Problem formulation: Information propagation in the presence of corrupting noise

We consider a model in which a vector of “peripheral” neural population responses, x, is determined by two components. The first is the set of tuning curves, f(s), which define the cells’ mean responses to any particular stimulus (typical tuning curves are shown in Fig 2A). Here we consider a one dimensional stimulus, denoted s, which may represent, for example, the direction of motion of a visual object. In that case, a natural interpretation of our model is that it describes the transmission of motion information by direction selective retinal ganglion cells to the visual cortex (Fig 1) [5, 6, 47]. Extension to multi-dimensional stimuli is straightforward. The second component of the neural population responses, ξ, represents the trial-to-trial variability. This results in the usual “tuning curve plus noise” model, (1) where ξ is a zero mean random variable with covariance Σ_ξ.

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Fig 2. Not all population codes are equally robust against corruption by noise.

We constructed two model populations, each with the same 100 tuning curves for the first layer of cells but with different covariance structures, Σ_ξ (see text, especially Eq (4)). The covariance structures were chosen so that the two populations convey identical amounts of information I_x(s) about the stimulus. (A) 20 randomly-chosen tuning curves from the 100 cell population. (B) We corrupted the responses of each neural population by additional Gaussian noise (independently and identically distributed for all cells) of variance σ², to mimic corruption that might arise as the signals propagate through a multi-layered neural circuit, and computed the “output” information I_y(s) that these further-corrupted responses convey about the stimulus (blue and green curves). The population shown in green forms a relatively fragile code wherein modest amounts of noise strongly reduce the information, whereas the population shown in blue is more robust. (C) Input information I_x(s) in the two model populations (left; “correlated”) and information that would be conveyed by the model populations if they had their same tuning curves and levels of trial-to-trial variability, but no correlations between cells (right; “trial-shuffled”). For panels B and C, we computed the information for each of 100 equally spaced stimulus values, and averaged the information over those stimuli. See Methods for additional details (section titled “Details for Numerical Examples”).

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The neural activity, x, propagates to the second layer via feed-forward weights, W, as in the model of [38]. The activity in the second layer is given by passing the input, W · x, through a nonlinearity, g(⋅), and then corrupting it with noise, η (Fig 1B), (2) where the nonlinearity is taken component by component, and η is zero mean noise with covariance matrix Σ_η. The function g(⋅) need not be invertible, so this model can include spike generation.

While we have, in Fig 1, given one explicit interpretation of our model, the model itself is quite general. This means that our results apply more broadly than just to circuits in the peripheral visual system. Moreover, while our analysis (below) focuses on information loss between layers, this should not be taken to mean that there is no meaningful computation happening within the circuit: because we have considered arbitrary nonlinear transformations between layers, the same model can describe a wide range of possible computations [46]. Our results apply to information loss during those computations.

In the standard fashion [6, 12, 20–22], we quantify the information in the neural responses using the linear Fisher information. This measure quantifies the precision (inverse of the mean squared error) with which a locally optimal linear estimator can recover the stimulus from the neural responses [48, 49]. The linear Fisher information in the first and second layers, denoted I_x(s) and I_y(s), respectively, is given by (3a) (3b) where a prime denotes a derivative. Here W_eff are the effective weights—basically, the weights, W, multiplied by the average slope of the gain function, g(⋅)—and Σ_eff,η includes contributions from the noise in the second layer, η, and, if g(⋅) is nonlinear, from the noise in the first layer. (If g is linear, Σ_eff,η = Σ_η, so in this case Σ_eff,η depends only on the noise in the second layer). This expression is valid if is invertible; so long as there are more cells in the second layer than the first, this is typically the case. See Methods for details (section titled “Information in the output layer”).

Eq (3b) is somewhat intuitive, at least at a gross level: both large effective noise (Σ_eff,η) and small effective weights (W_eff) reduce the amount of information at the second layer. At a finer level, the relationship between the two covariance structures—corresponding to the first and second terms in brackets in Eq (3b)—can have a large effect on I_y(s), as we will see shortly.

Information content and information propagation put different constraints on neural population codes

We begin with an example to highlight the difference between the information contained in neural population codes and the information that propagates through subsequent layers. Here, we consider two different neuronal populations with identical tuning curves (Fig 2A), nearly-identical levels of trial-to-trial neural variability, and identical amounts of stimulus information encoded in their firing-rate responses; the populations’ correlational structures, however, differ. We then corrupt these two populations’ response patterns with noise, to mimic corruption that might arise in subsequent processing stages, and ask how much of the stimulus information remains. Surprisingly, the two population codes can show very different amounts of information after corruption by even modest amounts of noise (Fig 2B).

In more detail, there are 100 neurons in the first layer; those neurons encode an angle, denoted s, via their randomly-shaped and located tuning curves (Fig 2A). We consider two separate model populations. Both have the same tuning curves, but different covariance matrices. For reasons we discuss below, those covariance matrices, denoted and (blue and green correspond to the colors in Fig 2B and 2C), are given by (4a) (4b) where Σ₀ is a diagonal matrix with elements equal to the mean response, (5)

Here δ_ij is the Kronecker delta (δ_ij = 1 if i = j and 0 otherwise), and we use the convention that two adjacent vectors denote an outer product; for instance, the ij^th element if uu is u_i u_j. The vector u has the same magnitude as f′, but points in a slightly different direction (it makes an angle θ_u with f′), and ϵ and ϵ_u are chosen so that the information in the two populations, I_x(s), is the same (ϵ_u also depends on s; we suppress that dependence for clarity).

