Advertisement

Journal of Computational Neuroscience

, Volume 37, Issue 1, pp 9–28 | Cite as

Frequency preference in two-dimensional neural models: a linear analysis of the interaction between resonant and amplifying currents

  • Horacio G. Rotstein
  • Farzan Nadim
Article

Abstract

Many neuron types exhibit preferred frequency responses in their voltage amplitude (resonance) or phase shift to subthreshold oscillatory currents, but the effect of biophysical parameters on these properties is not well understood. We propose a general framework to analyze the role of different ionic currents and their interactions in shaping the properties of impedance amplitude and phase in linearized biophysical models and demonstrate this approach in a two-dimensional linear model with two effective conductances g L and g 1. We compute the key attributes of impedance and phase (resonance frequency and amplitude, zero-phase frequency, selectivity, etc.) in the g L  − g 1 parameter space. Using these attribute diagrams we identify two basic mechanisms for the generation of resonance: an increase in the resonance amplitude as g 1 increases while the overall impedance is decreased, and an increase in the maximal impedance, without any change in the input resistance, as the ionic current time constant increases. We use the attribute diagrams to analyze resonance and phase of the linearization of two biophysical models that include resonant (I h or slow potassium) and amplifying currents (persistent sodium). In the absence of amplifying currents, the two models behave similarly as the conductances of the resonant currents is increased whereas, with the amplifying current present, the two models have qualitatively opposite responses. This work provides a general method for decoding the effect of biophysical parameters on linear membrane resonance and phase by tracking trajectories, parametrized by the relevant biophysical parameter, in pre-constructed attribute diagrams.

Keywords

Resonance Subthreshold oscillations Zero phase frequency Impedance Model neuron 

Notes

Acknowledgments

The authors thank Diana Martinez and David Fox for their comments on this manuscript.

Grants

Supported by NSF DMS1313861 (HGR) and NIH MH060605, NS083319 (FN).

Conflict of interest statement

The authors declare that they have no conflict of interest.

