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**|**Front Comput Neurosci**|**v.4; 2010**|**PMC2857958

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Front Comput Neurosci. 2010; 4: 1.

PMCID: PMC2857958

Tatjana Tchumatchenko,^{1,}^{2,}^{3,}^{*} Theo Geisel,^{1,}^{2} Maxim Volgushev,^{4,}^{5,}^{6} and Fred Wolf^{1,}^{2}

Edited by: Matthias Bethge, Max Planck Institute for Biological Cybernetics, Germany

Reviewed by: Eric Shea-Brown, University of Washington, USA; Benjamin Lindner, Max Planck Institute, Germany

*Correspondence: Tatjana Tchumatchenko, Bernstein Center for Computational Neuroscience Göttingen, Bunsenstr. 10, 37073 Göttingen, Germany.e-mail: ed.gpm.sd.dln@anajtat

Received 2009 November 15; Accepted 2010 February 5.

Copyright © 2010 Tchumatchenko, Geisel, Volgushev and Wolf.

This is an open-access article subject to an exclusive license agreement between the authors and the Frontiers Research Foundation, which permits unrestricted use, distribution, and reproduction in any medium, provided the original authors and source are credited.

This article has been cited by other articles in PMC.

Concerted neural activity can reflect specific features of sensory stimuli or behavioral tasks. Correlation coefficients and count correlations are frequently used to measure correlations between neurons, design synthetic spike trains and build population models. But are correlation coefficients always a reliable measure of input correlations? Here, we consider a stochastic model for the generation of correlated spike sequences which replicate neuronal pairwise correlations in many important aspects. We investigate under which conditions the correlation coefficients reflect the degree of input synchrony and when they can be used to build population models. We find that correlation coefficients can be a poor indicator of input synchrony for some cases of input correlations. In particular, count correlations computed for large time bins can vanish despite the presence of input correlations. These findings suggest that network models or potential coding schemes of neural population activity need to incorporate temporal properties of correlated inputs and take into consideration the regimes of firing rates and correlation strengths to ensure that their building blocks are an unambiguous measures of synchrony.

Coordinated activity of neural ensembles contributes a multitude of cognitive functions, e.g., attention (Steinmetz et al., 2000), encoding of sensory information (Stopfer et al., 1997; Galan et al., 2006), stimulus anticipation and discrimination (Zohary et al., 1994; Vaadia et al., 1995). Novel experimental techniques allow simultaneous recording of activity from a large number of neurons (Greenberg et al., 2008) and offer new possibilities to relate the activity of neuronal populations to sensory processing and behavior. Yet, understanding the function of neural assembles requires reliable tools for quantification, analysis and interpretation of multiple simultaneously recorded spike trains in terms of underlying connectivity and interactions between neurons.

As a first step beyond the analysis of single neurons in isolation, much attention has focused on the pairwise spike correlations (Schneidman et al., 2006; Macke et al., 2009; Roudi et al., 2009), their temporal structure and the influence of topology (Kass and Ventura, 2006; Kriener et al., 2009; Ostojic et al., 2009; Tchumatchenko et al., 2010). Pairwise neuronal correlations are traditionally quantified using count correlations, e.g., correlation coefficients (Perkel et al., 1967). However, it remains largely elusive how correlations present in the input to pairs of neurons are reflected in the count correlations of their spike trains. What are the signatures of input correlations in the count correlations? And vice versa, what conclusions about input correlations and interactions can be drawn on the basis of count correlations and their changes?

Here we address these questions using a framework of Gaussian random functions. We find that correlation coefficients can be a poor indicator of input synchrony for some cases of input correlations. In particular, count correlations computed for time bins larger than the intrinsic temporal scale of correlations can vanish for some functional forms of input correlations. These potential ambiguities were not reported in previous studies of leaky integrate and fire models which focused on the analytically accessible choice of white noise input currents (de la Rocha et al., 2007; Shea-Brown et al., 2008).

The paper is organized as follows: we first introduce several common spike count measures (Section “Materials and Methods”) and the statistical framework (Section “Results”). Then we study the zero time lag correlations (Section “Spike Correlations with Zero Time Lag”) and the influence of the temporal structure of input correlations on measures of spike correlations (Section “Temporal Scale of Spike Correlations”). We show that spike count correlations can vanish despite the presence of input cross correlations (Section “Vanishing Count Covariance in the Presence of Cross Correlations”). Finally, we discuss potential consequences of our findings for the design of population models and the experimentally measured spike correlations.

