Stimulus-Dependent Correlations and Population...

LETTER Communicated by Bruno Averbeck

Stimulus-Dependent Correlations and Population Codes

Kresimir [email protected] of Mathematics, University of Houston, Houston,TX 77204-3008, U.S.A.

Eric [email protected] of Applied Mathematics University of Washington Seattle,WA 98195-2420, U.S.A.

Brent [email protected] of Mathematics, University of Pittsburgh, Pittsburgh, PA 15206, U.S.A.

Jaime de la [email protected] for Neural Science, New York University, New York, NY 10012, USA

The magnitude of correlations between stimulus-driven responses ofpairs of neurons can itself be stimulus dependent. We examine how thisdependence affects the information carried by neural populations aboutthe stimuli that drive them. Stimulus-dependent changes in correlationscan both carry information directly and modulate the information sepa-rately carried by the firing rates and variances. We use Fisher informationto quantify these effects and show that, although stimulus-dependentcorrelations often carry little information directly, their modulatory ef-fects on the overall information can be large. In particular, if the stimulusdependence is such that correlations increase with stimulus-induced fir-ing rates, this can significantly enhance the information of the populationwhen the structure of correlations is determined solely by the stimulus.However, in the presence of additional strong spatial decay of correla-tions, such stimulus dependence may have a negative impact. Oppositerelationships hold when correlations decrease with firing rates.

1 Introduction

Correlations in the neural response have the potential to both positivelyand negatively affect the ability of a population to carry information aboutstimuli. Intuitively, correlated fluctuations imply a common component in

Neural Computation 21, 2774–2804 (2009) C© 2009 Massachusetts Institute of Technology

Stimulus-Dependent Correlations and Population Codes 2775

the response noise of different neurons, which cannot be removed by aver-aging the population response (Johnson, 1980; Britten, Shadlen, Newsome,& Movshon, 1992).

The impact of correlations on information encoded in neural tissue isa subject with a substantial history. We start our discussion with Zohary,Shadlen, and Newsome (1994), which reported significant correlations be-tween neuronal responses in paired recordings of neurons in the visualarea of monkeys. Correlations were deemed undesirable, as they lead toa decrease in the signal-to-noise ratio of the summed population activity(Johnson, 1980; Britten et al., 1992). Despite this impact on the signal-to-noise ratio, correlations in the neural response can increase the informationthat a population of neurons carries about a signal (Abbott & Dayan, 1999).The impact of correlations on coding depends in a complex way on theirdistribution over the neuronal population (Romo, Hernandez, Zainos, &Salinas, 2003; Chen, Geisler, & Seidemann, 2006; Poort & Roelfsema, 2009;Oram, Foldiak, Perrett, & Sengpiel, 1998; Averbeck, Latham, & Pouget,2006; Series, Latham, & Pouget, 2004; Kohn, Smith, & Movshon, 2004; Wu,Nakahara, & Amari, 2001; Abbott & Dayan, 1999; Shamir & Sompolinsky,2004, 2006; Sompolinsky, Yoon, Kang, & Shamir, 2001). As the range of po-tential patterns of correlation is vast and has not been characterized in mostneurobiological systems, the effect of correlations is not fully understood.

In many studies to date, the correlation coefficient between the responsesof pairs of neurons was assumed to be independent of the stimulus drivingthe response. In particular, it was assumed that covariances between cellresponses change in proportion to the variance so that the correlation coeffi-cient remained constant. Information about stimulus identity could then beencoded solely in the rate and variability of single cell responses (Abbott &Dayan, 1999; Shamir & Sompolinsky, 2004, 2006; Sompolinsky et al., 2001).However, experimental findings suggest that correlations themselves varywith stimuli (Gray, Engel, & Singer, 1989; deCharms & Merzenich, 1996;Samonds, Allison, Brown, & Bonds, 2003; Kohn & Smith, 2005; de la Rocha,Doiron, Shea-Brown, Josic, & Reyes, 2007; Biederlack et al., 2006; Chacron& Bastian, 2008). More specifically, it has been shown in Kohn and Smith(2005) that correlations in the visual cortex (V1) at short timescales vary withthe stimulus orientation and contrast. Biederlack et al. (2006) demonstratedexperimentally that in certain situations, changes in perceived brightnessare related to changes in neural correlations. Responses to preylike ver-sus conspecific-like stimuli in electric fish have also been demonstrated toevoke responses with different correlation structure (Chacron & Bastian,2008). A more detailed overview of the impact of correlated responses onthe information carried by neural tissue can be found in Averbeck (2009).

Here, we concentrate on a particular form of stimulus dependence, inwhich correlations depend on stimulus-evoked firing rates (although manyof our formulas hold more generally). In recent work, we have shown thatspike-to-spike correlations due to common inputs increase with firing rate

2776 K. Josic, E. Shea-Brown, B. Doiron, and J. de la Rocha

for neural models and in vitro neurons (de la Rocha et al., 2007). This effectwas observed in vivo in the anesthetized visual cortex (Kohn & Smith, 2005;Greenberg, Houweling, & Kerr, 2008) and, in certain experimental regimes,for motoneurons in vitro (Binder & Powers, 2001). In the oculomotor neuralintegrator, the opposite effect was observed: correlations decreased withrate (Aksay, Baker, Seung, & Tank, 2003), perhaps due to recurrent networkinteractions. We will study both of these cases, illustrating strongly differingeffects of stimulus dependence in each.

The goal of this letter is to examine, from a theoretical perspective, theimpact of stimulus-dependent correlations on population coding. Previ-ously, changes in discriminability due to changes in the covariance matrixof pairs of cells and small (three to eight cells) ensembles were examined byAverbeck and Lee (2006). Also, a series expansion of mutual informationto isolate and quantify the effects of stimulus-dependent correlations hasbeen developed (Panzeri, Schultz, Treves, & Rolls, 1999). Similarly, Montani,Kohn, Smith, and Schultz (2007) use mutual information to assess the im-pact of tuned correlations measured in primate V1. We take a somewhatdifferent approach based on computing the impact of the stimulus depen-dence of correlations on the Fisher information (IF ) for populations of neu-rons whose response is described by tuning curves (Seung & Sompolinsky,1993).

There are at least two distinct ways in which the stimulus dependenceof correlations can affect Fisher information. First, the fact that patterns ofcorrelation across a population are adjusted as stimuli change can have astrong modulatory impact on the information that other features of the neu-ral response, such as firing rates, carry about the stimulus (Montani et al.,2007; Gutnisky & Dragoi, 2008). We refer to this effect as correlation shap-ing. To better understand this, note that a stimulus-independent correlationstructure may be optimized for one stimulus. However, stimulus depen-dence offers the possibility that the correlation structure is adjusted andoptimized for a range of stimuli (Averbeck et al., 2006). In a related effect,adaptation has been shown to modify correlation structure and increase IF

(Kohn et al., 2004; Gutnisky & Dragoi, 2008).Second, information may be encoded directly by changes in the level of

correlation between neurons, in addition to encoding by changes in firingrate and variance. We refer to this mechanism as correlation coding. Onescenario where correlation coding clearly dominates is if stimuli only affectthe correlation structure, leaving rates and variances relatively constant,as has been observed experimentally (Vaadia et al., 1995; Biederlack et al.,2006; Chacron & Bastian, 2008).

The balance of the letter proceeds as follows. We start by defining ourstatistical description of the neural response to stimuli in section 2. Theinformation in the response of two cells is studied in section 3. As weshow, the insights gained from this case can be extended to small pop-ulations but do not always apply to larger populations. In section 4, we


study the information in the response of a large population. Here, we findthat correlation shaping effects can be substantial and often dominate overcorrelation coding. In section 5, we extend the model to address the addi-tional structure of correlations across the population, by including decayof correlations that depends explicitly on the spatial, or “functional,” dis-tance between preferred stimuli of neurons, as shown experimentally. Wefind that the impact of correlation shaping in the presence of such a decaycontinues to be strong, but that correlation coding also plays a significantrole. We conclude with a discussion of the results. A number of analyticalresults used in the main body of the letter, which may be of independentinterest, are derived in the appendixes.

