Receptive Field Properties of the Macaque Second Somatosensory Cortex: Nonlinear Mechanisms Underlying the Representation of Orientation Within a Finger Pad

Recording methods.

For a detailed description of the recording methods and an illustration of the stimulator that was used to indent the bar into the skin, see Fitzgerald et al. (2004). Briefly, we recorded from single neurons in the SII hand region of four hemispheres of two rhesus monkeys (Macaca mulatta). Thirty-five to 45 d were spent recording in each hemisphere. Single-neuron data were collected while the monkeys performed a visual detection task that maintained them in a nearly constant state of alertness. The monkeys were trained to allow their hands to be restrained during the single-unit recording sessions, because this allowed for accurate and repeatable stimulation of the digits by the motorized tactile stimulator.

Stimulation protocols.

We used two tactile stimulation protocols, protocol 1 and protocol 2, and protocol 2 is the focus of this study. Both protocols used the same motorized, computer-controlled stimulator that could position and indent a bar at arbitrary orientations and locations on the restrained hand. In protocol 1, the stimulator indented a small oriented bar (Ultem plastic) onto the center of the distal, middle, and proximal finger pads of digits two through five (of the hand contralateral to the recorded hemisphere). Stimuli were presented in a pseudorandom order to minimize the time spent traveling between finger pads. After stimulating a pad with a random sequence of two repetitions of each of the eight orientations (22.5° increments), the bar was moved to a randomly chosen neighboring finger pad. This sequence was repeated until each pad had been presented with eight repetitions of each of the eight orientations. Stimulus duration was 500 ms, with an indentation force of 10 g. The bar was approximately the width of a monkey's finger and had rounded ends; its short axis was a 90° wedge, and its long axis was circular with an 8 mm radius, effectively producing a length of ∼7 mm. The responses obtained from this protocol were used to determine the receptive field (RF) size of the neuron and to determine which finger pads were orientation tuned (Fitzgerald et al., 2004, 2006a,b).

Protocol 2 was performed on a single distal finger pad for a neuron that exhibited tuning in the center of that pad during protocol 1. By adjusting the hand holder, the stimulated finger was inclined slightly above the other digits to prevent the oriented bar from touching them. For this protocol, the stimulus bar (Ultem plastic) consisted of a 25-mm-long bar raised on a smooth, flat disk. The ends of the bar and edge of the disk never contacted the finger pad. The short axis of the bar was a 90° wedge, with an apex 500 μm above the disk surface. During each trial, the bar was rotated to one of eight orientations (22.5° increments) and indented into the skin at one of nine positions per orientation (Fig. 1 a). The center position of the nine positions per orientation was the center of the finger pad, and the other four positions on either side were spaced in 1 mm increments orthogonal to their orientation, such that the most offset pair of bars was 4 mm from the center of the pad. Thus, the stimuli spanned the contact patch of the finger pad. The bar was indented 1500 μm into the skin (from the point of contact of the apex of the bar), with a 500 ms period of steady indentation and 50 ms fall and rise times. A 1500 μm indentation was chosen because it allowed the bar to be indented into the uniformly flattened finger pad produced by the disk. The disk produced a slightly elliptically shaped contact patch, whose long axis was aligned with the distal–proximal axis of the finger. The interstimulus interval was 2 s. After stimulating the finger pad with a random sequence of all 72 orientation/position combinations, the stimulator repeated this process four more times (for a total of five samples at each combination), each time moving through all 72 combinations in a different random sequence.

• Download figure
• Open in new tab
• Download powerpoint

Figure 1.

Stimulation protocol and example rasters. a , Stimulation protocol, which stimulated a single distal finger pad with an indented oriented bar. We refer to this as protocol 2. For each stimulus presentation (see Materials and Methods), the bar was rotated to one of eight orientations (22.5° increments) and indented into the skin at one of nine positions per orientation. The center position of the nine positions per orientation was centered within the finger pad, and the other four positions on either side were spaced in 1 mm increments orthogonal to their orientation. b , Example raster (bottom right), position plots (left), and orientation tuning plots (top) for neuron CJ02O_8 that exhibited a divergent vector field that could be explained as a spot of excitation on the distal pad. In the raster, the data are sorted by the eight orientations and nine positions. In the position plots, each curve represents the mean firing rates of one orientation for the nine positions. In the orientation tuning plots, each curve represents the mean firing rates of one position for the eight orientations. c , Example raster, position plots, and orientation tuning plots for neuron CJ001_11 that exhibited an invariant vector field and thereby showed position invariant tuning throughout the distal pad. s/s, Spikes per second.

