When and Why Noise Correlations Are Important in Neural Decoding

Exact measure of the minimum information loss

Previous estimators fail to tightly bound the minimum information loss of NI decoders. The reasons for the failure depend on the estimator, as shown in the previous section. In the case of ΔI_NI, the failure is due to the fact that the optimal decoder is not searched among all possible NI decoders but, at best, among subsets of limited size. However, the failure can be avoided by extending the search to all possible NI decoders. To that end, in this section, we first introduce the notion of canonical NI decoders: the set of all decoders that comply with the NI assumption (Eq. 3). Using a fundamental theorem in information theory, the coding theorem, we then determine exactly the amount of information lost by the best canonical NI decoder. This information loss is smaller than the bounds analyzed in the previous section (Fig. 1).

All probabilistic NI decoders, here called canonical NI decoders (Eq. 4), can be described as a 2-stage process (Fig. 3). Without loss of generality, consider the population response R = [R₁, …, R_N] (where N is the number of neurons) elicited by a stimulus S_k (1 ≤ k ≤ K, where K is the number of stimuli). In the first stage, the population response R is internally represented as a vector R^NIL of NI likelihoods (defined in Eq. 3), given by the following:

This step, and only this step, embodies the NI assumption. The second stage represents the transformation of R^NIL into the decoded stimulus S^NI and embodies the estimation criterion used to decode the stimulus.

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Figure 3.

The canonical NI decoder is modeled as a sequence of transformations of the population response. The first stage involves the NI assumption, transforming the population response R into a vector R^NIL of NI likelihoods. The second stage involves the stimulus estimation, transforming R^NIL into the decoded stimulus S^NI. At each stage, information may be lost.

As stated by the data-processing inequality (see Materials and Methods), each transformation in the sequence cannot increase the information about the stimulus and may potentially induce an information loss. The information lost in each stage of the decoding process can be determined using standard methods previously developed for the analysis of neural codes (Borst and Theunissen, 1999; Panzeri et al., 2007; Eyherabide and Samengo, 2010). In particular, the actual information loss ΔI_NI induced by canonical NI decoders can be separated as follows:

Here, ΔI_NI^NIL is the information loss induced by the NI assumption (first stage) and ΔI_NI^Est is the information loss induced by the estimation process (second stage).

The NI assumption (first stage) is common to all NI decoders, and therefore ΔI_NI^NIL constitutes a lower bound to the information loss induced by all NI decoders. Mathematically, Equation 20 decomposes the actual information loss ΔI_NI into two non-negative terms, thereby proving that ΔI_NI^NIL is a lower bound of ΔI_NI. Nevertheless, ΔI_NI^NIL could still underestimate the minimum information loss induced by all NI decoders. To prove that ΔI_NI^NIL is tight, we need to prove that a NI decoder exists for which ΔI_NI coincides with ΔI_NI^NIL, as we do next.

Different estimation criteria induce different information losses ΔI_NI^Est. However, the coding theorem (Shannon, 1948; Cover and Thomas, 1991) demonstrates the existence of a decoding procedure that, operating on long sequences of responses, can make ΔI_NI^Est negligible, thus extracting all the information I(S;R^NIL) that is preserved after the NI assumption. Because the NI assumption is common to all NI decoders, I(S;R^NIL) constitutes the maximum amount of information that can be extracted by any canonical NI decoder, and the difference:

constitutes the minimum information loss induced by any decoder embodying the NI assumption; that is, the single fundamental property needed to evaluate the relevance of correlations (Nirenberg and Latham, 2003).

We have invoked the coding theorem to demonstrate Equation 21. This theorem was demonstrated by Shannon (1948) and promoted the development of information theory as a full discipline. In the context of our work, the theorem applies to the mapping S → R → R^NIL, which can be abbreviated as S → R^NIL. This mapping, in the notation of Shannon, constitutes a channel that transforms each S into R^NIL. Repeated uses of the channel transform sequences of Q stimuli [S₁,…, S_Q] into sequences [R₁^NIL, …, R_Q^NIL]. Shannon's proof involved actual decoders that mapped sequences of Q responses R^NIL into sequences of Q-decoded stimuli S^NI. By making Q sufficiently large, he showed that there is at least one decoder for which the information transmission rate I(S, S^NI) can be made as close as desired to I(S, R^NIL) with negligible decoding error, thus yielding Equation 21. Two things, however, should be noticed. First, Shannon did not display his decoder explicitly—and neither do we. He simply demonstrated its existence. Second, in order for the theorem to hold, long sequences of stimuli and responses may be required. This requirement may be undesirable when studying the neural code under behavioral contexts, so later we introduce the minimum decoding error to circumvent this inconvenience.

