A Bayesian approach for inferring neuronal connectivity from calcium fluorescent imaging data. (English) Zbl 1223.62162

Summary: Deducing the structure of neural circuits is one of the central problems of modern neuroscience. Recently-introduced calcium fluorescent imaging methods permit experimentalists to observe network activity in large populations of neurons, but these techniques provide only indirect observations of neural spike trains, with limited time resolution and signal quality. We present a Bayesian approach for inferring neural circuitry given this type of imaging data. We model the network activity in terms of a collection of coupled hidden Markov chains, with each chain corresponding to a single neuron in the network and the coupling between the chains reflecting the network’s connectivity matrix. We derive a Monte Carlo Expectation-Maximization algorithm for fitting the model parameters; to obtain the sufficient statistics in a computationally-efficient manner, we introduce a specialized blockwise-Gibbs algorithm for sampling from the joint activity of all observed neurons given the observed fluorescence data. We perform large-scale simulations of randomly connected neuronal networks with biophysically realistic parameters and find that the proposed methods can accurately infer the connectivity in these networks given reasonable experimental and computational constraints. In addition, the estimation accuracy may be improved significantly by incorporating prior knowledge about the sparseness of connectivity in the network, via standard \(L_{1}\) penalization methods.


62P10 Applications of statistics to biology and medical sciences; meta analysis
92C20 Neural biology
62F15 Bayesian inference
92C55 Biomedical imaging and signal processing
65C05 Monte Carlo methods
65C60 Computational problems in statistics (MSC2010)


Full Text: DOI arXiv


[1] Abeles, M. (1991). Corticonics . Cambridge Univ. Press, Cambridge.
[2] Ahmadian, Y., Pillow, J. and Paninski, L. (2011). Efficient Markov chain Monte Carlo methods for decoding population spike trains. Neural Comput. 23 . · Zbl 1217.92022 · doi:10.1162/NECO_a_00059
[3] Andrieu, C., Doucet, A. and Holenstein, A. (2007). Particle Markov chain Monte Carlo. Working paper. · Zbl 1184.65001
[4] Bickel, P., Li, B. and Bengtsson, T. (2008). Sharp failure rates for the bootstrap particle filter in high dimensions. In Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh (Clarke, B. and Ghosal, S., eds.) 318-329. IMS, Beachwod, OH. · doi:10.1214/074921708000000228
[5] Binzegger, T., Douglas, R. J. and Martin, K. A. C. (2004). A quantitative map of the circuit of cat primary visual cortex. J. Neurosci. 24 8441-8453.
[6] Boyd, S. and Vandenberghe, L. (2004). Convex Optimization . Oxford Univ. Press. · Zbl 1058.90049
[7] Boyden, E. S., Zhang, F., Bamberg, E., Nagel, G. and Deisseroth, K. (2005). Millisecond-timescale, genetically targeted optical control of neural activity. Nat. Neurosci. 8 1263-1268.
[8] Braitenberg, V. and Schuz, A. (1998). Cortex: Statistics and Geometry of Neuronal Connectivity . Springer, Berlin.
[9] Brenowitz, S. D. and Regehr, W. G. (2007). Reliability and heterogeneity of calcium signaling at single presynaptic boutons of cerebellar granule cells. J. Neurosci. 27 7888-7898.
[10] Briggman, K. L. and Denk, W. (2006). Towards neural circuit reconstruction with volume electron microscopy techniques. Curr. Opin. Neurobiol. 16 562.
[11] Brillinger, D. (1988). Maximum likelihood analysis of spike trains of interacting nerve cells. Biol. Cybern. 59 189-200. · Zbl 0646.92007 · doi:10.1007/BF00318010
[12] Brillinger, D. (1992). Nerve cell spike train data analysis: A progression of technique. J. Amer. Statist. Assoc. 87 260-271.