References

  1. Acker, C. D., Kopell, N., & White, J. A. (2003). Synchronization of strongly coupled excitatory neurons: Relating network behavior to biophysics. Journal of Computational Neuroscience, 15, 71–90.PubMedCrossRefGoogle Scholar
  2. Broicher, T., Malerba, P., Dorval, A. D., Borisyuk, A., Fernandez, F. R., & White, J. A. (2012). Spike phase locking in CA1 pyramidal neurons depends on background conductance and firing rate. Journal of Neuroscience, 32, 14374–14388.PubMedCrossRefGoogle Scholar
  3. Carandini, M., Mechler, F., Leonard, C. S., & Movshon, J. A. (1996). Spike train encoding by regular-spiking cells of the visual cortex. Journal of Neurophysiology, 76, 3425–3441.PubMedGoogle Scholar
  4. Castro-Alamancos, M. A., Rigas, P., & Tawara-Hirata, Y. (2007). Resonance (approximately 10 Hz) of excitatory networks in motor cortex: Effects of voltage-dependent ion channel blockers. Journal of Physiology, 578, 173–191.PubMedCentralPubMedCrossRefGoogle Scholar
  5. Dembrow, N. C., Chitwood, R. A., & Johnston, D. (2010). Projection-specific neuromodulation of medial prefrontal cortex neurons. Journal of Neuroscience, 30(50), 16922–16937.PubMedCentralPubMedCrossRefGoogle Scholar
  6. Dickson, C. T., & Alonso, A. A. (1997). Muscarinic induction of synchronous population activity in the entorhinal cortex. Journal of Neuroscience, 17(7), 6729–6744.PubMedGoogle Scholar
  7. 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.PubMedCentralPubMedCrossRefGoogle Scholar
  8. 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.PubMedCentralPubMedCrossRefGoogle Scholar
  9. Ermentrout, G. B., & Terman, D. (2010). Mathematical foundations of neuroscience. New York: Springer.CrossRefGoogle Scholar
  10. Fernandez, F. R., & White, J. A. (2008). Artificial synaptic conductances reduce subthreshold oscillations and periodic firing in stellate cells of the entorhinal cortex. Journal of Neuroscience, 28, 3790–3803.PubMedCrossRefGoogle Scholar
  11. Fernandez, F. R., Malerba, P., Bressloff, P. C., & White, J. A. (2013). Entorhinal stellate cells show preferred spike phase-locking to theta inputs that is enhanced by correlations in synaptic activity. Journal of Neuroscience, 33(14), 6027–6040.PubMedCentralPubMedCrossRefGoogle Scholar
  12. Fuhrmann, G., Markram, H., & Tsodyks, M. (2002). Spike frequency adaptation and neocortical rhythms. Journal of Neurophysiology, 88(2), 761–770.PubMedGoogle Scholar
  13. Gray, C. M. (1994). Synchronous oscillations in neuronal systems: mechanisms and functions. Journal of Computational Neuroscience, 1, 11–38.PubMedCrossRefGoogle Scholar
  14. Gutfreund, Y., Yarom, Y., & Segev, I. (1995). Subthreshold oscillations and resonant frequency in guinea pig cortical neurons: Physiology and modeling. Journal of Physiology, 483, 621–640.PubMedCentralPubMedGoogle Scholar
  15. Haas, J. S., & White, J. A. (2002). Frequency selectivity of layer II stellate cells in the medial entorhinal cortex. Journal of Neurophysiology, 88, 2422–2429.PubMedCrossRefGoogle Scholar
  16. Hodgkin, A. L., & Huxley, A. F. (1952). A quantitative description of membrane current and its application to conductance and excitation in nerve. Journal of Physiology, 117, 500–544.PubMedCentralPubMedGoogle Scholar
  17. 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. Journal of Physiology, 545(3), 783–805.PubMedCentralPubMedCrossRefGoogle Scholar
  18. Hutcheon, B., & Yarom, Y. (2000). Resonance oscillations and the intrinsic frequency preferences in neurons. Trends in Neurosciences, 23, 216–222.PubMedCrossRefGoogle Scholar
  19. Hutcheon, B., Miura, R. M., & Puil, E. (1996). Subthreshold membrane resonance in neocortical neurons. Journal of Neurophysiology, 76, 683–697.PubMedGoogle Scholar
  20. Izhikevich, E. M., Desay, N. S., Walcott, E. C., & Hoppensteadt, F. C. (2003). Bursts as a unit of neural information: Selective communication via resonance. Trends in Neurosciences, 26, 161–167.PubMedCrossRefGoogle Scholar
  21. Kispersky, T. J., Fernandez, F. R., Economo, M. N., & White, J. A. (2012). Spike resonance properties in hippocampal O-LM cells are dependent on refractory dynamics. Journal of Neuroscience, 32, 3637–3651.PubMedCentralPubMedCrossRefGoogle Scholar
  22. Lampl, I., & Yarom, Y. (1997). Subthreshold oscillations and resonant behaviour: Two manifestations of the same mechanism. Neuron, 78, 325–341.Google Scholar
