Periodic Forcing of Inhibition-Stabilized Networks: Nonlinear Resonances and Phase-Amplitude Coupling

Inhibition-stabilized networks (ISNs) are neural architectures with strong positive feedback among pyramidal neurons balanced by strong negative feedback from inhibitory interneurons, a circuit element found in the hippocampus and the primary visual cortex. In their working regime, ISNs produce damped oscillations in the -range in response to inputs to the inhibitory population. In order to understand the properties of interconnected ISNs, we investigated periodic forcing of ISNs. We show that ISNs can be excited over a range of frequencies and derive properties of the resonance peaks. In particular, we studied the phase-locked solutions, the torus solutions, and the resonance peaks. Periodically forced ISNs respond with (possibly multistable) phase-locked activity, whereas networks with sustained intrinsic oscillations respond more dynamically to periodic inputs with tori. Hence, the dynamics are surprisingly rich, and phase effects alone do not adequately describe the network response. This strengthens the importance of phase-amplitude coupling as opposed to phase-phase coupling in providing multiple frequencies for multiplexing and routing information.

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

[2]  F. Takens,et al.  Occurrence of strange AxiomA attractors near quasi periodic flows onTm,m≧3 , 1978 .

[3]  J. Neu.  Coupled Chemical Oscillators , 1979 .

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

[5]  Vladimir Igorevich Arnold,et al.  Geometrical Methods in the Theory of Ordinary Differential Equations , 1983 .

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

[7]  J. Gambaudo.  Perturbation of a Hopf Bifurcation by an External Time- Periodic Forcing , 1985 .

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

[9]  C. Elphick,et al.  Normal form reduction for time-periodically driven differential equations , 1987 .

[10]  G. Ermentrout,et al.  Amplitude response of coupled oscillators , 1990 .

[11]  R M Borisyuk,et al.  Dynamics and bifurcations of two coupled neural oscillators with different connection types , 1995, Bulletin of mathematical biology.

[12]  David Ruelle,et al.  Occurrence of Strange Axiom A Attractors Near Quasi Periodic Flows on T m , m ≧ 3 , 1995 .

[13]  G. Buzsáki,et al.  Temporal structure in spatially organized neuronal ensembles: a role for interneuronal networks , 1995, Current Opinion in Neurobiology.

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

[15]  B. McNaughton,et al.  Paradoxical Effects of External Modulation of Inhibitory Interneurons , 1997, The Journal of Neuroscience.

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

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

[18]  V. W. Noonburg,et al.  A Periodically Forced Wilson--Cowan System , 2003, SIAM J. Appl. Math..

[19]  P. Landa.  Mechanism of stochastic resonance , 2004 .

[20]  Roman Borisyuk,et al.  Bifurcation analysis of a neural network model , 1992, Biological Cybernetics.

[21]  G. Buzsáki,et al.  Neuronal Oscillations in Cortical Networks , 2004, Science.

[22]  D. Lewis,et al.  Cortical inhibitory neurons and schizophrenia , 2005, Nature Reviews Neuroscience.

[23]  Frank Schilder,et al.  Continuation of Quasi-periodic Invariant Tori , 2005, SIAM J. Appl. Dyn. Syst..

[24]  D. Hansel,et al.  Role of delays in shaping spatiotemporal dynamics of neuronal activity in large networks. , 2005, Physical review letters.

[25]  Robert Desimone,et al.  Parallel and Serial Neural Mechanisms for Visual Search in Macaque Area V4 , 2005, Science.

[26]  T. Sejnowski,et al.  Network Oscillations: Emerging Computational Principles , 2006, The Journal of Neuroscience.

[27]  Eugene M. Izhikevich,et al.  Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting , 2006 .

[28]  F. Wagener.  Semi-global analysis of periodic and quasi-periodic k:1 and k:2 resonances , 2006 .

[29]  P. Jonas,et al.  Synaptic mechanisms of synchronized gamma oscillations in inhibitory interneuron networks , 2007, Nature Reviews Neuroscience.

[30]  O. Jensen,et al.  Cross-frequency coupling between neuronal oscillations , 2007, Trends in Cognitive Sciences.

[31]  Nicolas Brunel,et al.  Synchronization properties of networks of electrically coupled neurons in the presence of noise and heterogeneities , 2009, Journal of Computational Neuroscience.

[32]  Peter J. Siekmeier,et al.  Modeling GABA alterations in schizophrenia: a link between impaired inhibition and altered gamma and beta range auditory entrainment. , 2008, Journal of neurophysiology.

