Deforming the metric of cognitive maps distorts memory

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Abstract

Environmental boundaries anchor cognitive maps that support memory. However, trapezoidal boundary geometry distorts the regular firing patterns of entorhinal grid cells, proposedly providing a metric for cognitive maps. Here we test the impact of trapezoidal boundary geometry on human spatial memory using immersive virtual reality. Consistent with reduced regularity of grid patterns in rodents and a grid-cell model based on the eigenvectors of the successor representation, human positional memory was degraded in a trapezoid environment compared with a square environment—an effect that was particularly pronounced in the narrow part of the trapezoid. Congruent with changes in the spatial frequency of eigenvector grid patterns, distance estimates between remembered positions were persistently biased, revealing distorted memory maps that explained behaviour better than the objective maps. Our findings demonstrate that environmental geometry affects human spatial memory in a similar manner to rodent grid-cell activity and, therefore, strengthen the putative link between grid cells and behaviour along with their cognitive functions beyond navigation.

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Fig. 1: Task design and SR grid patterns.
Fig. 2: Distorted positional memory in the trapezoid.
Fig. 3: Distortion of distance estimates.
Fig. 4: Reconstruction of remembered positions.

Data availability

The data that support the findings of this study are available from the corresponding authors on reasonable request.

Code availability

The custom code that supports the findings of this study is available from the corresponding authors on reasonable request.

References

  1. 1.

    Cheng, K. & Newcombe, N. S. Is there a geometric module for spatial orientation? Squaring theory and evidence. Psychon. Bull. Rev. 12, 1–23 (2005).

  2. 2.

    Julian, J. B., Keinath, A. T., Marchette, S. A. & Epstein, R. A. The neurocognitive basis of spatial reorientation. Curr. Biol. 28, R1059–R1073 (2018).

  3. 3.

    Cheng, K. A purely geometric module in the rat’s spatial representation. Cognition 23, 149–178 (1986).

  4. 4.

    Margules, J. & Gallistel, C. R. Heading in the rat: determination by environmental shape. Anim. Learn. Behav. 16, 404–410 (1988).

  5. 5.

    Hermer, L. & Spelke, E. S. A geometric process for spatial reorientation in young children. Nature 370, 57–59 (1994).

  6. 6.

    Kelly, J. W., McNamara, T. P., Bodenheimer, B., Carr, T. H. & Rieser, J. J. The shape of human navigation: how environmental geometry is used in maintenance of spatial orientation. Cognition 109, 281–286 (2008).

  7. 7.

    Doeller, C. F. & Burgess, N. Distinct error-correcting and incidental learning of location relative to landmarks and boundaries. Proc. Natl Acad. Sci. USA 105, 5909–5914 (2008).

  8. 8.

    Doeller, C. F., King, Ja & Burgess, N. Parallel striatal and hippocampal systems for landmarks and boundaries in spatial memory. Proc. Natl Acad. Sci. USA 105, 5915–5920 (2008).

  9. 9.

    Lee, S. A. et al. Electrophysiological signatures of spatial boundaries in the human subiculum. J. Neurosci. 38, 3265–3272 (2018).

  10. 10.

    Krupic, J., Bauza, M., Burton, S., Barry, C. & O’Keefe, J. Grid cell symmetry is shaped by environmental geometry. Nature 518, 232–235 (2015).

  11. 11.

    Krupic, J., Bauza, M., Burton, S. & O’Keefe, J. Local transformations of the hippocampal cognitive map. Science 359, 1143–1146 (2018).

  12. 12.

    Stensola, T., Stensola, H., Moser, M.-B. & Moser, E. I. Shearing-induced asymmetry in entorhinal grid cells. Nature 518, 207–212 (2015).

  13. 13.

    Hafting, T., Fyhn, M., Molden, S., Moser, M.-B. & Moser, E. I. Microstructure of a spatial map in the entorhinal cortex. Nature 436, 801–806 (2005).

  14. 14.

    Moser, E. I., Moser, M.-B. & McNaughton, B. L. Spatial representation in the hippocampal formation: a history. Nat. Neurosci. 20, 1448–1464 (2017).

  15. 15.