In our simulations, both ϵ and ϵ_u are small (on the order of 10⁻³; see Methods), so the variance of the i^th neuron is approximately equal to its mean. This makes the variability Poisson-like, as is typically observed when counting neural spikes in finite time windows [1–6]. (More precisely, the average Fano factors—averaged over neurons and stimuli—were 1.01 for the “blue” population and 1.04 for the “green” one.) Both model populations also have the same average correlation coefficients, which are near-zero (see Methods, section titled “Details for Numerical Examples”).

To determine how much of the information in the two populations propagates to the second layer, we computed I_y(s) for both populations using Eq (3b). For simplicity, we used the identity matrix for the feed-forward weights, W, a linear gain function, g(⋅), and independently and identically distributed (iid) noise with variance σ². Later we consider the more general case: arbitrary feedforward weights, nonlinear gain functions, and arbitrary covariance for the second layer noise. Those complications don’t, however, change the basic story.

Fig 2B shows the information in the output layer versus the level of output noise, σ², for the two populations. Blue and green curves correspond to the different covariance structures. Although the two populations have identical tuning curves, nearly-identical levels of trial-to-trial neural variability, and contain identical amounts of information about the stimulus, they differ markedly in the robustness of that information to corruption by noise in the second layer. Thus, quantifying the information content of neural population codes is not sufficient to characterize them: recordings from the first-layer cells of the two example populations in Fig 2 would yield identical information about the stimulus, but the blue population has a greater ability to propagate that information downstream.

One possible explanation for the difference in robustness is that the information in the green population relies heavily on correlations, which are destroyed by a small amount of noise. To check this, we compared the information of the correlated neural populations to the information that would be obtained with the same tuning curves and levels of single neuron trial-to-trial variability, but no inter-neuronal correlations [11, 50, 51] (Fig 2C). We find that removing the correlations actually increases the information in both populations (Fig 2C; “Trial-Shuffled”), and by about the same amount, so this possible explanation cannot account for the difference in robustness. We also considered the case where the correlated responses carry more information than would be obtained from independent cells. We again found (similar to Fig 2C) that there could be substantial differences in the amount of information propagated by equally informative population codes (see Methods, section titled “Details for numerical examples”, and the figure therein).

These examples illustrate that merely knowing the amount of information in a population, or how that information depends on correlations in neural responses, doesn’t tell us how much of that information will propagate to the next layer. In the remainder of this paper, we provide a theoretical explanation of this observation, and identify the covariance structures at the first layer that maximize robustness to information loss during propagation through downstream circuits.

Geometry of robust versus fragile population codes

To understand, from a geometrical point of view, why some population codes are more sensitive to noise than others, we need to consider the relationship between the noise covariance ellipse and the “signal direction,” f′(s)—the direction the mean neural response changes when the stimulus s changes by a small amount. Fig 3A and 3B show this relationship for two different populations. The noise distribution in the first layer is indicated by the magenta ellipses, and the signal direction by the green arrows. The uncertainty in the stimulus after observing the neural response is indicated by the overlap of the green line with the magenta ellipse. Because the overlap is the same for the two populations, they have the same amount of stimulus uncertainty, and thus the same amount of information—at least in the first layer.

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Fig 3. Geometry of robust versus fragile population codes.

Cartoons showing the interaction of signal and noise for two populations with the same information in the input layer. The dimension of the space is equal to the number of cells in the population; we show a two dimensional projection. Within this space, when the stimulus changes by an amount Δs (with Δs small), the average neural response changes by f′(s)Δs. Thus, f′(s) is the “signal direction” (green arrows). Trial-by-trial fluctuations in the neural responses in the first layer are described by the ellipses; these correspond to 1 standard-deviation probability contours of the conditional response distributions. The impact of the neural variability on the encoding of stimulus s is determined by the projection of the response distributions onto the signal direction (magenta double-headed arrows). By construction, these are identical in the first layer. Accordingly, an observer of the neural activity in the first layer of either population would have the same level of uncertainty about the stimulus, and so both populations encode the same amount of stimulus information. When additional iid noise is added to the neural responses, the response distributions grow; the dashed ellipses show the resultant response distributions at the second layer. Even though the same amount of iid noise is added to both populations, the one in panel A shows greater stimulus uncertainty after the addition of noise than does the one in panel B. Consequently, the information encoded by the population in panel B is more robust against corruption by noise.

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Although the two populations have the same amount of information, the covariance ellipses are very different: one long and skinny but slightly tilted relative to the signal direction (Fig 3A), the other shorter and fatter and parallel to the signal direction (Fig 3B). Consequently, when iid noise is added, as indicated by the dashed lines, stimulus uncertainty increases by very different amounts: there’s a much larger increase for the long skinny ellipse than for the short fat one. This makes the population code in Fig 3A much more sensitive to added noise than the one in Fig 3B.

To more rigorously support this intuition, in Methods, section titled “Analysis behind the geometry of information loss”, we derive explicit expressions for the stimulus uncertainty in the first and second layers as a function of the angle between the long axis of the covariance ellipse and the signal direction. Those expressions corroborate the phenomenon shown in Fig 3.