We studied neurons on the four distal pads of D2–D5, with D5d being the smallest of the four pads and the pad that we least frequently studied; its compressed width was ∼8 mm and its length was ∼10 mm. These dimensions are slightly larger than those of the uncompressed distal pad. We centered the bar at the midpoint between the interphalangeal crease and the distal edge of the pad; the bar was never so proximal as to touch the crease.

Histology.

After the recordings were completed, each monkey was deeply anesthetized and perfused transcardially. Electrode tracks, which had been marked with fluorescent dyes (DiCarlo et al., 1996), were reconstructed in Neurolucida and AutoCAD to confirm that the recordings were made in the SII region. We classified each neuron as belonging to the SII region posterior, central, or anterior field based on the subjective multiunit response properties (Fitzgerald et al., 2004).

Data analysis.

In a previous study (Fitzgerald et al., 2004), we statistically tested for orientation tuning in the center of the 12 finger pads stimulated during protocol 1. Based on the results from this protocol, we selected the distal finger pad that showed the best tuning for additional study using protocol 2.

Vector fields.

For each neuron that was studied using protocol 2, we created a vector field representing the orientation tuning profile of the finger pad. To do so, the firing rates evoked by the five presentations of the bar at a given orientation (θ) measured with respect to the mediolateral axis in the counterclockwise direction and distance (d) from the center of the finger pad were averaged to obtain a sample of the rate function r(θ, d) at the particular θ and d. The samples were spline interpolated along the d dimension to a density of 10 μm increments to obtain a high-resolution rate function, which was then used to estimate the rates corresponding to the eight orientations at 509,131 points on the finger pad. Specifically, the distance d _θ from the center of the finger pad for a bar oriented at an angle θ and passing through a point P(x, y) is as follows: Thus, for each of the eight bar orientations, the corresponding distance d _θ was computed at each point P(x,y) using the above equation, and the rates corresponding to the eight orientations at the point P(x,y) were obtained from r(θ, d).

A vector field was then constructed to quantify the local tuning throughout the pad. A vector V(x,y) was calculated at each point P(x,y), having a magnitude equal to one minus the circular variance (CV) (Fisher, 1993) at that point and a direction equal to the preferred orientation. The circular variance and phase angle of V(x,y) are, respectively, as follows: where r _θ is the rate corresponding to orientation (θ) at the point P(x,y).

Next, we estimated how well the vector field of each neuron matched each of three theoretical vector fields on the finger pad: an invariant (I) field, a divergent (D) field, and a curled (C) field (see supplemental data, available at www.jneurosci.org as supplemental material). The invariant field has all vectors pointing in the same direction and provides a direct measure of position invariant orientation tuning. The divergent field has vectors radiating from an untuned subregion and indicates that a neuron has higher firing rates in this subregion than in the surround. The simplest divergent RF structure is a spot of excitation in the subregion, which “attracts” the vectors. As such, bars placed at the center of the subregion will produce high firing rates but no orientation tuning, and bars centered outside the subregion but which touch it will also elicit high firing rates.

The curled field has vectors that are tangential to an untuned central subregion, and this indicates that a neuron has lower firing rates when the bar is placed in this subregion than in the surround. The simplest curled RF structure is a spot of inhibition in the subregion, which “repels” the vectors. Thus, bars placed at the center of the subregion will produce low firing rates and no orientation tuning, and bars centered outside the subregion but which touch it will also elicit low firing rates.