2 Setup

2.1 Structure of Correlations. We consider a population of N neuronsresponding to a stimulus described by a scalar variable θ (e.g., the orienta-tion of a visual grating). The number of spikes fired by neuron i in responseto stimulus θ during a fixed time interval is given by

ri (θ ) = fi (θ ) + ηi (θ ), (2.1)

where fi (θ ) is the mean response of neuron i across trials and ηi (θ ) mod-els the trial–to–trial variability of the response. We use boldface nota-tion for vectors, so that r(θ ) denotes the multivariate random variabler(θ ) = [r1(θ ), r2(θ ), . . . , rN(θ )]T . For simplicity, we sometimes suppress de-pendences on θ .

We assume that η follows a multivariate distribution with zero mean andcovariance matrix Q(θ ) defined by

Qi, j (θ ) = δi, jvi (θ ) + (1 − δi, j )ρi, j (θ )√

vi (θ )v j (θ ). (2.2)

Here vi (θ ) is the variance of the response of cell i , and −1 ≤ ρi, j (θ ) ≤ 1 is thecorrelation coefficient of the response of cells i and j . Although most of ourresults will be discussed in the range of small to intermediate correlations,ρi, j � 0.5, a similar analysis can be used to study the behavior of populationsclose to perfect correlations, ρi, j ≈ 1. When needed, we will assume that theresponse follows a multivariate gaussian distribution.

For studies of stimulus-dependent correlations in small-to-intermediatepopulations (see section 3), we allow general forms of ρi, j (θ ). When westudy large populations in sections 4 and 5, we will assume that

ρi, j (θ ) = Si, j (θ ) c(φi − φ j ), (2.3)


where φi and φ j are the preferred stimuli of neurons i and j , respectively.The stimulus-independent term c(φi − φ j ) represents the spatial or func-tional structure of correlations in the population. It describes how cor-relations vary across the population according to their preferred stimuli,perhaps due to hardwired differences in the level of shared inputs. Forinstance, neurons that prefer similar stimuli are frequently close by in thecortex and may share a larger number of common inputs than neurons thatexhibit different preferences (Holmgren, Harkany, Svennenfors, & Zilberter,2003). Moreover, the set of neurons upstream of two cells with similar stim-ulus preferences may also undergo common fluctuations in their activity.Therefore, c(φi − φ j ) = c(�φ) is frequently assumed to decrease with thefunctional distance �φ. We will refer to this simply as spatial decay (Dayan& Abbott, 2001; Sompolinsky et al., 2001; Wilke & Eurich, 2002; Smith &Kohn, 2008).

We emphasize that it is the stimulus dependence of the correlation co-efficient, ρi, j (θ ), that distinguishes this work from several previous inves-tigations (Abbott & Dayan, 1999; Wu et al., 2001; Sompolinsky et al., 2001;Shamir & Sompolinsky, 2004). This dependence enters through the termSi, j (θ ) (Kohn & Smith, 2005; de la Rocha et al., 2007; Greenberg et al., 2008).We mainly investigate cases in which correlations between pairs of cellsincrease, decrease, or have a single maximum with respect to the evokedfiring rates fi and f j (de la Rocha et al., 2007; Shea-Brown, Josic, Doiron,& de la Rocha, 2008; Kohn & Smith, 2005; Binder & Powers, 2001; Aksayet al., 2003). However, our results could also be applied to cases with otherrelations between ρi, j (θ ), fi , f j , vi , and v j such as those arising for differentcircuit and nonlinear spike generation mechanisms (cf. Figure 4 of de laRocha et al., 2007).

For large populations, we extend the multiplicative model in Shamirand Sompolinsky (2001) to the case of stimulus-dependent correlations byassuming that

Si, j (θ ) = si (θ )s j (θ ), (2.4)

where −1 < si (θ ), s j (θ ) < 1. Here si (θ ) may be thought of as the propensityof a neuron’s response to be correlated and s2

i (θ ) as the correlation betweentwo neurons that respond equivalently to the stimulus. There are severalreasons for adopting the form given in equation 2.4. First, this form of ρi j

arises for small to intermediate correlation in neuron models producing aspike train with renewal statistics (de la Rocha et al., 2007; Shea-Brown et al.,2008). Moreover, in this case, correlation has also been shown to vary withthe geometric mean of the firing rate of pairs of cells in vivo (Kohn & Smith,2005; de la Rocha et al., 2007), which can be modeled using equation 2.4. Inaddition, this form keeps the computations at hand analytically tractablefor large population sizes and limits the number of cases under study.


2.2 Fisher Information. To quantify the fidelity with which a neuronalpopulation represents a signal, we use Fisher information (Seung & Som-polinsky, 1993; Dayan & Abbott, 2001). For the probability distributionp[r | θ ] of the spike count vector r given stimulus θ , the Fisher informationis defined as

IF (θ ) =⟨− d2

dθ2 log p[r | θ ]⟩,

where 〈·〉 denotes expectation over the responses r. The inverse of the Fisherinformation, 1/IF (θ ), provides a lower bound on the variance (i.e., an upperbound on the accuracy) of an unbiased decoding estimate of θ from thepopulation response (Cover & Thomas, 1991; Dayan & Abbott, 2001). Fisherinformation is directly related to the discriminability d ′ between two stimuliθ and θ + �θ , since d ′ ≈ �θ

√IF (θ ) for small �θ (Dayan & Abbott, 2001).

The Fisher information can be written as (Kay, 1993)

IF = I meanF + I cov

F . (2.5)

Here

I meanF = f′T Q−1f′ (2.6)

is known as the linear approximation to the Fisher information, or the linearFisher information. Specifically, the inverse of f′T Q−1f′ gives the asymptotic1

error of the optimal linear estimator of the stimulus, for a response to thestimulus that follows any response distribution that has mean f(θ ) andcovariance Q(θ ) (Rao, 1945; Cramer, 1946; Series et al., 2004). In particular,this applies to gaussian or nongaussian distributions.

The second term, I covF , does depend on the the form of the response

distribution, beyond its covariance. In the following, whenever computingI cov

F , we assume that η follows a multivariate gaussian distribution, so that(Kay, 1993)

I covF = 1

2Tr

[Q′Q−1Q′Q−1] . (2.7)

As we explain below, correlation coding affects only I covF , while correlation

shaping affects both I meanF and I cov

F .

1Here, asymptotic implies that the optimal linear estimator is constructed based on fullknowledge of the mean and covariance of the underlying stimulus-response distributions.


3 The Cases of Cell Pairs and Small Populations

We start by considering the impact of correlations on the information carriedby cell pairs and small populations (N < 1/ρ). This was the setting of manyexperimental studies that addressed the role of correlations in the neuralcode (Petersen, Panzeri, & Diamond, 2001; Rolls, Franco, Aggelopoulos, &Reece, 2003; Averbeck & Lee, 2003; Samonds et al., 2004; Poort & Roelfsema,2009; Gutnisky & Dragoi, 2008). We use analytical expressions to show that,depending on correlation structure, correlation shaping can have either apositive or negative impact on I mean

F . Most beneficial are high correlationsbetween neurons with different stimulus preferences and low correlationsbetween neurons with similar preferences. For small to intermediate cor-relations, I mean

F ≈ IF , and hence correlation coding has little effect. Theseresults are in agreement with previous observations (Rolls et al., 2003; Aver-beck & Lee, 2004; Averbeck et al., 2006). We emphasize that these results canbe expected to hold only when N < 1/ρ. In subsequent sections, we showthat the intuition gained from studying cell pairs may not always extend tolarger populations.

