## Abstract

The present study asks in what way the activity of a neuronal population responding to a sensory stimulus could be most efficiently decoded, or ‘read off’, by the target neurons. A simple solution to this problem has been proposed – pooling the activity of responding neurons. However, pooling can be inefficient if sensory information is encoded by the ‘label’ of each neuron firing a spike. We have tested the efficiency of pooling by quantifying the extent to which information about a sensory stimulus is diminished when the identity of the individual neurons is lost by pooling. Analyzing the response of small groups of neurons in rat barrel cortex to single-whisker deflection, we found that pooling neurons within the same column is efficient for representing stimulus position; it causes a loss of only 1% of the information about whether the principal whisker was stimulated, and a loss of 5–12% of the finer information about which of nine possible whiskers (the principal and its neighbors) was stimulated. Cross-column pooling led to larger information losses, in the range of 25–55%. Thus, to decode stimulus position from the discharge of barrel cortex populations, ‘downstream’ neurons could pool the activity arising from neurons of the same column, while maintaining the activity arising from neurons of separate columns at least partially segregated. Since such parcellation is present in some of the projections from barrel cortex, these findings suggest that columnar organization of barrel cortex serves to facilitate decoding of the location of the stimulated whisker.

## Introduction

Most studies of neural coding have addressed the problem of how neurons encode sensory events. However, the brain also faces the complementary problem — that of decoding: given the activity in a certain population of cells, what was the sensory event that evoked it? In practice, decoding must occur for the successful transmission of information from one group of neurons to a second group. Consider a target population which receives input from a set of neurons with widely disparate tuning properties. To conserve all sensory information during synaptic transmission, the target neuron must conserve the ‘label’ of the spikes arriving from multiple input neurons at different places on its dendritic tree.

A simple solution to the decoding problem has been proposed, namely pooling (Darian-Smith et al., 1973; Shadlen et al., 1996). The idea is that target neurons simply sum up, or ‘pool’, the activity of the afferent population. Pooling can lead to a significant improvement in signal reliability, provided that the neurons being pooled are at most weakly cross-correlated (Darian-Smith et al., 1973; Zohary et al., 1994). However, it is sub-optimal if the pooled neurons have diverse tuning properties.

Cortical neurons with similar (but not identical) stimulus selectivity are generally organized in columns that span the whole cortical depth (Mountcastle, 1997). If the differences in the response properties of neurons located within the same column are not essential for representing salient stimulus features, then columnar organization could act as a framework to facilitate pooling with minimal information loss. To support this hypothesis, it is important to establish exactly how much information would be lost if the activity originating in a single column were pooled by its target population. Then, it is important to look at the effect of pooling across multiple cortical columns, in order to constrain the spatial scale across which neurons could be grouped before causing substantial loss of sensory information.

In this paper we used an information-theoretic formalism to study how groups of simultaneously recorded neurons, distributed within one or more barrel-columns of rat barrel cortex, code for stimulus position. We compared the information transmitted by the two decoding schemes: ‘labeled line’ and ‘pooling’ (Reich et al., 2001). According to labeled line decoding, downstream neurons maintain the identity of the neuron from which each spike arises; according to pooled decoding, all spikes are assumed to have an equal effect on the postsynaptic neuron. We used a mathematical approach (Panzeri and Schultz, 2001) that makes explicit what aspect of population spike trains — independent spikes or correlated firing — underlie the information transmitted by labeled line and pooling schemes.

Our experimental paradigm was the decoding of stimulus location in the primary somatosensory cortex of the rat. For several reasons, this is an excellent model for studying the role of columnar organization in cortical coding. First, in this system it is possible to obtain accurate estimates of the information conveyed by small populations of neurons (Panzeri et al., 2001; Petersen et al., 2001). Second, the barrel cortex offers the opportunity to relate neuronal population encoding and decoding to the anatomically (Woolsey and Van der Loos, 1970) and functionally (Simons, 1978) defined columnar structure of this cortical area. Finally, the selected sensory feature, the location of the stimulated whisker, has true behavioral significance (Harris et al., 1999) — under natural conditions rats must solve the very same decoding problem that we have posed here.

Our results suggest that pooling is an effective strategy for decoding stimulus location when restricted to individual cortical columns: all the information conveyed about principal whisker stimulation is preserved by the operation of within-column pooling. In contrast, across-column pooling leads to information loss. Thus cells in rat somatosensory cortex may convey information about the site of whisker stimulation precisely by virtue of the column in which they are located.

