Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

  • Article
  • Published:

Single-neuronal predictions of others’ beliefs in humans

Nature volume 591, pages 610–614 (2021)Cite this article

Subjects

An Author Correction to this article was published on 01 June 2023

This article has been updated

Abstract

Human social behaviour crucially depends on our ability to reason about others. This capacity for theory of mind has a vital role in social cognition because it enables us not only to form a detailed understanding of the hidden thoughts and beliefs of other individuals but also to understand that they may differ from our own1,2,3. Although a number of areas in the human brain have been linked to social reasoning4,5 and its disruption across a variety of psychosocial disorders6,7,8, the basic cellular mechanisms that underlie human theory of mind remain undefined. Here, using recordings from single cells in the human dorsomedial prefrontal cortex, we identify neurons that reliably encode information about others’ beliefs across richly varying scenarios and that distinguish self- from other-belief-related representations. By further following their encoding dynamics, we show how these cells represent the contents of the others’ beliefs and accurately predict whether they are true or false. We also show how they track inferred beliefs from another’s specific perspective and how their activities relate to behavioural performance. Together, these findings reveal a detailed cellular process in the human dorsomedial prefrontal cortex for representing another’s beliefs and identify candidate neurons that could support theory of mind.

This is a preview of subscription content, access via your institution

Access options

Buy this article

  • Purchase on Springer Link
  • Instant access to full article PDF
Buy now

Prices may be subject to local taxes which are calculated during checkout

Fig. 1: Tracking single-cell representations of another’s beliefs in the human dmPFC.
Fig. 2: Single-neuronal predictions of another’s true and false beliefs.
Fig. 3: Neuronal responses to self beliefs versus others’ beliefs and perspective.
Fig. 4: Population predictions of another’s belief contents and their relation to behavioural performance.

Similar content being viewed by others

Data availability

Details of the participants’ demographics and task conditions are provided in Extended Data Tables 1, 2. The behavioural and neuronal data that support the findings of this study are available from the corresponding author upon reasonable request.

Code availability

All software used in this study are listed in the Reporting Summary along with their versions. The primary MATLAB codes used to perform the statistical and data analyses in this study are available from the corresponding author upon reasonable request.

Change history

References

  1. Wimmer, H. & Perner, J. Beliefs about beliefs: representation and constraining function of wrong beliefs in young children’s understanding of deception. Cognition 13, 103–128 (1983).

    Article  CAS  PubMed  Google Scholar 

  2. Koster-Hale, J. & Saxe, R. Theory of mind: a neural prediction problem. Neuron 79, 836–848 (2013).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  3. Stone, V. E., Baron-Cohen, S. & Knight, R. T. Frontal lobe contributions to theory of mind. J. Cogn. Neurosci. 10, 640–656 (1998).

    Article  CAS  PubMed  Google Scholar 

  4. Saxe, R. & Kanwisher, N. People thinking about thinking people. The role of the temporo-parietal junction in “theory of mind”. Neuroimage 19, 1835–1842 (2003).

    Article  CAS  PubMed  Google Scholar 

  5. van Veluw, S. J. & Chance, S. A. Differentiating between self and others: an ALE meta-analysis of fMRI studies of self-recognition and theory of mind. Brain Imaging Behav. 8, 24–38 (2014).

    Article  PubMed  Google Scholar 

  6. Baron-Cohen, S., Jolliffe, T., Mortimore, C. & Robertson, M. Another advanced test of theory of mind: evidence from very high functioning adults with autism or asperger syndrome. J. Child Psychol. Psychiatry 38, 813–822 (1997).

    Article  CAS  PubMed  Google Scholar 

  7. Brent, E., Rios, P., Happé, F. & Charman, T. Performance of children with autism spectrum disorder on advanced theory of mind tasks. Autism 8, 283–299 (2004).

    Article  PubMed  Google Scholar 

  8. Amaral, D., Dawson, G. & Geschwind, D. H. Autism Spectrum Disorders (Oxford Univ. Press, 2011).

  9. Carruthers, P. & Smith, P. K. Theories of Theories of Mind (Cambridge Univ. Press, 1996).

  10. Frith, U. & Frith, C. D. Development and neurophysiology of mentalizing. Phil. Trans. R. Soc. Lond. B 358, 459–473 (2003).