3.1 Fisher Information in Cell Pairs. We first consider two cells whoseresponse follows a bivariate gaussian distribution given by equations 2.1and 2.2. For two neurons, we write the correlation coefficient as ρ1,2 = ρ2,1 =ρ, and obtain

IF = 11 − ρ2

[f ′1√v1

− f ′2√v2

]2

+ 21 + ρ

[f ′1 f ′

2√v1v2

]︸︷︷︸

I meanF

+ 2 − ρ2

4(1 − ρ2)

[(v′

1

v1

)2

+(

v′2

v2

)2]

− ρ2

2(1 − ρ2)v′

1v′2

v1v2+

[(1 + ρ2)ρ ′

1 − ρ2

] [ρ ′

1 − ρ2 − ρ

1 + ρ2

(v′

1

v1+ v′

2

v2

)]︸︷︷︸

I covF

,

(3.1)where all derivatives are taken with respect to the stimulus θ .

Intuitively, I meanF and I cov

F represent the contribution of changes in the fir-ing rate and covariance, respectively, to the Fisher information. While I mean

Fhas been studied previously, I cov

F has been examined only for stimulus-independent correlation coefficients—when ρ ′ = 0 (Abbott & Dayan, 1999;Sompolinsky et al., 2001; Shamir & Sompolinsky, 2001, 2004). We sepa-rate the influence of stimulus-dependent changes in correlation on IF asfollows:

� Correlation coding. The last of the five terms in the sum 3.1 is presentonly when ρ ′ = 0, and captures the amount of information directlydue to changes in correlations (Vaadia et al., 1995; deCharms &Merzenich, 1996; Chacron & Bastian, 2008). We refer to terms in IF

that are nonzero only when ρ ′ = 0 as the contribution of correlation


coding. If f ′1 = f ′

2 = v′1 = v′

2 = 0, then all information is due to corre-lation coding. It is necessary to use a nonlinear readout (decoding)scheme to recover this information (Shamir & Sompolinsky, 2004).

� Correlation shaping. I meanF is affected significantly by the level of corre-

lation, ρ. As (I meanF )−1 measures the error in the optimal linear estimate

of the stimulus, the impact of changes in correlation structure on I meanF

represents the amount by which correlations shape the informationavailable from linear readouts of the response. We refer to this effectas correlation shaping.

This terminology anticipates the discussion of larger populations, wherewe will be interested in how the spatial structure, together with stimulus-dependent changes of ρi j , affects IF . We note that stimulus-dependent cor-relations can also affect the information available from the variance of theneural response (see the third term in equation 3.1). This is another form ofcorrelation shaping with a marginal impact in the cases we discuss.

We first examine the effect of correlation shaping. A number of previousstudies concluded that an increase in correlation, ρ, can have a positiveimpact I mean

F for pairs of neurons that have different “normalized" meanresponses to the stimulus ( f ′

1/√

v1 = f ′2/

√v2). The effect tends to be negative

if the responses are similar ( f ′1/

√v1 ≈ f ′

2/√

v2). Intuitively, correlations canbe used to remove uncertainty from noisy responses of neuron pairs withdiffering response characteristics (Oram et al., 1998; Averbeck et al., 2006;Abbott & Dayan, 1999; Sompolinsky et al., 2001).

Indeed, the first term in equation 3.1,[

f ′1/

√v1 − f ′

2/√

v2]2

/(1 − ρ2), in-creases with ρ unless f ′

1/√

v1 = f ′2/

√v2. The resulting increase in discrim-

inability is illustrated in Figure 1 where we show the bivartiate distribu-tion p(r1, r2) of the response to two nearby stimuli θA and θB . In Figures1a and 1b, v′

1 = v′2 = ρ ′ = f ′

2 = 0, but f ′1 = 0, so that only the first term in

I meanF contributes to IF . In this example, an increase in correlation leads

to a large increase in IF . In Figure 1b, this increase results in improveddiscriminability between the stimuli, that is, a reduction of the probabilitythat the two stimuli will lead to the same response. However, when thetwo neurons respond similarly to the stimulus, f ′

1/√

v1 ≈ f ′2/

√v2, the sec-

ond term, 2[

f ′1 f ′

2/√

v1v2]/(1 + ρ), dominates. An increase in correlations

leads to a decrease in I meanF (Sompolinsky et al., 2001; Averbeck et al., 2006),

which is reflected in decreased discriminability between the stimuli (seeFigures 1c and 1d). High values of the correlation coefficients have beenused in Figures 1b and 1d for easier visualization.

In contrast, correlation coding typically has a small effect in the caseof two neurons, as the term I cov

F is far smaller than I meanF . There are two

reasons for this. The first holds only in the small correlation regime. Notethat ρ enters I cov

F at O(ρ2), while it enters I meanF at O(ρ). The second holds for

a larger range of correlation strengths: v′i/vi and ρ ′ are typically far smaller


r2

r2

r1 r1

1.4

1.2

1.0

0.8

0.6

1.41.21.00.80.6

1.4

1.2

1.0

0.8

0.61.41.21.00.80.6

a) b)

c) d)

Figure 1: Illustration of correlation shaping for neuron pairs. Each panel shows50% level curves of the joint density p(r1, r2) in response to two nearby stimuliθA (dashed line) and θB (solid line). In all cases, v1 = v2 = 1. A change fromstimulus θA to θB is assumed to affect only the fi , so that I cov

F = 0. The bene-ficial effect of correlations on I mean

F (first term in equation 3.1) is illustrated inpanels a and b. Here f ′

1(θA) = f ′2(θA) and increased correlations improve dis-

criminability. In contrast, f ′1(θA) = f ′

2(θA) in panels c and d, and increased cor-relations reduce discriminability. In panels a and b, f1(θA) = f2(θA) = 1, whilef1(θB) = 1, f2(θB) = 1.1. In panel a, ρ = 0.2, while in panel b, ρ = 0.99. In panelsc and d, f1(θA) = f2(θA) = 1, and f1(θB) = f2(θB) = 1.1. In panel c, ρ = 0.1, andin panel d, ρ = 0.99.

than f ′i /

√vi and, as a result,2 I cov

F I meanF . Therefore, under fairly general

assumptions, the dominant effect of correlations on Fisher information forcell pairs is by correlation shaping of I mean

F .

2In detail, if responses are given by counting spikes over approximately 1 second, thentypically f takes values substantially greater than 1. If firing is Poisson-like, then v ≈ f .This leads to the stated dominance of f ′/

√v among these terms.


Only close to perfect correlation, where ρ ≈ 1, is the impact of correlationcoding potentially significant. Assuming that ρ ′ = O(1) as ρ approaches 1and letting ε = 1 − ρ2, we have IF = 2ε−2(ρ ′)2 + O(ε−1). Therefore, when ρ

is close to 1, most information about a stimulus can be carried by correlationchanges. The balance between I mean

F and I corrF close to perfect correlations

strongly depends on the behavior of ρ ′ as ρ approaches 1. If ρ ′ approaches0 as ρ approaches 1, as in de la Rocha et al. (2007), I mean

F may continue todominate.

3.2 Fisher Information in Small Populations. We next use an asymp-totic expansion to show that many of these observations extend to smallpopulations of neurons with low correlations. The following results aregeneral, under the assumption that the response follows a multivariategaussian distribution. However, the range over which the approximation isvalid decreases with population size, and the approximation will typicallybreak down when N exceeds 1/ρi, j . (See appendix A and Figure 6.)

Let

(IF ) meani = ( f ′

i )2

vi, (IF )var

i = 12

(v′

i

vi

)2

, and

(IF )corri, j = ρi, jρ

′i, j

([ρ ′

i, j

ρi, j−

(v′

i

vi+ v′

j

v j

)]).

We show in appendix A that

IF =∑

i

(IF )meani −

∑i, j

i = j

f ′i f ′

jρi, j√viv j

+∑i, j,kk =i, j

f ′i f ′

jρi,kρk, j√viv j︸︷︷︸

I meanF

+∑

i

(IF )vari +

∑i, j

i = j

ρ2i, j

8

(v′

i

vi− v′

j

v j

)2

+∑i< j

(IF )corri, j

︸︷︷︸I cov

F

+O(ρ3

i, j

). (3.2)

Here (IF )meani and (IF )var

i are O(1), while (IF )corri, j is O(ρ2

i, j ). Therefore, IF isa sum of contributions from individual neuron responses ((IF )mean

i and(IF )var

i ) and corrections of higher order in ρ due to correlations in theresponse.

