# The effect of dendritic voltage-gated conductances on the neuronal impedance: a quantitative model

- 427 Downloads
- 2 Citations

## Abstract

Neuronal impedance characterizes the magnitude and timing of the subthreshold response of a neuron to oscillatory input at a given frequency. It is known to be influenced by both the morphology of the neuron and the presence of voltage-gated conductances in the cell membrane. Most existing theoretical accounts of neuronal impedance considered the effects of voltage-gated conductances but neglected the spatial extent of the cell, while others examined spatially extended dendrites with a passive or spatially uniform quasi-active membrane. We derived an explicit mathematical expression for the somatic input impedance of a model neuron consisting of a somatic compartment coupled to an infinite dendritic cable which contained voltage-gated conductances, in the more general case of non-uniform dendritic membrane potential. The validity and generality of this model was verified through computer simulations of various model neurons. The analytical model was then applied to the analysis of experimental data from real CA1 pyramidal neurons. The model confirmed that the biophysical properties and predominantly dendritic localization of the hyperpolarization-activated cation current *I* _{h} were important determinants of the impedance profile, but also predicted a significant contribution from a depolarization-activated fast inward current. Our calculations also implicated the interaction of *I* _{h} with amplifying currents as the main factor governing the shape of the impedance-frequency profile in two types of hippocampal interneuron. Our results provide not only a theoretical advance in our understanding of the frequency-dependent behavior of nerve cells, but also a practical tool for the identification of candidate mechanisms that determine neuronal response properties.

## Keywords

Impedance Cable theory Quasi-active membrane Hippocampus## Notes

### Acknowledgements

We are grateful to Dr Norbert Hájos for initiating the project and for helpful discussions and comments regarding the manuscript. We thank Katalin Lengyel and Erzsébet Gregori for their excellent technical assistance. This work was supported by the Hungarian Scientific Research Fund (OTKA T049517, OTKA K60927, OTKA K83251).