## Materials and Methods

### Electrophysiology

Methodology has been described in detail previously (Lebedev et al., 2000). All procedures conformed to NIH and international standards concerning the use of experimental animals. Adult male Wistar rats (n = 22) were anesthetized with urethane (1.5 g/kg body wt, i.p.) and the left somatosensory cortex exposed. An array of six tungsten electrodes, configured either as a row or a 2 × 3 matrix, with 300 ± 50 mm horizontal separation between adjacent electrode tips was advanced into the cortical barrel field, centered on barrel-column D2. Since the typical diameter of a barrel-column is 300–400 mm (Rice, 1995), in a typical experiment one or two electrodes (more rarely three) penetrated any single barrel-column under the array. However, only groups of neurons where each cell was recorded at a different electrode were considered for this study. The great majority of neurons were likely to have been located between 300 and 950 mm, the depths at which thalamic axons from the ventral posterior medial nucleus terminate (Lu and Lin, 1993). The columnar location of electrode penetration sites were identified relative to histological sections.

Neuronal activity was amplified and band-pass filtered in the range 300–7500 Hz. Action potentials were digitized at 25 kHz, 32 points per waveform, and time-stamped with 0.1 ms precision (Datawave, Boulder, CO). Off-line, single-unit action potentials were discriminated by differences in shape and amplitude.

Individual whiskers were stimulated 3 mm from their base by a piezoelectric wafer (Morgan Matroc, Bedford, OH). Vibrissae C1, C2, C3, D1, D2, D3, E1, E2 and E3 were deflected one at a time in order to study how populations of cortical neurons encode stimulus location. The stimulus was an up–down step function of 80 mm amplitude and 100 ms duration, delivered once per second, 50 times for each vibrissa.

We analyzed 171 pairs of simultaneously recorded cells. In 52 cases, both neurons in the pair were located in barrel-column D2. In other cases, the two neurons in the pair were located in different barrel-columns in the same barrel row: 80 D1–D2 pairs and 39 D1–D3 pairs. In addition, we analyzed 77 triplets of simultaneously recorded cells: eight triplets in which all three neurons were located in D2, and 69 triplets in which the neurons were located in D1, D2 and D3.

### Data Analysis

To calculate neuronal responses, we considered a time window 0–T ms after the onset of whisker deflection. We divided the spike train of each cell into bins of size Δt. Each possible spike train of each cell hence became a ‘word’ of length L = Tt. To consider the effect of pooling over a given set of cells, the responses in corresponding bins were summed on a trial-by-trial basis. All results reported here use a time bin size Δt of 10 ms.

In order to quantify how units code stimulus location, we applied information theoretic methods. Mutual information (Shannon, 1948) quantifies how well an ideal observer of neuronal responses (R) can discriminate between all the different stimulus locations (S), based on a single response trial:

(1)
$I(S;R)=\left\langle{{\sum}_{S}}P(n{\vert}s)\mathrm{log_{2}}\frac{P(n{\vert}s)}{P(n)}\right\rangle_{S}$
where P(n\s) is the conditional probability of observing a neuronal response n given a deflection of whisker s, P(n) is the unconditional probability of response n [the average of P(n\s) across stimuli], and <…>s denotes an average across stimuli weighted by the probability P(s) of deflecting whisker s. In general n is the ensemble of spike sequences across all neurons in the population.

Mutual information quantifies the amount of information about the stimulus available in the neuronal population response, thereby placing an upper bound on the performance of any decoder (Cover and Thomas, 1991). Thus, the analysis presented in this study quantifies the performance of ideal labeled line decoders and ideal pooling decoders

In general, it is difficult to estimate the information conveyed by neuronal populations, due to sampling bias (Panzeri and Treves, 1996). However, the fact that barrel cortical neurons emit very few spikes per stimulus (typically 0–2) substantially reduces this problem. We showed previously (Panzeri et al., 2001; Petersen et al., 2001) that it is possible to reliably estimate the mutual information conveyed by the responses of pairs of such neurons using the series expansion method (Panzeri et al., 1999; Panzeri and Schultz, 2001). This method approximates the mutual information by a second-order power series expansion in the time window T, and is accurate providing that the number of spikes in each bin (averaged across stimuli and trials) is small and spike times are not correlated with infinite timing precision. The approximated information estimates depend only on post-stimulus time histograms (PSTHs) and pairwise correlations between spikes at different times.

Our previous work (Panzeri and Schultz, 2001) made the approximation that at most one spike occurs in any given bin, a safe assumption for the labeled line analysis given the low firing rates observed. In fact, by comparison with the new method reported below, we checked that the one-spike-per-bin approximation leads, for the labeled line response, to an error on the information estimate which is <0.3% of the total information (averaged across all analyzed cell pairs). However, it might lead to a significant underestimate of the information conveyed by pooling. Hence we generalized the series expansion formalism to allow for multiple spikes per bin — the equations are given below.

