Journal of Computational Neuroscience

, Volume 41, Issue 1, pp 29–43 | Cite as

Flexible models for spike count data with both over- and under- dispersion



A key observation in systems neuroscience is that neural responses vary, even in controlled settings where stimuli are held constant. Many statistical models assume that trial-to-trial spike count variability is Poisson, but there is considerable evidence that neurons can be substantially more or less variable than Poisson depending on the stimuli, attentional state, and brain area. Here we examine a set of spike count models based on the Conway-Maxwell-Poisson (COM-Poisson) distribution that can flexibly account for both over- and under-dispersion in spike count data. We illustrate applications of this noise model for Bayesian estimation of tuning curves and peri-stimulus time histograms. We find that COM-Poisson models with group/observation-level dispersion, where spike count variability is a function of time or stimulus, produce more accurate descriptions of spike counts compared to Poisson models as well as negative-binomial models often used as alternatives. Since dispersion is one determinant of parameter standard errors, COM-Poisson models are also likely to yield more accurate model comparison. More generally, these methods provide a useful, model-based framework for inferring both the mean and variability of neural responses.


Spike count variability Tuning curves Poisson Conway-Maxwell-Poisson 

Supplementary material

10827_2016_603_MOESM1_ESM.docx (30 kb)
ESM 1(DOCX 30 kb)


  1. Amarasingham, A., Chen, T.-L., Geman, S., Harrison, M. T., & Sheinberg, D. L. (2006). Spike count reliability and the Poisson hypothesis. The Journal of Neuroscience : The Official Journal of the Society for Neuroscience, 26(3), 801–809. doi:10.1523/JNEUROSCI.2948-05.2006.CrossRefGoogle Scholar
  2. Arieli, A., Sterkin, A., Grinvald, A., & Aertsen, A. (1996). Dynamics of ongoing activity: explanation of the large variability in evoked cortical responses. Science, 273(5283), 1868–1871. doi:10.1126/science.273.5283.1868.CrossRefPubMedGoogle Scholar
  3. Averbeck, B. B., Latham, P. E., & Pouget, A. (2006). Neural correlations, population coding and computation. Nature Reviews. Neuroscience, 7(5), 358–366.CrossRefPubMedGoogle Scholar
  4. Azouz, R., & Gray, C. M. (1999). Cellular mechanisms contributing to response variability of cortical neurons in vivo. Journal of Neuroscience, 19(6), 2209.PubMedGoogle Scholar
  5. Bair, W., & Koch, C. (1996). Temporal precision of spike trains in extrastriate cortex of the behaving macaque monkey. Neural Computation, 8(6), 1185–1202. doi:10.1162/neco.1996.8.6.1185.CrossRefPubMedGoogle Scholar
  6. Barbieri, R., Quirk, M. C., Frank, L. M., Wilson, M. A., & Brown, E. N. (2001). Construction and analysis of non-Poisson stimulus-response models of neural spiking activity. Journal of Neuroscience Methods, 105(1), 25–37. doi:10.1016/S0165-0270(00)00344-7.CrossRefPubMedGoogle Scholar
  7. Berry, M. J., & Meister, M. (1998). Refractoriness and neural precision. The Journal of Neuroscience : The official Journal of the Society for Neuroscience, 18(6), 2200–2211.Google Scholar
