# Discovering Low-Dimensional Descriptions of Multineuronal Dependencies

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Copulas

#### 2.2. Pair Copula Constructions

#### 2.3. Copula Flows

#### 2.4. Sequential Estimation and Model Selection

#### 2.5. Weighted Non-Negative Matrix Factorization

#### 2.6. Synthetic Data

#### 2.7. Experimental Data

## 3. Results

#### 3.1. Validation on Synthetic Data

#### 3.2. WNMF Identifies Shared Latent Structures of Neural Dependencies in Visual Cortex

#### 3.3. WNMF Identifies Shared Latent Structures of Neural Dependencies in Macaque Motor Cortex

## 4. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Saxena, S.; Cunningham, J.P. Towards the neural population doctrine. Curr. Opin. Neurobiol.
**2019**, 55, 103–111. [Google Scholar] [CrossRef] - Vyas, S.; Golub, M.D.; Sussillo, D.; Shenoy, K.V. Computation through neural population dynamics. Annu. Rev. Neurosci.
**2020**, 43, 249–275. [Google Scholar] [CrossRef] - Urai, A.E.; Doiron, B.; Leifer, A.M.; Churchland, A.K. Large-scale neural recordings call for new insights to link brain and behavior. Nat. Neurosci.
**2022**, 25, 11–19. [Google Scholar] [CrossRef] - Chen, X.; Leischner, U.; Varga, Z.; Jia, H.; Deca, D.; Rochefort, N.L.; Konnerth, A. LOTOS-based two-photon calcium imaging of dendritic spines in vivo. Nat. Protoc.
**2012**, 7, 1818–1829. [Google Scholar] [CrossRef] [PubMed] - Jun, J.J.; Steinmetz, N.A.; Siegle, J.H.; Denman, D.J.; Bauza, M.; Barbarits, B.; Lee, A.K.; Anastassiou, C.A.; Andrei, A.; Aydın, Ç.; et al. Fully integrated silicon probes for high-density recording of neural activity. Nature
**2017**, 551, 232–236. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Wu, M.C.K.; David, S.V.; Gallant, J.L. Complete functional characterization of sensory neurons by system identification. Annu. Rev. Neurosci.
**2006**, 29, 477–505. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Rolls, E.T.; Treves, A. The neuronal encoding of information in the brain. Prog. Neurobiol.
**2011**, 95, 448–490. [Google Scholar] [CrossRef] - Kass, R.E.; Amari, S.I.; Arai, K.; Brown, E.N.; Diekman, C.O.; Diesmann, M.; Doiron, B.; Eden, U.T.; Fairhall, A.L.; Fiddyment, G.M.; et al. Computational neuroscience: Mathematical and statistical perspectives. Annu. Rev. Stat. Appl.
**2018**, 5, 183–214. [Google Scholar] [CrossRef] - Hurwitz, C.; Kudryashova, N.; Onken, A.; Hennig, M.H. Building population models for large-scale neural recordings: Opportunities and pitfalls. Curr. Opin. Neurobiol.
**2021**, 70, 64–73. [Google Scholar] [CrossRef] - Ohiorhenuan, I.E.; Mechler, F.; Purpura, K.P.; Schmid, A.M.; Hu, Q.; Victor, J.D. Sparse coding and high-order correlations in fine-scale cortical networks. Nature
**2010**, 466, 617–621. [Google Scholar] [CrossRef] [Green Version] - Yu, S.; Yang, H.; Nakahara, H.; Santos, G.S.; Nikolić, D.; Plenz, D. Higher-order interactions characterized in cortical activity. J. Neurosci.
**2011**, 31, 17514–17526. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Shimazaki, H.; Amari, S.I.; Brown, E.N.; Grün, S. State-space analysis of time-varying higher-order spike correlation for multiple neural spike train data. PLoS Comput. Biol.
**2012**, 8, e1002385. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Montangie, L.; Montani, F. Higher-order correlations in common input shapes the output spiking activity of a neural population. Phys. A Stat. Mech. Appl.
**2017**, 471, 845–861. [Google Scholar] [CrossRef] - Zohary, E.; Shadlen, M.N.; Newsome, W.T. Correlated neuronal discharge rate and its implications for psychophysical performance. Nature
**1994**, 370, 140–143. [Google Scholar] [CrossRef] - Brown, E.N.; Kass, R.E.; Mitra, P.P. Multiple neural spike train data analysis: State-of-the-art and future challenges. Nat. Neurosci.
**2004**, 7, 456–461. [Google Scholar] [CrossRef] [PubMed] - Moreno-Bote, R.; Beck, J.; Kanitscheider, I.; Pitkow, X.; Latham, P.; Pouget, A. Information-limiting correlations. Nat. Neurosci.