To illustrate how Equations 20 and 21 improve the analysis of the role of noise correlations in neural decoding, consider the examples shown in Figure 4. Each panel shows how the population response is transformed throughout the decoding process by the canonical NI decoder. These examples were analyzed in Figure 1, in which we showed that, for a wide range of stimulus and response probabilities, all previous estimators indicated that noise correlations are important. However, as we show next, these estimators include not only the information loss induced by the NI assumption, but also the information loss induced by underlying assumptions constraining the estimation criteria, thus overestimating the importance of noise correlations.

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Figure 4.

Examples of population activities decoded using the canonical NI decoder. The arrows show the transformation of population responses into vectors of NI likelihoods (R^NIL) induced by the NI assumption (left to middle) and optimal estimation algorithms (middle to right). In A, P(S₁) is set to 0.75, and P(M,L|S₁) and P(H,H|S₂) are both set to 0.5. In B and C, stimuli are equally likely and P(H,L|S₁) and P(H,H|S₂) are both set to 0.5 in B and to 0.66 in C. A, After the NI assumption, the distinction between responses elicited by different stimuli is preserved (middle panel). Therefore, ΔI_NI^NIL is zero and noise correlations are unimportant for decoding. B, After the NI assumption, all responses are identical. No information about the stimulus remains and noise correlations are crucial for decoding. C, However, whenever population responses are not equally likely given each stimulus, the NI assumption preserves all the encoded information and noise correlations are unimportant for decoding. The case shown in B, in which noise correlations are important, therefore constitutes an isolated example.

In the example shown in Figure 4A (previously analyzed in Fig. 1A), ΔI_NI^NIL is zero regardless of the values of the stimulus and response probabilities. Indeed, after the first stage (where the NI assumption takes place) population responses elicited by different stimuli remain different (i.e., they are associated with different R^NIL; Fig. 4A, middle), as we formally show after the next section. Therefore, the losses reported by previous estimators necessarily occur during the estimation stage. However, among all mappings between the representation R^NIL and the stimulus, there is at least one capable of correctly estimating the stimulus. Therefore, all the encoded information can be extracted without any loss and noise correlations are irrelevant for decoding. Notice, nevertheless, that finding an optimal estimation criterion explicitly is unnecessary. As we showed above, the minimum inefficiency of NI decoders, and the importance of noise correlations in neural decoding, can both be assessed by using ΔI_NI^NIL even before considering any estimation criterion.

Of course, information may be lost before the stimulus estimation takes place due to the NI assumption. Consider the examples shown in Figure 4, B and C (previously analyzed in Fig. 1B). When both P(H,L|S₁) and P(H,H|S₂) are set to 0.5 (Fig. 4B), ΔI_NI^NIL is equal to the encoded information. Noise correlations are thus crucial for decoding. Indeed, after the NI assumption, all population responses become indistinguishable (Fig. 4B, middle) and no estimation criterion is capable of extracting any information about the stimulus (Gawne and Richmond, 1993). These carefully chosen response probabilities, however, constitute an isolated case. For other values of the response probabilities (Fig. 4C), all population responses are represented differently after the NI assumption (as we formally show after the next section), and therefore ΔI_NI^NIL is zero. Therefore, except for the isolated case of Figure 4B, noise correlations are irrelevant for decoding.

When and why information is lost throughout the decoding process

In the previous section, we first modeled the NI decoder as a sequence of transformations of the population response (Fig. 3) and then quantified the average information loss induced by each stage of the NI-decoding process (Eq. 20) over all population responses (see Materials and Methods). However, information losses need not be evenly distributed among responses. In this section, we determine which are the specific responses that induce losses and the amount and type of information that is lost. We demonstrate that losses may only appear in those responses in which the decoding mapping is not injective. To localize the loss, we analyze how each successive transformation in the decoding process merges distinct representations of two or more responses into a single one so that their distinction is lost. The approach is similar to previous studies of the neural code (Eyherabide and Samengo, 2010, and references therein), revealing what sort of information about the stimulus is preserved or lost and which response features encode such information.

