Biological Cybernetics

, Volume 108, Issue 4, pp 495–516 | Cite as

Estimating parameters and predicting membrane voltages with conductance-based neuron models

  • C. Daniel MelizaEmail author
  • Mark Kostuk
  • Hao Huang
  • Alain Nogaret
  • Daniel Margoliash
  • Henry D. I. Abarbanel
Original Paper


Recent results demonstrate techniques for fully quantitative, statistical inference of the dynamics of individual neurons under the Hodgkin–Huxley framework of voltage-gated conductances. Using a variational approximation, this approach has been successfully applied to simulated data from model neurons. Here, we use this method to analyze a population of real neurons recorded in a slice preparation of the zebra finch forebrain nucleus HVC. Our results demonstrate that using only 1,500 ms of voltage recorded while injecting a complex current waveform, we can estimate the values of 12 state variables and 72 parameters in a dynamical model, such that the model accurately predicts the responses of the neuron to novel injected currents. A less complex model produced consistently worse predictions, indicating that the additional currents contribute significantly to the dynamics of these neurons. Preliminary results indicate some differences in the channel complement of the models for different classes of HVC neurons, which accords with expectations from the biology. Whereas the model for each cell is incomplete (representing only the somatic compartment, and likely to be missing classes of channels that the real neurons possess), our approach opens the possibility to investigate in modeling the plausibility of additional classes of channels the cell might possess, thus improving the models over time. These results provide an important foundational basis for building biologically realistic network models, such as the one in HVC that contributes to the process of song production and developmental vocal learning in songbirds.


Data assimilation Neuronal dynamics Ion channel properties Song system 



B. Toth contributed software used to generate IPOPT code. We acknowledge many productive conversations with P. E. Gill on numerical optimization, and we thank A. Daou for conversations about neuron classes in HVC. Support from the US Department of Energy (Grant DE-SC0002349 ) and the National Science Foundation (Grants IOS-0905076, IOS-0905030, and PHY-0961153) is gratefully acknowledged. Partial support from the NSF sponsored Center for Theoretical Biological Physics is also appreciated.

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  1. Abarbanel HDI (2009) Effective actions for statistical data assimilation. Phys Lett A 373:4044–4048CrossRefGoogle Scholar
  2. Abarbanel HDI (2013) Predicting the future: completing models of complex systems. Springer, New YorkCrossRefGoogle Scholar
  3. Abarbanel HDI, Creveling D, Jeanne J (2008) Estimation of parameters in nonlinear systems using balanced synchronization. Phys Rev E 77(016):208Google Scholar
  4. Abarbanel HDI, Creveling DR, Farsian R, Kostuk M (2009) Dynamical state and parameter estimation. SIAM J Appl Dyn Syst 8(4):1341–1381CrossRefGoogle Scholar
  5. Abarbanel HDI, Kostuk M, Whartenby W (2010) Data assimilation with regularized nonlinear instabilities. Q J Meteor Soc 136(648):769–783Google Scholar
  6. 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. In: Glanzman D, Ding M (eds) The dynamic brain: an exploration of neuronal variability and its functional significance, chap 8. Oxford University Press, New YorkGoogle Scholar
  7. Achard P, Schutter ED (2006) Complex parameter landscape for a complex neuron model. PLoS Comput Biol 2(7):e94. doi: 10.1371/journal.pcbi.0020094 PubMedCentralPubMedCrossRefGoogle Scholar
  8. Amador A, Perl YS, Mindlin GB, Margoliash D (2013) Elemental gesture dynamics are encoded by song premotor cortical neurons. Nature 495(7439):59–64. doi: 10.1038/nature11967 PubMedCrossRefGoogle Scholar
