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Semi-orthogonal subspaces for value mediate a binding and generalization trade-off

Nature Neuroscience (2024)Cite this article

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Abstract

When choosing between options, we must associate their values with the actions needed to select them. We hypothesize that the brain solves this binding problem through neural population subspaces. Here, in macaques performing a choice task, we show that neural populations in five reward-sensitive regions encode the values of offers presented on the left and right in distinct subspaces. This encoding is sufficient to bind offer values to their locations while preserving abstract value information. After offer presentation, all areas encode the value of the first and second offers in orthogonal subspaces; this orthogonalization also affords binding. Our binding-by-subspace hypothesis makes two new predictions confirmed by the data. First, behavioral errors should correlate with spatial, but not temporal, neural misbinding. Second, behavioral errors should increase when offers have low or high values, compared to medium values, even when controlling for value difference. Together, these results support the idea that the brain uses semi-orthogonal subspaces to bind features.

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Fig. 1: Task outline, behavioral model and brain areas.
Fig. 2: Diverse value–response functions produce semi-orthogonal subspaces for value.
Fig. 3: Subspace correlation mediates a trade-off between the reliability of binding and generalization.
Fig. 4: The theory predicts misbinding and generalization rates for each region.
Fig. 5: The representational geometry of the offer sequence.
Fig. 6: The representation of past and current offers predicts key elements of animal behavior.

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Data availability

The raw data analyzed in this work are fully available on Figshare via https://doi.org/10.6084/m9.figshare.26065600 (ref. 94).

Code availability

The code underlying this work relies on the Python scientific computing environment, including: python (3.8.2), numpy (1.23.5), scipy (1.10.1), sklearn (1.3.0), rsatoolbox and matplotlib (3.7.2). The custom code written to generate the figures and analyze the data is available at https://github.com/wj2/subspace_binding. The version of the code used to generate the figures is available on Zenodo via https://doi.org/10.5281/zenodo.12194146 (ref. 95).

References

  1. Kable, J. W. & Glimcher, P. W. The neurobiology of decision: consensus and controversy. Neuron 63, 733–745 (2009).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  2. Samejima, K., Ueda, Y., Doya, K. & Kimura, M. Representation of action-specific reward values in the striatum. Science 310, 1337–1340 (2005).

    Article  CAS  PubMed  Google Scholar 

  3. Rangel, A., Camerer, C. & Montague, P. R. A framework for studying the neurobiology of value-based decision making. Nat. Rev. Neurosci. 9, 545–556 (2008).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  4. Wunderlich, K., Rangel, A. & O’Doherty, J. P. Neural computations underlying action-based decision making in the human brain. Proc. Natl Acad. Sci. USA 106, 17199–17204 (2009).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  5. Cai, X. & Padoa-Schioppa, C. Contributions of orbitofrontal and lateral prefrontal cortices to economic choice and the good-to-action transformation. Neuron 81, 1140–1151 (2014).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  6. Hare, T. A., Schultz, W., Camerer, C. F., O’Doherty, J. P. & Rangel, A. Transformation of stimulus value signals into motor commands during simple choice. Proc. Natl Acad. Sci. USA 108, 18120–18125 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  7. Hayden, B. Y. & Moreno-Bote, R. A neuronal theory of sequential economic choice. Brain Neurosci. Adv. 2, 2398212818766675 (2018).

    Article  PubMed  PubMed Central  Google Scholar 

  8. Knudsen, E. B. & Wallis, J. D. Taking stock of value in the orbitofrontal cortex. Nat. Rev. Neurosci. 23, 428–438 (2022).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  9. Padoa-Schioppa, C. & Assad, J. A. The representation of economic value in the orbitofrontal cortex is invariant for changes of menu. Nat. Neurosci. 11, 95–102 (2008).

    Article  CAS  PubMed  Google Scholar 

  10. Yim, M. Y., Cai, X. & Wang, X. J. Transforming choice outcome to action plan in monkey lateral prefrontal cortex: a neural circuit model. Neuron 103, 520–532 (2019).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  11. Stoet, G. & Hommel, B. Action planning and the temporal binding of response codes. J. Exp. Psychol. Hum. Percept. Perform. 25, 1625–1640 (1999).

