Neurons in sensory systems gather information about the environment , and transmit that information to other parts of the nervous system ., This information is encoded in the activity of neural populations , and that activity is variable: repeated presentations of the same stimulus lead to different neuronal responses 1–7 ., This variability can degrade the ability of neural populations to encode information about stimuli , leading to the question: which features of population codes help to combat—or exacerbate—information loss ?, This question is typically addressed by assessing the amount of information that is encoded in the periphery as a function of the covariance structure 6 , 8–24 , the shapes of the tuning curves 25 , 26 , or both 27 , 28 ., However , the informativeness of the population responses at the periphery is not the only relevant quantity for understanding sensory coding; of potentially equal importance is the amount of information that propagates through the neural circuit to downstream structures 29 , 30 ., To illustrate the ideas , consider the case of retinal ganglion cells transmitting information about visual stimuli to the cortex via the thalamus , as shown in Fig 1 ., To quantify the performance of the retina , one must consider not only the informativeness of the optic nerve responses ( Ix ( s ) in Fig 1A ) , but also how much of that information is transmitted by the lateral geniculate nucleus ( LGN ) to the cortex ( Iy ( s ) in Fig 1A ) 31 ., The two may be very different , as only information that survives the LGN’s spike-generating nonlinearity and noise corruption will propagate to downstream cortical structures ., Despite its importance , the ability of information to propagate through neural circuits remains relatively unexplored 31 ., One notable exception is the literature on how synchrony among the spikes of different cells affects responses in downstream populations 32–36 ., This is , however , distinct from the information propagation question we consider here , as there is no guarantee that those downstream spikes will be informative ., Other work 25 , 29 , 30 , 37 , 38 investigated the question of optimal network properties ( tuning curves and connection matrices ) for information propagation in the presence of noise ., No prior work , however , has isolated the impact of correlations on the ability of population-coded information to propagate ., Given the frequent observations of correlations in the sensory periphery 6 , 8 , 17 , 39–45 , and the importance of the information propagation problem , this is a significant gap in our knowledge ., To fill that gap , we consider a model ( Fig 1B; described in more detail below ) , in which there are two layers ( retina and LGN , for example ) ., The first layer contains a fixed amount of information , Ix ( s ) , which is encoded in the noisy , stimulus-dependent responses of the cells in that layer ., The information is passed to the second layer via feedforward connections followed by a nonlinearity , with noise added along the way ., We ask how the covariance structure of the trial-to-trial variability in the first layer affects the amount of information in the second ., Although we focus on information propagation , the problem we consider applies to more general scenarios ., In essence , we are asking: how does the noise in the input to a network interact with noise added to the output ?, Because we consider linear feedforward weights followed by a nonlinearity , the possible transformations from input to output , and thus the computations the network could perform , is quite broad 46 ., Thus , the conclusions we draw apply not just to information propagation , but also to many computations ., Moreover , it may be possible to extend our analysis to recurrent , time-dependent neural networks ., That is , however , beyond the scope of this work ., Our results indicate that the amount of information that successfully propagates to the second layer depends strongly on the structure of correlated responses in the first ., For linear neural gain functions , and some classes of nonlinear ones , we identify analytically the covariance structures that optimize information propagation through noisy downstream circuits ., Within the optimal family of covariance structures , we find variability with so-called differential correlations 22—correlations that are proven to minimize the information in neural population activity ., Thus , covariance structures that maximize the information content of neural population codes , and those that maximize the ability of this information to propagate , can be very different ., Importantly , we also find that redundancy is neither necessary nor sufficient for the population code to be robust against corruption by noise ., Consequently , to understand how correlated neural activity affects the function of neural systems , we must not only consider the impact of those correlations on information , but also the ability of the encoded information to propagate robustly through multi-layer circuits ., 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 ,, x = f ( s ) + ξ , ( 1 ), where ξ is a zero mean random variable with covariance Σξ ., 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 ) ,, y = g ( W · x ) + η , ( 2 ), where the nonlinearity is taken component by component , and η is zero mean noise