To survive, organisms must extract information from the past that is relevant for their future. How this process is expressed at the neural level remains unclear. We address this problem by developing a novel approach from first principles. We show here how to generate low-complexity representations of the past that produce optimal predictions of future events. We then illustrate this framework by studying the coding of ‘oddball’ sequences in auditory cortex. We find that for many neurons in primary auditory cortex, trial-by-trial fluctuations of neuronal responses correlate with the theoretical prediction error calculated from the short-term past of the stimulation sequence, under constraints on the complexity of the representation of this past sequence. In some neurons, the effect of prediction error accounted for more than 50% of response variability. Reliable predictions often depended on a representation of the sequence of the last ten or more stimuli, although the representation kept only few details of that sequence.
Organisms often operate in unknown and uncertain environments. Therefore, extracting aspects of past observations, which are maximally predictive of the relevant future, is essential for survival. It has been suggested that the sensory cortex evolved to extract the statistical regularities of the world [1]. Adaptation of the nervous system to the statistical structure of the input is reflected in studies of neuronal responses to natural stimuli. For example, in the auditory system, auditory nerve fibers–part of the auditory periphery–have been shown to achieve high coding efficiency by implementing a “tuned” nonlinear filter that selectively amplifies the anticipated signal [2]. Similarly, in the visual system, Laughlin [3] showed that the contrast-response function of interneurons in the fly' s compound eye approximates the cumulative probability distribution of contrast levels in natural scenes. The central auditory system shows sensitivity to stimulus statistics as well. Event-related potentials recorded in humans show sensitivity to deviant stimuli. This sensitivity may occur rather early, at the mid-latency potentials range [4], and has been intensively studied in the context of the mismatch negativity (MMN), peaking about 150 ms after the point of deviance [5]. Similar sensitivity occurs also in the responses of single neurons in auditory cortex: using oddball sequences composed of two frequencies with different probabilities, Ulanovsky et al. [6] found that neurons in cat auditory cortex responded more strongly to a given tone frequency when it was rare than when it was common. This sensitivity, named stimulus-specific adaptation (SSA) by Ulanovsky et al. [6], has been by now shown in multiple mammalian species and even in birds (See [7] for review). SSA may be linked to deviance detection in the time frame of the mid-latency potentials rather than to MMN (Grimm, Escera and Nelken 2015). Thus, we hypothesize that neurons in the auditory system encode some notion of a prediction error. The coding of prediction error is considered central to learning [8], memory formation [9], and decision-making [10]. Prediction errors are also known to be related to efficient coding where only the unexpected at one stage of processing should be transmitted to the next stage [11]. This paper has two goals. The first is to present a theory of prediction error from first principles. For an organism that operates in a statistically stationary world (as is generally the case in laboratory experiments), prediction quality is limited by the statistical structure of the data that the organism collects from the past. The theory depends only on this statistical structure, and not on any assumptions about specific brain mechanisms. We assume that the brain forms a reduced representation of the past, which serves to generate predictions of future events. We use information theory to quantify both the complexity of these reduced representations and the predictive information they carry with respect to future events [12]. The term complexity refers here to the rate of information (specifically, the mutual information in bits/s or, equivalently but more conveniently here, in bits/stimulus) that the representation carries about the past. Predictive information refers to the rate of information (again, mutual information in bits/s or bits/stimulus) that the reduced representation carries about future events. Both terms are defined precisely below. Extraction of the predictive aspects of the past stimuli can be formalized as an optimization problem: minimize the complexity of the reduced representation of the past while preserving a predefined level of predictive information. A reduced representation provides a predictive probability for every event–this is the probability assigned to the event just before it actually occurred, given the reduced representation of the past. Our theory uses these predictive probabilities to calculate prediction errors–events with low predictive probability generate large prediction error, while events with a high predictive probability generate small prediction errors. The constrained optimization problem we use to calculate the reduced representations is a special instance of the Information Bottleneck (IB) principle [13]. The IB principle applies to any two random variables X and Y with a known joint distribution. Like the special case described here in details, the IB principle provides a way of finding reduced representations (as defined later in the paper) of X that are maximally informative about Y. Here we apply it to the past of the stimulation sequence (X) and to its future (Y). Much of the detailed discussion below can be considered as a primer for the use of the IB principle. The second goal of the paper is to demonstrate the usefulness of the theory by applying it to the neuronal responses evoked by random tone sequences consisting of two frequencies with varying probabilities (‘oddball’ sequences). We show that prediction errors calculated by the theory correlate well with neuronal responses. Most importantly, we use the theory to extract parameters of the reduced representations that underlie these responses: we show that these reduced representations have a long duration (typically N ≥ 10 preceding stimuli) but keep only coarse details about the sequence that was presented (i. e. , the reduced representations have low complexity). These results show how to establish properties of the neural code rigorously from first principles.
