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

Abstract

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.

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References

  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.

  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.

  3. Averbeck, B. B., Latham, P. E., & Pouget, A. (2006). Neural correlations, population coding and computation. Nature Reviews. Neuroscience, 7(5), 358–366.

  4. Azouz, R., & Gray, C. M. (1999). Cellular mechanisms contributing to response variability of cortical neurons in vivo. Journal of Neuroscience, 19(6), 2209.

  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.

  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.

  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.

  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.

  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.

  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.

  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.

  12. Cameron, A. C., & Trivedi, P. K. (2001). Essentials of count data regression. In A companion to theoretical econometrics (Vol. 331). Blackwell Publishing Ltd.

  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.

  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.

  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.

  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.

  17. Cohen, M. R., & Kohn, A. (2011). Measuring and interpreting neuronal correlations. Nature Neuroscience, 14(7), 811–819. doi:10.1038/nn.2842.

  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.

  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.

  20. De Boor, C. (1978). A practical guide to splines. Applied mathematical sciences 27. Verlag: Springer.

  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.

  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.

  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.

  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.

  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.

  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.

  27. Ermentrout, G. B., Galán, R. F., & Urban, N. N. (2008). Reliability, synchrony and noise. Trends in Neurosciences, 31(8), 428–434.

  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.

  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.

  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. http://projecteuclid.org/euclid.aoas/1231424214. Accessed 30 July 2015.

  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.

  32. Gourieroux, C., Monfort, A., & Trognon, A. (1984). Pseudo maximum likelihood methods: applications to Poisson models. Econometrica, 52(3), 701–720.

  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.

  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.

  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.

  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.

  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. http://projecteuclid.org/euclid.ba/1340371067. Accessed 11 December 2015.

  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.

  39. Kass, R. E., & Ventura, V. (2001). A spike-train probability model. Neural Computation, 13(8), 1713–1720.

  40. Kass, R. E., Ventura, V., & Cai, C. (2003). Statistical smoothing of neuronal data. Network (Bristol, England), 14(1), 5–15. http://www.ncbi.nlm.nih.gov/pubmed/12613549. Accessed 29 October 2015.

  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.

  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.

  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.

  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.

  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.

  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.

  47. Lansky, P., & Vaillant, J. (2000). Stochastic model of the overdispersion in place cell discharge. Biosystems, 58(1), 27–32.

  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. http://www.jneurosci.org/content/18/3/1161.abstract?ijkey=bd8ccb3d3a84873b46d8a3414a579c19586b02c6&keytype2=tf_ipsecsha. Accessed 11 November 2015.

  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.

  50. Mainen, Z. F., & Sejnowski, T. J. (1995). Reliability of spike timing in neocortical neurons. Science, 268(5216), 1503–1506.

  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.

  52. Masquelier, T. (2013). Neural variability, or lack thereof. Frontiers in Computational Neuroscience, 7(February), 7. doi:10.3389/fncom.2013.00007.

  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). http://lib.stat.cmu.edu/cmu-stats/tr/tr776/tr776.pdf

  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.

  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.

  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.

  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.

  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.

  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. http://jn.physiology.org/content/77/5/2836.abstract. Accessed 12 November 2015.

  60. Rubin, D. B. (1981). The Bayesian bootstrap. The Annals of Statistics, 9(1), 130–134. http://projecteuclid.org/euclid.aos/1176345338. Accessed 30 October 2015.

  61. Sanger, T. D. (1996). Probability density estimation for the interpretation of neural population codes. Journal of Neurophysiology, 76(4), 2790–2793.

  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.

  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.

  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). http://papers.nips.cc/paper/4567-fully-bayesian-inference-for-neural-models-with-negative-binomial-spiking. 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).

  66. Sellers, K. F., & Shmueli, G. (2010). A flexible regression model for count data. The Annals of Applied Statistics, 943–961.

  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.

  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.

  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.

  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.

  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.

  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.

  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.

  74. Stan: A C++ Library for probability and sampling, version 2.8.0. (2015). Retrieved from http://mc-stan.org/

  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.

  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.

  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.

  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.

  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.

  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.

  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.

  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.

  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.

  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.

  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. http://www.jneurosci.org/content/23/6/2394.full. Accessed 14 December 2015.

  86. Zador, A. (1998). Impact of synaptic unreliability on the information transmitted by spiking neurons. Journal of Neurophysiology, 79(3), 1219–1229.

  87. Zhao, M., & Iyengar, S. (2010). Nonconvergence in logistic and poisson models for neural spiking. Neural Computation, 22(5), 1231–1244.

  88. Zhu, L., Morris, D. S., Sellers, K. F., & Shmueli, G. (2015). Bridging the gap: a generalized stochastic process for count data. Under Review.

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Acknowledgments

Thanks to Mike DeWeese, Heather Read, and Monty Escabi for helpful comments and discussions. IHS was supported by an NSF Computing Innovation Fellowship (NSF-0937060 CIF-D-018).

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Stevenson, I.H. Flexible models for spike count data with both over- and under- dispersion. J Comput Neurosci 41, 29–43 (2016). https://doi.org/10.1007/s10827-016-0603-y

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Keywords

  • Spike count variability
  • Tuning curves
  • Poisson
  • Conway-Maxwell-Poisson