After each transformation in the decoding process, two or more population responses whose distinction is informative may be represented in identical manner so their distinction is no longer available for subsequent transformations. Whenever that happens, information is lost. To assess whether the distinction between two (or more) population responses R_A and R_B is informative or constitutes noise, we first construct a representation $\tilde{R}$ that treats those responses as if they were the same, but keeps the distinction between all other responses and then we compare the encoded information with and without the distinction (Eyherabide and Samengo, 2010)

Whenever ΔI_{R→$\tilde{R}$} is zero, the distinction between R_A and R_B provides no additional information and only constitutes noise. For example, in Figure 4B, this is the case for population responses [R_1,R₂] = [L,H] and [H,L] or responses [L,L] and [H,H]. Notice that, when responses vary in a continuum, single responses are typically associated with a probability density. In that case, a representation $\tilde{R}$ that treats two single responses R_A and R_B as equivalent induces no information loss. In the continuous case, an information loss occurs only when $\tilde{R}$ treats a set of population responses as equivalent and, in addition, the probability of the set is nonzero.

The condition for the distinction between responses to be informative (Eq. 22) can also be written as a direct comparison between the real posterior probabilities P(S|R). The distinction between two (or more) population responses R_A and R_B constitutes noise if and only if:

for all stimuli S_k (this comes directly from Eq. 22). Otherwise, their distinction is informative, and ignoring it induces an information loss ΔI_{R→$\tilde{R}$}. Examples of informative variations and noise when population responses are represented as continuous variables are given in the last section of Results.

We can now formally state when and why noise correlations are important in neural decoding. During the decoding process, the population response R is first transformed into R^NIL immediately after the NI assumption takes place (Fig. 3). This transformation induces an information loss when (and only when) the three following conditions are fulfilled:

• (24a) The mapping R → R^NIL is not injective, and, as a consequence, there are two or more responses {R₁, … R_Q} mapped onto the same R^NIL,
• (24b) Equation 23 is not fulfilled for at least two responses complying with condition 24a and one stimulus S_k, and
• (24c) The probability of the set of population responses fulfilling both conditions 24a and 24b is nonzero.

Noise correlations are important for decoding specifically those responses satisfying these three conditions: the cause of the loss (the “why”) relies on the fact that the NI assumption no longer allows the decoder to take into account the differences in their information content. The losses can be linked to specific stimulus and response features by interpreting R^NIL as a reduced representation of the population response (Eyherabide and Samengo, 2010). In such a paradigm, population responses are represented as a vector of response features, each conveying information about specific stimulus features. Only some of those response features (and their information content) are preserved after the NI assumption (first stage). The analysis of the preserved and lost features allows one to determine the effect of the NI assumption on the neural code.

With conditions 24a, 24b, and 24c in mind, we can provide an approximate estimation of the likelihood that correlations be relevant. For discrete responses, noise correlations are often irrelevant because condition 24a is often violated. This condition requires that at least two different responses, R_A and R_B, be mapped onto the same vector, R^NIL. This is unlikely, though, because the mapping from R to R^NIL goes from a discrete set to a continuous space (Eq. 19). For continuous responses, the mapping from R to R^NIL goes from a continuous space of dimension N (where N is the number of neurons) to a continuous space of dimension K (where K is the number of stimuli). Intuitively, one would therefore expect that correlations would be relevant more often in the case of continuous responses than in the case of discrete responses and that the importance of correlations should tend to increase with N and decrease with K.

This approximate estimation, however, should not be taken as a hard rule. One can construct an infinite number of counterexamples in which noise correlations are crucial for discrete responses (Figs. 2, ​,4)4) or in which the importance of noise correlations, for discrete and continuous responses, decreases with N and increases with K, exactly opposite to the approximate estimation mentioned above. Nevertheless, one can prove that, for examples with a finite number of discrete responses, these counterexamples, though infinite in number, constitute a set of measure zero in the space of all possible examples, at least when using a counting measure (i.e., a measure that simply counts the number of cases regardless of how likely they occur in nature; Tao, 2011). Unfortunately, for examples with continuous responses or with an infinite number of discrete responses, such proof remains elusive.