A family of optimal noise structures

The geometrical picture in the previous section tells us that a code is robust against added noise if the covariance ellipse lines up with the signal direction. Taken to its extreme, this suggests that when all the noise is concentrated along the f′(s) direction, so that the covariance matrix is given by (6) the resulting code should be optimally robust. While this may be intuitively appealing, the arguments that led to it were based on several assumptions: iid noise added in the second layer, feedforward weights, W, set to the identity matrix, and a linear neural response function g(⋅). In real neural circuits, none of these assumptions hold. It turns out, though, that the only one that matters is the linearity of g(⋅). In this section we demonstrate that the covariance matrix given by Eq (6) optimizes information transmission for neurons with linear gain functions (although we find, perhaps surprisingly, that this optimum is not unique). In the next section we consider nonlinear gain functions; for that case the covariance matrix given by Eq (6) can be, but is not always guaranteed to be, optimal.

To determine what covariance structures maximize information propagation, we simply maximize information in the second layer, I_y(s), with respect to the noise covariance matrix in the first layer, Σ_ξ, with the information in the first layer held fixed. When the gain function, g(⋅), is linear (the focus of this section), this is relatively straightforward. Details of the calculation are given in Methods, section titled “Identifying the family of optimal covariance matrices”; here we summarize the results.

The main finding is that there exists a family of first-layer covariance matrices Σ_ξ, not just one, that maximizes the information in the second layer. That family, parameterized by α, is given by (7) where Σ_y is the effective covariance matrix in the second layer, (8) and I_η(s) is the information the second layer would have if there were no noise in the first layer, (9) (see in particular Methods, Eq (46)). For this whole family of distributions—that is, for any value of α for which Σ_ξ is positive semi-definite—the output information, I_y(s), has exactly the same value, (10) (see Methods, Eq (76)). This is the maximum possible output information given the input information, I_x(s).

Two members of this family are of particular interest. One is α = 0, for which the covariance matrix corresponds to differential correlations (Eq (6)); that covariance matrix is illustrated in Fig 4A. This covariance matrix aligns the noise direction with the signal direction. Accordingly, as for the geometrical picture in Fig 3, it makes the encoded information maximally robust.

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Fig 4. Family of optimal covariance matrices.

For all panels, green arrows indicate the signal direction, f′(s). Magenta ellipses indicate the noise in the first layer (with corresponding covariance matrix Σ_ξ), and grey ellipses indicate the effective noise in the second layer (with corresponding covariance matrix Σ_y). (A) The covariance ellipse in the first layer has its long axis aligned with the signal direction; this configuration (which corresponds to differential correlations) optimizes information robustness for any distribution of second layer noise. (B) The covariance ellipse in the first layer does not have its long axis aligned with the signal direction. However, the covariance ellipse of the effective noise in the second layer, Σ_y, has the same shape as the covariance ellipse in the first. In this case, the blue “good” projection—which is aligned both with a low-variance direction of the first-layer distribution (magenta), and with the signal curve (green), and thus is relatively informative about the stimulus (see text)—is corrupted by relatively little noise at the second layer. This “matched” noise configuration is among those that optimize robustness to noise. The optimal family of covariance matrices interpolates between the configurations shown in panels A and B. (C) Again the covariance ellipse in the first layer does not have its long axis aligned with the signal direction. But now the “good” projection is heavily corrupted by noise at the second layer. In this configuration, all projections are substantially corrupted by noise at some point in the circuit, and thus relatively little information can propagate.

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The other family member we highlight is α = 1, for which Σ_ξ ∝ Σ_y. For this case, the covariance matrix in the first layer matches the effective covariance matrix in the second layer; we thus refer to this as “matched covariance”. To understand why this covariance optimizes information in the second layer, we start with the observation that the population activities can be decomposed into their principal components: each principal component corresponds to a different axis along with the population activities can be projected. The information contained in each such projection (principal component) adds up to give the total Fisher information (see Methods, Eq 71). The most informative of these projections are those that have low noise variance, and which align somewhat with the signal curve—like the blue line in Fig 4B. When Σ_ξ ∝ Σ_y, the projections that are most informative in the first layer are corrupted by relatively little noise in the second layer. Consequently, this configuration enables robust information propagation. In contrast, when the covariance structures in the first and second layers are less well matched, all projections are heavily corrupted by noise at some point (i.e., either in the first or the second layer), and hence very little information propagates (Fig 4C).

The family of optima interpolates between the two configurations shown in Fig 4A and 4B (see also Eq (7)). Almost all members of this optimal covariance family depend on the details of the downstream circuit: for α ≠ 0 in Eq (7), the optimal noise covariance at the first layer depends on the feed-forward weights, W, and the structure of the downstream noise. The one exception to this is the covariance matrix given by Eq (6): that one is optimal regardless of the downstream circuit. These are so-called “differential correlations”—the only correlations that lead to information saturation in large populations [22], and the correlations that minimize information in general (see Methods, section titled “Minimum information”, for proof). The fact that correlations can minimize information content and at the same time maximize robustness highlights the fact that optimizing the amount of information in a population code versus optimizing the ability of that information to be transmitted put very different constraints on neural population codes.

The existence of an optimum where the covariance matrices are matched across layers emphasizes that not all optimally robust population codes are necessarily redundant. (By redundant we mean the population encodes less information than would be encoded by a population of independent cells with the same tuning curves and levels of single neuron trial-to-trial variability [12, 21]; see Fig 2). Notably, if the effective second layer covariance matrix, Σ_y, admits a synergistic population code—wherein more information is encoded in the correlated population versus an uncorrelated one with the same tuning curves and levels of trial-to-trial response variability—then the matched case, Σ_ξ ∝ Σ_y, will also admit a synergistic population code, and be optimally robust.

