Estimation of the time-varying information

At a given time t, the instantaneous mutual information between the stimulus S and the response Y_t can be written as a difference in Shannon entropies: Here, H(Y_t) is the response entropy for a window at time t, while H(Y_t|S) corresponds to the entropy of the response given the stimulus or the conditional response entropy. H(Y_t|S) can also be called the neural noise since it represents the variability in the neural response to the same stimulus. For spiking neurons, y_t represents the number of spikes in the window at time t (Note: in our notation, capitals are used for random variables and lower case for a sample from that random variable).

Similarly, the cumulative mutual information in neural responses that are discretized into time intervals is given by:

The entropies now include the time course of the neural responses starting at t = 0 and up to time t. The reference time t = 0 is set to the stimulus onset in our analyses but could be any arbitrary reference point.

The conditional response entropy and the response entropy are obtained from the distribution of the conditional probability of neural responses given the stimulus, p(y_t|s), and the distribution of probability of each stimulus p(s_i): with y_t, the neural response at time t, is measured in spike counts and takes values from zero to a maximum rate value, R_Max(for example, as dictated by the neuron’s refractory period or numerically as p log p becomes infinitely small for high spike counts that have very small probability of occurring). The probability of the stimulus p(s_i) is usually taken as 1/n_s, where n_s is the number of stimuli, unless the study incorporates natural stimulus statistics. The probability of spike counts at time t given a stimulus, (y_t|s_i), could be estimated empirically by recording hundreds or thousands of responses of the same neuron to the same stimulus. Although this approach has been shown to be possible in certain preparations [15], it severely limits the number of stimuli that can be investigated in most neurophysiological experiments. Here we propose a parametric approach where we model the distribution of neural responses to a given stimulus s_i as an inhomogeneous Poisson process. The conditional probability of response (spike count) given the stimulus is then: where is the mean response at time t for stimulus s_i. This mean rate was estimated empirically using the time varying kernel density estimation (KDE) proposed by Shimazaki and Shinomoto [42]. The instantaneous information estimated in this fashion is relatively straightforward, as long as the Poisson assumption is valid and a sufficient number of trials (i.e. stimulus presentations) is obtained to estimate (see Methods). The validity of the Poisson assumption for our dataset is assessed and discussed further below. Note that the Poisson distribution corresponds to the distribution with maximum entropy for unbounded count probabilities given a mean rate [43]. Thus, the value of entropy calculated for the empirical data, p(y_t|s_i), constitutes an upper-bound when using the Poisson distribution assumption. If the empirical data follows a different distribution, the true conditional response entropy, H(Y_t|S), will be lower and, thus, the actual mutual information higher. In other words, the assumption of a Poisson distribution yields a lower bound for the estimation of the mutual information.

The estimation of the cumulative information then extends this approach to joint probabilities of responses in successive time windows, (y_t, y_t−1, y_t−2, …). Due to what has been labelled as the “curse of dimensionality”, the numerical estimation of the unconditional probability becomes exponentially more expensive as the integration time increases. We evaluated multiple approaches based on different assumptions and found that Monte Carlo with importance sampling yielded the best results (see Methods and S3 Fig for comparison to alternative approaches).

If the Poisson assumption also holds for successive time-windows, the conditional probability of response at t is independent of the conditional response at previous times.

Given this probabilistic independence, it can be shown that the joint conditional response entropy is simply the sum of the conditional response entropies at each time point (see Methods):

Thus, the estimation of the conditional response entropy is straightforward and not affected by the integration time. The problem of dimensionality arises in the estimation of the unconditional probabilities of response and the corresponding response entropy.

The probability of the time varying response is the joint probability of observing (y_t, y_t−1, y_t−2, …). This joint probability cannot be expressed as the product of the probabilities at different times because these are not independent. More intuitively observing a particular y_t−1 will affect the probability of observing y_t. This is true because the time varying means of the Poisson distributions are correlated in time; for example, if is high we might expect to also have high values. These high values could be true for one particular stimulus s_i but not for the other stimuli. Then observing a high value of y_t−1 would predict a higher value than expected for y_t (and an increase in probability that it was caused by s_i). The joint unconditional probability distribution is:

Given the lack of independence, the response entropy must then be calculated from the joint probability distribution:

The estimation of this entropy was performed using Monte Carlo with importance sampling. In Monte Carlo, random samples of a vector (y_t, y_t−1, y_t−2, …) are drawn from a proposal distribution q(y_t, y_t−1, y_t−2, …) and used to estimate the expected value of log₂ p(y_t, y_t−1, y_t−2, …) by an algebraic average weighted by the likelihood ratio of p(y_t, y_t−1, y_t−2, …)/q(y_t, y_t−1, y_t−2, …). The sampling stops when entropy estimations reach an equilibrium. Information estimations are also known to suffer from positive bias [44]. Here, biased-corrected estimates and errors on information values were obtained from Jackknifing the estimation of the time-varying rates and bootstrapping the Monte Carlo samples.

For these calculations, one has also to determine the size of the time window used for estimating the instantaneous information and correspondingly the steps for the cumulative information. This time window is used to estimate the spike counts and the average rate and depends both on the dynamics of the stimulus and on the response properties of the neurons. By performing a coherence analysis on spike trains and a power spectral analysis on time varying rate in response to natural stimuli, we found that 10 ms (or 50 Hz) captured between 97% and 99% of the dynamics in our system (see Methods and S1 Fig).

Finally, we also estimated the information values for stimulus categories by performing the weighted sum of probabilities for stimuli belonging to each category. The information about categories at time t: is obtained from the conditional probability of response given the category c_k, which is in turn calculated as the average conditional probability of response for the stimuli belonging to that category: Here, p(s_k) is the probability of occurrence of stimulus s_k within the category c_k. In controlled playback experiments (as here), p(s_k) is where is the number of stimuli used to sample the category c_k. Similarly, p(c_k), the probability of occurrence of vocalizations in category c_k was taken as 1/n_c where n_c is the number of categories. In our system, the stimuli are individual renditions of vocal communication calls that fall into 9 call categories of the zebra finch vocal repertoire. We will contrast stimulus information to categorical information, both instantaneous and cumulative for these behaviorally relevant categories of call-types.

Time-varying information for model neurons

To validate our approach and to illustrate the behavior of time-varying information values, we calculated instantaneous and cumulative information for model neurons with simple and stereotyped response properties. Fig 1 shows the firing rates, raster plots and information values for 3 model neurons in response to 4 stimuli (S1-S4). One model neuron responds to the 4 stimuli with different mean firing rates that are constant in time (Rate Neuron). A second model neuron responds to the four stimuli with the same fixed firing rate but with different latencies (Onset Neuron). The third model neuron also responds with equal average firing rates to the four stimuli but the response occurs at different times (Temporal Neuron).

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Fig 1. Instantaneous and cumulative information for 3 model poisson neurons.