  9. Ayali A, Lange AB (2010) Rhythmic behaviour and pattern-generating circuits in the locust: key concepts and recent updates. J Insect Physiol 56(8):834–843. doi: 10.1016/j.jinsphys.2010.03.015 PubMedCrossRefGoogle Scholar
  10. Baldi P, Vanier MC, Bower JM (1998) On the use of Bayesian methods for evaluating compartmental neural models. J Comput Neurosci 5:285–314PubMedCrossRefGoogle Scholar
  11. Bean BP (2007a) The action potential in mammalian central neurons. Nat Rev Neurosci 8(6):451–465. doi: 10.1038/nrn2148 PubMedCrossRefGoogle Scholar
  12. Bean BP (2007b) The action potential in mammalian central neurons. Nat Rev Neurosci 8(6):451–465. doi: 10.1038/nrn2148 PubMedCrossRefGoogle Scholar
  13. Briggman KL, Abarbanel HDI, Kristan WB (2005) Optical imaging of neuronal populations during decision-making. Science 307:896–901PubMedCrossRefGoogle Scholar
  14. Buhry L, Pace M, Saighi S (2012) Global parameter estimation of an Hodgkin–Huxley formalism using membrane voltage recordings: application to neuro-mimetic analog integrated circuits. Neurocomputing 81:75–85. doi: 10.1016/j.neucom.2011.11.002 CrossRefGoogle Scholar
  15. Cerda O, Trimmer JS (2010) Analysis and functional implications of phosphorylation of neuronal voltage-gated potassium channels. Neurosci Lett 486(2):60–7. doi: 10.1016/j.neulet.2010.06.064 PubMedCentralPubMedCrossRefGoogle Scholar
  16. Clewley R (2011) Inferring and quantifying the role of an intrinsic current in a mechanism for a half-center bursting oscillation: a dominant scale and hybrid dynamical systems analysis. J Biol Phys 37(3):285–306PubMedCentralPubMedCrossRefGoogle Scholar
  17. Daou A, Ross M, Johnson F, Hyson RL, Bertram R (2013) Electrophysiological characterization and computational models of HVC neurons in the zebra finch. J Neurophysiol. doi: 10.1152/jn.00162.2013 PubMedGoogle Scholar
  18. Druckmann S, Banitt Y, Gidon A, Schürmann F, Markram H, Segev I (2007) A novel multiple objective optimization framework for constraining conductance-based neuron models by experimental data. Front Neurosci 1(1):7–18. doi: 10.3389/neuro. PubMedCentralPubMedCrossRefGoogle Scholar
  19. Druckmann S, Berger TK, Hill S, Schürmann F, Markram H, Segev I (2008) Evaluating automated parameter constraining procedures of neuron models by experimental and surrogate data. Biol Cybern 99(4–5):371–9. doi: 10.1007/s00422-008-0269-2 PubMedCrossRefGoogle Scholar
  20. Dutar P, Vu HM, Perkel DJ (1998) Multiple cell types distinguished by physiological, pharmacological, and anatomic properties in nucleus HVc of the adult zebra finch. J Neurophysiol 80(4):1828 –1838Google Scholar
  21. Fortune ES, Margoliash D (1995) Parallel pathways and convergence onto HVc and adjacent neostriatum of adult zebra finches (Taeniopygia guttata). J Comp Neurol 360(3):413–441. doi: 10.1002/cne.903600305 PubMedCrossRefGoogle Scholar
  22. Foster WR, Ungar LH, Schwaber JS (1993) Significance of conductances in Hodgkin–Huxley models. J Neurophysiol 70(6):2502–2518PubMedGoogle Scholar
  23. Geit WV, Schutter ED, Achard P (2008) Automated neuron model optimization techniques: a review. Biol Cybern 99:241–251. doi: 10.1007/s00422-008-0257-6 PubMedCrossRefGoogle Scholar
  24. Gill PE, Murray W, Wright MH (1981) Practical optimization. Academic Press, LondonGoogle Scholar
  25. Gleeson P, Crook S, Cannon RC, Hines ML, Billings GO, Farinella M, Morse TM, Davison AP, Ray S, Bhalla US, Barnes SR, Dimitrova YD, Silver RA (2010) NeuroML: a language for describing data driven models of neurons and networks with a high degree of biological detail. PLoS Comput Biol 6(6):e1000,815. doi: 10.1371/journal.pcbi.1000815 CrossRefGoogle Scholar
  26. Golowasch J, Goldman MS, Abbott LF, Marder E (2002) Failure of averaging in the construction of a conductance-based neuron model. J Neurophysiol 87(2):1129–31PubMedGoogle Scholar