with covariance matrix Ση ., 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 Ix ( s ) and Iy ( s ) , respectively , is given by, I x ( s ) = f ′ ( s ) · Σ ξ − 1 · f ′ ( s ) ( 3a ) I y ( s ) = f ′ ( s ) ·  Σ ξ + ( W eff T · Σ eff , η − 1 · W eff ) − 1  − 1 · f ′ ( s ) ( 3b ), where a prime denotes a derivative ., Here Weff are the effective weights—basically , the weights , W , multiplied by the average slope of the gain function , g ( ⋅ ) —and Σeff , 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 , Σeff , η = Ση , so in this case Σeff , η depends only on the noise in the second layer ) ., This expression is valid if W eff T · Σ eff , η - 1 · W eff 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 ( Σeff , η ) and small effective weights ( Weff ) 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 Iy ( s ) , as we will see shortly ., 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 Σ ξ blue and Σ ξ green ( blue and green correspond to the colors in Fig 2B and 2C ) , are given by, Σ ξ blue = Σ 0 + ϵ f ′ ( s ) f ′ ( s ) ( 4a ) Σ ξ green = Σ 0 + ϵ u u ( s ) u ( s ) ( 4b ), where Σ0 is a diagonal matrix with elements equal to the mean response ,, Σ 0 , i j = f i ( s ) δ i j ., ( 5 ) Here δij is the Kronecker delta ( δij = 1 if i = j and 0 otherwise ) , and we use the convention that two adjacent vectors denote an outer product; for instance , the ijth element if uu is ui uj ., The vector u has the same magnitude as f′ , but points in a slightly different direction ( it makes an angle θu with f′ ) , and ϵ and ϵu are chosen so that the information in the two populations , Ix ( s ) , is the same ( ϵu also depends on s; we suppress that dependence for clarity ) ., In our simulations , both ϵ and ϵu are small ( on the order of 10−3; see Methods ) , so the variance of the ith 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 Iy ( 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 ., 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 ., 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 ., 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, Σ ξ ( s ) ∝ f ′ ( s ) f ′ ( s ) , ( 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 , Iy ( s ) , with respect to the noise covariance matrix in the first layer , Σξ , 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 Σξ , not just one , that maximizes the information in the second layer ., That family , parameterized by α , is given by, Σ ξ ( s ) = α I x ( s ) I η ( s ) Σ y + 1 - α I x ( s ) f ′ ( s ) f ′ ( s ) , ( 7 ), where Σy is the effective covariance matrix in the second layer ,, Σ y ≡ ( W eff T · Σ η - 1 · W eff ) - 1 , ( 8 ), and Iη ( s ) is the information the second layer would have if there were no noise in the first layer ,, I η ( s ) = f ′ ( s ) · Σ y - 1 · f ′ ( s ) ( 9 ), ( see in particular Methods , Eq ( 46 ) ) ., For this whole family of distributions—that is , for any value of α for which Σξ is positive semi-definite—the output information , Iy ( s ) , has exactly the same value ,, I y ( s ) = I x ( s ) 1 + I x ( s ) / I η ( s ) ( 10 ), ( see Methods , Eq ( 76 ) ) ., This is the maximum possible output information given the input information , Ix ( 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 ., The other family member we highlight is α = 1 , for which Σξ ∝ Σy ., 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 Σξ ∝ Σy , 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 , Σy , 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 , Σξ ∝ Σy , 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 ., 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 , Σeff , 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 , Ix ( 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 , yi , are given by, y i = Θ ( x i - θ i ) ( 11 ), where Θ is the Heaviside step function ( Θ ( x ) = 1 if x ≥ 0 and 0 otherwise ) , and θi is the spiking threshold of the ith 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, Σ ξ = γ u Σ 0 + ϵ u u ( s ) u ( s ) ( 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 , γu , chosen to ensure that the information in the input layer is independent of both ϵu 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 θu = 0 ( recall that θu 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 ϵu ., In other words , information propagated through the step function nonlinearity increases as “upstream” correlations become more like pure differential correlations ., In contrast , when θu is nonzero ( as in Fig 3A ) , information does not propagate well: information decreases as ϵu 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 ., 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 ,, y i = Θ ( x i + ζ i - θ i ) ( 13 ), where ζi is zero-mean noise with covariance matrix Σζ ., We computed information for this model using the same input tuning curves , spike thresholds , and covariance matrix , Σξ , 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 ζi to be iid , and for simplicity we took them to be Gaussian distributed with variance σ ζ 2 ., We computed the amount of stimulus information , Iy ( s ) , for several different levels of the added input noise σ ζ 2 ., We found that for all levels of noise , differential correlations increase information transmission ( Iy ( s ) increases monotonically with ϵu in Fig 6A , for which θu = 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 θu = 0 . 