    Barry, C., Hayman, R., Burgess, N. & Jeffery, K. J. Experience-dependent rescaling of entorhinal grids. Nat. Neurosci. 10, 682–684 (2007).

  16. 16.

    Brun, V. H. et al. Progressive increase in grid scale from dorsal to ventral medial entorhinal cortex. Hippocampus 18, 1200–1212 (2008).

  17. 17.

    Stensola, H. et al. The entorhinal grid map is discretized. Nature 492, 72–78 (2012).

  18. 18.

    Jacobs, J. et al. Direct recordings of grid-like neuronal activity in human spatial navigation. Nat. Neurosci. 16, 1188–1190 (2013).

  19. 19.

    Nadasdy, Z. et al. Context-dependent spatially periodic activity in the human entorhinal cortex. Proc. Natl Acad. Sci. USA 114, E3516–E3525 (2017).

  20. 20.

    Doeller, C. F., Barry, C. & Burgess, N. Evidence for grid cells in a human memory network. Nature 463, 657–661 (2010).

  21. 21.

    McNaughton, B. L., Battaglia, F. P., Jensen, O., Moser, E. I. & Moser, M.-B. Path integration and the neural basis of the ‘cognitive map’. Nat. Rev. Neurosci. 7, 663–678 (2006).

  22. 22.

    Fiete, I. R., Burak, Y. & Brookings, T. What grid cells convey about rat location. J. Neurosci. 28, 6858–6871 (2008).

  23. 23.

    Burak, Y. & Fiete, I. R. Accurate path integration in continuous attractor network models of grid cells. PLoS Comput. Biol. 5, e1000291 (2009).

  24. 24.

    Mathis, A., Herz, A. V. M. & Stemmler, M. Optimal population codes for space: grid cells outperform place cells. Neural Comput. 24, 2280–2317 (2012).

  25. 25.

    Bush, D., Barry, C., Manson, D. & Burgess, N. Using grid cells for navigation. Neuron 87, 507–520 (2015).

  26. 26.

    Herz, A. V., Mathis, A. & Stemmler, M. Periodic population codes: from a single circular variable to higher dimensions, multiple nested scales, and conceptual spaces. Curr. Opin. Neurobiol. 46, 99–108 (2017).

  27. 27.

    Banino, A. et al. Vector-based navigation using grid-like representations in artificial agents. Nature 557, 429–433 (2018).

  28. 28.

    Carpenter, F. & Barry, C. Distorted grids as a spatial label and metric. Trends Cogn. Sci. 20, 164–167 (2016).

  29. 29.

    Sun, C. et al. Distinct speed dependence of entorhinal island and ocean cells, including respective grid cells. Proc. Natl Acad. Sci. USA 112, 9466–9471 (2015).

  30. 30.

    Chen, X., He, Q., Kelly, J. W., Fiete, I. R. & McNamara, T. P. Bias in human path integration Is predicted by properties of grid cells. Curr. Biol. 25, 1771–1776 (2015).

  31. 31.

    Dayan, P. Improving generalization for temporal difference learning: the successor representation. Neural Comput. 5, 613–624 (1993).

  32. 32.

    Stachenfeld, K. L., Botvinick, M. M. & Gershman, S. J. The hippocampus as a predictive map. Nat. Neurosci. 20, 1643–1653 (2017).

  33. 33.

    Momennejad, I. et al. The successor representation in human reinforcement learning. Nat. Hum. Behav. 1, 680 (2017).

  34. 34.

    Russek, E. M., Momennejad, I., Botvinick, M. M., Gershman, S. J. & Daw, N. D. Predictive representations can link model-based reinforcement learning to model-free mechanisms. PLoS Comput. Biol. 13, e1005768 (2017).

  35. 35.

    Gershman, S. J. The successor representation: its computational logic and neural substrates. J. Neurosci. 38, 7193–7200 (2018).

  36. 36.

    Towse, B. W., Barry, C., Bush, D. & Burgess, N. Optimal configurations of spatial scale for grid cell firing under noise and uncertainty. Proc. R. Soc. B 369, 20130290 (2014).

  37. 37.

    Jacobs, J. et al. Direct electrical stimulation of the human entorhinal region and hippocampus impairs memory. Neuron 92, 983–990 (2016).

  38. 38.