  8. Berry, M. J., Warland, D. K., & Meister, M. (1997). The structure and precision of retinal spike trains. Proceedings of the National Academy of Sciences, 94(10), 5411–5416. doi:10.1073/pnas.94.10.5411.CrossRefGoogle Scholar
  9. Brette, R., & Gerstner, W. (2005). Adaptive exponential integrate-and-fire model as an effective description of neuronal activity. Journal of Neurophysiology, 94(5), 3637–3642. doi:10.1152/jn.00686.2005.CrossRefPubMedGoogle Scholar
  10. Britten, K. H., Shadlen, M. N., Newsome, W. T., & Movshon, J. A. (1992). The analysis of visual motion: a comparison of neuronal and psychophysical performance. Journal of Neuroscience, 12(12), 4745.PubMedGoogle Scholar
  11. Brown, E., Barbieri, R., Eden, U., & Frank, L. (2003). Likelihood methods for neural data analysis. In J. Feng (Ed.), Computational Neuroscience: A comprehensive approach (pp. 253–286). London: Chapman and Hall.Google Scholar
  12. Cameron, A. C., & Trivedi, P. K. (2001). Essentials of count data regression. In A companion to theoretical econometrics (Vol. 331). Blackwell Publishing Ltd.Google Scholar
  13. Carandini, M. (2004). Amplification of trial-to-trial response variability by neurons in visual cortex. PLoS Biology, 2(9), E264. doi:10.1371/journal.pbio.0020264.CrossRefPubMedPubMedCentralGoogle Scholar
  14. Churchland, M. M., Yu, B. M., Ryu, S. I., Santhanam, G., & Shenoy, K. V. (2006). Neural variability in premotor cortex provides a signature of motor preparation. Journal of Neuroscience, 26(14), 3697.CrossRefPubMedGoogle Scholar
  15. Churchland, M. M., Yu, B. M., Cunningham, J. P., Sugrue, L. P., Cohen, M. R., Corrado, G. S., et al. (2010). Stimulus onset quenches neural variability: a widespread cortical phenomenon. Nature Neuroscience, 13(3), 369–378. doi:10.1038/nn.2501.CrossRefPubMedPubMedCentralGoogle Scholar
  16. Churchland, A. K., Kiani, R., Chaudhuri, R., Wang, X. J., Pouget, A., & Shadlen, M. N. (2011). Variance as a signature of neural computations during decision making. Neuron, 69(4), 818–831. doi:10.1016/j.neuron.2010.12.037.CrossRefPubMedPubMedCentralGoogle Scholar
  17. Cohen, M. R., & Kohn, A. (2011). Measuring and interpreting neuronal correlations. Nature Neuroscience, 14(7), 811–819. doi:10.1038/nn.2842.CrossRefPubMedPubMedCentralGoogle Scholar
  18. Cronin, B., Stevenson, I. H., Sur, M., & Kording, K. P. (2010). Hierarchical Bayesian modeling and Markov Chain Monte Carlo sampling for tuning-curve analysis. Journal of Neurophysiology, 103(1), 591.CrossRefPubMedGoogle Scholar
  19. Czanner, G., Eden, U. T., Wirth, S., Yanike, M., Suzuki, W. A., & Brown, E. N. (2008). Analysis of between-trial and within-trial neural spiking dynamics. Journal of Neurophysiology, 99(5), 2672–2693. doi:10.1152/jn.00343.2007.CrossRefPubMedPubMedCentralGoogle Scholar
  20. De Boor, C. (1978). A practical guide to splines. Applied mathematical sciences 27. Verlag: Springer.CrossRefGoogle Scholar
  21. del Castillo, J., & Pérez-Casany, M. (2005). Overdispersed and underdispersed Poisson generalizations. Journal of Statistical Planning and Inference, 134(2), 486–500. doi:10.1016/j.jspi.2004.04.019.CrossRefGoogle Scholar
  22. Deweese, M. R., & Zador, A. M. (2004). Shared and private variability in the auditory cortex. Journal of Neurophysiology, 92(3), 1840–1855. doi:10.1152/jn.00197.2004.CrossRefPubMedGoogle Scholar
