On the Phase Reduction and Response Dynamics of Neural Oscillator Populations

We undertake a probabilistic analysis of the response of repetitively firing neural populations to simple pulselike stimuli. Recalling and extending results from the literature, we compute phase response curves (PRCs) valid near bifurcations to periodic firing for Hindmarsh-Rose, Hodgkin-Huxley, Fitz Hugh-Nagumo, and Morris-Lecar models, encompassing the four generic (codimension one) bifurcations. Phase density equations are then used to analyze the role of the bifurcation, and the resulting PRC, in responses to stimuli. In particular, we explore the interplay among stimulus duration, baseline firing frequency, and population-level response patterns. We interpret the results in terms of the signal processing measure of gain and discuss further applications and experimentally testable predictions.

[1]  J. Cohen,et al.  The role of locus coeruleus in the regulation of cognitive performance. , 1999, Science.

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

[3]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

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

[5]  John Rinzel,et al.  Dynamics of Spiking Neurons Connected by Both Inhibitory and Electrical Coupling , 2003, Journal of Computational Neuroscience.

[6]  Nicolas Brunel,et al.  Dynamics of the Firing Probability of Noisy Integrate-and-Fire Neurons , 2002, Neural Computation.

[7]  T. Sejnowski,et al.  Reliability of spike timing in neocortical neurons. , 1995, Science.

[8]  A. Winfree.  The geometry of biological time , 1991 .

[9]  A. Hodgkin,et al.  A quantitative description of membrane current and its application to conduction and excitation in nerve , 1990 .

[10]  C. Gray.  The Temporal Correlation Hypothesis of Visual Feature Integration Still Alive and Well , 1999, Neuron.

[11]  D. Hansel,et al.  Phase Dynamics for Weakly Coupled Hodgkin-Huxley Neurons , 1993 .

[12]  G. Aston-Jones,et al.  Locus coeruleus and regulation of behavioral flexibility and attention. , 2000, Progress in brain research.

[13]  Germán Mato,et al.  Synchrony in Excitatory Neural Networks , 1995, Neural Computation.

[14]  G. Aston-Jones,et al.  Locus coeruleus neurons in monkey are selectively activated by attended cues in a vigilance task , 1994, The Journal of neuroscience : the official journal of the Society for Neuroscience.

[15]  Michael Shub,et al.  The local theory of normally hyperbolic, invariant, compact manifolds , 1977 .

[16]  F E Bloom,et al.  Central catecholamine neuron systems: anatomy and physiology of the norepinephrine and epinephrine systems. , 1979, Annual review of neuroscience.

[17]  Yan Zhu,et al.  A neural circuit for circadian regulation of arousal , 2001, Nature Neuroscience.

[18]  Stephen Coombes,et al.  Dynamics of Strongly Coupled Spiking Neurons , 2000, Neural Computation.

[19]  Yoshiki Kuramoto,et al.  Phase- and Center-Manifold Reductions for Large Populations of Coupled Oscillators with Application to Non-Locally Coupled Systems , 1997 .

[20]  Wulfram Gerstner,et al.  Spiking Neuron Models , 2002 .

[21]  Frances S. Chance,et al.  Effects of synaptic noise and filtering on the frequency response of spiking neurons. , 2001, Physical review letters.

[22]  Eugene M. Izhikevich,et al.  Phase Equations for Relaxation Oscillators , 2000, SIAM J. Appl. Math..

[23]  J D Cohen,et al.  A network model of catecholamine effects: gain, signal-to-noise ratio, and behavior. , 1990, Science.

[24]  A. Winfree.  Patterns of phase compromise in biological cycles , 1974 .

[25]  P. Holmes,et al.  Globally Coupled Oscillator Networks , 2003 .

[26]  J. Cowan,et al.  Excitatory and inhibitory interactions in localized populations of model neurons. , 1972, Biophysical journal.

[27]  James P. Keener,et al.  Mathematical physiology , 1998 .

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

[29]  Robert J Butera,et al.  Neuronal oscillators in aplysia californica that demonstrate weak coupling in vitro. , 2005, Physical review letters.

[30]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[31]  R. Stein.  A THEORETICAL ANALYSIS OF NEURONAL VARIABILITY. , 1965, Biophysical journal.

[32]  Duane Q. Nykamp,et al.  A Population Density Approach That Facilitates Large-Scale Modeling of Neural Networks: Extension to Slow Inhibitory Synapses , 2001, Neural Computation.

