# An *L* _{1}-regularized logistic model for detecting short-term neuronal interactions

- 264 Downloads
- 10 Citations

## Abstract

Interactions among neurons are a key component of neural signal processing. Rich neural data sets potentially containing evidence of interactions can now be collected readily in the laboratory, but existing analysis methods are often not sufficiently sensitive and specific to reveal these interactions. Generalized linear models offer a platform for analyzing multi-electrode recordings of neuronal spike train data. Here we suggest an *L* _{1}-regularized logistic regression model (*L* _{1} *L* method) to detect short-term (order of 3 ms) neuronal interactions. We estimate the parameters in this model using a coordinate descent algorithm, and determine the optimal tuning parameter using a Bayesian Information Criterion. Simulation studies show that in general the *L* _{1} *L* method has better sensitivities and specificities than those of the traditional shuffle-corrected cross-correlogram (covariogram) method. The *L* _{1} *L* method is able to detect excitatory interactions with both high sensitivity and specificity with reasonably large recordings, even when the magnitude of the interactions is small; similar results hold for inhibition given sufficiently high baseline firing rates. Our study also suggests that the false positives can be further removed by thresholding, because their magnitudes are typically smaller than true interactions. Simulations also show that the *L* _{1} *L* method is somewhat robust to partially observed networks. We apply the method to multi-electrode recordings collected in the monkey dorsal premotor cortex (PMd) while the animal prepares to make reaching arm movements. The results show that some neurons interact differently depending on task conditions. The stronger interactions detected with our *L* _{1} *L* method were also visible using the covariogram method.

## Keywords

Multi-electrode recording Model selection Coordinate descent BIC Premotor cortex## Notes

### Acknowledgements

We thank Trevor Hastie and Erin Crowder for their advice during the early stages of this work. We thank Ashwin Iyengar for help with the scalable vector figures. The simulations were done using PITTGRID. We also thank the Action Editor and reviewers for their thoughtful comments.