The top row (A,B,C) shows the firing rate and simulated spike rasters of 3 model neurons to 4 hypothetical stimuli (S1-S4). The onset and offset of all stimuli are identical and shown by the top solid black line. The model rates for the four stimuli are shown in colored dotted lines. These rates are used to generate spike patterns using an inhomogeneous Poisson process. Ten realizations for S1-S4 are shown as spike rasters. The rate recovered from those rasters using a time-varying kernel density estimate (KDE) are shown as colored solid lines. The shaded area around each line indicate the error on the KDE rate estimated by the standard error on the Jackknife estimates. The bottom row (D,E,F) shows the instantaneous and cumulative information obtained using either the true model rate or the recovered rate from KDE. The error bars in the instantaneous information calculations were obtained by jackknifing the KDE estimates. For the cumulative information calculations, the Monte Carlo simulations were repeated for each jackknife estimate. The error bars therefore include both the errors in the estimation of the true rate from spike rasters (data limitation) and the errors in the estimation of the cumulative information due to limited sampling (computational limitation). The time-varying response rates were chosen to illustrate neural representation for stimuli using different neural codes labelled here as Rate (A, D), Onset (B, E), and Temporal (C, F).

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These simulations illustrate some very basic principles of neural coding. First, many different response profiles can lead to very similar rates of information: in all three simulations, the cumulative information approaches the maximum possible value (2 bits). Second, the coding capacities of neurons are a function of both the range of firing rates that can be achieved (as in the rate neuron) and the modulations in time of this neural activity. Third, the estimation of the instantaneous information gives an incomplete picture of the neural coding of a neuron as it does not incorporate the redundancy or independence of the neural representation over time. For example, on the one hand, when comparing the instantaneous information in the Rate neuron to that of the Onset neuron, one might erroneously conclude that the Rate neuron has more information while, in fact, the cumulative information shows that the Onset neuron is more informative at short time scales. On the other hand, one can also observe that the cumulative information in the Rate neuron continues to increase while the firing rate and the instantaneous information are constant; additional time points allows for a better assessment of that firing rate by time-averaging out neural noise.

These simulations also allowed us to validate our methods. The KDE for the empirical estimation of the time-varying rate based on the generated spike rasters gave very good predictions (solid lines Fig 1A–1C) of the actual model rates (dashed lines Fig 1A–1C): over all stimuli and model neurons, the average error was less than 0.02 spikes/ms. Not surprisingly then, using the actual rate versus the estimated rate yielded practically identical results in the information calculations (dashed vs solid lines, Fig 1D–1F). We also checked that the bias corrected estimates were accurate: the instantaneous information was indeed centered at zero when the response to the 4 stimuli was identical. We verified that the actual values of instantaneous and cumulative information were correct. For example, in the Temporal neuron (Fig 1C & 1F) a peak instantaneous information of 0.5 bits is expected as 1 out 4 stimuli will be almost perfectly discriminated. Finally, we also assessed the limitations of Monte Carlo with importance sampling for the estimation of the cumulative information. For neurons, with continuously high firing rates, this estimation can become unreliable at longer integration times as illustrated by the calculation for the Rate neuron (Fig 1A & 1D). However, in those cases, the estimate of the standard error also increased drastically and allowed us to define end points for the calculation of the cumulative information.

Avian auditory neurons have approximate Poisson statistics

Our estimation of the time-varying information values is based on a Poisson parametrization for the distribution of spike counts in each time window of analysis and on the independence of the conditional probability of spike counts, p(y_t|s_i), between successive time windows. Here, we assessed the limits of these assumptions.

First, we examined for each neuron, for each stimulus and for each time bin of 10 ms, whether the empirical distribution of spike counts across stimulus presentations was different than the one expected from a Poisson distribution whose mean was given by the kernel density estimation (resampling test; see Methods). For each neuron, we obtained the proportion of time bins where the Poisson assumption was rejected. Fig 2A shows the distribution of this proportion across the neural population. For the large majority of cells (240/404), the empirical spike count could not be distinguished from a Poisson distribution for 90% of the time bins. For a small number of neurons (5/404) non-Poisson spike count statistics could be observed in 40% of the time bins. The mean number of rejected time bins was 10.5%. Thus, the Poisson statistics assumption for spike is warranted in more than half of our neurons. For the other neurons, the deviation is relatively small. As stated above, the Poisson assumption used for non-Poisson data yields a lower bound on information. Given the small number of presentations of each stimulus (n = 10), one can argue that the test of conformity to a Poisson distribution is relatively conservative (i.e. we might fail to reject the Poisson statistic because of small sample size). Thus, we also assessed whether the first and second moments (mean and variance) obeyed Poisson statistics. Fig 2C shows the Fano Factor estimated at each consecutive time bins after stimulus onset, both for individual stimuli (insets) and average across stimuli (solid line). The mean Fano factor for the population was 1.04 with SEM of 0.04 (Fig 2D). This second analysis further quantifies the degree with which our neural data follow Poisson statistics and also shows that for a majority of neurons the assumption is warranted.

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Fig 2. Poisson statistics.

A. Distribution in number of neurons (n = 404) of the proportion of time bins where the Poisson assumption can be rejected. B. Distribution of the values of stimulus correlation (yellow) and noise correlation in pairs of successive time windows for the neural population (n = 404). Noise correlation values are represented in a stacked histogram depicting values significantly different from zero in red and non-significant values in blue. C. Time-varying Fano Factor (Variance/Mean) obtained from the empirical distribution of spike counts estimated in successive 10 ms windows. Spike count distributions are obtained from 10 trials and the average Fano Factor from repeating the estimation of mean and variance calculation for all stimuli presented (min = 54, max = 104). The Fano Factor is shown for one example neuron (solid black line) and also averaged across the entire population (n = 404). Error bars are ± two SEMs. The insets show the mean and variance spike count for the example neuron at a given time shown by the arrows. On those plots, each red dot corresponds to one stimulus (many points overlap). D. Distribution of time averaged Fano Factors for the population of neurons (n = 404). Fano Factors are not significantly different from 1.