Optimally robust, however, does not necessarily mean the majority of the information is transmitted; for that we need another condition. We show in the Methods section titled “Variances of neural responses, and robustness to added noise, for different coding strategies” that for non-redundant codes, a large fraction of the information is transmitted only if there are many more neurons in the second layer than in the first. This is typically the case in the periphery. For differential correlations, that condition is not necessary—so long as there are a large number of neurons in both the input and output layers, most of the information is transmitted.

Nonlinear gain functions

So far we have focused on linear gain functions g(⋅); here we consider nonlinear ones. This case is much harder to analyze, as the effective covariance structure in the second layer, Σ_eff,η, depends on the noise in the first layer (see Methods, Eq (22)). We therefore leave the analysis to Methods (section titled “Nonlinear gain functions”); here we briefly summarize the main results. After that we consider two examples of nonlinear gain functions—both involving a thresholding nonlinearity to mimic spike generation.

For linear gain functions we were able to find a whole family of optimal covariance structures, for nonlinear ones we did not even try. Instead, we asked: under what circumstances are differential correlations optimal? Even for this simplified question a definitive answer does not appear to exist. Nevertheless, we can make progress in special cases. When there is no added noise in the second layer (e.g., η = 0 for the model in Fig 1B), differential correlations maximize the amount of information that propagates through the nonlinearity, so long as the tuning curves are sufficiently dense relative to the steepness of the tuning curves (meaning that whenever the stimulus changes, the average stimulus-evoked response of at least one neuron also changes; see Methods). If there is added noise at the second layer, differential correlations tend to be optimal in cases where the addition of noise at the first layer, ξ, causes reductions in information, I_x(s). (This means that, so long as there are no stochastic resonance effects causing added noise to increase information, then differential correlations are optimal.)

We first check, with simulations, the prediction that differential correlations are optimal if there is no added noise. For that we use a thresholding nonlinearity, chosen for two reasons: it is an extreme nonlinearity, and so should be a strong test of our theory, and it is somewhat realistic in that it mimics spike generation. For this model, the responses at that second layer, y_i, are given by (11) where Θ is the Heaviside step function (Θ(x) = 1 if x ≥ 0 and 0 otherwise), and θ_i is the spiking threshold of the i^th neuron. This is the popular dichotomized Gaussian model [52–56], which has been shown to provide a good description of population responses in visual cortex, at least in short time windows [54], and to provide high-fidelity descriptions of the responses of integrate-and-fire neurons, again in short time windows [57].

In our simulations with the step function nonlinearity, as for all of the other cases we considered above, the first layer responses are given by the tuning curve plus noise model (Eq (1)). The tuning curves, f(s), of the 100-neuron population are again heterogeneous (similar to those in Fig 2A but with a different random draw from the tuning curve distribution), and the trial-to-trial variability is given by (12) with Σ₀ given by Eq (5). This is the same covariance matrix as in Eq (4b), except that we have included an overall scale factor, γ_u, chosen to ensure that the information in the input layer is independent of both ϵ_u and u(s) (see Methods, Eq (99)).

Because these (step function) nonlinearities are infinitely steep, the tuning curves are not sufficiently dense for our mathematical analysis to guarantee that differential correlations are optimal for information propagation. However, we argue in Methods (section titled “Nonlinear gain functions”), that this should be approximately true for large populations. And indeed, that’s what we find with our numerical simulation, as shown in Fig 5B. When θ_u = 0 (recall that θ_u is the angle between u(s) and f′(s)), so that u(s) = f′(s), the second term in Eq (12) corresponds to differential correlations; in this case, information increases monotonically with ϵ_u. In other words, information propagated through the step function nonlinearity increases as “upstream” correlations become more like pure differential correlations. In contrast, when θ_u is nonzero (as in Fig 3A), information does not propagate well: information decreases as ϵ_u increases. This is consistent with our findings for the linear gain function considered in Fig 2. Thus, differential correlations can optimize information transmission even for a nonlinearity as extreme as a step function.

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Fig 5. Differential correlations enhance information propagation through “spike-generating” nonlinearities.

Responses in the second layer were generated using the dichotomized Gaussian model of spike generation, in which the input from the first layer was simply binarized via a step function (see Eq (11)). We varied the correlations in these inputs (see Eq (12)) while keeping the input information and input tuning curves fixed. (A) Heterogeneous tuning curves in the second layer, evaluated at ϵ_u = 0; we show a random subset of 20 cells out of the 100-neuron population studied in panel B. (B) Information transmitted by the 100-cell spiking population as a function of ϵ_u, which is the strength of the noise in the u(s) direction, for different angles, θ_u, between u and f′(s) (see Eq (12)). The input information was held fixed as ϵ_u was varied. The information is averaged over 20 evenly spaced stimuli (see Methods, section titled “Details for Numerical Examples”).

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The lack of explicit added noise at the second layer makes this case somewhat unrealistic. In neural circuits, we expect noise to be added at each stage of processing—if nothing else, due to synaptic failures. We thus considered a model in which noise is added before the spike-generation process, (13) where ζ_i is zero-mean noise with covariance matrix Σ_ζ.

We computed information for this model using the same input tuning curves, spike thresholds, and covariance matrix, Σ_ξ, as without the additional noise (i.e., as in Fig 5). To mimic the kind of independent noise expected from synaptic failures, we chose the ζ_i to be iid, and for simplicity we took them to be Gaussian distributed with variance . We computed the amount of stimulus information, I_y(s), for several different levels of the added input noise . We found that for all levels of noise, differential correlations increase information transmission (I_y(s) increases monotonically with ϵ_u in Fig 6A, for which θ_u = 0). And we again found that when the long axis of the covariance ellipse makes a small angle with the signal direction, information propagates poorly (Fig 6B, for which θ_u = 0.1 rad.).