The spike train *s _{i}*(

The spike timing correlations of two spike trains *s _{i}*(

$${\nu}_{cond,ij}\left(\tau \right)=\langle {s}_{i}\left(t\right){s}_{j}\left(t+\tau \right)\rangle /\sqrt{{\nu}_{i}{\nu}_{j}}$$

(1)

$${\text{\nu}}_{\text{cond}}(\text{\tau})={\text{\nu}}_{\text{cond,}ii}(\text{\tau})=\langle {s}_{i}(t){s}_{i}(t+\text{\tau})\rangle /{\text{\nu}}_{i}.$$

(2)

Here ν* _{i}* and ν

An alternative measure based on count correlations is the correlation coefficient ρ* _{ij}* (Perkel et al., 1967; de la Rocha et al., 2007; Greenberg et al., 2008; Shea-Brown et al., 2008; Tetzlaff et al., 2008):

$${\rho}_{ij}=\frac{\text{Cov}\left({n}_{i}\left(T\right),{n}_{j}\left(T\right)\right)}{\sqrt{\text{Var}\left({n}_{i}\left(T\right),{n}_{i}\left(T\right)\right)\cdot \text{Var}\left({n}_{j}\left(T\right),{n}_{j}\left(T\right)\right)}}$$

(3)

where *n _{i}*(

$${c}_{ij}=\frac{\text{Cov}\left({n}_{i}(T),{n}_{j}(T)\right)}{\langle {n}_{i}(T)\rangle \langle {n}_{j}(T)\rangle}=\frac{\text{Cov}\left({n}_{i}(T),{n}_{j}(T)\right)}{{\text{\nu}}_{i}{\text{\nu}}_{j}{T}^{2}}$$

(4)

$${J}_{ij}=\mathrm{log}\left(1+{c}_{ij}\right)+O(N\overline{\nu}T).$$

(5)

Covariance can be obtained via the integration of cross conditional firing rate ν_{cond,ij} (τ) over the time bin *T*:

$$\begin{array}{c}\text{Cov}\left({n}_{i}(T),{n}_{j}(T)\right)=\langle {n}_{i}(T),{n}_{j}(T)\rangle -\langle {n}_{i}(T)\rangle \text{\hspace{0.17em}}\langle {n}_{j}(T)\rangle \\ =\langle \text{\hspace{0.05em}}{\int}_{0}^{T}{s}_{i}({x}_{1})d{x}_{1}{\int}_{0}^{T}{s}_{j}({x}_{2})d{x}_{2}\rangle -{\text{\nu}}_{i}{\text{\nu}}_{j}{T}^{2}\end{array}$$

(6)

$$\begin{array}{l}={\displaystyle {\int}_{-T}^{T}\sqrt{{\text{\nu}}_{i}{\text{\nu}}_{j}}\left({\text{\nu}}_{\text{cond},ij}(t)-\sqrt{{\text{\nu}}_{i}{\text{\nu}}_{j}}\right)(T-|t|)dt}.\\ \end{array}$$

(7)

The count variance can be obtained from the auto conditional firing rate ν_{cond}(τ):

$$Var\left({n}_{i}(T),{n}_{i}(T)\right)={\text{\nu}}_{i}\cdot T+2\cdot {\displaystyle \underset{0}{\overset{T}{\int}}{\text{\nu}}_{i}\left({\text{\nu}}_{cond}(t)-{\text{\nu}}_{i}\right)(T-|t|)dt.}$$

(8)

For bin sizes smaller than the intrinsic time constant (*T* <τ_{s}, see Eq. 14), we can directly relate conditional firing rate ν_{cond,ij}(τ) and the correlation coefficient ρ_{ij}

$${\text{\rho}}_{ij,T<{\text{\tau}}_{s}}\approx \frac{\sqrt{{\text{\nu}}_{i}{\text{\nu}}_{j}}\cdot \left({\text{\nu}}_{\text{cond},\text{\hspace{0.17em}}ij}(0)-\sqrt{{\text{\nu}}_{i}{\text{\nu}}_{j}}\right){T}^{2}}{\sqrt{{\text{\nu}}_{i}{\text{\nu}}_{j}}T\sqrt{\left(\text{1}-{\text{\nu}}_{i}\cdot T\right)\left(1-{\text{\nu}}_{i}\cdot T\right)}}=\left({\text{\nu}}_{\text{cond},\text{}ij}(0)-\sqrt{{\text{\nu}}_{i}{\text{\nu}}_{j}}\right)T$$

(9)

$${c}_{ij,T<{\tau}_{s}}\approx \frac{\sqrt{{\text{\nu}}_{i}{\text{\nu}}_{j}}\cdot \left({\nu}_{\text{cond},\text{\hspace{0.17em}}ij}(0)-\sqrt{{\text{\nu}}_{i}{\text{\nu}}_{j}}\right){T}^{2}}{{\text{\nu}}_{i}{\text{\nu}}_{j}{T}^{2}}=\frac{{\nu}_{\text{cond},\text{\hspace{0.17em}}ij}(0)-\sqrt{{\text{\nu}}_{i}{\text{\nu}}_{j}}}{\sqrt{{\text{\nu}}_{i}{\text{\nu}}_{j}}}.$$

(10)

In this limit, the properties of ρ* _{ij}*,

The quantities presented here all measure different aspects of spike correlations and can potentially have different computational properties. Furthermore, each of the quantities can exhibit a nonlinear dependence on firing rate, input statistics or bin size. Below, we consider these measures of spike correlations, as well as their dependence on firing rate, input statistics and bin size.

