* the perturbation $q$ induces no change in the prediction of the network: $q \cdot w_{hp} = 0$, where $w_{hp}$ is the weight vector of the output unit of the network;

* the perturbation has a constant intensity $c$ across simulations, which we formalize as the norm of the perturbation: $||q|| = c$.

We describe below the algorithm that we used to generate random perturbations $q$ that satisfy these constraints. The idea is to decompose $q$ into two components: both components leave the prediction unaffected, the first ($q_ψ$) is used to induce a controlled change in precision, the second ($q_r$) does not change the precision but is added to ensure a constant intensity of the perturbation across simulations.

1. To ensure no change in precision, we compute $Q$, the subspace of the activation space spanned by all vectors $q$ that are orthogonal to the prediction weight vector $w_{hp}$, as the null space of $w_{hp}$ (i.e. the orthogonal complement of the subspace spanned by $w_{hp}$, dimension N-1).

2. We compute $q_ψ$, the vector component of $Q$ that affects precision, as the orthogonal projection of $w_{hp}$ onto $Q$ ($q_ψ$ is thus collinear to the orthogonalized precision axis shown in Figure 4b and described above).

3. We compute $β_ψ$, the coefficient to assign to $q_ψ$ in the perturbation vector to produce the desired change in precision $δψ$, as $β_ψ = \frac{δψ}{||q_ψ \cdot w_{hp}||}$.

4. We compute $R$, the subspace spanned by all vector components of $Q$ that do not affect precision, as the null space of $q_ψ$ (dimension N-2). A perturbation vector in $R$ therefore leaves both the prediction and the precision unchanged.

5. We draw a random unit vector $q_r$ within $R$ (by drawing from all N-2 components).

6. We compute $β_r$, the coefficient to assign to $q_r$ in the perturbation vector so as to ensure that the final perturbation's norm equals $c$, as $β_r = \sqrt{c^2 - β_ψ^2 ||q_ψ||^2}$.

7. We combine $q_ψ$ and $q_r$ into the final perturbation vector as $q = β_ψq_ψ + β_rq_r$.

The experiment was run on a set of 1000 sample time points randomly drawn from 300 sequences. First, the unperturbed learning rate was measured by running the network on all of the sequences. Second, for each sample time point, the network was run unperturbed up until that point, a perturbation vector was randomly generated for the desired change of precision and applied to the network at that point, then the perturbed network was run on the next time point and its perturbed learning rate was measured. This was repeated for each level of change in precision. Finally, for a given change in precision, the change in learning rate was calculated as the difference between the perturbed and the unperturbed learning rate.

For statistical analysis, we ran a one-tailed paired t-test to test whether the population's mean change in learning rate was higher at one level of precision change than at the next level of precision change. This was done for each of the four consecutive pairs of levels of change in precision.

## Test of higher-level inference about changes

For a given network architecture, higher-level inference about changes was assessed by comparing the population of 20 networks trained in the environment with coupled change points to the population of 20 networks trained in the environment with independent change points.

In Figure 7c, the change in unobserved bigram prediction for a given streak length $m$ was computed as follows. First, prior sequences were generated and each network was run on each of the sequences. We generated initial sequences of 74 observations each with a probability of 0.2 for the 'observed' bigram (which will render its repetition surprising) and a probability $p$ for the 'unobserved' bigram equal to 0.2 or 0.8 (such probabilities, symmetric and substantially different from the default prior 0.5, should render a change in their inferred value detectable). We crossed all possibilities (0|0 or 1|1 as observed bigram, 0.2 or 0.8 for $p$) and generated 100 sequences for each (hence 400 sequences total). Second, at the end of each of these initial sequences, the prediction for the unobserved bigram, $p_{before}$, was queried by retrieving the output of the network after giving it as input '0' if the unobserved bigram was 0|0 or '1' otherwise. Third, the network was further presented with $m$ repeated observations of the same value: '1' if the observed bigram was 1|1 or '0' otherwise. Finally, after this streak of repetition, the new prediction for the unobserved bigram, $p_{after}$, was queried (as before) and we measured its change with respect to the previous query, $|p_{after} - p_{before}|$. This procedure was repeated for $m$ ranging from 2 and 75.
For statistics, we ran a one-tailed two independent samples t-test to test whether the mean change in unobserved bigram prediction of the population trained on coupled change points was higher than that of the population trained on independent change points.

## Complexity analyses

The complexity analysis reported in Figure 8 consisted in measuring, for each network architecture and each environment, the performance of optimally trained networks as a function of the number of units N. For optimal training, hyperparameter optimization was repeated at several values of N, for each type of network and each environment (the resulting values are listed in Table 1). For the complexity analysis, a grid of equally spaced N values in logarithmic space between 1 and 45 was generated, an additional value of 474 was included specifically for the networks without recurrent weight training so as to match their number of trained parameters to that of an 11-unit gated recurrent network, and an additional value of 1,000 was included specifically for the networks without gating to facilitate the extrapolation. For every value on this grid, 20 networks of a given architecture in a given environment were randomly initialized with the set of hyperparameter values that was determined to be optimal for the nearest neighboring N value in logarithmic space. The performance of these networks after training was evaluated using a new couple of training and test datasets per environment, each consisting of 400 minibatches of 20 sequences for training and 1000 sequences for testing.

## Statistics

To assess the variability between different agent solutions, we trained 20 agents for each type of agent and each environment. These agents have different random seeds (which changes their parameter initialization and how their training data is shuffled). Throughout the article, we report mean or median over these agents, and individual data points when possible or 95% confidence intervals (abbreviated as "CI") otherwise, as fully described in the text and figure legends. No statistical methods were used to pre-determine sample sizes but our sample sizes are similar to those reported in previous publications (Masse et al., 2019; Yang et al., 2019). Data analysis was not performed blind to the conditions of the experiments. No data were excluded from the analyses. All statistical tests were two-tailed unless otherwise noted. The data distribution was assumed to be normal, but this was not formally tested. The specific details of each statistical analysis are reported directly in the text.

## Code availability

The code to reproduce exhaustively the analyses of this paper is available at https://github.com/cedricfoucault/networks_for_sequencereddited on Zenodo with DOI: 10.5281/zenodo.5707498. This code also enables to train new networks equipped with any number of units and generate Figures 2-7 with those networks.

## Data availability

This paper presents no experimental data. All synthetic data are available in the code repository at https://github.com/cedricfoucault/networks_for_sequenceæreredited and archived on Zenodo with DOI: 10.5281/zenodo.5707498.

## Acknowledgements

We thank Yair Lakretz for useful feedback, advice, and discussions throughout the project, Alexandre Pouget for his input when starting this project, and Charles Findling for comments on a previous version of the manuscript.
For statistics, we ran a one-tailed two independent samples t-test to test whether the mean change in unobserved bigram prediction of the population trained on coupled change points was higher than that of the population trained on independent change points.