In order to demonstrate the relevance of our approach to neuronal processing, we studied responses of single neurons in primary auditory cortex (A1) to oddball sequences. Oddball sequences are generated by selecting two stimuli (two pure tones in our case) and then forming a sequence composed of these two stimuli in which one of the two is common and the other is rare (Fig 3A). Usually, the overall number of times each of the stimuli occur in the sequence is fixed. We will however test here a slightly different statistical model, in which the probability of each of the tones is fixed. We will furthermore assume that successive tones are selected independently of each other with a given probability (‘Bernoulli sequences’). This assumption, which is a very good approximation to the experimental setting when the number of tones in the sequence is large, makes the application of the theory particularly transparent. Neurons in auditory cortex are sensitive to stimulus probability in such sequences [6,19–23]. The majority of neurons in cat auditory cortex produced a larger average response to the same tone frequency when it was rare than when it was common (Fig 3B). These responses often had substantial sustained components that could outlast stimulus duration (e. g. [24]), resulting in large spike counts within the counting window we used (0–330 ms after stimulus onset, for stimuli whose total duration was 230 ms, same as in [6,20]). We interpret these responses as encoding a prediction error: in the oddball sequences, the probability of the next stimulus to be either of the two frequencies is roughly given by the probability with which it occurred in the past. The prediction error, −log2 (p), for the tone when rare is therefore larger than for the same tone when common. Thus, responses of neurons in auditory cortex seem to fit qualitatively the notion of prediction error as defined here. Using the theory presented above, we aimed to quantify this intuition rigorously. We start by describing in substantial detail the family of optimal reduced representations in the case of the oddball sequences, and then we show how we used these representations to extract information about the memory duration and complexity of the reduced representation underlying the dependence of neuronal responses in auditory cortex on tone probability. In the experiments, sequences with a number of different probability values have been used: each stimulus was used both as common (with probabilities of 90% and 70%) and as rare (with probabilities of 10% and 30%), as well as in equiprobable sequences. Formally, the stimulus sequence can be modeled as a Bernoulli process, where one stimulus is drawn with probability p (0<p<1) and the other with probability 1-p. We assume that the parameter p is drawn from a uniform distribution prior to generating each stimulus block. This assumption is not crucial–it changes the exact numerical values, but not the trends that will be discussed below, as long as all the tone probability conditions that occurred in the experiment are allowed under the prior. These assumptions determine the joint probability PN (past, future) characterizing the statistical structure of the stimulus sequence, where past denotes the sequence of the last N stimuli and future denotes the next stimulus in the sequence (see Methods). The memory duration, N, is a parameter of the model and will be discussed in more details below. As we discussed above, the most detailed (and complex) representation in this case would consist of a perfect encoding of the past sequence by identifying each of the 2N possible configurations of the stimulation sequence with a unique state m (Fig 2A; case (i) is this representation for N = 4). This is, however, an unnecessarily detailed representation since the number of occurrences of the B tone among the last N is a sufficient statistic. The IB method identifies the relevant information for predicting the next stimulus: in our case, it assigns a single and unique state to all sequences that share the same number of occurrences of A tones and B tones (Fig 2A; case (ii) ). This reduced representation filters out non-relevant details of the past (i. e. the exact order of the tones): it maintains the maximal predictive information possible (for a given N), but at a lower complexity (2. 3 bits instead of 3. 6 bits for N = 4). Fig 2B (gray line) displays the maximal predictive power at each complexity of the reduced representation for N = 4, and cases (i) and (ii) can be directly compared to each other. Using the IB method enabled us to reduce the complexity of the representation even further. While representations that are simpler than the minimal sufficient statistic (case ii) do reduce the quality of the prediction of the next stimulus, the loss in predictive power may be relatively small. For example, for a memory of N = 4 tones, constraining the reduced representation complexity to 1 bit results in a ‘noisy’ representation with 3 unique states (Fig 2A and 2B, case (iii) ). In this case, given the reduced representation alone, it is impossible to recover the exact number of B tones that occurred in the past (their exact order was already lost in the reduction to the minimal sufficient statistic). Nevertheless, this solution is optimal in the sense that its predictive power is maximal among all possible reduced representations with a complexity of 1 bit (for N = 4). For longer past durations and large complexities, the low sensitivity of predictive power to complexity is even more prominent. For a memory of N = 10 tones, the complexity reduction when using the sufficient statistic is larger than for N = 4, and further reducing the complexity of the representation by another 57% (from 3. 