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Journal of Computational Neuroscience

, Volume 46, Issue 2, pp 169–195 | Cite as

Membrane potential resonance in non-oscillatory neurons interacts with synaptic connectivity to produce network oscillations

  • Andrea Bel
  • Horacio G. RotsteinEmail author
Article

Abstract

Several neuron types have been shown to exhibit (subthreshold) membrane potential resonance (MPR), defined as the occurrence of a peak in their voltage amplitude response to oscillatory input currents at a preferred (resonant) frequency. MPR has been investigated both experimentally and theoretically. However, whether MPR is simply an epiphenomenon or it plays a functional role for the generation of neuronal network oscillations and how the latent time scales present in individual, non-oscillatory cells affect the properties of the oscillatory networks in which they are embedded are open questions. We address these issues by investigating a minimal network model consisting of (i) a non-oscillatory linear resonator (band-pass filter) with 2D dynamics, (ii) a passive cell (low-pass filter) with 1D linear dynamics, and (iii) nonlinear graded synaptic connections (excitatory or inhibitory) with instantaneous dynamics. We demonstrate that (i) the network oscillations crucially depend on the presence of MPR in the resonator, (ii) they are amplified by the network connectivity, (iii) they develop relaxation oscillations for high enough levels of mutual inhibition/excitation, and (iv) the network frequency monotonically depends on the resonators resonant frequency. We explain these phenomena using a reduced adapted version of the classical phase-plane analysis that helps uncovering the type of effective network nonlinearities that contribute to the generation of network oscillations. We extend our results to networks having cells with 2D dynamics. Our results have direct implications for network models of firing rate type and other biological oscillatory networks (e.g, biochemical, genetic).

Keywords

Preferred frequency response Latent time scales Inhibitory networks Neuronal filters 

Notes

Acknowledgments

This work was partially supported by the National Science Foundation grant DMS-1608077 (HGR) and the Universidad Nacional del Sur grant PGI 24/L096 (AB). The authors thank Eran Stark for useful comments and discussions. HGR is grateful to the Courant Institute of Mathematical Sciences at NYU and the Department of Mathematics at Universidad Nacional del Sur, Argentina.

