Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Fractional differentiation by neocortical pyramidal neurons

Nature Neuroscience volume 11, pages 1335–1342 (2008)Cite this article

Abstract

Neural systems adapt to changes in stimulus statistics. However, it is not known how stimuli with complex temporal dynamics drive the dynamics of adaptation and the resulting firing rate. For single neurons, it has often been assumed that adaptation has a single time scale. We found that single rat neocortical pyramidal neurons adapt with a time scale that depends on the time scale of changes in stimulus statistics. This multiple time scale adaptation is consistent with fractional order differentiation, such that the neuron's firing rate is a fractional derivative of slowly varying stimulus parameters. Biophysically, even though neuronal fractional differentiation effectively yields adaptation with many time scales, we found that its implementation required only a few properly balanced known adaptive mechanisms. Fractional differentiation provides single neurons with a fundamental and general computation that can contribute to efficient information processing, stimulus anticipation and frequency-independent phase shifts of oscillatory neuronal firing.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Figure 1: The time constant of rate adaptation increases with step duration or period T.
Figure 2: The effect of the fractional differentiating filter H(f) = (i2πf)α.
Figure 3: The gain of the neuronal firing response can be described by a power law and the phase lead is frequency independent.
Figure 4: Neuronal response to sine-wave noise envelopes is similar when periods T = 4, 8, 16 and 32 s are presented simultaneously (with phases φ = 0, 1, 2 and 3 rad) or individually.
Figure 5: Responses can be predicted using fractional differentiation with the parameter α, which was estimated by several methods to be ∼0.15.
Figure 6: Fractional differentiation depends on a balance of multiple adaptation mechanisms with different time scales.
Figure 7: Single-compartment Hodgkin-Huxley model neurons with two adaptation time scales can approximate a fractional differentiator over a 30-fold range of stimulus period T.

Similar content being viewed by others

References

  1. Adrian, E.D. & Zotterman, Y. The impulses produced by sensory nerve endings: part 2. The response of a single end-organ. J. Physiol. (Lond.) 61, 151–171 (1926).

    Article  CAS  Google Scholar 

  2. Barlow, H.B. Possible principles underlying the transformation of sensory messages. in Sensory Communication (ed. Rosenblith, W.) 217–234 (MIT Press, Cambridge, Massachusetts, 1961).

    Google Scholar 

  3. Brenner, N., Bialek, W. & de Ruyter van Steveninck, R. Adaptive rescaling maximizes information transmission. Neuron 26, 695–702 (2000).

    Article  CAS  PubMed  Google Scholar 

  4. Fairhall, A.L., Lewen, G.D. & Bialek, W. de Ruyter Van Steveninck, R.R. Efficiency and ambiguity in an adaptive neural code. Nature 412, 787–792 (2001).

    Article  CAS  PubMed  Google Scholar 

  5. Dean, I., Harper, N.S. & McAlpine, D. Neural population coding of sound level adapts to stimulus statistics. Nat. Neurosci. 8, 1684–1689 (2005).

    Article  CAS  PubMed  Google Scholar 

  6. Diaz-Quesada, M. & Maravall, M. Intrinsic mechanisms for adaptive gain rescaling in barrel cortex. J. Neurosci. 28, 696–710 (2008).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  7. Higgs, M.H., Slee, S.J. & Spain, W.J. Diversity of gain modulation by noise in neocortical neurons: regulation by the slow after-hyperpolarization conductance. J. Neurosci. 26, 8787–8799 (2006).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  8. Kim, K.J. & Rieke, F. Temporal contrast adaptation in the input and output signals of salamander retinal ganglion cells. J. Neurosci. 21, 287–299 (2001).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  9. Maravall, M., Petersen, R.S., Fairhall, A.L., Arabzadeh, E. & Diamond, M.E. Shifts in coding properties and maintenance of information transmission during adaptation in barrel cortex. PLoS Biol. 5, e19 (2007).

    PubMed  PubMed Central  Google Scholar 

  10. Nagel, K.I. & Doupe, A.J. Temporal processing and adaptation in the songbird auditory forebrain. Neuron 51, 845–859 (2006).

    Article  CAS  PubMed  Google Scholar 

  11. Sanchez-Vives, M.V., Nowak, L.G. & McCormick, D.A. Cellular mechanisms of long-lasting adaptation in visual cortical neurons in vitro. J. Neurosci. 20, 4286–4299 (2000).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  12. Smirnakis, S.M., Berry, M.J., Warland, D.K., Bialek, W. & Meister, M. Adaptation of retinal processing to image contrast and spatial scale. Nature 386, 69–73 (1997).

    Article  CAS  PubMed  Google Scholar 

  13. Kvale, M.N. & Schreiner, C.E. Short-term adaptation of auditory receptive fields to dynamic stimuli. J. Neurophysiol. 91, 604–612 (2004).