## Complexity analyses

The complexity analysis reported in Figure 8 consisted in measuring, for each network architecture and each environment, the performance of optimally trained networks as a function of the number of units N. For optimal training, hyperparameter optimization was repeated at several values of N, for each type of network and each environment (the resulting values are listed in Table 1). For the complexity analysis, a grid of equally spaced N values in logarithmic space between 1 and 45 was generated, an additional value of 474 was included specifically for the networks without recurrent weight training so as to match their number of trained parameters to that of an 11-unit gated recurrent network, and an additional value of 1,000 was included specifically for the networks without gating to facilitate the extrapolation. For every value on this grid, 20 networks of a given architecture in a given environment were randomly initialized with the set of hyperparameter values that was determined to be optimal for the nearest neighboring N value in logarithmic space. The performance of these networks after training was evaluated using a new couple of training and test datasets per environment, each consisting of 400 minibatches of 20 sequences for training and 1000 sequences for testing.

## Statistics

To assess the variability between different agent solutions, we trained 20 agents for each type of agent and each environment. These agents have different random seeds (which changes their parameter initialization and how their training data is shuffled). Throughout the article, we report mean or median over these agents, and individual data points when possible or 95% confidence intervals (abbreviated as "CI") otherwise, as fully described in the text and figure legends. No statistical methods were used to pre-determine sample sizes but our sample sizes are similar to those reported in previous publications (Masse et al., 2019; Yang et al., 2019). Data analysis was not performed blind to the conditions of the experiments. No data were excluded from the analyses. All statistical tests were two-tailed unless otherwise noted. The data distribution was assumed to be normal, but this was not formally tested. The specific details of each statistical analysis are reported directly in the text.

## Code availability

The code to reproduce exhaustively the analyses of this paper is available at https://github.com/cedricfoucault/networks_for_sequencereddited on Zenodo with DOI: 10.5281/zenodo.5707498. This code also enables to train new networks equipped with any number of units and generate Figures 2-7 with those networks.

## Data availability

This paper presents no experimental data. All synthetic data are available in the code repository at https://github.com/cedricfoucault/networks_for_sequenceæreredited on Zenodo with DOI: 10.5281/zenodo.5707498.

## Acknowledgements

We thank Yair Lakretz for useful feedback, advice, and discussions throughout the project, Alexandre Pouget for his input when starting this project, and Charles Findling for comments on a previous version of the manuscript.
## Additional information

### Funding

<table><thead><tr><td>Funder</td><td>Grant reference number</td><td>Author</td></tr></thead><tbody><tr><td>École normale supérieure Paris-Saclay</td><td>PhD fellowship "Contrat doctoral spécifique normalien"</td><td>Cédric Foucault</td></tr><tr><td>Agence Nationale de la Recherche</td><td>18-CE37-0010-01 "CONFI LEARN"</td><td>Florent Meyniel</td></tr><tr><td>H2020 European Research Council</td><td>ERC StG 947105 "NEURAL PROB"</td><td>Florent Meyniel</td></tr></tbody></table>

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

### Author contributions

Cédric Foucault, Florent Meyniel, Conceptualization, Formal analysis, Funding acquisition, Methodology, Project administration, Supervision, Visualization, Writing - original draft, Writing - review and editing

### Author ORCIDs

Cédric Foucault http://orcid.org/0000-0002-7247-6927

Florent Meyniel http://orcid.org/0000-0002-6992-678X

### Decision letter and Author response

Decision letter https://doi.org/10.7554/eLife.71801.sa1

Author response https://doi.org/10.7554/eLife.71801.sa2

## Additional files

### Supplementary files

* Transparent reporting form

### Data availability

This paper presents no experimental data. All synthetic data are available in the code repository at https://github.com/cedricfoucault/networks_for_sequence prediction and archived on Zenodo with https://doi.org/10.5281/zenodo.5707498.

The following dataset was generated:

<table><thead><tr><td>Author(s)</td><td>Year</td><td>Dataset title</td><td>Dataset URL</td><td>Database and Identifier</td></tr></thead><tbody><tr><td>Foucault C</td><td>2021</td><td>Networks for sequence prediction</td><td>https://github.com/cedricfoucault/networks_for_sequencequi</td><td>Github, prediction</td></tr><tr><td>Foucault C</td><td>2021</td><td>Networks for sequence prediction</td><td>http://dx.doi.org/10.5281/zenodo.5707498</td><td>Zenodo, 10.5281/zenodo.5707498</td></tr></tbody></table>

## References

* **Agnihotri A**, Batra N. 2020. Exploring Bayesian Optimization. *Distill* 5:e26. DOI: https://doi.org/10.23915/distill.00026

* **Amini A**, Schwarting W, Soleimany A, Rus D. 2020. Deep Evidential Regression. Advances in Neural Information Processing Systems. 14927–14937.

* **Aston-Jones G**, Rajkowski J, Kubiak P. 1997. Conditioned responses of monkey locus coeruleus neurons anticipate acquisition of discriminative behavior in a vigilance task. *Neuroscience* 80:697–715. DOI: https://doi.org/10.1016/s0306-4522(97)00060-2, PMID: 9276487
# Additional information

## Funding

<table><thead><tr><td>Funder</td><td>Grant reference number</td><td>Author</td></tr></thead><tbody><tr><td>École normale supérieure Paris-Saclay</td><td>PhD fellowship "Contrat doctoral spécifique normalien"</td><td>Cédric Foucault</td></tr><tr><td>Agence Nationale de la Recherche</td><td>18-CE37-0010-01 "CONFI LEARN"</td><td>Florent Meyniel</td></tr><tr><td>H2020 European Research Council</td><td>ERC StG 947105 "NEURAL PROB"</td><td>Florent Meyniel</td></tr></tbody></table>

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

## Author contributions

Cédric Foucault, Florent Meyniel, Conceptualization, Formal analysis, Funding acquisition, Methodology, Project administration, Supervision, Visualization, Writing - original draft, Writing - review and editing

## Author ORCIDs

Cédric Foucault http://orcid.org/0000-0002-7247-6927

Florent Meyniel http://orcid.org/0000-0002-6992-678X

## Decision letter and Author response

Decision letter https://doi.org/10.7554/eLife.71801.sa1

Author response https://doi.org/10.7554/eLife.71801.sa2

# Additional files

## Supplementary files

* Transparent reporting form

## Data availability

This paper presents no experimental data. All synthetic data are available in the code repository at https://github.com/cedricfoucault/networks_for_sequence prediction and archived on Zenodo with https://doi.org/10.5281/zenodo.5707498.

The following dataset was generated:

<table><thead><tr><td>Author(s)</td><td>Year</td><td>Dataset title</td><td>Dataset URL</td><td>Database and Identifier</td></tr></thead><tbody><tr><td>Foucault C</td><td>2021</td><td>Networks for sequence prediction</td><td>https://github.com/cedricfoucault/networks_for_sequencequi</td><td>Github, prediction</td></tr><tr><td>Foucault C</td><td>2021</td><td>Networks for sequence prediction</td><td>http://dx.doi.org/10.5281/zenodo.5707498</td><td>Zenodo, 10.5281/zenodo.5707498</td></tr></tbody></table>

# References

* Agnihotri A, Batra N. 2020. Exploring Bayesian Optimization. Distill 5:e26. DOI: https://doi.org/10.23915/distill.00026

* Amini A, Schwarting W, Soleimany A, Rus D. 2020. Deep Evidential Regression. Advances in Neural Information Processing Systems. 14927–14937.