References

  1. Ambrosio-Mouser, C., Nadim, F., Bose, A. (2006). The effects of varying the timing of inputs on a neural oscillator. SIAM Journal on Applied Dynamical Systems, 5, 108–139.CrossRefGoogle Scholar
  2. Art, J.J., Crawford, A.C., Fettiplace, R. (1986). Electrical resonance and membrane currents in turtle cochlear hair cells. Hearing Research, 22, 31–36.CrossRefGoogle Scholar
  3. Baroni, F., Burkitt, A.N., Grayden, D.B. (2014). Interplay of intrinsic and synaptic conductances in the generation of high-frequency oscillations in interneuronal networks with irregular spiking. PLoS Computational Biology, 10, e1003574.CrossRefGoogle Scholar
  4. Beatty, J., Song, S.C., Wilson, C.J. (2015). Cell-type-specific resonances shape the response of striatal neurons to synaptic inputs. Journal of Neurophysiology, 113, 688–700.CrossRefGoogle Scholar
  5. Beer, R. (1995). On the dynamics of small continuous-time recurrent neural networks. Adaptive Behavior, 4, 471–511.Google Scholar
  6. Boehlen, A., Heinemann, U., Erchova, I. (2010). The range of intrinsic frequencies represented by medial entorhinal cortex stellate cells extends with age. The Journal of Neuroscience, 30, 4585–4589.CrossRefGoogle Scholar
  7. Boehlen, A., Henneberger, C., Heinemann, U., Erchova, I. (2013). Contribution of near-threshold currents to intrinsic oscillatory activity in rat medial entorhinal cortex layer II, stellate cells. Journal of Neurophysiology, 109, 445–463.CrossRefGoogle Scholar
  8. Borgers, C. (2017). An introduction to modeling neuronal dynamics. Berlin: Springer.CrossRefGoogle Scholar
  9. Brea, J.N., Kay, L.M., Kopell, N.J. (2009). Biophysical model for gamma rhythms in the olfactory bulb via subthreshold oscillations. Proceedings of the National Academy of Sciences of the United States of America, 106, 21954–21959.CrossRefGoogle Scholar
  10. Burden, R.L., & Faires, J.D. (1980). Numerical analysis. Boston: PWS Publishing Company.Google Scholar
  11. Chen, Y., Li, X., Rotstein, H.G., gap, F. Nadim. (2016). Membrane potential resonance frequency directly influences network frequency through junctions. Journal of Neurophysiology, 116, 1554–1563.CrossRefGoogle Scholar
  12. Curtu, R., & Rubin, J. (2011). Interaction of canard and singular Hopf mechanisms in a neural model. SIAM Journal on Applied Dynamical Systems, 4, 1443–1479.CrossRefGoogle Scholar
  13. D’angelo, E., Nieus, T., Maffei, A., Armano, S., Rossi, P., Taglietti, V., Fontana, A., Naldi, G. (2001). Theta-frequency bursting and resonance in cerebellar granule cells: Experimental evidence and modeling of a slow K+ - dependent mechanism. The Journal of Neuroscience, 21, 759–770.CrossRefGoogle Scholar
  14. D’Angelo, E., Koekkoek, S.K.E., Lombardo, P., Solinas, S., Ros, E., Garrido, J., Schonewille, M., De Zeeuw, C.I. (2009). Timing in the cerebellum: oscillations and resonance in the granular layer. Neuroscience, 162, 805–815.CrossRefGoogle Scholar
  15. David, F., Courtiol, E., Buonviso, N., Fourcaud-Trocme, N. (2015). Competing mechanisms of gamma and beta oscillations in the olfactory bulb based on multimodal inhibition of mitral cells over a respiratory cycle. eNeuro, 2, e0018–15.2015.CrossRefGoogle Scholar
  16. Dayan, P., & Abbott, L.F. (2001). Theoretical Neuroscience. Cambridge: MIT Press.Google Scholar
  17. Dhooge, A., Govaerts, W., Kuznetsov, Yu.A. (2003). MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs. ACM Transactions on Mathematical Software, 29(2), 141–164.CrossRefGoogle Scholar
  18. Engel, T.A., Schimansky-Geier, L., Herz, A.V., Schreiber, S., Erchova, I. (2008). Subthreshold membrane-potential resonances shape spike-train patterns in the entorhinal cortex. Journal of Neurophysiology, 100, 1576–1588.CrossRefGoogle Scholar
