–847.

    Article  PubMed  CAS  Google Scholar 

  • Haenschel C, Linden DE, Bittner RA, Singer W, Hanslmayr S (2010) Alpha phase locking predicts residual working memory performance in schizophrenia. Biol Psychiatry 68:595–598.

    Article  PubMed  Google Scholar 

  • Harris-Warrick RM, Marder E, Selverston AI, Moulins M (1992) Dynamic Biological Networks. The Stomatogastric Nervous System. Cambridge: MIT Press

    Google Scholar 

  • Hartline D (1979) Pattern generation in the lobster (Panulirus) stomatogastric ganglion. II. Pyloric network simulation. Biol Cybern 33:223–236.

    CAS  Google Scholar 

  • Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117:500–544.

    PubMed  CAS  Google Scholar 

  • Hooper SL, DiCaprio RA (2004) Crustacean motor pattern generator networks. Neurosignals 13:50–69.

    Article  PubMed  CAS  Google Scholar 

  • Hoppensteadt FC, Izhikevich EM (1997) Weakly connected neural networks. New York: Springer-Verlag.

    Book  Google Scholar 

  • Huerta R, Sánchez-Montañés MA, Corbacho F, Sigüenza JA (2000) A Central pattern generator to control a pyloric-based system. Biological Cybernetics 82:85–94.

    Article  PubMed  CAS  Google Scholar 

  • Ijspeert AJ (2008) Central pattern generators for locomotion control in animals and robots: A review. Neural Networks 21:642–653.

    Article  PubMed  Google Scholar 

  • Izhikevich EM (2007) Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting: MIT Press, Cambridge, Massachusetts & London, England.

    Google Scholar 

  • Kalb SS, Dobrovolny HM, Tolkacheva EG, Idriss SF, Krassowska W, Gauthier DJ (2004) The restitution portrait: a new method for investigating rate-dependent restitution. J Cardiovasc Electrophysiol 15:698–709.

    Article  PubMed  Google Scholar 

  • Kopell N, Ermentrout B (1988) Coupled oscillators and the design of central pattern generators. Mathematical Biology 90:87–109.

    Google Scholar 

  • Marder E, Bucher D (2001) Central pattern generators and the control of rhythmic movements. Current Biology 11:R986–R996.

    Article  PubMed  CAS  Google Scholar 

  • Marder E, Bucher D (2007) Understanding circuit dynamics using the stomatogastric nervous system of lobsters and crabs. Annu Rev Physiol 69:291–316.

    Article  PubMed  CAS  Google Scholar 

  • Miller JP, Selverston AI (1982) Mechanisms underlying pattern generation in lobster stomatogastric ganglion as determined by selective inactivation of identified neurons. II. Oscillatory properties of pyloric neurons J Physiol 48:1378–1391.

    CAS  Google Scholar 

  • Morris C, Lecar H (1981) Voltage oscillations in the barnacle giant muscle fiber. Biophysical Journal 35:193–213.

    Article  PubMed  CAS  Google Scholar 

  • Nakanishi J, Morimoto J, Endo G, Cheng G, Schaal S, Kawato M (2004) Learning from demonstration and adaptation of biped locomotion. Robotics and Autonomous Systems 47:79–91.

    Article  Google Scholar 

  • Oprisan S. A, Canavier CC (2005) Stability criterion for a two-neuron reciprocally coupled network based on the phase and burst resetting curves. Neurocomputing 65:733–739.

    Article  Google Scholar 

  • Oprisan SA (2009) Stability of Synchronous Oscillations in a Periodic Network. International Journal of Neuroscience 119:482–491.

    Article  PubMed  Google Scholar 

  • Oprisan SA (2010) Existence and stability criteria for phase-locked modes in ring neural networks based on the spike time resetting curve method. Journal of Theoretical Biology 262:232–244.

    Article  PubMed  Google Scholar 

  • Oprisan SA, Canavier CC (2000) Phase response curve via multiple time scale analysis of limit cycle behavior of type I and type II excitability. Biophysical Journal 78:218–230.

    Google Scholar 

  • Oprisan SA, Canavier CC (2001) Stability Analysis of Rings of Pulse-Coupled Oscillators: The Effect of Phase Resetting in the Second Cycle After the Pulse Is Important at Synchrony and For Long Pulses. Differential Equations and Dynamical Systems 9:243–258.

    Google Scholar 

  • Oprisan SA, Canavier CC (2002) The Influence of Limit Cycle Topology on the Phase Resetting Curve. Neural Computation 14:1027–1057.

    Article  PubMed  Google Scholar 

  • Oprisan SA, Canavier CC (2003) Stability analysis of entrainment by two periodic inputs with a fixed delay. Neurocomputing 52–54:59–63.

    Article  Google Scholar 

  • Oprisan SA, Canavier CC (2006) Technique for eliminating nonessential components in the refinement of a model of dopamine neurons. Neurocomputing 69:1030–1034.