https://doi.org/10.1371/journal.pcbi.1006698.g002

The second assumption used in the calculation of cumulative information is that the conditional probability of spike counts, p(y_t|s_i), is independent across successive time bins. To assess the validity of this assumption, we estimated the Pearson correlation coefficient of spike counts obtained in successive time windows in responses to the same stimulus, also known as the noise correlation. Fig 2B shows the distribution of noise correlations for the 404 neurons examined. Significant noise correlations were observed (red bars versus non-significant values shown in blue), but they were small, centered around zero and their distribution showed a slight positive skew. These positive values correspond to cases were a slight increase of the spike count above average at a given stimulus presentation and time bin is somewhat giving a higher chance of greater than average spike count in the successive time bin of the same stimulus presentation. Ignoring noise correlations can lead to either under or over estimates of mutual information depending on the functions that describe the mean time-varying response to each stimulus [45, 46]. In the majority of cases, not taking into account noise correlations leads to an under-estimate of mutual information. For instance, if responses in successive time bins decrease for one stimulus and increase for another stimulus, positive or negative noise correlations will result in lower conditional response entropies, H(Y_t, Y_t−1, Y_t−2, …|S), and thus higher mutual information. In the general case of stimulus time-varying rates that have different and random shapes depending on the stimulus, one can also show that noise correlations increase mutual information. In some cases, however, noise correlations can increase the conditional response entropies and not taking them into account lead to an over-estimate of the mutual information. Over-estimation of mutual information commonly happens when the responses for stimuli change in the same manner in successive time bins (increase or decrease in mean rates for all sounds), then positive noise correlations can lead to an increase in conditional response entropies and thus a decrease in mutual information. In our data, the time varying rates of the different stimuli are positively correlated at the onset response, but mostly uncorrelated at other times (see Figs 3–5). We might therefore slightly overestimate mutual information at stimulus onset, but for most of our analysis we might be slightly underestimating cumulative information. Given that the noise correlations are small, this estimation bias is small.

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Fig 3. Time-varying firing rates and information values for example neuron 1.

A. Time-varying firing rate obtained from kernel density estimation (KDE) of ten repetitions of different example stimuli. The plot shows the responses to two examples of calls from each call-type; the color scheme is as in B; 0 ms corresponds to stimulus onset. B. Averaged time-varying firing rate obtained for each call-type category. These average responses are obtained by averaging the KDE estimates of firing rates for each stimulus belonging to the same call-type (~ 10 example stimuli per call-type). Shaded error bars show ± one SEM. C. The overall (averaged over all stimuli) time-varying firing rate for the same example neuron is shown by the solid black line (right y-axis) with ± SEM as shaded error bars. The green line and yellow line (left y-axis) correspond to the instantaneous information for stimuli and call-type categories calculated in successive 10 ms windows. Shaded error bars were obtained by a jackknife procedure D. Cumulative information for stimuli and call-type categories for the same example neuron. The shaded error bars are ± SE obtained by jackknifing and resampling (see Fig 1 and Methods). Each cumulative information curve is also fitted using an exponential function (dashed lines) characterized by three parameters: a latency (dt, in ms), an exponential rate (tau, in ms) and the infinite time limit value expressed as a fraction of the maximum information achievable (k). The MSE is the mean square error of the fit in bits. This example neuron had a very high stimulus evoked mean firing rate with rapid and reliable time-varying dynamics leading to high information rates. Spontaneous firing was very low (~0 spikes/s). Spike rasters for this neuron are shown on S4 Fig.

https://doi.org/10.1371/journal.pcbi.1006698.g003

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Fig 4. Time-varying firing rates and information values for example neuron 2.

As in Fig 3 but for a neuron with lower firing rates and displaying selectivity for Distance Calls. Spike rasters for this neuron are shown on S5 Fig. A. Time-varying firing rate obtained from KDE of ten repetitions of different example stimuli. B. Averaged time-varying firing rate obtained for each call-type category. C. Overall time-varying firing rate and instantaneous information for stimuli and call-type categories. D. Cumulative information for stimuli and call-type categories. See details in legend of Fig 3.

https://doi.org/10.1371/journal.pcbi.1006698.g004

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Fig 5. Time-varying firing rates and information values for example neuron 3.

As in Fig 3 but for a neuron with intermediate firing rates and displaying selectivity for Wsst (Ws) or Aggressive Calls. Note also the longer latency relative to the neuron shown in 3 and 4. Spike rasters for this neuron are shown on S6 Fig. A. Time-varying firing rate obtained from KDE of ten repetitions of different example stimuli. B. Averaged time-varying firing rate obtained for each call-type category. C. Overall time-varying firing rate and instantaneous information for stimuli and call-type categories. D. Cumulative information for stimuli and call-type categories. See details in legend of Fig 3.

https://doi.org/10.1371/journal.pcbi.1006698.g005

While we can neglect noise correlations in our calculations and assume independence of conditional probability of spike counts, p(y_t|s_i), across successive time bins, we are not assuming independence of the unconditional probability, p(y_t), across successive time bins. Indeed, the distribution of stimulus correlation (correlation of mean rate between pairs of consecutive time bins in the trial average response of each stimulus) across the neural population show high positive values centered around 0.4 (Fig 1B). These values indicate that the change in the spike counts in successive time bins is highly driven by the stimulus identity. As such, the unconditional probability distributions in successive time windows are not independent. We do take into account these large stimulus correlations in our estimation of the cumulative mutual information.

Time-varying Information for 3 example neurons

In Figs 3–5, we show the time-varying firing rates, the time-varying instantaneous information and the cumulative information for 3 zebra finch auditory neurons with distinct response properties. The neuron in Fig 3 responded robustly to all communication calls with high and reliable firing rates. It also responded in a time locked fashion sometimes at multiple time points for single calls. Although this neuron was not selective for a particular stimulus or call-type category, by combining rate and temporal codes it reached very high levels of instantaneous information. Moreover, this instantaneous information showed little redundancy yielding very high cumulative information. One can also observe, that for this particular neuron, the average time-varying firing rate is not correlated with instantaneous information. This neuron shows a strong onset response in its firing rate while the instantaneous information is almost constant and even slightly lower during the onset response. The neurons in Figs 4 and 5 have much lower firing rates and exhibit selectivity for a call-type, the Distance Call (DC) and the Wsst Call (Ws) respectively. The neuron in Fig 4 exhibits both an onset and sustained response both of which are selective for DC. The neuron in Fig 5 has a much longer latency for response with correlated peaks in firing rate and instantaneous information found between 100 and 300 ms after stimulus onset (Example of spike rasters from single trials for these three neurons are shown in S4, S5 and S6 Figs).

The cumulative information curves were fitted with an exponential function that allows us to quantify the time constant of information accumulation (τ), the saturation level (k) relative to the maximum information that could be achieved (I_Max) and the latency (dt):

Most of the calls in the zebra finch repertoire have durations that are shorter than 300 ms [39]. In order to investigate the cumulative information accumulated at this behaviorally relevant fixed point in time, we estimated the relative cumulative information as the value of the model at 300 ms relative to I_Max.;

The results of the fit are shown in dashed lines in the cumulative information plot for each neuron. One can observe that the neuron in Fig 3 has exceptionally high values of saturation: with sufficiently long integrations time, single spike trials could be used to perfectly assess what stimulus (out of those used in the experiment) was heard and, thus, which call-type category it belonged to. The selective neurons in Figs 4 and 5 have much lower saturation levels as expected since they mostly respond to stimuli from a single call-type category. The neuron in Fig 5 has longer latency in its categorical cumulative information but this is not the case for the neuron in Fig 4 that has a rapid yet selective onset response. The high firing neuron (Fig 3) also has a faster time constant τ for the stimulus information than the selective neurons shown in Figs 4 and 5. All three neurons have similar and relatively slow time constants for the categorical information in the 200–300 ms range.