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Fig 6. Information propagation through spike-generating nonlinearities with additive input noise.

As with Fig 5, responses in the second layer were generated using the dichotomized model of spike generation, in which the input from the first layer was simply binarized. Here, though, Gaussian noise was added before thresholding; see Eq (13). We varied the correlations in the input layer (see Eq (12)) while keeping the input information and input tuning curves fixed for the 100-cell population (same tuning curves and covariance matrices as in Fig 5). The additive noise at the second layer (the ζ_i) was iid Gaussian, with variance ; different colored lines correspond to different values of . (A) Output information versus ϵ_u for populations with differential correlations (u = f′(s)). (B) Same as panel A, but for populations that concentrate noise along an axis, u, that makes an angle of 0.1 rad with the f′(s) direction. For both panels, the input information was held fixed as ϵ_u was varied, and the information was averaged over 20 evenly spaced stimuli.

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These numerical findings for a spike-generating nonlinearity with added noise are similar to the previous cases of a linear transfer function, g(⋅), with added input noise (Figs 2 and 3), for which we have analytical results, or a spike generating nonlinearity with no added input noise (Fig 5), for which we do not. We further argue in Methods (section titled “Nonlinear gain functions”), that for nonlinear gain functions differential correlations are likely to be optimal if the tuning curves are optimal (in the case of Eq (13), if the thresholds θ_i are chosen optimally). Taken together, our findings demonstrate that differential correlations in upstream populations generally increase the information that can be propagated downstream through noisy, nonlinear neural circuits.

Information in the output layer

Our analysis focuses on information loss through one layer of circuitry; to compute the loss, we need expressions for the linear Fisher information in the first and second layers. Expressions for those two quantities are given in Eqs (3a) and (3b). The first is standard; here we derive the second.

To make the result as general as possible, we include noise inside the nonlinearity as well as outside it; if nothing else, that’s probably a reasonable model for the spiking nonlinearity given in Eq (13). We thus generalize slightly Eq (2), and write (14) where ζ is zero mean noise with covariance matrix Σ_ζ, and here and in what follows we use the convention that g is a pointwise nonlinearity, so for any vector v, the i^th element of g(v) is g(v_i). When Σ_ζ = 0, we recover exactly the model in Eq (2).

Using Eq (1) for x, Eq (14) becomes (15) where, recall, ξ and η are zero mean noise with covariance matrices Σ_ξ and Σ_η, respectively, and h(s) is the mean drive to neuron i, (16)

To compute the linear Fisher information in the second layer, we start with the usual expression, (17) where E and Cov denote mean and covariance, respectively. The mean value of y given s is, via Eq (15)), (18)

Like g(⋅), is a pointwise nonlinearity. To compute the covariance, we assume, as in the main text, that ξ and η are independent; in addition, we assume that both are independent of ζ. Thus, the covariance of is the sum of the covariances of the first and second terms in Eq (15). The covariance of the second term is just Σ_η. The covariance of the first term is harder. To make progress, we start by implicitly defining the quantity δΣ_g(s) via (19) where W_eff(s) is the actual feedforward weight multiplied by the average slope of g, (20) and is the a diagonal matrix with entries corresponding to the average slope of g, (21)

As in the main text, δ_ij is the Kronecker delta and a prime denotes a derivative. The above implicit definition of δΣ_g is motivated by the observation that when g is linear, δΣ_g vanishes. Below, in Sec., we show that if ξ is Gaussian, δΣ_g is positive semi-definite. Here we assume that the noise is sufficiently close to Gaussian that δΣ_g remains positive semi-definite, and thus can be treated as the covariance matrix of an effective noise source. This last assumption is needed below, in the section titled “Nonlinear gain functions”, where we argue that information loss is small when δΣ_g is small (see text following Eq (64)).

Making the additional definition (22) and using Eqs (15) and (19) and the fact that η is independent of both ξ and ζ, we see that (23)

Combining this with the expression for the mean value of y, Eq (18), the linear Fisher information, Eq (17) becomes (24) where we used Eqs (16) and (20) to replace ∂_sE(y|s) with W_eff · f′ and, to reduce clutter, we have suppresed any dependence on s. To pull the effective weights inside the inverse, we use the Woodbury matrix identity to write (25)

Then, using the fact that [A + B]⁻¹ = A⁻¹ · [A⁻¹ + B⁻¹]⁻¹ · B⁻¹, and applying a very small amount of algebra, this becomes (26) where I is the identity matrix. It is then straightforward to show that (27)

Inserting this into Eq (24), we see that the right hand side of that equation is equal to the expression given in Eq (3b) of the main text.

Identifying the family of optimal covariance matrices

Here we address the question: what noise covariance matrix optimizes information transmission? In other words, what covariance matrix Σ_ξ maximizes the information given in Eq (3b)? That is hard to answer when g is nonlinear, because in that case Σ_eff,η depends on Σ_ξ via δΣ_g (see Eqs (19) and (22)). In this section, then, we consider linear gain functions; in the next we consider nonlinear ones. To make our expressions more readable, we generally suppress the dependence on s.

