Dynamical Mechanism of Hyperpolarization-Activated Non-specific Cation Current Induced Resonance and Spike-Timing Precision in a Neuronal Model

Hyperpolarization-activated cyclic nucleotide-gated cation current (Ih) plays important roles in the achievement of many physiological/pathological functions in the nervous system by modulating the electrophysiological activities, such as the rebound (spike) to hyperpolarization stimulations, subthreshold membrane resonance to sinusoidal currents, and spike-timing precision to stochastic factors. In the present paper, with increasing gh (conductance of Ih), the rebound (spike) and subthreshold resonance appear and become stronger, and the variability of the interspike intervals (ISIs) becomes lower, i.e., the enhancement of spike-timing precision, which are simulated in a conductance-based theoretical model and well explained by the nonlinear concept of bifurcation. With increasing gh, the stable node to stable focus, to coexistence behavior, and to firing via the codimension-1 bifurcations (Hopf bifurcation, saddle-node bifurcation, saddle-node bifurcations on an invariant circle, and saddle homoclinic orbit) and codimension-2 bifurcations such as Bogdanov-Takens (BT) point related to the transition between saddle-node and Hopf bifurcations, are acquired with 1- and 2-parameter bifurcation analysis. The decrease of variability of ISIs with increasing gh is induced by the fast decrease of the standard deviation of ISIs, which is related to the increase of the capacity of resisting noisy disturbance due to the firing becomes far away from the bifurcation point. The enhancement of the rebound (spike) with increasing gh builds up a relationship to the decrease of the capacity of resisting disturbance like the hyperpolarization stimulus as the resting state approaches the bifurcation point. The “typical”-resonance and non-resonance appear in the parameter region of the stable focus and node far away from the bifurcation points, respectively. The complex or “strange” dynamics, such as the “weak”-resonance for the stable node near the transition point between the stable node and focus and the non-resonance for the stable focus close to the codimension-1 and −2 bifurcation points, are discussed.

[1]  Gergo Orbán,et al.  Intrinsic and synaptic mechanisms determining the timing of neuron population activity during hippocampal theta oscillation. , 2006, Journal of neurophysiology.

[2]  Jean Paul Maidana,et al.  Hyperpolarization-Activated Current Induces Period-Doubling Cascades and Chaos in a Cold Thermoreceptor Model , 2017, Front. Comput. Neurosci..

[3]  M. Danik,et al.  The Hippocamposeptal Pathway Generates Rhythmic Firing of GABAergic Neurons in the Medial Septum and Diagonal Bands: An Investigation Using a Complete Septohippocampal Preparation In Vitro , 2008, The Journal of Neuroscience.

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

[5]  J. Kwag,et al.  Frequency dependence of CA3 spike phase response arising from h-current properties , 2013, Front. Cell. Neurosci..

[6]  Emilie Campanac,et al.  The role of hyperpolarization‐activated cationic current in spike‐time precision and intrinsic resonance in cortical neurons in vitro , 2011, The Journal of physiology.

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

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

[9]  Kazuyuki Aihara,et al.  Ergodicity of Spike Trains: When Does Trial Averaging Make Sense? , 2003, Neural Computation.

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

[11]  Chun-I Yeh,et al.  Temporal precision in the neural code and the timescales of natural vision , 2007, Nature.

[12]  P. Jonas,et al.  Shunting Inhibition Improves Robustness of Gamma Oscillations in Hippocampal Interneuron Networks by Homogenizing Firing Rates , 2006, Neuron.

[13]  J. Kwag,et al.  Dendritic-targeting interneuron controls spike timing of hippocampal CA1 pyramidal neuron via activation of I h , 2012, Neuroscience Letters.

[14]  Boris S. Gutkin,et al.  Dynamics of Membrane Excitability Determine Interspike Interval Variability: A Link Between Spike Generation Mechanisms and Cortical Spike Train Statistics , 1998, Neural Computation.

[15]  L. Colgin.  Mechanisms and functions of theta rhythms. , 2013, Annual review of neuroscience.

[16]  H. Pape,et al.  Queer current and pacemaker: the hyperpolarization-activated cation current in neurons. , 1996, Annual review of physiology.

[17]  D. DiFrancesco,et al.  Action of the hyperpolarization-activated current (Ih) blocker ZD 7288 in hippocampal CA1 neurons , 1997, Pflügers Archiv.

[18]  G. Buzsáki,et al.  Memory, navigation and theta rhythm in the hippocampal-entorhinal system , 2013, Nature Neuroscience.

[19]  G Bard Ermentrout,et al.  Phase-response curves and synchronized neural networks , 2010, Philosophical Transactions of the Royal Society B: Biological Sciences.

[20]  Xiao-Jing Wang,et al.  Pacemaker neurons for the theta rhythm and their synchronization in the septohippocampal reciprocal loop. , 2002, Journal of neurophysiology.

