Biological Cybernetics

, Volume 105, Issue 3–4, pp 217–237 | Cite as

Dynamical estimation of neuron and network properties I: variational methods

  • Bryan A. Toth
  • Mark Kostuk
  • C. Daniel Meliza
  • Daniel Margoliash
  • Henry D. I. AbarbanelEmail author
Original Paper


We present a method for using measurements of membrane voltage in individual neurons to estimate the parameters and states of the voltage-gated ion channels underlying the dynamics of the neuron’s behavior. Short injections of a complex time-varying current provide sufficient data to determine the reversal potentials, maximal conductances, and kinetic parameters of a diverse range of channels, representing tens of unknown parameters and many gating variables in a model of the neuron’s behavior. These estimates are used to predict the response of the model at times beyond the observation window. This method of \({{\tt data\, assimilation}}\) extends to the general problem of determining model parameters and unobserved state variables from a sparse set of observations, and may be applicable to networks of neurons. We describe an exact formulation of the tasks in nonlinear data assimilation when one has noisy data, errors in the models, and incomplete information about the state of the system when observations commence. This is a high dimensional integral along the path of the model state through the observation window. In this article, a stationary path approximation to this integral, using a variational method, is described and tested employing data generated using neuronal models comprising several common channels with Hodgkin–Huxley dynamics. These numerical experiments reveal a number of practical considerations in designing stimulus currents and in determining model consistency. The tools explored here are computationally efficient and have paths to parallelization that should allow large individual neuron and network problems to be addressed.


Data assimilation Neuronal dynamics Ion channel properties 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abarbanel HD (2009) Effective actions for statistical data assimilation. Phys Lett A 373(44): 4044–4048CrossRefGoogle Scholar
  2. Abarbanel HD (2011) Self consistent model errors. Q J Roy Meteor Soc submittedGoogle Scholar
  3. Abarbanel HDI, Creveling DR, Farsian R, Kostuk M (2009) Dynamical state and parameter estimation. SIAM J Appl Dyn Syst 8(4): 1341–1381CrossRefGoogle Scholar
  4. Abarbanel HDI, Bryant P, Gill PE, Kostuk M, Rofeh J, Singer Z, Toth B, Wong E (2011) Dynamical parameter and state estimation in neuron models, Chap 8. In: Ding M, Glanzman DL (eds) The Dynamic Brain, Oxford University Press, pp 139–180Google Scholar
  5. Brette R, Rudolph M, Carnevale T, Hines M, Beeman D, Bower J, Diesmann M, Morrison A, Goodman P, Harris F, Zirpe M, Natschläger T, Pecevski D, Ermentrout B, Djurfeldt M, Lansner A, Rochel O, Vieville T, Muller E, Davison A, El Boustani S, Destexhe A (2007) Simulation of networks of spiking neurons: a review of tools and strategies. J Comp Neurosci 23: 349–398CrossRefGoogle Scholar
  6. Creveling DR, Gill PE, Abarbanel HD (2008) State and parameter estimation in nonlinear systems as an optimal tracking problem. Phys Lett A 372(15): 2640–2644CrossRefGoogle Scholar
  7. Evensen G (2009) Data assimilation: the ensemble Kalman filter. 2. Springer, BerlinGoogle Scholar
  8. Fano R (1961) Transmission of information: a statistical theory of communications. The MIT Press, CambridgeGoogle Scholar
  9. Gill P, Barclay A, Rosen JB (1998) Sqp methods and their application to numerical optimal control. In: Bulirsch R, Bittner L, Schmidt WH, Heier K (eds) Variational calculus, optimal control and applications, international series of numerical mathematics, vol 124. Birkhauser, Basel, Boston and Berlin, pp 207–222Google Scholar
  10. Gill P, Murray W, Saunders M (2005) Snopt: an sqp algorithm for large-scale constrained optimization. SIAM Rev 47(1): 99–131CrossRefGoogle Scholar
  11. Gill PE, Murray W, Wright MH (1981) Practical optimization. Academic Press, LondonGoogle Scholar
  12. Graham L (2002) Modelling neuronal biophysics. In: Arbib MA (eds) The handbook for brain theory and neural networks. MIT Press, Cambridge, pp 164–170Google Scholar
  13. Hamill OP, Marty A, Neher E, Sakmann B, Sigworth FJ (1981) Improved patch-clamp techniques for high-resolution current recording from cells and cell-free membrane patches. Pflugers Arch 391(2): 85–100PubMedCrossRefGoogle Scholar
  14. Huys QJM, Ahrens MB, Paninski L (2006) Efficient estimation of detailed single-neuron models. J Neurophysiol 96(2): 872–890PubMedCrossRefGoogle Scholar
  15. Johnston D, Wu SMS (1995) Foundations of cellular neurophysiology. MIT Press, CambridgeGoogle Scholar
  16. Kirk DE (2004) Optimal control theory: an introduction. Dover Publications, MineolaGoogle Scholar
  17. Koch C (1999) Biophysics of computation: information processing in single neurons. Oxford University Press, New YorkGoogle Scholar
  18. Kostuk M, Toth B, Meliza CD, Abarbanel HDI, Margoliash D (2011) Dynamical estimation of neuron and network properties II: Monte carlo methods. Biol Cybern (in preparation)Google Scholar
  19. Laurent G, Stopfer M, Friedrich RW, Rabinovich MI, Volkovskii A, Abarbanel HDI (2001) Odor encoding as an active dynamical process: experiments, computation, and theory. Annu Rev Neurosci 24: 293–297CrossRefGoogle Scholar
  20. Quinn JC, Abarbanel HD (2010) State and parameter estimation using monte carlo evaluation of path integrals. Q J Roy Meteor Soc 136(652): 1855–1867CrossRefGoogle Scholar
  21. Stein PSG, Grillner S, Selverston AI, Stuart DG (eds) (1997) Neurons, Networks, and Motor Behavior. MIT Press, CambridgeGoogle Scholar
  22. Toth B (2010) Dynamical estimation of neuron and network properties. SIAG/OPT Views-and-News 21(1): 1–8Google Scholar
  23. Wächter A, Biegler LT (2006) On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math Prog 106(1): 25–57CrossRefGoogle Scholar
  24. Zinn-Justin J (2002) Quantum field theory and critical phenomena. 4. Oxford University Press, OxfordCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2011

Authors and Affiliations

  • Bryan A. Toth
    • 1
  • Mark Kostuk
    • 1
  • C. Daniel Meliza
    • 2
  • Daniel Margoliash
    • 2
  • Henry D. I. Abarbanel
    • 1
    • 3
    Email author
  1. 1.Department of PhysicsUniversity of CaliforniaSan Diego, La JollaUSA
  2. 2.Department of Organismal Biology and AnatomyUniversity of ChicagoChicagoUSA
  3. 3.Marine Physical Laboratory (Scripps Institution of Oceanography), Center for Theoretical Biological PhysicsUniversity of CaliforniaSan DiegoUSA

Personalised recommendations