  19. Erchova, I., Kreck, G., Heinemann, U., Herz, A.V.M. (2004). Dynamics of rat entorhinal cortex layer II and III cells: Characteristics of membrane potential resonance at rest predict oscillation properties near threshold. Journal of Physiology, 560, 89–110.CrossRefGoogle Scholar
  20. Ermentrout, G.B., & Terman, D. (2010). Mathematical foundations of neuroscience. Berlin: Springer.CrossRefGoogle Scholar
  21. Fox, D.M., Tseng, H., Smolinsky, T., Rotstein, H.G., Nadim, F. (2017). Mechanisms of generation of membrane potential resonance in a neuron with multiple resonant ionic currents. PLoS Computational Biology, 13, e1005565.CrossRefGoogle Scholar
  22. Gastrein, P., Campanac, E., Gasselin, C., Cudmore, R.H., Bialowas, A., Carlier, E., Fronzaroli-Molinieres, L., Ankri, N., Debanne, D. (2011). The role of hyperpolarization-activated cationic current in spike-time precision and intrinsic resonance in cortical neurons in vitro. The Journal of Physiology, 589, 3753–3773.CrossRefGoogle Scholar
  23. Gillies, M.J., Traub, R.D., LeBeau, F.E.N., Davies, C.H., Gloveli, T., Buhl, E.H., Whittington, M.A. (2002). A model of atropine-resistant theta oscillations in rat hippocampal area CA1. Journal of Physiology, 543.3, 779–793.CrossRefGoogle Scholar
  24. Guckenheimer, J., & Holmes, P. (1983). Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. New York: Springer.CrossRefGoogle Scholar
  25. Gutfreund, Y., Yarom, Y., Segev, I. (1995). Subthreshold oscillations and resonant frequency in Guinea pig cortical neurons: Physiology and modeling. The Journal of Physiology, 483, 621–640.CrossRefGoogle Scholar
  26. Heys, J.G., Giacomo, L.M., Hasselmo, M.E. (2010). Cholinergic modulation of the resonance properties of stellate cells in layer II, of the medial entorhinal. Journal of Neurophysiology, 104, 258–270.CrossRefGoogle Scholar
  27. Heys, J.G., Schultheiss, N.W., Shay, C.F., Tsuno, Y., Hasselmo, M.E. (2012). Effects of acetylcholine on neuronal properties in entorhinal cortex. Frontiers in Behavioral Neuroscience, 6, 32.CrossRefGoogle Scholar
  28. Higgs, M.H., & Spain, W.J. (2009). Conditional bursting enhances resonant firing in neocortical layer 2-3 pyramidal neurons. Australasian Journal of Neuroscience, 29, 1285–1299.CrossRefGoogle Scholar
  29. Hodgkin, A.L., & Huxley, A.F. (1952). A quantitative description of membrane current and its application to conductance and excitation in nerve. The Journal of Physiology, 117, 500–544.CrossRefGoogle Scholar
  30. Hu, H., Vervaeke, K., Storm, J.F. (2002). Two forms of electrical resonance at theta frequencies generated by M-current, h-current and persistent Na+ current in rat hippocampal pyramidal cells. The Journal of Physiology, 545.3, 783–805.CrossRefGoogle Scholar
  31. Hu, H., Vervaeke, K., Graham, J.F., Storm, L.J. (2009). Complementary theta resonance filtering by two spatially segregated mechanisms in CA,1 hippocampal pyramidal neurons. Journal of Neuroscience, 29, 14472–14483.CrossRefGoogle Scholar
  32. Hutcheon, B., Miura, R.M., Puil, E. (1996). Subthreshold membrane resonance in neocortical neurons. Journal of Neurophysiology, 76, 683–697.CrossRefGoogle Scholar
  33. Hutcheon, B., & Yarom, Y. (2000). Resonance, oscillations and the intrinsic frequency preferences in neurons. Trends in Neurosciences, 23, 216–222.CrossRefGoogle Scholar
  34. Izhikevich, E. (2006). Dynamical Systems in Neuroscience: The geometry of excitability and bursting. Cambridge: MIT Press.CrossRefGoogle Scholar
  35. Lampl, I, & Yarom, Y. (1997). Subthreshold oscillations and resonant behaviour: Two manifestations of the same mechanism. Neuroscience, 78, 325–341.CrossRefGoogle Scholar
  36. Lau, T., & Zochowski, M. (2011). The resonance frequency shift, pattern formation, and dynamical network reorganization via sub-threshold input. PLoS ONE, 6, e18983.CrossRefGoogle Scholar
  37. Ledoux, E., & Brunel, N. (2011). Dynamics of networks of excitatory and inhibitory neurons in response to time-dependent inputs. Frontiers in Computational Neuroscience, 5, 1–17.CrossRefGoogle Scholar
