Abstract
Characteristics of action potential generation are important to understanding brain functioning and, thus, must be understood and modeled. It is still an open question what model can describe concurrently the phenomena of sharp spike shape, the spike threshold variability, and the divisive effect of shunting on the gain of frequency-current dependence. We reproduced these three effects experimentally by patch-clamp recordings in cortical slices, but we failed to simulate them by any of 11 known neuron models, including one- and multi-compartment, with Hodgkin-Huxley and Markov equation-based sodium channel approximations, and those taking into account sodium channel subtype heterogeneity. Basing on our voltage-clamp data characterizing the dependence of sodium channel activation threshold on history of depolarization, we propose a 3-state Markov model with a closed-to-open state transition threshold dependent on slow inactivation. This model reproduces the all three phenomena. As a reduction of this model, a leaky integrate-and-fire model with a dynamic threshold also shows the effect of gain reduction by shunt. These results argue for the mechanism of gain reduction through threshold dynamics determined by the slow inactivation of sodium channels.
Similar content being viewed by others
References
Azouz, R., & Gray, C. M. (1999). Cellular mechanisms contributing to response variability of cortical neurons in vivo. Journal of Neuroscience, 19, 2209–2223.
Badel, L., Lefort, S., Brette, R., Petersen, C. C. H., Gerstner, W., & Richardson, M. J. E. (2008). Dynamic I-V curves are reliable predictors of naturalistic pyramidal-neuron voltage traces. Journal of Neurophysiology, 99(2), 656–666.
Benda, J., Maler, L., & Longtin, A. (2010). Linear versus nonlinear signal transmission in neuron models with adaptation currents or dynamic thresholds. Journal of Neurophysiology, 104(5), 2806–2820.
Borg-Graham, L. (1999). Interpretations of data and mechanisms for hippocampal pyramidal cell models. Cerebral Cortex, 13, 19–138.
Brette, R., & Gerstner, W. (2005). Adaptive exponential integrate-and-fire model as an effective description of neuronal activity. Journal of Neurophysiology, 94(5), 3637–3642.
Carter, B. C., Giessel, A. J., Sabatini, B. L., & Bean, B. P. (2012). Transient sodium current at subthreshold voltages: activation by EPSP waveforms. Neuron, 75(6), 1081–1093.
Chacron, M. J., Longtin, A., St-Hilaire, M., & Maler, L. (2000). Suprathreshold stochastic firing dynamics with memory in P-type electroreceptors. Physical Review Letters, 85(7), 1576–1579.
Chacron, M. J., Pakdaman, K., & Longtin, A. (2003). Interspike interval correlations, memory, adaptation, and refractoriness in a leaky integrate-and-fire model with threshold fatigue. Neural Computation, 15(2), 253–278.
Chacron, M. J., Lindner, B., & Longtin, A. (2007). Threshold fatigue and information transfer. Journal of Computational Neuroscience, 23(3), 301–311.
Chance, F. S., Abbott, L. F., & Reyes, A. D. (2002). Gain modulation from background synaptic input. Neuron, 35, 773–782.
Chizhov, A. V. (2013). Conductance-based refractory density model of primary visual cortex. Journal of Computational Neuroscience PMID: 23888313 (Epub ahead of print). http://link.springer.com/article/10.1007%2Fs10827-013-0473-5
Chizhov, A. V., & Graham, L. J. (2007). Population model of hippocampal pyramidal neurons, linking a refractory density approach to conductance-based neurons. Physical Review E, 75, 011924.
Chizhov, A. V., & Graham, L. J. (2008). Efficient evaluation of neuron populations receiving colored-noise current based on a refractory density method. Physical Review E, 77, 011910.
Chizhov A.V., Smirnova E.Yu., Karabasov I.N., Simonov A.Yu., Marinazzo D., Schramm A., Graham L.J. (2011). Spike thresholds dynamics explains the ability of a neuron to divide. Proceedings of the conference. Neuroinformatics, 2, 205–213.
Colwell, L. J., & Brenner, M. P. (2009). Action potential initiation in the Hodgkin-Huxley model. PLoS Computational Biology, 5, e1000265.
Fernandez, F. R., & White, J. A. (2009). Reduction of spike after depolarization by increased leak conductance alters interspike interval variability. Journal of Neuroscience, 29(4), 973–986.