* Aston-Jones G, Rajkowski J, Kubiak P. 1997. Conditioned responses of monkey locus coeruleus neurons anticipate acquisition of discriminative behavior in a vigilance task. Neuroscience 80:697–715. DOI: https://doi.org/10.1016/s0306-4522(97)00060-2, PMID: 9276487
Aston-Jones G, Cohen JD. 2005. An integrative theory of locus coeruleus-norepinephrine function: adaptive gain and optimal performance. *Annual Review of Neuroscience* **28**:403-450. DOI: https://doi.org/10.1146/annurev.neuro.28.061604.135709, PMID: 16022602

Bach DR, Hulme O, Penny WD, Dolan RJ. 2011. The known unknowns: neural representation of second-order uncertainty, and ambiguity. *The Journal of Neuroscience* **31**:4811-4820. DOI: https://doi.org/10.1523/JNEUROSCI.1452-10.2011, PMID: 21451019

Barron AR. 1993. Universal approximation bounds for superpositions of a sigmoidal function. *IEEE Transactions on Information Theory* **39**:930-945. DOI: https://doi.org/10.1109/18.256500

Baumgarten TJ, Schnitzler A, Lange J. 2016. Prestimulus Alpha Power Influences Tactile Temporal Perceptual Discrimination and Confidence in Decisions. *Cerebral Cortex* **26**:891-903. DOI: https://doi.org/10.1093/cercor/bhu247, PMID: 25331603

Behrens TEJ, Woolrich MW, Walton ME, Rushworth MFS. 2007. Learning the value of information in an uncertain world. *Nature Neuroscience* **10**:1214-1221. DOI: https://doi.org/10.1038/nn1954, PMID: 17676057

Beiran M, Dubreuil A, Valente A, Mastrogiuseppe F, Ostojic S. 2021. Shaping Dynamics With Multiple Populations in Low-Rank Recurrent Networks. *Neural Computation* **33**:1572-1615. DOI: https://doi.org/10.1162/neco_a_01381, PMID: 34496384

Bengio Y, Simard P, Frasconi P. 1994. Learning long-term dependencies with gradient descent is difficult. *IEEE Transactions on Neural Networks* **5**:157-166. DOI: https://doi.org/10.1109/72.279181, PMID: 18267787

Berniker M, Kording K. 2008. Estimating the sources of motor errors for adaptation and generalization. *Nature Neuroscience* **11**:1454-1461. DOI: https://doi.org/10.1038/nn.2229, PMID: 19011624

Bhui R, Lai L, Gershman SJ. 2021. Resource-rational decision making. *Current Opinion in Behavioral Sciences* **4**:15-21. DOI: https://doi.org/10.1016/j.cobeha.2021.02.015

Bill J, Pailian H, Gershman SJ, Drugowitsch J. 2020. Hierarchical structure is employed by humans during visual motion perception. *PNAS* **117**:24581-24589. DOI: https://doi.org/10.1073/pnas.2008961117, PMID: 32938799

Blalock D, Ortiz JJG, Frankle J, Guttag J. 2020. What Is the State of Neural Network Pruning. [arXiv]. http://arxiv.org/abs/2003.03033

Boldt A, Blundell C, De Martino B. 2019. Confidence modulates exploration and exploitation in value-based learning. *Neuroscience of Consciousness* **2019**:niz004. DOI: https://doi.org/10.1093/nc/niz004, PMID: 31086679

Bornstein AM, Daw ND. 2013. Cortical and hippocampal correlates of deliberation during model-based decisions for rewards in humans. *PLOS Computational Biology* **9**:e1003387. DOI: https://doi.org/10.1371/journal.pcbi.1003387, PMID: 24339770

Bowers JS, Davis CJ. 2012. Bayesian just-so stories in psychology and neuroscience. *Psychological Bulletin* **138**:389-414. DOI: https://doi.org/10.1037/a0026450, PMID: 22545686

Busch NA, Dubois J, VanRullen R. 2009. The phase of ongoing EEG oscillations predicts visual perception. *The Journal of Neuroscience* **29**:7869-7876. DOI: https://doi.org/10.1523/JNEUROSCI.0113-09.2009, PMID: 19535598

Caucheteux C, King JR. 2021. Language Processing in Brains and Deep Neural Networks: Computational Convergence and Its Limits. [bioRxiv]. DOI: https://doi.org/10.1101/2020.07.03.186288, PMID: 34420675

Chater N, Tenenbaum JB, Yuille A. 2006. Probabilistic models of cognition: conceptual foundations. *Trends in Cognitive Sciences* **10**:287-291. DOI: https://doi.org/10.1016/j.tics.2006.05.007, PMID: 16807064

Chechik G. 1999. Neuronal regulation: A mechanism for synaptic pruning during brain maturation. *Neural Computation* **11**:2061-2080. DOI: https://doi.org/10.1162/089976699300016089, PMID: 10578044

Cho K, van Merrienboer B, Gulcehre C, Bahdanau D, Bougares F, Schwenk H, Bengio Y. 2014. Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation. Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing. 1724-1734. DOI: https://doi.org/10.3115/v1/D14-1179

Chung J, Gulcehre C, Cho K, Bengio Y. 2014. Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling. [ArXiv:1412.3555 [Cs]]. http://arxiv.org/abs/1412.3555

Cooper GF. 1990. The computational complexity of probabilistic inference using bayesian belief networks. *Artificial Intelligence* **42**:393-405. DOI: https://doi.org/10.1016/0004-3702(90)90060-D

Costa R, Assael IA, Shillingford B, de Freitas N, Vogels T. 2017. Cortical microcircuits as gated-recurrent neural networks. Advances in Neural Information Processing Systems.