Our goal is to maximize I_y with I_x fixed. Using the definition of Σ_y given in Eq (8), for linear gain functions the information in the second layer (Eq (3b)) is written (35)

We use Lagrange multipliers, (36) where λ is a Lagrange multiplier that enforces the constraint . Taking the derivative and setting it to zero yields (37)

In deriving this expression we used the fact that the gain functions are linear, which implies that Σ_y does not depend on Σ_ξ. Multiplying by Σ_ξ + Σ_y on both the left and right, we arrive at (38)

This is satisfied when (39)

There are two ways this can happen, (40a) (40b) where a is an arbitrary vector. Combining these linearly, taking into account that Σ_ξ is a covariance matrix and thus symmetric, and enforcing equality in Eq (38), we arrive at (41) where I_η is the information the output layer would have if there was no noise in the input layer, (42) (this is the same expression as in Eq (9), it’s repeated here for convenience), Ω is an arbitrary symmetric matrix, P is a projection operator, chosen so that P · f′ = 0, (43) and α is arbitrary (but subject to the constraint that Σ_ξ has no negative eigenvalues). Note that P is a linear combination of the right hand sides of Eqs (40a) and (40b)), with a = f′ in the latter equation. It is straightforward to verify that when Σ_ξ is given by Eqs (41) and (38) is satisfied.

To find an explicit expression for Σ_ξ, not just its inverse, we apply the Woodbury matrix identity to Eq (41); that gives us (44) where (45)

Inserting this into Eq (44), we arrive at (46)

This is the same as Eq (7) in the main text, except in that equation we let Ω go to ∞, so we ignore the projection-related term. Ignoring that term is reasonable, as it just puts noise in a direction perpendicular to f′, and so has no effect on the information.

By choosing different scalars α and matrices Ω, a family of optimal Σ_ξ is obtained. These all have the same input information, I_x, and the same output information, I_y, after corruption by noise. An especially interesting covariance matrix is found in the limit α = 0, in which case (47)

These are so-called differential correlations [22]. Importantly, the choice α = 0 is the only one for which the optimal correlational structure is independent of the correlations in the output layer, Σ_y. Note that pure differential correlations don’t satisfy Eq (39). As such, they represent a singular limit, in the sense that Σ_ξ in Eq (46) satisfies Eq (39) with alpha arbitrarily small, but not precisely zero.

The other covariance that we highlight in the text is found for α = 1 and Ω → ∞, in which case . This is the matched covariance case.

Nonlinear gain functions

We now focus on differential correlations, and determine conditions under which they are optimal for information propagation when the gain function, g(⋅), is nonlinear. In this regime, the effective noise in the second layer (the second term in brackets in Eq (3b)) depends on Σ_ξ. This greatly complicates the analysis, and to make headway we need to reformulate our mathematical description of differential correlations. This reformulation is based on the observation that differential correlations correspond to trial-to-trial variability in the value of the stimulus, s[22]. Consequently, the encoding model in the input layer can be written as a multi-step process, (48a) (48b) (48c)

Here s₀ is the value of the stimulus that is actually presented. However, the neurons in the input layer, x, encode s—a corrupted version of s₀. This is indicated by Eq (48a), which tells us that s deviates on a trial-to-trial basis from s₀, with deviations that are described by a zero-mean random variable, δ_s.

To see that this model does indeed exhibit differential correlations, we Taylor expand Eq (48b) around s₀, yielding a model of the form (49) for which the covariance matrix is (50)

The first term corresponds to differential correlations.

Eqs (48b) and (48c) correspond exactly to our previous model (Eq (4a)). Consequently, the information about s in the first and second layers are still given by Eqs (3a) and (3b) of the main text. However, we can’t use those equations for the information about s₀. For that, we focus on the variance of its optimal estimator given x, which we denote . Because of the Markov structure of our model (s₀ ↔ s ↔ x), we can construct by first considering the optimal estimator of s₀ given s, and then the optimal estimator of s given x. The variance of given x is then simply the sum of the variances of these two (independent) noise sources.

The optimal estimator of s₀ given s is simply s, with conditional variance . The optimal estimator of s given x is , with variance . Consequently, (51)

As usual, we approximate the variance of given s by the linear Fisher information, yielding an approximation for the total Fisher information about s₀ given x, (52)

Similarly, the Fisher information about s₀ given y is approximated by (53)

Note that we are slightly abusing notation here: above, I_x(s) and I_y(s) referred to the total information about the stimulus; now they refer to the information about the stimulus that is encoded in the first layer, which is different from the actual stimulus, s₀. However, it is a convenient abuse, as it allows us to take over our previous results without introducing much new notation.

Our first step is to parametrize the covariance matrix, ξ, and Var[δ_s], in a way that ensures that the information in the first layer remains fixed while we vary ξ and Var[δ_s]. A convenient choice is (54a) (54b) where (55)

Inserting Eq (54) into Eq (52), we see that , independent of Σ₀(s).

The information in the second layer about s, I_y(s), is given by Eq (3b), with Σ_eff,η given in Eq (22). It is convenient to make the definition (56)

This is the analog of Eq (8), but for nonlinear gain functions. It is clear from Eqs (22) and (19) that Σ_eff,η depends on Σ_ξ; consequently, it depends on ϵ.

To maximize information with respect to ϵ, we take a two step approach. We write (57)

Here Σ_ξ(s,ϵ) and Σ_y(s,ϵ₀) are the same as in Eqs (54b) and (56); we have just made the dependence on ϵ explicit. The two steps are to maximize first with respect to ϵ, then with respect to ϵ₀. If the two maxima occurr in the same place, then we have identified the covariance structure that optimizes information transmission.