Rate and time-varying information: Population analysis

Fig 6A shows the time course of the firing rate averaged across all stimuli and its relationship with the instantaneous information. On average, avian cortical auditory neurons show an onset response followed by a sustained response as observed at many stages of auditory processing and in many sensory systems. The instantaneous information, however, remains almost constant during the entire time. This is true both for the information about individual stimuli or the information about categories. Thus, the onset response is only less informative than the sustained response in bits/spike but not in bits/s; when processing natural vocalizations, the onset and sustained response are equally informative. Moreover, the information in the sustained response continues to provide new information as reflected by the continuous and relatively fast increase in cumulative information shown in Fig 6B. The non-redundancy of the information in the sustained period is also well captured by examining the derivative or slope of the cumulative information shown in Fig 6C. That plot shows how much new information is acquired at each time bin. The new information for categories is relatively constant (red line Fig 6C) and similar to the instantaneous information about category (yellow line Fig 6A), while the spike rate is drastically decreasing between 50 and 150ms. Thus, as time progresses, individual spikes carry more new information for categories: there is very little redundancy in the code for call-type categories up to 200 ms and during that time the coding efficiency in bits/spike increases for the new information that is acquired about categories. In contrast, the new information for individual stimuli decreases and is, on average, remarkably correlated with spike rate (Fig 6C). Thus, while, in terms of instantaneous information, the coding efficiency (in bits/spike) is greater in the sustained versus onset period, in terms of non-redundant information, the coding efficiency is remarkably constant throughout the response. This result suggests that an integrating average spike rate measure (i.e. the time running average spike count over stimuli) could serve as a relatively good proxy of cumulative information on individual stimuli. However, note that this cumulative information needs to take into account time varying spike rates (or stimulus locked spike patterns) as in our calculation.

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Fig 6. Time-varying firing rates and information values for the population.

A. The solid black line (right y-axis) shows the time-varying firing rate averaged across all neurons (n = 337) and all stimuli played. The green and yellow lines (left y-axis) show the instantaneous information for stimuli and call-type categories respectively. B. The time-varying mean firing rate is shown as in A. The blue and red solid lines show the population average cumulative information for stimuli and call-type categories respectively. The blue and red dashed lines show the corresponding average cumulative information values calculated with constant averaged firing rates for each stimulus for each neuron over the first 600ms after stimulus onset (a fixed rate code). C. The same data as in panel B is plotted but with the derivative (or slope) of the cumulative information values. In all plots, the shaded error bars are ± 2 SEM.

https://doi.org/10.1371/journal.pcbi.1006698.g006

Indeed, taking into account time varying spike rates for each stimulus as opposed to the mean firing rate across the entire time-window for each stimulus is crucial in the accurate calculation of time-varying cumulative information. As seen in Fig 6B, the increases in the cumulative information for the time-varying rate code (solid lines) is much larger than the increases obtained with the fixed rate code (dashed lines; the information still increases because of noise averaging as explained above). The time-varying rates observed in neural responses (as illustrated in Fig 3A) provide additional information. How much more? For the coding of individual stimuli (comparing the dashed blue line to the solid blue line in Fig 6B), a fixed rate code (or assumption) captures only 24% of the information at 100 ms and 21% at 300 ms. The effect for categorical information is smaller because some of the coding dynamics in time-varying responses to stimuli belonging to the same category effectively become neural noise: for categorical information, a fixed rate code captures 50% of the information at 100 ms and 41% at 300 ms.

The distributions of time constants, τ for the cumulative information for stimuli and call-type categories are shown on Fig 7A. The distributions of relative cumulative information at 300ms (k₃₀₀) are shown on Fig 7B. The range of time constants observed across the population of neurons was large (100 ms-600 ms) with average time constants of 459 ms for cumulative stimulus information and 372 ms for cumulative categorical information. This difference in means of time constants is statistically significant (Paired t-test t(214) = 3.49, p = 0.00058); the ongoing time-varying rate changes continue to provide more information for decoding stimuli and less so for decoding categories of stimuli. There is also a wide distribution of relative cumulative information values (k₃₀₀ ranging from close to zero to 0.6). On average across neurons, the k₃₀₀ was 0.2 (or 20%) for stimuli and 0.15 (or 15%) for categories. These differences in relative information values are highly significant suggesting that, single neurons, capture more variability in stimuli than in categories. Note however that measures of relative information depend on the number of stimuli or categories. Therefore, a direct statistical comparison is not warranted. The comparison between stimulus representation and category representation requires estimations of expected values of categorical information given stimulus information which is performed below. Relative cumulative information values can, however, be used for stimulus and categories independently to assess other coding properties: although average time-varying rates and instantaneous time varying information are not well correlated within single neurons (as shown in Figs 3 and 6), the relative cumulative information is correlated with average firing rates. For cumulative stimulus information, one finds an increase in k₃₀₀ of 1% per spike/s (Adj R² = 0.36, F(1,213) = 120, p = 1.82 10⁻²²) and, for cumulative categorical information, an increase in k₃₀₀ of 0.8% per spike/s (Adj R² = 0.34, F(1,213) = 112, p = 2.26 10⁻²¹).

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Fig 7. Time constants (tau) for cumulative information and relative cumulative information estimated at 300 ms (K₃₀₀).

For a large fraction of neurons (214/337 neurons), cumulative information for stimuli and call-type categories were well fitted with exponential curves. The time constants of the exponential fits were used as a time integration estimate for cumulative information (A). The exponential fits were also used to estimate the cumulative information obtained at 300 ms relative to the maximum achievable information (log₂(n_stim) or log₂(n_cat)), k₃₀₀. k₃₀₀ is a number between 0 and 1 (B).A. Histogram of the distributions of time constants of the exponential fit for stimuli and call categories. Dashed lines indicate the mean of each distribution (Stim = 459 ms; Cat = 372 ms). B. Histogram of the two distributions of k₃₀₀. Dashed lines indicate the mean of each distribution (Stim = 0.2; Cat = 0.15).