  23. DeWeese, M. R., Wehr, M., & Zador, A. M. (2003). Binary spiking in auditory cortex. The Journal of Neuroscience : The Official Journal of the Society for Neuroscience, 23(21), 7940–7949.Google Scholar
  24. Dimatteo, I., Genovese, C. R., & Kass, R. E. (2001). Bayesian curve-fitting with free-knot splines. Biometrika, 88(4), 1055–1071. doi:10.1093/biomet/88.4.1055.CrossRefGoogle Scholar
  25. Eden, U. T., & Kramer, M. a. (2010). Drawing inferences from Fano factor calculations. Journal of Neuroscience Methods, 190(1), 149–152. doi:10.1016/j.jneumeth.2010.04.012.CrossRefPubMedGoogle Scholar
  26. Eden, U. T., Frank, L. M., Barbieri, R., Solo, V., & Brown, E. N. (2004). Dynamic analysis of neural encoding by point process adaptive filtering. Neural Computation, 16(5), 971–998. doi:10.1162/089976604773135069.CrossRefPubMedGoogle Scholar
  27. Ermentrout, G. B., Galán, R. F., & Urban, N. N. (2008). Reliability, synchrony and noise. Trends in Neurosciences, 31(8), 428–434.CrossRefPubMedPubMedCentralGoogle Scholar
  28. Faisal, A. A., Selen, L. P. J., & Wolpert, D. M. (2008). Noise in the nervous system. Nature Reviews. Neuroscience, 9(4), 292–303. doi:10.1038/nrn2258.CrossRefPubMedPubMedCentralGoogle Scholar
  29. Gao, Y., Buesing, L., Shenoy, K. V, & Cunningham, J. P. (2015). High-dimensional neural spike train analysis with generalized count linear dynamical systems. In NIPS.Google Scholar
  30. Gelman, A., Jakulin, A., Pittau, M. G., & Su, Y.-S. (2008). A weakly informative default prior distribution for logistic and other regression models. The Annals of Applied Statistics, 2(4), 1360–1383. Accessed 30 July 2015.CrossRefGoogle Scholar
  31. Goris, R. L. T., Movshon, J. A., & Simoncelli, E. P. (2014). Partitioning neuronal variability. Nature Neuroscience, 17(6), 858–65. doi:10.1038/nn.3711.CrossRefPubMedPubMedCentralGoogle Scholar
  32. Gourieroux, C., Monfort, A., & Trognon, A. (1984). Pseudo maximum likelihood methods: applications to Poisson models. Econometrica, 52(3), 701–720.CrossRefGoogle Scholar
  33. Harris, K. D., Csicsvari, J., Hirase, H., Dragoi, G., & Buzsáki, G. (2003). Organization of cell assemblies in the hippocampus. Nature, 424(6948), 552–556.CrossRefPubMedGoogle Scholar
  34. Hoffman, M., & Gelman, A. (2014). The no-U-turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo. Journal of Machine Learning Research, 15, 30.Google Scholar
  35. Hoyer, P. O., Hyvarinen, A., & Hyvärinen, A. (2003). Interpreting neural response variability as Monte Carlo sampling of the posterior (Vol. 15, pp. 277–284,). MIT Press.Google Scholar
  36. Hussar, C., & Pasternak, T. (2010). Trial-to-trial variability of the prefrontal neurons reveals the nature of their engagement in a motion discrimination task. Proceedings of the National Academy of Sciences of the United States of America, 107(50), 21842–7. doi:10.1073/pnas.1009956107.CrossRefPubMedPubMedCentralGoogle Scholar
  37. Kadane, J. B., Shmueli, G., Minka, T. P., Borle, S., & Boatwright, P. (2006). Conjugate analysis of the Conway-Maxwell-Poisson distribution. Bayesian Analysis, 1(2), 363–374. Accessed 11 December 2015.CrossRefGoogle Scholar