In the first step we differentiate with respect to ϵ. To simplify the expressions, we make the definition (58)

Combining Eqs (53), (54) and (57), we have (59) where we used the fact that for any square matrix A(x), (d/dx)A⁻¹ = −A⁻¹ · d A/dx · A⁻¹, and we suppressed much of the s, ϵ and ϵ₀ dependence for clarity. Using Eq (54b) for Σ_ξ(s,ϵ), the derivative with respect to ϵ in the second term is straightforward, (60)

Then, applying the definition (see Eqs (57) and (58)), and making the new definition (61) we arrive at the expression (62)

The right hand side of Eq (62) is negative or zero if the term in brackets is negative semi-definite; that is, if all its eigenvalues are non-positive. Since the term in square brackets is a rank one matrix minus the identity, all but one of its eigenvalues are equal to -1. The remaining eigenvalue is 0, with corresponding eigenvector (see Eq (55)). Thus, , and must have a global maximum at ϵ = ∞. If g is linear, Σ_eff,y doesn’t depend on ϵ₀, and ϵ = ∞ corresponds to pure differential correlations. We have, therefore, recovered the α = 0 limit of Eq (46).

When ϵ = ∞, Σ_ξ vanishes, and so the expression for the information in the second layer simplifies considerably. Combining Eqs (53) and (54a), we have, in the ϵ → ∞ limit, (63) where (64)

The latter equation follows by combining the fact that Σ_ξ(s, ∞) = 0 (Eq (54b)) with the definitions of Σ_eff,y and Σ_eff,η (Eqs (56) and (22), respectively).

The total information in the output layer is maximized when I_y(s₀; ∞,ϵ₀) is maximized. That quantity depends on ϵ₀ via Σ_ξ(s,ϵ₀), the noise covariance in the input layer. As can be seen from Eq (54b), larger ϵ₀ implies smaller Σ_ξ(s,ϵ₀). That has two effects. First, when Σ_ξ(s,ϵ₀) is small enough, the covariance matrix δΣ_g becomes small (see Eq (19), and note that δΣ_g is positive definite, as shown in Sec.). This tends to increase I_y(s). However, the effective tuning curves, W_eff(s;ϵ) · f(s), also depend on Σ_ξ(s,ϵ₀) (see Eq (20)). It is possible that increasing Σ_ξ(s,ϵ₀) modifes the tuning curves such that I_y(s) increases. Consequently, it is impossible to make completely general statements.

Nevertheless, we can identify two regimes. First, if there is no added noise in the output layer (η = ζ = 0), then I_y(s; ∞, ϵ) goes to ∞ as ϵ₀ goes to ∞, thus maximizing the total information. This holds, however, only if the tuning curves are sufficiently dense relative to the steepness of the tuning curves; otherwise, the Fisher information is no longer a good approximation to the true information. For smooth tuning curves this is generally satisfied, but it is not satisfied for the noise-free spike generating mechanism we consider in the main text (Eq (11)), since for that nonlinearity f′(s) = 0 with probability 1. We expect, though, that in the absence of noise, this particular nonlinearity introduces an error that is , implying that I_y(s; ∞, ϵ) ∝ n². Numerical simulations corroborated this scaling. Thus, for sufficiently large populations, differential correlations are optimal for the noise-free spike-generating nonlinearity. Note, though, that the thresholds must be chosen so that there are always both active and silent neurons; otherwise, in the limit that Σ_ξ vanishes, the activity will contain no information at all about the stimulus.

The second regime is one in which the tuning curves have been optimized. In this case, modifying the tuning curves by adding noise decreases information, and again differential correlations optimize information transmission.

To summarize, we have analyzed the scenario considered in the main text (section titled “Nonlinear gain functions”)—namely, the neural activities at the second layer, y, are given by a nonlinear function of the neural activities at the first layer, x, with noise added both before and after the nonlinearity. In this case, whether or not differential correlations in the first layer optimize information transmission depends on the details. They do if g is linear, the tuning curves are optimal, or there is no added noise in the second layer and the tuning curves are sufficiently dense relative to the steepness of the tuning curves. If none of these are satisfied, however, differential correlations may be sub-optimal.

Analysis behind the geometry of information loss

Our goal in this section is to make more rigorous the geometrical arguments in Fig 3. We start with the observation that, for Gaussian distributed neural responses, the 1 standard-deviation probability contours for the responses in the first layer (magenta ellipses in Fig 3) are defined by (65) where Δr ≡ f(s) − r represents fluctuations around the mean response to stimulus s. In two dimensions, which we’ll focus on here, Eq (65) becomes (66) where σ₁ and σ₂ are the lengths of the principal axes of the covariance ellipse (so and are the eigenvalues of Σ_ξ) and Δr₁ and Δr₂ are distances spanned by the magenta ellipses along those axes.

As shown in Fig 3, the intersection between the magenta ellipse (the one defined in Eq (66)) and the signal curve tells us the uncertainty in the value of the stimulus. To quantify this uncertainty, we simply set Δr to f′(s)Δs_x (the subscript x indicates that this is the uncertainty in the input layer), insert that into Eq (66), and solve for Δs_x. Defining θ to be the angle between f′(s) and the long principal axis (see Fig 3, and note that θ = 0 in panel B), and letting σ₁ correspond to the length of the ellipse’s major axis (so σ₁ > σ₂), we have (67)

The left hand side is the linear Fisher information in the first layer [10], a fact that is useful primarily because it validates our (relatively informal) derivation. More importantly, we can now see how iid noise affects information. The addition of iid noise simply increases the eigenvalues by σ², so the ratio of the information in the output layer to that in the input layer is (68)

We can identify two limits. First, if θ = 0 (as it is in Fig 3B), this ratio reduces to (69)

Second, if tan θ ≫ σ₂/σ₁ (which essentially means the green line in Fig 3 intersects the covariance ellipse on the side, as in panel A, rather than somewhere near the end, as in panel B), the ratio of the informations becomes (70)

Because σ₁ > σ₂, the information loss is larger in the second case than in the first. And the longer and skinnier the covariance ellipse, the larger the difference in information loss. Thus, this analysis quantifies the geometrical picture given in Fig 3, in which there is larger information loss in panel A (where θ > 0) than in panel B (where θ = 0).