46 to 1. 49 bits) results in a loss of merely 12% in its predictive power (from 0. 223 to 0. 196 bits; Fig 2B, black). We examined the tradeoff between the complexity and the predictive power of the reduced representation for memory durations from N = 1 to N = 50 (Fig 4A) by applying the IB method to the probability distributions of the oddball paradigm PN (past, future). Note that these probability distributions depend on N, the duration of the past. We used the IB algorithm to find, for each N, 200 reduced representations whose complexity spanned the full allowed range (see below), and whose predictive power was maximal for each value of the complexity. Each curve in Fig 4A is produced by linearly interpolating between these 200 points. The 200 points, however, sample the relevant ranges so densely that the linear interpolation is imperceptible. Thus, each point along the curves shown in Fig 4A corresponds to a unique optimal reduced representation: as described above, such reduced representation is given by the two conditional distributions P (m|past) and P (future|m). The curves in Fig 4A plot the predictive power as a function of the complexity for these 200 reduced representations (the N = 4 and N = 10 cases are the same lines plotted in Fig 2B). These convex curves describe quantitatively the tradeoff between complexity and predictive power. Their convexity is a general property of the tradeoff postulated by the IB principle (see [13]). For each N, complexity (abscissa of Fig 4A) spans the range between 0 and the complexity of the sufficient statistic. The predictive power (ordinate in Fig 4A) spans the range between 0 and the mutual information between past and future, which is equal to the mutual information between the sufficient statistic and the future. Thus, the rightmost point of each line shows the complexity of the sufficient statistic (abscissa) and the predictive power of the sufficient statistic (ordinate). More complex representations do exist (e. g. the full past), but do not result in an increased predictive power. As complexity is lowered below that of the sufficient statistic (moving leftward along a curve), predictive power is lost, but at least initially the loss of predictive power is rather minor. The curves separate achievable and non-achievable combinations of complexity and predictive power. Any point on or below the curve is achievable, in the sense that there is (at least one) pair of conditional distributions P (m|past) and P (future|m) with the corresponding complexity and predictive power. Any point above the curve is non-achievable in this sense–there is no way to process the information from the past, keeping that level of complexity, and still get better predictions than those specified by the curve. The dependence of optimal predictive power on N, the duration of the past, is illustrated in Fig 4B for different constraints imposed on the complexity (0. 5,1, 2 bits and with no constraints). Two effects are readily apparent. First, for each level of complexity, increasing past duration much above N = 10 does not result in a major increase in predictive power. Second, as hinted above, although the complexity of the sufficient statistic for long memory duration may be as high as 4–5 bits (Fig 4A), there is a strong effect of ‘diminishing returns’–complexity above 2 bits does not add substantial amount of predictive power to the reduced representation, even for a past duration of N = 50 stimuli (Fig 4B). One way of understanding these effects is by noting that the predictive power reflects the precision of the probability estimates of the B tone. While this precision increases with both N and complexity, beyond a certain point increasing the precision at which the probability of the B tone is known does not improve much the predictions anymore. To test the hypothesis that neurons in the auditory cortex represent the prediction error derived from a reduced representation of the recent past, we correlated the prediction errors, derived from the reduced representations we computed, with the neuronal responses to oddball sequences (Fig 5) [6,20]. We calculated separately the prediction errors for each one of the reduced representations we computed above (defined by memory duration N = 1 to N = 50 and 200 complexity values ranging from 0 to the maximum possible for each N, for a total of 200*50 = 10,000 reduced representations). For each reduced representation, for