  38. Llinás, R. R., & Yarom, Y. (1986). Oscillatory properties of Guinea pig olivary neurons and their pharmachological modulation: An in vitro study. The Journal of Physiology, 376, 163–182.CrossRefGoogle Scholar
  39. Loewenstein, Y., Yarom, Y., Sompolinsky, H. (2001). The generation of oscillations in networks of electrically coupled cells. Proceedings of the National Academy of Sciences of the United States of America, 98, 8095–8100.CrossRefGoogle Scholar
  40. Manor, Y., Rinzel, J., Segev, I., Yarom, Y. (1997). Low-amplitude oscillations in the inferior olive A model based on electrical coupling of neurons with heterogeneous channel densities. Journal of Neurophysiology, 77, 2736–2752.CrossRefGoogle Scholar
  41. Manor, Y., Nadim, F., Epstein, S., Ritt, J., Marder, E., Kopell, N. (1999). Network oscillations generated by balancing graded asymmetric reciprocal inhibition in passive neurons. Journal of Neuroscience, 19, 2765–2779.CrossRefGoogle Scholar
  42. Marcelin, B., Becker, A., Migliore, M., Esclapez, M., Bernard, C. (2009). H channel-dependent deficit of theta oscillation resonance and phase shift in temporal lobe epilepsy. Neurobiology of Disease, 33, 436–447.CrossRefGoogle Scholar
  43. Mikiel-Hunter, J., Kotak, V., Rinzel, J. (2016). High-frequency resonance in the gerbil medial superior olive. PLoS Computational Biology, 12, 1005166.CrossRefGoogle Scholar
  44. Moca, V.V., Nicolic, D., Singer, W., Muresan, R. (2014). Membrane resonance enables stable robust gamma oscillations. Cerebral Cortex, 24, 119–142.CrossRefGoogle Scholar
  45. Morris, H., & Lecar, C. (1981). Voltage oscillations in the barnacle giant muscle fiber. Biophysical Journal, 35, 193–213.CrossRefGoogle Scholar
  46. Muresan, R., & Savin, C. (2007). Resonance or integration? self-sustained dynamics and excitability of neural microcircuits. Journal of Neurophysiology, 97, 1911–1930.CrossRefGoogle Scholar
  47. Narayanan, R., & Johnston, D. (2007). Long-term potentiation in rat hippocampal neurons is accompanied by spatially widespread changes in intrinsic oscillatory dynamics and excitability. Neuron, 56, 1061–1075.CrossRefGoogle Scholar
  48. Narayanan, R., & Johnston, D. (2008). The h channel mediates location dependence and plasticity of intrinsic phase response in rat hippocampal neurons. The Journal of Neuroscience, 28, 5846–5850.CrossRefGoogle Scholar
  49. Nolan, M.F., Dudman, J.T., Dodson, P.D., Santoro, B. (2007). HCN1 channels control resting and active integrative properties of stellate cells from layer II of the entorhinal cortex, have been effectively employed for different. Journal of Neuroscience, 27, 12440–12551.CrossRefGoogle Scholar
  50. Pike, F.G., Goddard, R.S., Suckling, J.M., Ganter, P., Kasthuri, N., Paulsen, O. (2000). Distinct frequency preferences of different types of rat hippocampal neurons in response to oscillatory input currents. Journal of Physiology, 529, 205–213.CrossRefGoogle Scholar
  51. Prinz, A.A., Abbott, L.F., Marder, E. (2004). The dynamic clamp comes of age. Trends in Neurosciences, 27, 218–224.CrossRefGoogle Scholar
  52. Rathour, R.K., & Narayanan, R. (2012). Inactivating ion channels augment robustness of subthreshold intrinsic response dynamics to parametric variability in hippocampal model neurons. Journal of Physiology, 590, 5629–5652.CrossRefGoogle Scholar
  53. Rathour, R.K., & Narayanan, R. (2014). Homeostasis of functional maps in inactive dendrites emerges in the absence of individual channelostasis. Proceedings of the National Academy of Sciences of the United States of America, 111, E1787–E1796.CrossRefGoogle Scholar
  54. Rau, F., Clemens, J., Naumov, V., Hennig, R.M., Schreiber, S. (2015). Firing-rate resonances in the peripheral auditory system of the cricket, gryllus bimaculatus. Journal of Computational Physiology, 201, 1075–1090.CrossRefGoogle Scholar