[13] Buhl, E., Halasy, K. and Somogyi, P. (1994). Diverse sources of hippocampal unitary inhibitory postynaptic potentials and the number of synaptic release sites. Nature 368 823-828.
[14] Candes, E. J. and Wakin, M. (2008). An introduction to compressive sampling. IEEE Signal Proc. Mag. 25 21-30.
[15] Chornoboy, E., Schramm, L. and Karr, A. (1988). Maximum likelihood identification of neural point process systems. Biol. Cybern. 59 265-275. · Zbl 0658.92007 · doi:10.1007/BF00332915
[16] Cocco, S., Leibler, S. and Monasson, R. (2009). Neuronal couplings between retinal ganglion cells inferred by efficient inverse statistical physics methods. Proc. Nat. Acad. Sci. 106 14058-14062.
[17] Cossart, R., Aronov, D. and Yuste, R. (2003). Attractor dynamics of network up states in the neocortex. Nature 423 283-288.
[18] Dempster, A., Laird, N. and Rubin, D. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Statist. Soc. Ser. B 39 1-38. · Zbl 0364.62022
[19] Djurisic, M., Antic, S., Chen, W. R. and Zecevic, D. (2004). Voltage imaging from dendrites of mitral cells: EPSP attenuation and spike trigger zones. J. Neurosci. 24 6703-6714.
[20] Dombeck, D. A., Khabbaz, A. N., Collman, F., Adelman, T. L. and Tank, D. W. (2007). Imaging large-scale neural activity with cellular resolution in awake, mobile mice. Neuron 56 43-57.
[21] Donoho, D. and Elad, M. (2003). Optimally sparse representation in general (nonorthogonal) dictionaries via L 1 minimization. PNAS 100 2197-2202. · Zbl 1064.94011 · doi:10.1073/pnas.0437847100
[22] Douc, R., Cappe, O. and Moulines, E. (2005). Comparison of resampling schemes for particle filtering. In Proc. 4th Int. Symp. Image and Signal Processing and Analysis 64-69. ISPA.
[23] Doucet, A., de Freitas, N. and Gordon, N., eds. (2001). Sequential Monte Carlo in Practice . Springer, New York. · Zbl 0967.00022
[24] Doucet, A., Godsill, S. and Andrieu, C. (2000). On sequential Monte Carlo sampling methods for Bayesian filtering. Stat. Comput. 10 197-208.
[25] Escola, S. and Paninski, L. (2011). Hidden Markov models applied toward the inference of neural states and the improved estimation of linear receptive fields. Neural Comput.
[26] Feldmeyer, D., Egger, V., Lubke, J. and Sakmann, B. (1999). Reliable synaptic connections between pairs of excitatory layer 4 neurones within a single “barrel” of developing rat somatosensory cortex. J. Physiol. 1 169-90.
[27] Feldmeyer, D. and Sakmann, B. (2000). Synaptic efficacy and reliability of excitatory connections between the principal neurones of the input (layer 4) and output (layer 5) of the neocortex. J. Physiol. 525 31-39.
[28] Gamerman, D. (1997). Sampling from the posterior distribution in generalized linear mixed models. Statist. Comput. 7 57-68.
[29] Gamerman, D. (1998). Markov chain Monte Carlo for dynamic generalised linear models. Biometrika 85 215-227. · Zbl 0904.62083 · doi:10.1093/biomet/85.1.215
[30] Garofalo, M., Nieus, T., Massobrio, P. and Martinoia, S. (2009). Evaluation of the performance of information theory-based methods and cross-correlation to estimate the functional connectivity in cortical networks. PLoS ONE 4 e6482.
[31] Godsill, S., Doucet, A. and West, M. (2004). Monte Carlo smoothing for non-linear time series. J. Amer. Statist. Assoc. 99 156-168. · Zbl 1089.62517 · doi:10.1198/016214504000000151
[32] Gomez-Urquijo, S. M., Reblet, C., Bueno-Lopez, J. L. and Gutierrez-Ibarluzea, I. (2000). Gabaergic neurons in the rabbit visual cortex: Percentage, distribution and cortical projections. Brain Res. 862 171-179.