each neuron in the dataset, and for each of the two frequencies with which the neurons were tested, we used the actual sequence of tone presentations used in the experiment to calculate a corresponding sequence of prediction errors. We followed the prescription described above. For each tone presentation along the sequence, we used the previous N tone presentations as the past. Each past leads to a state m of the reduced representation, with an associated estimate P (future|m) for the probability of the upcoming stimulus. The prediction error associated with that stimulus was −log2 P (future|m). As explained above, the prediction error itself is a random variable. The transformation from past to reduced representation, P (m|past), provided a set of probabilities to be in each of the states of the reduced representation, and through them to each of the possible prediction errors (see Methods). Therefore, instead of a single prediction error for each stimulus, we calculated a set of prediction errors, one for each possible state of the reduced representation m, together with their probabilities. To compare the prediction errors with the neuronal responses, we used linear regression of the neuronal responses against the prediction errors, weighted by the corresponding probabilities (see Methods for details). For illustration purposes, in Fig 5 we plot neuronal responses (spike counts evoked by individual tone presentations, ordinate) against the average prediction error (−log2 P (future|m) averaged over all possible values for the state m of the reduced presentation, each with its probability P (m|past) ) for that same stimulus presentation. In the experiments, neurons were tested using blocks composed of two tones whose frequency separation was about half octave (fhigh/flow = 1. 44; this is the Δf = 0. 37 case of [6,20]). The probabilities of the two tones in each block were fixed, but their order was random. For the main analysis (Figs 5 and 6), we used a conservative counting window (0–330 ms after stimulus onset; tone duration was 230 ms and onset-to-onset interval was 730 ms), and we used the responses of all 68 neurons tested with 3 blocks in which the tone probabilities were 90%/10%, 50%/50%, and 10%/90% (for the low frequency and high frequency tones respectively). The use of responses from these 3 blocks ensured sampling of the full range of relevant values of prediction errors. The analysis was performed separately for each frequency, resulting in 136 combinations of neuron and frequency, with 117/136 combinations showing responses to the corresponding tone that were significantly larger than the spontaneous rate. Prediction error was significantly correlated with neuronal responses in many cases. Fig 5A compares responses of one neuron (spike counts evoked by single presentations of one of the two frequencies with which it was tested) with the expected prediction errors calculated for each tone presentation using three different optimal reduced representations. These representations had a complexity of 1 bit and past durations of N = 5, N = 10 and N = 15 stimuli. Each block had 400 stimuli, of which 90% (360), 50% (200) or 10% (40) consisted of the relevant frequency. To avoid using stimuli whose past sequence was not well defined, the first 50 stimuli of each block were removed from the analysis. Due to the randomized nature of the sequences, for each combination of neuron and test frequency we ended with 311–322 stimuli in the p = 90% condition (blue points), 168–182 stimuli in the p = 50% condition (black points), and 28–39 stimuli in the p = 10% condition. As the tones presented in the 10% condition were, by definition, rare, they were associated with large prediction errors, and therefore the red points are mostly concentrated at the right of the scatter plots. Similarly, the tones presented in the 90% condition were, by definition, common, and were associated with small prediction errors. Therefore the blue points are mostly concentrated at the left of the scatter plots. On the other hand, for the 50% condition, prediction errors varied quite substantially, leading to a larger dispersion along the abscissa between (and partially overlapping) the two extremes. While the correlation with the prediction errors corresponding to N = 10 was slightly larger than for shorter or longer memory durations, this example mainly illustrates the finding that the correlation between the prediction error and neuronal responses was sometimes only weakly dependent on past duration, a finding we will return to later. To illustrate the range of goodness of fit that could be achieved, Fig 5B compares single-trial responses of three neurons (columns; the rows show the two frequencies used for testing each neuron) versus their expected prediction errors, calculated using reduced representations with complexity of 2 bits and past duration of N = 10 stimuli. Neurons 1 and 2 had strong dependence of their responses on prediction error, while Neuron 3 was typical for the data. To quantify these observations for the entire data set, we performed