  55. Remme, W.H., Donato, R., Mikiel-Hunter, J., Ballestero, J.A., Foster, S., Rinzel, J., McAlpine, D. (2014). Subthreshold resonance properties contribute to the efficient coding of auditory spatial cues. Proceedings of the National Academy of Sciences of the United States of America, 111, E2339–E2348.CrossRefGoogle Scholar
  56. Richardson, M.J.E., Brunel, N., Hakim, V. (2003). From subthreshold to firing-rate resonance. Journal of Neurophysiology, 89, 2538–2554.CrossRefGoogle Scholar
  57. Rotstein, H.G., Pervouchine, D., Gillies, M.J., Acker, C.D., White, J.A., Buhl, E.H., Whittington, M.A., Kopell, N. (2005). Slow and fast inhibition and h-current interact to create a theta rhythm in a model of CA1 interneuron networks. Journal of Neurophysiology, 94, 1509–1518.CrossRefGoogle Scholar
  58. Rotstein, H.G. (2014a). Frequency preference response to oscillatory inputs in two-dimensional neural models: a geometric approach to subthreshold amplitude and phase resonance. The Journal of Mathematical Neuroscience, 4, 11.Google Scholar
  59. Rotstein, H.G., & Nadim, F. (2014b). Frequency preference in two-dimensional neural models: a linear analysis of the interaction between resonant and amplifying currents. Journal of Computational Neuroscience, 37, 9–28.Google Scholar
  60. Rotstein, H.G. (2015). Subthreshold amplitude and phase resonance in models of quadratic type: nonlinear effects generated by the interplay of resonant and amplifying currents. Journal of Computational Neuroscience, 38, 325–354.CrossRefGoogle Scholar
  61. Rotstein, H.G. (2017a). Resonance modulation, annihilation and generation of antiresonance and antiphasonance in 3d neuronal systems: interplay of resonant and amplifying currents with slow dynamics. J. Comp. Neurosci., 43, 35–63.Google Scholar
  62. Rotstein, H.G. (2017b). The shaping of intrinsic membrane potential oscillations: positive/negative feedback, ionic resonance/amplification, nonlinearities and time scales. Journal of Computational Neuroscience, 42, 133–166.Google Scholar
  63. Rotstein, H.G. (2017c). Spiking resonances in models with the same slow resonant and fast amplifying currents but different subthreshold dynamic properties. Journal of Computational Neuroscience, 43, 243–271.Google Scholar
  64. Rotstein, H.G., Ito, T., stark, E. (2017d). Inhibition based theta spiking resonance in a hippocampal network. Society for Neuroscience Abstracts, 615, 11.Google Scholar
  65. Schmidt, S.L., Dorsett, C.R., Iyengar, A.K., Frölich, F. (2016). Interaction of intrinsic and synaptic currents mediate network resonance driven by layer V pyramidal cells. Cereb. Cortex, page  https://doi.org/10.1093/cercor/bhw242.
  66. Schreiber, S., Erchova, I, Heinemann, U., Herz, A.V. (2004). Subthreshold resonance explains the frequency-dependent integration of periodic as well as random stimuli in the entorhinal cortex. Journal of Neurophysiology, 92, 408–415.CrossRefGoogle Scholar
  67. Sciamanna, G., & Wilson, C.J. (2011). The ionic mechanism of gamma resonance in rat striatal fast-spiking neurons. Journal of Neurophysiology, 106, 2936–2949.CrossRefGoogle Scholar
  68. Sharp, A.A., O’Neil, M.B., Abbott, L.F., Marder, E. (1993). The dynamic clamp: artificial conductances in biological neurons. Trends in Neurosciences, 16, 389–394.CrossRefGoogle Scholar
  69. Shpiro, A., Moreno-Bote, R., Rubin, N., Rinzel, J. (2009). Balance between noise and adaptation in competition models of perceptual bistability. Journal of Computational Neuroscience, 27, 37–54.CrossRefGoogle Scholar
  70. Skinner, F.K. (2006). Conductance-based models. Scholarpedia, 1, 1408.CrossRefGoogle Scholar
  71. Solinas, S., Forti, L., Cesana, E., Mapelli, J., De Schutter, E., D’Angelo, E. (2007). Fast-reset of pacemaking and theta-frequency resonance in cerebellar Golgi cells: simulations of their impact in vivo. Frontiers in Cellular Neuroscience, 1, 4.CrossRefGoogle Scholar