    Article  Google Scholar 

  11. Richardson, H., Lisandrelli, G., Riobueno-Naylor, A. & Saxe, R. Development of the social brain from age three to twelve years. Nat. Commun. 9, 1027 (2018).

    Article  ADS  PubMed  PubMed Central  Google Scholar 

  12. Williams, Z. M. & Eskandar, E. N. Selective enhancement of associative learning by microstimulation of the anterior caudate. Nat. Neurosci. 9, 562–568 (2006).

    Article  CAS  PubMed  Google Scholar 

  13. Saxe, R. & Powell, L. J. It’s the thought that counts: specific brain regions for one component of theory of mind. Psychol. Sci. 17, 692–699 (2006).

    Article  PubMed  Google Scholar 

  14. Saxe, R., Moran, J. M., Scholz, J. & Gabrieli, J. Overlapping and non-overlapping brain regions for theory of mind and self reflection in individual subjects. Soc. Cogn. Affect. Neurosci. 1, 229–234 (2006).

    Article  PubMed  PubMed Central  Google Scholar 

  15. Moessnang, C. et al. Differential responses of the dorsomedial prefrontal cortex and right posterior superior temporal sulcus to spontaneous mentalizing. Hum. Brain Mapp. 38, 3791–3803 (2017).

    Article  PubMed  PubMed Central  Google Scholar 

  16. Martin, A. K., Dzafic, I., Ramdave, S. & Meinzer, M. Causal evidence for task-specific involvement of the dorsomedial prefrontal cortex in human social cognition. Soc. Cogn. Affect. Neurosci. 12, 1209–1218 (2017).

    Article  PubMed  PubMed Central  Google Scholar 

  17. Döhnel, K. et al. Functional activity of the right temporo-parietal junction and of the medial prefrontal cortex associated with true and false belief reasoning. Neuroimage 60, 1652–1661 (2012).

    Article  PubMed  Google Scholar 

  18. Bardi, L., Desmet, C., Nijhof, A., Wiersema, J. R. & Brass, M. Brain activation for spontaneous and explicit false belief tasks overlaps: new fMRI evidence on belief processing and violation of expectation. Soc. Cogn. Affect. Neurosci. 12, 391–400 (2017).

    PubMed  Google Scholar 

  19. Fletcher, P. C. et al. Other minds in the brain: a functional imaging study of “theory of mind” in story comprehension. Cognition 57, 109–128 (1995).

    Article  CAS  PubMed  Google Scholar 

  20. Apperly, I. A., Samson, D., Chiavarino, C., Bickerton, W. L. & Humphreys, G. W. Testing the domain-specificity of a theory of mind deficit in brain-injured patients: evidence for consistent performance on non-verbal, “reality-unknown” false belief and false photograph tasks. Cognition 103, 300–321 (2007).

    Article  PubMed  Google Scholar 

  21. Dodell-Feder, D., Koster-Hale, J., Bedny, M. & Saxe, R. fMRI item analysis in a theory of mind task. Neuroimage 55, 705–712 (2011).

    Article  PubMed  Google Scholar 

  22. Sabbagh, M. A. & Taylor, M. Neural correlates of theory-of-mind reasoning: an event-related potential study. Psychol. Sci. 11, 46–50 (2000).

    Article  CAS  Google Scholar 

  23. Dayan, P. & Abbott, L. F. Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems (Massachusetts Institute of Technology Press, 2001).

  24. Arslan, B., Verbrugge, R., Taatgen, N. & Hollebrandse, B. Accelerating the development of second-order false belief reasoning: a training study with different feedback methods. Child Dev. 91, 249–270 (2020).

    Article  PubMed  Google Scholar 

  25. Baron-Cohen, S., Tager-Flusberg, H. & Lombardo, M. Understanding Other Minds: Perspectives from Developmental Social Neuroscience 3rd edn (Oxford Univ. Press, 2013).