**2014**, 17, 1410–1417. [Google Scholar] [CrossRef] [Green Version] - Kohn, A.; Coen-Cagli, R.; Kanitscheider, I.; Pouget, A. Correlations and neuronal population information. Annu. Rev. Neurosci.
**2016**, 39, 237–256. [Google Scholar] [CrossRef] [Green Version] - Panzeri, S.; Moroni, M.; Safaai, H.; Harvey, C.D. The structures and functions of correlations in neural population codes. Nat. Rev. Neurosci.
**2022**, 23, 551–567. [Google Scholar] [CrossRef] - Onken, A.; Grünewälder, S.; Munk, M.H.; Obermayer, K. Analyzing short-term noise dependencies of spike-counts in macaque prefrontal cortex using copulas and the flashlight transformation. PLoS Comput. Biol.
**2009**, 5, e1000577. [Google Scholar] [CrossRef] - Kudryashova, N.; Amvrosiadis, T.; Dupuy, N.; Rochefort, N.; Onken, A. Parametric Copula-GP model for analyzing multidimensional neuronal and behavioral relationships. PLoS Comput. Biol.
**2022**, 18, e1009799. [Google Scholar] [CrossRef] - Pillow, J.W.; Shlens, J.; Paninski, L.; Sher, A.; Litke, A.M.; Chichilnisky, E.; Simoncelli, E.P. Spatio-temporal correlations and visual signalling in a complete neuronal population. Nature
**2008**, 454, 995–999. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Michel, M.M.; Jacobs, R.A. The costs of ignoring high-order correlations in populations of model neurons. Neural Comput.
**2006**, 18, 660–682. [Google Scholar] [CrossRef] [PubMed] - Jaworski, P.; Durante, F.; Härdle, W.K. Copulae in Mathematical and Quantitative Finance: Proceedings of the Workshop Held in Cracow, 10–11 July 2012; Springer: Berlin/Heidelberg, Germany, 2012; Volume 10, p. 11. [Google Scholar]
- Jenison, R.L.; Reale, R.A. The shape of neural dependence. Neural Comput.
**2004**, 16, 665–672. [Google Scholar] [CrossRef] [PubMed] - Berkes, P.; Wood, F.; Pillow, J. Characterizing neural dependencies with copula models. Adv. Neural Inf. Process. Syst.
**2008**, 21, 129–136. [Google Scholar] - Onken, A.; Grünewälder, S.; Munk, M.; Obermayer, K. Modeling short-term noise dependence of spike counts in macaque prefrontal cortex. Adv. Neural Inf. Process. Syst.
**2008**, 21, 85117. [Google Scholar] - Sacerdote, L.; Tamborrino, M.; Zucca, C. Detecting dependencies between spike trains of pairs of neurons through copulas. Brain Res.
**2012**, 1434, 243–256. [Google Scholar] [CrossRef] [Green Version] - Onken, A.; Panzeri, S. Mixed vine copulas as joint models of spike counts and local field potentials. Adv. Neural Inf. Process. Syst.
**2016**, 29, 910122. [Google Scholar] - Faugeras, O.P. Inference for copula modeling of discrete data: A cautionary tale and some facts. Depend. Model.
**2017**, 5, 121–132. [Google Scholar] [CrossRef] - Genest, C.; Nešlehová, J. A primer on copulas for count data. ASTIN Bull. J. IAA
**2007**, 37, 475–515. [Google Scholar] [CrossRef] [Green Version] - Nagler, T. A generic approach to nonparametric function estimation with mixed data. Stat. Probab. Lett.
**2018**, 137, 326–330. [Google Scholar] [CrossRef] [Green Version] - Aas, K.; Czado, C.; Frigessi, A.; Bakken, H. Pair-copula constructions of multiple dependence. Insur. Math. Econ.
**2009**, 44, 182–198. [Google Scholar] [CrossRef] [Green Version] - Song, P.X.K.; Li, M.; Yuan, Y. Joint regression analysis of correlated data using Gaussian copulas. Biometrics
**2009**, 65, 60–68. [Google Scholar] [CrossRef] [PubMed] - de Leon, A.R.; Wu, B. Copula-based regression models for a bivariate mixed discrete and continuous outcome. Stat. Med.
**2011**, 30, 175–185. [Google Scholar] [CrossRef] - Smith, M.S.; Khaled, M.A. Estimation of copula models with discrete margins via Bayesian data augmentation. J. Am. Stat. Assoc.
**2012**, 107, 290–303. [Google Scholar] [CrossRef] - Panagiotelis, A.; Czado, C.; Joe, H. Pair copula constructions for multivariate discrete data. J. Am. Stat. Assoc.