    Article  PubMed  Google Scholar 

  14. Hosoya, T., Baccus, S.A. & Meister, M. Dynamic predictive coding by the retina. Nature 436, 71–77 (2005).

    Article  CAS  PubMed  Google Scholar 

  15. Toib, A., Lyakhov, V. & Marom, S. Interaction between duration of activity and time course of recovery from slow inactivation in mammalian brain Na+ channels. J. Neurosci. 18, 1893–1903 (1998).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  16. Schwindt, P.C., Spain, W.J. & Crill, W.E. Long-lasting reduction of excitability by a sodium-dependent potassium current in cat neocortical neurons. J. Neurophysiol. 61, 233–244 (1989).

    Article  CAS  PubMed  Google Scholar 

  17. Abel, H.J., Lee, J.C., Callaway, J.C. & Foehring, R.C. Relationships between intracellular calcium and after-hyperpolarizations in neocortical pyramidal neurons. J. Neurophysiol. 91, 324–335 (2004).

    Article  CAS  PubMed  Google Scholar 

  18. Fleidervish, I.A., Friedman, A. & Gutnick, M.J. Slow inactivation of Na+ current and slow cumulative spike adaptation in mouse and guinea-pig neocortical neurones in slices. J. Physiol. (Lond.) 493, 83–97 (1996).

    Article  CAS  Google Scholar 

  19. La Camera, G. et al. Multiple time scales of temporal response in pyramidal and fast-spiking cortical neurons. J. Neurophysiol. 96, 3448–3464 (2006).

    Article  PubMed  Google Scholar 

  20. Schwindt, P.C., Spain, W.J., Foehring, R.C., Chubb, M.C. & Crill, W.E. Slow conductances in neurons from cat sensorimotor cortex in vitro and their role in slow excitability changes. J. Neurophysiol. 59, 450–467 (1988).

    Article  CAS  PubMed  Google Scholar 

  21. Destexhe, A., Rudolph, M., Fellous, J.M. & Sejnowski, T.J. Fluctuating synaptic conductances recreate in vivo–like activity in neocortical neurons. Neuroscience 107, 13–24 (2001).

    Article  CAS  PubMed  Google Scholar 

  22. Richardson, M.J. Effects of synaptic conductance on the voltage distribution and firing rate of spiking neurons. Phys. Rev. E 69, 051918 (2004).

    Article  Google Scholar 

  23. Crochet, S. & Petersen, C.C. Correlating whisker behavior with membrane potential in barrel cortex of awake mice. Nat. Neurosci. 9, 608–610 (2006).

    Article  CAS  PubMed  Google Scholar 

  24. Hasenstaub, A., Sachdev, R.N. & McCormick, D.A. State changes rapidly modulate cortical neuronal responsiveness. J. Neurosci. 27, 9607–9622 (2007).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  25. Thorson, J. & Biederman-Thorson, M. Distributed relaxation processes in sensory adaptation. Science 183, 161–172 (1974).

    Article  CAS  PubMed  Google Scholar 

  26. French, A.S. & Torkkeli, P.H. The power law of sensory adaptation: simulation by a model of excitability in spider mechanoreceptor neurons. Ann. Biomed. Eng. 36, 153–161 (2008).

    Article  PubMed  Google Scholar 

  27. Kleinz, M. & Osler, T.J.A. A child's garden of fractional derivatives. Coll. Math. J. 31, 82–88 (2000).

    Article  Google Scholar 

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

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  29. Kondgen, H. et al. The dynamical response properties of neuocortical neurons to temporally modulated noisy inputs in vitro. Cereb. Cortex 18, 2086–2097 (2008).

    Article  PubMed  PubMed Central  Google Scholar 

  30. Anastasio, T.J. Nonuniformity in the linear network model of the oculomotor integrator produces approximately fractional-order dynamics and more realistic neuron behavior. Biol. Cybern. 79, 377–391 (1998).

    Article  CAS  PubMed  Google Scholar 

  31. Gilboa, G., Chen, R. & Brenner, N. History-dependent multiple time-scale dynamics in a single-neuron model. J. Neurosci. 25, 6479–6489 (2005).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  32. Drew, P.J. & Abbott, L.F. Models and properties of power-law adaptation in neural systems. J. Neurophysiol. 96, 826–833 (2006).

    Article  PubMed  Google Scholar 

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

    Article  CAS  Google Scholar 

  34. Benda, J. & Herz, A.V. A universal model for spike-frequency adaptation. Neural Comput. 15, 2523–2564 (2003).

    Article  PubMed  Google Scholar 

  35. Mainen, Z.F. & Sejnowski, T.J. Reliability of spike timing in neocortical neurons. Science 268, 1503–1506 (1995).

    Article  CAS  PubMed  Google Scholar 

  36. Middleton, J.W., Longtin, A., Benda, J. & Maler, L. The cellular basis for parallel neural transmission of a high-frequency stimulus and its low-frequency envelope. Proc. Natl. Acad. Sci. USA 103, 14596–14601 (2006).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  37. Lundstrom, B.N. & Fairhall, A.L. Decoding stimulus variance from a distributional neural code of interspike intervals. J. Neurosci. 26, 9030–9037 (2006).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  38. Jolivet, R., Rauch, A., Luscher, H.R. & Gerstner, W. Predicting spike timing of neocortical pyramidal neurons by simple threshold models. J. Comput. Neurosci. 21, 35–49 (2006).