Only the term −∑i, j,i = j f ′

i f ′jρi, j/

√viv j in I mean

F is of first order in ρ.This term therefore dominates the correction when correlations are small tointermediate. In this case, correlations between differently tuned neuronsagain increase IF , and those between similarly tuned neurons decrease IF . If


correlations ρi, j across the (small) population are stronger between neuronsi and j for which f ′

i and f ′j have opposite signs and weaker when these

signs are the same, they increase IF . This is in agreement with the two-cellcase discussed above, as well as previous results (Averbeck & Lee, 2006;Romo et al., 2003; Averbeck et al., 2006; Sompolinsky et al., 2001).

4 Large Populations with No Spatial Correlation Decay

In general, for large populations, it is difficult to obtain a closed-form ex-pression for IF in terms of the variances, correlation coefficients, and firingrates. Results are available under different simplifying assumptions thatmake the problem mathematically tractable (Abbott & Dayan, 1999; Wilke& Eurich, 2002; Shamir & Sompolinsky, 2004). In most of these cases, it wasassumed that correlation coefficients, ρi, j , are independent of the stimulus θ ,so that ρ ′

i, j = 0. In the following, we refer to this as the stimulus-independent(SI) case and contrast it to the stimulus-dependent (SD) case. The assump-tion that we make is that correlations between cell pairs, ρi, j , are given byequation 2.3 and that stimulus dependence of correlations, Si, j (θ ), takes theproduct form in equation 2.4.

In this section, we let c(φi − φ j ) = 1. Therefore, the correlation structureis completely determined by the stimulus. In this case, an analytical ex-pression for Q−1 and IF can be found using the Sherman-Morrison formula(Meyer, 2000). We derive the exact expression for IF for arbitrary popula-tion sizes N, arbitrary response characteristics vi (θ ), fi (θ ), and si (θ ), as wellas an approximation valid for large populations, in appendixes B and C.

To give concrete examples of how the stimulus dependence of correla-tions affects IF in large populations, in the remainder of the letter, we furtherassume (as in, e.g., Seung & Sompolinsky, 1993; Shamir & Sompolinsky,2001; Sompolinsky et al., 2001; Butts & Goldman, 2006) that cell responsesfollow tuning curves that differ only by a phase shift, so that we can write

fi (θ ) = f (θ − φi ), vi (θ ) = v(θ − φi ), and si (θ ) = s(θ − φi ), (4.1)

where θ, φi ∈ [0, 2π ). We take all functions to be periodic. The response,fi (θ ), is chosen so that neuron i responds preferentially (with maximumrate) to stimulus θ = φi , where φi is fixed. These are common assumptionsthat simplify the analysis considerably (Sompolinsky et al., 2001; Wilke& Eurich, 2002). Correlations are therefore determined by ρi j (θ ) = s(θ −φi )s(θ − φ j ).

Assuming the neurons sample the stimulus space uniformly and suffi-ciently densely, we can use the continuum limit to approximate IF . In thiscase, an arbitrary vector a(θ ) with components a (θ − φi ) tends to a functiona (θ ) of the stimulus θ . By symmetry, neither I mean

F , I covF , nor IF depends on θ

in the large population limit, since the response provides equal information


about any stimulus. Therefore, we fix θ = π in the following and write thefiring rates, variances, and correlations as functions of the neurons’ pre-ferred stimuli, φ. As we show in appendix C, IF can then be approximatedas the sum of

I meanF ∼ D

(f ′(φ)√v(φ)

, s(φ)

), and

I covF ∼ N

π

∫ 2π

0

(v′(φ)2v(φ)

− s ′(φ)s(φ)1 − s2(φ)

)2

dφ + D(G(φ)s(φ), s(φ)), (4.2)

where G(φ) = (s ′(φ) + v′(φ)2v(φ) s(φ)) and

D(a (φ), s(φ)) ≈ N2π

[∫ 2π

0

a2(φ)1 − s2(φ)

dφ

−∫ 2π

0

a (φ)s(φ)1 − s2(φ)

dφ

/∫ 2π

0

s2(φ)1 − s2(φ)

dφ

]. (4.3)

Note again that I meanF and I cov

F are independent of the stimulus.The correlation between two neurons with preferred stimuli φ and φ′ will

be denoted by ρ(φ, φ′), and ρ(φ) = ρ(φ, φ) = s2(φ) will be the correlationcoefficient between two neurons with equal stimulus preference.

In the remainder of the letter, we make one final assumption: that thefunctions f , v, and s are even (i.e., symmetric around preferred orien-tations), as in, for example, Sompolinsky et al. (2001), Wilke and Eurich(2002), and many other studies.

4.1 Effects of Stimulus-Dependent Correlations on ImeanF . To illustrate

how stimulus dependence of correlations can influence the informationcontained in the population response, we first consider I mean

F . Even whencorrelations are small, this stimulus dependence can have a strong effect bycorrelation shaping.

Since f (φ) and v(φ) are even, f ′(φ)/√

v(φ) is odd. Therefore, settinga (φ) = f ′(φ)/

√v(φ), the second term in equation 4.3 vanishes, and

I meanF = N

2π

∫ 2π

0

( f ′(φ))2

v(φ)1

1 − s2(φ)dφ. (4.4)

Although I meanF is the average of the Fisher information [ f ′

i ]2/vi of sin-gle neurons, with a weighting factor, caution needs to be exercised wheninterpreting this result. Equation 4.4 is the result of simplifying an expres-sion derived from all pairwise interactions across the population.


In the SI case, s(φ) is constant across the population: s(φ) = s. We focuson comparisons between SI and SD cases matched to have the same averagecorrelation coefficient across the population. We therefore assess the effectsof the stimulus dependence of correlation, as opposed to the level of corre-lations. Specifically, we ensure that the average correlation coefficient acrossthe population in the SD case, (4π2)−1

∫ 2π

0

∫ 2π

0 s(φ1)s(φ2) dφ1 dφ2, equals thatin the SI case by setting s = 1/(2π )

∫ 2π

0 s(φ)dφ. Examples of typical matchedcorrelation matrices, ρi j , in the SD and SI cases, are shown in the right-handcolumn of Figure 3.

Figures 2a and 2b illustrate how correlation shaping may increase I meanF

in the SD case over the SI case. In each, stimulus dependence of correlationsarises from a different relationship between stimulus-induced firing rateand correlation (see the insets). In Figure 2a, ρ(φ) increases with f (φ), as inde la Rocha et al. (2007) and certain regimes in Binder and Powers (2001),Kohn and Smith (2005), and Greenberg et al. (2008). In Figure 2b, ρ(φ) firstincreases with f (φ), and then decreases, as in feedforward networks withrefractory effects (Shea-Brown et al., 2008). Importantly, for both Figures 2aand 2b, correlations are high between neurons that individually carry themost information about the stimulus (between neurons with large values of( f ′(φ))2/v(φ)). Therefore, the weighting factor 1/(1 − s2(φ)) assigns a greatercontribution of these more informative cells to the weighted average inequation 4.4 for the SD case, leading to the increase in I mean

F .On the other hand, Figure 2c illustrates a case in which correlations

decrease with firing rates, as observed in Aksay et al. (2003). As a result,correlations between the most informative neurons are smaller than aver-age, and correlation shaping has a negative impact on IF . We note that inall panels, maximum pairwise correlations satisfy ρmax � 0.45, within therange typically reported (e.g., Gutnisky & Dragoi, 2008; Poort & Roelfsema,2009; Zohary et al., 1994). Increasing this maximum without changingthe mean correlation can make these correlation shaping effects morepronounced.

