Fig 1.
(A) Task design. In the experiments, monkeys chose between different juices offered in variable amounts. The two juices were labeled A and B, with A preferred. Offers were presented as visual stimuli on a computer monitor. Different juice types were associated with different colors, and the number of squares represented the juice quantity. After a randomly variable delay, the animal indicated its choice with an eye movement. (B) Example offer value A response. In this panel, the x-axis represents different offer types ranked by the ratio qB/qA, where qJ is the quantity of juice J offered. For each offer type, a black dot indicates the percent of trials in which the animal chose juice B (y-axis on the right). The relative value of the two juices (ρ) was obtained from a logistic fit. For this session, we measured ρ = 1.9. Red symbols illustrate one neuronal response. Diamonds and circle refer to trials in which the animal chose juice A and juice B, respectively. Vertical error bars indicate SEM. The activity of this cell increased almost linearly with the quantity of juice A offered, and did not depend on the quantity of juice B offered. (C) Example offer value B response. In this case, the response increased with the quantity of juice B offered, independently of juice A. (D) Example chosen juice B response. This response was nearly binary–high when the animal chose juice B and low when the animal chose juice A, independently of the quantity. (E) Example chosen value response. This response increased with the value of the chosen option, independently of the chosen juice. For chosen juice response, negative encoding for one juice is indistinguishable from positive encoding of the other juice (with this task design). Conventions in panels (C)-(E) are as in panel (B).
Fig 2.
Illustration of the procedure for category discovery.
For the original rates (left), each axis of the space denotes the firing rate of the cells in the various trial types. Each data point in this space represents a cell. For illustration, we present only 3 of the 9 dimensions. The original rates are then centered and normalized to unit length. The normalization effectively moves the points to the surface of a hyper-sphere. The points are then clustered using spherical k-means for a given number of clusters and centroid clustering for given variables. In the illustration, variables are represented as larger points. The resulting partitions are compared using the adjusted mutual information measure as a function of the number of clusters and number of variables (right).
Table 1.
Considered list of variables that are potentially encoded by the population.
Fig 3.
Two examples of how the ΔR2 metrics can fail.
(A,B) A dip in the distribution of ΔR2 does not necessarily imply categorical encoding. The clustering algorithm yields two clusters. However, the analyst might have erroneously concluded that there are three variables, including two variables located in the tips of the banana cloud (red and black). The dip in the ΔR2 histogram suggests that these two variables are encoded by categorically distinct populations, but this is in fact not the case. (C,D) Categorical encoding does not always result in a dip in the distribution of ΔR2. In this case, we assume that the analyst correctly concluded that there are two variables, but might have defined these variables such that one is on the north pole (gray) and the other is on the east end of the banana cluster (black). Inspection of the ΔR2 histogram does not reveal any dip. The reason is that data points on the west end of the banana cluster are equally far from the two variables.
Fig 4.
Silhouette comparison of clustering algorithms on synthetic categorical data.
Synthetic data consist of firing rates from a total of 400 simulated cells representing the variables chosen value, offer value A, offer value B and chosen juice (100 cells each). Independent Gaussian noise with a standard deviation of 0.25 and a mean given by the variable rates was used to simulate the activity of a cell. Each color corresponds to one cluster. Clustering algorithms were Mini-Batch k-means (A), Spectral Clustering (B), Ward (C), Agglomerative Clustering (D), Birch (E) and Spherical k-means (F).The number of clusters was fixed to 4.
Fig 5.
Silhouette comparison of spherical k-means clustering for different numbers of clusters on synthetic data.
Synthetic data were either categorical (top row) simulated like in Fig 3 or non-categorical consisting of 400 samples uniformly distributed over the unit hyper-sphere (bottom row). For each data set, the number of clusters was varied between 2 and 8 (A-G) and (H-N) respectively. Each color corresponds to one cluster.
Fig 6.
Comparison of different cluster similarity measures for spherical k-means partitions on synthetic data.
Data were either categorical (top row) or non-categorical (bottom row) and simulated like in Figs 3 and 4. The similarity measure was either mutual information (A, D), normalized mutual information (B, E) or adjusted mutual information (C, F). The gray scale indicates the strength of similarity for given number of clusters and number of variables. Corresponding numbers of clusters and numbers of variables are marked in red.
Fig 7.
Cluster results for real data recorded from macaque orbitofrontal cortex limited to the post-juice time window.
(A)-(G) Silhouette plots for the spherical k-means partitions of one example pool (pool 1). Each color corresponds to one cluster. The number of clusters was varied between 2 (A) and 8 (G). (H) Adjusted mutual information cluster similarity between spherical k-means clustering and variable-based centroid clustering as a function of the number of clusters and number of variables over all pools. Corresponding numbers of clusters and numbers of variables are marked in red. (I) Maximum adjusted mutual information for each number of clusters where each bar shows the result of one Jackknife fold.
Fig 8.
Visualization of four post-juice clusters in the 9-dimensional trial type space.
Each color corresponds to one cluster. Each panel shows the centered and normalized firing rates of a pair of trial types and each point in a panel represents a cell from pool 1. Cluster centers are marked with black circles.
Fig 9.
Tuning curves of post-juice response prototypes as defined by 8 cluster centers.
The x-axis represents offer types ranked by the ratio #B:#A. The y-axis in red represents normalized response rates of pool 1. The y-axis in black shows monkey behavior. Encoded variables are denoted in the panel titles. Red diamonds represent the responses to chosen juice A whereas red dots represent the responses to chosen juice B. The separate red diamond and red dot show forced choices.
Table 2.
Selected variables.