To access spike correlations in a pair of neurons, we use the framework of correlated, stationary Gaussian processes to model the voltage potential *V*(*t*) of each neuron. This approach generates voltage traces with statistical properties consistent with cortical neurons (Azouz and Gray, 1999; Destexhe et al., 2003). The simplest conceivable model of spike generation from a fluctuating voltage *V*(*t*) identifies the spike times *t _{j}* with upward crossings of a threshold voltage (Rice, 1954; Jung, 1995; Burak et al., 2009). The times

$$s(t)={\sum}_{j}\text{\delta}\left(t-{t}_{j}\right)=\text{\delta}\left(V(t)-{\text{\psi}}_{0}\right)|\dot{V}(t)|\text{\theta}\left(\dot{V}(t)\right),$$

(11)

where ψ_{0} is the threshold voltage, and δ(·) and θ(·) are the Dirac delta and Heaviside theta functions, respectively. Each neuron has a stationary firing rate ν= *s*(*t*). We model *V*(*t*) by a random realization of a stationary continuous correlated Gaussian process *V*(*t*) (Azouz and Gray, 1999; Destexhe et al., 2003) with zero mean and a temporal correlation function *C*(τ), which decays for larger time lags τ.

$$C(\text{\tau})=\langle V(t)V(t+\text{\tau})\rangle =\langle V(0)V(\text{\tau})\rangle ,\langle V(t)\rangle =0$$

(12)

· denotes the ensemble average. We assume a smooth *C*(τ) such that *C ^{n}*(0) exist for

$$C(\text{\tau})={\text{\sigma}}_{V}^{2}\mathrm{cosh}{\left(\text{\tau}/{\text{\tau}}_{s}\right)}^{-1}.$$

(13)

In cortical neurons *in vivo* the temporal width of *C*(τ) can from 10 to 100ms (Azouz and Gray, 1999; Lampl et al., 1999). We characterize the temporal width of *C*(τ) using the correlation time constant τ* _{s}*:

$${\text{\tau}}_{s}=\sqrt{C(0)/|{C}^{\u2033}(0)|}.$$

(14)

Note, that the correlation time τ* _{s}* as defined in Eq. 14 is close to a commonly used definition of autocorrelation time ${\text{\tau}}_{a}={\int}_{0}^{\infty}C(\text{\tau})/{\text{\sigma}}_{V}^{2}.$ For

$$\text{\nu}=\frac{\mathrm{exp}\left[-{\text{\psi}}_{0}^{2}/\left(2{\text{\sigma}}_{V}^{2}\right)\right]}{2\text{\pi}{\text{\tau}}_{s}}.$$

(15)

The firing rate ν is the rate of positive threshold crossings, which is equivalent to half of the Rice rate of a Gaussian process (Rice, 1954). For non-Gaussian processes the rate of threshold crossings can deviate from Eq. 15 and there is no general approach for obtaining ν in this case (Leadbetter et al., 1983). We note, that the firing rate ν of a neuron depends only on two parameters: the correlation time and the threshold-to-variance ratio, but not on the specific functional choice of the correlation function. Hence, processes with the same correlation time but with a different functional form of *C*(τ) will have the same mean rate of spikes, though their spike auto and cross correlations can differ significantly. Our framework can be expected to capture neural activity in the regime where the mean time between the subsequent spikes is much longer than the decay time of the spike triggered currents. This occurs if the spikes are sufficiently far apart and the spike decision is primarily determined by the stationary voltage statistics rather than spike evoked currents. Therefore, this model should only be used in the fluctuation driven, low firing rate ν <1/(2πτ* _{s}*) regime, which is important for cortical neurons (Greenberg et al., 2008).

The leaky integrate and fire (LIF) model (Brunel and Sergi, 1998; Fourcaud and Brunel, 2002) has a similar spike generation mechanism. To compare both models, we study the transformation of input current to spikes. The LIF neuron driven by Ornstein–Uhlenbeck current *I*(*t*) with time constant τ* _{I}* can be described by

$${\text{\tau}}_{M}\dot{V}(\text{\tau})=-V+{I}_{0}+I(t),$$

(16)

where τ* _{M}* is the membrane time constant and

We include cross correlation between two spike trains *i* and *j* via a common component in *V _{i}*(

$$\begin{array}{l}{V}_{i}(t)=\sqrt{1-r}{\text{\xi}}_{i}(t)+\sqrt{r}{\text{\xi}}_{c}(t)\\ {V}_{j}(t)=\sqrt{1-r}{\text{\xi}}_{j}(t)+\sqrt{r}{\text{\xi}}_{c}(t).\end{array}$$

