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From Spiking Neuron Models to Linear-Nonlinear Models

Author

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  • Srdjan Ostojic
  • Nicolas Brunel

Abstract

Neurons transform time-varying inputs into action potentials emitted stochastically at a time dependent rate. The mapping from current input to output firing rate is often represented with the help of phenomenological models such as the linear-nonlinear (LN) cascade, in which the output firing rate is estimated by applying to the input successively a linear temporal filter and a static non-linear transformation. These simplified models leave out the biophysical details of action potential generation. It is not a priori clear to which extent the input-output mapping of biophysically more realistic, spiking neuron models can be reduced to a simple linear-nonlinear cascade. Here we investigate this question for the leaky integrate-and-fire (LIF), exponential integrate-and-fire (EIF) and conductance-based Wang-Buzsáki models in presence of background synaptic activity. We exploit available analytic results for these models to determine the corresponding linear filter and static non-linearity in a parameter-free form. We show that the obtained functions are identical to the linear filter and static non-linearity determined using standard reverse correlation analysis. We then quantitatively compare the output of the corresponding linear-nonlinear cascade with numerical simulations of spiking neurons, systematically varying the parameters of input signal and background noise. We find that the LN cascade provides accurate estimates of the firing rates of spiking neurons in most of parameter space. For the EIF and Wang-Buzsáki models, we show that the LN cascade can be reduced to a firing rate model, the timescale of which we determine analytically. Finally we introduce an adaptive timescale rate model in which the timescale of the linear filter depends on the instantaneous firing rate. This model leads to highly accurate estimates of instantaneous firing rates.Author Summary: Deciphering the encoding of information in the brain implies understanding how individual neurons emit action potentials (APs) in response to time-varying stimuli. This task is made difficult by two facts: (i) although the biophysics of AP generation are well understood, the dynamics of the membrane potential in response to a time-varying input are highly complex; (ii) the firing of APs in response to a given stimulus is inherently stochastic as only a fraction of the inputs to a neuron are directly controlled by the stimulus, the remaining being due to the fluctuating activity of the surrounding network. As a result, the input-output transform of individual neurons is often represented with the help of simplified phenomenological models that do not take into account the biophysical details. In this study, we directly relate a class of such phenomenological models, the so called linear-nonlinear models, with more biophysically detailed spiking neuron models. We provide a quantitative mapping between the two classes of models, and show that the linear-nonlinear models provide a good approximation of the input-output transform of spiking neurons, as long as the fluctuating inputs from the surrounding network are not exceedingly weak.

Suggested Citation

  • Srdjan Ostojic & Nicolas Brunel, 2011. "From Spiking Neuron Models to Linear-Nonlinear Models," PLOS Computational Biology, Public Library of Science, vol. 7(1), pages 1-16, January.
  • Handle: RePEc:plo:pcbi00:1001056
    DOI: 10.1371/journal.pcbi.1001056
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    References listed on IDEAS

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    1. Jonathan W. Pillow & Jonathon Shlens & Liam Paninski & Alexander Sher & Alan M. Litke & E. J. Chichilnisky & Eero P. Simoncelli, 2008. "Spatio-temporal correlations and visual signalling in a complete neuronal population," Nature, Nature, vol. 454(7207), pages 995-999, August.
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    Cited by:

    1. Skander Mensi & Olivier Hagens & Wulfram Gerstner & Christian Pozzorini, 2016. "Enhanced Sensitivity to Rapid Input Fluctuations by Nonlinear Threshold Dynamics in Neocortical Pyramidal Neurons," PLOS Computational Biology, Public Library of Science, vol. 12(2), pages 1-38, February.
    2. Yuxin Zhang & Yifei Yang & Xiaosi Li & Zijing Yuan & Yuki Todo & Haichuan Yang, 2023. "A Dendritic Neuron Model Optimized by Meta-Heuristics with a Power-Law-Distributed Population Interaction Network for Financial Time-Series Forecasting," Mathematics, MDPI, vol. 11(5), pages 1-20, March.
    3. Evan S Schaffer & Srdjan Ostojic & L F Abbott, 2013. "A Complex-Valued Firing-Rate Model That Approximates the Dynamics of Spiking Networks," PLOS Computational Biology, Public Library of Science, vol. 9(10), pages 1-11, October.
    4. Richard Naud & Wulfram Gerstner, 2012. "Coding and Decoding with Adapting Neurons: A Population Approach to the Peri-Stimulus Time Histogram," PLOS Computational Biology, Public Library of Science, vol. 8(10), pages 1-14, October.
    5. Omri Harish & David Hansel, 2015. "Asynchronous Rate Chaos in Spiking Neuronal Circuits," PLOS Computational Biology, Public Library of Science, vol. 11(7), pages 1-38, July.
    6. Michelle F Craft & Andrea K Barreiro & Shree Hari Gautam & Woodrow L Shew & Cheng Ly, 2021. "Differences in olfactory bulb mitral cell spiking with ortho- and retronasal stimulation revealed by data-driven models," PLOS Computational Biology, Public Library of Science, vol. 17(9), pages 1-28, September.
    7. Pengcheng Zhou & Shawn D Burton & Adam C Snyder & Matthew A Smith & Nathaniel N Urban & Robert E Kass, 2015. "Establishing a Statistical Link between Network Oscillations and Neural Synchrony," PLOS Computational Biology, Public Library of Science, vol. 11(10), pages 1-25, October.
    8. James Trousdale & Yu Hu & Eric Shea-Brown & Krešimir Josić, 2012. "Impact of Network Structure and Cellular Response on Spike Time Correlations," PLOS Computational Biology, Public Library of Science, vol. 8(3), pages 1-15, March.
    9. Julian Rossbroich & Daniel Trotter & John Beninger & Katalin Tóth & Richard Naud, 2021. "Linear-nonlinear cascades capture synaptic dynamics," PLOS Computational Biology, Public Library of Science, vol. 17(3), pages 1-27, March.
    10. Victor J Barranca & Gregor Kovačič & Douglas Zhou & David Cai, 2014. "Sparsity and Compressed Coding in Sensory Systems," PLOS Computational Biology, Public Library of Science, vol. 10(8), pages 1-11, August.

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