Predicting spike timing of neocortical pyramidal neurons by simple threshold models

Abstract

Neurons generate spikes reliably with millisecond precision if driven by a fluctuating current—is it then possible to predict the spike timing knowing the input? We determined parameters of an adapting threshold model using data recorded in vitro from 24 layer 5 pyramidal neurons from rat somatosensory cortex, stimulated intracellularly by a fluctuating current simulating synaptic bombardment in vivo. The model generates output spikes whenever the membrane voltage (a filtered version of the input current) reaches a dynamic threshold. We find that for input currents with large fluctuation amplitude, up to 75% of the spike times can be predicted with a precision of ±2 ms. Some of the intrinsic neuronal unreliability can be accounted for by a noisy threshold mechanism. Our results suggest that, under random current injection into the soma, (i) neuronal behavior in the subthreshold regime can be well approximated by a simple linear filter; and (ii) most of the nonlinearities are captured by a simple threshold process.

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References

  1. Abeles M (1991) Corticonics. Cambridge, Cambridge University Press.

  2. Arcas B, Fairhall A (2003) What causes a neuron to spike? Neural Comp. 15: 1789–1807.

    Article  Google Scholar 

  3. Arcas B, Fairhall A, Bialek W (2003) Computation in a single neuron: Hodgkin and Huxley revisited. Neural Comp. 15: 1715–1749.

    Article  Google Scholar 

  4. Arieli A, Sterkin A, Grinvald A, A, A (1996) Dynamics of ongoing activity: explanation of the large variability in evoked cortical responses. Science 273: 1868–1871.

    PubMed  CAS  Google Scholar 

  5. Azouz R, Gray C (2000) Dynamic spike threshold reveals a mechanism for synaptic coincidence detection in cortical neurons in vivo. Proc. Natl. Acad. Sci. USA 97: 8110–8115.

    PubMed  CAS  Article  Google Scholar 

  6. Bair W, Koch C (1996) Temporal precision of spike trains in extrastriate cortex of the behaving macaque monkey. Neural Comp. 8: 1185–1202.

    CAS  Google Scholar 

  7. Bair W, Zohary E, Newsome W (2001) Correlated firing in macaque visual area MT: Time scales and relationship to behavior. J. Neurosci. 21: 1676–1697.

    PubMed  CAS  Google Scholar 

  8. Benda J, Herz A (2003) A universal model for spike-frequency adaptation. Neural Comp. 15: 2523–2564.

    Article  Google Scholar 

  9. Berry M, Warland D, Mesiter M (1997) The structure and precision of retinal spike trains. Proc. Natl. Acad. Sci. USA 94: 5411–5416.

    PubMed  CAS  Article  Google Scholar 

  10. Bialek W, Rieke F, de Ruyter van Stevenick R, Warland D (1991) Reading a neural code. Science 252: 1854–1857.

    PubMed  CAS  Google Scholar 

  11. Borg-Graham L, Monier C, Fregnac Y (1998) Visual input evokes transient and strong shunting inhibition in visual cortical neurons. Nature 393: 369–373.

    PubMed  CAS  Article  Google Scholar 

  12. Braitenberg V, Schütz A (1991) Anatomy of the cortex. Berlin, Springer-Verlag.

  13. Brette R, Gerstner W (2005) Adaptive exponential integrate-and-fire model as an effective description of neuronal activity. J. Neurophysiol. 94: 3637–3642.

    PubMed  Article  Google Scholar 

  14. Brillinger D (1988) The maximum likelihood approach to the identification of neuronal firing systems. Ann. Biomed. Eng. 16: 3–16.

    PubMed  CAS  Article  Google Scholar 

  15. Brillinger D, Segundo J (1979) Empirical examination of the threshold model of neuronal firing. Biol. Cyber. 35: 213–220.

    CAS  Article  Google Scholar 

  16. Bryant H, Segundo J (1976) Spike initiation by transmembrane current: a white noise analysis. J. Physiol. 260: 279–314.

    PubMed  CAS  Google Scholar 

  17. Bugmann G, Christodoulou C, Taylor J (1997) Role of temporal integration and fluctuation detection in the highly irregular firing of leaky integrator neuron model with partial reset. Neural Comp. 9: 985–1000.