[33] Greenberg, D. S., Houweling, A. R. and Kerr, J. N. D. (2008). Population imaging of ongoing neuronal activity in the visual cortex of awake rats. Nat. Neurosci 11 749-751.
[34] Gupta, A., Wang, Y. and Markram, H. (2000). Organizing principles for a diversity of gabaergic interneurons and synapses in the neocortex. Science 287 273-278.
[35] Harris, K., Csicsvari, J., Hirase, H., Dragoi, G. and Buzsaki, G. (2003). Organization of cell assemblies in the hippocampus. Nature 424 552-556.
[36] Hatsopoulos, N., Ojakangas, C., Paninski, L. and Donoghue, J. (1998). Information about movement direction obtained by synchronous activity of motor cortical neurons. PNAS 95 15706-15711.
[37] Helmchen, F., Imoto, K. and Sakmann, B. (1996). Ca 2+ buffering and action potential-evoked Ca 2+ signaling in dendrites of pyramidal neurons. Biophys. J. 70 1069-1081.
[38] Ikegaya, Y., Aaron, G., Cossart, R., Aronov, D., Lampl, I., Ferster, D. and Yuste, R. (2004). Synfire chains and cortical songs: Temporal modules of cortical activity. Science 304 559-564.
[39] Ishwaran, H. (1999). Applications of hybrid Monte Carlo to Bayesian generalized linear models: Quasicomplete separation and neural networks. J. Comput. Graph. Statist. 8 779-799.
[40] Iyer, V., Hoogland, T. M. and Saggau, P. (2006). Fast functional imaging of single neurons using random-access multiphoton (RAMP) microscopy. J. Neurophysiol. 95 535-545.
[41] Koch, C. (1999). Biophysics of Computation . Oxford Univ. Press, Oxford.
[42] Kulkarni, J. and Paninski, L. (2007). Common-input models for multiple neural spike-train data. Network Comput. Neural Syst. 18 375-407.
[43] Lefort, S., Tomm, C., Floyd Sarria, J.-C. and Petersen, C. C. H. (2009). The excitatory neuronal network of the c2 barrel column in mouse primary somatosensory cortex. Neuron 61 301-316.
[44] Lei, N., Watson, B., MacLean, J., Yuste, R. and Shepard, K. (2008). A 256-by-256 cmos microelectrode array for extracellular stimulation of acute brain slices. In Proceedings to the International Solid-State Circuits Conference , ISSCC.
[45] Li, K. and Duan, N. (1989). Regression analysis under link violation. Ann. Statist. 17 1009-1052. · Zbl 0753.62041 · doi:10.1214/aos/1176347254
[46] Litke, A., Bezayiff, N., Chichilnisky, E., Cunningham, W., Dabrowski, W., Grillo, A., Grivich, M., Grybos, P., Hottowy, P., Kachiguine, S., Kalmar, R., Mathieson, K., Petrusca, D., Rahman, M. and Sher, A. (2004). What does the eye tell the brain? Development of a system for the large scale recording of retinal output activity. IEEE Trans. Nucl. Sci. 51 1434-1440.
[47] Livet, J., Weissman, T., Kang, H., Draft, R., Lu, J., Bennis, R., Sanes, J. and Lichtman, J. (2007). Transgenic strategies for combinatorial expression of fluorescent proteins in the nervous system. Nature 450 56-62.
[48] Luczak, A., Bartho, P., Marguet, S., Buzsaki, G. and Harris, K. (2007). Sequential structure of neocortical spontaneous activity in vivo. PNAS 104 347-352.
[49] MacLean, J., Watson, B., Aaron, G. and Yuste, R. (2005). Internal dynamics determine the cortical response to thalamic stimulation. Neuron 48 811-823.