weighted linear regression analysis, separately for each combination of a neuron and frequency that evoked significant responses (n = 117 combinations), using the entire set of pre-calculated optimal solutions. The distribution of the resulting best goodness-of-fit scores (rmax2) is displayed in Fig 6A. We tested the significance of the fit separately for each combination of neuron and frequency. The goodness of fit scores were calculated as the maximum r2 over all possible reduced representations, so that standard significance tests could not be applied due to the inherent multiple comparisons involved. We therefore tested the significance of a fit using a permutation test. We repeated the same analysis for the measured spike counts but replacing the stimuli by 20 randomly permuted sequences with the same tone probabilities, thus breaking the short-term relationships between responses and associated prediction errors. The effect of prediction error on the neuronal responses was considered to be significant (p<0. 05) if rmax2 for the actual stimulation sequence was larger than rmax2 calculated using 20 different random permutations. The prediction error significantly influenced the neuronal responses in about 2/3 of cases (78/117 combinations, 67%; see Fig 6A, red bars), supporting our hypothesis that auditory cortical neurons represent this formal notion of prediction error in their responses. Most neurons contributed to Fig 6A twice, so the figure could overstate the degree to which prediction error and neuronal responses are related to each other. Fig 6B is a scatter plot of the goodness of fit values for the two frequencies, with the corresponding histograms for the two frequencies separately. There was a mild correlation between the goodness of fit values for the two frequencies (r = 0. 34, n = 68, p<0. 05, using the normal approximation to the Fisher z-transformed correlation), suggesting that the representation of prediction error in the neuronal responses was at least to some degree a property of the neuron itself. Considering each frequency by itself, 41/62 neurons that responded significantly to the low frequency and 37/55 of those that responded significantly to the high frequency showed a significant dependence of neuronal responses on prediction error. More conservatively, 52 neurons had significant responses to both frequencies, and 27 of these neurons showed significant dependence of the responses to both frequencies on prediction error. Thus, depending on the criterion, between 1/2 and 2/3 of the neurons showed significant dependence (p<0. 05) of the neuronal responses on prediction error. In order to characterize the past duration and the complexity of the representations that best fitted the neuronal responses, we analyzed the parameters of these representations that best explained the data. For this analysis we used only combinations of neuron and test frequency with explanatory power of rmax2≥0. 1 (adjusted correlation coefficient > 0. 33, n = 34/117, see Methods). We found that the dependence of the goodness of fit scores on the parameters (memory duration and complexity) could be rather weak once either past duration or complexity were large enough, so that many different models achieved approximately the same goodness of fit. Therefore, we defined, for each combination of neuron and test frequency, a set of good representations consisting of those that achieved goodness-of-fit scores higher than an (admittedly somewhat arbitrary) threshold: r2≥0. 9∙rmax2. The parameters of these ‘good representations’ (past duration, complexity and predictive power) are summarized in two-dimensional, color-coded histograms (Figs 6C–6E). For each reduced representation, the histograms show the fraction of cases for which that specific representation belonged to the set of good representations. The reduced representations are index in three different ways (by complexity and predictive power, Fig 6C; complexity and duration, Fig 6D; predictive power and duration, Fig 6E). Thus, Fig 6D shows that reduced representations with duration N < 10 stimuli were outside the set of good representations for most cases. Thus, for a reduced representation to be good for a large fraction of cases, it had to have a relatively long past duration, N ≥ 10 stimuli (corresponding to 7. 3 seconds or longer). Similarly, Fig 6D shows that for a reduced representation to be good for a large fraction of cases, it could have a complexity as low as about 2–3 bits (white line in Fig 6D) –reduced representation with lower complexities tended to be outside the set of good representations for most cases. Moreover, although the resulting reduced representations were relatively coarse (low complexity) their predictive power almost matched the predictive power attainable with models that have maximal complexity (Fig 6C and 6E; compare with Fig 4A and 4B). Thus, the good reduced representation with shortest memory tended to have a long memory (N>10), coarse (complexity of 2–3 bits relative to maximal complexity of ~5 bits), but keep a high degree of predictive power.