An important feature of the series expansion is that it separates the contribution of individual spikes (IPSTH) from that of correlated spike patterns (Icorr):

(2)
$I=I_{\mathrm{PSTH}}+I_{\mathrm{corr}}$

The series expansion approximation for the labeled line code is as follows (Panzeri and Schultz, 2001). The first term, IPSTH, quantifies the contribution of individual spikes. If spike times are independent given the time-dependent firing rate, then the PSTH is a complete description of the response. The contribution of PSTH stimulus-modulations to information transmission was computed as follows (Panzeri and Schultz 2001):

(3)
\begin{eqnarray*}&&I_{\mathrm{PSTH}}={{\sum}_{a,t}}\left\{\left\langle{\bar{n}}_{a}(t{\vert}s)\mathrm{log_{2}}\left[{\bar{n}}_{a}(t{\vert}s)\right]\right\rangle_{s}{-}\left\langle{\bar{n}}_{a}(t{\vert}s)\right\rangle_{s}\mathrm{log_{2}}\left\langle{\bar{n}}_{a}(t{\vert}s)\right\rangle_{s}\right\}\\&&+\frac{\mathrm{1}}{\mathrm{2ln(2)}}{{\sum}_{a,b,t_{\mathrm{1}},t_{\mathrm{2}}}}\left\{\left\langle{\bar{n}}_{a}(t_{1}{\vert}s){\bar{n}}_{b}(t_{2}{\vert}s)\right\rangle_{s}\left[\mathrm{1}+\mathrm{ln}\frac{\left\langle{\bar{n}}_{a}(t_{1}{\vert}s)\right\rangle_{s}\left\langle{\bar{n}}_{b}(t_{2}{\vert}s)\right\rangle}{\left\langle{\bar{n}}_{a}(t_{1}{\vert}s){\bar{n}}_{b}(t_{2}{\vert}s)\right\rangle}\right]\right.\ \\&&\left.\ {-}\left\langle{\bar{n}}_{a}(t_{1}{\vert}s)\right\rangle_{s}\left\langle{\bar{n}}_{b}(t_{2}{\vert}s)\right\rangle_{s}\right\}\end{eqnarray*}
where na(t\s) is the number of spikes in time bin t of cell a to stimulus s on a particular trial. The bar …̅ denotes an average over trials; thus a(t\s) is the corresponding PSTH.

The other part of the series expansion equation, Icorr, quantifies the information contributed by the presence of correlated patterns of spikes (either within- or across-cell):

(4)
In the above equation, cab(t1,t2) is the joint PSTH of bin t1 of cell a and bin t2 of cell b given stimulus s (either combinations a = b or ab are possible). It is equal to )̅, unless a = b and t1= t2. caa(t1,t1\s) measures the probability of more than one spike occurring in time bin t1 of cell a, and is computed as (t1\s)−(t1\s). [This is the only difference in the equations compared to the case of a maximum of one spike per bin, in which case caa(t1,t1/s) is simply zero.] The form of caa(t1,t1\s) in the ‘multiple spike per bin’ case can be derived by observing that considering more than one spike per time bin becomes equivalent to considering a spike count code in each particular bin, and making use of the expression of the auto-correlation terms of the series expansion for the spike count case, reported by Panzeri et al (Panzeri et al., 1999).
$$ce_{ab}(t_{\mathrm{1}},t_{\mathrm{2}}{\vert}s)={\bar{n}}_{a}(t_{\mathrm{1}}{\vert}s){\bar{n}}_{b}(t_{\mathrm{2}}{\vert}s)$$
is the expected value of cab(t1,t2\s) in the case that the spikes are statistically independent. Note that the relative contribution of within-neuron and cross-neuron spike patterns can be assessed simply by considering separately the a = b and ab components in the equations.

The information conveyed by a pooled population code was estimated in the following way. On each trial, the pooled response N(t\s), which is the number of spikes occurring in time bin t to presentation of stimulus s, summed across all neurons in the population, was computed. The PSTH of the pooled response (t\s) was obtained, as before, by averaging the pooled single-trial response N(t\s) over presentations of stimulus s. The correlational component of the information was then computed from the pair-wise spike timing correlations of the pooled spike train: if t1t2,

$$C(t_{\mathrm{1}},t_{\mathrm{2}}{\vert}s)\ =\ \overline{N(t_{\mathrm{1}}{\vert}s)N(t_{\mathrm{2}}{\vert}s)}$$
is the joint PSTH of bin t1 and bin t2 of the pooled spike train, given stimulus s. C(t1,t1\s) is given by
$${\bar{N}}^{2}(t_{\mathrm{1}}{\vert}s){-}{\bar{N}}(t_{\mathrm{1}}{\vert}s)$$
, and Ce(t1,t2\s) is the expected value of C(t1,t2\s) in the case that the spike times are statistically independent.

The series expansion equations for the pooled code were then evaluated exactly as above, with the only differences that (i) the quantities , c and ce were replaced by the corresponding quantities for the pooled spike train: , C and Ce, and (ii) in the pooled series expansion equations there was no sum over cells (a,b) because pooling collapsed the responses of all the cells into a single spike train.