  72. Song, S.C., Beatty, J.A., Wilson, C.J. (2016). The ionic mechanism of membrane potential oscillations and membrane resonance in striatal lts interneurons. Journal of Neurophysiology, 116, 1752–1764.CrossRefGoogle Scholar
  73. Stark, E., Eichler, R., Roux, L., Fujisawa, S., Rotstein, H.G., Buzsáki, G. (2013). Inhibition-induced theta resonance in cortical circuits. Neuron, 80, 1263–1276.CrossRefGoogle Scholar
  74. Szucs, A., Ráktak, A., Schlett, K., Huerta, R. (2017). Frequency-dependent regulation of intrinsic excitability by voltage-activated membrane conductances, computational modeling and dynamic clamp. European Journal of Neuroscience, 46, 2429–2444.CrossRefGoogle Scholar
  75. Tabak, J., Rinzel, J., Bertram, R. (2011). Quantifying the relative contributions of divisive and substractive feedback to rhythm generation. PLoS Computational Biology, 7, e1001124.CrossRefGoogle Scholar
  76. Tchumatchenko, T., & Clopath, C. (2014). Oscillations emerging from noise-driven steady state in networks with electrical synapses and subthreshold resonance. Nature Communications, 5, 5512.CrossRefGoogle Scholar
  77. Tikidji-Hamburyan, R.A., Martínez, J. J., White, J.A., Canavier, C. (2015). Resonant interneurons can increase robustness of gamma oscillations. Journal of Neuroscience, 35, 15682–15695.CrossRefGoogle Scholar
  78. Tohidi, V., & Nadim, F. (2009). Membrane resonance in bursting pacemaker neurons of an oscillatory network is correlated with network frequency. The Journal of Neuroscience, 29, 6427–6435.CrossRefGoogle Scholar
  79. Torben-Nielsen, B., Segev, I., Yarom, Y. (2012). The generation of phase differences and frequency changes in a network model of inferior olive subthreshold oscillations. PLoS Computational Biology, 8, 31002580.CrossRefGoogle Scholar
  80. Tseng, H., & Nadim, F. (2010). The membrane potential waveform on bursting pacemaker neurons is a predictor of their preferred frequency and the network cycle frequency. The Journal of Neuroscience, 30, 10809–10819.CrossRefGoogle Scholar
  81. van Brederode, J.F.M., & Berger, A.J. (2008). Spike-firing resonance in hypoglossal motoneurons. Journal of Neurophysiology, 99, 2916–2928.CrossRefGoogle Scholar
  82. Wang, X.-J., & Rinzel, J. (1992). Alternating and synchronous rhythms in reciprocally inhibitory model neurons. Neural Computation, 4, 84–97.CrossRefGoogle Scholar
  83. Wilson, H.R., & Cowan, J.D. (1972). Excitatory and inhibitory interactions in localized populations of model neurons. Biophysical Journal, 12, 1–24.CrossRefGoogle Scholar
  84. Wu, N., Hsiao, C. -F., Chandler, S.H. (2001). Membrane resonance and subthreshold membrane oscillations in mesencephalic V neurons: Participants in burst generation. The Journal of Neuroscience, 21, 3729–3739.CrossRefGoogle Scholar
  85. Yang, S., Lin, W., Feng, A. A. (2009). Wide-ranging frequency preferences of auditory midbrain neurons Roles of membrane time constant and synaptic properties. The European Journal of Neuroscience, 30, 76–90.CrossRefGoogle Scholar
  86. Zemankovics, R., Káli, S., Paulsen, O., Freund, T.F., Hájos, N. (2010). Differences in subthershold resonance of hippocampal pyramidal cells and interneurons: The role of h-current and passive membrane characteristics. The Journal of Physiology, 588, 2109–2132.CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Departamento de MatemáticaUniversidad Nacional del SurBahía BlancaArgentina
  2. 2.Instituto de Matemática de Bahía Blanca - INMABB (UNS - CONICET)Bahía BlancaArgentina
  3. 3.Federated Department of Biological SciencesNew Jersey Institute of Technology and Rutgers UniversityNewarkUSA
  4. 4.Institute for Brain and Neuroscience ResearchNew Jersey Institute of TechnologyNewarkUSA
  5. 5.Graduate Faculty, Behavioral Neuroscience ProgramRutgers UniversityNewarkUSA

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