https://doi.org/10.1371/journal.pcbi.1006698.g007

Analysis of the categorical information

We are interested in identifying neurons that could play an important role in categorizing vocalizations. We had previously identified example neurons that were highly selective for particular call-types and showed a high degree of invariance such as those shown in Figs 4 and 5 [40]. Here, we attempted to quantify the neural invariance for call renditions within call-type categories along time. For this purpose, we computed a Categorical Information Index (CII). The CII compares the actual categorical cumulative information to three potential values (see cartoon probability distributions of neural responses in Fig 8A): a floor or minimum value (set at 0), an expected value for shared information between stimuli and categories (set at 1) and a ceiling or maximum value (set at 2). The floor is the categorical information that one would obtain if stimuli are randomly grouped. The shared-information value is the information that one would obtain if the information about stimuli is equally shared across all stimuli and the neural responses for stimuli are perfectly sorted for each natural call-type category; for example, at a given point in time the 10 renditions of DC would elicit the 10 highest rates, the 10 renditions of LT Call the next 10 higher firing rates and so forth. This shared-information value does therefore assume that, for a given neuronal signal to noise ratio, neural responses segregate categories maximally while also preserving the maximum discrimination between stimuli within categories. The ceiling value is the categorical information value that one would obtain if a maximum amount of information about stimuli was used for the discrimination of categories and a minimum for discriminating stimuli within categories; it assumes maximum invariance to variations within a category. The ceiling value is equal to the stimulus information until it reaches log₂(n_c), where n_c is the number of categories, corresponding here to the 9 call-types. Fig 8B shows the time-varying floor (dashed-green), shared (dashed-orange) and ceiling (dashed-red) values of cumulative categorical information along with the actual stimulus (solid blue) and categorical (solid red) cumulative information for 3 neurons chosen to illustrate CIIs that are below 1, around 1 and above 1.

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Fig 8. Cumulative information and the categorical information index.

A. Schematic of hypothetical response probability distributions that yield different values of the Categorical Information Index (CII). CII is defined by comparing the cumulative categorical information to a floor (CII = 0), an expected value predicted from the stimulus information (CII = 1) and a ceiling value (CII = 2) as described in the text and methods. Each plot represents the distributions of hypothetical neural responses for different stimuli (S1 to S4) that yield different values of CII. When stimuli are grouped randomly into categories (S₁ and S₂ belong to C₁, while S₃ and S₄ belong to C₂), CII is close to 0 (left panel). When stimulus categories preserve the order of the stimulus conditioned response distributions and these distributions are equally separated, effectively coding for both stimuli and categories, CII is equal to 1 (middle panel). When stimuli belonging to the same category have identical responses, the stimulus information is equal to the category information, the response being effectively invariant within categories and CII is equal to 2 (right panel). Note that the actual response probability distributions are multidimensional (one dimension for each time lag) with Poisson marginals. B. Example of the cumulative information for stimuli and call-type categories and CII for 3 different representative neurons. The blue and red lines are the cumulative information for stimuli and categories respectively. Estimates of the floor (dashed-green), expected (dashed-orange) and ceiling (dashed-purple, here overlapping with blue) values for the categorical cumulative information as described in A are given as well. The CII is shown as a solid line in the second row for each example neuron. Cell 1 has long periods of time with CII < 1, Cell 2 has long periods of time with CII ~ 1 and Cell 3 has long periods of time with CII > 1 as shown by the arrows.

https://doi.org/10.1371/journal.pcbi.1006698.g008

Fig 9A shows the distribution of CII values across the neural population (see also S7 Fig for results of this analysis in absolute information units). The thin colored lines correspond to CII curves for single neurons and they are colored according to the time average CII (see legend Fig 9A). The average CII over neurons (bold black line) is very close to the shared value of 1 as one might expect if acoustical differences across stimuli drive neural responses in a linear fashion along some acoustical feature and call categories segregate perfectly along that same acoustical feature. However, the average CII is also slightly (and significantly) above 1 between 120 and 260 ms showing some small degree of average invariance for call renditions within a call-type category during those times. In addition it is clear that there is a wide distribution of CII around the shared value of 1. And this distribution is positively skewed towards values higher than 1 as exemplified by the density plots calculated at 150ms. This distribution includes many neurons that exhibit a high degree of invariance for call renditions within call-type categories as also shown by the average CII for the top quartile (red solid line). Focusing on these 25% of neurons with the highest CII, we found a maximum value of CII at 175 ms, indicating that, relative to the maximum invariance achievable given their individual stimulus information values, these neurons are maximally invariant around that time. In terms of absolute value of additional categorial cumulative information relative to the expected shared information value, these same neurons reach a maximum of 0.16 bits at 320 ms (S7 Fig).

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Fig 9. Categorical cumulative information index and its relationship to time constants tau and relative cumulative information estimate K₃₀₀.

As explained in Fig 8 and in the text, the cumulative information (CI) for call-type categories can be analyzed in terms of a floor or minimum value, a shared value and a ceiling or maximum value. A Value of the Categorical Information Index (CII) as a function of time for all neurons in the population. Each line color corresponds to the time average CII of the neuron obtained between 50 and 300ms. The solid red line is the average obtained for the quartile of neurons with the highest CII and the blue solid line is average obtained for the quartile of neurons with the lowest CII. Error bars on the average CII (solid bold black line) are 2 SEM. The right vertical panel shows the distribution of CIIs at 150 ms. B Scatter plot of the time constant of the exponential fit for stimuli (Stim—x axis) versus call categories (Cat—y axis). C Scatter plot of k₃₀₀ for stimuli (Stim—x axis) versus categories (Cat—y axis). Single data points in B and C are colored according to the time average CII of each neuron as in A.

https://doi.org/10.1371/journal.pcbi.1006698.g009

Do these high CII neurons exhibit other characteristic response properties? In the scatter plots of Fig 9B and 9C, we examined the relationship between CII (color coded) and, respectively, time constants and relative level of the cumulative information. It can be seen from Fig 9B, that neurons with high CII have relatively long stimulus time constants in comparison to their corresponding time constant observed for categorical information. This relationship is also significant for the entire population of neurons (Linear Regression explaining CII from τ_cat/τ_stim: Adj R² = 0.11, F(1,213) = 28.5, p = 2.37 10⁻⁷). As shown in Fig 9C, neurons with high CII also have higher relative levels of categorical information in comparison to their relative values for stimulus information although this result is expected given our definition of CII (Linear Regression explaining CII from k_300,cat—k_300,stim: Adj R² = 0.76, F(1,213) = 675, p = 5.5 10⁻⁶⁸). Finally, one can also notice that neurons with high CII have low values of relative stimulus cumulative information (Linear Regression explaining CII from k_stim(300): Adj R² = 0.28, F(1,213) = 84, p = 3.96 10⁻¹⁷). This effect is caused by the correlation between invariance and selectivity as we have shown previously [40]: neurons that show the highest degrees of invariance also tend to respond to a small number of call-type categories and thus a small number of stimuli. As such, high invariance and high selectivity goes hand in hand with lower values of information.

Anatomical organization

We examined whether neurons were organized in the avian auditory cortex based on their CII, cumulative information time constants (τ) and the relative cumulative information at 300 ms relative to the maximum (k₃₀₀). Most of our recording sites were identified histologically and could be assigned to avian cortical areas that had been segregated into the thalamic recipient area, L2; intermediate primary auditory regions, L1, L3, CLM and L; and secondary areas, CMM and NCM [47–50] (n = 303/337 neurons). We also obtained spatial x,y,z coordinates of the recording sites relative to the midline, the position along the rostral-caudal axis where the lamina pallio-subpallialis (LPS) is the most dorsal, and the top of the brain (Fig 10).

