Skip to main content

Advertisement

Springer Nature Link
Log in

A nonlinear autoregressive Volterra model of the Hodgkin–Huxley equations

  • Published:
Journal of Computational Neuroscience Aims and scope Submit manuscript

Abstract

We propose a new variant of Volterra-type model with a nonlinear auto-regressive (NAR) component that is a suitable framework for describing the process of AP generation by the neuron membrane potential, and we apply it to input-output data generated by the Hodgkin–Huxley (H–H) equations. Volterra models use a functional series expansion to describe the input-output relation for most nonlinear dynamic systems, and are applicable to a wide range of physiologic systems. It is difficult, however, to apply the Volterra methodology to the H–H model because is characterized by distinct subthreshold and suprathreshold dynamics. When threshold is crossed, an autonomous action potential (AP) is generated, the output becomes temporarily decoupled from the input, and the standard Volterra model fails. Therefore, in our framework, whenever membrane potential exceeds some threshold, it is taken as a second input to a dual-input Volterra model. This model correctly predicts membrane voltage deflection both within the subthreshold region and during APs. Moreover, the model naturally generates a post-AP afterpotential and refractory period. It is known that the H–H model converges to a limit cycle in response to a constant current injection. This behavior is correctly predicted by the proposed model, while the standard Volterra model is incapable of generating such limit cycle behavior. The inclusion of cross-kernels, which describe the nonlinear interactions between the exogenous and autoregressive inputs, is found to be absolutely necessary. The proposed model is general, non-parametric, and data-derived.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

References

  • Barahona, M., & Poon, C. (1996). Detection of nonlinear dynamics in short, noisy time series. Nature, 381, 215–217.

    Article  CAS  Google Scholar 

  • Berger, T. W., Song, D., Chan, R., & Marmarelis, V. Z. (2010). The neurobiological basis of cognition: Identification by multi-input, multi-output nonlinear dynamic modeling. Proceedings of IEEE, 98, 356–374.

    Article  CAS  Google Scholar 

  • Bryant, H. L., & Segundo, J. P. (1976). Spike initiation by transmembrane current: A white-noise analysis. The Journal of Physiology, 260, 279–314.

    PubMed  CAS  Google Scholar 

  • Buño, W., Bustamante, J., & Fuentes, J. (1984). White noise analysis of pace-maker-response interactions and non-linearities in slowly adapting crayfish stretch receptor. The Journal of Physiology, 350, 55–80.

    PubMed  Google Scholar 

  • Bustamante, J., & Buño, W. (1992). Signal transduction and nonlinearities revealed by white noise inputs in the fast adapting crayfish stretch receptor. Experimental Brain Research, 88, 303–312.

    Article  CAS  Google Scholar 

  • Chon, K. H., Richard, C. J., & Holstein-Rathlou, N.-H. (1997). Compact and accurate linear and nonlinear autoregressive moving average model parameter estimation using Laguerre functions. Annals of Biomedical Engineering, 25, 731–738.

    Article  PubMed  CAS  Google Scholar 

  • Diaz, H., & Desrochers, A. A. (1988). Modeling of nonlinear discrete-time systems from input-output data. Automatica, 24, 629–641.

    Article  Google Scholar 

  • Gerstner, W. (1995). Time structure of the activity in neural network models. Physical Review E, 51, 738–758.

    Article  CAS  Google Scholar 

  • Gerstner, W., & van Hemmen, J. L. (1992). Associative memory in a network of ‘spiking’ neurons. Network, 3, 139–164.

    Article  Google Scholar 

  • Gerstner, W., & Naud, R. (2009). How good are neuron models? Science, 326, 379–380.

    Article  PubMed  CAS  Google Scholar 

  • Guttman, R., & Feldman, L. (1975). White noise measurement of squid axon membrane impedance. Biochemical and Biophysical Research Communications, 67, 427–432.

    Article  PubMed  CAS  Google Scholar 

  • Guttman, R., Feldman, L., & Lecar, H. (1974). Squid axon membrane response to white noise stimulation. Biophysical Journal, 14, 941–955.

    Article  PubMed  CAS  Google Scholar 

  • Guttman, R., Grisell, R., & Feldman, L. (1977). Strength-frequency relationship for white noise stimulation of squid axons. Mathematical Biosciences, 33, 335–343.

    Article  Google Scholar 

  • Hampson R. E., Song, D., Chan, R. H. M., Sweatt, A. J., Fuqua, J., Gerhardt, G. A., et al. (2012a). A nonlinear model for hippocampal cognitive prostheses: Memory facilitation by hippocampal ensemble stimulation. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 20, 184–197.