Minimum information

Here we ask: what correlational structure minimizes linear Fisher information? To answer that, we use the multi-dimensional analog of Eq (67), (71) where is the k^th eigenvalue of the noise covariance matrix and θ_k is the angle between f′(s) and the k^th eigenvector [10]. We would like to minimize I_x(s) with respect to the angles, θ_k, and the eigenvalues, . Without constraints, this problem is trivial: information is minimized by having infinite variances for the neural activities. To make the problem better-formulated, we add a constraint that prevents the optimization procedure from simply identifying that trivial solution.

We’ll come to the constraint shortly, but first we’ll minimize information with respect to the angles, θ_k. That minimum occurs when the eigenvector corresponding to the largest eigenvalue is parallel to f′(s); ordering the eigenvalues so that is the largest eigenvalue, we have cos θ₀ = 1 and cos θ_{k > 0} = 0. Consequently, the information at the minimum is (72)

The next step is to minimize I_x(s) with respect to the eigenvalues, subject to a constraint on the covariance matrix. We consider constraints of the form (73) where, to avoid the trivial solution (of infinite neural variances), C is an increasing function of each of it’s arguments: for all k, (74)

Examples of are the trace of the covariance matrix (the sum of the eigenvalues) and the Frobenius norm (the square root of the sum of the squares of the eigenvalues).

Because of Eq (74), the information, Eq (72), is minimized and the constraint, Eq (73), is satisfied when all the eigenvalues except are zero. At this global minimum, the covariance matrix, Σ_ξ, displays purely differential correlations, (75) where v₀ is the eigenvector associated with the largest eigenvalue. The last term in this expression follows because the above minimization with respect to the angles forced v₀ to be parallel to f′(s). Thus, for a broad, and reasonable, class of constraints on the covariance matrix, differential correlations minimize information.

Variances of neural responses, and robustness to added noise, for different coding strategies

Throughout most of our analysis we focused on optimality of information transmission. However, also important is how much information is transmitted at the optimum. That’s the subject of this section. For simplicity we consider a linear gain function, which we set, without loss of generality, to the identity. That allows us to use the analysis above, in the section titled “Identifying the family of optimal covariance matrices”, and in particular Eq (46), which links the noise in the input and output layers.

Our starting point is the derivation of an expression for the ratio of the information in the output layer to that in the input layer. To do that, we dot both sides of Eq (37) by f′ on the left and right sides and solve for λ; we then do the same, except we dot with on the left and its transpose on the right. This yields, after a small amount of algebra, (76) where I_x, I_y and I_η are given by Eqs (3a), (3b) and (9), respectively. For information to be transmitted efficiently, I_x, the information in the input layer, must be small compared to I_η, the information associated with the added noise in the output layer. Below, we investigate the conditions under which I_x ≪ I_η, and thus when information loss is small.

Our strategy is to express I_x/I_η in terms of the single neuron variability, quantified as the average variance—something that has an easy interpretation. We consider two cases: the weights are set to the identity (W = I), and the weights are more realistic (each neuron in the input layer connects to a large number of neurons in the output layer). The first case, identity weights, is not very realistic; we include it because it is much simpler than the second.

While the analysis is straightforward, it is somewhat heavy on the algebra, so we summarize the results here. We consider two extremes in the family of optimal covariance structures: the “matched” case (α = 1 in Eq (46), and, for simplicity, Ω = ∞) and differential correlations (α = 0). For matched covariances, near complete information transfer (I_x ≪ I_η) requires the effective variance of the noise in the second layer to be small. For identity feedforward weights, the effective variance in the input and output layers is about the same, so information loss is large. However, identity feedforward weights are never observed in the brain; instead, each neuron in the input layer connects to a large number of neurons in the output layer. Using N_x and N_y to denote the number of neurons in the input and output layers, respectively, and K the average number of connections per neuron, the effective noise is reduced by a factor or (see Eq 95 below). Thus, if the number of neurons in the output layer is larger than the number in the input layer by a factor much larger than K^1/2, near complete information transmission is possible. For pure differential correlations, the story is much simpler: so long as the number of neurons in both layers is large, and the added noise doesn’t have a strong component in the f′(s) direction, near complete information transmission always occurs.

Information in a population with a rank 1 perturbation to the covariance matrix

In the analysis of nonlinear gain functions in the main text (section titled “Nonlinear gain functions”), it was necessary to construct a covariance matrix such that the information in the first layer was independent of ϵ_u and u. For that we included a prefactor γ_u in the definition of the covariance matrix, Σ_ξ (see Eq (12)). Here we determine how γ_u should depend on ϵ_u and u. Our starting point is an expression for the inverse of Σ_ξ. As is straightforward to show, via direct substitution, that’s given by (97)

Thus, the information in the input layer, , is given by (98)

To ensure that this information is independent of γ_u, we let (99)

Note that γ_u depends on s as well as ϵ_u and u.

Details for numerical examples

In this section we provide details for the numerical simulations for each relevant figure.