## References

- Aertsen, A. M. H. J., Gerstein, G. L., Habib, M. K., & Palm, G. (1989). Dynamics of neuronal firing correlation: Modulation of ‘effective connectivity’.
*Journal of Neurophysiology, 61*, 900–917.PubMedGoogle Scholar - Avalos, M., Grandvalet, Y., & Ambroise C. (2003). Regularization methods for additive models. In
*Advances in intelligent data analysis V*.Google Scholar - Batista, A. P., Santhanam, G., Yu, B. M., Ryu, S. I., Afshar, A., & Shenoy, K. V. (2007). Reference frames for reach planning in macaque dorsal premotor cortex.
*Journal of Neurophysiology, 98*, 966–983.PubMedCrossRefGoogle Scholar - Brillinger, D. R. (1988). Maximum likelihood analysis of spike trains of interacting nerve cells.
*Biological Cybernetics, 59*, 189–200.PubMedCrossRefGoogle Scholar - Brody, C. D. (1999). Correlations without synchrony.
*Neural Computation, 11*, 1537–1551.PubMedCrossRefGoogle Scholar - Brown, E. N., Kass, R. E., & Mitra, P. P. (2004). Multiple neural spike train data analysis: State-of-the-art and future challenges.
*Nature Neuroscience, 7*(5), 456–461.PubMedCrossRefGoogle Scholar - Chen, Z., Putrino, D. F., Ghosh, S., Barbieri, R., & Brown, E. N. (2010). Statistical inference for assessing functional connectivity of neuronal ensembles with sparse spiking data. In
*IEEE transactions on neural systems and rehabilitation engineering*.Google Scholar - Chestek, C. A., Batista, A. P., Santhanam, G., Yu, B. M., Afshar, A., Cunningham, J. P., et al. (2007). Single-neuron stability during repeated reaching in macaque premoter cortex.
*Journal of Neuroscience, 27*(40), 10742–10750.PubMedCrossRefGoogle Scholar - Czanner, G., Grun, S., & Iyengar, S. (2005). Theory of the snowflake plot and its relations to higher-order analysis methods.
*Neural Computation, 17*, 1456–1479.PubMedCrossRefGoogle Scholar - Ecker, A. S., Berens, P., Keliris, G. A., Bethge, M., Logothetis, N. K., & Tolias, A. S. (2010). Decorrelated neuronal firing in cortical microcircuits.
*Science, 327*(5965), 584–587.PubMedCrossRefGoogle Scholar - Efron, B., Hastie, T., Johnstone, I., & Tibshirani, R. (2004). Least angle regression.
*Annals of Statistics, 32*, 407–499.CrossRefGoogle Scholar - Eldawlatly, S., Jin, R., & Oweiss, K. G. (2009). Identifying functional connectivity in large-scale neural ensemble recordings: A multiscale data mining approach.
*Neural Computation, 21*, 450–477.PubMedCrossRefGoogle Scholar - Fan, J., & Li, R. (2001). Variable selection via nonconcave penalized likelihood and its oracle properties.
*Journal of the American Statistical Association, 96*(456), 1348–1360.CrossRefGoogle Scholar - Friedman, J., Hastie, T., Hofling, H., & Tibshirani, R. (2007). Pathwise coordinate optimization.
*Annals of Applied Statistics, 1*(2), 302–332.CrossRefGoogle Scholar - Friedman, J., Hastie, T., & Tibshirani, R. (2010). Regularization paths for generalized linear models via coordinate descent.
*Journal of Statistical Software, 33*(1), 1–22.PubMedGoogle Scholar - Fujisawa, S., Amarasingham, A., Harrison, M. T., & Buzsaki, G. (2008). Behavior-dependent short-term assembly dynamics in the medial prefrontal cortex.
*Nature Neuroscience, 11*(7), 823–833.PubMedCrossRefGoogle Scholar - Gao, Y., Black, M. J., Bienenstock, E., Wei, W., & Donoghue, J. P. (2003). A quantitative comparison of linear and non-linear models of motor cortical activity for the encoding and decoding of arm motions. In
*First intl. IEEE/EMBS conf. on neural eng.*(pp. 189–192).Google Scholar - Gerstein, G. L., & Perkel, D. H. (1972). Mutual temporal relationships among neuronal spike trains: Statistical techniques for display and analysis.
*Biophysical Journal, 12*, 453–473.PubMedCrossRefGoogle Scholar - Harrison, M. T., & Geman, S. (2009). A rate and history-preserving resampling algorithm for neural spike trains.
*Neural Computation, 21*, 1244–1258.PubMedCrossRefGoogle Scholar - Hastie, T., Tibshirani, R., & Friedman, J. (2001).
*The elements of statistical learning: Data mining, inference and prediction*. New York: Springer-Verlag.Google Scholar - Kass, R. E., Kelly, R. C., & Loh, W. (2011). Assessment of synchrony in multiple neural spike trains using loglinear point process models.
*Annals of Applied Statistics, 5*(2B), 1262–1292. (Special Section on Statistics and Neuroscience)PubMedCrossRefGoogle Scholar - Kelly, R. C., Smith, M. A., Kass, R. E., & Lee, T. S. (2010). Accounting for network effects in neuronal responses using L1 regularized point process models. In
*Advances in Neural Information Processing Systems*(Vol. 23, pp. 1099–1107).Google Scholar - Kohn, A., & Smith, M. A. (2005). Stimulus dependence of neuronal correlation in primary visual cortex of the macaque.