  38. Kara, P., Reinagel, P., & Reid, R. C. (2000). Low response variability in simultaneously recorded retinal, thalamic, and cortical neurons. Neuron, 27(3), 635–646. doi:10.1016/S0896-6273(00)00072-6.CrossRefPubMedGoogle Scholar
  39. Kass, R. E., & Ventura, V. (2001). A spike-train probability model. Neural Computation, 13(8), 1713–1720.CrossRefPubMedGoogle Scholar
  40. Kass, R. E., Ventura, V., & Cai, C. (2003). Statistical smoothing of neuronal data. Network (Bristol, England), 14(1), 5–15. Accessed 29 October 2015.CrossRefGoogle Scholar
  41. Kaufman, C. G., Ventura, V., & Kass, R. E. (2005). Spline-based non-parametric regression for periodic functions and its application to directional tuning of neurons, 24(14), 2255–2265.Google Scholar
  42. Keat, J., Reinagel, P., Reid, R. C., & Meister, M. (2001). Predicting every spike a model for the responses of visual neurons. Neuron, 30(3), 803–817.CrossRefPubMedGoogle Scholar
  43. Kelly, R. C., Smith, M. A., Kass, R. E., & Lee, T. S. (2010). Local field potentials indicate network state and account for neuronal response variability. Journal of Computational Neuroscience, 29(3), 567–579. doi:10.1007/s10827-009-0208-9.CrossRefPubMedPubMedCentralGoogle Scholar
  44. Kohn, A., & Movshon, J. A. (2003). Neuronal adaptation to visual motion in area MT of the macaque. Neuron, 39(4), 681–691. doi:10.1016/S0896-6273(03)00438-0.CrossRefPubMedGoogle Scholar
  45. Kottas, A., Behseta, S., Moorman, D. E., Poynor, V., & Olson, C. R. (2012). Bayesian nonparametric analysis of neuronal intensity rates. Journal of Neuroscience Methods, 203(1), 241–53. doi:10.1016/j.jneumeth.2011.09.017.CrossRefPubMedGoogle Scholar
  46. Koyama, S. (2015). On the spike train variability characterized by variance-to-mean power relationship. Neural Computation, 27(7), 1530–48. doi:10.1162/NECO_a_00748.CrossRefPubMedGoogle Scholar
  47. Lansky, P., & Vaillant, J. (2000). Stochastic model of the overdispersion in place cell discharge. Biosystems, 58(1), 27–32.CrossRefPubMedGoogle Scholar
  48. Lee, D., Port, N. L., Kruse, W., & Georgopoulos, A. P. (1998). Variability and correlated noise in the discharge of neurons in motor and parietal areas of the primate cortex. Journal of Neuroscience, 18(3), 1161–1170. Accessed 11 November 2015.PubMedGoogle Scholar
  49. Maimon, G., & Assad, J. a. (2009). Beyond Poisson: increased spike-time regularity across primate parietal cortex. Neuron, 62(3), 426–440. doi:10.1016/j.neuron.2009.03.021.CrossRefPubMedPubMedCentralGoogle Scholar
  50. Mainen, Z. F., & Sejnowski, T. J. (1995). Reliability of spike timing in neocortical neurons. Science, 268(5216), 1503–1506.CrossRefPubMedGoogle Scholar
  51. Mandelblat-Cerf, Y., Paz, R., & Vaadia, E. (2009). Trial-to-trial variability of single cells in motor cortices is dynamically modified during visuomotor adaptation. The Journal of Neuroscience : The Official Journal of the Society for Neuroscience, 29(48), 15053–62. doi:10.1523/JNEUROSCI.3011-09.2009.CrossRefGoogle Scholar
  52. Masquelier, T. (2013). Neural variability, or lack thereof. Frontiers in Computational Neuroscience, 7(February), 7. doi:10.3389/fncom.2013.00007.PubMedPubMedCentralGoogle Scholar
  53. Minka, T. T. P., Shmueli, G., Kadane, J. B. J., Borle, S., & Boatwright, P. (2003). Computing with the COM-Poisson distribution., PA: Department of, (776).