Fernandez, F. R., & White, J. A. (2010). Gain control in CA1 pyramidal cells using changes in somatic conductance. Journal of Neuroscience, 30(1), 230–241.
Fernandez, F. R., Broicher, T., Truong, A., & White, J. A. (2011). Membrane voltage fluctuations reduce spike frequency adaptation and preserve output gain in CA1 pyramidal neurons in a high-conductance state. Journal of Neuroscience, 31(10), 3880–3893.
Fricker, D., Verheugen, J. A., & Miles, R. (1999). Cell-attached measurements of the firing threshold of rat hippocampal neurones. Journal of Physiology, 517(3), 791–804.
Graham, L. J., & Schramm, A. (2009). In vivo dynamic clamp: The functional impact of synaptic and intrinsic conductances in visual cortex. In A. Destexhe, & T. Bal (Eds.) Dynamic clamp: From principles to applications. Springer.
Gutkin, B., & Ermentrout, G. B. (2006). Neuroscience: spikes too kinky in the cortex? Nature, 440(7087), 999–1000.
Henze, D. A., & Buzsáki, G. (2001). Action potential threshold of hippocampal pyramidal cells in vivo is increased by recent spiking activity. Neuroscience, 105(1), 121–130.
Huang, M., Volgushev, M., & Wolf, F. (2012). A small fraction of strongly cooperative sodium channels boosts neuronal encoding of high frequencies. PLoS One, 7, e37629.
Johannesma, P. I. M. (1968). Diffusion models of the stochastic acticity of neurons. In E. R. Caianiello (Ed.), Neural networks (pp. 116–144). Berlin: Springer.
Liu, Y. H., & Wang, X. J. (2001). Spike-frequency adaptation of a generalized leaky integrate-and-fire model neuron. Journal of Computational Neuroscience, 10(1), 25–45.
McCormick, D. A., Shu, Y., & Yu, Y. (2007). Neurophysiology: Hodgkin and Huxley model–still standing? Nature, 445(E1–2), discussion E2–3.
Migliore, M., Hoffman, D. A., Magee, J. G., & Jonhston, D. (1999). Role of an A-type K + conductance in the back-propagation of action potentials in the dendrites of hippocampal pyramidal neurons. Journal of Computational Neuroscience, 7, 5–15.
Milescu, L. S., Yamanishi, T., Ptak, K., & Smith, J. C. (2010). Kinetic properties and functional dynamics of sodium channels during repetitive spiking in a slow pacemaker neuron. Journal of Neuroscience, 30(36), 12113–12127.
Naundorf, B., Wolf, F., & Volgushev, M. (2006). Unique features of action potential initiation in cortical neurons. Nature, 440(7087), 1060–1063.
Platkiewicz, J., & Brette, R. (2010). A threshold equation for action potential initiation. PLoS Computational Biology, 6(7), e1000850.
Platkiewicz, J., & Brette, R. (2011). Impact of fast sodium channel inactivation on spike threshold dynamics and synaptic integration. PLoS Computational Biology, 7, e1001129.
Priebe, N. J., & Ferster, D. (2012). Mechanisms of neuronal computation in mammalian visual cortex. Neuron, 75(2), 194–208.
Wilent, W. B., & Contreras, D. (2005). Stimulus-dependent changes in spike threshold enhance feature selectivity in rat barrel cortex neurons. Journal of Neuroscience, 25, 2983–2991.
Yu, Y., Shu, Y., & McCormick, D. A. (2008). Cortical action potential back propagation explains spike threshold variability and rapid-onset kinetics. Journal of Neuroscience, 28(29), 7260–7272.
Acknowledgments
The reported study was supported by RFBR, research projects 11-04-01281a and 13-04-01835a.
Conflict of interest
The authors declare that they have no conflict of interest.
Author information
Authors and Affiliations
Corresponding author
Additional information
Action Editor: David Golomb
Electronic supplementary material
Below is the link to the electronic supplementary material.
ESM 1
(DOC 1387 kb)
Rights and permissions
About this article
Cite this article
Chizhov, A.V., Smirnova, E.Y., Kim, K.K. et al. A simple Markov model of sodium channels with a dynamic threshold. J Comput Neurosci 37, 181–191 (2014). https://doi.org/10.1007/s10827-014-0496-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10827-014-0496-6