A multivariate method to determine the dimensionality of neural representation from population activity

Abstract

How do populations of neurons represent a variable of interest? The notion of feature spaces is a useful concept to approach this question: According to this model, the activation patterns across a neuronal population are composed of different pattern components. The strength of each of these components varies with one latent feature, which together are the dimensions along which the population represents the variable. Here we propose a new method to determine the number of feature dimensions that best describes the activation patterns. The method is based on Gaussian linear classifiers that use only the first d most important pattern dimensions. Using cross-validation, we can identify the classifier that best matches the dimensionality of the neuronal representation. We test this method on two datasets of motor cortical activation patterns measured with functional magnetic resonance imaging (fMRI), during (i) simultaneous presses of all fingers of a hand at different force levels and (ii) presses of different individual fingers at a single force level. As expected, the new method shows that the representation of force is low-dimensional; the neural activation for different force levels is scaled versions of each other. In comparison, individual finger presses are represented in a full, four-dimensional feature space. The approach can be used to determine an important characteristic of neuronal population codes without knowing the form of the underlying features. It therefore provides a novel tool in the building of quantitative models of neuronal population activity as measured with fMRI or other approaches.

Fig. 1

Latent feature spaces. (A) In the theoretical framework, the observed patterns of neural activity are explained by a set of latent features, each of which is linearly combined with an associated pattern component. The mapping between the experimental conditions (stimuli, movements, tasks, etc.) and the features (dashed line) can be non-linear. (B) Mathematically, each observed pattern (y_k,n) is a linear combination of different pattern components (u_d), each weighted by the corresponding dimension in the latent feature vector (f_k).

Fig. 2

Conditions can be differently arranged in feature space. Shown are the values of four conditions on two feature dimensions. Each feature dimension is associated with a pattern component, which determines the mean activity pattern for each condition. (A) When all four conditions only differ on a single feature dimension, only one dimension in pattern space will be necessary to distinguish them. (B) If the conditions are randomly arranged in feature space, the second pattern component will contribute less to the distinguishability of the associated patterns than the first feature dimension. (C) If the conditions are evenly spaced, then each associated dimensions in pattern space will contribute equal amounts to the distinguishability of the conditions.

Fig. 3

Proportion of accurate classifications for the 1, 2, and 3-dimensional classifiers (x-axis) on simulated datasets generated from a 1- (light gray), 2- (middle gray), or 3-dimensional (thick black line) feature space with random arrangement of the conditions. A simulation with features that were spaced maximally in 3-dimensional feature space is shown as a thin black line. Simulation runs are selected to have the same accuracy on the full classifier (0.58). Error bars indicate the 95% confidence interval when drawing and averaging over 40 independent simulation runs.

Fig. 4

Influence of voxel-selection on dimensionality analysis. Classification accuracy for the 1- to 3-dimensional classifier (x-axis) for simulated data, assuming a 1-, 2- or 3-dimensional representation. Simple voxel-selection (choosing the 15% most informative voxels from an area based on an overall F-test) seriously biases the accuracy curve. Voxel-selection nested within the cross-validation (dashed line) does not have the same adverse effect.

Fig. 5

Variation of activity patterns in primary motor cortex with different force levels (Experiment 1). (A) BOLD-activity patterns in the hand area of the primary motor cortex in a flat representation. Activity patterns for a single individual are shown. The dotted line indicates the fundus of the central sulcus while the bend in sulcus indicates the location of hand knob. Color maps indicate t-values of BOLD activation against rest, for low (left panel), medium (middle two panels) and high (right panel) force levels. (B) Bold-activity (red) averaged within each M1 ROI and averaged over participants, with the error bars indicating between-subject standard error. Activity increases nonlinearly with the level of exerted force. The linear (dashed line) and logarithmic (solid line) fits are shown. (C) Classification accuracy for classifiers of different dimensionality. Average results for 5 subjects are shown (red line). Predicted accuracies based on simulations with randomly spaced 1- to 3-dimensional representations are shown in gray. The simulations are matched to the data to have the same accuracy of the full-dimensional classifier. Error bars indicate between-subject standard error. (D) Average arrangement of activity patterns in an inferred feature space, defined by the 1st and 2nd most discriminating eigenvector. Ellipses indicate standard deviation after alignment of individual patterns using the Procrustes analysis.

Fig. 6

Variation of activity patterns in primary motor cortex for different finger movements (Experiment 2). (A) BOLD-activity patterns in the hand area of the primary motor cortex in a flat representation. Activity patterns for a single individual and hemisphere are shown. The dotted line indicates the fundus of the central sulcus, with the bend in the sulcus indicating the location of the hand knob. Color map indicate t-values of BOLD activation against rest, for digit 1–5 (left to right). (B) Classification accuracy for classifiers of different dimensionality. Average results from 12 hemispheres of 6 participants are shown in red. Predicted accuracies based on simulations with randomly spaced 1- to 4-dimensional representations are shown in gray, and matched for the accuracy of the full-dimensional classifier. Error bars indicate between-subject standard error. (C) Average arrangement of activity patterns in an inferred feature space, defined by the 1st and 2nd most discriminating eigenvector. Ellipses indicate standard deviation after affine alignment of individual patterns.