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The geometry of cortical representations of touch in rodents

Nature Neuroscience volume 26, pages 239–250 (2023)Cite this article

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Abstract

Neurons often encode highly heterogeneous non-linear functions of multiple task variables, a signature of a high-dimensional geometry. We studied the representational geometry in the somatosensory cortex of mice trained to report the curvature of objects touched by their whiskers. High-speed videos of the whiskers revealed that the task can be solved by linearly integrating multiple whisker contacts over time. However, the neural activity in somatosensory cortex reflects non-linear integration of spatio-temporal features of the sensory inputs. Although the responses at first appeared disorganized, we identified an interesting structure in the representational geometry: different whisker contacts are disentangled variables represented in approximately, but not fully, orthogonal subspaces of the neural activity space. This geometry allows linear readouts to perform a broad class of tasks of different complexities without compromising the ability to generalize to novel situations.

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Fig. 1: Different geometries of the neuronal representations and the whisker-based shape discrimination task.
Fig. 2: The whisker-based shape discrimination task can be solved by linearly integrating whisker contacts across time.
Fig. 3: Non-linear classifiers outperform linear classifiers on more complex tasks but not on more complex shapes.
Fig. 4: Populations of neurons in the mouse somatosensory cortex (S1) exhibit non-linear mixed selectivity for task variables.
Fig. 5: Contacts and whisker angular position are the variables that contribute the most to the prediction of S1 population activity.
Fig. 6: Neurons show non-linear mixed selectivity, mostly originating from non-linear interactions between contacts and whisker angular position.
Fig. 7: The geometry of representations observed in S1.
Fig. 8: The geometry of representations in RNNs is modulated by the difficulty of the task.

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Data availability

The datasets analyzed in this study have been deposited to Zenodo: https://doi.org/10.5281/zenodo.4743837.

Code availability

The code used for data acquisition and pre-processing is available at https://github.com/cxrodgers/Rodgers2021. The code used for all analyses in this study is available at https://github.com/ramonnogueira/TheGeometryOfS1.

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Acknowledgements

We would like to thank the members of the Center for Theoretical Neuroscience, Marcus K. Benna and Mattia Rigotti for all their insightful comments and suggestions. Support was provided by NINDS/NIH (R01NS094659, R01NS069679, F32NS096819, and U01NS099726); NSF 1707398 (Neuronex); the Gatsby Charitable Foundation (GAT3708); the Simons Foundation; the Swartz Foundation; Northrop Grumman; a Kavli Institute for Brain Science postdoctoral fellowship (to CR); and a Brain and Behavior Research Foundation Young Investigator Award (to CR).

Author information

Authors and Affiliations

  1. Center for Theoretical Neuroscience, Columbia University, New York, NY, USA

    Ramon Nogueira & Stefano Fusi

  2. Zuckerman Mind Brain Behavior Institute, Columbia University, New York, NY, USA

    Ramon Nogueira, Chris C. Rodgers, Randy M. Bruno & Stefano Fusi

  3. Department of Neuroscience, Columbia University, New York, NY, USA

    Ramon Nogueira, Chris C. Rodgers, Randy M. Bruno & Stefano Fusi

  4. Kavli Institute for Brain Science, Columbia University, New York, NY, USA

    Chris C. Rodgers, Randy M. Bruno & Stefano Fusi

  5. Department of Neurosurgery, Emory University, Atlanta, GA, USA

    Chris C. Rodgers

  6. Department of Physiology, Anatomy and Genetics, University of Oxford, Oxford, UK

    Randy M. Bruno

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  1. Ramon Nogueira
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  2. Chris C. Rodgers
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Contributions

RN and SF conceived the project and the analytic approach. CR developed the behavior, videography and performed the electrophysiological recordings. RN analyzed the data. RN, CR, RB, and SF decided how to interpret the results. RN and SF wrote, and CR and RB edited the manuscript.

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Correspondence to Ramon Nogueira or Stefano Fusi.

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Extended data

Extended Data Fig. 1 Contact rate for convex and concave shapes and for correct and error trials for all mice.

Contact rate (y-axis) as a function of time throughout the trial (x-axis) separately for convex and concave shapes (left), and for correct and error trials (right). Contacts were higher for convex than concave shapes for whisker C1 and C2, whereas whisker C3 showed the opposite trend. Contacts were higher for correct than error trials for all whiskers and animals. (a-j) Results for all mice. (k) Mean results across animals (n = 10 mice).