    Article  Google Scholar 

  18. Buracas G, Zador A, De Weese M, Albright T (1998) Efficient discrimination of temporal patterns by motion-sensitive neurons in primate visual cortex. Neuron 20: 959–969.

    PubMed  CAS  Article  Google Scholar 

  19. Cash S, Yuste R (1988) Input summation by cultured pyramidal neurons is linear and position-independent. J. Neurosci. 18: 10–15.

    Google Scholar 

  20. Cox D, Miller H (1965) The Theory of Stochastic Processes. New-York, Chapman & Hall.

  21. de Ruyter van Stevenick R, Lowen G, Strong S, Koberle R, Bialek W (1997) Reproducibility and variability in neural spike trains. Science 275: 1805.

    Article  Google Scholar 

  22. De Weese M, Zador A (2003) Binary spiking in auditory Cortex. J. Neurosci. 23: 7940–7949.

    CAS  Google Scholar 

  23. De Weese M, Zador A (2004) Shared and private variability in the auditory cortex. J. Neurophysiol. 92: 1840–1855.

    Article  Google Scholar 

  24. Destexhe A, Rudolph M, Paré D (2003) The high-conductance state of neocortical neurons in vivo. Nat. Rev. Neurosci. 4: 739–751.

    PubMed  CAS  Article  Google Scholar 

  25. Diesmann M, Gewaltig M, Aertsen A (1999) Stable propagation of synchronous spiking in cortical neural networks. Nature 402: 529–533.

    PubMed  CAS  Article  Google Scholar 

  26. Erisir A, Lau D, Rudy B, Leonard C (1999) Specific K+ channels are required to sustain high frequency firing in fast-spiking neocortical interneurons. J. Neurophysiol. 82: 2476–2489.

    PubMed  CAS  Google Scholar 

  27. Fourcaud-Trocmé N, Hansel D, van Vreeswijk C, Brunel N (2003) How spike generation mechanisms determine the neuronal response to fluctuating inputs. J. Neurosci. 23: 11628–11640.

    PubMed  Google Scholar 

  28. Fuortes M, Mantegazzini F (1962) Interpretation of the repetitive firing of nerve cells. J. Gen. Physiol. 45: 1163–1179.

    PubMed  CAS  Article  Google Scholar 

  29. Gawne T, Richmond B (1993) How independent are the messages carried by adjacent inferior temporal cortical neurons. J. Neurosci. 13: 2758–2771.

    PubMed  CAS  Google Scholar 

  30. Gerstner W, Kempter R, van Hemmen J, Wagner H (1996) A neuronal learning rule for sub-millisecond temporal coding. Nature 386: 76–78.

    Article  Google Scholar 

  31. Gerstner W, Kistler W (2002) Spiking Neurons Models: Single Neurons, Populations, Plasticity. Cambridge, Cambridge University Press.

  32. Häusser M, Roth A (1997) Estimating the time course of the excitatory synaptic conductance in neocortical pyramidal cells using a novel voltage jump method. J. Neurosci. 17: 7606–7625.

    PubMed  Google Scholar 

  33. Heggelund P, Albus K (1978) Response variability and orientation discrimination of single cells in striate cortex of cat. Exp. Brain Res. 32: 197–211.

    PubMed  CAS  Article  Google Scholar 

  34. Helmchen F, Svoboda K, Denk W, Tank D (1999) In vivo dendritic calcium dynamics in deep-layer cortical pyramidal neurons. Nat. Neurosci. 2: 989–996.

    PubMed  CAS  Article  Google Scholar 

  35. Hill A (1936) Excitation and accommodation in nerve. Proc. Roy. Soc. B 119: 305–355.

    Google Scholar 

  36. Ikegaya Y, Aaron G, Cossart R, Aronov D, Lampl I, Ferster D, Yuste R (2004) Synfire chains and cortical songs: temporal modules of cortical activity. Science 304: 559–564.

    PubMed  CAS  Article  Google Scholar 

  37. Izhikevich E (2003) Simple model of spiking neurons. IEEE Trans. Neural Net. 14: 1569–1572.

    Article  CAS  Google Scholar 

  38. Izhikevich E (2004) Which model to use for cortical spiking neurons? IEEE Trans. Neural Net. 15: 1063–1070.

    Article  Google Scholar 

  39. Johansson R, Birznieks I (2004) First spikes in ensembles of human tactile afferents code complex spatial fingertip events. Nat. Neurosci. 7: 170–177.