We performed a least-squares fit to obtain the best fitting version of each of these three theoretical vector fields for a given neuron. The angle between the vector field of the neuron and each of the three best fitting theoretical vector fields, treating each field as a single vector in a higher dimensional space (a vector field spanning N points can be treated as a single vector in 2N-dimensional space), is a measure of the goodness of fit. This angle, which we call the proximity measure, was computed by taking the inverse cosine of the Frobenius norm of the best fitting theoretical vector field divided by the Frobenius norm of the observed vector field of the neuron. Significance of the fit was determined by resampling (with replacement) the rates at each orientation and distance from the center of the pad and repeating the analysis 500 times to obtain a distribution of the proximity measures. A field was considered purely invariant (I) only if the proximity measure to an invariant field was lower than the proximity measure to a divergent or curled field and the overlap between the distributions of proximity measures to an invariant field and proximity measures to divergent and curled fields was not significant (p > 0.05). If the distribution had a significant overlap with the distribution of one of the other test fields, the field was considered mixed (ID, IC, or DC). If there was significant overlap between all of the distributions of the proximity measures for the three fields, the field was termed heterogeneous (H).

Proximity measure.

Let V ϵ C_n * _n be the observed vector field. (Here C is a complex vector field of dimensions nxn.) The matrix V can be written in terms of its real and imaginary components as follows: The vector field V can also be split into two mutually orthogonal components, I _θ and I _{θ + π/2}, such that I _θ is composed of vectors oriented along θ, and I _{θ + π/2} is composed of vectors oriented along θ + π/2. Thus, we have for ∀x,y ϵ [1,n], Thus, V = I _θ and I _{θ + π/2}.

Taking the l ₂ norm, Thus, this system can be represented as a right triangle, with the length of the hypotenuse equaling ‖V‖ and the length of the other two sides equaling ‖I _θ‖ and ‖I _θ+π/2‖, respectively. The angle between the hypotenuse and the side equaling ‖I _θ‖, given as α=cos−1‖Iθ‖‖V‖ , is a measure of how close the observed vector field is to an invariant field oriented at θ, where θ is chosen such that I _{θ+ π/2} is minimized, which in turn minimizes α. If a vector field is untuned, then the vectors will project randomly in every direction, and the distribution of the angles of vectors for the vector field will be uniform and the probability of a vector projecting in any direction will be equal. In other words, ‖I _θ‖ = constant for ∀θ ϵ [0,π], ‖I _θ‖ = ‖I _{θ+ π/2}‖, and α = 45°. For an invariant vector field, when I _{θ+ π/2} is minimized, ‖I _θ‖ = ‖V‖, and α = 0°. As the vector field varies from being invariant to random, α will vary from 0° to 45°.

Proximity to divergent and curled fields.

Instead of splitting the vector field into I _θ and I _{θ+ π/2}, the vector field V can also be split into a divergent field D_xc,_yc , and a curled field C_xc,_yc , with their centers located at (x_c,y_c ). Thus, we have for ∀x,y ϵ [1,n]: and Thus, Also, In this case, α=cos−1‖Dxc,yc‖‖V‖ measures the proximity of the observed vector field to a divergent vector field with center located at (x_c,y_c ). A measure of 0° implies a perfectly divergent field centered around (x_c,y_c ), whereas a measure of 45° implies no divergence. The position (x_c,y_c ) is chosen such that ‖C _{x c},_{y c} ‖² is minimized, thus maximizing α. Similarly, a measure of the proximity of the observed vector field to a curled vector field can be obtained by β=cos−1‖Dxc,yc‖‖V‖ . In this case, ‖D_{x c},_{y c} ‖² is minimized by varying (x_c,y_c ). A measure of 0° implies a perfectly curled field centered around (x_c,y_c ), whereas a measure of 45° implies no curling.

Linear receptive fields.