  23. Lau, T., & Zochowski, M. (2011). The resonance frequency shift, pattern formation, and dynamical network reorganization via sub-threshold input. PLoS ONE, 6, e18983.PubMedCentralPubMedCrossRefGoogle Scholar
  24. 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.Google Scholar
  25. Llinás, R. R., & Yarom, Y. (1986). Oscillatory properties of guinea pig olivary neurons and their pharmachological modulation: An in vitro study. Journal of Physiology, 376, 163–182.PubMedCentralPubMedGoogle Scholar
  26. Marder, E., & Calabrese, R. L. (1996). Principles of rhythmic motor pattern generation. Physiological Reviews, 76, 687–717.PubMedGoogle Scholar
  27. Moca, V. V., Nicolic, D., Singer, W., Muresan, R. (2012). Membrane resonance enables stable robust gamma oscillations. Cerebral Cortex, Epub ahead of print.Google Scholar
  28. Morris, C., & Lecar, H. (1981). Voltage oscillations in the barnacle giant muscle fiber. Biophysical Journal, 35, 193–213.PubMedCentralPubMedCrossRefGoogle Scholar
  29. 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.PubMedCentralPubMedCrossRefGoogle Scholar
  30. Reboreda, A., Sanchez, E., Romero, M., & Lamas, J. A. (2003). Intrinsic spontaneous activity and subthreshold oscillations in neurones of the rat dorsal column nuclei in culture. Journal of Physiology, 551(Pt 1), 191–205.PubMedCentralPubMedCrossRefGoogle Scholar
  31. Reinker, S., Puil, E., & Miura, R. M. (2004). Membrane resonance and stochastic resonance modulate firing patterns of thalamocortical neurons. Journal of Computational Neuroscience, 16, 15–25.PubMedCrossRefGoogle Scholar
  32. Richardson, M. J. E., Brunel, N., & Hakim, V. (2003). From subthreshold to firing-rate resonance. Journal of Neurophysiology, 89, 2538–2554.PubMedCrossRefGoogle Scholar
  33. Rinzel, J., & Ermentrout, G. B. (1998). Analysis of neural excitability and oscillations. In C. Koch & I. Segev (Eds.), Methods in neural modeling (2nd ed., pp. 251–292). Cambridge, Massachusetts: MIT Press.Google Scholar
  34. Rotstein, H. G., Oppermann, T., White, J. A., & Kopell, N. (2006). A reduced model for medial entorhinal cortex stellate cells: Subthreshold oscillations, spiking and synchronization. Journal of Computational Neuroscience, 21, 271–292.PubMedCrossRefGoogle Scholar
  35. Schmitz, D., Gloveli, T., Behr, J., Dugladze, T., & Heinemann, U. (1998). Subthreshold membrane potential oscillations in neurons of deep layers of the entorhinal cortex. Neuron, 85, 999–1004.Google Scholar
  36. 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.PubMedCrossRefGoogle Scholar
  37. Sciamanna, G., & Wilson, C. J. (2011). The ionic mechanism of gamma resonance in rat striatal fast-spiking neurons. Journal of Neurophysiology, 106, 2936–2949.PubMedCentralPubMedCrossRefGoogle Scholar
  38. Smith, G. D., Cox, C. L., Sherman, S. M., & Rinzel, J. (2000). Fourier analysis of sinusoidally driven thalamocortical relay neurons and a minimal integrate-and-fire-or-burst model. Journal of Neurophysiology, 83(1), 588–610.PubMedGoogle Scholar
  39. Stark, E., Eichler, R., Roux, L., Fujisawa, S., Rotstein, H. G., Buzsáki, G. (2013). Inhibitioninduced theta resonance in cortical circuits. Neuron, In Press.Google Scholar
  40. Thevenin, J., Romanelli, M., Vallet, M., Brunel, M., & Erneux, T. (2011). Resonance assisted synchronization of coupled-oscillators: frequency locking without phase locking. Physical Review Letters, 107, 104101.PubMedCrossRefGoogle Scholar
  41. Tohidi, V., & Nadim, F. (2009). Membrane resonance in bursting pacemaker neurons of an oscillatory network is correlated with network frequency. Journal of Neuroscience, 6427, 6435.Google Scholar
  42. Wang, X. J. (2010). Neurophysiological and computational principles of cortical rhythms in cognition. Physiological Review, 90, 1195–1268.CrossRefGoogle Scholar
  43. Wu, N., Hsiao, C.-F., & Chandler, S. H. (2001). Membrane resonance and subthreshold membrane oscillations in mesencephalic V neurons: Participants in burst generation. Journal of Neuroscience, 21, 3729–3739.PubMedGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNew Jersey Institute of TechnologyNewarkUSA
  2. 2.Department of Biological SciencesNew Jersey Institute of Technology, and Rutgers University-NewarkNewarkUSA

Personalised recommendations