Courville AC, Daw ND, Touretzky DS. 2006. Bayesian theories of conditioning in a changing world. *Trends in Cognitive Sciences* **10**:294-300. DOI: https://doi.org/10.1016/j.tics.2006.05.004, PMID: 16793323

Cybenko G. 1989. Approximation by superpositions of a sigmoidal function. *Mathematics of Control, Signals, and Systems* **2**:303-314. DOI: https://doi.org/10.1007/BF02551274

Dagum P, Luby M. 1993. Approximating probabilistic inference in Bayesian belief networks is NP-hard. *Artificial Intelligence* **60**:141-153. DOI: https://doi.org/10.1016/0004-3702(93)90036-B

de Lange FP, Heilbron M, Kok P. 2018. How Do Expectations Shape Perception? *Trends in Cognitive Sciences* **22**:764-779. DOI: https://doi.org/10.1016/j.tics.2018.06.002, PMID: 30122170

Dehaene S, Meynel F, Wacongne C, Wang L, Pallier C. 2015. The Neural Representation of Sequences: From Transition Probabilities to Algebraic Patterns and Linguistic Trees. *Neuron* **88**:2-19. DOI: https://doi.org/10.1016/j.neuron.2015.09.019, PMID: 26447569

den Ouden HEM, Kok P, de Lange FP. 2012. How prediction errors shape perception, attention, and motivation. *Frontiers in Psychology* **3**:548. DOI: https://doi.org/10.3389/fpsyg.2012.00548, PMID: 23248610
Aston-Jones G, Cohen JD. 2005. An integrative theory of locus coeruleus-norepinephrine function: adaptive gain and optimal performance. *Annual Review of Neuroscience* **28**:403-450. DOI: https://doi.org/10.1146/annurev.neuro.28.061604.135709, PMID: 16022602

Bach DR, Hulme O, Penny WD, Dolan RJ. 2011. The known unknowns: neural representation of second-order uncertainty, and ambiguity. *The Journal of Neuroscience* **31**:4811-4820. DOI: https://doi.org/10.1523/JNEUROSCI.1452-10.2011, PMID: 21451019

Barron AR. 1993. Universal approximation bounds for superpositions of a sigmoidal function. *IEEE Transactions on Information Theory* **39**:930-945. DOI: https://doi.org/10.1109/18.256500

Baumgarten TJ, Schnitzler A, Lange J. 2016. Prestimulus Alpha Power Influences Tactile Temporal Perceptual Discrimination and Confidence in Decisions. *Cerebral Cortex* **26**:891-903. DOI: https://doi.org/10.1093/cercor/bhu247, PMID: 25331603

Behrens TEJ, Woolrich MW, Walton ME, Rushworth MFS. 2007. Learning the value of information in an uncertain world. *Nature Neuroscience* **10**:1214-1221. DOI: https://doi.org/10.1038/nn1954, PMID: 17676057

Beiran M, Dubreuil A, Valente A, Mastrogiuseppe F, Ostojic S. 2021. Shaping Dynamics With Multiple Populations in Low-Rank Recurrent Networks. *Neural Computation* **33**:1572-1615. DOI: https://doi.org/10.1162/neco_a_01381, PMID: 34496384

Bengio Y, Simard P, Frasconi P. 1994. Learning long-term dependencies with gradient descent is difficult. *IEEE Transactions on Neural Networks* **5**:157-166. DOI: https://doi.org/10.1109/72.279181, PMID: 18267787

Berniker M, Kording K. 2008. Estimating the sources of motor errors for adaptation and generalization. *Nature Neuroscience* **11**:1454-1461. DOI: https://doi.org/10.1038/nn.2229, PMID: 19011624

Bhui R, Lai L, Gershman SJ. 2021. Resource-rational decision making. *Current Opinion in Behavioral Sciences* **4**:15-21. DOI: https://doi.org/10.1016/j.cobeha.2021.02.015

Bill J, Pailian H, Gershman SJ, Drugowitsch J. 2020. Hierarchical structure is employed by humans during visual motion perception. *PNAS* **117**:24581-24589. DOI: https://doi.org/10.1073/pnas.2008961117, PMID: 32938799

Blalock D, Ortiz JJG, Frankle J, Guttag J. 2020. What Is the State of Neural Network Pruning. [arXiv]. http://arxiv.org/abs/2003.03033

Boldt A, Blundell C, De Martino B. 2019. Confidence modulates exploration and exploitation in value-based learning. *Neuroscience of Consciousness* **2019**:niz004. DOI: https://doi.org/10.1093/nc/niz004, PMID: 31086679

Bornstein AM, Daw ND. 2013. Cortical and hippocampal correlates of deliberation during model-based decisions for rewards in humans. *PLOS Computational Biology* **9**:e1003387. DOI: https://doi.org/10.1371/journal.pcbi.1003387, PMID: 24339770

Bowers JS, Davis CJ. 2012. Bayesian just-so stories in psychology and neuroscience. *Psychological Bulletin* **138**:389-414. DOI: https://doi.org/10.1037/a0026450, PMID: 22545686

Busch NA, Dubois J, VanRullen R. 2009. The phase of ongoing EEG oscillations predicts visual perception. *The Journal of Neuroscience* **29**:7869-7876. DOI: https://doi.org/10.1523/JNEUROSCI.0113-09.2009, PMID: 19535598

Caucheteux C, King JR. 2021. Language Processing in Brains and Deep Neural Networks: Computational Convergence and Its Limits. [bioRxiv]. DOI: https://doi.org/10.1101/2020.07.03.186288, PMID: 34420675

Chater N, Tenenbaum JB, Yuille A. 2006. Probabilistic models of cognition: conceptual foundations. *Trends in Cognitive Sciences* **10**:287-291. DOI: https://doi.org/10.1016/j.tics.2006.05.007, PMID: 16807064

Chechik G. 1999. Neuronal regulation: A mechanism for synaptic pruning during brain maturation. *Neural Computation* **11**:2061-2080. DOI: https://doi.org/10.1162/089976699300016089, PMID: 10578044

Cho K, van Merrienboer B, Gulcehre C, Bahdanau D, Bougares F, Schwenk H, Bengio Y. 2014. Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation. Proceedings of the 2014 Conference on Empirical Methods in Natural Language Processing. 1724-1734. DOI: https://doi.org/10.3115/v1/D14-1179

Chung J, Gulcehre C, Cho K, Bengio Y. 2014. Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling. [ArXiv:1412.3555 [Cs]]. http://arxiv.org/abs/1412.3555

Cooper GF. 1990. The computational complexity of probabilistic inference using bayesian belief networks. *Artificial Intelligence* **42**:393-405. DOI: https://doi.org/10.1016/0004-3702(90)90060-D

Costa R, Assael IA, Shillingford B, de Freitas N, Vogels T. 2017. Cortical microcircuits as gated-recurrent neural networks. Advances in Neural Information Processing Systems.

