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Improved lower bound for the mutual information between signal and neural spike count

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Abstract

The mutual information between a stimulus signal and the spike count of a stochastic neuron is in many cases difficult to determine. Therefore, it is often approximated by a lower bound formula that involves linear correlations between input and output only. Here, we improve the linear lower bound for the mutual information by incorporating nonlinear correlations. For the special case of a Gaussian output variable with nonlinear signal dependencies of mean and variance we also derive an exact integral formula for the full mutual information. In our numerical analysis, we first compare the linear and nonlinear lower bounds and the exact integral formula for two different Gaussian models and show under which conditions the nonlinear lower bound provides a significant improvement to the linear approximation. We then inspect two neuron models, the leaky integrate-and-fire model with white Gaussian noise and the Na–K model with channel noise. We show that for certain firing regimes and for intermediate signal strengths the nonlinear lower bound can provide a substantial improvement compared to the linear lower bound. Our results demonstrate the importance of nonlinear input–output correlations for neural information transmission and provide a simple nonlinear approximation for the mutual information that can be applied to more complicated neuron models as well as to experimental data.

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Notes

  1. There is a strong similarity of the observed maximum of the spike count vs mean signal to a standard problem in statistical physics: the giant acceleration of diffusion in a tilted periodic potential (Reimann et al. 2001; Lindner and Sokolov 2016). A Brownian particle in an inclined washboard potential attains a pronounced maximum in its positional variance [quantified by the diffusion coefficient, \(D=\lim _{T\rightarrow \infty } \langle (x-\langle x\rangle )^2\rangle /(2T)\)] for a certain intermediate value of the bias force [equivalent to the mean signal in the LIF model]. The maximum is attained for a force (mean signal strength) close to the bifurcation value that determines the transition from a noise-induced transport (or firing) regime to a regime, in which movement (spiking) is possible without noise.

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Acknowledgements

This work was supported by the BMBF (FKZ: 01GQ1001A) and the DFG (Research Training Group GRK1589/2).

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Authors and Affiliations

  1. Bernstein Center for Computational Neuroscience Berlin, Philippstr. 13, Haus 2, 10115, Berlin, Germany

    Sergej O. Voronenko & Benjamin Lindner

  2. Physics Department, Humboldt University Berlin, Newtonstr. 15, 12489, Berlin, Germany

    Sergej O. Voronenko & Benjamin Lindner

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  1. Sergej O. Voronenko
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  2. Benjamin Lindner
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Correspondence to Sergej O. Voronenko.

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Communicated by Peter J. Thomas.

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Voronenko, S.O., Lindner, B. Improved lower bound for the mutual information between signal and neural spike count. Biol Cybern 112, 523–538 (2018). https://doi.org/10.1007/s00422-018-0779-5

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