  26. Bird, C. M., Castelli, F., Malik, O., Frith, U. & Husain, M. The impact of extensive medial frontal lobe damage on ‘theory of mind’ and cognition. Brain 127, 914–928 (2004).

    Article  PubMed  Google Scholar 

  27. Mukamel, R., Ekstrom, A. D., Kaplan, J., Iacoboni, M. & Fried, I. Single-neuron responses in humans during execution and observation of actions. Curr. Biol. 20, 750–756 (2010).

    Article  CAS  PubMed Central  Google Scholar 

  28. Rizzolatti, G., Fadiga, L., Gallese, V. & Fogassi, L. Premotor cortex and the recognition of motor actions. Brain Res. Cogn. Brain Res. 3, 131–141 (1996).

    Article  CAS  PubMed  Google Scholar 

  29. Haroush, K. & Williams, Z. M. Neuronal prediction of opponent’s behavior during cooperative social interchange in primates. Cell 160, 1233–1245 (2015).

    Article  CAS  PubMed Central  Google Scholar 

  30. Chang, S. W., Gariépy, J. F. & Platt, M. L. Neuronal reference frames for social decisions in primate frontal cortex. Nat. Neurosci. 16, 243–250 (2013).

    Article  CAS  PubMed  Google Scholar 

  31. Williams, Z. M., Bush, G., Rauch, S. L., Cosgrove, G. R. & Eskandar, E. N. Human anterior cingulate neurons and the integration of monetary reward with motor responses. Nat. Neurosci. 7, 1370–1375 (2004).

    Article  CAS  PubMed  Google Scholar 

  32. Patel, S. R. et al. Studying task-related activity of individual neurons in the human brain. Nat. Protoc. 8, 949–957 (2013).

    Article  CAS  PubMed  Google Scholar 

  33. Sheth, S. A. et al. Human dorsal anterior cingulate cortex neurons mediate ongoing behavioural adaptation. Nature 488, 218–221 (2012).

    Article  ADS  CAS  PubMed  PubMed Central  Google Scholar 

  34. Mian, M. K. et al. Encoding of rules by neurons in the human dorsolateral prefrontal cortex. Cereb. Cortex 24, 807–816 (2014).

    Article  PubMed  Google Scholar 

  35. Erdodi, L. A. et al. Wechsler Adult Intelligence Scale-Fourth Edition (WAIS-IV) processing speed scores as measures of noncredible responding: The third generation of embedded performance validity indicators. Psychol. Assess. 29, 148–157 (2017).

    Article  PubMed  Google Scholar 

  36. Holdnack, J. A., Xiaobin Zhou Larrabee, G. J., Millis, S. R. & Salthouse, T. A. Confirmatory factor analysis of the WAIS-IV/WMS-IV. Assessment 18, 178–191 (2011).

    Article  PubMed Central  Google Scholar 

  37. Amirnovin, R., Williams, Z. M., Cosgrove, G. R. & Eskandar, E. N. Experience with microelectrode guided subthalamic nucleus deep brain stimulation. Neurosurgery 58 (Suppl), ONS96–ONS102 (2006).

    PubMed  Google Scholar 

  38. Jamali, M. et al. Dorsolateral prefrontal neurons mediate subjective decisions and their variation in humans. Nat. Neurosci. 22, 1010–1020 (2019).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  39. Nicolelis, M. A. L. (ed.). Methods for Neural Ensemble Recordings 2nd edn (CRC Press/Taylor & Francis, 2008).

  40. Oby, E. R. et al. Extracellular voltage threshold settings can be tuned for optimal encoding of movement and stimulus parameters. J. Neural Eng. 13, 036009 (2016).

    Article  ADS  PubMed  PubMed Central  Google Scholar 

  41. Perel, S. et al. Single-unit activity, threshold crossings, and local field potentials in motor cortex differentially encode reach kinematics. J. Neurophysiol. 114, 1500–1512 (2015).

    Article  PubMed Central  Google Scholar 

  42. Braüner, T., Blackburn, P. & Polyanskaya, I. Being deceived: information asymmetry in second-order false belief tasks. Top. Cogn. Sci. 12, 504–534 (2020).

