title: Improved estimation and interpretation of correlations in neural circuits.
abstract: Ambitious projects aim to record the activity of ever larger and denser neuronal populations in vivo. Correlations in neural activity measured in such recordings can reveal important aspects of neural circuit organization. However, estimating and interpreting large correlation matrices is statistically challenging. Estimation can be improved by regularization, i.e. by imposing a structure on the estimate. The amount of improvement depends on how closely the assumed structure represents dependencies in the data. Therefore, the selection of the most efficient correlation matrix estimator for a given neural circuit must be determined empirically. Importantly, the identity and structure of the most efficient estimator informs about the types of dominant dependencies governing the system. We sought statistically efficient estimators of neural correlation matrices in recordings from large, dense groups of cortical neurons. Using fast 3D random-access laser scanning microscopy of calcium signals, we recorded the activity of nearly every neuron in volumes 200 μm wide and 100 μm deep (150-350 cells) in mouse visual cortex. We hypothesized that in these densely sampled recordings, the correlation matrix should be best modeled as the combination of a sparse graph of pairwise partial correlations representing local interactions and a low-rank component representing common fluctuations and external inputs. Indeed, in cross-validation tests, the covariance matrix estimator with this structure consistently outperformed other regularized estimators. The sparse component of the estimate defined a graph of interactions. These interactions reflected the physical distances and orientation tuning properties of cells: The density of positive 'excitatory' interactions decreased rapidly with geometric distances and with differences in orientation preference whereas negative 'inhibitory' interactions were less selective. Because of its superior performance, this 'sparse+latent' estimator likely provides a more physiologically relevant representation of the functional connectivity in densely sampled recordings than the sample correlation matrix.
authors.affiliation: Department of Neuroscience, Baylor College of Medicine, Houston, Texas, United States of America.
authors.forename: Dimitri
authors.initials: D
authors.lastname: Yatsenko
authors.affiliation: Department of Mathematics and Department of Biology and Biochemistry, University of Houston, Houston, Texas, United States of America.
authors.forename: Krešimir
authors.initials: K
authors.lastname: Josić
authors.affiliation: Department of Neuroscience, Baylor College of Medicine, Houston, Texas, United States of America; Werner Reichardt Center for Integrative Neuroscience and Institute for Theoretical Physics, University of Tübingen, Germany; Bernstein Center for Computational Neuroscience, Tübingen, Germany; Max Planck Institute for Biological Cybernetics, Tübingen, Germany.
authors.forename: Alexander S
authors.initials: AS
authors.lastname: Ecker
authors.affiliation: Department of Neuroscience, Baylor College of Medicine, Houston, Texas, United States of America.
authors.forename: Emmanouil
authors.initials: E
authors.lastname: Froudarakis
authors.affiliation: Department of Neuroscience, Baylor College of Medicine, Houston, Texas, United States of America.
authors.forename: R James
authors.initials: RJ
authors.lastname: Cotton
authors.affiliation: Department of Neuroscience, Baylor College of Medicine, Houston, Texas, United States of America; Department of Computational and Applied Mathematics, Rice University, Houston, Texas, United States of America.