    Article  Google Scholar 

  12. Treisman, A. M. & Gelade, G. A feature-integration theory of attention. Cogn. Psychol. 12, 97–136 (1980).

    Article  CAS  PubMed  Google Scholar 

  13. Roelfsema, P. R. Solving the binding problem: assemblies form when neurons enhance their firing rate-they don’t need to oscillate or synchronize. Neuron 111, 1003–1019 (2023).

    Article  CAS  PubMed  Google Scholar 

  14. Chung, S. & Abbott, L. F. Neural population geometry: an approach for understanding biological and artificial neural networks. Curr. Opin. Neurobiol. 70, 137–144 (2021).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  15. Ebitz, R. B. & Hayden, B. Y. The population doctrine in cognitive neuroscience. Neuron 109, 3055–3068 (2021).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  16. Mante, V., Sussillo, D., Shenoy, K. V. & Newsome, W. T. Context-dependent computation by recurrent dynamics in prefrontal cortex. Nature 503, 78–84 (2013).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  17. Saxena, S. & Cunningham, J. P. Towards the neural population doctrine. Curr. Opin. Neurobiol. 55, 103–111 (2019).

    Article  CAS  PubMed  Google Scholar 

  18. Blanchard, T. C., Piantadosi, S. T. & Hayden, B. Y. Robust mixture modeling reveals category-free selectivity in reward region neuronal ensembles. J. Neurophysiol. 119, 1305–1318 (2018).

    Article  PubMed  Google Scholar 

  19. Fusi, S., Miller, E. K. & Rigotti, M. Why neurons mix: high dimensionality for higher cognition. Curr. Opin. Neurobiol. 37, 66–74 (2016).

    Article  CAS  PubMed  Google Scholar 

  20. Rigotti, M. et al. The importance of mixed selectivity in complex cognitive tasks. Nature 497, 585–590 (2013).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  21. Barak, O., Rigotti, M. & Fusi, S. The sparseness of mixed selectivity neurons controls the generalization–discrimination trade-off. J. Neurosci. 33, 3844–3856 (2013).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  22. Babadi, B. & Sompolinsky, H. Sparseness and expansion in sensory representations. Neuron 83, 1213–1226 (2014).

    Article  CAS  PubMed  Google Scholar 

  23. Litwin-Kumar, A., Harris, K. D., Axel, R., Sompolinsky, H. & Abbott, L. F. Optimal degrees of synaptic connectivity. Neuron 93, 1153–1164 (2017).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  24. Johnston, W. J., Palmer, S. E. & Freedman, D. J. Nonlinear mixed selectivity supports reliable neural computation. PLoS Comput. Biol. 16, e1007544 (2020).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  25. Matthey, L., Bays, P. M. & Dayan, P. A probabilistic palimpsest model of visual short-term memory. PLoS Comput. Biol. 11, e1004003 (2015).

    Article  PubMed  PubMed Central  Google Scholar 

  26. Parthasarathy, A. et al. Mixed selectivity morphs population codes in prefrontal cortex. Nat. Neurosci. 20, 1770–1779 (2017).

    Article  CAS  PubMed  Google Scholar 

  27. Bernardi, S. et al. The geometry of abstraction in the hippocampus and prefrontal cortex. Cell. 183, 954–967 (2020).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  28. Johnston, W. J. & Fusi, S. Abstract representations emerge naturally in neural networks trained to perform multiple tasks. Nat. Commun. 14, 1040 (2023).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  29. Dosher, B. A., Jeter, P., Liu, J. & Lu, Z. L. An integrated reweighting theory of perceptual learning. Proc. Natl Acad. Sci. USA 110, 13678–13683 (2013).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  30. Dosher, B. & Lu, Z. L. Visual perceptual learning and models. Annu. Rev. Vis. Sci. 3, 343–363 (2017).

    Article  PubMed  PubMed Central  Google Scholar 

  31. Yoo, S. B. M. & Hayden, B. Y. The transition from evaluation to selection involves neural subspace reorganization in core reward regions. Neuron 105, 712–724 (2020).