The results reported in this paper use a time bin size Δt of 10 ms in all cases, and a post-stimulus time window of 0–40 ms pairs and of 0–30 ms for triplets. The series expansion information analysis must be applied to triplets for a shorter time period for sampling reasons. In fact, a rule of thumb for effective removal of finite sampling biases with the series expansion is that the number of trials per stimulus (50 in this case) should be roughly twice as large as the number of pairwise combinations of time bins (Panzeri and Schultz, 2001). This series expansion has two components: an auto-correlation component from within single cells, and a cross-correlation component from across pairs of cells. Auto-correlation contributes CL(L – 1)/2 responses, cross-correlation L2C(C – 1)/2 responses, where L is the number of time bins and C the number of cells. A simple calculation shows that, given these constraints and the number of trials available, the number of time bins that can be reliably analyzed is up to four for pairs and up to three for triplets.

We performed several analyses to verify that the series expansion approximation was accurate for triplets [the checks performed for single cells and pairs have been reported previously (Panzeri et al., 2001; Petersen et al., 2001)]. Results are reported only for D2–D2–D2 triplets, but are similar also for the cross-columnar case. (i) We estimated the spike count information for each triplet, both by ‘brute force’ directly from the mutual information definition, equation (1), and using the series expansion. For the post-stimulus time window of 0–10 ms, these estimates differed by just 4.5% averaged across all triplets. (ii) We compared response entropy in spike counts, estimated using the series expansion, to that estimated by ‘brute force’ from the response probabilities. For the 0–30 ms post-stimulus window, these estimates differed by just 1.5% averaged across all triplets. (iii) We compared response entropy in the spike times, estimated using the series expansion, to that estimated by ‘brute force’. For the 0–10 ms post-stimulus window, these estimates differed by just 2% averaged across all triplets. Collectively, these results indicate that the series expansion approximation is accurate under the conditions of our dataset.

## Results

### Decoding the Information About Nine Whiskers

Analyses were carried out to find out how information might be decoded from the activity of groups of neurons responding to stimulation of nine individual whiskers. The question was how neuronal activity specifies stimulus site among these nine whiskers (C1, C2, C3, D1, D2, D3, E1, E2, E3). To elucidate what aspects of neuronal population firing are important for decoding the location of the stimulated whisker, we computed how much information the spike emission times of neuronal groups transmitted according to two decoding schemes: (i) labeled line, which conserved the identity of which neuron fired each spike, and (ii) pooling, which ignored which neuron fired each spike. We quantified the efficiency of pooling as a decoding mechanism by computing the difference between the labeled line and the pooled information. Since labeled line decoding is an upper bound on the information transmitted by a neuronal population, this difference must be positive or zero (Cover and Thomas, 1991). This is analogous to the way in which Nirenberg and co-workers (Nirenberg et al., 2001) recently assessed the information cost of neglecting cross-correlations in the retina.

Results for a typical pair of cells recorded from barrel column D2 are illustrated in Figure 1a. The left panel shows the times at which spikes occurred for each of 50 deflections of whiskers D1, D2 and D3 (here and in the following, stimulus onset was defined as time = 0 ms). Both cells responded strongly and rapidly to the principal whisker D2, but weakly and with longer latency to non-principal whiskers D1 and D3. This observation is consistent with the well-known functional properties of barrel cortex (Armstrong-James and Fox, 1987). Since the individual cells have similar response properties, the pooled response maintains these characteristics. Pooled decoding was thus only marginally less informative than labeled decoding for the nine-whisker set.

We also computed information transmission for populations of simultaneously recorded triplets of neurons. For D2–D2–D2 triplets, the information lost in pooling was 11 ± 2%, 14 ± 3% and 12 ± 5% at 10, 20 and 30 ms respectively. These values, for each time window considered, were slightly higher than those obtained for the corresponding time windows of D2–D2 pairs, but the difference did not reach statistical significance (t-test, P > 0.3).

For D1–D2–D3 triplets, the information lost in pooling was very large: 43 ± 1% at 0–30 ms. Loss of information due to pooling was thus higher for triplets than for pairs. However, consistent with the results for cell pairs, there was markedly less information loss for within-column pooling compared to crosscolumn pooling.

### Within-column Pooling: Effect on Stimulus-specific Information

Is the loss of information due to pooling specific to particular elements of the stimulus set, or does it reflect an overall degradation of the information conveyed by the population? We addressed this question by estimating the information that neurons convey about whether the stimulus site was, or was not, the selected whisker, (for example, D2 versus ‘not D2’). For this analysis, all other eight whiskers were considered to be in the same category. By computing this quantity selecting one whisker at a time, we obtained a ‘whisker-specific’ information function for both labeled line and pooled decoding mechanisms (Fig. 3). For neuron pairs located in the barrel column D2 (Fig. 3; left panel), the most reliably encoded whisker was the principal whisker, D2, for both decoding procedures. At 40 ms poststimulus the information about stimulation of whisker D2 was 0.185 ± 0.012 bits, which accounted for 65% of the total information about all nine whiskers. Neuronal pairs located in the D2 barrel-column also conveyed some information about non-principal whiskers, consistent with previous physiological studies (Petersen and Diamond, 2000). However, the specific information for non-principal whiskers was at least six times smaller than that for the principal one. In the context of stimulus localization, the neuronal population within a given barrelcolumn appears to report mainly whether the principal whisker was stimulated or not.