  54. Moshitch, D., & Nelken, I. (2014). Using Tweedie distributions for fitting spike count data. Journal of Neuroscience Methods, 225, 13–28. doi:10.1016/j.jneumeth.2014.01.004.CrossRefPubMedGoogle Scholar
  55. Nawrot, M. P. (2010). Analysis and interpretation of interval and count variability in neural spike trains. In Analysis of parallel spike trains (pp. 37–58). Springer.Google Scholar
  56. Paninski, L., Ahmadian, Y., Ferreira, D. G., Koyama, S., Rahnama Rad, K., Vidne, M., et al. (2010). A new look at state-space models for neural data. Journal of Computational Neuroscience, 29(1), 107–126. doi:10.1007/s10827-009-0179-x.CrossRefPubMedGoogle Scholar
  57. Pillow, J. W., Paninski, L., Uzzell, V. J., Simoncelli, E. P., & Chichilnisky, E. J. (2005). Prediction and decoding of retinal ganglion cell responses with a probabilistic spiking model. Journal of Neuroscience, 25(47), 11003–11013.CrossRefPubMedGoogle Scholar
  58. Pillow, J. W., Shlens, J., Paninski, L., Sher, A., Litke, A. M., Chichilnisky, E. J., & Simoncelli, E. P. (2008). Spatio-temporal correlations and visual signalling in a complete neuronal population. Nature, 454(7207), 995–999.CrossRefPubMedPubMedCentralGoogle Scholar
  59. Reich, D. S., Victor, J. D., Knight, B. W., Ozaki, T., & Kaplan, E. (1997). Response variability and timing precision of neuronal spike trains in vivo. Journal of Neurophysiology, 77(5), 2836–41. Accessed 12 November 2015.PubMedGoogle Scholar
  60. Rubin, D. B. (1981). The Bayesian bootstrap. The Annals of Statistics, 9(1), 130–134. Accessed 30 October 2015.CrossRefGoogle Scholar
  61. Sanger, T. D. (1996). Probability density estimation for the interpretation of neural population codes. Journal of Neurophysiology, 76(4), 2790–2793.PubMedGoogle Scholar
  62. Scaglione, A., Moxon, K. A., Aguilar, J., & Foffani, G. (2011). Trial-to-trial variability in the responses of neurons carries information about stimulus location in the rat whisker thalamus. Proceedings of the National Academy of Sciences of the United States of America, 108(36), 14956–61. doi:10.1073/pnas.1103168108.CrossRefPubMedPubMedCentralGoogle Scholar
  63. Schölvinck, M. L., Saleem, A. B., Benucci, A., Harris, K. D., & Carandini, M. (2015). Cortical state determines global variability and correlations in visual cortex. The Journal of Neuroscience : The Official Journal of the Society for Neuroscience, 35(1), 170–8. doi:10.1523/JNEUROSCI.4994-13.2015.CrossRefGoogle Scholar
  64. Scott, J., & Pillow, J. W. (2012). Fully Bayesian inference for neural models with negative-binomial spiking. In Advances in Neural Information Processing Systems (pp. 1898–1906). Accessed 27 July 2015.
  65. Sellers, K. F., & Shmueli, G. (2009). A regression model for count data with observation-level dispersion. In 24th International Workshop on Statistical Modelling (IWSM).Google Scholar
  66. Sellers, K. F., & Shmueli, G. (2010). A flexible regression model for count data. The Annals of Applied Statistics, 943–961.Google Scholar
  67. Sellers, K. F., & Shmueli, G. (2013). Data dispersion: Now you see it… now you don’t. Communications in Statistics-Theory and Methods, 42(17), 3134–3147.CrossRefGoogle Scholar
  68. Sellers, K. F., Borle, S., & Shmueli, G. (2012). The COM-Poisson model for count data: a survey of methods and applications. Applied Stochastic Models in Business and Industry, 28(2), 104–116.CrossRefGoogle Scholar
  69. Shadlen, M. N., & Newsome, W. T. (1998). The variable discharge of cortical neurons: implications for connectivity, computation, and information coding. Journal of Neuroscience, 18(10), 3870–3896.PubMedGoogle Scholar
  70. Shidara, M., Mizuhiki, T., & Richmond, B. J. (2005). Neuronal firing in anterior cingulate neurons changes modes across trials in single states of multitrial reward schedules. Experimental Brain Research, 163(2), 242–5. doi:10.1007/s00221-005-2232-y.CrossRefPubMedGoogle Scholar
  71. Shinomoto, S., Kim, H., Shimokawa, T., Matsuno, N., Funahashi, S., Shima, K., et al. (2009). Relating neuronal firing patterns to functional differentiation of cerebral cortex. PLoS Computational Biology, 5(7), e1000433. doi:10.1371/journal.pcbi.1000433.CrossRefPubMedPubMedCentralGoogle Scholar