[50] McLachlan, G. and Krishnan, T. (1996). The EM Algorithm and Extensions . Wiley, New York. · Zbl 0882.62012
[51] Meyer, A. H., Katona, I., Blatow, M., Rozov, A. and Monyer, H. (2002). In vivo labeling of parvalbumin-positive interneurons and analysis of electrical coupling in identified neurons. J. Neurosci. 22 7055-7064.
[52] Micheva, K. and Smith, S. (2007). Array tomography: A new tool for imaging the molecular architecture and ultrastructure of neural circuits. Neuron 55 25-36.
[53] Mishchenko, Y. (2009). Strategies for identifying exact structure of neural circuits with broad light microscopy connectivity probes. Preprint. Available at .
[54] Mishchenko, Y., Spacek, J., Mendenhall, J., Chklovskii, D. and Harris, K. M. (2009). Reconstruction of hippocampal CA1 neuropil at nanometer resolution reveals disordered packing of processes and dependence of synaptic connectivity on local environment and dendritic caliber.
[55] Neal, R., Beal, M. and Roweis, S. (2003). Inferring state sequences for non-linear systems with embedded hidden Markov models. In NIPS 16 401-408. MIT Press, Cambridge.
[56] Ng, A. (2004). Feature selection, L 1 vs. L 2 regularization, and rotational invariance. In Proceedings of the Twenty-First International Conference on Machine Learning. ICML 21.
[57] Nguyen, Q. T., Callamaras, N., Hsieh, C. and Parker, I. (2001). Construction of a two-photon microscope for video-rate Ca 2+ imaging. Cell Calcium 30 383-393.
[58] Nikolenko, V., Poskanzer, K. and Yuste, R. (2007). Two-photon photostimulation and imaging of neural circuits. Nature Methods 4 943-950.
[59] Nikolenko, V., Watson, B., Araya, R., Woodruff, A., Peterka, D. and Yuste, R. (2011). SLM microscopy: Scanless two-photon imaging and photostimulation using spatial light modulators. Frontiers in Neural Circuits .
[60] Nykamp, D. Q. (2005). Revealing pairwise coupling in linear-nonlinear networks. SIAM J. Appl. Math. 65 2005-2032. · Zbl 1077.92011 · doi:10.1137/S0036139903437072
[61] Nykamp, D. Q. (2007). A mathematical framework for inferring connectivity in probabilistic neuronal networks. Math. Biosci. 205 204-251. · Zbl 1109.92004 · doi:10.1016/j.mbs.2006.08.020
[62] 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.
[63] Paninski, L. (2004). Maximum likelihood estimation of cascade point-process neural encoding models. Network Comput. Neural Syst. 15 243-262.
[64] Paninski, L., Ahmadian, Y., Ferreira, D., Koyama, S., Rahnama, K., Vidne, M., Vogelstein, J. and Wu, W. (2009). A new look at state-space models for neural data. J. Comput. Neurosci.
[65] Paninski, L., Fellows, M., Shoham, S., Hatsopoulos, N. and Donoghue, J. (2004). Superlinear population encoding of dynamic hand trajectory in primary motor cortex. J. Neurosci. 24 8551-8561.
[66] Petersen, C. C. and Sakmann, B. (2000). The excitatory neuronal network of rat layer 4 barrel cortex. J. Neurosci. 20 7579-7586.
[67] Petrusca, D., Grivich, M. I., Sher, A., Field, G. D., Gauthier, J. L., Greschner, M., Shlens, J., Chichilnisky, E. J. and Litke, A. M. (2007). Identification and characterization of a Y-like primate retinal ganglion cell type. J. Neurosci. 27 11019-11027.
[68] Pillow, J., Shlens, J., Paninski, L., Sher, A., Litke, A., Chichilnisky, E. and Simoncelli, E. (2008). Spatiotemporal correlations and visual signaling in a complete neuronal population. Nature 454 995-999.