    Article  CAS  PubMed  Google Scholar 

  32. Strait, C. E., Blanchard, T. C. & Hayden, B. Y. Reward value comparison via mutual inhibition in ventromedial prefrontal cortex. Neuron 82, 1357–1366 (2014).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  33. Farashahi, S., Azab, H., Hayden, B. & Soltani, A. On the flexibility of basic risk attitudes in monkeys. J. Neurosci. 38, 4383–4398 (2018).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  34. Farashahi, S., Donahue, C. H., Hayden, B. Y., Lee, D. & Soltani, A. Flexible combination of reward information across primates. Nat. Hum. Behav. 3, 1215–1224 (2019).

    Article  PubMed  PubMed Central  Google Scholar 

  35. Heilbronner, S. & Hayden, B. Contextual factors explain risk-seeking preferences in rhesus monkeys. Front. Neurosci. https://doi.org/10.3389/fnins.2013.00007 (2013).

  36. Fine, J. M. et al. Abstract value encoding in neural populations but not single neurons. J. Neurosci. 43, 4650–4663 (2023).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  37. Strait, C. E. et al. Neuronal selectivity for spatial positions of offers and choices in five reward regions. J. Neurophysiol. 115, 1098–1111 (2016).

    Article  PubMed  Google Scholar 

  38. Hayden, B. Y. & Platt, M. L. Neurons in anterior cingulate cortex multiplex information about reward and action. J. Neurosci. 30, 3339–3346 (2010).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  39. Strait, C. E., Sleezer, B. J. & Hayden, B. Y. Signatures of value comparison in ventral striatum neurons. PLoS Biol. 13, e1002173 (2015).

    Article  PubMed  PubMed Central  Google Scholar 

  40. Dean, H. L. & Platt, M. L. Allocentric spatial referencing of neuronal activity in macaque posterior cingulate cortex. J. Neurosci. 26, 1117–1127 (2006).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  41. Libby, A. & Buschman, T. J. Rotational dynamics reduce interference between sensory and memory representations. Nat. Neurosci. 24, 715–726 (2021).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  42. Pu, S., Dang, W., Qi, X. L. & Constantinidis, C. Prefrontal neuronal dynamics in the absence of task execution. Nat. Commun. https://doi.org/10.1101/2022.09.16.508324 (2022).

  43. Panichello, M. F. & Buschman, T. J. Shared mechanisms underlie the control of working memory and attention. Nature 592, 601–605 (2021).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  44. Piwek, E. P., Stokes, M. G. & Summerfield, C. A recurrent neural network model of prefrontal brain activity during a working memory task. PLoS Comput. Biol. 19, e1011555 (2023).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  45. Sorscher, B., Ganguli, S. & Sompolinsky, H. Neural representational geometry underlies few-shot concept learning. Proc. Natl Acad. Sci. USA 119, e2200800119 (2022).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  46. Alleman, M., Panichello, M., Buschman, T. J. & Johnston, W. J. The neural basis of swap errors in working memory. Proc. Natl Acad. Sci. USA 121, e2401032121 (2024).

    Article  PubMed  PubMed Central  Google Scholar 

  47. Schneegans, S. & Bays, P. M. Neural architecture for feature binding in visual working memory. J. Neurosci. 37, 3913–3925 (2017).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  48. Crist, R. E., Kapadia, M. K., Westheimer, G. & Gilbert, C. D. Perceptual learning of spatial localization: specificity for orientation, position, and context. J. Neurophysiol. 78, 2889–2894 (1997).

    Article  CAS  PubMed  Google Scholar 

  49. Luck, S. J. & Vogel, E. K. The capacity of visual working memory for features and conjunctions. Nature 390, 279–281 (1997).

    Article  CAS  PubMed  Google Scholar 

  50. Cowan, N. The magical number 4 in short-term memory: a reconsideration of mental storage capacity. Behav. Brain Sci. 24, 87–114 (2001).

    Article  CAS  PubMed  Google Scholar 

  51. Zhang, W. & Luck, S. J. Discrete fixed-resolution representations in visual working memory. Nature 453, 233–235 (2008).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  52. Ma, W. J., Husain, M. & Bays, P. M. Changing concepts of working memory. Nat. Neurosci. 17, 347–356 (2014).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  53. Bays, P. M., Schneegans, S., Ma, W. J. & Brady, T. F. Representation and computation in visual working memory. Nat. Hum. Behav. https://doi.org/10.1038/s41562-024-01871-2 (2024).