Extended Data Fig. 2 Linear and non-linear decoders performed similarly for correct and error trials, different stopping locations, and flatter shapes.

(a) A multi-layer feedforward network model is trained to use the full spatio-temporal pattern of contacts and angle of contacts to predict the stimulus and the choice of the animal on a trial-by-trial basis (see Fig. 2). Models were trained using all trials and tested on correct (black) and incorrect (red) trials. Only linear and non-linear models with one hidden layer are shown. Stimulus decoding (left) produced higher decoding performance for correct trials than errors, probably due to the higher number of contacts made by mice on correct trials. On the contrary, correct trials conveyed much more information about animals’ choice than incorrect trials (right). One possible explanation of these effects is that in approximately 60% of the trials animals make very accurate choices that are based on properly sampled sensory cues. In the other 40% of the trials, animals still sample information properly but their choice is inaccurate and based on a hidden variable we do not have access to51,52. (b) Decoding performance (y-axis) for the different decoders (x-axis) for the easy (linearly separable; left panel) and very complex (non-linearly separable, 3D-parity; right panel) tasks (see Methods). Non-linear cue integration is only advantageous when the task itself requires complex sensory integration across time and whiskers. (c) Linear and non-linear classifiers performed equally well on the shape discrimination task from the real spatio-temporal pattern of whisker contacts when conditioned on trials corresponding to far, medium and close stopping locations. Error bars in panels (a-c) correspond to s.e.m. across mice (n = 10). (d) Similar to behavior (see5), flatter shapes were more difficult to discriminate than the standard ones from the real spatio-temporal pattern of whisker contacts. Errorbars correspond to s.e.m. across mice that were presented both standard and flatter shapes (n = 3).

Extended Data Fig. 3 Different feedforward and recurrent architectures were used to decode behavior and fit the S1 encoding models.

(a) Stimulus (green) and choice (blue) were decoded on a trial-by-trial basis from the spatio-temporal pattern of whisking contacts using RNNs with different number of hidden units. On each time step and trial the input to the RNN decoder was the number of contacts and angle of contact for all the whiskers. Errorbars correspond to s.e.m. across mice (n = 10). Feed-forward networks performed in general better than RNNs for decoding behavior (see Fig. 2). (b) S1 population activity was regressed against whisker and task variables (encoding model) using RNNs with different number of hidden units. Errorbars correspond to s.e.m. across neurons (n = 584). See Extended Data Fig. S5a for the distribution of CV R2 across neurons for the linear and the best non-linear (feed-forward with one hidden layer of hundred units) encoding models. Feed-forward networks performed in general better than RNNs on explaining S1 population activity (see Fig. 4). (c) Linear and non-linear feed-forward networks with different number of units (artificial neurons) in the hidden layers are equally good at predicting the presented shape (stimulus; green) and animal’s choice (choice; blue) on a trial-by-trial basis from the spatio-temporal pattern of whisker contacts and angle of contacts (see Fig. 2). Errorbars correspond to s.e.m. across mice (n = 10). (d) The profile of explanatory power for S1 population activity across models is qualitatively equivalent for 20, 100 (see Fig. 4), and 200 units in the hidden layers of the feed-forward encoding models. Errorbars correspond to s.e.m. across neurons (n = 584). See Extended Data Fig. S5a for the distribution of CV R2 across neurons for the linear and the best non-linear (feed-forward with one hidden layer of hundred units) encoding models.

Extended Data Fig. 4 Linear and non-linear classifiers perform equally well on other shapes used in the simulated whisker discrimination task.

(a-e) Linear and non-linear classifiers performed equally well from the spatio-temporal pattern of simulated whisker contacts and angles of contact when different general parameters of the shapes were used in the simulations: more distant stopping locations (a); smaller shapes (b); further shapes (c); closer shapes (d); and flatter shapes (e) (see Methods). (f) Flatter shapes (larger radius) were more difficult to discriminate than more curved shapes on a simulation of the whisker-based discrimination task. (g) Curvature discrimination task: discriminate between curved (green) or flat (orange) shapes (first panel). Non-linear classifiers substantially outperform linear ones on the simulated curvature discrimination task when the full spatio-temporal pattern of contacts and angle of contacts is used to discriminate between flatter (orange) and more curved (green) shapes (second panel). Total number of contacts for the pair (C1, C2) (third panel), (C1, C3) (fourth panel) and (C2, C3) (fifth panel). The boundary between the two categories is non-linear. (h) Example of another simulated non-linear discrimination task (green vs orange bars). Non-linear classifiers substantially outperform linear ones on this task when the full spatio-temporal pattern of contacts and angle of contacts is used. The boundary between the two categories is non-linear, especially for C1 vs C2. Errorbars in all panels correspond to s.e.m. across independent simulations (n = 5).