  27. Günay C, Edgerton JR, Jaeger D (2008) Channel density distributions explain spiking variability in the globus pallidus: a combined physiology and computer simulation database approach. J Neurosci 28(30):7476–7491. doi: 10.1523/JNEUROSCI.4198-07.2008 PubMedCrossRefGoogle Scholar
  28. Hahnloser RHR, Kozhevnikov AA, Fee MS (2002) An ultra-sparse code underlies the generation of neural sequences in a songbird. Nature 419:65–70PubMedCrossRefGoogle Scholar
  29. Hastings WK (1970) Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57(1):97CrossRefGoogle Scholar
  30. Hendrickson EB, Edgerton JR, Jaeger D (2011) The use of automated parameter searches to improve ion channel kinetics for neural modeling. J Comput Neurosci 31(2):329–346. doi: 10.1007/s10827-010-0312-x PubMedCrossRefGoogle Scholar
  31. Herz AVM, Gollisch T, Machens CK, Jaeger D (2006) Modeling single-neuron dynamics and computations: a balance of detail and abstraction. Science 314(5796):80–85. doi: 10.1126/science.1127240 PubMedCrossRefGoogle Scholar
  32. Hines ML, Carnevale NT (1997) The NEURON simulation environment. Neural Comput 9(6):1179–1209PubMedCrossRefGoogle Scholar
  33. Hobbs KH, Hooper SL (2008) Using complicated, wide dynamic range driving to develop models of single neurons in single recording sessions. J Neurophysiol 99(4):1871–83. doi: 10.1152/jn.00032.2008 PubMedCrossRefGoogle Scholar
  34. Hochberg D, Molina-París C, Pérez-Mercader J, Visser M (1999) Effective action of stochastic partial differential equations. Phys Rev E 60(6):6343–6360CrossRefGoogle Scholar
  35. Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol (Lond) 117(4):500–44Google Scholar
  36. Huijberts HJC, Lilge T, Nijmeijer H (2001) Nonlinear discrete-time synchronization via extended observers. Int J Bifurcat Chaos 11(7):1997–2006CrossRefGoogle Scholar
  37. Huys QJM, Paninski L (2009) Smoothing of, and parameter estimation from, noisy biophysical recordings. PLoS Comput Biol 5(5):e1000,379. doi: 10.1371/journal.pcbi.1000379 CrossRefGoogle Scholar
  38. Huys QJM, Ahrens MB, Paninski L (2006) Efficient estimation of detailed single-neuron models. J Neurophysiol 96(2):872–90. doi: 10.1152/jn.00079.2006 PubMedCrossRefGoogle Scholar
  39. Jin D, Ramazanoğlu F, Seung H (2007) Intrinsic bursting enhances the robustness of a neural network model of sequence generation by avian brain area HVC. J Comput Neurosci 23(3):283–299PubMedCrossRefGoogle Scholar
  40. Jin L, Han Z, Platisa J, Wooltorton JR, Cohen LB, Pieribone VA (2012) Single action potentials and subthreshold electrical events imaged in neurons with a fluorescent protein voltage probe. Neuron 75(5):779–785. doi: 10.1016/j.neuron.2012.06.040 PubMedCentralPubMedCrossRefGoogle Scholar
  41. Johnston J, Forsythe ID, Kopp-Scheinpflug C (2010) Going native: voltage-gated potassium channels controlling neuronal excitability. J Physiol (Lond) 588(Pt 17):3187–3200. doi: 10.1113/jphysiol.2010.191973 CrossRefGoogle Scholar
  42. Jolivet R, Lewis TJ, Gerstner W (2004) Generalized integrate-and-fire models of neuronal activity approximate spike trains of a detailed model to a high degree of accuracy. J Neurophysiol 92(2):959–76. doi: 10.1152/jn.00190.2004 PubMedCrossRefGoogle Scholar
  43. Jolivet R, Kobayashi R, Rauch A, Naud R, Shinomoto S, Gerstner W (2008a) A benchmark test for a quantitative assessment of simple neuron models. J Neurosci Methods 169(2):417–424. doi: 10.1016/j.jneumeth.2007.11.006 PubMedCrossRefGoogle Scholar
  44. Jolivet R, Schürmann F, Berger TK, Naud R, Gerstner W, Roth A (2008b) The quantitative single-neuron modeling competition. Biol Cybern 99(4–5):417–426. doi: 10.1007/s00422-008-0261-x PubMedCrossRefGoogle Scholar
  45. Jouvet B, Phythian R (1979) Quantum aspects of classical and statistical fields. Phys Rev A 19(3):1350–1355CrossRefGoogle Scholar
  46. Kew JNC, Davies CH (eds) (2010) Ion channels: from structure to function. Oxford University Press, New YorkGoogle Scholar