Furthermore, one should observe that, in experimental conditions, neither responses nor response probabilities can be measured with infinite precision. Experimental errors and limited sampling may both produce broad distributions of responses, each associated with distributions of response probabilities. Mathematically, each population response R is associated not with a probability P(R|S), as would occur if probabilities could be estimated with infinite precision, but with a distribution of probabilities Q[P(R|S)] for each stimulus S. The distribution Q[P(R|S)] can be estimated either using Bayesian approaches or resampling methods (Bishop, 2006; Hastie et al., 2009). In other words, due to experimental errors and limited sampling, P(R|S) becomes a random variable with probability Q[P(R|S)]. Furthermore, the mapping from R to R^NIL (Eq. 19) becomes a probabilistic one-to-many mapping as opposed to the deterministic one-to-one mapping that would be obtained if response probabilities were measured with infinite precision. The change in the nature of the mapping R → R^NIL should be taken into account when evaluating ΔI_NI^NIL, resulting in a distribution of information losses rather than in a unique deterministic value. Moreover, the equalities in conditions 24a and 24b should be interpreted in statistical terms (i.e., equalities should be assessed through hypothesis testing). In these circumstances, NI decoders become lossy more frequently than predicted by the approximate prediction above. Therefore, the relevance of correlations and the certainty with which such relevance is assessed depend on the quality of the measured data.

Finally, notice that, in many applications, discrete responses arise as quantizations of continuous responses. It would be desirable to recover, for increasingly small bins, the results obtained with the original continuous responses. The typical procedure is to assign a single probability to each bin as, for example, the mean value of the original continuous probability distribution inside the bin. This approach leads to a purely discrete model that represents poorly the importance of noise correlations in the underlying continuous responses. To solve this problem, a different quantization procedure should be used. One possibility consists in associating each bin R^Bin with a probability distribution P(R^NIL|R^Bin), representing the spread of the conditional response probabilities of the continuous case and thereby including some uncertainty in the value of R^NIL. The probabilistic mapping between R^Bin and R^NIL connects two continuous spaces and the results obtained with such quantization procedure becomes consistent with those obtained with the original continuous responses.

Impact of the choice of NI decoder

Within the family of canonical NI decoders (Fig. 3), specific NI decoders differ in the estimation criteria used to decode the stimulus. For example, in classical NI decoders, the estimation process involves the calculation of the probability of each stimulus given the population response (using Bayes' rule) as if neurons were noise independent, and then the selection of the most likely stimulus. In this section, we show in detail why this estimation strategy need not be optimal when applied after the NI assumption.

Classical NI decoders can be modeled as a three-stage process (Fig. 5A). In the first stage, the population response R is transformed into a vector of NI likelihoods R^NIL (Eq. 19). In the second stage, R^NIL is further transformed into a vector of NI posterior probabilities as follows:

through Bayes' rule (Eq. 5). The final stage is the estimation of the stimulus from R^NIP through the maximum-posterior criterion (Wu et al., 2001; Nirenberg and Latham, 2003; Latham and Nirenberg, 2005; Ince et al., 2010; Eq. 7).

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Figure 5.

The impact of the choice of a NI decoder. A, The classical NI decoder is modeled as a three-stage process involving: the NI assumption (first stage), in which the population response is transformed into a vector of NI likelihoods (R^NIL); Bayes' rule (second stage), in which R^NIL is transformed into a vector of NI posteriors (R^NIP); and the estimation criterion (third stage), in which the decoded stimulus is inferred from R^NIP. Each stage may induce an information loss. B, C, Population responses of Figure 4, A and C, decoded with the classical NI decoder. B, Both R^NIL and R^NIP keep responses elicited by different stimuli segregated and therefore all information is preserved. However, the stimulus cannot always be correctly inferred by simply choosing the one corresponding to the maximum NI posterior (argmax criterion, dotted line, and arrows; right). Nevertheless, there is an estimation criterion capable of correctly estimating the stimulus (continuous lines and arrows; right). C, Although the NI assumption preserves all the encoded information, after Bayes' rule, responses associated with different stimuli are merged, and thus some (but not all) information is lost (ΔI_NI^NIB is greater than zero). As a result, no estimation criterion is capable of perfectly decoding the stimulus. However, other NI decoders may still be optimal for decoding (Fig. 4C).