Extended Data Fig. 5 Mean goodness-of-fit across neurons, correct and incorrect trials, neuronal types, and S1 layers.

(a) Mean goodness-of-fit across neurons (y-axis; R2) for the different encoding models (x-axis) on held-out data (left), and the distribution of R2 across neurons for the non-linear (one hidden layer) (y-axis) and the linear encoding models (x-axis) (right). All models were fit on simultaneously recorded populations. For held-out data, the best model is a feed-forward fully connected network with only one hidden layer (NonLin-1). (b) Mean goodness-of-fit across neurons (y-axis; R2) for the different encoding models (x-axis) on the data used for training. For the train data, the more complex the model (more parameters), the better the prediction. (c) Mean CV R2 (y-axis) across S1 neurons for the different encoding models (x-axis) when evaluated in correct (black) and incorrect trials (red). Encoding models explained S1 activity better on correct than error trials. Errorbars in panels (a-c) correspond to s.e.m. across neurons (n = 584). (d) Mean CV R2 across neurons for the different encoding models (x-axis) on held-out data for all neurons (left), only excitatory (middle) and only inhibitory neurons (right). (e) Mean CV R2 across neurons for the different encoding models on held-out data for neurons across layers for all (top), excitatory (middle) and inhibitory neurons (bottom). (f-g) Encoding fits when using Poisson loss instead of mean squared error (MSE). The y-axis shows the Poisson-loss equivalent of the R2, the Pseudo-R2. The Pseudo-R2 is calculated as 1 - PLoss/Variance, where PLoss is the negative Log-likelihood of the Poisson model. All models were fit on simultaneously recorded populations. Errorbars in all panels correspond to s.e.m. across neurons (all n = 584; excitatory n = 491; inhibitory n = 93; layer 2/3 n = 68; layer 4 n = 157; layer 5 n = 249; layer 6 n = 96).

Extended Data Fig. 6 Contribution of the different regressors to S1 activity across neuronal types, layers, and columns in S1.

(a) Distribution of R2 for the full model (x-axis; R2 Full Model) vs R2 for the different ablations (y-axis; R2 Reduced Model) across all recorded neurons. The R2 for the different ablations is calculated by testing the model on held-out trials where those features (or groups of features) have been set to zero. (b) Mean ΔR2 across all neurons (y-axis; \(\Delta {{{{\rm{R}}}}}^{2}={{{{{\rm{R}}}}}^{2}}_{{{{\rm{Full}}}}}-{{{{{\rm{R}}}}}^{2}}_{{{{\rm{Reduced}}}}}\)) for the different groups of variables. Whisker contacts (Contacts) and continuous angle (Angle) are the most important groups of regressors in predicting S1 activity. Errorbars correspond to s.e.m. across neurons (n = 584). (c) ΔR2 separately for excitatory and inhibitory neurons. Inhibitory populations show a higher ΔR2 because their R2 is overall higher (see Extended Data Fig. S5e). (d) ΔR2 across layers of the somatosensory cortex. Whisker contacts have a stronger effect on superficial layers, while whisker angle has a stronger effect on deeper layers. (e) The metric ΔR2 reveals that neurons in S1 do not strictly obey somatotopy during the whisker-based discrimination task (see also5). The contribution to the encoding model’s performance (y-axis; ΔR2) for the groups of variables associated with whiskers’ contacts (C0, C1, C2 and C3 for all time steps) and angular position (θC0, θC1, θC2, and θC3 for all time steps) for all neurons. (f-h) Whisker and angular position encoding strength for C1 column (b), C2 column (c) and C3 column (d). While C1 contacts is the strongest driver in C1 column, C2 and C3 columns are not dominated by C2 and C3 contacts, respectively. Errorbars in all panels correspond to s.e.m. across neurons (all n = 584; excitatory n = 491; inhibitory n = 93; layer 2/3 n = 68; layer 4 n = 157; layer 5 n = 249; layer 6 n = 96; C1 column n = 117; C2 column n = 208; C3 column n = 204).

Extended Data Fig. 7 Non-linear mixed selectivity across neuronal types and cortical layers for the different groups of task variables.