A different way of seeing how I meanF can be greater in the SD than the SI

case is given in Figure 3a. Here, I meanF and I cov

F are computed numericallyand plotted as a function of the population size N for both the SD and SIcases that correspond to the example of correlations increasing with rate(see Figure 2a). Note that I mean

F dominates I covF over a wide range of N and

that the total Fisher information, not just I meanF , is greater in the SD versus

SI case. Moreover, the continuum limit given in equation 4.4 appears valideven for moderate population sizes.

Care needs to be taken when trying to intuitively understand thesepopulation-level effects of stimulus-dependent correlations on I mean

F by in-voking the case of two neurons studied in section 3. Consider the case ofcorrelations increasing with firing rate (see Figures 2a and 3a). As notedin the discussion of equation 3.1, an increase in correlations between twosimilarly tuned neurons will typically have a negative impact on I mean

F due


Figure 2: Examples of different correlation tuning curves and their impact onI mean

F for large populations. The top panels show mean response f (φ) (dashed),and covariance ρ(φ) f (φ) of two identically responding neurons for the SD(black) and SI (gray) cases. The middle panels show the correlation tuningcurves, ρ(φ) = s2(φ) for the SD (black) and SI (gray) cases. Average correla-tions are matched to equal 0.1 in all cases. Insets illustrate the ρ − f rela-tionship for each choice of the correlation tuning. Bottom panels show theintegrand of equation 4.4 for the SD (black) and SI (gray) cases. (a) ρ − f fol-lows a concave increasing curve, and ρ(φ) shows a slightly broader tuningthan f (φ) in the SD case, resulting in a substantial increase in I mean

F with re-spect to the SI case (increase of ∼ 21%). (b) ρ − f is nonmonotonic, and ρ(φ)is bimodal and matches ( f ′(φ))2/v(φ) in the SD case. This yields a larger en-hancement of I mean

F with respect to the SI case (increase ∼ 29%). (c) Corre-lations that decrease with rate have a negative impact on I mean

F (decrease of∼ 7% compared to the SI case). In all cases, I mean

F was computed in the largeN limit using equation 4.2. Parameters: Average correlation coefficient s2 = 0.1in all cases (larger values, e.g., 0.2, will typically more than double the differ-ence in I mean

F between SD and SI cases). In all cases, f (φ) = 5 + 45a 6(φ), witha (φ) = 1/2(1 − cos(φ)) and v(φ) = f (φ) (Poisson). (a) s(φ) = kρ + bρa 2(φ) wherekρ = 0.135 and bρ = 0.5. (b) s(φ) = 4rmax f (θ )[ fmax − f (θ )]/ f 2

max with rmax = 0.65and fmax = 50. (c) s(φ) = kρ + bρa 2(φ) where kρ = 0.47 and bρ = −0.4. (See ap-pendix E.)


Figure 3: I meanF and I cov

F as a function of population size N, for matched SD and SIcorrelation cases and various correlation decay lengthscales. Here, correlationis assumed to increase with firing rate, as in Figure 2a. The coefficients kρ

and bρ defining s(φ) = kρ + bρa 2(φ) are chosen to keep the average correlationcoefficients over the population equal to 0.1 (see appendix E). The correspondingcovariance and correlation matrices, ρi, j

√fi f j , and ρi, j , respectively, are also

shown for the SD and SI cases (on-diagonal terms are set to 0 in these plots).

to the dominance of the second term of I meanF in equation 3.1. On the other

hand, equation 4.4 shows that increasing correlations between the mostinformative neurons in a large population, regardless of the similarity oftheir tuning, has a positive impact. The two results are not contradictory.Consider the pairwise sum of the two-neuron I mean

F from equation 3.1 over


Figure 4: Plots of the matrix f ′i f ′

j Q−1i, j whose double sum determines I mean

F .(a) No spatial correlation decay. (b) Spatial decay with α = 0.25. Top: On-diagonal terms of matrix. Bottom: Off-diagonal terms (with on-diagonal valuesset to 0 for ease of visualization).

all neuron pairs in the population. Note that the second term of I meanF in

equation 3.1 can be expected to be matched with one of equal and oppo-site sign in such a sum if the tuning curves are symmetric and correlationsdepend only on firing rate. Therefore, the typically dominant second termcancels, and it is the first term in equation 3.1, always positively affectedby the presence of correlation, that remains. Moreover, examination of thisfirst term in equation 3.1 does show similarity with equation 4.4. In bothcases, assigning largest correlations ρi, j or s(φ, φ′) to the most informativeneurons will yield the greatest total value of I mean

F .Figure 4a shows that this cancellation argument, while not directly appli-

cable, is at least analogous to what happens when computing I meanF for the

large population via the complete expression, equation 4.2. The sum of theterms f ′

i f ′j Q−1

i, j defines the linear Fisher information, I meanF = ∑

i, j f ′i f ′

j Q−1i, j

(see equation 2.6). Under the existing symmetry assumptions, the off-diagonal terms cancel, and only the diagonal terms contribute to the sum.In appendix C, we show that Q−1

i,i = [vi (1 − s2i )]−1, in agreement with the

remaining term in equation 4.4.


These observations are robust to the presence of weak asymmetry inthe functions f , v, and s. For instance, when the tuning curve f (θ ) is asum of a symmetric and small asymmetric part, fsym(θ ) + ε fasym(θ ), anexamination of equation 4.3 shows that the impact of the asymmetry onI mean

F = D( f ′(θ )√v(θ )

, s(θ )) is of order O(εN), while I meanF is O(N). However, we

show in the next section that the large population limit can be changedsignificantly when c(φi − φ j ) is not constant.

4.2 Effects of Stimulus-Dependent Correlations on I covF . We now turn

to the impact on I covF of the stimulus dependence of correlations. In ap-

pendix D, we show that this impact is negligible for small to intermediatecorrelations and that

I covF ≈ N

2π

∫ 2π

0

v(φ)2v′(φ)

dφ. (4.5)

Moreover, as discussed in section 3, values of v(φ)/2v′(φ) are typicallysmaller in magnitude than values of ( f ′(φ))2/v(φ). Therefore, for small tointermediate correlations, the major contribution of the stimulus depen-dence of correlations comes from I mean

F rather than I covF . This agrees with

the case of two cells (see section 3). Asymptotic estimates of the integralsin I cov

F show that this remains true even for correlation coefficients close toone. The dominance of I mean

F over I covF is apparent in Figure 3a. As we show

in the next section, however, this dominance may no longer hold in thepresence of spatial decay of correlations (Sompolinsky et al., 2001; Shamir& Sompolinsky, 2004).

4.3 Summary of Section 4. Stimulus dependence may shape the struc-ture of correlations so that neurons that are most informative about thestimulus presented are most highly correlated. This can lead to an increasein overall information. This is possible even when the average correlationsacross the population are low, but not when correlations are fixed or if allneurons have identical mean responses.

5 Effects of Correlation Stimulus Dependence in the Presenceof Spatial Decay

In this section we examine how stimulus-dependent correlations affect IF inthe presence of spatial correlation decay. We again assume that correlationsand rates are described by equations 2.3 and 2.4, but we now assume that

c(φi − φ j ) = exp[−|φi − φ j |

α

].


The constant α determines the spatial range of correlations. Other parame-ters were chosen so that the average correlation across the population 〈ρi, j 〉remains constant (for details, see appendix E). As an exact expression forthe inverse of the covariance matrix is difficult to obtain, we study this casenumerically and give an intuitive explanation of the results.

Experimental data on the spatial range of correlations are consistentwith values of α between 0.16 and 4 (Smith & Kohn, 2008). These valuesappear to depend on the cortical distance between the neurons, as well asthe difference between their preferred stimuli. In the following, we examinetwo cases in this range: α = 0.25, corresponding to sharp correlation decay,and α = 2, corresponding to a more gradual decay.

5.1 Effect of Correlation Shaping on ImeanF . When α = ∞, there is no

spatial decay, and we are in the situation discussed in the previous section:IF is typically dominated by I mean

F , which grows linearly with populationsize N (see Figure 3a). However, for finite values of α, I mean

F generallysaturates with increasing N (see Figures 3b and 3c). This agrees with earlierfindings for stimulus-independent correlations (Shamir & Sompolinsky,2004, 2006).