[21]  Horacio G. Rotstein,et al.  Mechanisms of generation of membrane potential resonance in a neuron with multiple resonant ionic currents , 2017, bioRxiv.

[22]  Ruben A Tikidji-Hamburyan,et al.  Resonant Interneurons Can Increase Robustness of Gamma Oscillations , 2015, The Journal of Neuroscience.

[23]  Boris S. Gutkin,et al.  Dopamine Neurons Change the Type of Excitability in Response to Stimuli , 2016, bioRxiv.

[24]  Willy Govaerts,et al.  MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs , 2003, TOMS.

[25]  A. Bacci,et al.  Enhancement of Spike-Timing Precision by Autaptic Transmission in Neocortical Inhibitory Interneurons , 2006, Neuron.

[26]  Shyan-Shiou Chen,et al.  Application of a Two-Dimensional Hindmarsh-Rose Type Model for bifurcation Analysis , 2013, Int. J. Bifurc. Chaos.

[27]  J. Storm,et al.  Two forms of electrical resonance at theta frequencies, generated by M‐current, h‐current and persistent Na+ current in rat hippocampal pyramidal cells , 2002, The Journal of physiology.

[28]  Lisa M. Giocomo,et al.  Temporal Frequency of Subthreshold Oscillations Scales with Entorhinal Grid Cell Field Spacing , 2007, Science.

[29]  C. Reid,et al.  HCN channelopathies: pathophysiology in genetic epilepsy and therapeutic implications , 2012, British journal of pharmacology.

[30]  Szabolcs Káli,et al.  Differences in subthreshold resonance of hippocampal pyramidal cells and interneurons: the role of h-current and passive membrane characteristics , 2010, The Journal of physiology.

[31]  P. Somogyi,et al.  Defined types of cortical interneurone structure space and spike timing in the hippocampus , 2005, The Journal of physiology.

[32]  A. Hess,et al.  HCN2 channels account for mechanical (but not heat) hyperalgesia during long-standing inflammation , 2014, PAIN®.

[33]  Hiroyuki Kitajima,et al.  Bifurcations in Morris-Lecar neuron model , 2006, Neurocomputing.

[34]  S. Franceschetti,et al.  Recessive Loss-of-Function Mutation in the Pacemaker HCN2 Channel Causing Increased Neuronal Excitability in a Patient with Idiopathic Generalized Epilepsy , 2011, The Journal of Neuroscience.

[35]  L. Pinneo.  On noise in the nervous system. , 1966, Psychological review.

[36]  P. McNaughton,et al.  HCN2 ion channels: an emerging role as the pacemakers of pain. , 2012, Trends in pharmacological sciences.

[37]  Germán Mato,et al.  Analysis of the role of the low threshold currents IT and Ih in intrinsic delta oscillations of thalamocortical neurons , 2015, Front. Comput. Neurosci..

[38]  Hiroshi Kawakami,et al.  Bifurcations in Two-Dimensional Hindmarsh-rose Type Model , 2007, Int. J. Bifurc. Chaos.

[39]  Eve Marder,et al.  Ion channel degeneracy enables robust and tunable neuronal firing rates , 2015, Proceedings of the National Academy of Sciences.

[40]  Horacio G. Rotstein,et al.  Subthreshold amplitude and phase resonance in models of quadratic type: Nonlinear effects generated by the interplay of resonant and amplifying currents , 2015, Journal of Computational Neuroscience.

[41]  Erik De Schutter,et al.  Impact of Neuronal Properties on Network Coding: Roles of Spike Initiation Dynamics and Robust Synchrony Transfer , 2013, Neuron.

[42]  Rita Zemankovics,et al.  The presence of pacemaker HCN channels identifies theta rhythmic GABAergic neurons in the medial septum , 2008, The Journal of physiology.

[43]  Terrence J. Sejnowski,et al.  Biophysical Basis for Three Distinct Dynamical Mechanisms of Action Potential Initiation , 2008, PLoS Comput. Biol..

[44]  P. Somogyi,et al.  Brain-state- and cell-type-specific firing of hippocampal interneurons in vivo , 2003, Nature.

[45]  T. Albright,et al.  Efficient Discrimination of Temporal Patterns by Motion-Sensitive Neurons in Primate Visual Cortex , 1998, Neuron.

[46]  N. Kraus,et al.  Learning to Encode Timing: Mechanisms of Plasticity in the Auditory Brainstem , 2009, Neuron.

[47]  Jordan D. T. Engbers,et al.  Bistability in Purkinje neurons: Ups and downs in cerebellar research , 2013, Neural Networks.

[48]  Reza Tadayonnejad,et al.  Distinct roles for IT and IH in controlling the frequency and timing of rebound spike responses , 2011, The Journal of physiology.

[49]  G. Buzsáki,et al.  Inhibition-Induced Theta Resonance in Cortical Circuits , 2013, Neuron.