    Article  Google Scholar 

  • Oprisan SA, Boutan C (2008) Prediction of Entrainment And 1:1 Phase-Locked Modes in Two-Neuron Networks Based on the Phase Resetting Curve Method. International Journal of Neuroscience 118:867–890.

    Article  PubMed  Google Scholar 

  • Oprisan SA, Thirumalai V, Canavier CC (2003) Dynamics from a Time Series: Can We Extract the Phase Resetting Curve from a Time Series? Biophys J 84:2919–2928.

    Article  PubMed  CAS  Google Scholar 

  • Oprisan SA, Prinz AA, Canavier CC (2004) Phase resetting and phase locking in hybrid circuits of one model and one biological neuron. Biophys J 87:2283–2298.

    Article  PubMed  CAS  Google Scholar 

  • Proctor J, Holmes P (2010) Reflexes and preflexes: on the role of sensory feedback on rhythmic patterns in insect locomotion. Biological Cybernetics 102:513–531.

    Article  PubMed  CAS  Google Scholar 

  • Proctor J, Kukillaya RP, Holmes P (2010) A phase-reduced neuro-mechanical model for insect locomotion: feed-forward stability and proprioceptive feedback. Phil Trans R Soc A 368:5087–5104

    Article  PubMed  CAS  Google Scholar 

  • Rutishauser U, Ross IB, Mamelak AN, Schuman EM (2010) Human memory strength is predicted by theta-frequency phase-locking of single neurons. Nature 464:903–907.

    Article  PubMed  CAS  Google Scholar 

  • Taiga Y, Taishin N, Shunsuke S (2003) Possible functional roles of phase resetting during walking. Biological Cybernetics 88:468–496.

    Google Scholar 

  • Tazaki K, Cooke IM (1990) Characterization of Ca current underlying burst formation in lobster cardiac ganglion motorneurons. Journal of Neurophysiology 63:370–384.

    PubMed  CAS  Google Scholar 

  • Thuma JB, Morris LG, Weaver AL, Hooper SL (2003) Lobster (Panulirus interruptus) Pyloric Muscles Express the Motor Patterns of Three Neural Networks, Only One of Which Innervates the Muscles. The Journal of Neuroscience 23:8911–8920.

    PubMed  CAS  Google Scholar 

  • Tolkacheva EG, Schaeffer DG, Gauthier DJ, Krassowska W (2003) Condition for alternans and stability of the 1:1 response pattern in a “memory” model of paced cardiac dynamics. Phys Rev E 67(3 Pt 1):031904.

    Article  CAS  Google Scholar 

  • Wingerden MV, Vinck M, Lankelma JV, Pennartz CMA (2010) Learning-Associated Gamma-Band Phase-Locking of Action–Outcome Selective Neurons in Orbitofrontal Cortex. Journal of Neuroscience 30:10025–10038.

    Article  PubMed  Google Scholar 

  • Xu X (2008) Complicated dynamics of a ring neural network with time delays. Journal of Physics A: Mathematical and Theoretical 41:035102.

    Google Scholar 

  • Zacksenhouse M, Ahissar E (2006) Temporal decoding by phase-locked loops: unique features of circuit-level implementations and their significance for vibrissal information processing. Neural Computation 18:1611–1636.

    Article  PubMed  Google Scholar 

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Acknowledgements

I would like to thank my former postdoc advisor, Dr. Carmen C. Canavier, for introducing me to the fascinating field of phase resetting, and for the freedom she has allowed me in pursuing my own ideas and interests. I deeply appreciate the guidance, support, and inspiration she has given me.

I am especially grateful to my wife, Dr. Ana Oprisan, for standing beside me throughout my career. Her patience and unwavering support made this work possible. I also thank my wonderful children: Andrei and Andra for their understanding and good humor. I dedicate this work to the memory of mother-in-law and both my parents.

I gratefully acknowledge the helpful comments and feedback I received from reviewers while preparing this chapter.

This work was partly supported by the National Science Foundation CAREER award IOS – 1054914 to SAO.

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Authors and Affiliations

  1. Department of Physics and Astronomy, College of Charleston, 66 George Street, Charleston, SC, 29424, USA

    Sorinel Adrian Oprisan

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  1. Sorinel Adrian Oprisan
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Correspondence to Sorinel Adrian Oprisan .

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  1. 2 Cumminton St., Rm 109, Boston, 02215, Massachusetts, USA

    Nathan W. Schultheiss

  2. Dept. Biology, Emory University, Clifton Road 1510, Atlanta, 30322, Georgia, USA

    Astrid A. Prinz

  3. School of Electrical &, Computer Engineering, Georgia Institute of Technology, Ferst Drive 313, Atlanta, 30332-0535, Georgia, USA

    Robert J. Butera

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Oprisan, S.A. (2012). Existence and Stability Criteria for Phase-Locked Modes in Ring Networks Using Phase-Resetting Curves and Spike Time Resetting Curves. In: Schultheiss, N., Prinz, A., Butera, R. (eds) Phase Response Curves in Neuroscience. Springer Series in Computational Neuroscience, vol 6. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-0739-3_17

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