Additionally, effects of stimulus dependence in correlations on I meanF can

be reversed for finite values of α. For example, assume that si (φ) increaseswith the firing rate, as in Figure 2a. When α = ∞, the stimulus dependenceof correlations increases I mean

F (see Figure 3a). However, for finite α, thisstimulus dependence has a negative impact on I mean

F (see Figures 3b and3c).

Intuitively, this may be due to spatial correlation decay-reducing corre-lations between neurons with differing stimulus preferences. The negativeimpact of correlations between similarly tuned neurons on IF is no longerbalanced by the positive impact on differently tuned neurons. Indeed, thestronger the spatial decay of correlations, the more this balance is broken.Therefore, the cancellation arguments presented in the previous section nolonger hold (compare Figures 3b and 3c), and it is no longer the case thatsimply increasing correlations for more informative neurons will increaseI mean

F . Instead, correlation structures that increase correlation for similarlyversus differently tuned neurons can again be expected to decrease I mean

F .Figure 4 shows that this is the precisely the effect of no spatial decay (α = ∞)versus spatial decay (α = 0.25) correlation structures.

As a second example, assume that correlations decrease, rather than in-crease, with firing rate, as in Figure 2c. In this case, correlations betweensimilarly tuned, strongly responding neurons are decreased. As expectedfrom the arguments above, stimulus-dependent correlations then increaseI mean

F over its value in the stimulus-independent case (see Figure 5a). More-over, absolute levels of IF increase twofold compared to the analogous casewhere correlations increase with rate (compare Figures 3c and 5a).


a)

N

8

4

00 5000 10000

IF

x103

N

6

3

x104

0 5000 10000

IF

b)

0.5

0.25

0

index

IF

, SDcov

IF

, SIcov

IF

, SDmean

IF

, SImean

de

gre

es

2

0 1000

Figure 5: Examples where the Fisher information is larger in the SD than the SIcase, despite strong correlation decay α = 0.25. In each case (as in all of Figures2–6), s(φ) is set so that the average correlation coefficient ρ across the populationis 0.1. (a) Correlations decrease with firing rate, as in Figure 2c. (b) The responseof a population with two subpopulations each tuned to the stimulus. In theright panel, as in Figure 2, ( f ′(φ))2/v(φ) is scaled and represented by the solidline, while ρ(φ) = s(φ)2 is represented by the dotted line. The effects of spatialcorrelation decay are not shown. The response of each cell in the populationfollows a unimodal tuning curve; however, there are two sets of cells at differentspatial locations that share the same stimulus preference. The left panel showsthe effect of this arrangement on the Fisher information. Other parameters areas in Figure 3.

However, in all of these cases, note that levels of IF are lower in thepresence of correlation decay for both SD and SI cases. We now mentionone way in which this can be mitigated. As illustrated in Figure 5b, weincrease the number of areas or subpopulations that respond strongly to agiven stimulus. The response of each cell still follows a unimodal tuningcurve, as above. However, the entire population has a number of cells atdifferent spatial locations that share the same stimulus preference. There-fore, cells in different subpopulations are only weakly correlated and canbe thought of as members of different, nearly independent populations.As Figure 5b shows, this boosts overall levels of I mean

F , while maintain-ing the benefit of stimulus dependence in correlations within individualsubpopulations.

In sum, the spatial decay of correlations has a strong negative effecton linear Fisher information I mean

F . If correlations depend on stimuli byan increasing relationship with firing rate, this effect can be accentuated,with levels of I mean

F decreasing by a further factor of two for SD versus SIcases. However, the opposite effect occurs if correlations decrease with rate:stimulus dependence can then approximately double I mean

F .


5.2 Effect of Correlation Coding on I covF . For large populations, Fig-

ures 3 and 5 show that information can be carried predominantly by I covF ,

and this dominance is more pronounced as the correlation length scale α de-creases. This agrees with earlier findings (Sompolinsky et al., 2001; Shamir& Sompolinsky, 2004; Chelaru & Dragoi, 2008). Moreover, we see that theeffects of stimulus dependence of correlations on I cov

F have the same “sign"as those on I mean

F . Specifically, when correlations increase with rate, as inFigure 3, both I mean

F and I covF are lower in the SD than in the SI cases for

finite values of correlation length α. On the other hand, when correlationsdecrease with rate, as in Figure 5, corresponding values of both I mean

F andI cov

F are higher for the SD than the SI case.The effects of stimulus dependence on the (dominant) I cov

F terms can beattributed to correlation coding. In detail, the contribution of ρ ′

i j (θ ) terms toI cov

F can be isolated numerically by simply computing I covF twice: once with

these terms at the nonzero values expected from stimulus dependence andonce after “artificially" setting all of these terms equal to zero. The differenceis the contribution to IF attributable directly to changes in correlation withthe stimulus (i.e., correlation-coding, as opposed to the correlation-shaping,effects that have been the focus of much of the previous discussion). Ourcalculations (not shown) indicate that almost the entire increase, or decrease,of I cov

F in the SD relative to the SI cases is due to this correlation coding.

6 Discussion

The presence of correlations in a neural response can either increase ordecrease the level of information a population carries. Intuitively, this canbe attributed to the following competing effects. Correlated fluctuationsimply a common component in the response noise of different neurons.Similarly tuned, correlated neurons then provide redundant information,as the common noise cannot be averaged away (Johnson, 1980; Britten et al.,1992; Zohary et al., 1994). However, for differently tuned neurons, correlatednoise can be removed by, for example, considering the difference in theiractivity (Abbott & Dayan, 1999). The net effect of correlations depends onthe balance among different effects.

An overview of the extensive research is given in section 1. In the ma-jority of these studies, correlations did not depend on the stimulus, butrather on the “spatial” distance between neurons (i.e., between their pre-ferred stimuli). Here we wish to generalize two simplified messages thatstand out in this case. The first is that the presence of correlations on short“length” scales (mostly between similarly tuned neurons) decreases themean component of the Fisher information, I mean

F ; if these correlations havelong length scales, they increase I mean

F . The second message is that in thepresence of correlations that decline with distance, I mean

F can be dominatedby the “covariance term” in the Fisher information, I cov

F (cf. Sompolinskyet al., 2001).


Here we ask a fundamentally different question: What is the effect ofstimulus dependence in correlations on population Fisher information? Inseveral cases, the results we find can be interpreted in the light of obser-vations made previously. We stress that these are analogies, and the samemechanisms may not be at play.

We find two ways in which such stimulus dependence influences Fisherinformation. The first, correlation coding, refers to the information di-rectly carried by changes in correlation structure in response to stimuli.The second, correlation shaping, refers to the impact of stimulus depen-dence on information carried by the mean and variance of neural re-sponses. In different cases, we derive expressions for the Fisher informa-tion that isolate correlation shaping and correlation coding effects. For cellpairs, and small-to-intermediate populations equations 3.1 and 3.2 are validfor general correlation structures. For correlations with product structure,ρi j (θ ) = si (θ )s j (θ ), expressions are derived for populations of arbitrary sizeN, with simplifications in the continuum limit N → ∞ (see equations 4.4and 4.5).

These expressions allow us to make a number of general observations.For typical firing regimes, we find that the effects of correlation shapingdominate those of correlation coding for pairs of neurons or small popu-lations with weak to moderate correlations, with most information beingcarried by I mean

F . Correlation coding becomes significant only for strongcorrelations. However, for large populations, the answer is different. In theabsence of spatial decay, correlation shaping and I mean

F dominate (cf. Shamir& Sompolinsky, 2004, 2006) regardless of correlation strength. However, cor-relation coding and I cov

F become important in the presence of decay. Thesepoints are consistent with the findings from the literature on stimulus-independent correlations.