    Kubie, J. L. & Fenton, A. A. Linear look-ahead in conjunctive cells: an entorhinal mechanism for vector-based navigation. Front. Neural Circuits 6, 20 (2012).

  39. 39.

    Erdem, U. M. & Hasselmo, M. A goal-directed spatial navigation model using forward trajectory planning based on grid cells: forward linear look-ahead trajectory model. Eur. J. Neurosci. 35, 916–931 (2012).

  40. 40.

    Gil, M. et al. Impaired path integration in mice with disrupted grid cell firing. Nat. Neurosci. 21, 81–91 (2018).

  41. 41.

    Stangl, M. et al. Compromised grid-cell-like representations in old age as a key mechanism to explain age-related navigational deficits. Curr. Biol. 28, 1108–1115 (2018).

  42. 42.

    Kunz, L. et al. Reduced grid-cell–like representations in adults at genetic risk for Alzheimer’s disease. Science 350, 430–433 (2015).

  43. 43.

    Sturz, B. R., Gurley, T. & Bodily, K. D. Orientation in trapezoid-shaped enclosures: implications for theoretical accounts of geometry learning. J. Exp. Psychol. 37, 246–253 (2011).

  44. 44.

    Twyman, A. D., Holden, M. P. & Newcombe, N. S. First direct evidence of cue integration in reorientation: a new paradigm. Cogn. Sci. 42, 923–936 (2018).

  45. 45.

    O’Keefe, J. & Dostrovsky, J. The hippocampus as a spatial map. Preliminary evidence from unit activity in the freely-moving rat. Brain Res. 34, 171–175 (1971).

  46. 46.

    Hartley, T., Trinkler, I. & Burgess, N. Geometric determinants of human spatial memory. Cognition 94, 39–75 (2004).

  47. 47.

    Schuck, N. W., Doeller, C. F., Polk, T. A., Lindenberger, U. & Li, S.-C. Human aging alters the neural computation and representation of space. NeuroImage 117, 141–150 (2015).

  48. 48.

    Taube, J. S., Valerio, S. & Yoder, R. M. Is navigation in virtual reality with fMRI really navigation? J. Cogn. Neurosci. 25, 1008–1019 (2013).

  49. 49.

    Campbell, M. G. et al. Principles governing the integration of landmark and self-motion cues in entorhinal cortical codes for navigation. Nat. Neurosci. 21, 1096–1106 (2018).

  50. 50.

    Chen, G., Lu, Y., King, J. A., Cacucci, F. & Burgess, N. Differential influences of environment and self-motion on place and grid cell firing. Nat. Commun. 10, 630 (2019).

  51. 51.

    Boto, E. et al. Moving magnetoencephalography towards real-world applications with a wearable system. Nature 555, 657–661 (2018).

  52. 52.

    Cadwallader, M. Problems in cognitive distance: implications for cognitive mapping. Environ. Behav. 11, 559–576 (1979).

  53. 53.

    Sadalla, E. K., Burroughs, W. J. & Staplin, L. J. Reference points in spatial cognition. J. Exp. Psychol. 6, 516–528 (1980).

  54. 54.

    Thorndyke, P. W. Distance estimation from cognitive maps. Cogn. Psychol. 13, 526–550 (1981).

  55. 55.

    McNamara, T. P. Mental representations of spatial relations. Cogn. Psychol. 18, 87–121 (1986).

  56. 56.

    McNamara, T. P. & Diwadkar, V. A. Symmetry and asymmetry of human spatial memory. Cogn. Psychol. 34, 160–190 (1997).

  57. 57.

    Newcombe, N., Huttenlocher, J., Sandberg, E., Lie, E. & Johnson, S. What do misestimations and asymmetries in spatial judgement indicate about spatial representation? J. Exp. Psychol. 25, 986–996 (1999).

  58. 58.

    Brunec, I. K., Javadi, A.-H., Zisch, F. E. L. & Spiers, H. J. Contracted time and expanded space: the impact of circumnavigation on judgements of space and time. Cognition 166, 425–432 (2017).

  59. 59.

    Jafarpour, A. & Spiers, H. Familiarity expands space and contracts time. Hippocampus 27, 12–16 (2017).

  60. 60.