## References

- Angelo, K., London, M., Christensen, S. R., & Häusser, M. (2007). Local and global effects of I(h) distribution in dendrites of mammalian neurons.
*Journal of Neuroscience, 27*, 8643–8653.PubMedCrossRefGoogle Scholar - Biel, M., Wahl-Schott, C., Michalakis, S., & Zong, X. (2009). Hyperpolarization-activated cation channels: From genes to function.
*Physiological Reviews, 89*, 847–885.PubMedCrossRefGoogle Scholar - Bishop, C. M. (2006).
*Pattern recognition and machine learning*. Singapore: Springer.Google Scholar - Borg-Graham, L. (1999). Interpretations of data and mechanisms for hippocampal pyramidal cell models. In P. S. Ulinski, E. G. Jones, & A. Peters (Eds.),
*Cortical models ed.*(pp. 19–138). New York: Plenum Press.Google Scholar - Bower, J. M., & Beeman, D. (1998).
*The book of GENESIS: Exploring realistic neural models with the GEneral NEural SImulation System*(2nd ed.). New York, Inc.: Springer.Google Scholar - Bressloff, P. C. (1999). Resonantlike synchronization and bursting in a model of pulse-coupled neurons with active dendrites.
*Journal of Computational Neuroscience, 6*, 237–249.PubMedCrossRefGoogle Scholar - Cole, K. S. (1949). Some physical aspects of bioelectric phenomena.
*Proceedings of the National Academy of Sciences of the United States of America, 35*, 558–566.PubMedCrossRefGoogle Scholar - Coombes, S., Timofeeva, Y., Svensson, C.-M., et al. (2007). Branching dendrites with resonant membrane: A “sum-over-trips” approach.
*Biological Cybernetics, 97*, 137–149.PubMedCrossRefGoogle Scholar - Engel, T. A., Schimansky-Geier, L., Herz, A. V. M., et al. (2008). Subthreshold membrane-potential resonances shape spike-train patterns in the entorhinal cortex.
*Journal of Neurophysiology, 100*, 1576–1589.PubMedCrossRefGoogle Scholar - 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 (London), 560*, 89–110.CrossRefGoogle Scholar - Golding, N. L., Mickus, T. J., Katz, Y., et al. (2005). Factors mediating powerful voltage attenuation along CA1 pyramidal neuron dendrites.
*Journal of Physiology (London), 568*, 69–82.CrossRefGoogle Scholar - Gutfreund, Y., Yarom, Y., & Segev, I. (1995). Subthreshold oscillations and resonant frequency in guinea-pig cortical neurons: Physiology and modelling.
*Journal of Physiology (London), 483*(Pt 3), 621–640.Google Scholar - Hines, M. (1984). Efficient computation of branched nerve equations.
*International Journal of Bio-Medical Computing, 15*, 69–76.PubMedCrossRefGoogle Scholar - Hu, H., Vervaeke, K., Graham, L. J., & Storm, J. F. (2009). Complementary theta resonance filtering by two spatially segregated mechanisms in CA1 hippocampal pyramidal neurons.
*Journal of Neuroscience, 29*, 14472–14483.PubMedCrossRefGoogle Scholar - 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 (London), 545*, 783–805.CrossRefGoogle Scholar - Hutcheon, B., Miura, R. M., & Puil, E. (1996a). Models of subthreshold membrane resonance in neocortical neurons.
*Journal of Neurophysiology, 76*, 698–714.PubMedGoogle Scholar - Hutcheon, B., Miura, R. M., & Puil, E. (1996b). Subthreshold membrane resonance in neocortical neurons.
*Journal of Neurophysiology, 76*, 683–697.PubMedGoogle Scholar - Hutcheon, B., Miura, R. M., Yarom, Y., & Puil, E. (1994). Low-threshold calcium current and resonance in thalamic neurons: A model of frequency preference.
*Journal of Neurophysiology, 71*, 583–594.PubMedGoogle Scholar - Hutcheon, B., & Yarom, Y. (2000). Resonance, oscillation and the intrinsic frequency preferences of neurons.
*Trends in Neurosciences, 23*, 216–222.PubMedCrossRefGoogle Scholar - Jack, J. B., Noble, D., & Tsien, R. W. (1983).
*Electric current flow in excitable cells*. USA: Oxford University Press.Google Scholar - Johnston, D., Christie, B. R., Frick, A., et al. (2003). Active dendrites, potassium channels and synaptic plasticity.
*Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences, 358*, 667–674.PubMedCrossRefGoogle Scholar - Johnston, D., & Narayanan, R. (2008). Active dendrites: Colorful wings of the mysterious butterflies.
*Trends in Neurosciences, 31*, 309–316.PubMedCrossRefGoogle Scholar - Káli, S., & Freund, T. F. (2005). Distinct properties of two major excitatory inputs to hippocampal pyramidal cells: A computational study.
*European Journal of Neuroscience, 22*, 2027–2048.PubMedCrossRefGoogle Scholar - Koch, C. (1984). Cable theory in neurons with active, linearized membranes.