1 rad . ) ., 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 θi 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 ., Much work in systems neuroscience has investigated the factors that influence the amount of information about a stimulus that is encoded in neural population activity patterns ., Here we addressed a related question that is often overlooked: how do correlations between neurons affect the ability of information to propagate robustly through subsequent stages of neural circuitry ?, The question of robustness is potentially quite important , as the ability of information to propagate determines how much information from the periphery will reach the deeper neural structures that affect decision making and behavior ., To investigate this issue , we considered a model with two cell layers ., We varied the covariance matrix of the noise in the first layer ( while keeping the tuning curves and information in the first layer fixed ) , and asked how much information could propagate to the second layer ., Our main findings were threefold ., First , population codes with different covariance structures but identical tuning curves and equal amounts of encoded information can differ substantially in their robustness to corruption by additional noise ( Figs 2 , 5 , 6 and 7 ) ., Consequently , measurements of information at the sensory periphery are insufficient to understand the ability of those peripheral structures to propagate information to the brain , as that propagation process inevitably adds noise ., For instance , populations of independent neurons can be much worse at transmitting information than can populations displaying correlated variability ( Fig 5B ) ., Thus , to understand how the brain efficiently encodes information , we must concern ourselves not just with the amount of information in a population code , but also with the robustness of that encoded information against corruption by noise ., Second , for linear gain functions , or noise-free nonlinear ones with sufficiently dense tuning curves , populations with so-called differential correlations 22 are maximally robust against noise induced by information propagation ., This fact may seem surprising given that differential correlations are the only ones that lead to information saturation in large populations 22 , and the correlations that minimize information in general ., However , in hindsight it makes sense: differential correlations correspond to a covariance ellipse aligned with the signal direction ( see Fig 3B ) , and added noise simply doesn’t make it much longer ., For nonlinear gain 
Sensory neurons give highly variable responses to stimulation , which can limit the amount of stimulus information available to downstream circuits ., Much work has investigated the factors that affect the amount of information encoded in these population responses , leading to insights about the role of covariability among neurons , tuning curve shape , etc ., However , the informativeness of neural responses is not the only relevant feature of population codes; of potentially equal importance is how robustly that information propagates to downstream structures ., For instance , to quantify the retina’s performance , one must consider not only the informativeness of the optic nerve responses , but also the amount of information that survives the spike-generating nonlinearity and noise corruption in the next stage of processing , the lateral geniculate nucleus ., Our study identifies the set of covariance structures for the upstream cells that optimize the ability of information to propagate through noisy , nonlinear circuits ., Within this optimal family are covariances with “differential correlations” , which are known to reduce the information encoded in neural population activities ., Thus , covariance structures that maximize information in neural population codes , and those that maximize the ability of this information to propagate , can be very different ., Moreover , redundancy is neither necessary nor sufficient to make population codes robust against corruption by noise: redundant codes can be very fragile , and synergistic codes can—in some cases—optimize robustness against noise .
Information about the outside world , which originates in sensory neurons , propagates through multiple stages of processing before reaching the neural structures that control behavior ., While much work in neuroscience has investigated the factors that affect the amount of information contained in peripheral sensory areas , very little work has asked how much of that information makes it through subsequent processing stages ., That’s the focus of this paper , and it’s an important issue because information that fails to propagate cannot be used to affect decision-making ., We find a tradeoff between information content and information transmission: neural codes which contain a large amount of information can transmit that information poorly to subsequent processing stages ., Thus , the problem of robust information propagation—which has largely been overlooked in previous research—may be critical for determining how our sensory organs communicate with our brains ., We identify the conditions under which information propagates well—or poorly—through multiple stages of neural processing .
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