    Article  PubMed  Google Scholar 

  • Hampson, R. E., Song, D., Chan, R. H. M., Sweatt, A. J., Riley, M. R., Goonawardena, A. V., et al. (2012b). Closing the loop for memory prosthesis: Detecting the role of hippocampal neural ensembles using nonlinear models. IEEE Transactions on Neural Systems and Rehabilitation Engineering. In press.

  • Hodgkin, A. L., & Huxley, A. L. (1952). A quantitative description of membrane current and its application to conduction and excitation in nerve. Journal of Physiolology, 117, 500–544.

    CAS  Google Scholar 

  • Jolivet, R., Kobayashi, R., Rauch, A., Naud, R., Shinomoto, S., & Gerstner, W. (2008). A benchmark test for a quantitative assessment of simple neuron models. Journal of Neuroscience Methods, 169, 417–424.

    Article  PubMed  Google Scholar 

  • Jolivet, R., Rauch, A., Lüscher, H. R., & Gerstner, W. (2006). Predicting spike timing of neocortical pyramidal neurons by simple threshold models. Journal of Computational Neuroscience, 21, 35–49.

    Article  PubMed  Google Scholar 

  • Jones, J. C. P., & Billings, S. A. (1989). Recursive algorithm for computing the frequency response of a class of non-linear difference equation models. International Journal of Control, 50, 1925–1940.

    Article  Google Scholar 

  • Kistler, W. M., Gerstner, W., & van Hemmen, J. L. (1997). Reduction of Hodgkin–Huxley equations to a single-variable threshold model. Neural Computation, 9, 1015–1045.

    Article  Google Scholar 

  • Korenberg, M. J., French, A. S., & Voo, S. K. L. (1988). White-noise analysis of nonlinear behavior in an insect sensory neuron: Kernel and cascade approaches. Biological Cybernetics, 58, 313–320.

    Article  PubMed  CAS  Google Scholar 

  • Lee, Y. W., & Schetzen, M. (1965). Measurement of the Wiener kernels of a nonlinear system by cross-correlation. International Journal of Control, 2, 237–254.

    Article  Google Scholar 

  • Lewis, E. R., Henry, K. R., & Yamada, W. M. (2000). Essential roles of noise in neural coding and in studies of neural coding. BioSystems, 58, 109–115.

    Article  PubMed  CAS  Google Scholar 

  • Marmarelis, V. Z. (1993). Identification of nonlinear biological systems using laguerre expansions of kernels. Annals of Biomedical Engineering, 21, 573–589.

    Article  PubMed  CAS  Google Scholar 

  • Marmarelis, V. Z. (1997). Modeling methodology for nonlinear physiological systems. Annals of Biomedical Engineering, 25, 239–251.

    Article  PubMed  CAS  Google Scholar 

  • Marmarelis, V. Z. (2004). Nonlinear dynamic modeling of physiological systems. Hoboken: Wiley-IEEE Press.

    Book  Google Scholar 

  • Marmarelis, V. Z. (1989). Signal transformation and coding in neural systems. IEEE Transactions on Biomedical Engineering, 36, 15–24.

    Article  PubMed  CAS  Google Scholar 

  • Marmarelis, P. Z., & McCann, G. D. (1973). Development and application of white-noise modeling techniques for studies of insect visual nervous system. Biological Cybernetics, 12, 74–89.

    CAS  Google Scholar 

  • Marmarelis, P. Z., & Naka, K. I. (1973). Nonlinear analysis and synthesis of receptive-field responses in the catfish retina. 3. Two-input white-noise analysis. Journal of Neurophysiology, 36, 634–648.

    PubMed  CAS  Google Scholar 

  • Marmarelis, V. Z., & Orme, M. E. (1993). Modeling of neural systems by use of neuronal modes. IEEE Transactions on Biomedical Engineering, 40, 1149–1158.

    Article  PubMed  CAS  Google Scholar 

  • Mensi, S., Naud, R., Pozzorini, C., Avermann, M., Petersen, C. C., & Gerstner, W. (2012). Parameter extraction and classification of three cortical neuron types reveals two distinct adaptation mechanisms. Journal of Neurophysiology, 107, 1756–1775.

    Article  PubMed  Google Scholar 

  • Naud, R., Gerhard, F., Mensi, S., & Gerstner, W. (2011). Improved similarity measures for small sets of spike trains. Neural Computation, 23, 3016–3069.