We present here an information theoretic formulation for the problem of sensory perception and prediction in the brain. We suggest that the representation of past stimuli reflects a tradeoff between the complexity of the reduced representation and its predictive power. The prediction errors, calculated from the statistics of the stimulation sequence using first principles, were significantly associated with the measured neuronal responses. Importantly, these results suggest that the relevant time scale for the calculation of the prediction error is very long relative to the time scale usually considered in sensory coding–on the order of several seconds or longer. Our results therefore suggest that neurons in auditory cortex rely on surprisingly long time scales to calculate prediction errors, although the representations of the past, which underlie the computation of the prediction errors over these time scales, were found to be rather coarse. Attneave [28] and Barlow [29] already suggested that neural information processing might follow principles of information theory. Information theory provides universal bounds on the minimal expected prediction error that can be achieved, independent of other assumptions on the implementation of a predictive process by the brain. In this sense the theory is normative: it specifies absolute bounds that cannot be improved in any way, and that are achieved by specific reduced representations of the past sequence. These bounds are governed by only two parameters, the complexity of the reduced representation (or, alternatively, its predictive power) and the duration of the past memory used for perception. The notion of predictive coding is not new. The importance of the interplay between the representation complexity and the accuracy of predictions has been noticed before (e. g. [12]). In recent years, the tradeoff has been used extensively by Friston [30]. Friston used an approximation to the exact Bayesian inference problem, which is very difficult. Our approach circumvent this difficulty: crucially, it differs from previous attempts to formally use the interplay between complexity and prediction quality in that the prediction error in our formulation is derived from a reduced representation rather than directly from the explicit past stimulus sequence. Most importantly, one unique aspect of our approach is that we use a model-independent (information theoretic) bound. In spite of the generality of this approach, we could apply it to real experimental situations: we show how to use it for studying rigorously the neuronal code in one simple case. It turns out that the neural responses reflect the prediction error derived from efficient reduced representations, allowing us to extract bounds on the duration of the memory that underlie the observed neuronal responses as well as on its complexity: memory duration is remarkably long (longer than 10 stimuli back) but rather coarse (with a complexity less than 2 bits). The theory presented here can be considered as part of a more general theory of neural function. Since the reduced representation is assumed to be internal to the organism, its complexity may be related to the metabolic cost required to maintain and update it. As we illustrated in the case of the oddball sequences, less complex representations in our sense are also simpler to implement biologically–they require a smaller number of states, and incorporate noisy assignments of past sequences to states of the representation. On the other hand, constraints on expected prediction error may stand for general constraints on future value. Thus, our theory can be seen as an application of the general principle that organisms attempt to minimize metabolic or other costs subject to future value constraints. This principle unifies most known learning and control theoretic models, potentially linking information theoretic measures with general biological constraints directly [31]. Our approach provides a principled way to study long-term dependencies in neuronal responses, which are inaccessible to many models of auditory responses [32–34]. These models relate the neuronal responses with the preceding content of the stimulus within a short time window (typically 50–100 ms). Attempts to include context in such models end up with analyzing relatively short-term contextual effects, from a few ms [16] to a few tens of ms [35] and up to a few hundreds of ms [36,37]. Only a few attempts to quantify longer-range dependencies have been published. For example, Ulanovsky et al. [20] demonstrated (in a previous analysis of the data presented here) that the larger responses to rare tones could not be accounted for by taking into account only a recent past (N ≤ 4 preceding stimuli, 3 seconds). They concluded that the larger responses to rare stimuli depended on longer segments of the stimulation sequence, although the effective memory duration and its content were left unspecified in these studies. Similarly, studies that fitted more mechanistic models of synaptic depression to such data [19,23] concluded that such models cannot fully account for the data, but such conclusions were limited by the restricted range of models that have been considered. Our main experimental observation in this paper is that the dependence of neuronal responses on prediction error calculated over long time scales can be made precise. This observation supports the notion that auditory cortex neurons carry a prediction error signal. Furthermore, we specify for the first time potential properties of the reduced representation underlying auditory cortical responses to oddball sequences at such long time constants. In particular, we demonstrate that the neuronal responses are compatible with reduced representations that have long memory duration but low complexity. Importantly, the descriptive power of our theory is not restricted to statistically simple sequences such as the oddball paradigm presented here. The full power of the theory resides in more complex stimulation sequences that may lack sufficient statistics (e. g. [21]). Of particular interest are the statistical regularities related to the syntax and semantics of language and music, which span multiple temporal scales. The tools developed here allow for a formal examination of the sensitivity of neurons to the complex statistical regularities of real-world soundscapes and therefore present a broad framework for characterizing sensory perception both qualitatively and quantitatively.