(17)

where ξ* _{c}* denotes the common component and ξ

$$C=\left(\begin{array}{cccc}{\text{\sigma}}_{{V}_{i}}^{2}& 0& {C}_{ij}(\text{\tau})& {C}_{ij}^{\prime}(\text{\tau})\\ 0& {\text{\sigma}}_{{\dot{V}}_{i}}^{2}& -{C}_{ij}^{\prime}(\text{\tau})& -{C}_{ij}^{\u2033}(\text{\tau})\\ {C}_{ij}(\text{\tau})& -{C}_{ij}^{\prime}(\text{\tau})& {\text{\sigma}}_{{V}_{i}}^{2}& 0\\ {C}_{ij}^{\prime}(\text{\tau})& -{C}_{ij}^{\u2033}(\text{\tau})& 0& {\text{\sigma}}_{{\dot{V}}_{j}}^{2}\end{array}\right)\text{\hspace{0.17em}}.$$

(18)

Matrix entries are covariances *C _{xy}* =

The above framework allows one to derive an analytical expression for the cross conditional firing rate with zero time lag, ν_{cond,ij}(0). Via Eqs 5, 9and 10 ν_{cond,ij}(0) can be related to *c _{ij}*, ρ

$$\begin{array}{l}\sum =\frac{{V}_{1}(0)+{V}_{2}(\text{\tau})}{\sqrt{2}\sqrt{{\text{\sigma}}_{V}^{2}+rC(\text{\tau})}},\text{\hspace{0.17em}}\dot{\sum}=\frac{{\dot{V}}_{1}(0)+{\dot{V}}_{2}(\text{\tau})}{\sqrt{2}\sqrt{{\text{\sigma}}_{\dot{V}}^{2}-r{C}^{\u2033}(\text{\tau})}},\\ \Delta =\frac{{V}_{1}(0)-{V}_{2}(\text{\tau})}{\sqrt{2}\sqrt{{\text{\sigma}}_{V}^{2}-rC(\text{\tau})}},\text{\hspace{0.05em}\hspace{0.17em}}\dot{\Delta}=\frac{{\dot{V}}_{1}(0)-{\dot{V}}_{2}(\text{\tau})}{\sqrt{2}\sqrt{{\text{\sigma}}_{\dot{V}}^{2}+r{C}^{\u2033}(\text{\tau})}}.\end{array}$$

The matrix *C* is then the identity matrix for τ=0, and $\sum =\sqrt{2}{\text{\psi}}_{0}/\sqrt{{\text{\sigma}}_{V}^{2}+r{\text{\sigma}}_{V}^{2}},\Delta =0.$ We obtain:

$$\begin{array}{l}{\text{\nu}}_{\text{cond},ij}(0)={\displaystyle \underset{-\infty}{\overset{\infty}{\int}}\text{\hspace{0.17em}}{\displaystyle \underset{-\infty}{\overset{\infty}{\int}}d\dot{\sum},d\dot{\Delta}\mathrm{exp}}}\text{\hspace{0.17em}}\left(-\left(\frac{{\text{\psi}}_{0}^{2}}{{\text{\sigma}}_{V}^{2}(1+r)}+\frac{{\dot{\Delta}}^{2}+{\dot{\sum}}^{2}}{2}\right)\right)\\ \times \text{\hspace{0.17em}}\frac{{\text{\sigma}}_{\dot{V}}^{\text{4}}\sqrt{\left(1-{r}^{2}\right)}}{\text{\nu 8}{\text{\pi}}^{2}\sqrt{\text{Det}C}}\left({\dot{\sum}}^{2}(1+r)-{\dot{\Delta}}^{2}(1-r)\right)\\ \times \text{\hspace{0.17em}\theta}\left(\frac{{\text{\sigma}}_{\dot{V}}}{\sqrt{2}}\left(\dot{\sum}\sqrt{(1+r)}+\dot{\Delta}\sqrt{(1-r)}\right)\right)\times \text{\hspace{0.17em}\theta}\left(\frac{{\text{\sigma}}_{\dot{V}}}{\sqrt{2}}\left(\dot{\sum}\sqrt{(1+r)}-\dot{\Delta}\sqrt{(1-r)}\right)\right)\\ =\frac{1}{4{\text{\pi}}^{2}\text{\nu}{\text{\tau}}_{s}^{2}}\mathrm{exp}\left(\frac{-{\text{\psi}}_{0}^{2}}{{\text{\sigma}}_{V}^{2}(1+r)}\right)\text{\hspace{0.17em}}\left[1+\frac{2r\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}arctan\hspace{0.17em}}\left(\sqrt{{\scriptscriptstyle \frac{1+r}{1-r}}}\right)}{\sqrt{1-{r}^{2}}}\right].\end{array}$$

(19)

Equation 19 (Figure (Figure3A)3A) shows, as expected, that ν_{cond,ij}(0) increases with increasing strength of input correlations *r*. Since both correlation coefficients ρ* _{ij}*, and normalized correlation coefficient

Equation 19 further exposes one important feature of ν_{cond,ij}(0), and thus of *c _{ij}* and ρ

In the linear *r*-regime, the analytical expression for ν_{cond,ij}(0) can be further simplified:

$${\text{\nu}}_{\text{cond},ij}(0)\approx \text{\nu}\left(\text{1}+\frac{r}{2}\left(\text{\pi}+\text{4\hspace{0.05em}|log(}\nu 2\pi {\tau}_{s}\text{)|}\right)\right).$$

(20)

In this limit, ν_{cond,ij}(0) shows a strong dependence on the firing rate ν (Figure (Figure3A,3A, right, Figure Figure2A,2A, top). This dependence is remarkably similar to the firing rate dependence found previously *in vitro* and *in vivo* in cortical neurons and LIF models (de la Rocha et al., 2007; Greenberg et al., 2008; Shea-Brown et al., 2008).