Next, we compared these results to those obtained with pooling. Pooling did not lead to any statistically significant loss about whether the stimulus was D2 or not (1 ± 1%; P > 0.95; one-way ANOVA ). Hence, information about whether or not the principal whisker was stimulated is extremely robust to pooling. In contrast, pooling led to substantial loss of information specific to the surround whiskers. The loss of whisker-specific information was highest for surround whiskers — particularly those located in the same row as the principal one. For neuron pairs located in barrel-column D2, pooling caused 32 ± 7% loss of information specific to whisker D1 and 27 ± 15% loss specific to D3 (Fig. 3, left panel).

We repeated the same analysis for triplets of neurons located in D2. We found again, that the information about the principal whisker that was lost in pooling these triplets was negligible: 1 ± 4% (0–30 ms post-stimulus; P > 0.95; one way ANOVA).

### What Features of Population Activity Are Lost in Pooling?

Pooling has three effects. First, if cells differ in their pattern of PSTH profiles across stimuli, pooling will blur informative differences and hence cause information loss. Second, within-neuron phenomena like refractoriness can lead to informative structure in auto-correlations — this will also be degraded by pooling. Third, since the origin of each spike is lost, cross-correlations (and any information they might convey) no longer exist. Since these three mechanisms correspond to separate terms of the series expansion, we were able to consider the effect of pooling on correlation-dependent information. Previous work on labeled line and pooled decoding schemes (Reich et al., 2001) considered mainly the first mechanism. Results are shown, averaged across all pairs of cells, in Figure 4. For cell pairs within the same column, the information loss due to pooling was nearly entirely attributable to the individual spike component of the code. The results were similar for pairs located in different columns: for D1–D2 pairs individual spikes accounted for 92 ± 9% of total pooling loss; for D1–D3 pairs the result was 83 ± 12%. These results are consistent with our previous report that the fundamental coding unit in this system is the individual spike (Panzeri et al., 2001; Petersen et al., 2001).

The total amount of individual spike information lost due to pooling depended on the relative locations of the neurons. At 40 ms post-stimulus, the loss was 0.03 ± 0.01 bits for D2-D2 pairs, 0.08 ± 0.02 bits for D1–D2 pairs and 0.13 ± 0.03 bits for D1–D3 pairs. This finding can be explained by the fact that neurons in different columns differ more in their PSTH profiles across stimuli than do neurons within the same column. Hence pooling destroys more PSTH structure for cross-column neuron pairs than for within-column neuron pairs.

Although small compared to the role of individual spikes, a significant (15%) contribution to the information conveyed by barrel cortex populations about stimulus location comes from within-cell spike patterns, provided that neuronal ‘labels’ are conserved (Petersen et al., 2001). This effect is due to a negative dip in the auto-correlation, perhaps reflecting refractoriness or inhibitory feedback (Berry and Meister, 1998) (Petersen et al. give a detailed examination (Petersen et al., 2001)]. Figure 4 shows that for D2–D2 pairs part of this within-cell pattern information is lost by pooling. This type of information loss was smaller than that due to pooling independent spikes in all cases. Cross-columnar pooling led to a greater loss of within-cell spike pattern information. This was because negative auto-correlations are particularly effective in carrying information when the PSTHs to different stimuli are diverse (Oram et al., 1998; Panzeri and Schultz, 2001).

For coding of stimulus location, there was very little information contributed by cross-cell spike patterns (Fig. 4) (Petersen et al., 2001). Hence, there was no information present that could be lost by pooling.

## Discussion

Can ‘downstream’ neurons decode the messages in afferent spike trains simply by pooling over inputs, or is it also necessary to take into account the identity of the afferent neurons? A recent study addressed this issue for populations of 2–6 nearby neurons in primary visual cortex of anesthetized macaques (Reich et al., 2001). The result was that neuron identity (‘labeled line’) conveyed information about the nature of the visual stimuli, and that pooling led to loss of information. The aim of our study was to explore this issue in a different neural system and to address three further questions. (i) Is it truly neuronal identity that matters, or is the critical label the columnar identity in which the neuron resides? (ii) Is all sensory information equally degraded by pooling, or are certain aspects robust to pooling? (iii) What features of the population code are destroyed, or conserved, by pooling?