  54. Cueva, C. J. et al. Low-dimensional dynamics for working memory and time encoding. Proc. Natl Acad. Sci. USA 117, 23021–23032 (2020).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  55. Gallego, J. A., Perich, M. G., Miller, L. E. & Solla, S. A. Neural manifolds for the control of movement. Neuron 94, 978–984 (2017).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  56. Jazayeri, M. & Ostojic, S. Interpreting neural computations by examining intrinsic and embedding dimensionality of neural activity. Curr. Opin. Neurobiol. 70, 113–120 (2021).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  57. Nogueira, R., Rodgers, C. C., Bruno, R. M. & Fusi, S. The geometry of cortical representations of touch in rodents. Nat. Neurosci. 26, 239–250 (2023).

    Article  CAS  PubMed  Google Scholar 

  58. Boyle, L., Posani, L., Irfan, S., Siegelbaum, S. A. & Fusi, S. Tuned geometries of hippocampal representations meet the computational demands of social memory. Neuron 112, 1358–1371 (2024).

    Article  CAS  PubMed  Google Scholar 

  59. Koay, S. A., Charles, A. S., Thiberge, S. Y., Brody, C. D. & Tank, D. W. Sequential and efficient neural-population coding of complex task information. Neuron 110, 328–349 (2022).

    Article  CAS  PubMed  Google Scholar 

  60. Gore, F. et al. Orbitofrontal cortex control of striatum leads economic decision-making. Nat. Neurosci. 26, 1566–1574 (2023).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  61. Yoo, S. B. M., Sleezer, B. J. & Hayden, B. Y. Robust encoding of spatial information in orbitofrontal cortex and striatum. J Cogn. Neurosci. 30, 898–913 (2018).

    Article  PubMed  Google Scholar 

  62. Feierstein, C. E., Quirk, M. C., Uchida, N., Sosulski, D. L. & Mainen, Z. F. Representation of spatial goals in rat orbitofrontal cortex. Neuron 51, 495–507 (2006).

    Article  CAS  PubMed  Google Scholar 

  63. Jonikaitis, D. & Zhu, S. Action space restructures visual working memory in prefrontal cortex. Preprint at bioRxiv https://doi.org/10.1101/2023.08.13.553135 (2023).

  64. Krajbich, I., Armel, C. & Rangel, A. Visual fixations and the computation and comparison of value in simple choice. Nat. Neurosci. 13, 1292–1298 (2010).

    Article  CAS  PubMed  Google Scholar 

  65. Padoa-Schioppa, C. & Assad, J. A. Neurons in the orbitofrontal cortex encode economic value. Nature 441, 223–226 (2006).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  66. Padoa-Schioppa, C. Neurobiology of economic choice: a good-based model. Annu. Rev. Neurosci. 34, 333–359 (2011).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  67. Fine, J. M. & Hayden, B. Y. The whole prefrontal cortex is premotor cortex. Philos. Trans. R. Soc. Lond. B Biol. Sci. 377, 20200524 (2022).

    Article  PubMed  Google Scholar 

  68. Hayden, B. Y. & Niv, Y. The case against economic values in the orbitofrontal cortex (or anywhere else in the brain). Behav. Neurosci. 135, 192–201 (2021).

    Article  PubMed  Google Scholar 

  69. Paxinos, G., Petrides, M. & Evrard, H. C. The Rhesus Monkey Brain in Stereotaxic Coordinates (Academic Press, 2009).

  70. Blanchard, T. C., Hayden, B. Y. & Bromberg-Martin, E. S. Orbitofrontal cortex uses distinct codes for different choice attributes in decisions motivated by curiosity. Neuron 85, 602–614 (2015).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  71. Heilbronner, S. R. & Hayden, B. Y. Dorsal anterior cingulate cortex: a bottom-up view. Annu. Rev. Neurosci. 39, 149–170 (2016).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  72. Wang, M. Z., Hayden, B. Y. & Heilbronner, S. R. A structural and functional subdivision in central orbitofrontal cortex. Nat. Commun. 13, 3623 (2022).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  73. Azab, H. & Hayden, B. Y. Correlates of decisional dynamics in the dorsal anterior cingulate cortex. PLoS Biol. 15, e2003091 (2017).

    Article  PubMed  PubMed Central  Google Scholar 

  74. Sleezer, B. J., Castagno, M. D. & Hayden, B. Y. Rule encoding in orbitofrontal cortex and striatum guides selection. J. Neurosci. 36, 11223–11237 (2016).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  75. Blanchard, T. C. & Hayden, B. Y. Neurons in dorsal anterior cingulate cortex signal postdecisional variables in a foraging task. J. Neurosci. 34, 646–655 (2014).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  76. Blanchard, T. C., Wolfe, L. S., Vlaev, I., Winston, J. S. & Hayden, B. Y. Biases in preferences for sequences of outcomes in monkeys. Cognition 130, 289–299 (2014).

    Article  PubMed  PubMed Central  Google Scholar 

  77. Heilbronner, S. R. & Hayden, B. Y. The description-experience gap in risky choice in nonhuman primates. Psychon. Bull. Rev. 23, 593–600 (2016).

    Article  PubMed  Google Scholar 

  78. Ebitz, R. B., Sleezer, B. J., Jedema, H. P., Bradberry, C. W. & Hayden, B. Y. Tonic exploration governs both flexibility and lapses. PLoS Comput. Biol. 15, e1007475 (2019).

    Article  PubMed  PubMed Central  Google Scholar 

  79. Ferro, D., Cash-Padgett, T., Wang, M. Z., Hayden, B. & Moreno-Bote, R. Gaze-centered gating and re-activation of value encoding in orbitofrontal cortex. Nat. Commun. https://doi.org/10.1101/2023.04.20.537677 (2023).

  80. Daunizeau, J., Adam, V. & Rigoux, L. VBA: a probabilistic treatment of nonlinear models for neurobiological and behavioural data. PLoS Comput. Biol. 10, e1003441 (2014).

    Article  PubMed  PubMed Central  Google Scholar 

  81. Kimmel, D. L., Elsayed, G. F., Cunningham, J. P. & Newsome, W. T. Value and choice as separable and stable representations in orbitofrontal cortex. Nat. Commun. 11, 3466 (2020).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  82. Elsayed, G. F., Lara, A. H., Kaufman, M. T., Churchland, M. M. & Cunningham, J. P. Reorganization between preparatory and movement population responses in motor cortex. Nat. Commun. 7, 13239 (2016).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  83. Yao, Y., Vehtari, A., Simpson, D. & Gelman, A. Using stacking to average bayesian predictive distributions (with discussion).Bayesian Anal. 13, 917–1007 (2018).

    Article  Google Scholar 

  84. Watanabe, S. A widely applicable Bayesian information criterion. J. Mach. Learn. Res. 14, 867–897 (2013).

    Google Scholar 

  85. Gao, P. et al. A theory of multineuronal dimensionality, dynamics and measurement. Preprint at BioRxiv https://doi.org/10.1101/214262 (2017).

  86. Xie, Y. et al. Geometry of sequence working memory in macaque prefrontal cortex. Science 375, 632–639 (2022).

    Article  CAS  PubMed  Google Scholar 

  87. Nieder, A. The neuronal code for number. Nat. Rev. Neurosci. 17, 366–382 (2016).

    Article  CAS  PubMed  Google Scholar 

  88. Kim, J. H. J., Fiete, I. & Schwab, D. J. Superlinear precision and memory in simple population codes. Preprint at https://arxiv.org/abs/2008.00629 (2020).

  89. Johnston, W. J. & Freedman, D. J. Redundant representations are required to disambiguate simultaneously presented complex stimuli. PLoS Comput. Biol. 19, e1011327 (2023).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  90. Pedregosa, F. et al. Scikit-learn: machine learning in Python. J. Mach. Learn. Res. 12, 2825–2830 (2011).

    Google Scholar 

  91. Walther, A. et al. Reliability of dissimilarity measures for multi-voxel pattern analysis. Neuroimage 137, 188–200 (2016).

    Article  PubMed  Google Scholar 

  92. Nili, H. et al. A toolbox for representational similarity analysis. PLoS Comput. Biol. 10, e1003553 (2014).

    Article  PubMed  PubMed Central  Google Scholar 

  93. Kriegeskorte, N., Mur, M. & Bandettini, P. A. Representational similarity analysis—connecting the branches of systems neuroscience. Front. Syst. Neurosci. https://doi.org/10.3389/neuro.06.004.2008 (2008).

  94. Johnston, J. Semi-orthogonal subspaces for value mediate a binding and generalization tradeoff. figshare https://doi.org/10.6084/m9.figshare.26065600.v1 (2024).

  95. Johnston, J. et al. wj2/subspace_binding: Version corresponding to paper (paper). Zenodo https://doi.org/10.5281/zenodo.12194146 (2024).

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Acknowledgements

We thank B. Vinje, who led an excellent summer journal club on the binding problem at UC Berkeley in 2001. We thank S. Fusi for useful discussions of previous versions of this paper. We thank M. Wang, T. Cash-Padgett, M. Mancarella, C. Strait, T. Blanchard and B. Sleezer for assistance with data collection. We also thank L. Mickiewicz, A. Ong and A. Silcott for administrative support. This research was supported by NIDA R01 DA038615 (to B.Y.H.) and MH124687 (to B.Y.H.) W.J.J. was supported by NSF 1707398, Simons Foundation 542983SPI, Gatsby Charitable Foundation GAT3708, NIMH R01 MH129031 and the Kavli Foundation. We acknowledge computing resources from Columbia University’s Shared Research Computing Facility project, which is supported by NIH Research Facility Improvement Grant 1G20RR030893-01, and associated funds from the New York State Empire State Development, Division of Science Technology and Innovation (NYSTAR) Contract C090171, both awarded 15 April 2010.

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Author notes

  1. These authors contributed equally: W. Jeffrey Johnston, Justin M. Fine.

Authors and Affiliations

  1. Center for Theoretical Neuroscience and Mortimer B. Zuckerman Mind, Brain, and Behavior Institute, Columbia University, New York, NY, USA

    W. Jeffrey Johnston

  2. Department of Neurosurgery, Baylor College of Medicine, Houston, TX, USA

    Justin M. Fine & Benjamin Y. Hayden

  3. Department of Biomedical Engineering, Sunkyunkwan University, and Center for Neuroscience Imaging Research, Institute of Basic Sciences, Suwon, Republic of Korea

    Seng Bum Michael Yoo

  4. Department of Neuroscience, Université de Montréal, Montreal, Quebec, Canada

    R. Becket Ebitz

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Contributions

W.J.J., J.M.F., S.B.M.Y., R.B.E. and B.Y.H. conceived of the project. S.B.M.Y. and B.Y.H. designed the experiments. S.B.M.Y. and R.B.E. collected the data. W.J.J. and J.M.F. designed and performed the analyses. W.J.J. developed the theoretical approach. W.J.J. and J.M.F. created the figures. W.J.J., J.M.F. and B.Y.H. wrote and edited the paper.

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Correspondence to W. Jeffrey Johnston.

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Extended data

Extended Data Fig. 1 Behavioral consistency across subjects and regions.

A. The time-average proportion of optimal choices made by each subject (columns) during experiments in each region (rows). The proportion is smoothed with a 50 trial-wide boxcar filter. The black traces (n = 8 across all regions and animals) are sessions where performance dropped below a threshold of 60% optimal choices for at least one time bin. B. The subspace correlation analysis performed for the subset of sessions where performance did not drop below the 60% threshold. The results are similar to the full set of sessions.

Extended Data Fig. 2 Example neurons, model comparison, and subspace correlations for the linear models with a nonlinear value representation.

A. The firing rates of example neurons from each region during the offer window, shown for high and low value offers presented on the left or right side (100 ms boxcar filter, shaded area is SEM). B. The value-response function for each neuron in A. The value-response function fit by the linear regression model with a b-spline value encoding and an interaction term is overlaid (dashed lines). The b-spline representation uses 4 knots and is degree 2. C. A simplex showing the weight given to each of the noise-only, linear, and interaction regression models by the Bayesian model stacking analysis. The points corresponding to the example neurons shown in A and B have dark outlines here. Both the linear and interaction categories include both linear and spline value representation models. D. Schematic of three different representational geometries that would lead to different subspace correlation results. (top) Two perfectly aligned value vectors vl and vr in population space (left) would produce a subspace correlation close to 1 (right). (middle) Two partially aligned value vectors vl and vr in would produce a subspace correlation between 0 and 1 (note there is an additional possibility: partially aligned but negatively correlated subspaces; not schematized). (bottom) Two unaligned value vectors vl and vr would produce a subspace correlation close to 0. E. Alignment indices for all regions for the offer presentation window (top) and the delay period (bottom). The upper gray point is the alignment index expected if the left- and right value representations were aligned and corrupted only due to noise – the lower gray point is the noise floor. In these models, pgACC, VS, and the combined population are semi-orthogonal, while PCC, OFC, and vmPFC are indistinguishable from orthogonal.