*Biological Cybernetics, 50*, 15–33.PubMedCrossRefGoogle Scholar - Koch, C. (1998).
*Biophysics of computation: Information processing in single neurons*. USA: Oxford University Press.Google Scholar - Koch, C., & Poggio, T. (1985). A simple algorithm for solving the cable equation in dendritic trees of arbitrary geometry.
*Journal of Neuroscience Methods, 12*, 303–315.PubMedCrossRefGoogle Scholar - Leung, L. S., & Yu, H. W. (1998). Theta-frequency resonance in hippocampal CA1 neurons
*in vitro*demonstrated by sinusoidal current injection.*Journal of Neurophysiology, 79*, 1592–1596.PubMedGoogle Scholar - Llinás, R. R. (1988). The intrinsic electrophysiological properties of mammalian neurons: Insights into central nervous system function.
*Science, 242*, 1654–1664.PubMedCrossRefGoogle Scholar - Lörincz, A., Notomi, T., Tamás, G., et al. (2002). Polarized and compartment-dependent distribution of HCN1 in pyramidal cell dendrites.
*Nature Neuroscience, 5*, 1185–1193.PubMedCrossRefGoogle Scholar - Magee, J. C. (1998). Dendritic hyperpolarization-activated currents modify the integrative properties of hippocampal CA1 pyramidal neurons.
*Journal of Neuroscience, 18*, 7613–7624.PubMedGoogle Scholar - Magee, J., Hoffman, D., Colbert, C., & Johnston, D. (1998). Electrical and calcium signaling in dendrites of hippocampal pyramidal neurons.
*Annual Review of Physiology, 60*, 327–346.PubMedCrossRefGoogle Scholar - Mauro, A. (1961). Anomalous impedance, a phenomenological property of time-variant resistance. An analytic review.
*Biophysical Journal, 1*, 353–372.PubMedCrossRefGoogle Scholar - Mauro, A., Conti, F., Dodge, F., & Schor, R. (1970). Subthreshold behavior and phenomenological impedance of the squid giant axon.
*Journal of General Physiology, 55*, 497–523.PubMedCrossRefGoogle Scholar - Megías, M., Emri, Z., Freund, T. F., & Gulyás, A. I. (2001). Total number and distribution of inhibitory and excitatory synapses on hippocampal CA1 pyramidal cells.
*Neuroscience, 102*, 527–540.PubMedCrossRefGoogle Scholar - 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.PubMedCrossRefGoogle Scholar - Narayanan, R., & Johnston, D. (2008). The h channel mediates location dependence and plasticity of intrinsic phase response in rat hippocampal neurons.
*Journal of Neuroscience, 28*, 5846–5860.PubMedCrossRefGoogle Scholar - Pike, F. G., Goddard, R. S., Suckling, J. M., et al. (2000). Distinct frequency preferences of different types of rat hippocampal neurones in response to oscillatory input currents.
*Journal of Physiology (London), 529*(Pt 1), 205–213.CrossRefGoogle Scholar - Rall, W. (1959). Branching dendritic trees and motoneuron membrane resistivity.
*Experimental Neurology, 1*, 491–527.PubMedCrossRefGoogle Scholar - Rall, W. (1962). Theory of physiological properties of dendrites.
*Annals of the New York Academy of Sciences, 96*, 1071–1092.PubMedCrossRefGoogle Scholar - Rall, W., & Agmon-Snir, H. (1998). Cable theory for dendritic neurons. In C. Koch, & I. Segev (Eds.),
*Methods in neuronal modeling: From ions to networks*(2 ed., pp. 27–92). Cambridge, MA: MIT Press.Google Scholar - Reyes, A. (2001). Influence of dendritic conductances on the input-output properties of neurons.
*Annual Review of Neuroscience, 24*, 653–675.PubMedCrossRefGoogle Scholar - Richardson, M. J. E., Brunel, N., & Hakim, V. (2003). From subthreshold to firing-rate resonance.
*Journal of Neurophysiology, 89*, 2538–2554.PubMedCrossRefGoogle Scholar - Sjöström, P. J., Rancz, E. A., Roth, A., & Häusser, M. (2008). Dendritic excitability and synaptic plasticity.
*Physiological Reviews, 88*, 769–840.PubMedCrossRefGoogle Scholar - Ströhmann, B., Schwarz, D. W., & Puil, E. (1994). Subthreshold frequency selectivity in avian auditory thalamus.
*Journal of Neurophysiology, 71*, 1361–1372.PubMedGoogle Scholar - Stuart, G., & Spruston, N. (1998). Determinants of voltage attenuation in neocortical pyramidal neuron dendrites.
*Journal of Neuroscience, 18*, 3501–3510.PubMedGoogle Scholar - Ulrich, D. (2002). Dendritic resonance in rat neocortical pyramidal cells.
*Journal of Neurophysiology, 87*, 2753–2759.PubMedGoogle Scholar - Zemankovics, R., Káli, S., Paulsen, O., et al. (2010). Differences in subthreshold resonance of hippocampal pyramidal cells and interneurons: The role of h-current and passive membrane characteristics.
*Journal of Physiology (London), 588*, 2109–2132.CrossRefGoogle Scholar