Abstract
Neuronal activity is measured by the number of stereotyped action potentials, called spikes, elicited in response to a stimulus or the behavioral conditions of an animal. Any nonparametric method for grasping the time-varying rate of spike firing contains a single parameter that controls the jaggedness of the estimated rate, such as the binsize of the time histogram or the bandwidth of the kernel smoother. In most neurophysiological studies, the parameter that determines the interpretation of neuronal activity has been selected subjectively by individual researchers. Recently, theories for objectively selecting the parameter have been developed. This chapter introduces the standard rate estimation tools, such as the peri-stimulus time histogram (PSTH), kernel density estimation, or Bayes estimation, and shows ways of selecting their parameters under the principles of minimizing the mean integrated squared error or maximizing the likelihood. We also sum up the methods in handy recipes that may be useful in practical data analysis.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Abeles M (1982) Quantification, smoothing, and confidence-limits for single-units histograms. J Neurosci Methods 5:317–325
Abramson I (1982) On bandwidth variation in kernel estimates—a square root law. Ann Statist 10:1217–1223
Adrian ED (1928) The basis of sensation: the action of the sense organs. Christophers, London
Akaike H (1980) Likelihood and Bayes procedure. In: Bernardo JM, DeGroot MH, Lindley DV, Smith AFM (eds) Bayesian statistics. University Press, Valencia, p 143
Baker SN, Lemon RN (2000) Precise spatiotemporal repeating patterns in monkey primary and supplementary motor areas occur at chance levels. J Neurophysiol 84:1770–1780
Barbieri R, Quirk MC, Frank LM, Wilson MA, Brown EN (2001) Construction and analysis of non-Poisson stimulus-response models of neural spiking activity. J Neurosci Methods 105:25–37
Brown EN, Frank LM, Tang D, Quirk MC, Wilson MA (1998) A statistical paradigm for neural spike train decoding applied to position prediction from ensemble firing patterns of rat hippocampal place cells. J Neurosci 18:7411–7425
Carlin BP Louis TA (2000) Bayes and empirical bayes methods for data analysis, 2nd edn. Chapman and Hall, New York
Cherif S, Cullen KE, Galiana HL (2008) An improved method for the estimation of firing rate dynamics using an optimal digital filter. J Neurosci Methods 173:165–181
Cox DR, Lewis PAW (1966) The statistical analysis of series of events. Wiley, New York
Cunningham JP, Yu BM, Shenoy KV, Sahani M (2008) Inferring neural firing rates from spike trains using Gaussian processes. Adv Neural Inf Process Syst 20:329–336
Daley D, Vere-Jones D (2003) An introduction to the theory of point processes, vol. 1: Elementary theory and methods, 2nd edn. Springer-Verlag, New York
Dayan P, Abbott L (2001) Theoretical neuroscience: computational and mathematical modeling of neural systems. MIT Press, Cambridge
Dempster AP, Laird NM, Rubin DB (1977) Maximum likelihood from incomplete data via the EM algorithm. J Roy Statist Soc Ser B 39:1–38
DiMatteo I, Genovese CR, Kass RE (2001) Bayesian curve-fitting with free-knot splines. Biometrika 88:1055–1071
Endres D, Oram M, Schindelin J, Földiák P (2008) Bayesian binning beats approximate alternatives: estimating peri-stimulus time histograms. Adv Neural Inf Process Syst 20:393–400
Gerstein GL, Kiang, NYS (1960) An approach to the quantitative analysis of electrophysiological data from single neurons. Biophys J 1:15–28
Gerstein GL, Mandelbrot B (1964) Random walk models for the spike activity of a single neuron. Biophys J 4:41–68
Good IJ (1965) The estimation of probabilities: an essay on modern Bayesian methods. MIT Press, Cambridge
Johnson DH (1996) Point process models of single-neuron discharges. J Comput Neurosci 3:275–299
Kass RE, Ventura V, Cai C (2003) Statistical smoothing of neuronal data. Network Comput Neural Syst 14:5–15
Kass RE, Ventura V, Brown EN (2005) Statistical issues in the analysis of neuronal data. J Neurophysiol 94:8–25
Kostal L, Lansky P (2006) Classification of stationary neuronal activity according to its information rate. Network Comput Neural Syst 17:193–210
Koyama S, Shinomoto S (2004) Histogram bin-width selection for time-dependent point processes. J Phys A Math Theor 37:7255–7265
Koyama S, Shinomoto S (2005) Empirical Bayes interpretations of random point events. J Phys A Math Theor 38:L531–L537