    Article  PubMed  Google Scholar 

  43. Shimazaki, H. & Shinomoto, S. Kernel bandwidth optimization in spike rate estimation. J. Comput. Neurosci. 29, 171–182 (2010).

    Article  MathSciNet  PubMed  MATH  Google Scholar 

  44. Bowman, A.W. & Azzalini, A. (eds). Applied Smoothing Techniques for Data Analysis: The Kernel Approach with S-Plus Illustrations (Oxford Science, 1997).

  45. Pagan, M., Urban, L. S., Wohl, M. P. & Rust, N. C. Signals in inferotemporal and perirhinal cortex suggest an untangling of visual target information. Nat. Neurosci. 16, 1132–1139 (2013).

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  46. Quian Quiroga, R., Snyder, L. H., Batista, A. P., Cui, H. & Andersen, R. A. Movement intention is better predicted than attention in the posterior parietal cortex. J. Neurosci. 26, 3615–3620 (2006).

    Article  PubMed  PubMed Central  Google Scholar 

  47. Hung, C. P., Kreiman, G., Poggio, T. & DiCarlo, J. J. Fast readout of object identity from macaque inferior temporal cortex. Science 310, 863–866 (2005).

    Article  ADS  CAS  PubMed  Google Scholar 

  48. Wasserman, L. All of Statistics: A Concise Course in Statistical Inference (Springer, 2005).

  49. Sarafyazd, M. & Jazayeri, M. Hierarchical reasoning by neural circuits in the frontal cortex. Science 364, eaav8911 (2019).

    Article  CAS  PubMed  Google Scholar 

  50. Cohen, R. & Elhadad, M. Syntactic dependency parsers for biomedical-NLP. AMIA Annu. Symp. Proc. 2012, 121–128 (2012).

    PubMed  PubMed Central  Google Scholar 

  51. Li, Z. et al. Integrating shortest dependency path and sentence sequence into a deep learning framework for relation extraction in clinical text. BMC Med. Inform. Decis. Mak. 19 (Suppl 1), 22 (2019).

    Article  PubMed  PubMed Central  Google Scholar 

Download references

Acknowledgements

M.J. is supported by the Banting Foundation, NARSAD Young Investigator Grant, and Foundations of Human Behavior Initiative, B.L.G. is supported by the NREF and NIH NRSA, E.F. is supported by NIH R01DC016607 and R01DC016950, R.B.-M. is supported by the MGH ECOR and NARSAD Young Investigator Grant, and Z.M.W. is supported by NIH R01HD059852, NIH R01NS091390 and the Presidential Early Career Award for Scientists and Engineers.

Author information

Authors and Affiliations

  1. Department of Neurosurgery, Massachusetts General Hospital, Harvard Medical School, Boston, MA, USA

    Mohsen Jamali, Benjamin L. Grannan, Raymundo Báez-Mendoza & Ziv M. Williams

  2. Brain and Cognitive Sciences, Massachusetts Institute of Technology, Boston, MA, USA

    Evelina Fedorenko & Rebecca Saxe

  3. Harvard-MIT Division of Health Sciences and Technology, Boston, MA, USA

    Ziv M. Williams

  4. Program in Neuroscience, Harvard Medical School, Boston, MA, USA

    Ziv M. Williams

Authors
  1. Mohsen Jamali
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Benjamin L. Grannan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Evelina Fedorenko
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Rebecca Saxe
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Raymundo Báez-Mendoza
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. Ziv M. Williams
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

M.J. performed the analysis, M.J., B.L.G. and R.B.-M. performed the experiments, E.F. and R.S. provided feedback and Z.M.W. conceived and designed the study, performed the experiments, obtained the recordings and oversaw the project.

Corresponding author

Correspondence to Ziv M. Williams.

Ethics declarations

Competing interests

The authors declare no competing interests.