In this model, Bayes' rule acts as a deterministic mapping that transforms the vector of NI likelihoods R^NIL into another vector of NI posteriors, R^NIP. This mapping can be injective or not. If it is injective, then Bayes' rule is obviously lossless. Otherwise, it may cause an information loss, as we show below when discussing the example of Figure 5C. Notice, however that, when using the real stimulus-response probabilities describing the data, Bayes' rule is always lossless because the responses that are confounded constitute noise, as shown in the previous section.

The actual information loss ΔI_NI induced by the classical NI decoder can be separated as follows:

This equation shows that the actual information loss induced by the classical NI decoder contains three different contributions:

1. the information loss ΔI_NI^NIL induced by the NI assumption (Eq. 21);
2. the information loss ΔI_NI^NIB induced by Bayes' rule; and
3. the information loss ΔI_NI^Est induced by the chosen stimulus-estimation criterion (in this case, the maximum posterior).

To understand how choosing the NI decoder affects the decoded information, consider the example shown in Figure 5B (previously analyzed in Fig. 4A). P(S₁) is set to 0.75 and both P(M,L|S₁) and P(H,H|S₂) are set to 0.5. Throughout the NI-decoding process, the population responses R = [R_1,R₂] are first transformed through Equation 19 into the representations R^NIL = [P_NI(R|S₁), P_NI(R|S₂)] and then through Equation 25 into R^NIP = [P_NI(S₁|R), P_NI(S₂|R)], resulting in:

These transformations are also shown in Figure 5B. The first stage only merges the population responses [L,M] and [M,L], but their distinction only constitutes noise (ΔI_R→R^NIL is zero; Eq. 22), and thus no information is lost (i.e., ΔI_NI^NIL is zero). The second stage is an injective mapping so it also does not affect the decoded information (ΔI_NI^NIB is zero).

Using Equations 19 and 25 in an analogous manner, we can generalize the previous results to arbitrary values of the stimulus and response probabilities: Responses associated with different stimuli are always represented in a different manner, both after the first stage and after the second stage. As a result, any information loss (Fig. 1G,J) is due to the estimation criterion. Nevertheless, among all possible estimation criteria, there is at least one capable of extracting all the information remaining in R^NIP, which is equal to the encoded information. This optimal estimation criterion, however, differs from the maximum-posterior criterion (Fig. 5B, right).

Another example is analyzed in Figure 5C (previously analyzed in Fig. 4C), in which losses are distributed differently throughout the decoding process. Stimuli are equally likely and both P(H,L|S₁) and P(H,H|S₂) are set to 0.66. Throughout the NI decoding process, the population responses R = [R₁,R₂] are first transformed (recall Eqs. 19 and 25) as follows:

These transformations are also shown in Figure 5C. The first transformation merges no population responses. Therefore, the distinction between population responses is preserved after the NI assumption and ΔI_NI^NIL is zero. The second transformation, however, merges response [L,H] with [H,H] and [H,L] with [L,L]. Unlike R^NIL, the representation R^NIP carries less information about the stimulus than the encoded information and ΔI_NI^NIB is greater than zero (Fig. 6A). Therefore, although there exists a canonical NI decoder capable of decoding without error (Fig. 4C), classical NI decoders are unable to extract all of the information preserved after the NI assumption.

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Figure 6.

Difference between assessing the role of noise correlations using mutual information or decoding error. The population response shown in Figure 5C is decoded using the classical NI decoder. Response probabilities are set according to P(H,L|S₁) = P(L,L|S₂). For different stimulus probabilities P(S₁), A shows the variation of the minimum information loss ΔI_NI^NIP relative to the encoded information I(R;S), and B shows the variation of the increment in the minimum decoding error Δξ_NI^NIP relative to the minimum decoding error ξ^Min(R;S). The decoding error is here measured as decoding-error probability. The curves for P(S₁) = p are identical to the curves for P(S₁) = 1 − p (0 ≤ p ≤ 1). A, Unlike the case shown in Figure 4B, here, information is only partially lost and the loss depends on the stimulus and response probabilities. The maximum loss, however, only occurs when P(H,L|S₁) reaches 0.5 regardless of the stimulus probability. B, Unlike ΔI_NI^NIP, Δξ_NI^NIP approaches its maximum value when P(H,L|S₁) is greater or equal to P(S₁).

