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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.

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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.

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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.

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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

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  1. Brian N Lundstrom
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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.

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Correspondence to Adrienne L Fairhall.

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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

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