With regard to the first question, our analyses revealed that the loss of information about stimulus location caused by within-column pooling is much smaller than that caused by cross-column pooling. This indicates that afferent spikes could be ‘labeled’ according to columnar identity (sacrificing neuronal label) with little loss in information about the stimulus site. With regard to the second question, the result was that pooling conserves essentially all the information that same-column neuronal populations convey about the stimulation of their principal whisker. The information loss was related exclusively to non-principal whiskers. If, for this stimulus set, the main message reported by a columnar population is whether or not its principal whisker has been deflected, then this message could be decoded by a pooling mechanism with almost no loss in information. With regard to the third question, our investigation showed that the main feature of the population code of stimulus location that is lost by pooling is the diversity in PSTH profiles. ‘Blurring’ of within-cell spike patterns causes only a small information loss.

In a previous study, we reported that only 20% of the information about stimulus location carried by pairs of within-column barrel cortex neurons was redundant (Petersen et al., 2001). This may seem to contradict the new findings presented here, which show that neurons in the same column have nearly identical responses: the within-column neuronal label carries little information about stimulus location. How can a cortical column containing very similarly responding neurons show such a small amount of redundancy? The results can be reconciled because barrel cortex neurons are characterized by low firing rates. Even if neurons have similar receptive fields, low firing rates cause each spike to carry nearly independent information since it is relatively uncommon for multiple neurons to emit spikes on the same trial.

How might our results generalize when somatosensory cortex is studied under different experimental conditions? The size of the pooled population is one issue to consider: the typical information transmitted by a cell pair or triplet is on the order of 0.2–0.4 bits, just a fraction of the information needed (3.16 bits) for 100% correct discrimination of stimulated site. Hence, the relevant population code for stimulus location must involve a larger population than we could examine. Although with the present methodology we could not detect any statistically significant trend, it would be important to determine in future work how the present results generalize to large population.

A second issue is whether within-column pooling would still be efficient in preserving information about more complex, naturalistic stimuli. This is an important topic for future research — we emphasize that the methodological approach introduced here is equally applicable to complex stimuli.

As shown in this paper, pooling is a plausible mechanism for decoding of stimulus location from the activity of barrel cortex neurons. Whether this strategy can be used by downstream neural systems will depend on the structure of the projections connecting barrel cortex to the decoder. Projections from barrel field to various brain areas such as the neostriatum (Alloway et al., 1999) and pontine nuclei (Leergaard et al., 2000) follow a crude somatotopic organization, with non-negligible overlaps of projections from different columns. Each target neuron receives strong inputs from one column and weaker inputs from a few neighboring columns. Is this structure compatible with decoding by pooling? Suppose that the target decoder neurons do not rely on sophisticated dendritic processing (Segev and London, 2000), but are only able to add up all inputs. If all columns project with the same weight, no decoding of stimulus position by pooling would be possible, since the decoder would respond exactly in the same way to all stimulus locations. Decoding stimulus location by a summating neuron hence requires that inputs from columns representing different whiskers have different weights. A ‘perfect’ one-to-one somatotopic organization of the projections is one such extreme possibility. However, the partial and relatively coarse segregation of inputs that seems to be realized by projections that leave the barrel cortex (Alloway et al., 1999; Leergaard et al., 2000) could also support a potentially information-rich read-out mechanism: pooling all its inputs, the target neuron would signal stimulation of its principal whisker (i.e. the principal whisker of its main afferent column) by emitting a strong or fast response, and would signal stimulation of surrounding whiskers by emitting a weaker, longer-latency response. These considerations suggest that pooling based on partial segregation of projections from different columns might be a plausible mechanism for decoding stimulus location from the activity of populations of neurons in barrel cortex.

One reason for studying whether pooling neuronal activity over a certain spatial scale distorts the information content of the neuronal signals is that pooling is closely related to the spatial signal averaging that underlies imaging techniques such as functional magnetic resonance imaging (Rees et al., 2000; Logothetis et al., 2001). Such techniques are used to investigate sensory tuning of responses to different stimuli in different spatial regions of the cortex (Wandell, 1999). Our findings suggest that it is important to understand the spatial scale of signal averaging compared to the size of cortical columns. If the spatial averaging spans several columns, much of the information present in the neuronal signal is likely to be lost. We believe that the present results points to important constraints on the spatial resolution of parametric imaging experiments.

In conclusion, we suggest that, despite the potentially enormous complexity of the cortical population code for stimulus location, the information carried about the location of a stimulated whisker may be read-off in a highly efficient manner by a simple mechanism — pooling of the afferent activity of neurons with similar sensory tuning. We speculate that the structural substrate supporting decoding of topographically mapped stimulus parameters is the cortical column.