[69] Plesser, H. and Gerstner, W. (2000). Noise in integrate-and-fire neurons: From stochastic input to escape rates. Neural Comput. 12 367-384.
[70] Rabiner, L. (1989). A tutorial on hidden Markov models and selected applications in speech recognition. Proc. IEEE 77 257-286.
[71] Ramon y Cajal, S. (1904). La Textura del Sistema Nerviosa del Hombre y los Vertebrados . Moya, Madrid.
[72] Ramon y Cajal, S. (1923). Recuerdos de mi vida: Historia de mi labor cientifica . Alianza Editorial, Madrid.
[73] Reddy, G., Kelleher, K., Fink, R. and Saggau, P. (2008). Three-dimensional random access multiphoton microscopy for functional imaging of neuronal activity. Nat. Neurosci. 11 713-720.
[74] Reyes, A., Lujan, R., Rozov, A., Burnashev, N., Somogyi, P. and Sakmann, B. (1998). Target-cell-specific facilitation and depression in neocortical circuits. Nat. Neurosci. 1 279-285.
[75] Rigat, F., de Gunst, M. and van Pelt, J. (2006). Bayesian modelling and analysis of spatio-temporal neuronal networks. Bayesian Anal. 1 733-764. · Zbl 1331.62161 · doi:10.1214/06-BA124
[76] Robert, C. and Casella, G. (2005). Monte Carlo Statistical Methods . Springer, New York. · Zbl 0935.62005
[77] Roxin, A., Hakim, V. and Brunel, N. (2008). The statistics of repeating patterns of cortical activity can be reproduced by a model network of stochastic binary neurons. J. Neurosci. 28 10734-10745.
[78] Salome, R., Kremer, Y., Dieudonne, S., Leger, J.-F., Krichevsky, O., Wyart, C., Chatenay, D. and Bourdieu, L. (2006). Ultrafast random-access scanning in two-photon microscopy using acousto-optic deflectors. J. Neurosci. Methods 154 161-174.
[79] Santhanam, G., Ryu, S. I., Yu, B. M., Afshar, A. and Shenoy, K. V. (2006). A high-performance brain-computer interface. Nature 442 195-198.
[80] Sayer, R. J., Friedlander, M. J. and Redman, S. J. (1990). The time course and amplitude of epsps evoked at synapses between pairs of CA3/CA1 neurons in the hippocampal slice. J. Neurosci. 10 826-836.
[81] Segev, R., Goodhouse, J., Puchalla, J. and Berry, M. (2004). Recording spikes from a large fraction of the ganglion cells in a retinal patch. Nat. Neurosci. 7 1154-1161.
[82] Shumway, R. and Stoffer, D. (2006). Time Series Analysis and Its Applications . Springer, New York. · Zbl 1096.62088
[83] Song, S., Sjostrom, P. J., Reiql, M., Nelson, S. and Chklovskii, D. B. (2005). Highly nonrandom features of synaptic connectivity in local cortical circuits. PLoS Biol. 3 e68.
[84] Stein, R. B., Weber, D. J., Aoyagi, Y., Prochazka, A., Wagenaar, J. B. M., Shoham, S. and Normann, R. A. (2004). Coding of position by simultaneously recorded sensory neurones in the cat dorsal root ganglion. J. Physiol. 560 883-896.
[85] Stevenson, I., Rebesco, J., Hatsopoulos, N., Haga, Z., Miller, L. and Koerding, K. (2008). Inferring network structure from spikes. In Statistical Analysis of Neural Data Meeting .
[86] Stevenson, I. H., Rebesco, J. M., Hatsopoulos, N. G., Haga, Z., Miller, L. E. and Kording, K. P. (2009). Bayesian inference of functional connectivity and network structure from spikes. IEEE Trans. Neural Syst. Rehab. 17 203-213.