Courville AC, Daw ND, Touretzky DS. 2006. Bayesian theories of conditioning in a changing world. *Trends in Cognitive Sciences* **10**:294-300. DOI: https://doi.org/10.1016/j.tics.2006.05.004, PMID: 16793323

Cybenko G. 1989. Approximation by superpositions of a sigmoidal function. *Mathematics of Control, Signals, and Systems* **2**:303-314. DOI: https://doi.org/10.1007/BF02551274

Dagum P, Luby M. 1993. Approximating probabilistic inference in Bayesian belief networks is NP-hard. *Artificial Intelligence* **60**:141-153. DOI: https://doi.org/10.1016/0004-3702(93)90036-B

de Lange FP, Heilbron M, Kok P. 2018. How Do Expectations Shape Perception? *Trends in Cognitive Sciences* **22**:764-779. DOI: https://doi.org/10.1016/j.tics.2018.06.002, PMID: 30122170

Dehaene S, Meynel F, Wacongne C, Wang L, Pallier C. 2015. The Neural Representation of Sequences: From Transition Probabilities to Algebraic Patterns and Linguistic Trees. *Neuron* **88**:2-19. DOI: https://doi.org/10.1016/j.neuron.2015.09.019, PMID: 26447569

den Ouden HEM, Kok P, de Lange FP. 2012. How prediction errors shape perception, attention, and motivation. *Frontiers in Psychology* **3**:548. DOI: https://doi.org/10.3389/fpsyg.2012.00548, PMID: 23248610
Deroy O, Spence C, Noppeney U. 2016. Metacognition in Multisensory Perception. *Trends in Cognitive Sciences* **20**:736-747. DOI: https://doi.org/10.1016/j.tics.2016.08.006, PMID: 27612983

Dolan RJ, Dayan P. 2013. Goals and habits in the brain. *Neuron* **80**:312–325. DOI: https://doi.org/10.1016/j.neuron.2013.09.007, PMID: 24139036

Douglas RJ, Martin KAC. 2007. Recurrent neuronal circuits in the neocortex. *Current Biology* **17**:R496-R500. DOI: https://doi.org/10.1016/j.cub.2007.04.024, PMID: 17610826

Dubreuil A, Valente A, Beiran M, Mastrogiuseppe F, Ostojic S. 2020. Complementary Roles of Dimensionality and Population Structure in Neural Computations. [bioRxiv]. DOI: https://doi.org/10.1101/2020.07.03.185942

Echeveste R, Aitchison L, Hennequin G, Lengyel M. 2020. Cortical-like dynamics in recurrent circuits optimized for sampling-based probabilistic inference. *Nature Neuroscience* **23**:1138–1149. DOI: https://doi.org/10.1038/s41593-020-0671-1, PMID: 32778794

Elman JL. 1990. Finding Structure in Time. *Cognitive Science* **14**:179–211. DOI: https://doi.org/10.1207/s15516709cog1402_1

Elman JL. 1991. Distributed representations, simple recurrent networks, and grammatical structure. *Machine Learning* **7**:195–225. DOI: https://doi.org/10.1007/BF00114844

Eshel N, Tian J, Uchida N. 2013. Opening the black box: dopamine, predictions, and learning. *Trends in Cognitive Sciences* **17**:430–431. DOI: https://doi.org/10.1016/j.tics.2013.06.010, PMID: 23830895

Fairhall AL, Lewen GD, Bialek W, de Ruyter Van Steveninck RR. 2001. Efficiency and ambiguity in an adaptive neural code. *Nature* **412**:787–792. DOI: https://doi.org/10.1038/35090500, PMID: 11518957

Farashahi S, Donahue CH, Khorsand P, Seo H, Lee D, Soltani A. 2017. Metaplasticity as a Neural Substrate for Adaptive Learning and Choice under Uncertainty. *Neuron* **94**:401-414. DOI: https://doi.org/10.1016/j.neuron.2017.03.044, PMID: 28426971

Findling C, Skvortsova V, Dromnelle R, Palminteri S, Wyart V. 2019. Computational noise in reward-guided learning drives behavioral variability in volatile environments. *Nature Neuroscience* **22**:2066–2077. DOI: https://doi.org/10.1038/s41593-019-0518-9, PMID: 31659343

Findling C, Wyart V. 2020. Computation Noise Promotes Cognitive Resilience to Adverse Conditions during Decision-Making. [bioRxiv]. DOI: https://doi.org/10.1101/2020.06.10.145300

Findling C, Chopin N, Koechlin E. 2021. Imprecise neural computations as a source of adaptive behaviour in volatile environments. *Nature Human Behaviour* **5**:99–112. DOI: https://doi.org/10.1038/s41562-020-00971-z, PMID: 33168951

Fiser J, Berkes P, Orbán G, Lengyel M. 2010. Statistically optimal perception and learning: from behavior to neural representations. *Trends in Cognitive Sciences* **14**:119–130. DOI: https://doi.org/10.1016/j.tics.2010.01.003, PMID: 20153683

Friston K, Rigoli F, Ognibene D, Mathys C, Fitzgerald T, Pezzulo G. 2015. Active inference and epistemic value. *Cognitive Neuroscience* **6**:187–214. DOI: https://doi.org/10.1080/17588928.2015.1020053, PMID: 25689102

Fusi S, Drew PJ, Abbott LF. 2005. Cascade models of synaptically stored memories. *Neuron* **45**:599–611. DOI: https://doi.org/10.1016/j.neuron.2005.02.001, PMID: 15721245

Fusi S, Asaad WF, Miller EK, Wang XJ. 2007. A neural circuit model of flexible sensorimotor mapping: learning and forgetting on multiple timescales. *Neuron* **54**:319–333. DOI: https://doi.org/10.1016/j.neuron.2007.03.017, PMID: 17442251

Gallistel CR, Krishan M, Liu Y, Miller R, Latham PE. 2014. The perception of probability. *Psychological Review* **121**:96–123. DOI: https://doi.org/10.1037/a0035232, PMID: 24490790

Gijsen S, Grundei M, Lange RT, Ostwald D, Blankenburg F. 2021. Neural surprise in somatosensory Bayesian learning. *PLOS Computational Biology* **17**:e1008068. DOI: https://doi.org/10.1371/journal.pcbi.1008068, PMID: 33529181

Gil Z, Connors BW, Amitai Y. 1997. Differential regulation of neocortical synapses by neuromodulators and activity. *Neuron* **19**:679–686. DOI: https://doi.org/10.1016/s0896-6273(00)80380-3, PMID: 9331357

Griffiths TL, Chater N, Norris D, Pouget A. 2012. How the Bayesians got their beliefs (and what those beliefs actually are): comment on Bowers and Davis (2012). *Psychological Bulletin* **138**:415–422. DOI: https://doi.org/10.1037/a0026884, PMID: 22545687

Hahn G, Ponce-Alvarez A, Deco G, Aertsen A, Kumar A. 2019. Portraits of communication in neuronal networks. *Nature Reviews. Neuroscience* **20**:117–127. DOI: https://doi.org/10.1038/s41583-018-0094-0, PMID: 30552403

Hauser MD, Newport EL, Aslin RN. 2001. Segmentation of the speech stream in a non-human primate: statistical learning in cotton-top tamarins. *Cognition* **78**:B53-B64. DOI: https://doi.org/10.1016/s0010-0277(00)00132-3, PMID: 11124355

Haxby JV, Connolly AC, Guntupalli JS. 2014. Decoding neural representational spaces using multivariate pattern analysis. *Annual Review of Neuroscience* **37**:435–456. DOI: https://doi.org/10.1146/annurev-neuro-062012-170325, PMID: 25002277

Heilbron M, Meyniel F. 2019. Confidence resets reveal hierarchical adaptive learning in humans. *PLOS Computational Biology* **15**:e1006972. DOI: https://doi.org/10.1371/journal.pcbi.1006972, PMID: 30964861