For other values of stimulus and response probabilities, almost always a canonical NI decoder exists capable of decoding without error (Fig. 4C), except in the isolated case shown in Figure 4B. However, classical NI decoders may still be incapable of extracting all the information preserved after the NI assumption. Unlike R^NIL, population responses associated with different stimuli are not always mapped after Bayes' rule onto different R^NIP. There are two cases in which responses are merged: (1) when P(L,H|S₁) = P(H,H|S₂), in which case response [L,H] is merged with [L,L] and [H,L] with [H,H]; and (2) when P(L,H|S₁) = P(L,L|S₂), in which case response [L,H] is merged with [H,H] and [H,L] with [L,L] (this case is shown in Fig. 5C). Therefore, the representation R^NIP carries less information about the stimulus than the encoded information and ΔI_NI^NIB is greater than zero (Fig. 6A). These cases constitute examples in which NI decoders can be optimal, but for achieving optimality, the estimation must be based purely on the NI assumption (i.e., on R^NIL).

Using Equations 21, 23, and 26 we can now prove why estimators based on ΔI_NI (criterion 8a) and ΔI_NI^LS (criterion 8b) overestimate the minimum information loss ΔI_NI^Min. Both methods have two occasions to include unnecessary losses. The first occasion appears when transforming R^NIL into R^NIP (Bayes' rule). When neurons are truly noise independent, ΔI_NI^NIL and ΔI_NI^NIP coincide; otherwise, as a result of Bayes' rule, some responses for which a distinction is informative may be merged (Fig. 5C), and therefore ΔI_NI^NIP > ΔI_NI^NIL. The second occasion appears when passing from R^NIP to the decoded stimulus S^NI. In the case of ΔI_NI, the estimation criterion usually coincides with the maximum posterior, which, as shown above, may be suboptimal when used under the NI assumption. In the case of ΔI_NI^LS, R^NIP is transformed into a ranking of stimuli. This stage, although more finely grained than the purely maximum-posterior criterion, may still lump distinct representations R^NIP into one single ranking and thereby perhaps lose information.

Minimum decoding error

The analysis of the effects of ignoring noise correlations in neural decoding was here performed using mutual information (Cover and Thomas, 1991; Borst and Theunissen, 1999). This quantity sets a limit to the decoding performance (e.g., in the number of stimulus categories that can be distinguished with negligible error probability), but this limit may only be achievable when decoding long sequences of population responses (i.e., comprising several consecutive population responses, also known as block coding; Cover and Thomas, 1991). Long sequences of responses inevitably have a long duration. To produce timely behavioral reactions, however, neural systems must process information in short time windows (tens or hundreds of milliseconds) (Hari and Kujala, 2009; Panzeri et al., 2010). Long sequences of responses may therefore be inconsistent with the fast behavioral responses observed in nature. In addition, mutual information may not adequately represent the cost of wrongly estimating the stimulus (Nirenberg and Latham, 2003). To overcome these issues, in this section, we also bound the inefficiency of NI decoders using the minimum decoding error.

The minimum decoding error ξ^Min(S;R) (Eq. 11) can be defined using different cost functions (Simoncelli, 2009), allowing one to assess the importance of noise correlations when decoding is performed on a single-response basis. Like mutual information, it is non-negative and depends on the representation R of the population response. Furthermore, ξ^Min(S;R) also follows the data-processing inequality (Cover and Thomas, 1991; Quian Quiroga and Panzeri, 2009), but it actually increases with transformations of R. Let $\tilde{R}$ = g(R) be one of such transformations (deterministic or stochastic); then:

To prove this, recall that EY[X] represents the weighted mean of X with weights Y. We derive Equation 27 as follows:

Because of these similarities, the mathematical framework and the interpretations obtained from the minimum decoding error are almost identical to those of mutual information, taking care of the change in the sign of the data-processing inequality (as shown below). However, when applied to experimental data, the results obtained using mutual information or minimum decoding error may differ both quantitatively and qualitatively depending on the case under study (as shown in the next section).

The increment in the decoding error Δξ_NI induced by a canonical NI decoder (Fig. 3) with respect to the minimum decoding error ξ^Min(R;S) (that would be achievable if noise correlations were taken into account) is given as follows:

where ξ_NI is the actual decoding error induced by the specific implementation of the canonical NI decoder. Analogously to Equation 20, Δξ_NI can be separated as follows:

Δξ_NI^NIL and Δξ_NI^Est represent the increment in the minimum decoding error induced by the NI assumption and the estimation criterion, respectively. Among all mappings between R^NIL and the decoded stimulus S^NI, there is one for which:

Such a decoder induces no additional increment in the minimum decoding error (Eq. 12). Therefore, Δξ_NI^NIL constitutes the minimum increment in the minimum decoding error attainable by at least one canonical NI decoder (i.e., those purely based on the NI assumption). Whenever Δξ_NI^NIL is zero, the NI assumption does not increase the minimum decoding error and noise correlations can be safely ignored.