*Journal of Neuroscience, 25*(14), 3661–3673.PubMedCrossRefGoogle Scholar - Kulkarni, J. E., & Paninski, L. (2007). Common-input models for multiple neural spike-train data.
*Network: Computation in Neural Systems, 18*(5), 375–407.CrossRefGoogle Scholar - Matsumura, M., Chen, D., Sawaguchi, T., Kubota, K., & Fetz, E. E. (1996). Synaptic interactions between primate precentral cortex neurons revealed by spike-triggered averaging of intracellular membrane potentials
*in vivo*.*Journal of Neuroscience, 16*(23), 7757–7767.PubMedGoogle Scholar - McCullagh, P., & Nelder, J. A. (1989).
*Generalized linear models*(2nd ed.). London: Chapman and Hall.Google Scholar - Meinshausen, N., & Yu, B. (2009). Lasso-type recovery of sparse representations for high-dimentional data.
*Annals of Statistics, 37*(1), 246–270.CrossRefGoogle Scholar - Mishchencko, Y., Vogelstein, J. T., & Paninski, L. (2011). Bayesian approach for inferring neuronal connectivity from calcium fluorescent imaging data.
*Annals of Applied Statistics, 5*(2B), 1229–1261. (Special Section on Statistics and Neuroscience)CrossRefGoogle Scholar - Moran, D. W., & Schwartz, A. B. (1999). Motor cortical representation of speed and direction during reaching.
*Journal of Neurophysiology, 82*, 2676–2692.PubMedGoogle Scholar - Paninski, L. (2004). Maximum likelihood estimation of cascade point-process neural encoding models.
*Network: Computation in Neural Systems, 15*, 243–262.CrossRefGoogle Scholar - Park, M. Y., & Hastie, T. (2007). L1-regularization path algorithm for generalized linear models.
*Journal of the Royal Statistical Society, Series B, 69*(4), 659–677.CrossRefGoogle Scholar - Peng, J., Wang, P., Zhou, N., & Zhu, J. (2009). Partial correlation estimation by joint sparse regression models.
*Journal of the American Statistical Association, 104*(486), 735–746.PubMedCrossRefGoogle Scholar - Perkel, D. H., Gerstein, G. L., & Moore, G. P. (1967). Neuronal spike trains and stochastic point process ii. Simultaneous spike trains.
*Biophysical Journal, 7*, 414–440.Google Scholar - Perkel, D. H., Gerstein, G. L., Smith, M. S., & Tatton, W. G. (1975). Nerve-impulse patterns: A quantitative display technique for three neurons.
*Brain Research, 100*, 271–296.PubMedCrossRefGoogle Scholar - Qian, G., & Wu, Y. (2006). Strong limit theorems on the model selection in generalized linear regression with binomial responses.
*Statistica Sinica, 16*, 1335–1365.Google Scholar - Reid, C. R., & Alonso, J. (1995). Specificty of monosynaptic connections from thalamus to visual cortex.
*Nature, 378*(16), 281–284.PubMedCrossRefGoogle Scholar - Rosset, S. (2004). Following curved regularized optimization solution paths.
*Advances in NIPS*.Google Scholar - Santhanam, G., Sahani, M., Ryu, S., & Shenoy, K. (2004). An extensible infrastructure for fully automated spike sorting during online experiments. In
*Conf. proc. IEEE eng. med. biol. soc.*(Vol. 6, pp. 4380–4384).Google Scholar - Stevenson, I. H., Rebesco, J. M., Hatsopoulos, N. G., Haga, Z., Miller, L. E., & Kording, K. P. (2009). Bayesian inference of functional connectivity and network structure from spikes.
*IEEE TNSRE (Special Issue on Brain Connectivity), 17*(3), 203–213.Google Scholar - Tibshirani, R. (1996). Regression shrinkage and selection via the lasso.
*Journal of the Royal Statistical Society, Series B, 58*, 267–288.Google Scholar - Tibshirani, R. (1997). The lasso method for variable selection in the cox model.
*Statistics in Medicine, 16*, 385–395.PubMedCrossRefGoogle Scholar - 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*, 1074–1089.PubMedCrossRefGoogle Scholar - Truccolo, W., Hochberg, L. R., & Donoghue, J. P. (2010). Collective dynamics in human and monkey sensorimotor cortex: Predicting single neuron spikes.
*Nature Neuroscience, 13*(1), 105–111.PubMedCrossRefGoogle Scholar - Wang, H., Li, B., & Leng, C. (2009). Shrinkage tuning parameter selection with a diverging number of parameters.
*Journal of the Royal Statistical Society, Series B, 71*(3), 671–683.CrossRefGoogle Scholar - Wasserman, L., & Roeder, K. (2009). High-dimensional variable selection.
*Annals of Statistics, 37*, 2178–2201.PubMedCrossRefGoogle Scholar - Wu, T., & Lange, K. (2008). Pathwise coordinate optimization.
*Annals of Applied Statistics, 2*(1), 224–244.CrossRefGoogle Scholar - Zhao, M., & Iyengar, S. (2010). Nonconvergence in logistic and poisson models for neural spiking.
*Neural Computation, 22*, 1231–1244.PubMedCrossRefGoogle Scholar - Zohary, E., Shadlen, N. M., & Newsome, W. T. (1994). Correlated neuronal discharge rate and its implications for psychophysical performance.
*Nature, 370*, 140–143.PubMedCrossRefGoogle Scholar - Zou, H., Hastie, T., & Tibshirani, R. (2007). On the “degrees of freedom” of the lasso.
*Annals of Statistics, 35*(5), 2173–2192.CrossRefGoogle Scholar