In the limit of strong input correlations, Eq. 19 can be simplified to:

$${\text{\nu}}_{\text{cond},ij}(0)\approx \frac{1}{2\sqrt{2}\sqrt{1-r}{\text{\tau}}_{s}}.$$

(21)

In this regime, ν_{cond,ij}(0) does not depend on the firing rate ν (Amari, 2009). Furthermore, for strong input correlations and small bin sizes *T* the correlation coefficient ρ* _{ij}* also changes only marginally over a range of firing rates (0 <ν <15 Hz, Figure Figure2A),2A), since it depends linearly on ν

So far we considered only spike correlations occurring with zero time lag. However, spike correlations can also span across significant time intervals (Azouz and Gray, 1999; Destexhe et al., 2003). The temporal structure of spike correlations, as reflected in the conditional firing rate ν_{cond,ij}(τ), can induce temporal correlations within and across time bins and could potentially alter count correlations. To capture correlations with a non-zero time lag, spike correlation measures are calculated for time bins *T* spanning tens to hundreds of milliseconds, e.g., 20ms (Schneidman et al., 2006), 30–70ms (Vaadia et al., 1995), 192ms (Greenberg et al., 2008) and 2s (Zohary et al., 1994). For time bins longer than the time constant of the input correlations, measures of correlations become sensitive to the temporal structure of ν_{cond,ij}(τ). Moreover, the values of ρ* _{ij}* and

$$\begin{array}{l}\text{Cov}\left({n}_{i}(T\text{,}t),{n}_{j}(T\text{,}t\text{+\tau})\right)=\langle {n}_{i}(T,0){n}_{j}(T,\text{\tau})\rangle -{\text{\nu}}_{i}{\text{\nu}}_{j}{T}^{2}\\ ={\displaystyle {\int}_{-T}^{T}\sqrt{{\text{\nu}}_{i}{\text{\nu}}_{j}}\left({\text{\nu}}_{\text{cond},ij}(\text{\tau +}t)-\sqrt{{\text{\nu}}_{i}{\text{\nu}}_{j}}\right)}\left(T-\left|t\right|\right)dt,\end{array}$$

(22)

where *n _{i}*(

For large time lags τ we expect the auto conditional firing rate to approach the stationary rate but to deviate from it significantly for small time lags. Of particular importance for population models is the limit of small but finite τ, which determines the time scale on which adjacent time bins are correlated. At τ=0, the auto conditional firing rate has a δ-peak reflecting the trivial auto correlation of each spike with itself. In the limit of small but finite time lag (0< τ< τ* _{s}*) we find a period of intrinsic silence, where the leading order τ

$$\begin{array}{l}\sum =\frac{V(0)+V(\text{\tau})}{\sqrt{2}\sqrt{{\text{\sigma}}_{V}^{2}+C(\text{\tau})}},\text{\hspace{0.17em}}\dot{\sum}=\frac{\dot{V}(0)+\dot{V}(\text{\tau})}{\sqrt{2}\sqrt{{\text{\sigma}}_{\dot{V}}^{2}-\text{C}\u2033\text{(}\tau \text{)}}},\\ \Delta =\frac{V(0)-V(\text{\tau})}{\sqrt{2}\sqrt{{\text{\sigma}}_{V}^{2}-C(\text{\tau})}},\text{\hspace{0.05em}\hspace{0.17em}}\dot{\Delta}=\frac{\dot{V}(0)-\dot{V}(\text{\tau})}{\sqrt{2}\sqrt{{\text{\sigma}}_{\dot{V}}^{2}+{C}^{\u2033}(\text{\tau})}}.\\ \end{array}$$

Then only few elements of the corresponding symmetric density matrix${C}_{\sum ,\dot{\Delta},\dot{\sum},\Delta}$ remain non-zero: the diagonal elements${C}_{\sum ,\dot{\Delta},\dot{\sum},{\Delta}_{ii}}=1,$ *i* {1, 2, 3, 4} and the non-diagonal elements

$$\begin{array}{c}{C}_{\sum ,\dot{\Delta},\dot{\sum},{\Delta}_{12}}=\frac{-{C}^{\prime}(\text{\tau})}{\sqrt{{\text{\sigma}}_{V}^{2}+C(\text{\tau})}\sqrt{{\sigma}_{\dot{V}}^{2}+{C}^{\u2033}(\text{\tau})}},\\ {C}_{\sum ,\dot{\Delta},\dot{\sum},{\Delta}_{34}}=\frac{{C}^{\prime}(\tau )}{\sqrt{{\text{\sigma}}_{V}^{2}-C(\text{\tau})}\sqrt{{\sigma}_{\dot{V}}^{2}-{C}^{\u2033}(\text{\tau})}}.\end{array}$$