[87] Stosiek, C., Garaschuk, O., Holthoff, K. and Konnerth, A. (2003). In vivo two-photon calcium imaging of neuronal networks. Proc. Natl. Acad. Sci. USA 100 7319-7324.
[88] Szobota, S., Gorostiza, P., Del Bene, F., Wyart, C., Fortin, D. L., Kolstad, K. D., Tulyathan, O., Volgraf, M., Numano, R., Aaron, H. L., Scott, E. K., Kramer, R. H., Flannery, J., Baier, H., Trauner, D. and Isacoff, E. Y. (2007). Remote control of neuronal activity with a light-gated glutamate receptor. Neuron 54 535-545.
[89] Thompson, A., Girdlestone, D. and West, D. (1988). Voltage-dependent currents prolong single-axon postsynaptic potentials in layer III pyramidal neurons in rat neocortical slices. J. Neurophysiol. 60 1896-1907.
[90] Tibshirani, R. (1996). Regression shrinkage and selection via the Lasso. J. Roy. Statist. Soc. Ser. B 58 267-288. · Zbl 0850.62538
[91] Tipping, M. (2001). Sparse Bayesian learning and the relevance vector machine. J. Mach. Learn. Res. 1 211-244. · Zbl 0997.68109 · doi:10.1162/15324430152748236
[92] Truccolo, W., Eden, U., Fellows, M., Donoghue, J. and Brown, E. (2005). A point process framework for relating neural spiking activity to spiking history, neural ensemble and extrinsic covariate effects. J. Neurophysiol. 93 1074-1089.
[93] Tsien, R. Y. (1989). Fluorescent probes of cell signaling. Ann. Rev. Neurosci. 12 227-253.
[94] Vakorin, V. A., Krakovska, O. A. and Mcintosh, A. R. (2009). Confounding effects of indirect connections on causality estimation. J. Neurosci. Methods 184 152-160.
[95] Vidne, M., Kulkarni, J., Ahmadian, Y., Pillow, J., Shlens, J., Chichilnisky, E., Simoncelli, E. and Paninski, L. (2009). Inferring functional connectivity in an ensemble of retinal ganglion cells sharing a common input. In Computational and Systems Neuroscience (COSYNE09) .
[96] Vogels, T. and Abbott, L. F. (2005). Signal propagation and logic gating in networks of integrate-and-fire neurons. J. Neurosci. 25 10786-10795.
[97] Vogelstein, J., Babadi, B., Watson, B., Yuste, R. and Paninski, L. (2008). Fast nonnegative deconvolution via tridiagonal interior-point methods, applied to calcium fluorescence data. In Statistical Analysis of Neural Data (SAND) Conference .
[98] Vogelstein, J., Watson, B., Packer, A., Jedynak, B., Yuste, R. and Paninski, L. (2009). Spike inference from calcium imaging using sequential monte carlo methods. Biophys. J. 97 636-655.
[99] Wallace, D., zum Alten Borgloh, S., Astori, S., Yang, Y., Bausen, M., Kugler, S., Palmer, A., Tsien, R., Sprengel, R., Kerr, J., Denk, W. and Hasan, M. (2008). Single-spike detection in vitro and in vivo with a genetic Ca 2+ sensor. Nat. Methods 5 797-804.
[100] Yaksi, E. and Friedrich, R. W. (2006). Reconstruction of firing rate changes across neuronal populations by temporally deconvolved Ca 2+ imaging. Nat. Methods 3 377-383.
[101] Yasuda, R., Nimchinsky, E. A., Scheuss, V., Pologruto, T. A., Oertner, T. G., Sabatini, B. L. and Svoboda, K. (2004). Imaging calcium concentration dynamics in small neuronal compartments. Sci. STKE 219 15.
[102] Yuste, R., Konnerth, A., Masters, B. et al. (2006). Imaging in Neuroscience and Development, A Laboratory Manual . Oxford, New York.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.