**2012**, 107, 1063–1072. [Google Scholar] [CrossRef] - Racine, J.S. Mixed data kernel copulas. Empir. Econ.
**2015**, 48, 37–59. [Google Scholar] [CrossRef] [Green Version] - Geenens, G.; Charpentier, A.; Paindaveine, D. Probit transformation for nonparametric kernel estimation of the copula density. Bernoulli
**2017**, 23, 1848–1873. [Google Scholar] [CrossRef] [Green Version] - Schallhorn, N.; Kraus, D.; Nagler, T.; Czado, C. D-vine quantile regression with discrete variables. arXiv
**2017**, arXiv:1705.08310. [Google Scholar] - Nagler, T.; Schellhase, C.; Czado, C. Nonparametric estimation of simplified vine copula models: Comparison of methods. Depend. Model.
**2017**, 5, 99–120. [Google Scholar] [CrossRef] - Mitskopoulos, L.; Amvrosiadis, T.; Onken, A. Mixed vine copula flows for flexible modelling of neural dependencies. Front. Neurosci.
**2022**, 16, 1645. [Google Scholar] [CrossRef] - Durkan, C.; Bekasov, A.; Murray, I.; Papamakarios, G. Neural spline flows. In Proceedings of the 33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Vancouver, BC, Canada, 8–14 December 2019. [Google Scholar]
- Rezende, D.; Mohamed, S. Variational inference with normalizing flows. In Proceedings of the International Conference on Machine Learning, Lille, France, 6–11 July 2015; pp. 1530–1538. [Google Scholar]
- Lee, D.D.; Seung, H.S. Learning the parts of objects by non-negative matrix factorization. Nature
**1999**, 401, 788–791. [Google Scholar] [CrossRef] [PubMed] - Russo, A.A.; Bittner, S.R.; Perkins, S.M.; Seely, J.S.; London, B.M.; Lara, A.H.; Miri, A.; Marshall, N.J.; Kohn, A.; Jessell, T.M.; et al. Motor cortex embeds muscle-like commands in an untangled population response. Neuron
**2018**, 97, 953–966. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Guillamet, D.; Vitria, J.; Schiele, B. Introducing a weighted non-negative matrix factorization for image classification. Pattern Recognit. Lett.
**2003**, 24, 2447–2454. [Google Scholar] [CrossRef] - Zhou, Q.; Feng, Z.; Benetos, E. Adaptive noise reduction for sound event detection using subband-weighted NMF. Sensors
**2019**, 19, 3206. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Sklar, M. Fonctions de repartition an dimensions et leurs marges. Publ. Inst. Stat. Univ. Paris
**1959**, 8, 229–231. [Google Scholar] - Bedford, T.; Cooke, R.M. Vines—A new graphical model for dependent random variables. Ann. Stat.
**2002**, 30, 1031–1068. [Google Scholar] [CrossRef] - Czado, C. Analyzing Dependent Data with Vine Copulas; Lecture Notes in Statistics; Springer: Berlin/Heidelberg, Germany, 2019. [Google Scholar]
- Haff, I.H.; Aas, K.; Frigessi, A. On the simplified pair-copula construction—Simply useful or too simplistic? J. Multivar. Anal.
**2010**, 101, 1296–1310. [Google Scholar] [CrossRef] [Green Version] - Bergstra, J.; Bengio, Y. Random search for hyper-parameter optimization. J. Mach. Learn. Res.
**2012**, 13, 281–305. [Google Scholar] - Fasano, G.; Franceschini, A. A multidimensional version of the Kolmogorov—Smirnov test. Mon. Not. R. Astron. Soc.
**1987**, 225, 155–170. [Google Scholar] [CrossRef] - Owen, A.B.; Perry, P.O. Bi-cross-validation of the SVD and the nonnegative matrix factorization. Ann. Appl. Stat.
**2009**, 3, 564–594. [Google Scholar] [CrossRef] [Green Version] - Nelsen, R.B. An Introduction to Copulas; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2007. [Google Scholar]
- Henschke, J.U.; Dylda, E.; Katsanevaki, D.; Dupuy, N.; Currie, S.P.; Amvrosiadis, T.; Pakan, J.M.; Rochefort, N.L. Reward association enhances stimulus-specific representations in primary visual cortex. Curr. Biol.
**2020**, 30, 1866–1880. [Google Scholar] [CrossRef] [PubMed] - Churchland, M.M.; Cunningham, J.P.; Kaufman, M.T.; Foster, J.D.; Nuyujukian, P.; Ryu, S.I.; Shenoy, K.V. Neural population dynamics during reaching. Nature
**2012**, 487, 51–56. [Google Scholar] [CrossRef] [PubMed] [Green Version]