For *C*(τ) as in Eq. 13 we obtain a simple analytical expression in the limit of 0< τ< τ* _{s}*:

$${\text{\nu}}_{\text{cond}}(\text{\tau})=\frac{{\text{\nu}}^{\text{1/4}}}{3\cdot {\left(2\text{\pi}{\text{\tau}}_{s}\right)}^{{\scriptscriptstyle \frac{3}{4}}}}\cdot {(\text{\tau /}{\text{\tau}}_{s})}^{4}.$$

(23)

This equation shows that ν_{cond}(τ) depends on the temporal structure of a neuron's input and firing rate, Figure Figure4B.4B. Respectively, the silence period after each spike depends on the functional form and time constant of the voltage correlation function *C*(τ) and firing rate (Figures (Figures4B4B and and5A).5A). Figure Figure4B4B illustrates ν_{cond}(τ) obtained using numerical integration of Gaussian probability densities (e.g., Wolfram Research, 2009), ν_{cond}(τ) obtained from simulations of digitally synthesized Gaussian processes (Prichard and Theiler, 1994) and the τ< τ* _{s}* approximation in Eq. 23. In this framework, the silence period after each spike mimics the refractoriness present in real neurons (Dayan and Abbott, 2001).

Here we study how the input correlations shape the temporal structure of spike autocorrelations. In particular, we focus on how the input correlations and spike autocorrelations are reflected in count correlations within a spike train. The silence period after a spike is reflected in vanishing ν_{cond}(τ) for 0< τ< τ* _{s}* and results in negative covariation of spike counts in adjacent time bins. We find that the relation between ν

We explore the temporal structure of spike correlations in a weakly correlated pair of statistically identical neurons (ν =ν_{1} =ν_{2}). This is an important regime for cortical neurons *in vivo* (Greenberg et al., 2008; Smith and Kohn, 2008). To solve ν_{cond,ij}(τ) (Eq. 1), we expand the probability density $p({V}_{1}(t),{\dot{V}}_{1}(t),{V}_{2}(t+\text{\tau}),{\dot{V}}_{2}(t+\text{\tau}))$ using a von Neumann series of the correlation matrix *C* in Eq. 18. We obtain ν_{cond,ij}(τ) up in linear order

$$\begin{array}{l}{\nu}_{\text{cond},ij}\left(\tau \right)=\nu \left(1+r\left(\tilde{c}\left(\tau \right){k}^{2}-\pi {\tau}_{s}^{2}{\tilde{c}}^{\u2033}\left(\tau \right)/2\right)\right),\text{or}\\ {\nu}_{\text{cond},ij}\left(\tau \right)=\nu \left(1+r\left(\tilde{c}\left(\tau \right)2\left|\mathrm{log}\left(2\pi \nu {\tau}_{s}\right)\right|-\pi {\tau}_{s}^{2}{\tilde{c}}^{\u2033}\left(\tau \right)/2\right)\right),\end{array}$$

(24)

where $\tilde{c}(\text{\tau})=C(\text{\tau})/{\text{\sigma}}_{V}^{2}\text{\hspace{0.17em}and\hspace{0.17em}}k={\text{\psi}}_{0}/{\text{\sigma}}_{V}$. Equation 24 shows that weak spike correlations are generally firing rate dependent and directly reflect the structure of input correlations *C*(τ). Figure Figure5A5A shows three examples of voltage correlations which have the same τ* _{s}*, but different functional form. All three functional dependencies are reflected in the cross conditional firing rate ν

We now use the spike correlation function obtained above to study the pairwise count covariance.

$$\begin{array}{l}\text{Cov}\left({n}_{\text{i}}\text{(}T\text{),}{n}_{\text{j}}\text{(}T\text{)}\right)\text{\hspace{0.17em}=\hspace{0.17em}}{\displaystyle {\int}_{-T}^{T}{\text{\nu}}^{2}r(\tilde{c}(t)2|\mathrm{log}\left(2\text{\pi}{\text{\nu}}_{s}\right)|}\\ \text{\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}}-\text{\pi}{\text{\tau}}_{s}^{2}{\tilde{c}}^{\u2033}(t)/2)(T-|t|)dt,\end{array}$$