We also used a linear regression model (DiCarlo et al., 1998) to obtain linear RFs (see Figs. 2, 3) that best explain the observed responses to the oriented bar in protocol 2. Specifically, we defined a 21 × 21 (8 × 8 mm) pixel grid of the distal finger pad, in which each pixel (400 μm square) had an associated weight (w) that defined the effect (excitatory if positive, inhibitory if negative) of that region on the overall firing rate of the neuron. Because we stimulated the finger pad at eight orientations and nine positions per orientation centered on the pad (Fig. 1 a), a central, 16-sided polygonal region of the pad was equally stimulated during the protocol, whereas the edges of the pad were stimulated with fewer orientations/positions. Therefore, we only included a central region of 329 pixels in the regression analysis. For a particular bar stimulus presentation, pixels that were stimulated were assigned a value of one (S = 1), and all other pixels were assigned a value of zero (S = 0). The rate predicted by the regression model can be written as follows: We used spline interpolation to infer the mean rates, obtained from the eight orientations (22.5° increments) and nine positions (1 mm spacings) per orientation, to a density of 24 orientations (7.5° increments) and 801 positions (10 μm spacings) per orientation, thus yielding a system of 19,224 equations (24 orientations × 801 positions) in 329 variables. The linear RF was then the least-squares solution to this system of equations. Note that we explicitly inverted the stimulus matrix, and, as such, the RF that we computed accounts for stimulus autocorrelation. Significance of each weight was determined using a bootstrap analysis. For each position and orientation, the rates were resampled with replacement, spline interpolated, and subjected to the regression analysis 400 times. The weight of a given pixel was considered significant if it had the same sign (excitatory or inhibitory) on >95% (380 of 400) of the iterations.

Goodness of fit was determined using R ², which was computed using the following equation: where r̂_j is the rate predicted by the regression model at the jth stimulus presentation, and r̄ is the mean rate over all the stimulus presentations (24 × 801). Regression analysis at a number of different levels of interpolation of the rate function yielded similar results.

R ² is a measure of the fraction of the total variability that can be attributed to a linear component in the neural response to the oriented bar. The remaining variability corresponds to variability in the neural response (noise) and to the nonlinear component of the neural response.

Nonlinearity.

Matching pursuit regression with an overcomplete dictionary of kernels was used to capture different types of nonlinearity in the neural responses (Friedman and Stuetzle, 1981; Mallat and Zhang, 1993). We defined three classes of kernels. The first class, C _α, corresponded to linear weights for each of the 329 pixels (similar to the linear RFs described above). The second class, C _β, contained 24 weights for the 24 orientations. The third class, C _γ, had 24 × 329 weights, each corresponding to a particular orientation in a pixel. Thus, C _γ captures the local tuning aspect of the response, whereas C _β captures the position invariant tuning aspect of the response. The three classes together represent an overcomplete dictionary of kernels, and, hence, any observed response can be completely explained by this set of kernels. We used an iterative regression method, in which the best fitting kernel was selected at every step. At the end of each iteration, the current rate vector was replaced by the residual component by subtracting the projection along the selected kernel. This ensures that the algorithm at the next iteration attempts to capture the component of the rate vector not already accounted for by the kernels selected up to the current iteration. The regression was computed over a response set that was spline interpolated to a density of 24 orientations and 801 positions, as in the case of linear regression mentioned above. However, unlike in the linear regression, the spline interpolation here was performed over individual samples instead of the mean response at every orientation and position. This yielded five samples of the rate at every orientation and position, which was used to compute the residual sum of squares. This was necessary to terminate the iterative regression before the kernels started capturing variability attributable to intertrial fluctuations. At the end of each iteration, a regression sum of squares was computed and compared with the error sum of squares after correcting for the residual sum of squares. The regression was terminated when the addition of another kernel did not yield a significant improvement in the fit (partial F test, p > 0.05). Typically, the first 200–300 kernels captured >95% of the variability of the response.

If w_i (C) is the ith kernel from class C, and S_j , _i (C) is the projection (1 or 0) of stimulus j on the ith kernel from the class, then the predicted response for the jth stimulus is as follows: We defined the predicted response that was accounted for by each class, by regressing the responses against the kernels from that class. Let r̂_j (C _α) and r̂_j (C _β) be the predicted responses accounted for by the linear and invariant classes, respectively. We then quantified the linearity and position invariant tuning of a neuron by the fraction of variability explained by the kernels from the first two classes, respectively, using the following equations: R _lin ² and R _inv ² quantify, on a scale of 0–1, the degree to which the purely linear model and the purely invariant model explain the data compared with a null model consisting of a uniform response. We defined another quantity, R _local ², to quantify the fraction of the response that was explained by the local tuning kernels, Here r̂_j (C _α, C _β) is the rate predicted by a composite model of linear and invariant weights and is obtained by regressing the response against both the linear and invariant class kernels.