Additional information

Peer review information Nature thanks Robert Knight and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

Publisher’s note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Extended data figures and tables

Extended Data Fig. 1 Recording location, waveform morphology and single-unit isolation.

a, Single-neuronal recordings were obtained from the superior frontal gyrus of the dmPFC using incrementally advancing microelectrode arrays. The region of recordings in MNI coordinates (x = –6, y = 49, z = 42) is shown in a canonical structure MRI. b, Examples of waveform morphologies displaying mean waveform ± 3× s.d. The top panel illustrates a single representative unit isolated from a fine-tip tungsten microelectrode. The bottom panel illustrates two representative units that were isolated from another microelectrode. The horizontal bar indicates a 500 μs interval for scale. c, Isolation patterns corresponding to the waveforms shown in b represented by principal component distributions. The grey areas in the PC space represent baseline noise. All putative units displayed significant separation by one-way MANOVA (P < 0.0001) and no overlap with baseline signal/noise.

Extended Data Fig. 2 Schematic depiction of experimental logic and narrative features across trial conditions.

On the left, other belief vs physical trials were used to identify neurons that responded selectively to another’s beliefs. Whereas both required the participant to consider false vs true representations, only the former required the participants to consider another’s specific beliefs. In the middle, other belief vs self-belief trials were used to further differentiate other- from self-related representations. Whereas both required the participant to consider a belief, only the former required the participants to consider another’s false vs true beliefs. Aware vs unaware trials were given to additionally differentiate other- from self-perspective. On the right, first- vs second-order belief trials were used to evaluate for the consistency of neuronal response across different depths of reasoning. High vs low degree of inference as well as high vs low task demand trials were used to evaluate for the consistency of neuronal response across different degrees of inference and cognitive demand.

Extended Data Fig. 3 Consistency of the results across different statistical methodology and neuronal isolation approaches.

a, A parametric two-sided unpaired t-test was used to evaluate whether cells displayed a significant difference in their responses. Comparisons were made between other belief vs physical trials (top, n = 62 neurons) and between false vs true other-belief trials (bottom, n = 47 neurons). The magnitude of effect (mean ± s.e.m.) over the course of the trial is displayed based on the t-statistic. Neuronal activity is aligned to the question onset (time zero). The insets display the t-statistic values for all neurons that displayed (n = 62 in the top and n = 47 in the bottom panel, coloured) and did not display (n = 150 in the top and n = 165 in the bottom panel, grey) significant selectivity. b, A two-sided unpaired non-parametric rank-sum test was used with the same conventions as above. Here, the magnitude of effect (mean ± s.e.m.) is displayed based on the z-value (n = 64 in the top and n = 45 in the bottom panel). c, These results also held when considering other neural isolation approaches. Decoding performances were obtained for MUA using the same modelling and decoding approach as for the single-neuronal data. The bar graphs provide the individual MUAs (n = 8) and their corresponding 95% CL. The horizontal line indicates chance performance (one-sided permutation test, P < 0.005).

Extended Data Fig. 4 Consistency of the results across subjects and clinical conditions.

a, The participants demonstrated a largely similar proportion of task-modulated neurons when considering belief vs physical trials (s.d., of 11.6%) as well as false- vs true-belief trials (s.d., of 10.0%). The arrows start from participant #1. b, Proportion of neurons displaying task modulations based on clinical conditions; PD and ET. The p-value by chi-square test is shown. We also found no difference in the firing rates of the neurons based on clinical diagnosis (1.61 ± 0.19 vs 1.70 ± 0.11 spikes s−1 for PD and ET, respectively; two-sided Wilcoxon rank-sum, z-value (1586) = 0.92, P = 0.36). c, A subject-dropping procedure was used to determine whether any of the participants disproportionately contributed to the population decoding performance. Here, individual participants were sequentially removed one at a time and the population decoding was repeated (200 iterations). Population decoding performances (mean ± s.e.m.) are separately presented after each participant was removed. Chance decoding based on random permutation of the neuronal data are provided in black for comparison. The decoding performances were largely unaffected by removal of any of the participants when decoding other-beliefs vs physical representations (top panel; one-way ANOVA: F(10,2189) = 1.2, P = 0.29) as well as when decoding other true- vs false-beliefs (bottom panel; one-way ANOVA: F(10,2189) = 0.68, P = 0.75). d, A subject-adding procedure was further used to determine how the participants cumulatively contributed to the population decoding by sequentially adding subjects contributing to the neuronal population from 1 to 11 and repeating the decoding analysis (200 iterations). Decoding performances are provided with the same convention as above (mean ± s.e.m.). As shown, adding subjects one at a time led to a consistent increase in the decoding performance suggesting that the participants made similar contributions.