(25)

which allows to obtain the correlation coefficient for a weakly correlated pair of neurons:

$${\rho}_{ij}=\frac{\text{Cov}\left({n}_{i}\left(T\right),{n}_{j}\left(T\right)\right)}{\sqrt{\text{Var}\left({n}_{i}\left(T\right),{n}_{i}\left(T\right)\right)\text{Var}\left({n}_{j}\left(T\right),{n}_{j}\left(T\right)\right)}}=\frac{{\displaystyle {\int}_{-T}^{T}\nu r\left(2\tilde{c}\left(t\right)\left|\mathrm{log}\left(2\pi \nu {\tau}_{s}\right)\right|-\pi /2{\tau}_{s}^{2}{\tilde{c}}^{\u2033}\left(t\right)\right)\left(T-\left|t\right|\right)/Tdt}}{\sqrt{\left(1+2\cdot {\displaystyle {\int}_{0}^{T}\left({\nu}_{\text{cond}}\left(t\right)-{\nu}_{i}\right)\left(T-\left|t\right|\right)/Tdt}\right)\left(1+2\cdot {\displaystyle {\int}_{0}^{T}\left({\nu}_{\text{cond}}\left(t\right)-{\nu}_{j}\right)\left(T-\left|t\right|\right)/Tdt}\right)}}$$

(26)

This offers the opportunity to study how changes in the input structure affect spike count correlations. Figure Figure55 shows that correlation coefficient ρ* _{ij}* depends on both bin size

Count covariances and correlation coefficients rely on the integral of the spike correlation function (Eqs 3 and 7). In cortical neurons, the spike correlation functions can exhibit oscillations and significant undershoots in addition to a correlation peak (Lampl et al., 1999; Galan et al., 2006), this may alter the correlation coefficients and their dependence on bin size *T*. In the weak correlation regime we obtained an analytic expression for ν_{cond,ij}(τ) (Eqs 24 and 26). This allows us to explore analytically how a change in the functional choice of voltage correlations will influence count correlations. To qualify as a reliable measure of synchrony, count cross correlations between two neurons should reflect primarily correlation strength and be independent of the functional form of input correlations. Our framework offers the possibility to test this hypothesis and explore whether previously reported finite correlation coefficients obtained for LIF model using white noise approximation (Shea-Brown et al., 2008) can be generalized to a larger class of input correlations.

Here we consider spike correlations generated by a voltage correlation function with a substantial undershoot (e.g., as in Figure 1E in Lampl et al., 1999). For illustration, we could use any voltage correlation function with a large undershoot and vanishing long-timescale variability (${\int}_{-\infty}^{\infty}C(\text{\tau})d\text{\tau}=0$). Besides variance and correlation time, the variability as quantified by ${\int}_{-\infty}^{\infty}C(\text{\tau})d\text{\tau}$ is an important characteristic of every noise process. For analytical tractability we chose the voltage correlation function C_{3}(τ) as the normalized second derivative of the function ${\tilde{C}}_{3}(\text{\tau})=-3{\text{\tau}}_{s}^{2}\text{exp(}-{\text{\tau}}^{2}/(6{\text{\tau}}_{s}^{2})\text{)}:$

$${C}_{3}(\text{\tau})={\text{\sigma}}_{V}^{2}\left(\text{exp}\left(\frac{-{\text{\tau}}^{2}}{6{\text{\tau}}_{s}^{2}}\right)-\frac{{\text{\tau}}^{2}}{3{\text{\tau}}_{s}^{2}}\text{exp}\left(\frac{-{\text{\tau}}^{2}}{6{\text{\tau}}_{s}^{2}}\right)\right).$$

(27)

Defined this way, the correlation time of *C*_{3}(τ) is τ* _{s}* and ${\int}_{-\infty}^{\infty}{C}_{3}(\text{\tau})d\text{\tau}=0$, which is equivalent to vanishing spectral power for zero frequency. Figure Figure55 illustrates functional form of

$$\text{Cov}\left({n}_{i}(T),{n}_{j}(T)\right)/T=\frac{{\nu}^{2}r{\tau}_{s}^{2}\left[12\left|\mathrm{log}\left(2\pi \nu {\tau}_{s}\right)\right|\left(1-\mathrm{exp}\left(\frac{-{T}^{2}}{6{\tau}_{s}^{2}}\right)\right)+\pi \left(1+\mathrm{exp}\left(\frac{-{T}^{2}}{6{\tau}_{s}^{2}}\right)\left(\frac{{T}^{2}}{3{\tau}_{s}^{2}}-1\right)\right)\right]}{T}$$

(28)

$$\Rightarrow \underset{T/{\text{\tau}}_{s}\to \infty}{\mathrm{lim}}\frac{\text{Cov}\left({n}_{i}(T),{n}_{j}(T)\right)}{T}\to 0,\text{\hspace{0.17em}}\underset{T/{\text{\tau}}_{s}\to \infty}{\mathrm{lim}}{\text{\rho}}_{ij}\to 0$$

(29)

We note that the correlation coefficients and count covariances calculated for this functional form of input correlations can be arbitrarily small if *T* >>τ* _{s}*. This means that the absence of long-timescale variability in the inputs (${\int}_{-\infty}^{\infty}{C}_{3}(\text{\tau})d\text{\tau}=0$) is equivalent to an absence of long-timescale co-variability in the spike counts. Notably, despite vanishing cross covariance, the variability of the single spike train is maintained and count variance of the single spike train (Eq. 8) is finite for

Notably, spike count correlations of cortical neurons *in vivo* can decrease or increase as the length of the time bin increases (Averbeck and Lee, 2003; Smith and Kohn, 2008). These results are consistent with our findings (Figure (Figure5C).5C). Thus, in contrast to the correlation coefficients computed for small *T* which are independent of *C*(τ) (Eqs 9 and 19), the count correlations computed for *T* ≥τ* _{s}* are a potentially unreliable measure of synchrony.