Herrero JL, Roberts MJ, Delicato LS, Gieselmann MA, Dayan P, Thiele A. 2008. Acetylcholine contributes through muscarinic receptors to attentional modulation in V1. *Nature* **454**:1110–1114. DOI: https://doi.org/10.1038/nature07141, PMID: 18633352

Hipp JF, Engel AK, Siegel M. 2011. Oscillatory synchronization in large-scale cortical networks predicts perception. *Neuron* **69**:387–396. DOI: https://doi.org/10.1016/j.neuron.2010.12.027, PMID: 21262474

Hochreiter S, Schmidhuber J. 1997. Long short-term memory. *Neural Computation* **9**:1735–1780. DOI: https://doi.org/10.1162/neco.1997.9.8.1735, PMID: 9377276
Deroy O, Spence C, Noppeney U. 2016. Metacognition in Multisensory Perception. *Trends in Cognitive Sciences* **20**:736-747. DOI: https://doi.org/10.1016/j.tics.2016.08.006, PMID: 27612983

Dolan RJ, Dayan P. 2013. Goals and habits in the brain. *Neuron* **80**:312–325. DOI: https://doi.org/10.1016/j.neuron.2013.09.007, PMID: 24139036

Douglas RJ, Martin KAC. 2007. Recurrent neuronal circuits in the neocortex. *Current Biology* **17**:R496-R500. DOI: https://doi.org/10.1016/j.cub.2007.04.024, PMID: 17610826

Dubreuil A, Valente A, Beiran M, Mastrogiuseppe F, Ostojic S. 2020. Complementary Roles of Dimensionality and Population Structure in Neural Computations. [bioRxiv]. DOI: https://doi.org/10.1101/2020.07.03.185942

Echeveste R, Aitchison L, Hennequin G, Lengyel M. 2020. Cortical-like dynamics in recurrent circuits optimized for sampling-based probabilistic inference. *Nature Neuroscience* **23**:1138–1149. DOI: https://doi.org/10.1038/s41593-020-0671-1, PMID: 32778794

Elman JL. 1990. Finding Structure in Time. *Cognitive Science* **14**:179–211. DOI: https://doi.org/10.1207/s15516709cog1402_1

Elman JL. 1991. Distributed representations, simple recurrent networks, and grammatical structure. *Machine Learning* **7**:195–225. DOI: https://doi.org/10.1007/BF00114844

Eshel N, Tian J, Uchida N. 2013. Opening the black box: dopamine, predictions, and learning. *Trends in Cognitive Sciences* **17**:430–431. DOI: https://doi.org/10.1016/j.tics.2013.06.010, PMID: 23830895

Fairhall AL, Lewen GD, Bialek W, de Ruyter Van Steveninck RR. 2001. Efficiency and ambiguity in an adaptive neural code. *Nature* **412**:787–792. DOI: https://doi.org/10.1038/35090500, PMID: 11518957

Farashahi S, Donahue CH, Khorsand P, Seo H, Lee D, Soltani A. 2017. Metaplasticity as a Neural Substrate for Adaptive Learning and Choice under Uncertainty. *Neuron* **94**:401-414. DOI: https://doi.org/10.1016/j.neuron.2017.03.044, PMID: 28426971

Findling C, Skvortsova V, Dromnelle R, Palminteri S, Wyart V. 2019. Computational noise in reward-guided learning drives behavioral variability in volatile environments. *Nature Neuroscience* **22**:2066–2077. DOI: https://doi.org/10.1038/s41593-019-0518-9, PMID: 31659343

Findling C, Wyart V. 2020. Computation Noise Promotes Cognitive Resilience to Adverse Conditions during Decision-Making. [bioRxiv]. DOI: https://doi.org/10.1101/2020.06.10.145300

Findling C, Chopin N, Koechlin E. 2021. Imprecise neural computations as a source of adaptive behaviour in volatile environments. *Nature Human Behaviour* **5**:99–112. DOI: https://doi.org/10.1038/s41562-020-00971-z, PMID: 33168951

Fiser J, Berkes P, Orbán G, Lengyel M. 2010. Statistically optimal perception and learning: from behavior to neural representations. *Trends in Cognitive Sciences* **14**:119–130. DOI: https://doi.org/10.1016/j.tics.2010.01.003, PMID: 20153683

Friston K, Rigoli F, Ognibene D, Mathys C, Fitzgerald T, Pezzulo G. 2015. Active inference and epistemic value. *Cognitive Neuroscience* **6**:187–214. DOI: https://doi.org/10.1080/17588928.2015.1020053, PMID: 25689102

Fusi S, Drew PJ, Abbott LF. 2005. Cascade models of synaptically stored memories. *Neuron* **45**:599–611. DOI: https://doi.org/10.1016/j.neuron.2005.02.001, PMID: 15721245

Fusi S, Asaad WF, Miller EK, Wang XJ. 2007. A neural circuit model of flexible sensorimotor mapping: learning and forgetting on multiple timescales. *Neuron* **54**:319–333. DOI: https://doi.org/10.1016/j.neuron.2007.03.017, PMID: 17442251

Gallistel CR, Krishan M, Liu Y, Miller R, Latham PE. 2014. The perception of probability. *Psychological Review* **121**:96–123. DOI: https://doi.org/10.1037/a0035232, PMID: 24490790

Gijsen S, Grundei M, Lange RT, Ostwald D, Blankenburg F. 2021. Neural surprise in somatosensory Bayesian learning. *PLOS Computational Biology* **17**:e1008068. DOI: https://doi.org/10.1371/journal.pcbi.1008068, PMID: 33529181

Gil Z, Connors BW, Amitai Y. 1997. Differential regulation of neocortical synapses by neuromodulators and activity. *Neuron* **19**:679–686. DOI: https://doi.org/10.1016/s0896-6273(00)80380-3, PMID: 9331357

Griffiths TL, Chater N, Norris D, Pouget A. 2012. How the Bayesians got their beliefs (and what those beliefs actually are): comment on Bowers and Davis (2012). *Psychological Bulletin* **138**:415–422. DOI: https://doi.org/10.1037/a0026884, PMID: 22545687

Hahn G, Ponce-Alvarez A, Deco G, Aertsen A, Kumar A. 2019. Portraits of communication in neuronal networks. *Nature Reviews. Neuroscience* **20**:117–127. DOI: https://doi.org/10.1038/s41583-018-0094-0, PMID: 30552403

Hauser MD, Newport EL, Aslin RN. 2001. Segmentation of the speech stream in a non-human primate: statistical learning in cotton-top tamarins. *Cognition* **78**:B53-B64. DOI: https://doi.org/10.1016/s0010-0277(00)00132-3, PMID: 11124355

Haxby JV, Connolly AC, Guntupalli JS. 2014. Decoding neural representational spaces using multivariate pattern analysis. *Annual Review of Neuroscience* **37**:435–456. DOI: https://doi.org/10.1146/annurev-neuro-062012-170325, PMID: 25002277