    PubMed  CAS  Article  Google Scholar 

  40. Jolivet R (2005) Effective minimal threshold models of neuronal activity. PhD Thesis, Lausanne, Ecole Polytechnique Fédérale de Lausanne (EPFL). http: //icwww.epfl.ch/∼rjolivet/publications/reports/PhDthesis.pdf.

  41. Jolivet R, Gerstner W (2004) Predicting spike times of a detailed conductance-based neuron model driven by stochastic spike arrival. J. Physiol.-Paris 98: 442–451.

    PubMed  Article  Google Scholar 

  42. Jolivet R, Lewis T, 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: 959–976.

    PubMed  Article  Google Scholar 

  43. Kara P, Reinagel P, Reid R (2000) Low response variabilities in simultaneously recorded retinal, thalamic, and cortical neurons. Neuron 27: 635–646.

    PubMed  CAS  Article  Google Scholar 

  44. Keat J, Reinagel P, Reid R, Meister M (2001) Predicting every spike: A model for the responses of visual neurons. Neuron 30: 803–817.

    PubMed  CAS  Article  Google Scholar 

  45. Kistler W, Gerstner W, van Hemmen J (1997) Reduction of Hodgkin-Huxley equations to a single-variable threshold model. Neural Comp. 9: 1015–1045.

    Article  Google Scholar 

  46. Koch C, Bernander O, Douglas R (1995) Do neurons have a voltage or a current threshold for action potential initiation? J. Comp. Neuro. 2: 63–82.

    CAS  Article  Google Scholar 

  47. Koch C, Rapp M, Segev I (1996) A brief history of time (constants). Cereb. cortex 6: 93–101.

    PubMed  CAS  Google Scholar 

  48. La Camera G, Rauch A, Lüscher H, Senn W, Fusi S (2004) Minimal models of adapted neuronal response to in vivo-like input currents. Neural Comp. 16: 2101–2124.

    Article  Google Scholar 

  49. Lapicque L (1907) Recherches quantitatives sur l’excitation électrique des nerfs traitée comme une polarization. J. Physiol. Pathol. Gen. 9: 620–635.

    Google Scholar 

  50. Larkum M, Zhu J, Sakmann B (2001) Dendritic mechanisms underlying the coupling of the dendritic with the axonal action potential initiation zone of adult rat layer 5 pyramidal neurons. J. Physiol. 533: 447–466.

    PubMed  CAS  Article  Google Scholar 

  51. Latham P, Richmond B, Nelson P, Nirenberg S (2000) Intrinsic dynamics in neuronal networks. I. Theory. J. Neurophysiol. 83: 808–827.

    PubMed  CAS  Google Scholar 

  52. Lee Y, Schetzen M (1965) Measurement of the wiener kernels of a non-linear system by cross-correlation. Int. J. Control 2: 237–254.

    Google Scholar 

  53. Mainen Z, Sejnowski T (1995) Reliability of spike timing in neocortical neurons. Science 268: 1503–1506.

    PubMed  CAS  Google Scholar 

  54. McCormick D, Connors B, Lighthall J, Prince D (1985) Comparative electrophysiology of pyramidal and sparsely stellate neurons of the neocortex. J. Neurophysiol. 54: 782–806.

    PubMed  CAS  Google Scholar 

  55. Paninski L, Pillow J, Simoncelli E (2005) Comparing integrate-and-fire models estimated using intracellular and extracellular data. Neurocomp. 65/66: 379–385.

    Article  Google Scholar 

  56. Polsky A, Mel B, Schiller J (2004) Computational subunits in thin dendrites of pyramidal cells. Nat. Neurosci. 7: 621–627.

    PubMed  CAS  Article  Google Scholar 

  57. Powers R, Binder M (1996) Experimental evaluation of input-output models of motoneuron discharges. J. Neurophysiol. 75: 367–379.

    PubMed  CAS  Google Scholar 

  58. Powers R, Sawczuk A, Musick J, Binder M (1999) Multiple mechanisms of spike-frequency adaptation in motoneurones. J. Physiol.-Paris 93: 101–114.

    PubMed  CAS  Article  Google Scholar 

  59. Rauch A, La Camera G, Lüscher H, Senn W, Fusi S (2003) Neocortical pyramidal cells respond as integrate-and-fire neurons to in-vivo-like input currents. J. Neurophysiol. 90: 1598–1612.