(a) Different coding scenarios would produce different values for \(\Delta {{{{\rm{R}}}}}^{2}-{\Delta {{{{\rm{R}}}}}^{2}}_{{{{\rm{Linear}}}}}\). Here we show different tuning schemes with respect to C1 and C3 for a fictional neuron (r1). The metric \(\Delta {{{{\rm{R}}}}}^{2}-{\Delta {{{{\rm{R}}}}}^{2}}_{{{{\rm{Linear}}}}}\) was used to evaluate to what extend the pure non-linear terms were important to predict the population’s firing rate. (Left) If the relationship between neuronal activity and encoding variables is linear, \({\Delta {{{{\rm{R}}}}}^{2}}_{{{{\rm{Linear}}}}}\ne 0,\Delta {{{{\rm{R}}}}}^{2}={\Delta {{{{\rm{R}}}}}^{2}}_{{{{\rm{Linear}}}}}\) and therefore \(\Delta {{{{\rm{R}}}}}^{2}-{\Delta {{{{\rm{R}}}}}^{2}}_{{{{\rm{Linear}}}}}=0\). (Middle) If the relationship between neuronal activity and encoding variables is purely non-linear, \({\Delta {{{{\rm{R}}}}}^{2}}_{{{{\rm{Linear}}}}}=0\), ΔR2 ≠ 0, and \(\Delta {{{{\rm{R}}}}}^{2}-{\Delta {{{{\rm{R}}}}}^{2}}_{{{{\rm{Linear}}}}} > 0\). (Right) If the encoding model is composed of both linear and non-linear components, \({\Delta {{{{\rm{R}}}}}^{2}}_{{{{\rm{Linear}}}}}\ne 0\), \(\Delta {{{{\rm{R}}}}}^{2}\ne {\Delta {{{{\rm{R}}}}}^{2}}_{{{{\rm{Linear}}}}}\), and \(\Delta {{{{\rm{R}}}}}^{2}-{\Delta {{{{\rm{R}}}}}^{2}}_{{{{\rm{Linear}}}}} > 0\). (b-c) Pure non-linear mixed selectivity contribution (\(\Delta {{{{\rm{R}}}}}^{2}-{\Delta {{{{\rm{R}}}}}^{2}}_{{{{\rm{Linear}}}}}\)) for the interaction between the different blocks of variables across neuronal types and S1 layers. Results were qualitatively equivalent for the excitatory (a) and the inhibitory (b) populations.

Extended Data Fig. 8 Non-linear mixed selectivity for whisker contacts and angular position for the different time steps and cortical columns in S1.

(a) Pure non-linear mixed selectivity (\(\Delta {{{{\rm{R}}}}}^{2}-{\Delta {{{{\rm{R}}}}}^{2}}_{{{{\rm{Linear}}}}}\)) contribution for the interaction between contacts for the different time steps (time lags) and whiskers separately by columnar location (mean across neurons). The strongest interactions occur at the time lags of 100 ms (1 time step). Even though C1 column shows that C1 terms have the strongest interaction, C2 and C3 columns present a more heterogeneous interaction pattern. (b) Equivalent plot for the interaction between angular position for the different time steps (time lags) and whiskers. The strongest interactions occurs at time lags of 0ms. All columns present strong interactions terms with the rest of whiskers. (c) Pure non-linear mixed selectivity contribution (\(\Delta {{{{\rm{R}}}}}^{2}-{\Delta {{{{\rm{R}}}}}^{2}}_{{{{\rm{Linear}}}}}\)) for the interaction between contacts and angular position for the different time steps (time lags) and whiskers (single regressors in the encoding models) (mean across neurons). The strongest non-linear contribution in whisker angular position occurs on the current time step for all whiskers. The strongest non-linear contribution for the interaction between contacts and angular position occurs on time lags between angular position (and neuronal activity) and contacts of 100 ms (1 time step). (d) Equivalent plot for the interaction between contacts for the different time steps (time lags) and whiskers. The strongest non-linear contribution in whisker contacts occurs on time lags between neuronal activity and contacts of 100 ms (1 time step) for all whiskers.

Extended Data Fig. 9 Different whiskers are represented across S1 columns in approximately orthogonal sub-spaces.

(a) Top: Decoding performance on whether the sum of whisker contacts across the current and previous four time steps corresponded to a high or a low number of contacts with respect to the median for each different whisker (decode high vs low number of contacts for each whisker) (C1 blue; C2 green; C3 red) and columns (see Methods). Information about whisker contacts for all whiskers was present in all columns. Decoding Performance was evaluated on surrogate activity generated by the best encoding model (NonLin-1) for each recording session. In each recording session, activity from only one column was recorded. In order to compare information across columns, surrogate activity was generated for 10 neurons, which corresponded to the smallest number of simultaneously recorded neurons across recording sessions (columns). Bottom: Qualitatively equivalent results were found when the CCGP was evaluated (see Methods). All columns encode information about all whiskers in approximately orthogonal spaces. Errorbars in all panels correspond to s.e.m. across recording sessions (C1 column n = 6; C2 column n = 9; C3 column n = 6). (b) Top: Decoding performance on whether the total number of contacts in a trial (2 sec.) corresponded to a high or a low number of contacts with respect to the median for each different whisker (decode high vs low number of contacts for each whisker) (C1 blue; C2 green; C3 red) and columns using pseudopopulations of neurons (see Methods). All bars are significantly above chance. Bottom: CCGP with respect to the total number of contacts for the different whiskers and columns. Even though there seems to be some somatotopic structure on CCGP for pseudopopulations, the differences are small and all bars are significantly above chance. Errorbars correspond to the standard deviation across cross-validation iterations (n = 10).

Extended Data Fig. 10 The geometry of representations and encoding properties in RNNs change for the easy, the complex, and the very complex tasks.

(a) Probability of correct response (y-axis) as a function of time (x-axis) when the RNN was trained on the easy task 1 (left panel), the complex task 1 (central panel) and the very complex task (right panel). RNNs were trained on low input noise (λlow = 0.23 and λhigh = 0.77) (see Methods). For each network, additional readout weights on the activity of the artificial units were trained to perform the rest of the tasks (solid lines). While an RNN trained on easy task 1 produced the best performance for easy task 1, the neuronal representations were not well suited for the rest of tasks (left). When an RNN was trained on the complex 1 (central) and the very complex (right) tasks, it produced representations that allowed the performance of many different tasks. This came at the expense of losing performance for the easy task 1. Generalization performance, defined as cross-condition generalization performance (CCGP), was also tested for the three easy tasks (dashed lines) when the RNN was trained on the easy task 1 (left panel), the complex task 1 (central panel) and the very complex task (right panel) (see Methods). While abstraction (CCGP) is high for easy task 1 when the RNN was trained on the easy task 1, it is low for easy tasks 2 and 3. Since complex task 1 is defined as the 2D-XOR between C1 and C2, CCGP was higher for easy task 1 and 2 (C1 and C2) than for easy task 3 (C3). For the RNN trained on the very complex task, CCGP was significantly above chance for all easy-task variables. (b-c) The same qualitative results were obtained when medium (b; λlow = 0.3 and λhigh = 0.7) and high (c; λlow = 0.35 and λhigh = 0.65) noise levels were used instead. For all panels in (a-c) the performance curves correspond to the mean across random realizations of input patterns and tasks (n = 50) (see Methods). (d) Similar to Fig. 2d, linear and non-linear classification models that read out the input (x-axis), were trained to perform the easy (left panel; easy task 1), the complex (central panel; complex task 1) and the very complex tasks (right panel). On the easy task, both linear and non-linear classifiers performed equally well, as shown by decoding performances (y-axis) of the different models. On the contrary, only non-linear classifiers that allow for complex cue combination, performed above chance on the complex (central) and very complex (right) tasks. The behavioral results obtained on the whisker-based discrimination task (see Fig. 2) are aligned with the easy task (left panel). In all panels errorbars correspond to the s.e.m. across different network realizations (n = 4). (e) Similar to Fig. 4d, CV R2 on explaining artificial units activity (y-axis), is plotted against the different encoding models (x-axis). RNNs trained to perform tasks that require non-linear integration of sensory cues are better explained by an encoding model that allows for non-linear mixed selectivity (central and right panels), while a non-linear encoding scheme provide marginal additional explanatory power for the easy task RNN (left panel). The higher the complexity of the trained task, the higher the advantage of non-linear encoding models on explaining the activity of the artificial units. In contrast to (d), the encoding properties of S1 are aligned with those of RNNs trained to perform tasks that require non-linear combination of sensory evidence (see Fig. 4). Errorbars correspond to the s.e.m. across artificial units (n = 240; 4 networks × 60 units).

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Nogueira, R., Rodgers, C.C., Bruno, R.M. et al. The geometry of cortical representations of touch in rodents. Nat Neurosci 26, 239–250 (2023). https://doi.org/10.1038/s41593-022-01237-9

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