    O’Keefe, J. & Nadel, L. The Hippocampus as a Cognitive Map (Clarendon Press, 1978).

  61. 61.

    Bellmund, J. L. S., Deuker, L., Navarro Schröder, T. & Doeller, C. F. Grid-cell representations in mental simulation. eLife 5, e17089 (2016).

  62. 62.

    Horner, A. J., Bisby, J. A., Zotow, E., Bush, D. & Burgess, N. Grid-like processing of imagined navigation. Curr. Biol. 26, 842–847 (2016).

  63. 63.

    Byrne, P., Becker, S. & Burgess, N. Remembering the past and imagining the future: a neural model of spatial memory and imagery. Psychol. Rev. 114, 340–375 (2007).

  64. 64.

    Buckner, R. L. The role of the hippocampus in prediction and imagination. Annu. Rev. Psychol. 61, 27–48 (2010).

  65. 65.

    Hasselmo, M. E. How We Remember: Brain Mechanisms of Episodic Memory (MIT Press, 2011).

  66. 66.

    Bellmund, J. L. S., Gärdenfors, P., Moser, E. I. & Doeller, C. F. Navigating cognition: spatial codes for human thinking. Science 362, eaat6766 (2018).

  67. 67.

    Constantinescu, A. O., O’Reilly, J. X. & Behrens, T. E. J. Organizing conceptual knowledge in humans with a gridlike code. Science 352, 1464–1468 (2016).

  68. 68.

    Delorme, A. & Makeig, S. EEGLAB: an open source toolbox for analysis of single-trial EEG dynamics including independent component analysis. J. of Neurosci. Methods 134, 9–21 (2004).

  69. 69.

    Lakens, D. Calculating and reporting effect sizes to facilitate cumulative science: a practical primer for t-tests and ANOVAs. Front. Psychol. 4, 863 (2013).

  70. 70.

    Hentschke, H. & Stüttgen, M. C. Computation of measures of effect size for neuroscience data sets. Eur. J. Neurosci. 34, 1887–1894 (2011).

  71. 71.

    Berens, P. CircStat: A MATLAB toolbox for circular statistics. J. Stat. Softw. 31, 10 (2009).

  72. 72.

    Cakmak, T. & Hager, H. Cyberith virtualizer: a locomotion device for virtual reality. In Proc. ACM SIGGRAPH 2014 Emerging Technologies https://doi.org/10.1145/2614066.2614105 (ACM, 2014).

  73. 73.

    Brainard, D. H. The Psychophysics Toolbox. Spatial Vision 10, 433–436 (1997).

  74. 74.

    Bellmund, J. L. S., Deuker, L. & Doeller, C. F. Mapping sequence structure in the human lateral entorhinal cortex. eLife 8, e45333 (2019).

  75. 75.

    Stemmler, M., Mathis, A. & Herz, A. V. M. Connecting multiple spatial scales to decode the population activity of grid cells. Sci. Adv. 1, e1500816 (2015).

  76. 76.

    Kass, R. E. & Raftery, A. E. Bayes factors. J. Am. Stat. Assoc. 90, 773–795 (1995).

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Acknowledgements

We thank J. N. Pereira for pilot work that led to the final experimental design. The research of C.F.D. is supported by the Max Planck Society, the European Research Council (ERC-CoG GEOCOG 724836), the Kavli Foundation, the Centre of Excellence scheme of the Research Council of Norway—Centre for Neural Computation (223262), The Egil and Pauline Braathen and Fred Kavli Centre for Cortical Microcircuits, the National Infrastructure scheme of the Research Council of Norway—NORBRAIN and the Netherlands Organisation for Scientific Research (NWO-Vidi 452-12-009; NWO-Gravitation 024-001-006; NWO-MaGW 406-14-114; NWO-MaGW 406-15-291). C.B. and W.C. are supported by a Wellcome Senior Research Fellowship (212281/Z/18/Z). The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript.

Author information

J.L.S.B., C.B. and C.F.D. conceived the experiment. J.L.S.B., T.A.R., M.N. and C.F.D. designed the experiment. T.A.R. collected the data. J.L.S.B. analysed the data and wrote the manuscript with input from M.N., C.B. and C.F.D. W.C. performed the model analysis under supervision of C.B. All of the authors discussed the results and contributed to the final manuscript.