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Fig 10. Anatomical organization.

A. The left panel is a schematic of the different processing stages in the ascending avian auditory system (grey areas) shown on a parasagittal view (the feedfoward pathways are in solid arrows, the feedback connections are in dotted arrows and motor areas are delineated by grey dotted lines). Our recording sites were from the primary and secondary avian auditory cortical areas (the auditory pallium) and are shown on the right panel. The recording sites from different birds were aligned in the rostral-caudal dimension using the oblique rostro-caudal long axis of L2 (dotted line). Recording site positions also varied along the medial-lateral dimension (in and out of the page) and are all superimposed here. Abbreviations: CN, cochlear nucleus; SO, superior olive; LL, lateral lemniscus; MLd, dorsolateral nucleus of the mesencephalon; Ov, nucleus ovoidalis; RA, robust nucleus of the arcopallium; HVC, used as a proper name; NCM, caudomedial nidopallium; CM, caudal mesopallium. B. Functional anatomical maps of the Categorical Information Index, the time constant τ and the relative value at 300ms k₃₀₀ of the cumulative information for stimuli for all single units with anatomical data (303/337 neurons). Each site is colored according to the value(s) of the variable being displayed. Transparency is used to reveal overlapping sites in this view as well as, potentially, combination of single units that were recorded from the same electrode and site.

https://doi.org/10.1371/journal.pcbi.1006698.g010

In all regions, we found a range of CII and thus only weak anatomical effects across areas. Using the L2 dorsal-ventral oblique axis as reference point, CII was slightly higher as one moved rostrally or caudally to higher regions of auditory processing (Linear regression: Adj R² = 0.036, F(2,195) = 8.39, p = 0.0042). However, an ANOVA also suggested that both regions NCM and L2 had slightly higher mean CII (Adj R² = 0.027, F(4,185) = 2.77, p = 0.042). We also observed an increase in the time constant (τ) for the stimulus cumulative information that parallel the increase in CII as one moved away from the L2 axis (Linear regression: Ajd R² = 0.04, F(2,195) = 9.32, p = 0.00258). The saturation constant at 300 ms (κ₃₀₀) for stimulus information was also higher in the central auditory region and smaller as one moved away from the L2 axis (Linear Regression: Ajd R² = 0.035, F(2,195) = 8.18, p = 0.0047). Thus, there is more invariance in higher auditory areas and longer time constants but less absolute information about the stimulus. Note again that these significant anatomical trends were characterized by very small effect sizes. We did not find, however, any effects of anatomy on the time constant, (τ), or saturation constants (κ₃₀₀) for the cumulative categorical information; in this data set, the subset of neurons that have high invariance have similar response properties in both lower and higher regions of the avian auditory pallium.

Ethics statement

All animal procedures were approved by the Animal Care and Use Committee of the University of California Berkeley (AUP-2016-09-9157) and were in accordance with the NIH guidelines regarding the care and use of animals for experimental procedures. Birds subjects were euthanized after neural recordings by overdose of Isoflurane.

Animals and stimuli

Four male and two female adult zebra finches (Taeniopygia guttata) were used for the electrophysiological experiments. The birds were bred and raised in family cages until they reached adulthood, and then maintained in uni-sex groups. Although birds could only freely interact with their cage-mates, all cages were in the same room allowing for visual and acoustical interactions between all birds in the colony. All birds were given seeds, water, grid and nest material ad libitum and were supplemented with eggs, lettuce and bath once a week.

Vocalizations used as stimuli during neurophysiological experiments were recorded from 15 adult birds and 8 chicks (20–30 days old). The vocalization bank obtained contains 486 vocalizations that included for each bird most of the calls in the Zebra finch repertoire: 7 call-types in adults and 2 in chicks. The adult calls included the following affiliative calls: Song (So), Distance Call (DC), Tet call (Te) and Nest call (Ne); and the following non-affiliative calls: Wsst or aggressive call (Ws), the Distress call (Di) and one of the two alarm calls, the Thuk (Th). The juvenile calls included the Begging call (Be) and the Long Tonal call (LT). Additional information about these stimuli and their behavioral meanings can be found in [39, 40, 97].

For the neurophysiological experiments, a new subset of the vocalization bank was used at each electrophysiological recording site. This subset was made from a representative number of vocalizations from the repertoire of individuals: three adult females, three adult males, two female chicks and two male chicks. From each individual caller, we randomly chose 3 call bouts from each category or fewer if fewer than 3 call bouts were obtained for that particular call-type and individual. The maximum number of stimuli that could be selected in that procedure was therefore 3x7x3 (adult males x repertoire x renditions) + 3x6x3 (adult females x repertoire x renditions) + 4x2x3 (juveniles x repertoire x renditions) = 141. The average number of stimuli played for each single unit was 114 (sd = 22, min = 34, max = 123). Ten trials were acquired for each stimulus with a few exceptions (min = 9, max = 11). Sounds were broadcasted in a random order using an RX8 processor (TDT System III, sample frequency 24414.0625 Hz) connected to a speaker (PCxt352, Blaupunkt, IL, USA) facing the bird at approximately 40cm. The sound level was calibrated on song stimuli to obtain playbacks at 75dB SPL measured at the bird’s location using a sound meter (Digital Sound Level Meter, RadioShack).

Neurophysiological and histological procedures

Extra-cellular electrophysiological recordings were performed in 6 urethane anesthetized adult zebra finches. The birds were placed in a sound-attenuated chamber (Acoustic Systems, MSR West, Louisville, CO, USA) and sound presentation and neural recording were performed using custom code written in TDT software language and TDT hardware (TDT System III). Sounds were broadcasted in a random order as described above. Neural responses were recorded using the signal of two (5 subjects) or one (1 subject) electrode arrays, band-pass filtered between 300Hz and 5kHz and collected by an RZ5-2 processor (TDT System III, sample frequency 24414.0625 Hz). The electrode arrays consisted of two rows of 8 tungsten electrodes with row separation of 500 μm and inter-electrode separation within row of 250 μm (Tucker Davis Technologies, FL). Electrode impedances were approximately 2 MOhms. When two electrode arrays were used, they were placed each in one hemisphere. Spike arrival times and spike shapes of multiple units were obtained by voltage threshold. The level of the threshold was set automatically by the TDT software using the variance of the voltage trace in absence of any stimuli. Electrodes were progressively lowered and neural responses were collected as soon as auditory responses to song, white noise, Distance call or limited modulation noise could be identified on half of the electrodes in each hemisphere (the stimuli used to identify auditory neurons were different from the stimuli used in the analysis). Several recording sites were randomly selected by progressively deepening the penetration of the electrodes and ensuring at least 100 μm between two sites. This distance between recordings insured that we did not record from the same neurons at separate sites. On average 4.2±2 sites (mean ± sd) were recorded per bird and per hemisphere at a depth ranging from 400 μm to 2550 μm.