Similarly to Equation 26, the increment in the decoding error induced by classical NI decoders (Fig. 5A) can be written as follows:

where Δξ_NI^NIL was defined in Equation 30. Here, Δξ_NI^NIB represents the increment in the minimum decoding error due to Bayes' rule. Δξ_NI^NIP is the minimum increment in the minimum decoding error attainable by all classical NI decoders and may be greater or equal to Δξ_NI^NIL (Eq. 27).

Any increment in the minimum decoding error occurs because, during the decoding process, some population responses are treated as identical. The importance of the distinction between two (or more) population responses, R_A and R_B, can be tested by first constructing a representation $\tilde{R}$ that treats them as identical (but keeps the distinction between all other responses) and then computing as follows:

Whenever Δξ_{R→$\tilde{R}$} is zero, the distinction between R_A and R_B does not increment the minimum decoding error and can be safely ignored. Otherwise, Δξ_{R→$\tilde{R}$} shows the increment in the minimum decoding error due to ignoring their distinction.

As an example, consider the population response shown in Figure 4, B and C. When responses are equally likely (Fig. 4B), Δξ_NI^NIL reaches its maximum value as follows:

Indeed, after the NI assumption, all population responses are represented in the same way and the NI decoder performs at chance level. However, in all cases in which responses are not equally likely (Fig. 4C), Δξ_NI^NIL is zero. In other words, if responses are not equally likely, a canonical NI decoder exists that is capable of decoding the stimulus with the same accuracy as if noise correlations were taken into account (such a decoder is shown in Fig. 4C). Classical NI decoders operate substantially worse (Fig. 5C) because population responses become indistinguishable before the estimation process and thus perform at chance level for a wide range of response probabilities (Fig. 6B). Even though some information still remains in R^NIP (Fig. 6A), it cannot be extracted using single responses.

Although, in general, ΔI_NI^NIL and Δξ_NI^NIL are not related deterministically (Thomson and Kristan, 2005), here we show some useful relations between these two quantities in specific cases as follows:

• (35a) If ΔI_NI^NIL is zero, then Δξ_NI^NIL is zero. Proof: If ΔI_NI^NIL = 0, then I(S;R|R^NIL) = 0 and hence R can be written as a transformation (deterministic or stochastic) of R^NIL. Following the data-processing inequality (Eq. 27), Δξ_NI^NIL = 0.
• (35b) Corollary: if Δξ_NI^NIL > 0, then ΔI_NI^NIL > 0.
• (35c) If ξ^Min(R^NIL,S) is zero, then ΔI_NI^NIL and Δξ_NI^NIL are zero. Proof: If ξ^Min(R^NIL,S) = 0, the data-processing inequality ensures that Δξ_NI^NIL = 0. In the absence of errors, such NI decoder extracts all the encoded information, and thus ΔI_NI^NIL = 0.
• (35d) If ΔI_NI^NIL is equal to the encoded information I(S;R), then ξ^Min(R^NIL,S) is given by Equation 34. Proof: In this case, S and R^NIL are independent. The result follows from introducing independence into Equation 11.

The analysis can also be generalized to other measures of transmitted information, such as those defined by Victor and Nirenberg (2008), with the condition that they comply with the data-processing inequality (Cover and Thomas, 1991). Moreover, it can be extended to other probabilistic mismatched decoders, that is, to decoders constructed using stimulus-response probability distributions that differ from the real ones (Quian Quiroga and Panzeri, 2009; Oizumi et al., 2010). One simply replaces the NI likelihoods with those corresponding to the probabilistic model under consideration. Decoding in the real brain may be subjected to additional constraints imposed by biophysics, connection length, metabolic cost, or robustness, which cannot simply be represented by a generalized measure of information. Our approach can be extended to these cases by computing the difference between the information transmitted by the optimal NI decoder that additionally satisfies these constraints with that of optimal decoders constructed with knowledge of noise correlations that operate under the same constraints.