Extended Data Fig. 5 Robustness of belief representations.

a, Reaction times (mean ± s.e.m.) from question offset to answer onset during the primary task conditions across participants (n = 11) were similar for other-belief vs physical trials (1,071 ± 135 vs 1,178 ± 201 ms) and for false- vs true-belief trials (1,130 ± 136 vs 1,028 ± 147 ms). The p-values obtained using a two-sided unpaired t-test. b, To evaluate how differences in neuronal decoding may relate to answer response time, decoding performances were first averaged across neurons that displayed significant selectivity and then sorted based on the participants’ reaction times (n = 18 time points). There was a slightly negative but non-significant correlation between RTs and decoding performances both when comparing other-belief to physical trials (r = −0.35, P = 0.16) and when comparing other false- to true-belief trials (r = −0.27, P = 0.28). The p-values by Spearman’s correlation test are shown. c, We found no relationship between neuronal activity (mean firing rates, n = 49 neurons) and trial difficulty (easy vs hard; Methods) based on the participants’ overall performances (two-sided rank-sum test, z-value = 0.85, P = 0.40). d–f, Neuronal activity was evaluated based on (d) the number of social agents presented to the participants (left: n = 4,024 trials, right: n = 4527 trials), (e) the number of items (for example, table, jar, cupboard, etc.) that had to be held in working memory before questioning (n = 4527 trials), and (f) the narrative length based on the number of words (n = 4527 trials). Activities were z-scored by removing the mean and dividing by the standard deviation. A lack of relation was demonstrated by correlation analysis in each condition (Pearson’s correlation, P > 0.1).

Extended Data Fig. 6 Decoding others’ beliefs based on variations in perspective and awareness.

a, Mean decoding profile with 95% CL for all neurons that accurately differentiated between false-belief vs true-belief trials (n = 49; one-sided permutation test, P < 0.025). Here, the trials were divided based on whether or not the social agent was made aware of events in the narratives. Since the state of reality was the same under these two conditions, demonstration of similar decoding performances on the standard other-belief and other-belief aware trials confirmed that neuronal predictions of the other’s beliefs reflected the other’s perspective of reality independently of the participant’s own. b, Decoding accuracies on other-belief aware trials were positively correlated with those decoded from the standard other-belief trials on a cell-by-cell basis (n = 49; Pearson’s correlation; P = 0.04).

Extended Data Fig. 7 Decoding others’ beliefs based on variations in the item’s identity or location being considered.

Above, the narratives and questions were varied in whether they required the participants to consider an items location or its identity. Below, the decoding performances of the individual neurons based on whether the social agent’s beliefs involved an item’s identity or its location are displayed. The Venn diagram (inset) shows the overlap between neurons.

Extended Data Fig. 8 Relation between neuronal predictions and performance for beliefs vs physical trials.

The histograms indicate decoding accuracies for neurons that predicted whether the participants were considering another’s beliefs vs physical representations on trials in which the participants provided the correct vs incorrect upcoming answer (one-sided permutation test, P = 0.001). The arrows indicate mean decoding performances.

Extended Data Table 1 Participants’ demographics and overall performances
Full size table
Extended Data Table 2 Narrative and question examples
Full size table

Supplementary information

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jamali, M., Grannan, B.L., Fedorenko, E. et al. Single-neuronal predictions of others’ beliefs in humans. Nature 591, 610–614 (2021). https://doi.org/10.1038/s41586-021-03184-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1038/s41586-021-03184-0

This article is cited by

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Advanced search

Quick links

Nature Briefing

Sign up for the Nature Briefing newsletter — what matters in science, free to your inbox daily.

Get the most important science stories of the day, free in your inbox. Sign up for Nature Briefing