We thank M. Lebedev for valuable scientific collaboration; I. Erchova and G. Mirabella for assisting with data collection; O. Lebedeva for histological processing; A. Nevado, J. Nicholls, G. Pola, S.R. Schultz, A. Thiele and M.P. Young for useful discussions. Supported by European Community IST-2000–28127, Telethon Foundation, J.S. McDonnell Foundation, Wellcome Trust grant 066372/Z/01/Z, Italian MURST, CNR and Regione Friuli Venezia Giulia. S.P. is supported by an MRC Research Fellowship, and F.P. is supported by a Wellcome Trust Prize Studentship.

Figure 1.

The importance of the neuronal label. (a) Cell pair located within barrel column D2. Left: raster plot for each cell (and for the pooled activity) in response to stimulation of whisker D1, D2 and D3. Right: mutual information between the stimulus set and the spike timing response evaluated with 10 ms precision. Two population codes are considered; the labeled line (solid line) and the pooled code (dashed line). In this case pooling led to little information loss. (b) Cell pair distributed across two barrel-columns, D1 and D3. Conventions as in (a). The labeled line code now conveyed twice more information than the pooled code.

Figure 1.

The importance of the neuronal label. (a) Cell pair located within barrel column D2. Left: raster plot for each cell (and for the pooled activity) in response to stimulation of whisker D1, D2 and D3. Right: mutual information between the stimulus set and the spike timing response evaluated with 10 ms precision. Two population codes are considered; the labeled line (solid line) and the pooled code (dashed line). In this case pooling led to little information loss. (b) Cell pair distributed across two barrel-columns, D1 and D3. Conventions as in (a). The labeled line code now conveyed twice more information than the pooled code.

Figure 2.

Labeled line and pooled population code. Labels above the graph refer to columnar location of the neuronal pair. Information in the labeled line (solid line) and pooled (dashed line) decoding procedures are plotted as a function of post-stimulus time, averaged over cell pairs. The information conveyed by which cell fired each spike increases both as a function of inter-neuronal distance and post-stimulus time. Bars denote SEM.

Figure 2.

Labeled line and pooled population code. Labels above the graph refer to columnar location of the neuronal pair. Information in the labeled line (solid line) and pooled (dashed line) decoding procedures are plotted as a function of post-stimulus time, averaged over cell pairs. The information conveyed by which cell fired each spike increases both as a function of inter-neuronal distance and post-stimulus time. Bars denote SEM.

Figure 3.

Effect of pooling on whisker-specific information. For each whisker of a set of nine, the information that responses of neuronal pairs carried about whether or not a specific whisker was stimulated is shown. Black histograms represent the labeled line information, and gray histograms represent the pooled information. Information values were computed from responses collected in the 0–40 post-stimulus window. Labels above the graph refer to the columnar location of the neuronal pair. Bars denote SEM.

Figure 3.

Effect of pooling on whisker-specific information. For each whisker of a set of nine, the information that responses of neuronal pairs carried about whether or not a specific whisker was stimulated is shown. Black histograms represent the labeled line information, and gray histograms represent the pooled information. Information values were computed from responses collected in the 0–40 post-stimulus window. Labels above the graph refer to the columnar location of the neuronal pair. Bars denote SEM.

Figure 4.

Contributions of different features of the population activity to information transmission for a labeled line and pooling decoding mechanism. The contribution of PSTH (dashed line) and correlation (dotted line) are plotted as a function of post-stimulus time, averaged over cell pairs. The contribution of PSTH information was computed from equation (3) using either the labeled line or the pooled response. The contribution of auto- and cross-correlations to the labeled line information was computed from equation (4) by considering separately the auto- (i.e. a = b) and cross- (ab) components. The contribution of auto-correlations to the pooled information was computed from equation (4) using the pooled response. In the last case no separation between a = b and ab was needed. Labels above the graph refer to the columnar location of the neuronal pair. Bars denote SEM.

Figure 4.

Contributions of different features of the population activity to information transmission for a labeled line and pooling decoding mechanism. The contribution of PSTH (dashed line) and correlation (dotted line) are plotted as a function of post-stimulus time, averaged over cell pairs. The contribution of PSTH information was computed from equation (3) using either the labeled line or the pooled response. The contribution of auto- and cross-correlations to the labeled line information was computed from equation (4) by considering separately the auto- (i.e. a = b) and cross- (ab) components. The contribution of auto-correlations to the pooled information was computed from equation (4) using the pooled response. In the last case no separation between a = b and ab was needed. Labels above the graph refer to the columnar location of the neuronal pair. Bars denote SEM.