The neuronal responses are those described in [6,20], which also contain the detailed experimental methods. In brief, extracellular recordings were made in primary auditory cortex (A1) of halothane-anesthetized cats. Anesthesia was induced by Ketamine and Xylazine and maintained with halothane using standard protocols authorized by the committee for animal care and ethics of the Hebrew University—Hadassah Medical School. Single units were spike sorted on-line using template-based sorting, and in most cases they were also sorted off-line, to improve unit isolation. Stimuli were presented to the animal through sealed, calibrated earphones. For the oddball paradigm, two frequencies were selected close to the best frequency of the neuron, with a frequency ratio fhigh/flow = 1. 44. Ulanovsky et al. (2003; 2004) defined the frequency difference slightly differently, and this is their Δf = 0. 37 condition. The two frequencies were presented in a number of blocks. Each block contained 400 pure tone stimuli of identical duration (230 ms), inter-stimulus interval (736 ms onset to onset) and tone level (approximately 40 dB above the neuron’s minimal threshold). The blocks differed by the number of times each frequency was presented. For example, in the 90%/10% block, 360 of the stimuli had frequency flow and 40 had frequency fhigh, presented using a random permutation. We also used blocks with probabilities 70%/30%, 50%/50%, 30%/70%, and 10%/90%. The dataset was composed of 99 neurons tested in the 90%/10% and the 10%/90% cases. Of these, 68 were also tested in the 50%/50% (these are the neurons used in the main analysis); and of those 68 neurons, 29 neurons were additionally tested in the 70%/30% and 30%/70% conditions. For a detailed presentation of the IB principle & algorithm see Tishby et al. [13]. In short, given a joint distribution P (x, y), the IB finds a compressed representation of x denoted by m that is most informative on the target variable y. The compression of the representation is quantified by the mutual information between x and m, given by I (x; m), and the information that m carries on the target variable y is quantified by I (m; y). Using the IB algorithm we effectively pass the information that x provides about y through a ‘bottleneck’ formed by the reduced representation m, defined by P (m|x). In practice, the reduced representation is determined by minimization of the Lagrangian, L[P (m|x) ]=I (x; m) −βI (m; y) with respect to P (m|x). The positive Lagrange multiplier β, associated with the constraint on I (m; y), controls smoothly the tradeoff between preserving relevant information and the compactness of the representation. The optimal assignment that minimizes the IB Lagrangian satisfies the following equations: P (m|x) =P (m) Z (x, β) exp (−βDKL[P (y|x) ||P (y|m) ]) P (m) =∑xP (x) P (m|x) P (y|m) =∑xP (y|x) P (x|m) where Z (x, β) is a normalization (partition) function. We used here an iterative algorithm (over the set of self consistent equations) to find the optimal solution P (m|x) for a given P (x, y) and β. In our case, x and y are the past and future of the stimulus, respectively, and m is the reduced representation of the past. We computed reduced representations for the oddball paradigm, for a given past duration of N stimuli. First, we calculated the joint distribution PN (past, future) corresponding to the oddball sequences. For any given value of the parameter p (the probability of the ‘high tone’ in the block) the events are independent by construction. Thus, for a uniform prior over p, the posterior probability can be calculated explicitly, PN (past, future=high) =∫01 (Nk) pk+1 (1−p) N−kdp=k+1 (N+1) (N+2) where k = 0,1, . . , N indicates the number ‘high tones’ in past. Then we applied the IB method over PN (past, future) to find reduced representations of the past that are predictive with respect to the future (i. e. , the next stimulus). We repeated this procedure with 200 different values of β to span the entire range of complexity and predictive power levels, and with 50 different values of N spanning past durations from N = 1 to N = 50. We ended up with a set of 200 × 50 reduced representations, where each one was defined through the conditional probability distributions: P (m|past) and P (future|m). We used the conditional probability distributions P (m|past) and P (future|m) to calculate prediction errors along the stimulation sequence, as follows. For each stimulus (future) we considered its previous N stimuli (past) to determine the probability of entering each state m of the reduced representation using P (m|past). The probability with which the future stimulus is then expected P (future|m) was used to calculate the prediction error −log2 P (future|m) associated with that stimulus. Since this quantity depends on the unknown state m, we used the expected value of the prediction error (with respect to m) to generate Fig 5: E (future|past) =−∑mP (m|past) log2P (future|m) Note that the expected prediction error with respect to the stimulus distribution P (past, future) is negatively related to the predictive power, I (m; future): ⟨E (future|past) ⟩P (past, future) =−∑past, m, futureP (past, future) P (m|past) log2P (future|m) =−∑past, m, futureP (past, m, future) log2P (future|m) =−∑m, futureP (m, future) log2P (future|m) =H (future|m) =H (future) −I (m; future) where H (Future) is the entropy of the future stimulus and does not depend on the reduced representation. It follows that maximizing the predictive power is equivalent to minimizing the expected prediction error. For the main analysis, spike counts were measured in a window of 330 ms, starting at stimulus onset and ending 100 ms after stimulus offset. For the analysis displayed in Fig 7, shorter windows were used as well. We collected the neuronal responses (represented as spike counts) for each neuron and for each test frequency (‘low’ or ‘high’) across the oddball stimuli blocks. For each combination of neuron, test frequency, and reduced representation (200 × 50 pre-calculated reduced representations) we calculated the prediction error at each stimulus along the actual stimulation sequence used in the experiments. Since the prediction error is calculated with respect to a past window of up to N = 50 stimuli, the first 50 stimuli in each block were omitted from the analysis, resulting in (400 − 50) × 3 = 1050 stimuli. Dividing the stimuli further into ‘low’ and ‘high’ tone-frequencies, resulted in about 525 stimuli for the analysis of each combination of neuron and test frequency. The reduced representation determines the prediction error associated with each stimulus for each state −log2 P (future|m). For each stimulus we considered the previous N stimuli, using them to estimate the probability of entering each state m by P (m|past). These probabilities served as weights in calculating the regression between spike counts and prediction error values. Using the method of weighted linear regression allowed us to take the uncertainty in the unknown state m into account. Finally, the fraction of explained variance based on this weighted linear regression (weighted r2) was used to measure the goodness-of-fit associated with each one of the reduced representations. For each combination of neuron and test frequency we denoted the highest fraction of explained variance over the entire set of reduced representations by rmax2. We considered reduced representations that achieved a score of at least 90% of rmax2 as ‘good representations’. For each combination of neuron and test frequency used in the main analysis, we constructed two-dimensional binary maps indicating the set of ‘good representations’ over the parameter space (complexity, predictive power and past duration: see Fig 6C–6E). For Fig 6, We calculated the averages of these binary maps over a subset of the population, which had high explanatory power (rmax2≥10%; n = 34/117).
A crucial aspect of all life is the ability to use past events in order to guide future behavior. To do that, creatures need the ability to predict future events. Indeed, predictability has been shown to affect neuronal responses in many animals and under many conditions. Clearly, the quality of predictions should depend on the amount and detail of the past information used to generate them. Here, by using a basic principle from information theory, we show how to derive explicitly the tradeoff between quality of prediction and complexity of the representation of past information. We then apply these ideas to a concrete case–neuronal responses recorded in auditory cortex during the presentation of oddball sequences, consisting of two tones with varying probabilities. We show that the neuronal responses fit quantitatively the prediction errors of optimal predictors derived from our theory, and use that result in order to deduce the properties of the representations of the past in the auditory system. We conclude that these memory representations have surprisingly long duration (10 stimuli back or more), but keep relatively little detail about this past. Our theory can be applied widely to other sensory systems.
sequencing techniques
auditory cortex
medicine and health sciences
computational biology
brain
random variables
neuroscience
learning and memory
probability distribution
mathematics
forecasting
statistics (mathematics)
cognition
computational neuroscience
memory
molecular biology techniques
research and analysis methods
sequence analysis
sensory physiology
animal cells
mathematical and statistical techniques
molecular biology
probability theory
auditory system
cellular neuroscience
cell biology
anatomy
physiology
neurons
single neuron function
biology and life sciences
cellular types
physical sciences
sensory systems
cognitive science
statistical methods
The Representation of Prediction Error in Auditory Cortex