Representation of Multidimensional Stimuli: Quantifying the Most Informative Stimulus Dimension from Neural Responses

Abstract

A common way to assess the function of sensory neurons is to measure the number of spikes produced by individual neurons while systematically varying a given dimension of the stimulus. Such measured tuning curves can then be used to quantify the accuracy of the neural representation of the stimulus dimension under study, which can in turn be related to behavioral performance. However, tuning curves often change shape when other dimensions of the stimulus are varied, reflecting the simultaneous sensitivity of neurons to multiple stimulus features. Here we illustrate how one-dimensional information analyses are misleading in this context, and propose a framework derived from Fisher information that allows the quantification of information carried by neurons in multidimensional stimulus spaces. We use this method to probe the representation of sound localization in auditory neurons of chinchillas and guinea pigs of both sexes, and show how heterogeneous tuning properties contribute to a representation of sound source position that is robust to changes in sound level.SIGNIFICANCE STATEMENT Sensory neurons' responses are typically modulated simultaneously by numerous stimulus properties, which can result in an overestimation of neural acuity with existing one-dimensional neural information transmission measures. To overcome this limitation, we develop new, compact expressions of Fisher information-derived measures that bound the robust encoding of separate stimulus dimensions in the context of multidimensional stimuli. We apply this method to the problem of the representation of sound source location in the face of changes in sound source level by neurons of the auditory midbrain.

Keywords: Fisher information; ILD; binaural hearing; multidimensional; neural coding.

Figure 1.

Information about a single stimulus dimension. a, Sound localization in azimuth: color map with respect to the ear ipsilateral (blue) and contralateral (red) to a sound source. b, Acoustical measurements of the ILD cue to sound location in the horizontal plane in a chinchilla. Data from Benichoux et al. (2016). c, Color-coded sound source azimuth as a function of level at each ear (ipsilateral and contralateral). A sound at the same source location but different intensity will result in a different combination of ipsilateral and contralateral levels.

Figure 2.

Tuning curves and Fisher information. a, Example tuning to ILD of a neuron in the chinchilla ICC at constant ABL = 60 dB. Shaded area is mean ± SEM. Solid curve is a sigmoidal fit to the tuning curve (r² = 0.96). Dashed curve is the one-dimensional FI computed from the tuning curve. b, Same as a for a different ILD-sensitive ICC neuron. c, Tuning curve (blue) and distribution of rates (green and yellow) for two stimulus values. When the distance between the distributions increases (as a result of increased tuning curve steepness), the distributions are more significantly separated. d, Similar to c, if the rate distributions get narrower, the distributions are also more significantly separated. As a result of c and d, the FI is proportional to the slope and inversely proportional to the rate variance.

Figure 3.

Fisher information in multiple dimensions. a, Tuning curves (shaded area), of the neuron presented in Figure 2a as measured at different ABL values (color) alongside sigmoidal fits to the tuning curves (all r² ≥ 0.9). b, Response map of the neuron, color coded rate as a function of ipsilateral and contralateral level. Right: gray shade represents spike rate in hertz (darker gray is higher spike rate). c, SD of the rate at each ILD, measured across ABLs. d, Fisher information measured independently at each ABL value. e–h, same as a–d for the neuron in Figure 2b. i, Blue curves: tuning curves to x at different y values. Shaded area on y-axis represents spike rate distributions. In multiple dimensions, there are now several tuning curves that contribute to the rate distribution at one x value. As a result, the actual distribution of spike rates widens. When the separation of tuning curves at different y values increases, the rate distribution increasingly widens and therefore the information decreases. j, Fisher information computed by taking into account the uncertainty about ABL (i.e., by marginalization) for the neuron of a–d (dashed line) and e–h (solid line). Inset, Color-coded distribution of rates as a function of ILD, where the influence of ILD has been marginalized.

Figure 4.

Directional information in single neurons. a, Intuition of the FIM. Locally around x,y, the response map is approximated as a plane (gray plane). The gradient (green arrow) points in the direction where the rate varies the most and magnitude of maximum information follows along the gradient (blue arrow). The gray arrow points in the direction where the response map does not vary, and therefore information vanishes. b, Magnitude of best information (color-coded) overlaid with direction of maximal information (arrows, length is proportional to the magnitude of information) for the neuron of Figure 2a. c, Same as b for the neuron in Figure 2b. d, Directions of dimensions of interest in the ipsilateral-contralateral level space (clockwise): ILD, contralateral level; ABL, ipsilateral level. Blue curve is a depiction of local marginal FI assuming that the maximal information vector (from the FIM) is along the ILD dimension (blue arrow). e, Each panel is the local marginal information about specific dimensions of the stimulus over the space color-coded over the same range as b. f, Local marginal FI averaged over space (center polar plot, blue curve), with the direction of maximal information (green line). g, h, Same as e, f except for the neuron in c.

Figure 5.

Information across the population. a, Directionality of the population tuning. A population (boxed) comprises two neurons with the same tuning (homogeneous): represented are the FIM vector (colored arrow) and local marginal FI (colored solid line) as a function of direction. The population mFI reflects information in a single direction (black line). b, Same as a for a population with heterogeneous tuning: neurons have different direction and magnitude (boxed). The population has information in all directions, and the principal directions of the FIM (black arrows) span the full stimulus space. c, Response maps of 28 auditory driven neurons recorded in the chinchilla ICC. d, same as c for 21 guinea pig ICC neurons. e, Top, Magnitude and direction of best information obtained from the local marginal FI metric (averaged across space). Each point is a different neuron: green, guinea pig; blue, chinchilla. Bottom, Histogram of best direction for all neurons in each species. f, Local marginal FI as a function of stimulus space direction for all neurons bearing ILD information, and (g) contralateral level information. Colors indicate the species. h, Averaged local marginal FI for all chinchilla (N = 28) ICC neurons (plain blue line) and model symmetrical ICC (dashed blue curve). Arrows represent the principal vectors of the pFIM. Black curve: averaged local marginal FI for the bilateral ICC. i, Same as h for guinea pig ICC neurons (N = 21; colored curve), as well as principal directions and magnitudes of the population FIM.

Figure 6.

Higher dimensional FI. a, Top, A neuron with best information along the x-axis (blue arrow), and its local marginal FI (blue line, arbitrary units). Bottom, the population consisting of one neuron has the same marginal information (gray line). b, A second neuron with best information along the y-axis (green arrow; and its marginal information profile, green line) is added to the population. The population now has information in all (two) stimulus directions, yet reduced in directions bisecting the maximal information of each neuron. c, Same as b with one additional neuron (red arrow and line). d, With a large number of neurons (black arrows) in the population, the information is equal in all directions, as a result the mFI profile is a circle. e, In 3D stimulus spaces, a neuron has information in the z direction (blue arrow). The population mFI is equal and its marginal information profile resembles that of the 2D case (top, blue surface; bottom, gray surface). It is nonzero in all directions except that orthogonal to the direction of best information (x–y, black plane). f, Adding a second neuron (green arrow and line) to the population reduces the dimensionality of the space with zero information to one (bottom, black line). g, A third neuron ensures that information is non-zero in all directions. h, Similar to the one dimensional case, a large population of heterogeneously tuned neurons (top, black arrows) has an equal amount of information in all directions (gray sphere).