Koyama S, Shimokawa T, Shinomoto S (2007) Phase transitions in the estimation of event rate: a path integral analysis. J Phys A Math Theor 40:F383–F390
Koyama S, Paninski L (2009) Efficient computation of the maximum a posteriori path and parameter estimation in integrate-and-fire and more general state-space models. J Comput Neurosci doi:10.1007/s10827-009-0179-x
Kuffler SW, Fitzhugh R, Barlow HB (1957) Maintained activity in the cat’s retina in light and darkness. J Gen Physiol 40:683–702
Loader CR (1999a) Bandwidth selection: classical or plug-in? Ann Statist 27:415–438
Loader CR (1999b) Local regression and likelihood. Springer-Verlag, New York
MacKay DJC (1992) Bayesian interpolation. Neural Comput 4:415–447
Nawrot M, Aertsen A, Rotter S (1999) Single-trial estimation of neuronal firing rates: from single-neuron spike trains to population activity. J Neurosci Methods 94:81–92
Nawrot MP, Boucsein C, Rodriguez-Molina V, Riehle A, Aertsen A, Rotter S (2008) Measurement of variability dynamics in cortical spike trains. J Neurosci Methods 169:374–390
Nemenman I, Bialek W (2002) Occam factors and model-independent Bayesian learning of continuous distributions. Phys Rev E 65:026137
Oram MW, Wiener MC, Lestienne R, Richmond BJ (1999) Stochastic nature of precisely timed spike patterns in visual system neuronal responses. J Neurophysiol 81:3021–3033
Parzen E (1962) Estimation of a probability density-function and mode. Ann Math Statist 33:1065
Paulin MG (1992) Digital filters for firing rate estimation. Biol Cybern 66:525–531
Paulin MG, Hoffman LF (2001) Optimal filtering rate estimation. Neural Networks 14:877–881
Reich DS, Victor JD, Knight BW (1998) The power ratio and the interval map: spiking models and extracellular recordings. J Neurosci 18:10090–10104
Richmond BJ, Optican LM, Spitzer H (1990) Temporal encoding of two-dimensional patterns by single units in primate primary visual cortex. I. Stimulus-response relations. J Neurophysiol 64:351–369
Rieke F, Warland D, de Ruyter van Steveninck R, Bialek W (1997) Spikes: exploring the neural code. MIT Press, Cambridge
Rudemo M (1982) Empirical choice of histograms and kernel density estimators. Scand J Statist 9:65–78
Sain S, Scott D (1996) On locally adaptive density estimation. J Amer Statist Assoc 91:1525–1534
Sain S, Scott D (2002) Zero-bias locally adaptive density estimators. Scand J Statist 29:441–460
Shimazaki H, Shinomoto S (2007a) A method for selecting the bin size of a time histogram. Neural Comput 19:1503–1527
Shimazaki H, Shinomoto S (2007b) Kernel width optimization in the spike-rate estimation. Budelli R, Caputi A, and Gomez L (eds) Neural coding 2007, pp 143–146
Shimazaki H, Shinomoto S (2010) Kernel bandwidth optimization in spike rate estimation. J Comput Neurosci, published on line. doi:10.1007/s10827-009-0180-4
Shimokawa T, Shinomoto S (2009) Estimating instantaneous irregularity of neuronal firing. Neural Comput 21:1931–1951
Shinomoto S, Shima K, Tanji J (2003) Differences in spiking patterns among cortical neurons. Neural Comput 15:2823–2842
Shinomoto S, Miyazaki Y, Tamura H, Fujita I (2005) Regional and laminar differences in in vivo firing patterns of primate cortical neurons. J Neurophysiol 94:567–575
Shinomoto S, Kim H, Shimokawa T, Matsuno N, Funahashi S, Shima K, Fujita I, Tamura H, Doi T, Kawano K, Inaba N, Fukushima K, Kurkin S, Kurata K, Taira M, Tsutsui K, Komatsu H, Ogawa T, Koida K, Tanji J, Toyama K (2009) Relating neuronal firing patterns to functional differentiation of cerebral cortex. PLoS Comput Biol 5:e1000433
Smith AC, Brown EN (2003) Estimating a state-space model from point process observations. Neural Comput 15:965–991
Snyder D (1975) Random point processes. Wiley, New York
Stein RB (1965) A theoretical analysis of neuronal variability. Biophys J 5:173–194
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer Science+Business Media, LLC
About this chapter
Cite this chapter
Shinomoto, S. (2010). Estimating the Firing Rate. In: Grün, S., Rotter, S. (eds) Analysis of Parallel Spike Trains. Springer Series in Computational Neuroscience, vol 7. Springer, Boston, MA. https://doi.org/10.1007/978-1-4419-5675-0_2
Download citation
DOI: https://doi.org/10.1007/978-1-4419-5675-0_2
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4419-5674-3
Online ISBN: 978-1-4419-5675-0
eBook Packages: Biomedical and Life SciencesBiomedical and Life Sciences (R0)