[50]  Takeshi Aihara,et al.  Mathematical Modeling of Subthreshold Resonant Properties in Pyloric Dilator Neurons , 2015, BioMed research international.

[51]  H. Sompolinsky,et al.  Bistability of cerebellar Purkinje cells modulated by sensory stimulation , 2005, Nature Neuroscience.

[52]  Zhiguo Zhao,et al.  Transitions between classes of neuronal excitability and bifurcations induced by autapse , 2017, Scientific Reports.

[53]  R. Reid,et al.  Precise Firing Events Are Conserved across Neurons , 2002, The Journal of Neuroscience.

[54]  Victoria Booth,et al.  Cellularly-Driven Differences in Network Synchronization Propensity Are Differentially Modulated by Firing Frequency , 2011, PLoS Comput. Biol..

[55]  T. Redman.  The Impact of , 1998 .

[56]  J. Hindmarsh,et al.  A model of intrinsic and driven spindling in thalamocortical neurons. , 1994, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

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

[58]  Arjen van Ooyen,et al.  H-Channels Affect Frequency, Power and Amplitude Fluctuations of Neuronal Network Oscillations , 2015, Front. Comput. Neurosci..

[59]  Zhian Hu,et al.  Neurophysiology of HCN channels: From cellular functions to multiple regulations , 2014, Progress in Neurobiology.

[60]  Zhiguo Zhao,et al.  Bifurcations and enhancement of neuronal firing induced by negative feedback , 2016 .

[61]  D. Kullmann,et al.  Ih-mediated depolarization enhances the temporal precision of neuronal integration , 2011, Nature communications.

[62]  Johannes Köhler,et al.  Persistent Discharges in Dentate Gyrus Perisoma-Inhibiting Interneurons Require Hyperpolarization-Activated Cyclic Nucleotide-Gated Channel Activation , 2015, The Journal of Neuroscience.

[63]  Michiel W. H. Remme,et al.  Subthreshold resonance properties contribute to the efficient coding of auditory spatial cues , 2014, Proceedings of the National Academy of Sciences.

[64]  Giorgio A. Ascoli,et al.  Local Control of Postinhibitory Rebound Spiking in CA1 Pyramidal Neuron Dendrites , 2010, The Journal of Neuroscience.

[65]  Bin Deng,et al.  Input-output relation and energy efficiency in the neuron with different spike threshold dynamics , 2015, Front. Comput. Neurosci..

[66]  Eduardo Ros,et al.  Spike Timing Regulation on the Millisecond Scale by Distributed Synaptic Plasticity at the Cerebellum Input Stage: A Simulation Study , 2013, Front. Comput. Neurosci..

[67]  C. Wahl-Schott,et al.  HCN channels: Structure, cellular regulation and physiological function , 2009, Cellular and Molecular Life Sciences.

[68]  H. Rotstein.  Preferred frequency responses to oscillatory inputs in an electrochemical cell model: linear amplitude and phase resonance. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[69]  Rodolphe Sepulchre,et al.  An Organizing Center in a Planar Model of Neuronal Excitability , 2012, SIAM J. Appl. Dyn. Syst..

[70]  A. Roque,et al.  Interplay of activation kinetics and the derivative conductance determines resonance properties of neurons. , 2017, Physical review. E.

[71]  Danko Nikolić,et al.  Membrane Resonance Enables Stable and Robust Gamma Oscillations , 2012, Cerebral cortex.

[72]  D. DiFrancesco,et al.  Dysfunctional HCN ion channels in neurological diseases , 2015, Front. Cell. Neurosci..

[73]  A. Hoffman,et al.  Impaired nigrostriatal function precedes behavioral deficits in a genetic mitochondrial model of Parkinson's disease , 2011, FASEB journal : official publication of the Federation of American Societies for Experimental Biology.

[74]  O. Paulsen,et al.  Spike Timing of Distinct Types of GABAergic Interneuron during Hippocampal Gamma Oscillations In Vitro , 2004, The Journal of Neuroscience.

[75]  A. Masi,et al.  MPP+‐dependent inhibition of Ih reduces spontaneous activity and enhances EPSP summation in nigral dopamine neurons , 2013, British journal of pharmacology.

[76]  Horacio G. Rotstein,et al.  Frequency preference in two-dimensional neural models: a linear analysis of the interaction between resonant and amplifying currents , 2013, Journal of Computational Neuroscience.

[77]  T. Sejnowski,et al.  Neurocomputational models of working memory , 2000, Nature Neuroscience.

[78]  Y. Yarom,et al.  Resonance, oscillation and the intrinsic frequency preferences of neurons , 2000, Trends in Neurosciences.

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

[80]  H. Robinson,et al.  Threshold firing frequency-current relationships of neurons in rat somatosensory cortex: type 1 and type 2 dynamics. , 2004, Journal of neurophysiology.