After the last recording site, the subject was euthanized by overdose of isoflurane and transcardially perfused. Coronal slices of 20μm obtained with a cryostat were then alternatively stained with Nissl staining or simply mounted in Fluoroshield medium (F-6057, Fluoroshield with DAPI, Sigma-Aldrich). While Fluoroshield slices were used to localize electrode tracks, Nissl stained slices were used to identify the position of the 6 auditory areas investigated here: the three regions of Field L (L1, L2 and L3), 2 regions of Mesopallium Caudale (CM): Mesopallium Caudomediale (CMM) and Mesopallium Caudolaterale (CLM); and Nidopallium Caudomediale (NCM). By aligning pictures, we were able to anatomically localize most of the recording sites and calculate the approximate coordinates of these sites. Since we could not localize the Y-sinus on slices, we used the position of the Lamina Pallio-Subpallialis (LPS) peak as the reference point for the rostro-caudal axis in all subjects. The surface of the brain and the midline were the reference for respectively the dorsal-ventral axis and the medial-lateral axis. The approximate coordinates of units were used to build 3-D reconstructions of all single unit positions in an hypothetic brain. We were not able to recover all the electrode traces in one of the six birds and excluded that data from the anatomical analysis. Table 1 shows the number of units from each bird for which we had an anatomical location and that had significant stimulus cumulative information (303/337 neurons; see below).

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Table 1. Number of units per bird and per anatomical area that was included in our anatomical analyses (n = 303).

The UNC column counts the units that we left as unclassified because they were found at the boundary between regions.

https://doi.org/10.1371/journal.pcbi.1006698.t001

Single unit isolation was performed off-line using custom software that used a combination of supervised and unsupervised clustering algorithms. These clustering algorithms used the spike-snippets shape as described by a PCA. Sorted units where declared to be single units based on spike shape reliability across snippets. The spike shape reliability measure was a signal to noise ratio (SNR) where the signal was the difference between the maximum and the minimum of the average snippet and the noise was the standard deviation of this measure across all snippets. Single units in our data set have an SNR > 5. Additional details on these experimental procedures can be found in [40].

The data of neural responses from 404 out of 914 isolated single units were used in this study. The 404 neurons were selected based on a prior analysis that showed that this subset of units were not only auditory but also contained information about call-types, in the sense that call-types could be decoded above chance level from neural responses (see [40]). Here, we analyzed the neural response in the first 600ms after stimulus onset.

Data analysis: Time-varying spike rate estimation

Calculations of cumulative information requires the estimation of very large distributions which sizes grow exponentially with the number of time points investigated. It is therefore imperative to find the optimal width of the time bin for the analysis: as small as possible to capture information in spike patterns but as large as possible to reduce the dimensionality for the estimation of mutual information. It is also well known that the relevant time scale can be different in different epochs of the responses (e.g. more precise spike timing at the onset phase of the response and less precise during the sustained phase) [98]. Here, we addressed this issue by first estimating the time-varying rate of our neurons using methods of density estimation based on adaptive kernels. In this methods, narrower kernels are used when spike precision is high and wider kernels at times when the spike precision is low. We then estimated the largest fixed time window that could capture most of the fastest changes in those average time-varying spike rate. Alternative methods based directly on measures of information have also been proposed [98].

For each stimulus, the 9 to 11 raw spike patterns of 600ms, sampled at 10kHz, were combined to obtain the corresponding time varying spike rate (sample frequency set at 1kHz) by applying a locally adaptive kernel bandwidth optimization method [42]. In cases where the neuron did not respond to any of the presentations of the stimulus or responded only once over all presentations, the rate was estimated as being constant for the 600ms duration of the neural response. For those two unresponsive cases, the rate was set to be 1/(2*Ntrials*NTimes) in the absence of any spike or 1/(Ntrials* NTimes) in case of one spike, with Ntrials the number of stimulus presentations (9–11) and NTimes the number of time points at which the rate was estimated (here 600, for a 600ms neural response section with a sampling rate set at 1kHz).

To investigate cumulative information values up to 600ms after stimulus onset, time-varying rates were sampled at 10ms (Nyquist limit frequency of 50Hz). To estimate, the amount of information potentially lost by this low-pass filtering, we estimated an information value based on coherence analysis of the signal to noise ratio in the raw spike train. The coherence between a single spike train (R) and the actual time-varying mean response (A) can be derived from the coherence between the peristimulus time histogram (PSTH) obtained from half of the trials and the PSTH obtained from the other half [99]. where M the total number of trials (presentations of the stimuli) and the coherence between the two PSTHs calculated on half of the trials. The coherence between two responses is a function of frequency (ω). An estimate of the mutual information (in bits per second) between R and A responses can then be obtained by integrating over all frequencies [4, 99]:

For each unit, we estimated the percentage of information preserved as the ratio between I_AR calculated up to 50Hz and I_AR calculated over all frequencies. Over all units, 96.7% ±6.9% (mean ± SD) of information was conserved by a lowpass filtering at 50Hz and only 88 out of 404 cells had information losses greater than 5%. For each unit, we also calculated the proportion of cumulative power across frequencies in the time varying spike rate estimation obtained with the KDE before low-pass filtering and down-sampling (averaged periodogram in overlapping 200 ms Hanning windows and 1 kHz sampling rate). The cumulative sum of the power was calculated across frequencies and normalized by the maximum power value to obtain the proportion of cumulative power. On average across units, the cumulative power reached 98.8%±1.6% (mean ± SD) at 50Hz, further validating our choice of the temporal resolution (S1 Fig).

Testing the poisson assumptions

To estimate values of mutual information, we assumed that spike counts in our 10 ms windows had a Poisson distribution. To assess the validity of this assumption, we calculated for each neuron, for each stimulus s_i and for each time bin t, the log-likelihood function from the 10 spike counts y_j each obtained in 1 trial j assuming a Poisson distribution with the mean response given by the delete one estimate.

We then generated a bootstrap distribution of 1000 of these likelihood measures each obtained for 10 samples from a Poisson with mean rate given by our KDE estimate. Finally, we calculated the number of times the log-likelihood value obtained for the data was greater or smaller than the bootstrap values to generate a p-value (two-tailed test). Given the very large number of statistical tests, a False Discovery Rate correction, the Benjamini–Hochberg procedure, was used for each neuron to estimate the number of bins for which the Poisson hypothesis could be rejected with p = 0.05.