    PubMed  Article  Google Scholar 

  60. Reich D, Victor J, Knight B, Ozaki T, Kaplan E (1997) Response variability and timing precision of Neuronal Spike trains in-vivo. J. Neurophysiol. 77: 2836–2841.

    PubMed  CAS  Google Scholar 

  61. Reinagel P, Reid R (2002) Precise firing events are conserved across neurons. J. Neurosci. 22: 6837–6841.

    PubMed  CAS  Google Scholar 

  62. Rieke F, Warland D, de Ruyter Van Stevenick R, Bialek W (1996) Spikes—Exploring the neural code. Cambridge, MIT Press.

  63. Robinson H, Kawai N (1993) Injection of digitally synthesized synaptic conductance transients to measure the integrative properties of neurons. J. Neurosci. Meth. 49: 157–165.

    CAS  Article  Google Scholar 

  64. Roth A, Häusser M (2001) Compartmental models of rat cerebellar Purkinje cells based on simultaneous somatic and dendritic patch-clamp recordings. J. Physiol. 535: 445–472.

    PubMed  CAS  Article  Google Scholar 

  65. Schneidman E, Freedman B, Segev I (1998) Ion channel stochasticity may be critical in determining the reliability and precision of spike timing. Neural Comp. 10: 1679–1703.

    CAS  Article  Google Scholar 

  66. Schwindt P, Crill W (1982) Factors influencing motoneuron rhythmic firing: results from a voltage-clamp study. J. Neurophysiol. 48: 875–890.

    PubMed  CAS  Google Scholar 

  67. Schwindt P, O’Brien J, Crill W (1997) Quantitative analysis of firing properties of pyramidal neurons from layer 5 of rat sensorimotor cortex. J. Neurophysiol. 77: 2484–2498.

    PubMed  CAS  Google Scholar 

  68. Shadlen M, Newsome W (1988) The variable discharge of cortical neurons: implications for connectivity, computation, and information coding. J. Neurosci. 18: 3870–3896.

    Google Scholar 

  69. Stein R (1967) Some models of neuronal variability. Biophys. J. 7: 37–68.

    Article  PubMed  CAS  Google Scholar 

  70. Steriade M, Timoveev I, Grenier F (2001) Natural waking and sleep states: a view from inside neocortical neurons. J. Neurophysiol. 85: 1969–1985.

    PubMed  CAS  Google Scholar 

  71. Stevens C, Zador A (1998) Novel integrate-and-fire like model of repetitive firing in cortical neurons. 5th Joint Symposium on Neural Computation, UCSD, La Jolla, CA, Institute for Neural Computation.

  72. Stuart G, Häusser M (2001) Dendritic coincidence detection of EPSPs and action potentials. Nat. Neurosci. 4: 63–71.

    PubMed  CAS  Article  Google Scholar 

  73. Stuart G, Sakmann B (1994) Active propagation of somatic action potentials into neocortical pyramidal cell dendrites. Nature 367: 69–72.

    PubMed  CAS  Article  Google Scholar 

  74. Theunissen F, Miller J (1995) Temporal encoding in nervous systems: a rigorous definition. J. Comp. Neuro. 2: 149–162.

    CAS  Article  Google Scholar 

  75. Troyer T, Miller K (1997) Physiological gain leads to high ISI variability in a simple model of a cortical regular spiking cell. Neural Comp. 9: 971–983.

    CAS  Article  Google Scholar 

  76. Tuckwell H (1988) Introduction to Theoretic Neurobiology. Cambridge, Cambridge University Press.

  77. Wehr M, Zador A (2003) Balanced inhibition underlies tuning and sharpens spike timing in auditory cortex. Nature 426: 442–446.

    PubMed  CAS  Article  Google Scholar 

  78. Wiener N (1958) Nonlinear Problems in Random Theory. Cambridge, MIT Press.

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Correspondence to Renaud Jolivet.

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Jolivet, R., Rauch, A., Lüscher, HR. et al. Predicting spike timing of neocortical pyramidal neurons by simple threshold models. J Comput Neurosci 21, 35–49 (2006). https://doi.org/10.1007/s10827-006-7074-5

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Keywords

  • Spike Response Model
  • Stochastic input
  • Adapting threshold
  • Spike-timing reliability
  • Predicting spike timing.