## References

Alloway KD, Crist J, Mutic JJ, Roy SA (
1999
) Corticostriatal projections from rat barrel cortex have an anisotropic organization that correlates with vibrissal behavior.
J Neurosci

19
:
10908
–10922.
Armstrong-James M, Fox K (
1987
) Spatiotemporal convergence and divergence in the rat S1 ‘barrel’ cortex.
J Comp Neurol

263
:
265
–281.
Berry MJ II, Meister MM (
1998
) Refractoriness and neuronal precision.
J Neurosci

18
:
2200
–2221.
Cover TM, Thomas JA (1991) Elements of information theory. New York: John Wiley.
Darian-Smith I, Johnson KO, Dykes R (
1973
) ‘Cold’ fiber population innervating palmar and digital skin of the monkey: responses to cooling pulses.
J Neurophysiol

36
:
325
–346.
Harris JA, Petersen RS, Diamond ME (
1999
) Distribution of tactile learning and its neural basis.

96
:
7587
–7591.
Lebedev MA, Mirabella G, Erchova I, Diamond ME (
2000
) Experience-dependent plasticity of rat barrel cortex: Redistribution of activity across barrel-columns.
Cereb Cortex

10
:
23
–31.
Leergaard TB, Alloway KD, Mutic JJ, Bjaalie JG (
2000
) Three-dimensional topography of corticopontine projections from rat barrel cortex: correlations with corticostriatal organization.
J Neurosci

15
:
8474
–8484.
Logothetis NK, Pauls J, Augath M, Trinath T, Oeltermann A (
2001
) Neurophysiological investigation of the basis of the fMRI signal.
Nature

412
:
150
–157.
Lu SM, Lin RC (
1993
). Thalamic afferents of the rat barrel cortex: a light- and electron-microscopic study using Phaseolus vulgaris leuco-agglutinin as an anterograde tracer.
Somatosens Mot Res

10
:
1
–16.
Mountcastle VB (
1997
) The columnar organization of the neocortex.
Brain

120
:
701
–722.
Nirenberg S, Carcieri SM, Jacobs AL, Latham PE (
2001
) Retinal ganglion cells act largely as independent encoders.
Nature

411
:
698
–701.
Oram MW, Foldiak P., Perrett DI, Sengpiel F (
1998
) The ‘ideal homunculus’: decoding neural population signals.
Trends Neurosci

21
:
259
–265.
Panzeri S, Treves A (
1996
) Analytical estimates of limited sampling biases in different information measures.
Network

7
:
87
–107.
Panzeri S, Schultz SR (
2001
) A unified approach to the study of temporal, correlational and rate coding.
Neural Comput

13
:
1311
–1349.
Panzeri S, Schultz SR, Treves A, Rolls ET (
1999
). Correlations and the encoding of information in the nervous system.
Proc R Soc Lond B Biol Sci

266
:
1001
–1012.
Panzeri S, Petersen RS, Schultz SR, Lebedev M, Diamond ME (
2001
) The role of spike timing in the coding of stimulus location in rat somatosensory cortex.
Neuron

29
:
769
–777.
Petersen RS, Diamond ME (
2000
) Spatio-temporal distribution of whisker-evoked activity in rat somatosensory cortex and the coding of stimulus location.
J Neurosci

20
:
6135
–6143.
Petersen RS, Panzeri S, Diamond ME (
2001
) Population coding of stimulus location in rat somatosensory cortex.
Neuron

32
:
503
–514.
Rees G, Friston KJ, Koch C (
2000
) A direct quantitative relationship between the functional properties of human and macaque V5.
Nat Neurosci

3
:
716
–723.
Reich DS, Mechler F, Victor JD (
2001
) Independent and redundant information in nearby cortical neurons.
Science

294
:
2566
–2568.
Rice FL (1995) Comparative aspects of barrel structure and development. In: The barrel cortex of rodents (Jones EG, Diamond IT, eds). New York: Plenum Press.
Segev I, London M (
2000
) Untangling dendrites with quantitative models.
Science

290
:
744
–750.
Shadlen MN, Britten KH, Newsome WT, Movshon JA (
1996
) A computational analysis of the relationship between neuronal and behavioral responses to visual motion.
J Neurosci

16
:
1486
–1510.
Shannon C (
1948
) A mathematical theory of communication.
Bell Sys Tech J

27
:
379
–423.
Simons DJ (
1978
) Response properties of vibrissa units in rat SI somatosensory neocortex.
J Neurophysiol

41
:
798
–820.
Vaadia E, Haalman I, Abeles M, Bergman H, Prut Y, Slovin H, Aertsen A (
1995
) Dynamics of neuronal interactions in monkey cortex in relation to behavioural events.
Nature

373
:
515
–518.
Wandell BA (
1999
) Computational neuroimaging of human visual cortex.
Annu Rev Neurosci

22
:
145
–173.
Woolsey TA, Van der Loos H (
1970
). The structural organization of layer IV in the somatosensory region (SI) of mouse cerebral cortex. The description of a cortical field composed of discrete cytoarchitectonic units.
Brain Res

17
:
205
–242.
Zohary E, Shadlen MN, Newsome WT (
1994
) Correlated neuronal discharge rate and its implications for psychophysical performance.
Nature

370
:
140
–143.