Advertisement

Estimating Neural Firing Rates: An Empirical Bayes Approach

  • Shinsuke Koyama
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7664)

Abstract

A lot of neurophysiological findings rely on accurate estimates of firing rates. In order to estimate an underlying rate function from sparse observations, i.e., spike trains, it is necessary to perform temporal smoothing over a short time window at each time point. In the empirical Bayes method, in which the assumption for the smoothness is incorporated in the Bayesian prior probability of underlying rate, the time scale of the temporal average, or the degree of smoothness, can be optimized by maximizing the marginal likelihood. Here, the marginal likelihood is obtained by marginalizing the complete-data likelihood over all possible latent rate processes. We carry out this marginalization using a path integral method. We show that there exists a lower bound of rate fluctuations below which the optimal smoothness parameter diverges. We also show that the optimal smoothness parameter obeys asymptotic scaling laws, the exponent of which depends on the smoothness of underlying rate processes.

Keywords

Neural firing rate empirical Bayes method Path integrals 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Softky, W.R., Koch, C.: The Highly Irregular Firing of Cortical Cells Is Inconsistent with Temporal Integration of Random EPSPs. J. Neurosci. 13, 334–350 (1993)Google Scholar
  2. 2.
    Kass, R.E., Ventura, V., Brown, E.N.: Statistical Issues in the Analysis of Neuronal Data. J. Neurophysiol. 94, 8–25 (2005)CrossRefGoogle Scholar
  3. 3.
    Shimazaki, H., Shinomoto, S.: A Method for Selecting the Bin Size of a Time Histogram. Neural Comp. 19, 1503–1700 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bishop, C.M.: Pattern Recognition and Machine Learning. Springer (2006)Google Scholar
  5. 5.
    Cunningham, J.P., Yu, B.M., Shenoy, K.V., Sahani, M.: Inferring Neural Firing Rates from Spike Trains Using Gaussian Processes. In: Neural Information Processing Systems, vol. 20, pp. 329–336 (2008)Google Scholar
  6. 6.
    Koyama, S., Shinomoto, S.: Empirical Bayes Interpretations of Random Point Events. J. Phys. A: Math. Gen. 38, L531–L537 (2005)Google Scholar
  7. 7.
    Koyama, S., Shinomoto, S.: Phase Transitions in the Estimation of Event Rate: a Path Integral Analysis. J. Phys. A. Math. Theor. 40, F383–F390 (2007)Google Scholar
  8. 8.
    Coleman, S.: Aspects of Symmetry. Cambridge University Press (1988)Google Scholar
  9. 9.
    Kleinert, H.: Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 5th edn. World Scientific Publishing Company (2009)Google Scholar
  10. 10.
    Berman, M.: Inhomogeneous and Modulated Gamma Processes. Biometrica 68, 143–152 (1981)zbMATHCrossRefGoogle Scholar
  11. 11.
    Karatzas, I., Shreve, S.E.: Brownian Motion and Stochastic Calculus, 2nd edn. Springer (1997)Google Scholar
  12. 12.
    Ramsay, J., Silverman, B.W.: Functional Data Analysis, 2nd edn. Springer (2010)Google Scholar
  13. 13.
    Cox, D.R.: Renewal Theory. Chapman and Hall (1962)Google Scholar
  14. 14.
    Shintani, T., Shinomoto, S.: Detection Limit for Rate Fluctuations in Inhomogeneous Poisson Processes. Phys. Rev. E 85, 041139 (2012)Google Scholar
  15. 15.
    Koyama, S., Shinomoto, S.: Histogram Bin Width Selection for Time-Dependent Poisson Processes. J. Phys. A: Math. Gen. 37, 7255–7265 (2004)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Shinsuke Koyama
    • 1
  1. 1.Department of Statistical ModelingThe Institute of Statistical MathematicsTachikawaJapan

Personalised recommendations