Unambiguous and concise measures of spike correlations are needed to quantify and decode neuronal activity (Abbott and Dayan, 1999; Greenberg et al., 2008; Krumin and Shoham, 2009). Pairwise spike count correlations are frequently used to describe interneuronal correlations (Averbeck and Lee, 2003; Kass and Ventura, 2006; Greenberg et al., 2008) and many population models are based on these measures (Schneidman et al., 2006; Shlens et al., 2006; Roudi et al., 2009). However, quantitative determinants of count correlations so far remained largely elusive. Here, we used a simple statistical model framework based on the threshold crossings and the flexible choice of temporal input structure to study the signatures of input correlations in count correlations. In general, the details of the spike generating model can have a strong effect on spike correlations, f.e. depending on the dynamical regime, two quadratic integrate and fire neurons or two LIF neurons can be more strongly correlated (Vilela and Lindner, 2009). Notably, we found that our statistical framework can replicate many important aspects of neuronal correlations, e.g., nonlinear dependence of spike correlations on the input correlation strength (Binder and Powers, 2001) (Eq. 19), firing rate dependence of weak spike correlations (Svirskis and Hounsgaard, 2003; de la Rocha et al., 2007) (Eq. 20), and independence of spike reliability of the threshold (Mainen and Sejnowski, 1995) (Eq. 21). Furthermore, spike correlations derived here are consistent with many recent results in the commonly used LIF model, e.g., firing rate dependence of weak cross correlations (de la Rocha et al., 2007; Shea-Brown et al., 2008) (Eqs 20 and 24), the influence of noise mean and variance on the firing rates and weak spike correlations (Brunel and Sergi, 1998; de la Rocha et al., 2007; Ostojic et al., 2009) (Eqs 15, 20 and 24), or sublinear dependence of correlation coefficients on input strength (Moreno-Bote and Parga, 2006; de la Rocha et al., 2007) (Eq. 19, Figure Figure3).3). While the analytical accessibility of the LIF model is limited by the technically demanding multi dimensional Fokker–Planck equations and provides solutions only in special limiting cases (Brunel and Sergi, 1998; de la Rocha et al., 2007; Shea-Brown et al., 2008), the framework presented here allows for an analytical description of spike correlations.

Measurements of correlation coefficients under different experimental conditions often aim to compare the input correlation strength in pairs of neurons (Greenberg et al., 2008; Mitchell et al., 2009). But is a change in count correlations always indicative of a change in input correlations? The tractability of our framework revealed that spike count correlations can be a poor indicator of input synchrony for some cases of input correlations. Count correlations computed for time bins smaller than the intrinsic scale of temporal correlations could be independent of the functional form of input correlations but depend on the firing rate and input correlation strength. This suggests that a change in the correlation coefficient can be related to a change in the input correlation strength, if a firing rate change and a change of intrinsic time scale can be excluded. On the other hand, a change in correlation coefficients computed for large time bins is indicative of a change in input correlation strength only if a change in firing rate, time scale and functional form of input correlations can be excluded. Furthermore, count correlations computed for large time bins can either increase or decrease with increasing time bin or even vanish in a correlated pair. This seemingly contradictory behavior is consistent with the functional dependence of spike count correlations observed in cortical neurons (Averbeck and Lee, 2003; Kass and Ventura, 2006; Smith and Kohn, 2008).

Our results suggest that emulating neuronal spike trains, building efficient population models or determining potential decoding algorithms requires the analysis of full spike correlation functions in order to compute unambiguous spike count correlations. In particular, spike count coefficients computed for time bins larger than intrinsic timescale of correlations can be an ambiguous estimate of input cross correlations in a neuronal population with potentially heterogeneous distribution of input structures. Furthermore, the details of the spike generation model can be very influential for the transfer of current correlations to spike correlations, and the analytical results obtained here could facilitate quantitative comparisons between different types of models and between models and real neurons, by providing a maximally tractable limiting case for future studies.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

We wish to thank M. Gutnick, I. Fleidervich, S. Ostojic and A. Malyshev for fruitful discussions and the Bundesministerium für Bildung und Forschung (#01GQ0430,#01GQ07113), Goettingen Graduate School for Neurosciences and Molecular Biosciences, German-Israeli Foundation (#I-906-17.1/2006), University of Connecticut and the Max Planck Society for financial support.

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