  72. Shmueli, G., Minka, T., Kadane, J., Borle, S., & Boatwright, P. (2004). A useful distribution for fitting discrete data:revival of the conway-Maxwell_Poisson distribution. Applied Statistic, 54(1), 127–142.Google Scholar
  73. Softky, W. R., & Koch, C. (1993). The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs. The Journal of Neuroscience : The Official Journal of the Society for Neuroscience, 13(1), 334–350.Google Scholar
  74. Stan: A C++ Library for probability and sampling, version 2.8.0. (2015). Retrieved from
  75. Stein, R. B., Gossen, E. R., & Jones, K. E. (2005). Neuronal variability: noise or part of the signal? Nature Reviews. Neuroscience, 6(5), 389–397. doi:10.1038/nrn1668.CrossRefPubMedGoogle Scholar
  76. Stevenson, I. H., Rebesco, J. M., Miller, L. E., & Körding, K. P. (2008). Inferring functional connections between neurons. Current Opinion in Neurobiology, 18(6), 582–588.CrossRefPubMedPubMedCentralGoogle Scholar
  77. Stevenson, I. H., Cherian, A., London, B. M., Sachs, N. A., Lindberg, E., Reimer, J., et al. (2011). Statistical assessment of the stability of neural movement representations. Journal of Neurophysiology, 106(2), 764–774. doi:10.1152/jn.00626.2010.CrossRefPubMedPubMedCentralGoogle Scholar
  78. Taouali, W., Benvenuti, G., Wallisch, P., Chavane, F., & Perrinet, L. U. (2016). Testing the odds of inherent vs. observed overdispersion in neural spike counts. Journal of Neurophysiology, 115(1), 434–44. doi:10.1152/jn.00194.2015.CrossRefPubMedGoogle Scholar
  79. Teich, M. C. (1989). Fractal character of the auditory neural spike train. IEEE Transactions on Bio-Medical Engineering, 36(1), 150–60. doi:10.1109/10.16460.CrossRefPubMedGoogle Scholar
  80. Truccolo, W., Eden, U. T., Fellows, M. R., Donoghue, J. P., & Brown, E. N. (2005). A point process framework for relating neural spiking activity to spiking history, neural ensemble, and extrinsic covariate effects. Journal of Neurophysiology, 93(2), 1074–1089.CrossRefPubMedGoogle Scholar
  81. Uzzell, V. J., & Chichilnisky, E. J. (2004). Precision of spike trains in primate retinal ganglion cells. Journal of Neurophysiology, 92(2), 780–789. doi:10.1152/jn.01171.2003.CrossRefPubMedGoogle Scholar
  82. van Steveninck, R. R. R., Lewen, G. D., Strong, S. P., Koberle, R., & Bialek, W. (1997). Reproducibility and variability in neural spike trains. Science, 275(5307), 1805–1808.CrossRefGoogle Scholar
  83. Vogel, A., Hennig, R. M., & Ronacher, B. (2005). Increase of neuronal response variability at higher processing levels as revealed by simultaneous recordings. Journal of Neurophysiology, 93(6), 3548–59. doi:10.1152/jn.01288.2004.CrossRefPubMedGoogle Scholar
  84. Werner, G., & Mountcastle, V. B. (1963). The variability of central neural activity in a sensory system, and its implications for the central reflection of sensory events. Journal of Neurophysiology, 26(6), 958–977.PubMedGoogle Scholar
  85. Wiener, M. C., & Richmond, B. J. (2003). Decoding spike trains instant by instant using order statistics and the mixture-of-poissons model. Journal of Neuroscience, 23(6), 2394–2406. Accessed 14 December 2015.PubMedGoogle Scholar
  86. Zador, A. (1998). Impact of synaptic unreliability on the information transmitted by spiking neurons. Journal of Neurophysiology, 79(3), 1219–1229.PubMedGoogle Scholar
  87. Zhao, M., & Iyengar, S. (2010). Nonconvergence in logistic and poisson models for neural spiking. Neural Computation, 22(5), 1231–1244.CrossRefPubMedGoogle Scholar
  88. Zhu, L., Morris, D. S., Sellers, K. F., & Shmueli, G. (2015). Bridging the gap: a generalized stochastic process for count data. Under Review.Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Department of Psychological SciencesUniversity of ConnecticutStorrsUSA
  2. 2.Department of Biomedical EngineeringUniversity of ConnecticutStorrsUSA
  3. 3.Connecticut Institute for Brain and Cognitive ScienceStorrsUSA

Personalised recommendations