    Article  PubMed  Google Scholar 

  • Ogura, H. (1985). Estimation of Wiener kernels of a nonlinear system and a fast algorithm using digital Laguerre filters. 15th NIBB Conference, Okazaki, Japan.

  • Pillow, J. W., Paninski, L., Uzzell, V. J., Simoncelli, E. P., & Chichilnisky, E. J. (2005). Prediction and decoding of retinal ganglion cell responses with a probabilistic spiking model. The Journal of Neuroscience, 25, 11003–11013.

    Article  PubMed  CAS  Google Scholar 

  • Pillow, J. W., Shlens, J., Paninski, L., Sher, A., Litke, A. M., Chichilnisky, E. J., et al. (2008). Spatio-temporal correlations and visual signalling in a complete neuronal population. Nature, 454, 995–999.

    Article  PubMed  CAS  Google Scholar 

  • Poggio, T., & Torre, V. (1977). A Volterra representation for some neuron models. Biological Cybernetics, 27, 113–124.

    Article  PubMed  CAS  Google Scholar 

  • Schiff, N. D., Victor, J. D., Canel, A., & Labar, D. R. (1995). Characteristic nonlinearities of the 3/s ictal electroencephalogram identified by nonlinear autoregressive analysis. Biological Cybernetics, 72, 519–526.

    Article  PubMed  CAS  Google Scholar 

  • Song, D., Chan, R. H., Marmarelis, V. Z., Hampson, R. E., Deadwyler, S. A., & Berger, T. W. (2007). Nonlinear dynamic modeling of spike train transformations for hippocampal-cortical prostheses. IEEE Transactions on Biomedical Engineering, 54, 1053–1066.

    Article  PubMed  Google Scholar 

  • Song, D., Chan, R. H. M., Marmarelis, V. Z., Hampson, R. E., Deadwyler, S. A., & Berger, T. W. (2009). Nonlinear modeling of neural population dynamics for hippocampal prostheses. Neural Networks, 22, 1340–1351.

    Article  PubMed  Google Scholar 

  • Takahata, T., Tanabe, S., & Pakdaman, K. (2002). White-noise stimulation of the Hodgkin–Huxley model. Biological Cybernetics, 86, 403–417.

    Article  PubMed  Google Scholar 

  • Victor, J. D., & Canel, A. (1992). A relation between the Akaike criterion and reliability of parameter estimates, with application to nonlinear autoregressive modelling of ictal EEG. Annals of Biomedical Engineering, 20, 167–180.

    Article  PubMed  CAS  Google Scholar 

  • Volterra, V. (1930). Theory of functionals and of integro-differential equations. New York: Dover Publications.

    Google Scholar 

  • Watanabe, A., & Stark, L. (1975). Kernel method for nonlinear analysis: Identification of a biological control system. Mathematical Biosciences, 27, 99–108.

    Article  Google Scholar 

  • Wiener, N. (1958). Nonlinear problems in random theory. Cambridge, MA: MIT Press.

    Google Scholar 

  • Zanos, T. P., Courellis, S. H., Berger, T. W., Hampson, R. E., Deadwyler, S. A., & Marmarelis, V. Z. (2008). Nonlinear modeling of causal interrelationships in neuronal ensembles. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 16, 336–352.

    Article  PubMed  Google Scholar 

  • Zhao, X., & Marmarelis, V. Z. (1997) On the relation between continuous and discrete nonlinear parametric models. Automatica, 33, 81–84.

    Article  Google Scholar 

  • Zhao, X., & Marmarelis, V. Z. (1998) Nonlinear parametric models from Volterra kernels measurements. Mathematical Computational Modeling, 27, 37–43.

    Article  CAS  Google Scholar 

Download references

Acknowledgements

This work was supported in part by the Biomedical Simulations Resource at the University of Southern California under NIH grant P41-EB001978. We also thank the anonymous reviewers for their careful reading and many helpful comments.

Author information

Authors and Affiliations

  1. Department of Biomedical Engineering, University of Southern California, 1042 Downey Way, Los Angeles, CA, 90089, USA

    Steffen E. Eikenberry & Vasilis Z. Marmarelis

Authors
  1. Steffen E. Eikenberry
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Vasilis Z. Marmarelis
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Steffen E. Eikenberry.

Additional information

Action Editor: Jonathan David Victor

Rights and permissions

Reprints and permissions

About this article

Cite this article

Eikenberry, S.E., Marmarelis, V.Z. A nonlinear autoregressive Volterra model of the Hodgkin–Huxley equations. J Comput Neurosci 34, 163–183 (2013). https://doi.org/10.1007/s10827-012-0412-x

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10827-012-0412-x

Keywords