Correspondence to Jacob L. S. Bellmund or Christian F. Doeller.

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Extended data

Extended Data Fig. 1 Successor representation eigenvectors.

A,B. The first 50 eigenvectors of the successor representation from a trapezoid and a square environment were used for analysis.

Extended Data Fig. 2 Position decoding based on the smallest rectangle enclosing the trapezoid.

A. To demonstrate that worse position decoding in the trapezoid is due to the distorted eigenvector grid patterns and not the elongated shape of the environment we analyzed the eigenvectors of the smallest rectangle enclosing the trapezoid. We repeated the position decoding on the area of the trapezoid based on the SR grid patterns from the smallest rectangle. B. Position decoding errors were larger when the analysis was based on the distorted grid patterns of the trapezoid rather than the regular grid patterns generated on the smallest rectangle enclosing the trapezoid (two-sample t-test: t(58)=64.52, p<0.001, d=16.44, 95%-CI: 14.29; 20.46). C. The 50 eigenvector grid patterns used in this analysis.

Extended Data Fig. 3 Spatial frequencies of eigenvector grid patterns.

A,B. Radial power spectra based on two-dimensional FFT averaged across the 50 SR grid patterns. Average spatial frequencies were higher in the square than the trapezoid (A) and higher in the narrow compared to the broad part of the trapezoid (B). Dotted lines indicate mean radial frequencies.

Extended Data Fig. 4 Positional memory.

A. Distribution of average memory scores across participants. Grey area indicates normal kernel density estimate, solid white line shows median and dashed white lines show upper and lower quartile of distribution. Black circles show memory scores of individual participants. B. Positional memory error difference between the two parts of the trapezoid. Higher values indicate larger errors in the narrow part of the trapezoid. Data points more than 1.5 times the interquartile range above or below the upper or lower quartile were excluded as outliers (grey dots) for the main analysis, but comparable results are obtained without outlier exclusion (t(36)=1.50, p=0.020, d=0.25, 95%-CI: -0.06; 0.50). Boxplot represents median as well as upper and lower quartile of distribution, whiskers show most extreme value within 1.5 times the interquartile range from the upper and lower quartile respectively. C. The positional memory error difference observed between the trapezoid parts (dashed line represents mean difference across participants) was significantly lower than the critical value (5th percentile, dotted line) of a shuffle distribution (blue) obtained from computing error difference between the square halves across 10000 iterations. D. Heatmaps showing response locations for all trials across all participants for objects in the broad (top) and narrow (bottom) part of the trapezoid. Dotted lines show correct location in x- and y-dimension with their intersection representing the true position. E. Relationships between the distance to the closest boundary and the memory score were quantified using Pearson correlation. Correlation coefficients were consistently negative in the square, indicating better memory for positions closer to the wall. No statistically significant difference from zero was observed for correlation coefficients in the trapezoid and correlations differed between environments. * p<0.05 *** p<0.001.

Extended Data Fig. 5 Navigation performance does not differ between environments.

A,B. There were no statistically significant differences in the excess path lengths of the trajectories from start to response positions between (A) square and trapezoid or (B) the two parts of the trapezoid. C,D. There were no statistically significant differences in walking speed between (C) square and trapezoid or (D) the two parts of the trapezoid. E,F. There was no statistically significant difference from zero in Spearman correlation coefficients between the Euclidean distance from the start positions to the correct object positions and replacement errors (E) in the square or trapezoid or (F) for objects located in the broad and narrow part of the trapezoid separately. Bars show mean±SEM and grey circles indicate individual subject data with lines connecting data points from the same participant.

Extended Data Fig. 6 Head and body orientation during navigation.

A,B. Circular means in degrees of (A) body and (B) head rotations centered on each trial’s direction from start to response position. Means were significantly clustered around 0° for both square and trapezoid and there was no statistically significant difference between them. C,D. Circular means of (C) body and (D) head rotations centered on each trial’s direction from start to response position. Means were significantly clustered around 0° for trials with target object positions in the broad and narrow part of the trapezoid, respectively, and there was no statistically significant difference between them. E. There was no statistically significant difference in the circular variance of body rotations over trials averaged for each participant between square and trapezoid. F. The circular variance of head rotations over trials averaged for each participant was larger in the trapezoid than in the square. G. There was no statistically significant difference in the circular variance of body rotations over trials averaged for each participant between navigation periods for target objects located in the broad or narrow portion of the trapezoid. H. The circular variance of head rotations over trials averaged for each participant was smaller when cued object position were in the broad compared to the narrow part of the trapezoid. *** p<0.001.

Extended Data Fig. 7 Angular and velocity sampling.

A. Average angular sampling for 10° bins during navigation from a trial’s start position to the remembered object location. Radial axis shows proportion of time points facing in a directional bin. For trial’s targeting objects in the broad part of the trapezoid, participants mostly faced towards the long base of the trapezoid (180°), whereas they more frequently faced towards the short base (0°) when targeting objects in the narrow part of the environment. B. Average movement speed (radial axis vm/s) for 10° directional bins for trials targeting objects in the broad and narrow part of the trapezoid. Navigation speed was higher along the long axis of the environment as indicated by higher movement speeds towards 0° and 180° for trials where participants targeted objects in the narrow and broad part of the trapezoid, respectively. In A and B, colored lines and shaded area show mean and SEM, respectively.

Extended Data Fig. 8 Distance estimates.

A. Long distances (i.e. the base of the isosceles triangle formed by a triplet of positions) were estimated to be longer than the shorter distances (i.e. the legs of the isosceles triangle). Only within-triplet distances were estimated in VR. Bars show mean±SEM and grey circles indicate individual subject data with lines connecting data points from the same participant. B. Grey area indicates distribution of Spearman correlation (mean±SD r=0.69±0.19) coefficients between correct and estimated distances based on normal kernel density estimate. Solid white line shows median and dashed white lines show upper and lower quartile. Black circles show correlation coefficients of individual participants. C. The difference between distance estimates for identical distances in the square and the trapezoid was highly correlated between the computer screen and the VR version of the task. D. Significant correlation of distance difference between the two parts of the trapezoid obtained from distance estimates on the computer screen and in VR. Circles in C and D denote individual participant data; solid line shows least squares line; dashed lines and shaded region highlight bootstrapped confidence intervals. E,F. The distance difference observed between the trapezoid parts (dashed line) was more extreme than the critical values (dotted line) of the shuffle distribution (blue) obtained from computing the distance difference between the square halves across 10000 iterations for the distance estimates in VR (E) and on the PC (F).

Extended Data Fig. 9 Two dimensions underlie distance estimates.

A. Model deviance of GLMs using pairwise Euclidean distances of coordinates obtained from MDS to predict estimated distances for different numbers of dimensions (solid line shows mean model deviance across participants, shaded area indicates SEM). In line with our a priori assumption that two dimensions underlie the distance estimates, model deviance sharply drops when using two rather than one dimension and there is no substantial benefit from including three or more dimensions. B. Heatmaps showing positions reconstructed using multi-dimensional scaling and Procrustes transform for objects in the broad (top) and narrow (bottom) part of the trapezoid. Dotted lines show correct position in x- and y-dimension with their intersection representing the true position.

Extended Data Fig. 10 No statistically significant differences in time estimates between environments.

A. Grey area indicates distribution of Spearman correlation coefficients (mean±SD r=0.77±0.23) between true and estimated times based on normal kernel density estimate. Solid white line shows median and dashed white lines show upper and lower quartile. Black circles show correlation coefficients of individual participants. B-D. There were no statistically significant differences between the two environments for (B) averaged time estimation errors, (C) averaged absolute time estimation errors or (D) the variability of time estimates as measured by their standard deviation. Bars show mean±SEM and grey circles indicate individual subject data with lines connecting data points from the same participant.

Supplementary information

Reporting Summary

Supplementary Video 1

Immersive VR setup. The immersive VR setup consisted of a HMD and a motion platform translating steps and rotations into virtual movement. The video illustrates how participants moved to navigate the virtual environments.

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Bellmund, J.L.S., de Cothi, W., Ruiter, T.A. et al. Deforming the metric of cognitive maps distorts memory. Nat Hum Behav (2019) doi:10.1038/s41562-019-0767-3

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