  47. Kobayashi R, Tsubo Y, Shinomoto S (2009) Made-to-order spiking neuron model equipped with a multi-timescale adaptive threshold. Front Comput Neurosci 3:9. doi: 10.3389/neuro.10.009.2009 PubMedCentralPubMedCrossRefGoogle Scholar
  48. Kole MH, Hallermann S, Stuart GJ (2006) Single Ih channels in pyramidal neuron dendrites: properties, distribution, and impact on action potential output. J Neurosci 26(6):1677–1687. doi: 10.1523/JNEUROSCI.3664-05.2006 PubMedCrossRefGoogle Scholar
  49. Kostuk M, Toth BA, Meliza CD, Margoliash D, Abarbanel HDI (2012) Dynamical estimation of neuron and network properties II: path integral monte carlo methods. Biol Cybern 106(3):155–167. doi: 10.1007/s00422-012-0487-5 PubMedCrossRefGoogle Scholar
  50. Kubota M, Saito N (1991) Sodium- and calcium-dependent conductances of neurones in the zebra finch hyperstriatum ventrale pars caudale in vitro. J Physiol (Lond) 440:131–142Google Scholar
  51. Kubota M, Taniguchi I (1998) Electrophysiological characteristics of classes of neuron in the HVc of the zebra finch. J Neurophysiol 80(2):914–923PubMedGoogle Scholar
  52. Lepora NF, Overton PG, Gurney K (2011) Efficient fitting of conductance-based model neurons from somatic current clamp. J Comput Neurosci. doi: 10.1007/s10827-011-0331-2 PubMedGoogle Scholar
  53. Long MA, Jin DZ, Fee MS (2010) Support for a synaptic chain model of neuronal sequence generation. Nature 468(7322):394–9. doi: 10.1038/nature09514 PubMedCentralPubMedCrossRefGoogle Scholar
  54. Marder E, Bucher D (2007) Understanding circuit dynamics using the stomatogastric nervous system of lobsters and crabs. Annu Rev Physiol 69:291–316. doi: 10.1146/annurev.physiol.69.031905.161516 PubMedCrossRefGoogle Scholar
  55. Metropolis N, Rosenbluth AW, Rosenbluth MN, Teller AH, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 21(6):1087–1092Google Scholar
  56. Mooney R (2000) Different subthreshold mechanisms underlie song selectivity in identified HVc neurons of the zebra finch. J Neurosci 20(14):5420–5436PubMedGoogle Scholar
  57. Mooney R, Prather JF (2005) The HVC microcircuit: the synaptic basis for interactions between song motor and vocal plasticity pathways. J Neurosci 25(8):1952–1964. doi: 10.1523/JNEUROSCI.3726-04.2005 PubMedCrossRefGoogle Scholar
  58. Nixdorf B, Davis S, DeVoogd T (1989) Morphology of Golgi-impregnated neurons in hyperstriatum ventralis, pars caudalis in adult male and female canaries. J Comp Neurol 284(3):337–349. doi: 10.1002/cne.902840302 PubMedCrossRefGoogle Scholar
  59. Olypher AV, Calabrese RL (2007) Using constraints on neuronal activity to reveal compensatory changes in neuronal parameters. J Neurophysiol 98(6):3749–58. doi: 10.1152/jn.00842.2007 PubMedCrossRefGoogle Scholar
  60. Pospischil M, Toledo-Rodriguez M, Monier C, Piwkowska Z, Bal T, Frégnac Y, Markram H, Destexhe A (2008) Minimal Hodgkin–Huxley type models for different classes of cortical and thalamic neurons. Biol Cybern 99(4–5):427–441. doi: 10.1007/s00422-008-0263-8 PubMedCrossRefGoogle Scholar
  61. Prinz AA, Billimoria CP, Marder E (2003) Alternative to hand-tuning conductance-based models: construction and analysis of databases of model neurons. J Neurophysiol 90(6):3998–4015. doi: 10.1152/jn.00641.2003 PubMedCrossRefGoogle Scholar
  62. Prinz AA, Bucher DEM (2004) Similar network activity from disparate circuit parameters. Nat Neurosci 7:1345–1352PubMedCrossRefGoogle Scholar
  63. Ransdell J, Nair S, Schulz D (2013) Neurons within the same network independently achieve conserved output by differentially balancing variable conductance magnitudes. J Neurosci 33(24):9950–9956. doi: 10.1523/JNEUROSCI.1095-13.2013 PubMedCrossRefGoogle Scholar
  64. Reid MS, Brown EA, DeWeerth SP (2007) A parameter-space search algorithm tested on a Hodgkin–Huxley model. Biol Cybern 96(6):625–634. doi: 10.1007/s00422-007-0156-2 PubMedCrossRefGoogle Scholar