Extended Data Fig. 3 Variations on the subspace correlation analysis.

A. Subspace correlation for the different time points of the experiment (that is, offer 1 and offer 2) as well as a combined set of trials from both offers, but given the same total number of trials as either offer individually. B. Subspace correlation for the best fitting models given the model comparison analysis. OFC, vmPFC, VS, and the combined population (“all”) have semi-orthogonal representation, while PCC has orthogonal representations and pgACC has parallel representations in the first time period, but orthogonal representations in the second time period. C. Subspace correlation for separate monkeys and regions. D. Subspace correlation for the set of sessions included in the simultaneous population error analysis in Fig. 5. E. The average subspace correlation for the full set of trials shown as a timecourse analysis; the shaded area represents the 95% confidence interval.

Extended Data Fig. 4 Accurate recovery of linear and nonlinear distances from simulated data.

The true (dashed line) and estimated (solid line with error bars) linear (left) and nonlinear (right) distances from simulated data. The error bars represent 95% intervals around the estimated values from n = 100 different synthetic datasets with matched statistics. Our decomposition accurately recovers the linear and nonlinear distances, as the true value is always within error bars of the estimated value and the estimate is typically unbiased.

Extended Data Fig. 5 Distance estimated from subsampled neural populations.

The distances are estimated from subsampled populations of 80 neurons. Otherwise, this plot is the same as Fig. 4c.

Extended Data Fig. 6 Estimated distances for individual subjects.

This figure is analogous to Fig. 4c, but where the data for each region is separated into the constituent subjects.

Extended Data Fig. 7 Value decoding, value generalization, and predicted value generalization of the code within each recorded region.

(Top) Pseudopopulation value decoding performance (circles), generalization performance (squares, trained on offers from one side, tested on offers from the other side), and predicted generalization performance (stars) shown for each region and the neural population combined across regions (“all”), shown for the left and right value comparison. The violin plot shows the values produced from two hundred bootstrap resamples of the trials. (Bottom) The same as (top) except shown for the offer 1 and offer 2 comparison.

Extended Data Fig. 8 Dependence of the behavioral decoding result (Fig. 6f) on the required number of neurons in the simultaneously recorded population.

The gray crosses indicate significance under a one-sided T-test. The individual circles show sessions, the error bar shows the average across cross-validation runs of those sessions. The grey stars indicate significance at the p < .05 level.

Extended Data Fig. 9 The main results replicate when safe trials are included.

This figure replicates Fig. 4 from the main text but including the safe trials in the dataset. The results are qualitatively similar in both conditions. A, Shows offer value decoding and generalization across left and right offers. B, Shows offer value decoding and generalization across offer 1 and offer 2. C, Shows the linear and nonlinear distances estimated for both left and right offers (left) and offer 1 and 2 (right). D, Shows the predicted binding error rate for each region in both the spatial (left) and temporal (right) configurations as a function of subspace correlation. E, The same as (D) but shows the predicted and empirical generalization error rates. F, Shows the position of each region on the binding and generalization error rate plane.

Extended Data Fig. 10 The available neurons for a particular required number of trials.

As more trials for each condition are required, fewer neurons are available for inclusion in the pseudopopulation. The default throughout the paper is 160, shown in grey.

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Supplementary Figs. 1–3, Table 1 and Discussion.

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Johnston, W.J., Fine, J.M., Yoo, S.B.M. et al. Semi-orthogonal subspaces for value mediate a binding and generalization trade-off. Nat Neurosci (2024). https://doi.org/10.1038/s41593-024-01758-5

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