[33]  W. Meck,et al.  Neuropsychological mechanisms of interval timing behavior. , 2000, BioEssays : news and reviews in molecular, cellular and developmental biology.

[34]  G. Ermentrout,et al.  Oscillator death in systems of coupled neural oscillators , 1990 .

[35]  G. Ermentrout,et al.  Frequency Plateaus in a Chain of Weakly Coupled Oscillators, I. , 1984 .

[36]  Wulfram Gerstner,et al.  Population Dynamics of Spiking Neurons: Fast Transients, Asynchronous States, and Locking , 2000, Neural Computation.

[37]  Kevin K. Lin,et al.  Entrainment and Chaos in a Pulse-Driven Hodgkin-Huxley Oscillator , 2005, SIAM J. Appl. Dyn. Syst..

[38]  E. Coddington,et al.  Theory of Ordinary Differential Equations , 1955 .

[39]  P. Dayan,et al.  Matching storage and recall: hippocampal spike timing–dependent plasticity and phase response curves , 2005, Nature Neuroscience.

[40]  P. Tass.  Phase Resetting in Medicine and Biology , 1999 .

[41]  Lawrence Sirovich,et al.  A Population Study of Integrate-and-Fire-or-Burst Neurons , 2002, Neural Computation.

[42]  Duane Q. Nykamp,et al.  A Population Density Approach That Facilitates Large-Scale Modeling of Neural Networks: Analysis and an Application to Orientation Tuning , 2004, Journal of Computational Neuroscience.

[43]  Wulfram Gerstner,et al.  What Matters in Neuronal Locking? , 1996, Neural Computation.

[44]  G. Ermentrout,et al.  Multiple pulse interactions and averaging in systems of coupled neural oscillators , 1991 .

[45]  Keith Turner.  Three Dimensional Model , 1991 .

[46]  Philip Holmes,et al.  The Influence of Spike Rate and Stimulus Duration on Noradrenergic Neurons , 2004, Journal of Computational Neuroscience.

[47]  T. Sejnowski,et al.  Dynamic Brain Sources of Visual Evoked Responses , 2002, Science.

[48]  Y. Kuznetsov.  Elements of applied bifurcation theory (2nd ed.) , 1998 .

[49]  Eugene M. Izhikevich,et al.  Neural excitability, Spiking and bursting , 2000, Int. J. Bifurc. Chaos.

[50]  Sang Gui Lee,et al.  Phase dynamics in the biological neural networks , 2000 .

[51]  Reinhard Eckhorn,et al.  Neural mechanisms of scene segmentation: recordings from the visual cortex suggest basic circuits for linking field models , 1999, IEEE Trans. Neural Networks.

[52]  Wulfram Gerstner,et al.  Spiking Neuron Models: An Introduction , 2002 .

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

[54]  J. Rinzel,et al.  Numerical calculation of stable and unstable periodic solutions to the Hodgkin-Huxley equations , 1980 .

[55]  G. Whitham,et al.  Linear and Nonlinear Waves , 1976 .

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

[57]  Michael C. Mackey,et al.  From Clocks to Chaos , 1988 .

[58]  Y. Kuznetsov.  Elements of Applied Bifurcation Theory , 2023, Applied Mathematical Sciences.

[59]  Neil Fenichel.  Persistence and Smoothness of Invariant Manifolds for Flows , 1971 .

[60]  E E Fetz,et al.  Relation between shapes of post‐synaptic potentials and changes in firing probability of cat motoneurones , 1983, The Journal of physiology.

[61]  Lawrence Sirovich,et al.  On the Simulation of Large Populations of Neurons , 2004, Journal of Computational Neuroscience.

[62]  Bard Ermentrout,et al.  When inhibition not excitation synchronizes neural firing , 1994, Journal of Computational Neuroscience.

[63]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[64]  J. Guckenheimer,et al.  Isochrons and phaseless sets , 1975, Journal of mathematical biology.

[65]  J. Hindmarsh,et al.  The assembly of ionic currents in a thalamic neuron I. The three-dimensional model , 1989, Proceedings of the Royal Society of London. B. Biological Sciences.

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

[67]  Wulfram Gerstner,et al.  Noise and the PSTH Response to Current Transients: I. General Theory and Application to the Integrate-and-Fire Neuron , 2001, Journal of Computational Neuroscience.

[68]  G. Ermentrout,et al.  Analysis of neural excitability and oscillations , 1989 .