[33]  S. Coombes,et al.  Delays in activity-based neural networks , 2009, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[34]  T. Hafting,et al.  Frequency of gamma oscillations routes flow of information in the hippocampus , 2009, Nature.

[35]  Michael J. Shelley,et al.  LFP spectral peaks in V1 cortex: network resonance and cortico-cortical feedback , 2010, Journal of Computational Neuroscience.

[36]  Jessica A. Cardin,et al.  Driving fast-spiking cells induces gamma rhythm and controls sensory responses , 2009, Nature.

[37]  Evan S. Schaffer,et al.  Inhibitory Stabilization of the Cortical Network Underlies Visual Surround Suppression , 2009, Neuron.

[38]  Nancy Kopell,et al.  The Dynamics of a Periodically Forced Cortical Microcircuit, With an Application to Schizophrenia , 2009, SIAM J. Appl. Dyn. Syst..

[39]  M. Golubitsky,et al.  The Feed-Forward Chain as a Filter-Amplifier Motif , 2009 .

[40]  Donald A. Glaser,et al.  Synaptic Plasticity Controls Sensory Responses through Frequency-Dependent Gamma Oscillation Resonance , 2010, PLoS Comput. Biol..

[41]  M. Haragus,et al.  Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems , 2010 .

[42]  Dimitri M. Kullmann,et al.  Oscillations and Filtering Networks Support Flexible Routing of Information , 2010, Neuron.

[43]  F. Wagener,et al.  Semi-global analysis of periodic and quasi-periodic normal-internal k : 1 and k : 2 resonances , 2010 .

[44]  David Terman,et al.  Mathematical foundations of neuroscience , 2010 .

[45]  Bard Ermentrout,et al.  A Model for the Origin and Properties of Flicker-Induced Geometric Phosphenes , 2011, PLoS Comput. Biol..

[46]  J. Maunsell,et al.  Different Origins of Gamma Rhythm and High-Gamma Activity in Macaque Visual Cortex , 2011, PLoS biology.

[47]  Yanyan Zhang,et al.  Periodically Forced Hopf Bifurcation , 2011, SIAM J. Appl. Dyn. Syst..

[48]  Nicolas Brunel,et al.  Dynamics of Networks of Excitatory and Inhibitory Neurons in Response to Time-Dependent Inputs , 2011, Front. Comput. Neurosci..

[49]  Karin Schwab,et al.  Modeling Brain Resonance Phenomena Using a Neural Mass Model , 2011, PLoS Comput. Biol..

[50]  W. Singer,et al.  Orientation selectivity and noise correlation in awake monkey area V1 are modulated by the gamma cycle , 2012, Proceedings of the National Academy of Sciences.

[51]  G. Green,et al.  Gamma Activation in Young People with Autism Spectrum Disorders and Typically-Developing Controls When Viewing Emotions on Faces , 2012, PloS one.

[52]  D. Kullmann,et al.  Oscillatory dynamics in the hippocampus support dentate gyrus–CA3 coupling , 2012, Nature Neuroscience.

[53]  V. W. Noonburg,et al.  A Periodically Forced Wilson-Cowan System with Multiple Attractors , 2012, SIAM J. Math. Anal..

[54]  K. Miller,et al.  Exaggerated phase–amplitude coupling in the primary motor cortex in Parkinson disease , 2013, Proceedings of the National Academy of Sciences.

[55]  Stephen Coombes,et al.  Phase-Amplitude Descriptions of Neural Oscillator Models , 2013, Journal of mathematical neuroscience.

[56]  Arvind Kumar,et al.  Communication through Resonance in Spiking Neuronal Networks , 2014, PLoS Comput. Biol..

[57]  S. Tonegawa,et al.  Successful Execution of Working Memory Linked to Synchronized High-Frequency Gamma Oscillations , 2014, Cell.

[58]  Li Lu,et al.  Coordination of entorhinal–hippocampal ensemble activity during associative learning , 2014, Nature.

[59]  T. Sejnowski,et al.  Cortical oscillations arise from contextual interactions that regulate sparse coding , 2014, Proceedings of the National Academy of Sciences.

[60]  Terrence J. Sejnowski,et al.  Regulating Cortical Oscillations in an Inhibition-Stabilized Network , 2014, Proceedings of the IEEE.

[61]  Daniel B. Rubin,et al.  The Stabilized Supralinear Network: A Unifying Circuit Motif Underlying Multi-Input Integration in Sensory Cortex , 2015, Neuron.

[62]  Balanced Amplification: A New Mechanism of Selective Amplification of Neural Activity Patterns , 2016, Neuron.