Additionally, for pairs of neurons or small populations with weak cor-relations, correlated responses between similarly tuned neurons typicallydecrease I mean

F , while correlations between oppositely tuned neurons in-crease I mean

F , as has been shown in related settings (cf. Averbeck & Lee,2006; Romo et al., 2003; Averbeck et al., 2006; Sompolinsky et al., 2001) andconsistent with observations described above. However, for large popula-tions with symmetric and uniformly distributed tuning curves, the situationmay be quite different. For correlations with product structure and withoutspatial decay, correlations between the most informative neurons (thosewith largest f ′

i (θ )/√

vi (θ )) have the greatest impact on I meanF , regardless of

similarity of tuning.Interestingly, in the presence of spatial decay of correlations, the effects

of stimulus dependence of correlations on Fisher information are typicallyreversed. We note one interpretation: since spatial decay tends to decreaseFisher information, the correct stimulus dependence of correlations cancounterbalance this effect. Moreover, as we argue in section 5, these ef-fects are consistent with the message from the literature that correlations


between similarly tuned neurons tend to decrease I meanF : the forms of stim-

ulus dependence that enhance I meanF also tend to correlate more differently

tuned neurons, and vice versa. This phenomenon also applies to the effectsof stimulus-dependent correlations on I cov

F , although the underlying mech-anisms could be different. We note that consistent with the messages abovefor stimulus-independent correlations, I cov

F can become dominant with suf-ficiently strong spatial decay of correlations. Moreover, we find that it is thechanges in correlation with stimulus themselves (correlation coding) thatcarry the majority of this information.

As discussed in section 2, it has been observed that correlations betweenneuronal responses decrease with the difference between their preferredstimuli (van Kan, Scobey, & Gabor, 1985; Lee, Port, Kruse, & Georgopoulos,1998; Holmgren et al., 2003; Smith & Kohn, 2008). This effect can also followfrom the stimulus dependence of correlations. When correlations increasewith firing rate, two neurons that both respond strongly to similar stimuliwill be more correlated than those of neurons whose preferences differ. Asneurons with similar preferences in stimuli can be expected to be physi-cally closer in the cortex, stimulus dependence can result in correlationsthat decay with physical distance (Shea-Brown et al., 2008). This is quitedifferent from the case where physically distant cells are less correlated dueto a smaller overlap in their inputs. With the stimulus dependence of corre-lations, two distant cells, one or both of which are responding strongly, maybe more correlated than two nearby cells that are both responding weakly(see Figure 3).

What biological mechanisms could underlie different patterns ofstimulus-dependent correlation? One is the co-tuning of correlation andresponse rate that has been observed in feedforward networks (de la Rochaet al., 2007; Shea-Brown et al., 2008). More complex network effects couldbe behind the decreasing trend of correlation with rates seen in Aksayet al. (2003). Moreover, stimulus-dependent adaptation of correlations hasbeen observed in the visual cortex (Kohn et al., 2004; Ghisovan, Nemri,Shumikhina, & Molotchnikoff, 2008; Gutnisky & Dragoi, 2008). Our studypoints to the potentially distinct impacts of the mechanisms on populationcodes.

Fisher information is only one of the possible metrics that can be usedto quantify the impact of correlations. However, its close connection withstimulus discriminability (Dayan & Abbott, 2001), relative ease of compu-tation compared to other metrics, and recent use in experimental settings(Gutnisky & Dragoi, 2008; Averbeck & Lee, 2006) make it a good startingpoint. Future work will extend our study of the impact of correlation stim-ulus dependence to other metrics, such as mutual information, adding tothe results of Montani et al. (2007) and Panzeri et al. (1999).

Another important question for future work comes from decoding:How can information encoded in correlation changes be read out? Forcases in which information is dominated by I mean

F terms, a linear readout


will suffice; however, when I covF dominates, as for large populations with

distance-dependent decay of correlations, nonlinear schemes are required(Wu et al., 2001; Shamir & Sompolinsky, 2004.

Appendix A: Fisher Information for Small Populationswith Small Correlations

The appendixes contain a number of exact expressions and approximationsof the Fisher information for both intermediate and large populations. Theseresults should be useful in the further analysis of the impact of correlationsin settings similar and distinct from those studied here.

The approximation in equation 3.2 is obtained from the assumption|ρi, j | 1. Defining ερi, j = ρi, j , we can write

Qi, j = δi, jvi + ε(1 − δi, j )ρi, j√

viv j .

Therefore, Q is a perturbation of a diagonal matrix R with entries Ri, j =δi, jvi (x), and the perturbation εS where Si, j = (1 − δi, j )ρi, j (x)

√vi (x)v j (x).

We can now use the standard matrix perturbation result (see also Wilke &Eurich, 2002; Demmel, 1997):

Q−1 = [R(I + εR−1S)]−1 = (I + εR−1S)−1R−1

=[ ∞∑

i=0

(−εR−1S)i

]R−1

= R−1 − εR−1SR−1 + ε2R−1SR−1SR−1 + (ε3).

(A.1)

The equality on the second line holds whenever ‖εR−1S‖ < 1 for a norm‖ · ‖ which is consistent with itself (Demmel, 1997, lemma 2.1). Usingequation A.1, we obtain

Q−1i, j = δi, j

1vi

− ε(1 − δi, j )ρi, j√viv j

+ ε2∑

kk =i, j

ρi,k ρk, j√viv j

. (A.2)

Using this equation, the first term in the expression for IF , fT Q−1f, can becomputed directly, to obtain the expression on the first line of equation 3.2.The second term, Tr[(Q′Q−1)2]/2, can be computed similarly through alengthier computation. This computation can be simplified using the ob-servations in the next section. This gives equation 3.2, keeping terms up tosecond order.

The convergence of the sum on the second line of equation A.1 is notguaranteed if ‖εR−1S‖ > 1. This implies that for fixed ε, the approxima-tion A.2 will break down for sufficiently large N (typically about whenN > 1/ε).


Appendix B: General Expression for IF in the Product Case

In this appendix, we use the Sherman-Morrison Formula (Meyer, 2000,p. 124) to derive a general expression for the Fisher information in theproduct case. Let

S =N∑

j=1

s2j(

1 − s2j

) . (B.1)

Then

Q−1i, j =

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

1vi(1 − s2

i

)(

1 − s2i

(1 + S)(1 − s2

i

))

if i = j

− si s j√viv j (1 + S)

(1 − s2

i

)(1 − s2

j

) if i = j

. (B.2)

Using this equation, we can obtain a compact expression for IF . The termresulting from changes in the mean number of spikes as the stimulus variesis given directly from definition 2.7 as

I meanF (x) =

N∑i, j=1

f ′i f ′

j Q−1i, j . (B.3)

The contribution to IF due to changes in the covariance, given by I covF =

Tr[(Q′Q−1)2]/2, can be expressed compactly by introducing

Ri = ddx

ln si = s ′i

si, and Zi = d

dxln(si

√vi ) = s ′

i

si+ 1

2v′

i

vi. (B.4)

Note that when ρi, j have the form given in equation 2.3, c(φi − φ j ) = 1, andthe stimulus dependence of correlations, Si, j (θ ) takes the product form inequation 2.4. We can write

Q′i, j = (Zi + Zj − 2δi, j Ri )Qi, j ,

where Zi and Ri are defined in equation B.4. Following this observation,we can follow the computations in Wilke and Eurich (2002, appendix A), toobtain

Tr[(Q′Q−1)2]2

=N∑

k=1

Z2k +

N∑k,l=1

Qk,l Zk Zl Q−1l,k − 4

N∑k=1

Qk,k Zk Rk Q−1k,k

+ 2N∑

k,l=1

Qk,k Ql,l Rk Rl Q−1k,l Q−1

l,k .