authors.forename: Andreas S
authors.initials: AS
authors.lastname: Tolias
mesh_terms: D000818:Animals; D001931:Brain Mapping / Q000379:methods; D002118:Calcium / Q000378:metabolism; D020013:Calcium Signaling; D051379:Mice; D008959:Models, Neurological; D009415:Nerve Net / Q000378:metabolism / Q000502:physiology*; D009434:Neural Pathways / Q000378:metabolism / Q000502:physiology; D009474:Neurons / Q000378:metabolism / Q000502:physiology*; D012044:Regression Analysis; D014793:Visual Cortex / Q000378:metabolism / Q000502:physiology
chemical_list: D002118:Calcium
references.citation: J Neurophysiol. 2010 Dec;104(6):3691-704
references.pmid: 20554834
references.citation: J Neurosci. 2012 Apr 18;32(16):5609-19
references.pmid: 22514322
references.citation: Proc Natl Acad Sci U S A. 2011 Mar 29;108(13):5419-24
references.pmid: 21383177
references.citation: Biophys J. 1970 Sep;10(9):876-900
references.pmid: 4322240
references.citation: IEEE Trans Biomed Eng. 1989 Jan;36(1):4-14
references.pmid: 2646211
references.citation: Stat Appl Genet Mol Biol. 2005;4:Article32
references.pmid: 16646851
references.citation: Front Integr Neurosci. 2010 Aug 19;4:
references.pmid: 20941381
references.citation: Nature. 2010 Apr 22;464(7292):1155-60
references.pmid: 20414303
references.citation: Science. 2010 Jan 29;327(5965):587-90
references.pmid: 20110507
references.citation: J Neurosci. 2011 Oct 5;31(40):14272-83
references.pmid: 21976512
references.citation: Neuron. 2008 Oct 9;60(1):162-73
references.pmid: 18940596
references.citation: Exp Brain Res. 1989;77(2):432-6
references.pmid: 2792290
references.citation: Neuroimage. 2012 Feb 15;59(4):3852-61
references.pmid: 22155039
references.citation: Nat Methods. 2010 May;7(5):399-405
references.pmid: 20400966
references.citation: J Neurosci. 2009 Sep 2;29(35):10890-9
references.pmid: 19726647
references.citation: Proc Natl Acad Sci U S A. 2011 Jun 7;108(23):9679-84
references.pmid: 21602497
references.citation: Science. 2010 Jan 29;327(5965):584-7
references.pmid: 20110506
references.citation: J Neurophysiol. 2008 Jun;99(6):2998-3008
references.pmid: 18417631
references.citation: Nat Methods. 2012 Jan 08;9(2):201-8
references.pmid: 22231641
references.citation: J Neurosci. 1998 May 15;18(10):3870-96
references.pmid: 9570816
references.citation: Proc Natl Acad Sci U S A. 2013 Apr 16;110(16):E1524-32
references.pmid: 23576737
references.citation: Nature. 1994 Jul 14;370(6485):140-3
references.pmid: 8022482
references.citation: J Neurosci. 2008 Jan 9;28(2):505-18
references.pmid: 18184793
references.citation: Front Neural Circuits. 2013 Oct 10;7:151
references.pmid: 24133414
references.citation: J Neurosci. 2009 Aug 19;29(33):10234-53
references.pmid: 19692598
references.citation: Front Neural Circuits. 2013 Apr 24;7:76
references.pmid: 23630470
references.citation: Nature. 2003 Oct 30;425(6961):954-6
references.pmid: 14586468
references.citation: Neural Comput. 2009 Oct;21(10):2774-804
references.pmid: 19635014
references.citation: PLoS Biol. 2005 Mar;3(3):e68
references.pmid: 15737062
references.citation: Neural Comput. 2011 May;23(5):1261-305
references.pmid: 21299426
references.citation: Biostatistics. 2008 Jul;9(3):432-41
references.pmid: 18079126
references.citation: Neural Comput. 2013 Aug;25(8):2172-98
references.pmid: 23607561
references.citation: Nature. 2011 May 5;473(7345):87-91
references.pmid: 21478872
references.citation: Science. 2013 Mar 15;339(6125):1284-5
references.pmid: 23470729
references.citation: Science. 1964 Mar 20;143(3612):1325-7
references.pmid: 17799237
references.citation: Nat Neurosci. 2009 Dec;12(12):1594-600
references.pmid: 19915566
references.citation: Nat Rev Neurosci. 2006 May;7(5):358-66
references.pmid: 16760916
references.citation: Neuron. 2011 Mar 24;69(6):1188-203
references.pmid: 21435562
references.citation: Neuron. 2011 Oct 20;72(2):231-43
references.pmid: 22017986
references.citation: J Comput Neurosci. 2010 Aug;29(1-2):309-325
references.pmid: 19529888
references.citation: Nat Neurosci. 1998 Sep;1(5):395-403