**Figure 1.**Mixed vine copula flows and NMF decomposition. (

**A**) Spike train samples from two neurons can be decomposed into their margins and a copula. Empirical copulas are extracted by transforming the samples to uniform through the distributional transform. (

**B**) Graphical illustration of a C-vine for 4 variables. Nodes and edges of the first tree denote the variables and bivariate dependencies, respectively. Edges of subsequent trees denote dependencies that condition on one or more variables. (

**C**) Decomposition of pair copulas into non-negative coefficients for neuron pairs and copula factors.

**Figure 2.**Overlapping tail dependencies are challenging for standard NMF. (

**A**) A matrix of 20 Frank copula flow densities and 20 Clayton copula flow densities (Top illustration) is reduced to 2 NMF factors with coefficients (blue bar plots) and copula modules. Bottom left tail dependence structures overlap, which does not affect the Clayton copula factor but leads to incorrect identification for the Frank copula left tail, which is highlighted with a red dashed circle. This is in contrast to (

**B**), where the rotated (90°) Frank copula tail regions do not overlap with those of the Clayton copulas (top illustration), allowing for correct detection of both copula types.

**Figure 3.**WNMF outperforms standard NMF in identifying dependence structures with overlapping tails. (

**A**) Train (blue) and validation (orange) MSE over 5 folds for 1 to 8 factors. Error is computed for the real data matrix against the reconstructed one for WNMF (left column of line plots) and for NMF (right column of line plots). Both methods were tested on data containing either 2, 4 or 6 different copulas. (

**B**) Bar plots depict MSE of the real copula densities versus the copula modules identified by WNMF (left column of bar plots) and NMF (right column of bar plots). Y axes are in logarithmic scale. Bar colors correspond to different copula families, namely Frank (brown) and Clayton (beige), as well as their rotated versions used in the cases with 4 and 6 copulas, namely Frank 90° (navy blue), Clayton 90° (red), Clayton 180° (yellow) and Clayton 270° (teal). (

**C**) Illustration of WNMF factorization for the case of 4 copulas. Bar plots depict WNMF coefficients and density plots depict the copula modules discovered by WNMF.

**Figure 4.**WNMF discovers structured and synergistic copula modules in mouse V1 responses. (

**A**) Illustration of mouse navigating a virtual environment with grating stimuli until a designated reward zone (at 120–140 cm), where it is required to lick in order to receive a water reward. Neurons were recorded across a number of trials on each day of the experiment and their activity was binned with respect to the position of the mouse in the virtual corridor. (

**B**) Average across 5 folds train (blue) and validation (orange) MSE for WNMF across different numbers of factors. (

**C**) WNMF 4-dimensional representation of visual cortex copula dependence structures. Blue Bar plots depict WNMF coefficients specific to each neuron pair. Density plots depict copula modules discovered by WNMF.

**Figure 5.**WNMF factors are distinctly grouped across different trees. (

**A**) WNMF coefficients from the same decomposition as Figure 4. The 4 factors have been plotted in a 2 by 2 arrangement where each block consists of rows of color-coded values of WNMF for a particular tree, starting from tree 1 at the top until tree 101 at the bottom. Warmer and colder colors illustrate the spatial divides across superficial and deeper trees with respect to copula modules per neuron pairs. Blank spaces denote independent neuron pairs. (

**B**) Same WNMF copula modules as in Figure 4 depicted again here for illustration purposes.

**Figure 6.**WNMF discovers main and distributed copula modules in macaque motor cortex. (

**A**) Illustration of a macaque monkey moving a lever as part of a delayed center-out reaching task [57]. Monitor has 8 different targets drawn as white squares, while the cued target is highlighted with yellow. (

**B**) Average firing rate (Hz) across trials of different target presentations. Horizontal axis indicate the time interval we chose for analysis, i.e., −0.5 to 1.5 s with respect to the go cue for movement initiation. (

**C**) Average across 5 folds train (blue) and validation (orange) MSE for WNMF across different numbers of factors. (

**D**) WNMF 4-dimensional representation of motor cortex copula dependence structures. Blue bar plots depict WNMF coefficients specific to each neuron pair. Density plots depict copula modules discovered by WNMF.

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**MDPI and ACS Style**

Mitskopoulos, L.; Onken, A.
Discovering Low-Dimensional Descriptions of Multineuronal Dependencies. *Entropy* **2023**, *25*, 1026.
https://doi.org/10.3390/e25071026

**AMA Style**

Mitskopoulos L, Onken A.
Discovering Low-Dimensional Descriptions of Multineuronal Dependencies. *Entropy*. 2023; 25(7):1026.
https://doi.org/10.3390/e25071026

**Chicago/Turabian Style**

Mitskopoulos, Lazaros, and Arno Onken.
2023. "Discovering Low-Dimensional Descriptions of Multineuronal Dependencies" *Entropy* 25, no. 7: 1026.
https://doi.org/10.3390/e25071026