Heilbron M, Meyniel F. 2019. Confidence resets reveal hierarchical adaptive learning in humans. *PLOS Computational Biology* **15**:e1006972. DOI: https://doi.org/10.1371/journal.pcbi.1006972, PMID: 30964861

Herrero JL, Roberts MJ, Delicato LS, Gieselmann MA, Dayan P, Thiele A. 2008. Acetylcholine contributes through muscarinic receptors to attentional modulation in V1. *Nature* **454**:1110–1114. DOI: https://doi.org/10.1038/nature07141, PMID: 18633352

Hipp JF, Engel AK, Siegel M. 2011. Oscillatory synchronization in large-scale cortical networks predicts perception. *Neuron* **69**:387–396. DOI: https://doi.org/10.1016/j.neuron.2010.12.027, PMID: 21262474

Hochreiter S, Schmidhuber J. 1997. Long short-term memory. *Neural Computation* **9**:1735–1780. DOI: https://doi.org/10.1162/neco.1997.9.8.1735, PMID: 9377276
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Iemi L, Busch NA, Laudini A, Haegens S, Samaha J, Villringer A, Nikulin VV. 2019. Multiple mechanisms link prestimulus neural oscillations to sensory responses. *eLife* **8**:e43620. DOI: https://doi.org/10.7554/eLife.43620, PMID: 31188126

Iglesias S, Mathys C, Brodersen KH, Kasper L, Piccirelli M, den Ouden HEM, Stephan KE. 2013. Hierarchical prediction errors in midbrain and basal forebrain during sensory learning. *Neuron* **80**:519–530. DOI: https://doi.org/10.1016/j.neuron.2013.09.009, PMID: 24139048

Ligaya K. 2016. Adaptive learning and decision-making under uncertainty by metaplastic synapses guided by a surprise detection system. *eLife* **5**:e18073. DOI: https://doi.org/10.7554/eLife.18073, PMID: 27504806

Jazayeri M, Movshon JA. 2006. Optimal representation of sensory information by neural populations. *Nature Neuroscience* **9**:690–696. DOI: https://doi.org/10.1038/nn1691, PMID: 16617339

Jazayeri M, Ostojic S. 2021. Interpreting neural computations by examining intrinsic and embedding dimensionality of neural activity. *Current Opinion in Neurobiology* **70**:113–120. DOI: https://doi.org/10.1016/j.conb.2021.08.002, PMID: 34537579

Jensen O, Mazaheri A. 2010. Shaping functional architecture by oscillatory alpha activity: gating by inhibition. *Frontiers in Human Neuroscience* **4**:186. DOI: https://doi.org/10.3389/fnhum.2010.00186, PMID: 21119777

Kaliukhovich DA, Vogels R. 2014. Neurons in macaque inferior temporal cortex show no surprise response to deviants in visual oddball sequences. *The Journal of Neuroscience* **34**:12801–12815. DOI: https://doi.org/10.1523/JNEUROSCI.2154-14.2014, PMID: 25232116

Khaw MW, Stevens L, Woodford M. 2017. Discrete adjustment to a changing environment: Experimental evidence. *Journal of Monetary Economics* **91**:88–103. DOI: https://doi.org/10.1016/j.jmoneco.2017.09.001

Khaw MW, Stevens L, Woodford M. 2021. Individual differences in the perception of probability. *PLOS Computational Biology* **17**:e1008871. DOI: https://doi.org/10.1371/journal.pcbi.1008871, PMID: 33793574

Kingma DP, Ba J. 2015. Adam: A Method for Stochastic Optimization. 3rd International Conference on Learning Representations, ICLR 2015.

Klimesch W. 1999. EEG alpha and theta oscillations reflect cognitive and memory performance: a review and analysis. *Brain Research. Brain Research Reviews* **29**:169–195. DOI: https://doi.org/10.1016/s0165-0173(98)00056-3, PMID: 10209231

Klimesch W, Sauseng P, Hanslmayr S. 2007. EEG alpha oscillations: the inhibition-timing hypothesis. *Brain Research Reviews* **53**:63–88. DOI: https://doi.org/10.1016/j.brainresrev.2006.06.003, PMID: 16887192

Knill DC, Pouget A. 2004. The Bayesian brain: the role of uncertainty in neural coding and computation. *Trends in Neurosciences* **27**:712–719. DOI: https://doi.org/10.1016/j.tins.2004.10.007, PMID: 15541511

Kriegeskorte N, Diedrichsen J. 2019. Peeling the Onion of Brain Representations. *Annual Review of Neuroscience* **42**:407–432. DOI: https://doi.org/10.1146/annurev-neuro-080317-061906, PMID: 31283895

LeCun Y, Denker J, Solla S. 1990. Optimal Brain Damage. Advances in Neural Information Processing Systems.

LeCun Y, Bengio Y, Hinton G. 2015. Deep learning. *Nature* **521**:436–444. DOI: https://doi.org/10.1038/nature14539, PMID: 26017442

LeCun Y. 2016. Predictive learning. Proc. Speech NIPS.

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Legenstein R, Maass W. 2014. Ensembles of spiking neurons with noise support optimal probabilistic inference in a dynamically changing environment. *PLOS Computational Biology* **10**:e1003859. DOI: https://doi.org/10.1371/journal.pcbi.1003859, PMID: 25340749

Lieder F, Griffiths TL. 2019. Resource-rational analysis: Understanding human cognition as the optimal use of limited computational resources. *The Behavioral and Brain Sciences* **43**:e1. DOI: https://doi.org/10.1017/S0140525X1900061X, PMID: 30714890

Lillicrap TP, Santoro A, Marris L, Akerman CJ, Hinton G. 2020. Backpropagation and the brain. *Nature Reviews. Neuroscience* **21**:335–346. DOI: https://doi.org/10.1038/s41583-020-0277-3, PMID: 32303713

Ma WJ, Beck JM, Latham PE, Pouget A. 2006. Bayesian inference with probabilistic population codes. *Nature Neuroscience* **9**:1432–1438. DOI: https://doi.org/10.1038/nn1790, PMID: 17057707

Ma WJ. 2010. Signal detection theory, uncertainty, and Poisson-like population codes. *Vision Research* **50**:2308–2319. DOI: https://doi.org/10.1016/j.visres.2010.08.035, PMID: 20828581

Ma WJ, Jazayeri M. 2014. Neural coding of uncertainty and probability. *Annual Review of Neuroscience* **37**:205–220. DOI: https://doi.org/10.1146/annurev-neuro-071013-014017, PMID: 25032495

Maheu M, Dehaene S, Meyniel F. 2019. Brain signatures of a multiscale process of sequence learning in humans. *eLife* **8**:e41541. DOI: https://doi.org/10.7554/eLife.41541, PMID: 30714904

Mante V, Sussillo D, Shenoy KV, Newsome WT. 2013. Context-dependent computation by recurrent dynamics in prefrontal cortex. *Nature* **503**:78–84. DOI: https://doi.org/10.1038/nature12742, PMID: 24201281

Masse NY, Yang GR, Song HF, Wang XJ, Freedman DJ. 2019. Circuit mechanisms for the maintenance and manipulation of information in working memory. *Nature Neuroscience* **22**:1159–1167. DOI: https://doi.org/10.1038/s41593-019-0414-3, PMID: 31182866

Mastrogiuseppe F, Ostojic S. 2017. Intrinsically-generated fluctuating activity in excitatory-inhibitory networks. *PLOS Computational Biology* **13**:e1005498. DOI: https://doi.org/10.1371/journal.pcbi.1005498, PMID: 28437436
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Iemi L, Busch NA, Laudini A, Haegens S, Samaha J, Villringer A, Nikulin VV. 2019. Multiple mechanisms link prestimulus neural oscillations to sensory responses. *eLife* **8**:e43620. DOI: https://doi.org/10.7554/eLife.43620, PMID: 31188126

Iglesias S, Mathys C, Brodersen KH, Kasper L, Piccirelli M, den Ouden HEM, Stephan KE. 2013. Hierarchical prediction errors in midbrain and basal forebrain during sensory learning. *Neuron* **80**:519-530. DOI: https://doi.org/10.1016/j.neuron.2013.09.009, PMID: 24139048

Ligaya K. 2016. Adaptive learning and decision-making under uncertainty by metaplastic synapses guided by a surprise detection system. *eLife* **5**:e18073. DOI: https://doi.org/10.7554/eLife.18073, PMID: 27504806

Jazayeri M, Movshon JA. 2006. Optimal representation of sensory information by neural populations. *Nature Neuroscience* **9**:690-696. DOI: https://doi.org/10.1038/nn1691, PMID: 16617339

Jazayeri M, Ostojic S. 2021. Interpreting neural computations by examining intrinsic and embedding dimensionality of neural activity. *Current Opinion in Neurobiology* **70**:113-120. DOI: https://doi.org/10.1016/j.conb.2021.08.002, PMID: 34537579

Jensen O, Mazaheri A. 2010. Shaping functional architecture by oscillatory alpha activity: gating by inhibition. *Frontiers in Human Neuroscience* **4**:186. DOI: https://doi.org/10.3389/fnhum.2010.00186, PMID: 21119777

Kaliukhovich DA, Vogels R. 2014. Neurons in macaque inferior temporal cortex show no surprise response to deviants in visual oddball sequences. *The Journal of Neuroscience* **34**:12801-12815. DOI: https://doi.org/10.1523/JNEUROSCI.2154-14.2014, PMID: 25232116

Khaw MW, Stevens L, Woodford M. 2017. Discrete adjustment to a changing environment: Experimental evidence. *Journal of Monetary Economics* **91**:88-103. DOI: https://doi.org/10.1016/j.jmoneco.2017.09.001

Khaw MW, Stevens L, Woodford M. 2021. Individual differences in the perception of probability. *PLOS Computational Biology* **17**:e1008871. DOI: https://doi.org/10.1371/journal.pcbi.1008871, PMID: 33793574

Kingma DP, Ba J. 2015. Adam: A Method for Stochastic Optimization. 3rd International Conference on Learning Representations, ICLR 2015.

Klimesch W. 1999. EEG alpha and theta oscillations reflect cognitive and memory performance: a review and analysis. *Brain Research. Brain Research Reviews* **29**:169-195. DOI: https://doi.org/10.1016/s0165-0173(98)00056-3, PMID: 10209231

Klimesch W, Sauseng P, Hanslmayr S. 2007. EEG alpha oscillations: the inhibition-timing hypothesis. *Brain Research Reviews* **53**:63-88. DOI: https://doi.org/10.1016/j.brainresrev.2006.06.003, PMID: 16887192

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LeCun Y, Denker J, Solla S. 1990. Optimal Brain Damage. Advances in Neural Information Processing Systems.

LeCun Y, Bengio Y, Hinton G. 2015. Deep learning. *Nature* **521**:436-444. DOI: https://doi.org/10.1038/nature14539, PMID: 26017442

LeCun Y. 2016. Predictive learning. Proc. Speech NIPS.

Lee TS, Mumford D. 2003. Hierarchical Bayesian inference in the visual cortex. *Journal of the Optical Society of America. A, Optics, Image Science, and Vision* **20**:1434-1448. DOI: https://doi.org/10.1364/josaa.20.001434, PMID: 12868647

Legenstein R, Maass W. 2014. Ensembles of spiking neurons with noise support optimal probabilistic inference in a dynamically changing environment. *PLOS Computational Biology* **10**:e1003859. DOI: https://doi.org/10.1371/journal.pcbi.1003859, PMID: 25340749

Lieder F, Griffiths TL. 2019. Resource-rational analysis: Understanding human cognition as the optimal use of limited computational resources. *The Behavioral and Brain Sciences* **43**:e1. DOI: https://doi.org/10.1017/S0140525X1900061X, PMID: 30714890

Lillicrap TP, Santoro A, Marris L, Akerman CJ, Hinton G. 2020. Backpropagation and the brain. *Nature Reviews. Neuroscience* **21**:335-346. DOI: https://doi.org/10.1038/s41583-020-0277-3, PMID: 32303713

Ma WJ, Beck JM, Latham PE, Pouget A. 2006. Bayesian inference with probabilistic population codes. *Nature Neuroscience* **9**:1432-1438. DOI: https://doi.org/10.1038/nn1790, PMID: 17057707

Ma WJ. 2010. Signal detection theory, uncertainty, and Poisson-like population codes. *Vision Research* **50**:2308-2319. DOI: https://doi.org/10.1016/j.visres.2010.08.035, PMID: 20828581

Ma WJ, Jazayeri M. 2014. Neural coding of uncertainty and probability. *Annual Review of Neuroscience* **37**:205-220. DOI: https://doi.org/10.1146/annurev-neuro-071013-014017, PMID: 25032495

Maheu M, Dehaene S, Meyniel F. 2019. Brain signatures of a multiscale process of sequence learning in humans. *eLife* **8**:e41541. DOI: https://doi.org/10.7554/eLife.41541, PMID: 30714904

Mante V, Sussillo D, Shenoy KV, Newsome WT. 2013. Context-dependent computation by recurrent dynamics in prefrontal cortex. *Nature* **503**:78-84. DOI: https://doi.org/10.1038/nature12742, PMID: 24201281

Masse NY, Yang GR, Song HF, Wang XJ, Freedman DJ. 2019. Circuit mechanisms for the maintenance and manipulation of information in working memory. *Nature Neuroscience* **22**:1159-1167. DOI: https://doi.org/10.1038/s41593-019-0414-3, PMID: 31182866

Mastrogiuseppe F, Ostojic S. 2017. Intrinsically-generated fluctuating activity in excitatory-inhibitory networks. *PLOS Computational Biology* **13**:e1005498. DOI: https://doi.org/10.1371/journal.pcbi.1005498, PMID: 28437436
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