references.pmid: 10196530
references.citation: Proc Natl Acad Sci U S A. 2003 Jun 10;100(12):7348-53
references.pmid: 12775756
references.citation: J Neurophysiol. 1995 May;73(5):2072-93
references.pmid: 7623099
references.citation: Nat Neurosci. 2008 Jul;11(7):749-51
references.pmid: 18552841
references.citation: J Neurosci Methods. 2013 Nov 15;220(2):149-66
references.pmid: 23639919
references.citation: Proc Natl Acad Sci U S A. 2013 Aug 6;110(32):13162-7
references.pmid: 23878209
references.citation: Neuron. 2011 Aug 25;71(4):750-61
references.pmid: 21867889
references.citation: J Neurosci. 2012 Nov 28;32(48):17108-19
references.pmid: 23197704
references.citation: PLoS Comput Biol. 2014 Jan;10(1):e1003408
references.pmid: 24391485
references.citation: Neuron. 2014 Oct 22;84(2):355-62
references.pmid: 25374359
references.citation: Biophys J. 1967 Jul;7(4):419-40
references.pmid: 4292792
references.citation: J Neurosci. 2008 Nov 26;28(48):12591-603
references.pmid: 19036953
references.citation: J Neurosci. 2011 Sep 14;31(37):13260-71
references.pmid: 21917809
references.citation: J Neurosci. 2012 Aug 1;32(31):10618-26
references.pmid: 22855811
references.citation: Nat Neurosci. 2011 Jul 17;14(8):1045-52
references.pmid: 21765421
references.citation: Nat Methods. 2013 May;10(5):413-20
references.pmid: 23524393
references.citation: Nature. 2013 Apr 4;496(7443):96-100
references.pmid: 23552948
references.citation: Nat Neurosci. 2009 Nov;12(11):1444-9
references.pmid: 19801988
references.citation: Curr Opin Neurobiol. 2009 Aug;19(4):434-8
references.pmid: 19608406
references.citation: Phys Rev E Stat Nonlin Soft Matter Phys. 2001 Nov;64(5 Pt 1):051904
references.pmid: 11735965
references.citation: Cereb Cortex. 2008 Dec;18(12):2891-901
references.pmid: 18400792
references.citation: Neuroimage. 2012 May 1;60(4):2096-106
references.pmid: 22343126
references.citation: Nature. 2006 Apr 20;440(7087):1007-12
references.pmid: 16625187
references.citation: Trends Neurosci. 2011 May;34(5):225-36
references.pmid: 21459463
references.citation: Nature. 2008 Aug 21;454(7207):995-9
references.pmid: 18650810
references.citation: PLoS One. 2011;6(6):e20996
references.pmid: 21698248
references.citation: J Neurosci. 2013 Mar 20;33(12):5422-32
references.pmid: 23516307
references.citation: J Neurosci. 2009 Apr 15;29(15):5022-31
references.pmid: 19369571
references.citation: Neuron. 2014 Apr 2;82(1):235-48
references.pmid: 24698278
references.citation: PLoS Comput Biol. 2009 May;5(5):e1000380
references.pmid: 19424487
references.citation: J Neurosci. 2014 Jun 4;34(23):7769-77
references.pmid: 24899701
references.citation: J Neurosci Methods. 1999 Dec 15;94(1):121-40
references.pmid: 10638820
references.citation: J Neurosci. 2005 Apr 6;25(14):3661-73
references.pmid: 15814797
references.citation: J Neurosci. 2001 Mar 1;21(5):1676-97
references.pmid: 11222658
references.citation: Cell. 2014 Mar 13;156(6):1139-1152
references.pmid: 24630718
references.citation: J Biomed Opt. 2005 Nov-Dec;10(6):064038
references.pmid: 16409103
references.citation: Neuron. 2012 Nov 8;76(3):590-602
references.pmid: 23141070
references.citation: Neural Comput. 1999 Jan 1;11(1):91-101
references.pmid: 9950724
references.citation: Nat Neurosci. 2010 Mar;13(3):353-60
references.pmid: 20118927
references.citation: Cereb Cortex. 2014 Oct;24(10):2707-20
references.pmid: 23689635
references.citation: J Physiol Paris. 2012 Sep-Dec;106(5-6):212-21
references.pmid: 22326672
references.citation: Nat Rev Neurosci. 2001 Aug;2(8):539-50
references.pmid: 11483997
references.citation: J Neurophysiol. 2013 Feb;109(4):940-7
references.pmid: 23197461
references.citation: J Neurophysiol. 1989 May;61(5):900-17
references.pmid: 2723733
references.citation: J Am Stat Assoc. 2009 Jun 1;104(486):735-746
references.pmid: 19881892
references.citation: Proc Natl Acad Sci U S A. 2011 Mar 15;108(11):4423-8
references.pmid: 21368193
references.citation: Neuron. 2002 Sep 12;35(6):1123-34
references.pmid: 12354401