  65. Restrepo JM (2008) A path integral method for data assimilation. Physica D 237:14–27CrossRefGoogle Scholar
  66. Roberts TF, Klein ME, Kubke MF, Wild JM, Mooney R (2008) Telencephalic neurons monosynaptically link brainstem and forebrain premotor networks necessary for song. J Neurosci 28(13):3479–3489. doi: 10.1523/JNEUROSCI.0177-08.2008 PubMedCentralPubMedCrossRefGoogle Scholar
  67. Sarkar AX, Christini DJ, Sobie EA (2012) Exploiting mathematical models to illuminate electrophysiological variability between individuals. J Physiol (Lond) 590(Pt 11):2555–67. doi: 10.1113/jphysiol.2011.223313 CrossRefGoogle Scholar
  68. Schenk O, Bollhoefer M, Gärtner K (2008) On large-scale diagonalization techniques for the Anderson model of localization. SIAM Rev 50:91–112CrossRefGoogle Scholar
  69. Schulz DJ, Goaillard JM, Marder E (2006) Variable channel expression in identified single and electrically coupled neurons in different animals. Nat Neurosci 9(3):356–362. doi: 10.1038/nn1639 PubMedCrossRefGoogle Scholar
  70. Shea SD, Koch H, Baleckaitis D, Ramirez JM, Margoliash D (2010) Neuron-specific cholinergic modulation of a forebrain song control nucleus. J Neurophysiol 103(2):733–745. doi: 10.1152/jn.00803.2009 PubMedCentralPubMedCrossRefGoogle Scholar
  71. Swensen AM, Bean BP (2005) Robustness of burst firing in dissociated Purkinje neurons with acute or long-term reductions in sodium conductance. J Neurosci 25(14):3509–20. doi: 10.1523/JNEUROSCI.3929-04.2005 PubMedCrossRefGoogle Scholar
  72. Szendro IG, Rodríguez MA, López JM (2009) On the problem of data assimilation by means of synchronization. J Geophys Rev. doi: 10.1029/2009JD012411 Google Scholar
  73. Tomaiuolo M, Bertram R, Leng G, Tabak J (2012) Models of electrical activity: calibration and prediction testing on the same cell. Biophys J 103(9):2021–2032. doi: 10.1016/j.bpj.2012.09.034 PubMedCentralPubMedCrossRefGoogle Scholar
  74. Toth BA, Kostuk M, Meliza CD, Margoliash D, Abarbanel HDI (2011) Dynamical estimation of neuron and network properties I: variational methods. Biol Cybern 105:217–237. doi: 10.1007/s00422-011-0459-1 PubMedCrossRefGoogle Scholar
  75. Trimmer J, Rhodes K (2004) Localization of voltage-gated ion channels in mammalian brain. Annu Rev Physiol 66(1):477–519. doi: 10.1146/annurev.physiol.66.032102.113328 PubMedCrossRefGoogle Scholar
  76. Vanier MC, Bower JM (1999) A comparative survey of automated parameter-search methods for compartmental neural models. J Comput Neurosci 7(2):149–171. doi: 10.1023/A:1008972005316 PubMedCrossRefGoogle Scholar
  77. Vavoulis DV, Straub VA, Aston JAD, Feng J (2012) A self-organizing state-space-model approach for parameter estimation in Hodgkin–Huxley-type models of single neurons. PLoS Comput Biol. doi: 10.1371/journal.pcbi.1002401
  78. Wächter A (2002) An interior point algorithm for large-scale nonlinear optimization with applications in process engineering. Phd thesis, Carnegie Mellon University Google Scholar
  79. Wolpert DH, Macready WG (1997) No free lunch theorems for optimization. IEEE T Evolut Comput 1(1):67–82CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • C. Daniel Meliza
    • 1
    • 2
    Email author
  • Mark Kostuk
    • 3
  • Hao Huang
    • 1
  • Alain Nogaret
    • 4
  • Daniel Margoliash
    • 1
  • Henry D. I. Abarbanel
    • 5
  1. 1.Department of Organismal Biology and AnatomyUniversity of ChicagoChicagoUSA
  2. 2.Department of PsychologyUniversity of VirginiaCharlottesvilleUSA
  3. 3.Department of PhysicsUniversity of California, San DiegoLa JollaUSA
  4. 4.Department of PhysicsUniversity of BathBathUK
  5. 5.Department of Physics and Marine Physical Laboratory (Scripps Institution of Oceanography), Center for Theoretical Biological PhysicsUniversity of California, San DiegoLa JollaUSA

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