Observing that Q−1 is self-adjoint, we obtain

I covF =

N∑i=1

(Zi )2 [1 + Q−1i,i vi

(1 − s2

i

)] +N∑i, j

Zi Zj si s j√

viv j Q−1i, j

+ 2N∑i, j

Ri Rj

[√viv j Q−1

i, j

]2− 4

N∑i=1

Zi Rivi Q−1i,i . (B.5)

Therefore, IF is the sum of equations B.3 and B.5.The contribution to IF due to only changes in the variances can be ob-

tained from equation B.5 by setting Ri = 0 and replacing Zi by v′i/(2vi ), so

that

I varF =

N∑i=1

(v′

i

2vi

)2 [1 + Q−1

i,i vi(1 − s2

i

)] +N∑i, j

v′iv

′j si s j

4√viv j

Q−1i, j . (B.6)

The contribution due to correlation stimulus dependence is therefore

I corrF = I cov

F − I varF .

Appendix C: Asymptotic Results

The expression for IF derived in appendix B can be simplified considerablyfor large cell populations. If N is large and 0 < ε < si < 1 − δ for some ε, δ >

0, then S = O(N), where S is defined in equation B.1. The assumptions onsi are not essential but make the derivation of the asymptotic expressionseasier.

Keeping only the leading-order terms in equation B.2, we can write

Q−1i, j ≈

⎧⎪⎪⎨⎪⎪⎩

1vi(1 − s2

i

) if i = j

− si s j√viv jS

(1 − s2

i

)(1 − s2

j

) if i = j. (C.1)

To obtain the asymptotic value of IF given in equation C.4 from equa-tions B.3 and B.5, first note that S = O(N). Therefore, for large N,

N∑i, j

Ri Rj

[√viv j Q−1

i, j

]2∼

N∑i

[Rivi Q−1

i,i

]2.


Using this observation together with the asymptotic value of Q−1i,i given in

equation C.1, the first and last two sums on the right-hand side of equa-tion B.5 behave asymptotically as

N∑i=1

(Zi )2 [1 + Q−1i,i vi

(1 − s2

i

)] + 2N∑i, j

Ri Rj

[√viv j Q−1

i, j

]2

− 4N∑

i=1

Zi Rivi Q−1i,i ∼ 2

N∑i=1

(Zi − Ri

1 − s2i

)2

.

By a slight abuse of notation, define the weighted average of the entriesin the vector a over the population as

1N

N∑i

a2i

1 − s2i

=⟨

a2

1 − s2

⟩,

and let

D(a, s) def= N

[⟨a2

1 − s2

⟩−

⟨as

1 − s2

⟩2 /⟨s2

1 − s2

⟩]. (C.2)

Then the observations above can be combined with

N∑i, j

ai a j√

viv j Q−1i, j ≈

⎡⎣( N∑

i

a2i

1 − s2i

)⎛⎝ N∑

j

s2j

1 − s2j

⎞⎠ −

(N∑i

ai si

1 − s2i

)2⎤⎦/

⎛⎝N∑

j

s2j

1 − s2j

⎞⎠= N

[⟨a2

1 − s2

⟩⟨s2

1 − s2

⟩−

⟨as

1 − s2

⟩2]/

⟨s2

1 − s2

⟩def= D(a, s) (C.3)

applied to the term I meanF , and the second sum on the right-hand side of

equation B.5 gives

I meanF (x) ∼ D

(f′

√v

, s)

, and

I covF (x) ∼ 2

N∑i

(v′

i

2vi− s ′

i si

1 − s2i

)2

+ D(Gs, s), (C.4)


IFmean Order 2IFCov Order 2IFmean ExactIFCov ExactIFmean Large NIFCov Large NIFmean cont limitIFCov cont limit

IF

x103

2

1

0

400 8000N

IFmean

IFcov

O(ρ2) approx.

Large N approx.

Continuum limit

Exact

Figure 6: Values of I meanF and I cov

F from (1) approximations for small ρ, validfor intermediate population sizes N, given by equation 3.2, (2) the “exact”value obtained by numerically inverting the correlation matrix Q, and usingequations 2.6 and 2.7, (3) the large N approximation given by equations C.4,and (4) the continuum limit given by equations 4.2 and 4.3. Here, f (φ) = 5 +45a (φ) with a (φ) = 1/2(1 + cos(φ)), and v(φ) = f (φ) (as for Poisson variability).Additionally, s(φ) = 0.2 + 0.5a (φ). Other parameter choices give similar results(not shown).

where Gi = ddx ln(si

√vi ) = s ′

i/si + 12v′

i/vi . As before, I meanF corresponds to

the linear Fisher information.The Cauchy inequality can be applied directly to show that⟨

a2

1 − s2

⟩ ⟨s2

1 − s2

⟩−

⟨ar

1 − s2

⟩2

≥ 0,

so that D(·, s) is always positive.Figure 6 shows that the approximations, together with the continuum

limit expressions found in the main text, are valid to high accuracy overbroad ranges of N.

Appendix D: Impact of Pure Correlation StimulusDependence on I cov

F

We show that the impact of stimulus dependence of correlations on I covF is

relatively small compared to the impact on I meanF in the situation discussed

in section 4. By invoking the symmetry of the tuning curves again,

D(Gs, s) = D(

s ′(θ ) + v′(θ )v(θ )

s(θ ), s(θ ))

∼ N2π

∫ 2π

0

(s ′(φ) + v′(φ)

v(φ)s(φ)

)2 11 − s2(φ)

dφ, (D.1)


where s ′(θ ) is typically much smaller than s(θ )v′(θ )/v(θ ). The term D(Gs, s)appearing in I cov

F is therefore of second order in s(θ ) and hence negligiblecompared to I mean

F . For typical parameters, the difference is greater than anorder of magnitude.

The last term in the Fisher information comes from the sum in I covF given

by equation C.4. In the continuum limit, this term is approximately

N2π

∫ 2π

0

(v(φ)

2v′(φ)− s ′(φ)s(φ)

1 − s2(φ)

)2

dφ

= N4π

∫ 2π

0

(d

dφ

[log(v(φ)(1 − s2(φ))

])2

dφ.

For the type of stimulus dependence that we assume, v(φ)2v′(φ) and − s ′(φ)s(φ)

1−s2(φ)have opposite signs. For small correlations, the first term will dominate,and the stimulus dependence of correlations will decrease this entry in I cov

F .When correlations are not perfect (near 1), the term I var

F is typically muchsmaller than I mean

F .

Appendix E: Details of the Numerical Implementations

Numerical values of Fisher Information in Figures 3 and 5 were found bydirectly inverting the correlation matrices Q and performing the requiredmatrix multiplications in Matlab. We authors are happy to provide thesecodes on request.

The procedure is as follows. We first fix the average value of correlations,〈ρi, j 〉, among all neurons in the population (the value 〈ρi, j 〉 = 0.1 was usedfor all figures in this letter). Next, we define correlation matrices consistentwith this value of 〈ρi, j 〉, for two cases, stimulus dependent (SD) and stimulusindependent (SI) (see the main text). We first define Qi, j via equation 2.2,assuming that the ρi, j (θ ) are given by equation 2.3. Here, for Figures 3 and 5,we used s(θ ) = kρ + bρa2(θ ), where a (θ ) = 1/2(1 + cos(θ )) and kρ and bρ areconstants chosen as follows: (1) the average correlation 〈ρi, j 〉 = 0.1 and (2)the ratio of largest so smallest pairwise correlations, (kρ + bρ)2/k2

ρ , shouldbe R = 10 for the SD case and R = 1 (i.e., bρ = 0) for the SI case.

To study the affects of heterogeneity, as a final step, we jitter the tuningcurves for s and v by ±20%.

Acknowledgments

We thank Bruno Averbeck, Jeff Beck, and Adam Kohn and the anony-mous reviewers for their insights, helpful comments, and suggestions.E.S.-B. holds a Career Award at the Scientific Interface from theBurroughs-Wellcome Fund. This research was also supported by NIH grant


DC005787-01A1 (J.R.), a Texas ARP/ATP, and NSF grant DMS-0604429 toK.J., and NSF grant DMS-0817649 to B. D., K.J., and E. S.-B.

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Received October 3, 2008; accepted March 10, 2009.

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