In our estimations of mutual information, we also assumed that the conditional probability of the response given a stimulus was independent between time bins. In other words, knowing that the spike count is greater (or smaller) than the mean in one time bin provides no information about probability of spike counts in the same trial in another time bin. To test this assumption, we calculated the noise correlation. The noise correlation is simply the Pearson correlation coefficient obtained from successive time bins. A single noise correlation was obtained for each neuron by averaging across pairs of time bins and stimuli.

Information theoretic calculations

As described in the results, the time-varying instantaneous mutual information between the stimulus S and the response Y_t can be written as a difference in Shannon entropies: and the cumulative mutual information for neural responses that are discretized into time intervals is given by:

In the present paper we calculated 4 different types of information: the stimulus instantaneous information, the categorical instantaneous information, the stimulus cumulative information and the categorical cumulative information. Stimulus instantaneous and cumulative information were calculated for all 404 units, while categorical instantaneous and cumulative information were calculated on a restricted set of 337 neurons that presented at least one time point with a significant value of stimulus cumulative information (significant threshold set as 3 times the local error, see below for error calculations). While we verified our assumptions (minimal information loss with spike rate binning, Poisson distributions of spike counts and maximum value of spike counts) on the full set of 404 units, the population analysis of time-varying information presented in the results section only include the relevant dataset of 337 neurons.

The custom Matlab code used to calculate time-varying information values is available at https://github.com/julieelie/PoissonTimeVaryingInfo along with a tutorial on how to use the core functions.

The cumulative information.

The conditional probability of a time varying response is the joint probability of observing (y_t, y_t−1, y_t−2,…) given s_i. Given our Poisson assumption, the conditional probability of response at t is independent of the conditional response at previous times.

We can show that the conditional entropy of the joint responses is the sum of the individual entropies:

The probability of the time varying response is the joint probability of observing (y_t, y_t−1, y_t−2,…). This joint probability cannot be expressed as the product of the probabilities at different times because these are not independent. The joint unconditional probability distribution is:

This joint probability distribution could be expressed as a product of probabilities by assuming that most of the interdependence can be calculated from the previous time point, as in the Markov chain assumption at a beginning of the time series at t = 0. In all cases, the joint probability distribution can be written as: or when it is approximated by a first-order Markov chain:

The conditional probability of y_t given y_t−1 is:

Note that this joint probability distribution can have very high dimensions. Assuming the number of spikes y_t ∈ [0, 19], the number of probabilities that must be estimated is 20^nt where nt is the number of windows in time. For example, calculating all the probability of all the outcomes for 10 windows (100ms) requires 20¹⁰~10¹³ calculations. On the other hand, the Markov chain approximation only requires the estimation of all pair-wise joint probability distributions: for 10 windows and 20 outcomes, the number of calculations is (10)(20²)~4000.

The response entropy is then calculated from the joint probability distribution:

To estimate this response entropy, we investigated various approaches: a time-running cumulative information, the Markov chain approximation and Monte Carlo with importance sampling. Monte Carlo with importance sampling gave the best results and was therefore used in our analyses. The Markov chain approximation overestimates the entropy and therefore can also be used to obtain an upper bound on the cumulative information, which could be useful in other applications. We briefly describe the three approaches. The time-running cumulative information consisted in calculating the full cumulative information (using all possible spike events) but only for a fixed number of successive time windows. We estimated that we could easily calculate all possible probabilities for 4 time-windows, corresponding to a 40 ms history. This approach gave the best estimate of the information in 40 ms windows but, in our system, grossly underestimated the cumulative information: some of the information in successive 40 ms is clearly independent (S3 Fig).

The second approximation was based on Markov chain of variable orders up to 4 (also 40 ms). With this approximation, we overestimated the cumulative information: the correlation time of the time-varying rates for different stimuli is clearly also greater than 40 ms. Using the first order Markov chain approximation:

Expanding that sum, the last term is:

The second to last term is:

The third to last term is: and, similarly, for all the other terms.

Thus, the response entropy using the Markov chain approximation is: with

This approximation can be extended using estimates at two previous time points, etc.: where the conditional probability based on two prior measurements is: where

Finally, we estimated cumulative information using Monte Carlo with importance sampling. In Monte Carlo in conjunction with importance sampling, N_i samples or, here, time varying responses , are taken from a proposal distribution, . The actual probability, , is calculated exactly at those samples and an estimate of the expected value of the measure of interest (here ) is obtained by the average of at the sample points, , weighted by the likelihood ratio p/q:

Our proposal distribution was based on the distribution at each time point and assuming independence across time:

For each sample obtained from the proposal distribution q we calculated the actual probability value using: and used that probability value in the estimation of the entropy. Monte Carlo samples were chunked in groups of 100,000 samples and at each additional sample chunk, a bootstrapped/jackknife bias corrected mean and standard error were estimated (see below). Sampling stopped when the standard error was below 0.2 bits or at a max number of samples set to 5,000,000 samples. If the error at the maximum number of samples was greater than 0.6 bits, the estimation was deemed unreliable and the calculation was not performed for any successive time points.

Calculation of the expected value of categorical information given the stimulus information

We computed a Categorical Information Index (CII) that compared the empirical categorical cumulative information for call-type categories, CCI, to three hypothetical values: a floor (CCI_Floor), an expected value (CCI_Exp) and a ceiling value (CCI_Ceil). The floor value is the categorical cumulative information obtained from random categories. The expected value is the categorical cumulative information that would be achieved if the stimulus information was 1) equally distributed for each stimulus and 2) could be used for classifying stimuli into groups. Note that the second assumption is not necessarily true in the actual data because the categorical information is based on averaging the probabilities for stimulus from the same category and thus effectively averaging time-varying rates. If time-varying rates are not grouped by categories, then it is possible that two stimuli from two different categories are distinguishable based on their time-varying rate, but that, the average time-varying rates for the two categories are not distinguishable, or less so than expected from the average pair-wise distances. The ceiling value corresponds to the case where all the cumulative information about stimuli is used for the categorization and none to discriminate stimuli belonging to the same category: CCI_Ceil. The CII is a number between 0 and 2 that is then calculated as:

The following three steps were taken to calculate the expected categorical information (CCI_Exp) from the stimulus cumulative information. First, the stimulus mutual information, mi, was expressed as the conditional probability of correct stimulus decoding, p, for any given stimulus (and assumed to be equal for all stimuli). Given a confusion matrix of size nxn obtained from a decoder for n stimuli, with p as the conditional probability given a stimulus of correct decoding (diagonal terms) and thus (1-p)/(n-1) as the conditional probability of error (off-diagonal terms), the mutual information is equal to:

The above equation was inverted numerically to solve for p given mi. Second, a new matrix was generated by grouping rows and columns of joint probabilities (and not conditional) to form a confusion matrix for categories. The number of stimulus in each category was matched to the actual values in the neurophysiological data on a unit per unit basis. Third, the expected mutual information for categories was then estimated from this new confusion matrix by subtracting the total entropy obtained from the joint probabilities, from the sum of the entropies of the marginal distributions for the rows and columns: