A common analytical problem in neuroscience is the interpretation of neural activity with respect to sensory input or behavioral output. This is typically achieved by regressing measured neural activity against known stimuli or behavioral variables to produce a “tuning function” for each neuron. Unfortunately, because this approach handles neurons individually, it cannot take advantage of simultaneous measurements from spatially adjacent neurons that often have similar tuning properties. On the other hand, sharing information between adjacent neurons can errantly degrade estimates of tuning functions across space if there are sharp discontinuities in tuning between nearby neurons. In this paper, we develop a computationally efficient block Gibbs sampler that effectively pools information between neurons to denoise tuning function estimates while simultaneously preserving sharp discontinuities that might exist in the organization of tuning across space. This method is fully Bayesian, and its computational cost per iteration scales sub-quadratically with total parameter dimensionality. We demonstrate the robustness and scalability of this approach by applying it to both real and synthetic datasets. In particular, an application to data from the spinal cord illustrates that the proposed methods can dramatically decrease the experimental time required to accurately estimate tuning functions.
Ann. Appl. Stat.
11(2):
598-637
(June 2017).
DOI: 10.1214/16-AOAS996
Afonso, M. V., Bioucas-Dias, J. M. and Figueiredo, M. A. T. (2010). Fast image recovery using variable splitting and constrained optimization. IEEE Trans. Image Process. 19 2345–2356.Afonso, M. V., Bioucas-Dias, J. M. and Figueiredo, M. A. T. (2010). Fast image recovery using variable splitting and constrained optimization. IEEE Trans. Image Process. 19 2345–2356.
Ahmadian, Y., Pillow, J. W. and Paninski, L. (2011). Efficient Markov chain Monte Carlo methods for decoding neural spike trains. Neural Comput. 23 46–96.Ahmadian, Y., Pillow, J. W. and Paninski, L. (2011). Efficient Markov chain Monte Carlo methods for decoding neural spike trains. Neural Comput. 23 46–96.
Ahrens, M., Orger, M., Robson, D., Li, J. and Keller, P. (2013). Whole-brain functional imaging at cellular resolution using light-sheet microscopy. Nat. Methods 10 413–420.Ahrens, M., Orger, M., Robson, D., Li, J. and Keller, P. (2013). Whole-brain functional imaging at cellular resolution using light-sheet microscopy. Nat. Methods 10 413–420.
Akay, T., Tourtellotte, W. G., Arber, S. and Jessell, T. M. (2014). Degradation of mouse locomotor pattern in the absence of proprioceptive sensory feedback. Proc. Natl. Acad. Sci. USA 111 16877–16882.Akay, T., Tourtellotte, W. G., Arber, S. and Jessell, T. M. (2014). Degradation of mouse locomotor pattern in the absence of proprioceptive sensory feedback. Proc. Natl. Acad. Sci. USA 111 16877–16882.
Barbero, A. and Sra, S. (2011). Fast Newton-type methods for total variation regularization. In Proceedings of the 28th International Conference on Machine Learning (ICML-11) 313–320.Barbero, A. and Sra, S. (2011). Fast Newton-type methods for total variation regularization. In Proceedings of the 28th International Conference on Machine Learning (ICML-11) 313–320.
Bardsley, J. M., Solonen, A., Haario, H. and Laine, M. (2014). Randomize-then-optimize: A method for sampling from posterior distributions in nonlinear inverse problems. SIAM J. Sci. Comput. 36 A1895–A1910.Bardsley, J. M., Solonen, A., Haario, H. and Laine, M. (2014). Randomize-then-optimize: A method for sampling from posterior distributions in nonlinear inverse problems. SIAM J. Sci. Comput. 36 A1895–A1910.
Bouchard, K. E., Mesgarani, N., Johnson, K. and Chang, E. F. (2013). Functional organization of human sensorimotor cortex for speech articulation. Nature 495 327–332.Bouchard, K. E., Mesgarani, N., Johnson, K. and Chang, E. F. (2013). Functional organization of human sensorimotor cortex for speech articulation. Nature 495 327–332.
Bouman, I. and Liu, B. (1988). A multiple resolution approach to regularization. In Proceedings SPIE 1001, Visual Communications and Image Processing ’88 512–520.Bouman, I. and Liu, B. (1988). A multiple resolution approach to regularization. In Proceedings SPIE 1001, Visual Communications and Image Processing ’88 512–520.
Boyd, S., Parikh, N., Chu, E., Peleato, B. and Eckstein, J. (2011). Distributed optimization and statistical learning via the alternating direction method of multipliers. Faund. Trends Mach. Learn. 3 1–122.Boyd, S., Parikh, N., Chu, E., Peleato, B. and Eckstein, J. (2011). Distributed optimization and statistical learning via the alternating direction method of multipliers. Faund. Trends Mach. Learn. 3 1–122.
Brezger, A., Fahrmeir, L. and Hennerfeind, A. (2007). Adaptive Gaussian Markov random fields with applications in human brain mapping. J. Roy. Statist. Soc. Ser. C 56 327–345.Brezger, A., Fahrmeir, L. and Hennerfeind, A. (2007). Adaptive Gaussian Markov random fields with applications in human brain mapping. J. Roy. Statist. Soc. Ser. C 56 327–345.
Candes, E. J. (1999b). Curvelets—a surprisingly effective nonadaptive representation for objects with edges. In Curve and Surface Fitting: Saint-Malo (A. Cohen, C. Rabut and L. L. Schumaker, eds.) Vanderbilt Univ. Press, Nashville, TN.Candes, E. J. (1999b). Curvelets—a surprisingly effective nonadaptive representation for objects with edges. In Curve and Surface Fitting: Saint-Malo (A. Cohen, C. Rabut and L. L. Schumaker, eds.) Vanderbilt Univ. Press, Nashville, TN.
Charbonnier, P., Blanc-Feraud, L., Aubert, G. and Barlaud, M. (1997). Deterministic edge-preserving regularization in computed imaging. IEEE Trans. Image Process. 6 298–311.Charbonnier, P., Blanc-Feraud, L., Aubert, G. and Barlaud, M. (1997). Deterministic edge-preserving regularization in computed imaging. IEEE Trans. Image Process. 6 298–311.
Cronin, B., Stevenson, I. H., Sur, M. and Kording, P. (2010). Hierarchical Bayesian modeling and Markov Chain Monte Carlo sampling for tuning-curve analysis. J. Neurophysiol. 103 591–602.Cronin, B., Stevenson, I. H., Sur, M. and Kording, P. (2010). Hierarchical Bayesian modeling and Markov Chain Monte Carlo sampling for tuning-curve analysis. J. Neurophysiol. 103 591–602.
Cunningham, J., Yu, B., Shenoy, K. and Sahani, M. (2008). Inferring neural firing rates from spike trains using Gaussian processes. In Advances in Neural Information Processing Systems 20 (J. C. Platt, D. Koller, Y. Singer and S. T. Roweis, eds.). Curran Associates, Red Hook, NY.Cunningham, J., Yu, B., Shenoy, K. and Sahani, M. (2008). Inferring neural firing rates from spike trains using Gaussian processes. In Advances in Neural Information Processing Systems 20 (J. C. Platt, D. Koller, Y. Singer and S. T. Roweis, eds.). Curran Associates, Red Hook, NY.
Cunningham, J. P., Gilja, V., Ryu, S. and Shenoy, K. (2009). Methods for estimating neural firing rates, and their application to brain-machine interface. Neural Netw. 22 1235–1246.Cunningham, J. P., Gilja, V., Ryu, S. and Shenoy, K. (2009). Methods for estimating neural firing rates, and their application to brain-machine interface. Neural Netw. 22 1235–1246.
Czanner, G., Eden, U., Wirth, S., Yanike, M., Suzuki, W. and Brown, E. (2008). Analysis of between-trial and within-trial neural spiking dynamics. J. Neurophysiol. 99 2672–2693.Czanner, G., Eden, U., Wirth, S., Yanike, M., Suzuki, W. and Brown, E. (2008). Analysis of between-trial and within-trial neural spiking dynamics. J. Neurophysiol. 99 2672–2693.
Dabov, K., Foi, A., Katkovnik, V. and Egiazarian, K. (2007). Image denoising by sparse 3-D transform-domain collaborative filtering. IEEE Trans. Image Process. 16 2080–2095.Dabov, K., Foi, A., Katkovnik, V. and Egiazarian, K. (2007). Image denoising by sparse 3-D transform-domain collaborative filtering. IEEE Trans. Image Process. 16 2080–2095.
Defrise, M., Vanhove, C. and Liu, X. (2011). An algorithm for total variation regularization in high-dimensional linear problems. Inverse Probl. 27 065002.Defrise, M., Vanhove, C. and Liu, X. (2011). An algorithm for total variation regularization in high-dimensional linear problems. Inverse Probl. 27 065002.
Doi, E., Gauthier, J. L., Field, G. D., Shlens, J., Sher, A., Greschner, M., Machado, T. A., Jepson, L. H., Mathieson, K., Gunning, D. E., Litke, A. M., Paninski, L., Chichilnisky, E. J. and Simoncelli, E. P. (2012). Efficient coding of spatial information in the primate retina. J. Neurosci. 32 16256–16264.Doi, E., Gauthier, J. L., Field, G. D., Shlens, J., Sher, A., Greschner, M., Machado, T. A., Jepson, L. H., Mathieson, K., Gunning, D. E., Litke, A. M., Paninski, L., Chichilnisky, E. J. and Simoncelli, E. P. (2012). Efficient coding of spatial information in the primate retina. J. Neurosci. 32 16256–16264.
Fahrmeir, L., Kneib, T. and Lang, S. (2004). Penalized structured additive regression for space-time data: A Bayesian perspective. Statist. Sinica 14 731–761.Fahrmeir, L., Kneib, T. and Lang, S. (2004). Penalized structured additive regression for space-time data: A Bayesian perspective. Statist. Sinica 14 731–761.
Gao, Y., Black, M., Bienenstock, E., Shoham, S. and Donoghue, J. (2002). Probabilistic inference of arm motion from neural activity in motor cortex. In Advances in Neural Information Processing Systems 14 (Z. G. Thomas, G. Dietterich and S. Becker, eds.) 213–220.Gao, Y., Black, M., Bienenstock, E., Shoham, S. and Donoghue, J. (2002). Probabilistic inference of arm motion from neural activity in motor cortex. In Advances in Neural Information Processing Systems 14 (Z. G. Thomas, G. Dietterich and S. Becker, eds.) 213–220.
Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6 721–741.Geman, S. and Geman, D. (1984). Stochastic relaxation, Gibbs distribution, and the Bayesian restoration of images. IEEE Trans. Pattern Anal. Mach. Intell. 6 721–741.
Gilavert, C., Moussaoui, S. and Idier, J. (2015). Efficient Gaussian sampling for solving large-scale inverse problems using MCMC. IEEE Trans. Signal Process. 63 70–80.Gilavert, C., Moussaoui, S. and Idier, J. (2015). Efficient Gaussian sampling for solving large-scale inverse problems using MCMC. IEEE Trans. Signal Process. 63 70–80.
Girman, S. V., Sauvé, Y. and Lund, R. D. (1999). Receptive field properties of single neurons in rat primary visual cortex. J. Neurophysiol. 82 301–311.Girman, S. V., Sauvé, Y. and Lund, R. D. (1999). Receptive field properties of single neurons in rat primary visual cortex. J. Neurophysiol. 82 301–311.
Girolami, M., Calderhead, B. and Chin, S. (2011). Riemann manifold Langevin and Hamiltonian Monte Carlo methods. J. Roy. Statist. Soc. Ser. B 73 123–214.Girolami, M., Calderhead, B. and Chin, S. (2011). Riemann manifold Langevin and Hamiltonian Monte Carlo methods. J. Roy. Statist. Soc. Ser. B 73 123–214.
Grosenick, L., Klingenberg, B., Katovich, K., Knutson, B. and Taylor, J. E. (2013). Interpretable whole-brain prediction analysis with GraphNet. NeuroImage 73 304–321.Grosenick, L., Klingenberg, B., Katovich, K., Knutson, B. and Taylor, J. E. (2013). Interpretable whole-brain prediction analysis with GraphNet. NeuroImage 73 304–321.
Groves, A. R., Chappell, M. A. and Woolrich, M. W. (2009). Combined spatial and non-spatial prior for inference on MRI time-scales. NeuroImage 45 795–809.Groves, A. R., Chappell, M. A. and Woolrich, M. W. (2009). Combined spatial and non-spatial prior for inference on MRI time-scales. NeuroImage 45 795–809.
Hafting, T., Fyhn, M., Molden, S., Moser, M. B. and Moser, E. I. (2005). Microstructure of a spatial map in the enthorhinal cortex. Nature 436 801–806.Hafting, T., Fyhn, M., Molden, S., Moser, M. B. and Moser, E. I. (2005). Microstructure of a spatial map in the enthorhinal cortex. Nature 436 801–806.
Hamel, E. J. O., Grewe, B. F., Parker, J. G. and Schnitzer, M. J. (2015). Cellular level brain imaging in behaving mammals: An engineering approach. Neuron 86 140–159. DOI:10.1016/j.neuron.2015.03.055.Hamel, E. J. O., Grewe, B. F., Parker, J. G. and Schnitzer, M. J. (2015). Cellular level brain imaging in behaving mammals: An engineering approach. Neuron 86 140–159. DOI:10.1016/j.neuron.2015.03.055.
Harrison, L. M., Penny, W., Ashburner, J., Trujillo-Barreto, N. and Friston, K. J. (2007). Diffusion-based spatial priors for imaging. NeuroImage 38 677–695.Harrison, L. M., Penny, W., Ashburner, J., Trujillo-Barreto, N. and Friston, K. J. (2007). Diffusion-based spatial priors for imaging. NeuroImage 38 677–695.
Harrison, S. J., Woolrich, M. W., Robinson, E. C., Glasser, M. F., Beckman, C. F., Jenkinson, M. and Smith, S. M. (2015). Large-scale probabilistic functional modes from resting state fMRI. NeuroImage 109 217–231.Harrison, S. J., Woolrich, M. W., Robinson, E. C., Glasser, M. F., Beckman, C. F., Jenkinson, M. and Smith, S. M. (2015). Large-scale probabilistic functional modes from resting state fMRI. NeuroImage 109 217–231.
Hoffman, Y. (2009). Gaussian fields and constrained simulations of the large-scale structure. In Data Analysis in Cosmology 565–583. Springer, Berlin.Hoffman, Y. (2009). Gaussian fields and constrained simulations of the large-scale structure. In Data Analysis in Cosmology 565–583. Springer, Berlin.
Hubel, D. H. and Wiesel, T. N. (1962). Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex. J. Physiol. 160 106–154.Hubel, D. H. and Wiesel, T. N. (1962). Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex. J. Physiol. 160 106–154.
Issa, J. B., Haeffele, B. D., Agarwal, A., Bergles, D. E., Young, E. D. and Yue, D. T. (2014). Multiscale optical Ca 2$+$ imaging of tonal organization in mouse auditory cortex. Neuron 83 944–959.Issa, J. B., Haeffele, B. D., Agarwal, A., Bergles, D. E., Young, E. D. and Yue, D. T. (2014). Multiscale optical Ca 2$+$ imaging of tonal organization in mouse auditory cortex. Neuron 83 944–959.
Kaschube, M., Schnabel, M., Löwel, S., Coppola, D. M., White, L. E. and Wolf, F. (2010). Universality in the evolution of orientation columns in the visual cortex. Science 330 1113–1116.Kaschube, M., Schnabel, M., Löwel, S., Coppola, D. M., White, L. E. and Wolf, F. (2010). Universality in the evolution of orientation columns in the visual cortex. Science 330 1113–1116.
Keil, W., Kaschube, M., Schnabel, M., Kisvarday, Z., Lowel, S., Coppola, D. M., White, L. E. and Wolf, F. (2012). Response to comment on “Universality in the evolution of orientation columns in the visual cortex.” Science 336.Keil, W., Kaschube, M., Schnabel, M., Kisvarday, Z., Lowel, S., Coppola, D. M., White, L. E. and Wolf, F. (2012). Response to comment on “Universality in the evolution of orientation columns in the visual cortex.” Science 336.
Krouchev, N., Kalaska, J. F. and Drew, T. (2006). Sequential activation of muscle synergies during locomotion in the intact cat as revealed by cluster analysis and direct decomposition. J. Neurophysiol. 96 1991–2010.Krouchev, N., Kalaska, J. F. and Drew, T. (2006). Sequential activation of muscle synergies during locomotion in the intact cat as revealed by cluster analysis and direct decomposition. J. Neurophysiol. 96 1991–2010.
Leyton, A. S. and Sherrington, C. S. (1917). Observations on the excitable cortex of the chimpanzee, orangutan, and gorilla. Q.j. Exp. Physiol. 11 135–222.Leyton, A. S. and Sherrington, C. S. (1917). Observations on the excitable cortex of the chimpanzee, orangutan, and gorilla. Q.j. Exp. Physiol. 11 135–222.
Machado, T. A., Pnevmatikakis, E., Paninski, L., Jessell, T. and Miri, A. (2015). Primacy of flexor locomotor pattern revealed by ancestral reversion of motor neuron identity. Cell 162 338–350.Machado, T. A., Pnevmatikakis, E., Paninski, L., Jessell, T. and Miri, A. (2015). Primacy of flexor locomotor pattern revealed by ancestral reversion of motor neuron identity. Cell 162 338–350.
MacKay, D. (1995). Probable networks and plausible predictions—a review of practical Bayesian methods for supervised neural networks. Network 6 469–505.MacKay, D. (1995). Probable networks and plausible predictions—a review of practical Bayesian methods for supervised neural networks. Network 6 469–505.
Macke, J. H., Gerwinn, S., White, L. E., Kaschube, M. and Bethge, M. (2010). Bayesian estimation of orientation preference maps. Adv. Neural Inf. Process. Syst. 22 1195–1203.Macke, J. H., Gerwinn, S., White, L. E., Kaschube, M. and Bethge, M. (2010). Bayesian estimation of orientation preference maps. Adv. Neural Inf. Process. Syst. 22 1195–1203.
Macke, J. H., Gerwinn, S., White, L. E., Kaschube, M. and Bethge, M. (2011). Gaussian process methods for estimating cortical maps. NeuroImage 56 570–581.Macke, J. H., Gerwinn, S., White, L. E., Kaschube, M. and Bethge, M. (2011). Gaussian process methods for estimating cortical maps. NeuroImage 56 570–581.
Metin, C., Godement, P. and Imbert, M. (1988). The primary visual cortex in the mouses: Receptive field properties and functional organization. Exp. Brain Res. 69.Metin, C., Godement, P. and Imbert, M. (1988). The primary visual cortex in the mouses: Receptive field properties and functional organization. Exp. Brain Res. 69.
Motwani, M. C., Gadiya, M. C., Motwani, R. C. and Harris, F. C. (2004). Survey of image denoising techniques. In Global Signal Processing Expo, Santa Clara, CA.Motwani, M. C., Gadiya, M. C., Motwani, R. C. and Harris, F. C. (2004). Survey of image denoising techniques. In Global Signal Processing Expo, Santa Clara, CA.
Murphy, E. H. and Berman, N. (1979). The rabit and the cat: A comparison of some features of response properties of single cells in the primary visual cortex. J. Comp. Neurol. 188.Murphy, E. H. and Berman, N. (1979). The rabit and the cat: A comparison of some features of response properties of single cells in the primary visual cortex. J. Comp. Neurol. 188.
Murray, I., Adams, R. P. and MacKay, D. (2010). Elliptical slice sampling. In The Proceedings of the 13th International Conference on Artificial Intelligence and Statistics (AISTATS) 9 541–548.Murray, I., Adams, R. P. and MacKay, D. (2010). Elliptical slice sampling. In The Proceedings of the 13th International Conference on Artificial Intelligence and Statistics (AISTATS) 9 541–548.
Ohki, K., Chung, S., Ch’ng, Y., Kara, P. and Reid, C. (2005). Functional imaging with cellular resolution reveals precise micro-architecture in visual cortex. Nature 433 597–603.Ohki, K., Chung, S., Ch’ng, Y., Kara, P. and Reid, C. (2005). Functional imaging with cellular resolution reveals precise micro-architecture in visual cortex. Nature 433 597–603.
Ohki, K., Chung, S., Kara, P., Hubener, M., Bonhoeffer, T. and Reid, R. C. (2006). Highly ordered arrangement of single neurons in orientation pinwheels. Nature 442 925–928. DOI:10.1038/nature05019.Ohki, K., Chung, S., Kara, P., Hubener, M., Bonhoeffer, T. and Reid, R. C. (2006). Highly ordered arrangement of single neurons in orientation pinwheels. Nature 442 925–928. DOI:10.1038/nature05019.
Oliveira, J., Bioucas-Dias, J. M. and Figueiredo, M. (2009). Adaptive total variation image deblurring: A majorization-minimization approach. Signal Process. 89 1683–1693.Oliveira, J., Bioucas-Dias, J. M. and Figueiredo, M. (2009). Adaptive total variation image deblurring: A majorization-minimization approach. Signal Process. 89 1683–1693.
Paninski, L., Ahmadian, Y., Ferreira, D. G., Koyama, S., Rahnama Rad, K., Vidne, M., Vogelstein, J. and Wu, W. (2010). A new look at state-space models for neural data. J. Comput. Neurosci. 29 107–126.Paninski, L., Ahmadian, Y., Ferreira, D. G., Koyama, S., Rahnama Rad, K., Vidne, M., Vogelstein, J. and Wu, W. (2010). A new look at state-space models for neural data. J. Comput. Neurosci. 29 107–126.
Papandreou, G. and Yuille, A. (2010). Gaussian sampling by local perturbations. In Advances in Neural Information Processing Systems 23 (J. D. Lafferty, C. K. I. Williams, J. Shawe-Taylor, R. S. Zemel and A. Culotta, eds.) 1858–1866. Curran Associates, Red Hook, NY.Papandreou, G. and Yuille, A. (2010). Gaussian sampling by local perturbations. In Advances in Neural Information Processing Systems 23 (J. D. Lafferty, C. K. I. Williams, J. Shawe-Taylor, R. S. Zemel and A. Culotta, eds.) 1858–1866. Curran Associates, Red Hook, NY.
Pnevmatikakis, E. A., Gao, Y., Soudry, D., Pfau, D., Lacefield, C., Poskanzer, K., Bruno, R., Yuste, R. and Paninski, L. (2014a). A structured matrix factorization framework for large scale calcium imaging data analysis. Preprint. Available at arXiv:1409.2903.Pnevmatikakis, E. A., Gao, Y., Soudry, D., Pfau, D., Lacefield, C., Poskanzer, K., Bruno, R., Yuste, R. and Paninski, L. (2014a). A structured matrix factorization framework for large scale calcium imaging data analysis. Preprint. Available at arXiv:1409.2903.
Pnevmatikakis, E. A., Rahnama Rad, K., Huggins, J. and Paninski, L. (2014b). Fast Kalman filtering and forward-backward smoothing via low-rank perturbative approach. J. Comput. Graph. Statist. 23 316–339.Pnevmatikakis, E. A., Rahnama Rad, K., Huggins, J. and Paninski, L. (2014b). Fast Kalman filtering and forward-backward smoothing via low-rank perturbative approach. J. Comput. Graph. Statist. 23 316–339.
Portilla, J., Strela, V., Wainwright, M. J. and Simoncelli, E. P. (2003). Image denoising using scale mixtures of Gaussians in the wavelet domain. IEEE Trans. Image Process. 12 1338–1351.Portilla, J., Strela, V., Wainwright, M. J. and Simoncelli, E. P. (2003). Image denoising using scale mixtures of Gaussians in the wavelet domain. IEEE Trans. Image Process. 12 1338–1351.
Portugues, R., Feierstein, C. E., Engert, F. and Orger, M. B. (2014). Whole-brain activity maps reveal stereotyped, distributed networks for visuomotor behavior. Neuron 81 1328–1343. DOI:10.1016/j.neuron.2014.01.019.Portugues, R., Feierstein, C. E., Engert, F. and Orger, M. B. (2014). Whole-brain activity maps reveal stereotyped, distributed networks for visuomotor behavior. Neuron 81 1328–1343. DOI:10.1016/j.neuron.2014.01.019.
Prevedel, R., Yoon, Y., Hoffmann, M., Pak, N., Wetzstein, G., Kato, S., Schrödel, T., Raskar, R., Zimmer, M., Boyden, E. et al. (2014). Simultaneous whole-animal 3D imaging of neuronal activity using light-field microscopy. Nat. Methods 11 727–730.Prevedel, R., Yoon, Y., Hoffmann, M., Pak, N., Wetzstein, G., Kato, S., Schrödel, T., Raskar, R., Zimmer, M., Boyden, E. et al. (2014). Simultaneous whole-animal 3D imaging of neuronal activity using light-field microscopy. Nat. Methods 11 727–730.
Rahnama Rad, K. and Paninski, L. (2010). Efficient estimation of two-dimensional firing rate surfaces via Gaussian process methods. Network 21 142–168.Rahnama Rad, K. and Paninski, L. (2010). Efficient estimation of two-dimensional firing rate surfaces via Gaussian process methods. Network 21 142–168.
Schnabel, M., Kaschube, M., Lowel, S. and Wolf, F. (2007). Random waves in the brain: Symmetries and defect generation in the visual cortex. Eur. Phys. J. Spec. Top. 145 137–157.Schnabel, M., Kaschube, M., Lowel, S. and Wolf, F. (2007). Random waves in the brain: Symmetries and defect generation in the visual cortex. Eur. Phys. J. Spec. Top. 145 137–157.
Shmuel, A. and Grinvald, A. (1996). Functional organization for direction of motion and its relationship to orientation maps in cat area 18. J. Neurosci. 16 6945–6964.Shmuel, A. and Grinvald, A. (1996). Functional organization for direction of motion and its relationship to orientation maps in cat area 18. J. Neurosci. 16 6945–6964.
Siden, P., Eklund, A., Bolin, D. and Villani, M. (2016). Fast Bayesian whole-brain fMRI analysis with spatial 3D priors. Preprint. Available at arXiv:1606.00980v1 [stat.CO].Siden, P., Eklund, A., Bolin, D. and Villani, M. (2016). Fast Bayesian whole-brain fMRI analysis with spatial 3D priors. Preprint. Available at arXiv:1606.00980v1 [stat.CO].
van Gerven, M. A. J., Cseke, B., de Lange, F. P. and Heskes, T. (2010). Efficient Bayesian multivariate fMRI analysis using a sparsifying spatio-temporal prior. NeuroImage 50 150–161.van Gerven, M. A. J., Cseke, B., de Lange, F. P. and Heskes, T. (2010). Efficient Bayesian multivariate fMRI analysis using a sparsifying spatio-temporal prior. NeuroImage 50 150–161.
Van Hooser, S. D., Heimel, J. A. F., Chung, S., Nelson, S. B. and Toth, L. J. (2005). Orientation selectivity without orientation maps in visual cortex of a highly visual mammal. J. Neurosci. 25 19–28.Van Hooser, S. D., Heimel, J. A. F., Chung, S., Nelson, S. B. and Toth, L. J. (2005). Orientation selectivity without orientation maps in visual cortex of a highly visual mammal. J. Neurosci. 25 19–28.
Vidne, M., Ahmadian, Y., Shlens, J., Pillow, J. W., Kulkarni, J., Litke, A. M., Chichilnisky, E. J., Simoncelli, E. and Paninski, L. (2012). Modeling the impact of common noise inputs on the network activity of retinal ganglion cells. J. Comput. Neurosci. 33 97–121.Vidne, M., Ahmadian, Y., Shlens, J., Pillow, J. W., Kulkarni, J., Litke, A. M., Chichilnisky, E. J., Simoncelli, E. and Paninski, L. (2012). Modeling the impact of common noise inputs on the network activity of retinal ganglion cells. J. Comput. Neurosci. 33 97–121.
Vogel, C. R. and Oman, M. E. (1998). Fast, robust total variation-based reconstruction of noisy, blurred images. IEEE Trans. Image Process. 7 813–824.Vogel, C. R. and Oman, M. E. (1998). Fast, robust total variation-based reconstruction of noisy, blurred images. IEEE Trans. Image Process. 7 813–824.
Wang, Y., Yang, J., Yin, W. and Zhang, Y. (2009). A new alternative minimization algorithm for total variation image reconstruction. SIAM J. Imaging Sci. 1 248–272.Wang, Y., Yang, J., Yin, W. and Zhang, Y. (2009). A new alternative minimization algorithm for total variation image reconstruction. SIAM J. Imaging Sci. 1 248–272.
Wipf, D. and Nagarajan, S. (2008). A new view of automatic relevance determination. In Advances in Neural Information Processing Systems 1625–1632. Curran Associates, Red Hook.Wipf, D. and Nagarajan, S. (2008). A new view of automatic relevance determination. In Advances in Neural Information Processing Systems 1625–1632. Curran Associates, Red Hook.
Woolrich, M. W., Jenkinson, M., Brady, J. M. and Smith, S. M. (2004). Fully Bayesian spatio-temporal modeling of fMRI data. IEEE Trans. Med. Imag. 23 213–231.Woolrich, M. W., Jenkinson, M., Brady, J. M. and Smith, S. M. (2004). Fully Bayesian spatio-temporal modeling of fMRI data. IEEE Trans. Med. Imag. 23 213–231.