    Article  PubMed  Google Scholar 

  39. Slee, S.J., Higgs, M.H., Fairhall, A.L. & Spain, W.J. Two-dimensional time coding in the auditory brainstem. J. Neurosci. 25, 9978–9988 (2005).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  40. Fusi, S., Drew, P.J. & Abbott, L.F. Cascade models of synaptically stored memories. Neuron 45, 599–611 (2005).

    Article  CAS  PubMed  Google Scholar 

  41. Paulin, M.G., Hoffman, L.F. & Assad, C. Dynamics and the single spike. IEEE Trans. Neural Netw. 15, 987–994 (2004).

    Article  PubMed  Google Scholar 

  42. Anastasio, T.J. The fractional-order dynamics of brainstem vestibulo-oculomotor neurons. Biol. Cybern. 72, 69–79 (1994).

    Article  CAS  PubMed  Google Scholar 

  43. Fairhall, A.L., Lewen, G.D., Bialek, W. & de Ruyter van Steveninck, R. in Advances in Neural Information Processing Systems 13 (eds. Leen, T.K., Dietterich, T.G. & Tresp, V.) 124–130 (MIT Press, Cambridge, Massachusetts, 2001).

    Google Scholar 

  44. Puccini, G.D., Sanchez-Vives, M.V. & Compte, A. Integrated mechanisms of anticipation and rate-of-change computations in cortical circuits. PLoS Comput. Biol. 3, e82 (2007).

    Article  PubMed  PubMed Central  Google Scholar 

  45. Wark, B., Lundstrom, B.N. & Fairhall, A. Sensory adaptation. Curr. Opin. Neurobiol. 17, 423–429 (2007).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  46. Wang, X.J., Liu, Y., Sanchez-Vives, M.V. & McCormick, D.A. Adaptation and temporal decorrelation by single neurons in the primary visual cortex. J. Neurophysiol. 89, 3279–3293 (2003).

    Article  PubMed  Google Scholar 

  47. Ruderman, D.L. & Bialek, W. Statistics of natural images: scaling in the woods. Phys. Rev. Lett. 73, 814–817 (1994).

    Article  CAS  PubMed  Google Scholar 

  48. Simoncelli, E.P. & Olshausen, B.A. Natural image statistics and neural representation. Annu. Rev. Neurosci. 24, 1193–1216 (2001).

    Article  CAS  PubMed  Google Scholar 

  49. Buzsaki, G. & Draguhn, A. Neuronal oscillations in cortical networks. Science 304, 1926–1929 (2004).

    Article  CAS  PubMed  Google Scholar 

  50. Hodgkin, A.L. & Huxley, A.F. A quantitative description of membrane current and its application to conduction and excitation in nerve. J. Physiol. (Lond.) 117, 500–544 (1952).

    Article  CAS  Google Scholar 

Download references

Acknowledgements

We are grateful to B. Aguera y Arcas, W. Bialek, F. Dunn, M. Famulare, M. Giugliano, S. Hong, M. Maravall, P. Murphy, F. Rieke, L. Sorensen and B. Wark for helpful discussions and/or comments on the manuscript. We thank S. Usher for excellent technical assistance. This work was supported by a Burroughs-Wellcome Careers at the Scientific Interface grant, and a McKnight Scholar Award and a Sloan Research Fellowship to A.L.F. B.N.L. was supported by grant number F30NS055650 from the National Institute of Neurological Disorders and Stroke, the University of Washington's Medical Scientist Training Program (supported by the National Institute of General Medical Sciences), and an Achievement Rewards for College Scientists (ARCS) fellowship. M.H.H. and W.J.S. were supported by a Veterans Affairs Merit Review to W.J.S.

Author information

Authors and Affiliations

  1. Department of Physiology and Biophysics, Box 357290, University of Washington, Seattle, 98195, Washington, USA

    Brian N Lundstrom, Matthew H Higgs, William J Spain & Adrienne L Fairhall

  2. Department of Neurology, Veterans Affairs Puget Sound Health Care System, 1660 South Columbian Way, Seattle, 98108, Washington, USA

    Matthew H Higgs & William J Spain

Authors
  1. Brian N Lundstrom
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matthew H Higgs
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. William J Spain
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Adrienne L Fairhall
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors conceived of and designed the experiments. B.N.L. and M.H.H. performed the experiments. B.N.L. analyzed the data, performed the modeling and wrote the initial draft. All authors revised the paper.

Corresponding author

Correspondence to Adrienne L Fairhall.

Supplementary information

Supplementary Text and Figures

Supplementary Figures 1–6, Supplementary Data and Discussion, Supplementary Equations and Supplementary Methods (PDF 678 kb)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Lundstrom, B., Higgs, M., Spain, W. et al. Fractional differentiation by neocortical pyramidal neurons. Nat Neurosci 11, 1335–1342 (2008). https://doi.org/10.1038/nn.2212

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/nn.2212

This article is cited by

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing