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CrossCorrelations and Joint Gaussianity in Multivariate Level Crossing Models
 Elena Di Bernardino^{1},
 José León^{2} and
 Tatjana Tchumatchenko^{3}Email author
https://doi.org/10.1186/21908567422
© E. Di Bernardino et al.; licensee Springer 2014
Received: 21 March 2013
Accepted: 16 December 2013
Published: 17 April 2014
Abstract
A variety of phenomena in physical and biological sciences can be mathematically understood by considering the statistical properties of level crossings of random Gaussian processes. Notably, a growing number of these phenomena demand a consideration of correlated level crossings emerging from multiple correlated processes. While many theoretical results have been obtained in the last decades for individual Gaussian levelcrossing processes, few results are available for multivariate, jointly correlated threshold crossings. Here, we address bivariate upward crossing processes and derive the corresponding bivariate Central Limit Theorem as well as provide closedform expressions for their joint levelcrossing correlations.
1 Introduction
Various phenomena in the biological or physical sciences are amenable to the description by level crossings of random Gaussian processes [1, 2]. Examples of these phenomena are spike coordination of neurons in the brain [3], insurance risk assessment [4] and stress levels generated by ocean waves [5]. Therefore a number of mathematical studies in recent decades have focused on the statistical properties of level crossings arising from stationary Gaussian processes [2]. However, largely this literature addresses the properties of one levelcrossing process and rarely deals with the coordinated level crossings of multivariate Gaussian processes. A prominent application where correlated level crossings are of particular importance is neuroscience. Recent work has shown that the spikes of a cortical neuron can be approximated by a Gaussian levelcrossing process [3, 6]. The assumption of Gaussianity is prompted by the experimental observation that cortical neurons are on average connected to ∼10000 neurons and therefore receive a barrage of inputs that together lead to a nearGaussian fluctuation at the cell body of any given cortical neuron [7]. The spikes of two neurons are then modeled as upward levelcrossing times of two crosscorrelated fluctuating Gaussian potentials.
In this article we aim to address two features of level crossings of multiple correlated Gaussian processes. First, we want to clarify whether levelcrossing counts derived from multiple correlated processes are jointly Gaussian. Second, we want to understand how many more coincident level crossings in a given time instance are expected if the underlying Gaussian random processes are correlated. Let us provide an intuitive reason for these questions. Starting with the first question, we recognize that if levelcrossing counts of two neurons were jointly Gaussian, then a simple measure of dependence is the covariance or the Pearson correlation coefficient. Measuring a vanishing correlation coefficient or vanishing covariance between two neuronal spike counts would in this case imply true statistical independence, because only in the case of multivariate Gaussian distribution is it permissible to conclude independence from vanishing count correlations. This implication is not permissible if the marginal distributions are not Gaussian or are Gaussian but the joint distribution is not a multivariate Gaussian distribution. While marginal Gaussianity has been shown for levelcrossing counts in [2] for large bin sizes, joint Gaussianity is still an open question. It might seem natural to imply joint Gaussianity from marginal Gaussianity for multivariate levelcrossing processes, however, numerous counter examples exist to prove this intuition wrong, see Sect. 5 in [8]. Here, we use a modified Breuer–Major Theorem to prove joint Gaussianity and show that any linear combination of levelcrossing counts of the two processes is also Gaussian.
The second question we address in this article deals with the conditional probabilities of two levelcrossing processes. We are interested in how the level crossings of one Gaussian process can be used to predict the levelcrossing probability of the partner process in a specific time interval relative to the observed level crossing in one process. In neuroscience, coordinated neuronal firing drives changes in synaptic connectivity and calculating the spike count dependencies across neurons is therefore a topic of current research efforts (e.g. Chap. 8 in [9]). The available mathematical results for conditional upward crossings in Gaussian processes currently comprise mostly variance and moments for one levelcrossing process (see Chaps. 3–5 in [2]) as well as the low and high correlation limit in pairs of processes [3, 10]. As yet, a comprehensive closedform solution covering the complete levelcrossing crosscorrelation function is currently lacking. Here, we use a regression approach to derive, for all correlation strengths, the conditional levelcrossing correlation functions in two continuous Gaussian processes. We hypothesize that the levelcrossing correlations we provide in this article could also be valuable in other fields outside of neuroscience for example in risk assessment calculations to predict the risk of joint default for insurance purposes.
The article is structured as follows. In Sect. 2 we define the mathematical model setting and introduce the concept of level crossings and specifically the upward crossings. In Sect. 3 we use a regression approach to obtain a general closedform solution for crosscorrelations of level crossings in two correlated Gaussian processes. In Sect. 4 we prove the joint Gaussianity (Central Limit Theorem) for the correlated joint upward crossings for two correlated Gaussian processes. In the section on materials and methods (Sect. 6) we provide detailed derivations of the reported results. We assume throughout this article that both levelcrossing processes arise from crossings of the same threshold level by two Gaussian processes with different variances. This is permissible because the number of level crossings, the Rice rate [11], depends only on the variancetothreshold ratio, but not on these quantities individually. We therefore work with a pair of levelcrossing processes where each process has a unique voltage variance and therefore the rate of crossings in the two neurons being considered are, unless stated otherwise, not the same. Let us note that this assumption is prompted by the observation that in a living brain typically no two neurons are identical in all their properties and differ at least in their firing rate.
2 Mathematical Definitions of Multivariate Level Crossings
2.1 Definitions of Multivariate Voltage Distributions
2.2 Upward Crossing Definitions
where ${\nu}_{j}=\frac{{\sigma}_{{V}_{j}^{\prime}}}{2\pi {\sigma}_{{V}_{j}}}exp(\frac{{\psi}^{2}}{2{\sigma}_{{V}_{j}}^{2}})$ is the firing rate of a neuron j, for $j=1,2$. In the next sections we provide closedform expressions for $\u3008{s}_{1}(t){s}_{2}(t+\tau )\u3009$ and ${\nu}_{\mathrm{cond}}(\tau )$.
3 CrossCorrelations of Two Upward Level Crossings
Here, we address $\u3008{s}_{1}(t){s}_{2}(t+\tau )\u3009$ and provide a closedform solution that is valid for any crosscorrelation strength r between two levelcrossing processes and any time delay τ.
ϕ and Φ are the standard Gaussian density and distribution, respectively, and $\overline{\Phi}=1\Phi $. ${H}_{n}(z)={(1)}^{n}\frac{{\mathrm{d}}^{n}}{\mathrm{d}{z}^{n}}({e}^{{z}^{2}/2}){e}^{{z}^{2}/2}$ are the Hermite polynomials. Note that the first two terms in Eq. (18) correspond to truncation orders $n=0$ and $n=1$, respectively.
Figure 2(a), (b) demonstrates ${\nu}_{\mathrm{cond}}(\tau )$ obtained using Eq. (17) for different truncation orders n alongside the zero lag correlation ${\nu}_{\mathrm{cond}}(0)$. Figure 2(c), (d) demonstrates ${\nu}_{\mathrm{cond}}(\tau )$ obtained using Eq. (17) as a function of the correlation strength r alongside the zero lag correlation ${\nu}_{\mathrm{cond}}(0)$. As previously, we chose $c(\tau )=cosh{(\tau /{\tau}_{s})}^{1}$ and $r\in [0,1)$. We note that for two identical neurons (${\sigma}_{{V}_{1}}={\sigma}_{{V}_{2}}$) ${\nu}_{\mathrm{cond}}(\tau )$ is a symmetric function. Yet, for a pair of neurons with different rates (${\sigma}_{{V}_{1}}\ne {\sigma}_{{V}_{2}}$) the spike correlation function ${\nu}_{\mathrm{cond}}(\tau )$ is asymmetric, indicating that the lower rate neuron spikes on average after the higher rate neuron.
3.1 Relation to the Leaky IntegrateandFire Model
where ${I}_{j}(t)$ is the input current of a neuron, $\xi (t)$ a white noise, unit variance drive. The voltage power spectrum for this model is a combination of lowpass filters ${f}_{V}(\lambda )\sim {[(1+{\tau}_{M}^{2}{\lambda}^{2})(1+{\tau}_{I}^{2}{\lambda}^{2})]}^{1}$ and its correlation function can be determined according to Eqs. (8). If the voltage ${V}_{j}$ reaches the threshold ϕ the neuron j emits a spike and the voltage is subsequently reset to a reset value ${V}_{r}$. The integrateandfire model differs only in one important detail from the levelcrossing approach—the presence of a reset after a spike. A recent article by Laurent Badel systematically compared the validity of upward levelcrossing approximation for the firing rate, spike correlations and frequency response of a leaky integrateandfire neuron [16]. This study reached the conclusion that the upward levelcrossing approach accurately represents the leaky integrateandfire model if two conditions are fulfilled: (1) the firing rate is much lower than the typical relaxation time of the voltage, (2) the synaptic filtering time constants remain of the same order of magnitude as the membrane time constant (${\tau}_{I}/{\tau}_{M}\approx 1$). Numerically, the validity of the approximation remained highly accurate even for synaptic time constants $0.4\lesssim {\tau}_{I}/{\tau}_{M}\lesssim 2.6$.
A number of spike correlation results have been derived in the leaky integrateandfire model for the limit of weak correlations [18–20]. They include the observation that the spike correlation coefficient increases with firing rate [18, 19]. The equivalent firing rate dependent increase in spike correlations and correlation coefficients for low correlation strengths has been reported for the levelcrossing model, see [3] and Fig. 3(A) (right) and Fig. 2(A) (top) in [10]. Furthermore, leaky integrateandfire model exhibits a sublinear dependence of correlation coefficients on input strength r [18, 21], which we see confirmed in Fig. 4.
4 Joint Gaussianity of Upcrossing Counts
Spike count cross correlations and correlation coefficient measurements in pairs of neurons are ubiquitous in neuroscience and are often used to measure the strength of interdependencies in a pair of neurons, e.g. in cortical neurons [18, 19, 22], in model neurons [23] and in theoretical and experimental studies of net correlations emerging in recurrent networks [23–28]. Spike counts and their cross correlations in neuroscience are often computed for a variety of bin sizes varying from $T=0.1\text{\u2013}1\text{ms}$ [22] to $T=2\text{s}$ [29]. Here, we are interested in the question when spike count correlations of two neurons computed in a bin size T are jointly Gaussian such that their cross correlations are unbiased measures of statistical dependence or independence.
will also converge to the respective ratio of the asymptotic covariances and variances.
4.1 Numerical Confirmation of Joint Gaussianity and Limit Covariances ${a}_{ij}$
is distributed according to a ${\chi}_{d}^{2}$distribution with d degrees of freedom (see, e.g., Sect. 3.1.4 and Eq. (3.16) in [4]). By numerically estimating the count sample average μ and Σ we calculate in our case ${D}_{i}^{2}$ and compare it with a ${\chi}_{2}^{2}$distribution, using the QQplot method (see Fig. 3(c)).
Figure 3 demonstrates the results of the joint Gaussianity tests for a bin size $T=25{\tau}_{s}$, where ${\tau}_{s}=1\text{ms}$. Figure 3(a) shows the empirical univariate distribution of spike counts in one levelcrossing process derived from $N=10000$ independent count realizations. Figure 3(b) demonstrates that in $N=10000$ independent count realizations of X pvalues for all θ are above the 10 % significance level. Figure 3(c) (left) illustrates that the Mahalanobis distance ${D}^{2}$ of a twodimensional spike count variable X are well approximated by the ${\chi}_{2}^{2}$distribution (solid line). Figure 3(c) (right) demonstrates in a QQplot of the empirically measured ${D}^{2}$quantiles and the theoretical ${\chi}_{2}^{2}$quantiles that they are linearly related. This is an indication that both distributions are equal.
5 Conclusions
Levelcrossing phenomena occur in a variety of physical and biological sciences. In many of these situations coordination between level crossings of multiple crosscorrelated Gaussian processes is of interest. Here, we focused on neuroscience and modeled the spikes of two crosscorrelated neurons by two crosscorrelated levelcrossing processes. While crossings and extrema of one levelcrossing process have been the focus of mathematical research, results describing the coordination of multiple levelcrossing processes are sparse and typically available only in specific and limited cases. Limits where levelcrossing crosscorrelations have been previously calculated are the weak and strong input correlation limit [3]. Here, we studied the case of two crosscorrelated upward crossing processes and derived closedform expressions for their joint levelcrossing coordination as well their joint count Gaussianity. Importantly, the results we present in this article are consistent with previously reported limits but we now extended and generalized them. The two main results of our article are (1) closedform explicit solution of the levelcrossing crosscorrelations and (2) the joint Gaussian limit of levelcrossing counts. Our first result provides an explicit solution to ${\nu}_{\mathrm{cond}}(\tau )=\u3008{s}_{1}(t){s}_{2}(t+\tau )\u3009/\sqrt{{\nu}_{1}{\nu}_{2}}$ that is valid for all correlation strengths and which comprises previously obtained limits, see discussion in Sect. 3. The rate of level crossings by a onedimensional Gaussian process is given by the prominent Rice’s equation derived by Rice in the 1950s [11]. The solution we obtained for the levelcrossing crosscorrelation ${\nu}_{\mathrm{cond}}(\tau )$ extends the Rice rate to the joint rate of two correlated processes. Our second result proves the joint Gaussianity of level crossings for large bin sizes. The joint Gaussianity of spike counts is a highly desired property because if and only if two levelcrossing counts are jointly Gaussian can zero count crosscorrelation imply statistical independence. Notably, marginal Gaussianity of spike counts in each neuron combined with zero count crosscorrelation is not sufficient to imply independence. Contrasting examples of where X and Y variables are both marginally but not jointly Gaussian, have a zero crosscorrelation but are not independent can be found in Sect. 5 in [8]. Count covariance and measures derived from it, such as the Pearson correlation coefficient, are computationally inexpensive and widely used as measures of statistical interdependencies [8]. Therefore, it is highly desirable to investigate the joint Gaussianity of level counts and thereby delimit the parameter space and mathematical conditions ensuring that independence can be implied from zero correlation coefficient. Notably, the joint Gaussianity of spike counts in bins of size T where T is much larger than the intrinsic time constant ${\tau}_{s}$ ($T\gg {\tau}_{s}$) also implies that models of multineuronal dynamics only need to consider the mean and variance of spike counts because all higher cumulants are zero.
6 Materials and Methods
6.1 Proof of Proposition 3.1

for $n=0$, ${c}_{0}(a)=\frac{{e}^{{a}^{2}/2}}{\sqrt{2\pi}}=\varphi (a)$, ${c}_{0}(b)=\varphi (b)$,

for $n=1$, ${c}_{1}(a)=a\varphi (a)\Phi (a)+1$, ${c}_{1}(b)=b\varphi (b)\Phi (b)+1$,

for $n\ge 2$, ${c}_{n}(a)=\frac{\varphi (a)}{n!}(a{H}_{n1}(a)+{H}_{n2}(a))$, ${c}_{n}(b)=\frac{\varphi (b)}{n!}(b{H}_{n1}(b)+{H}_{n2}(b))$.

for $n=0$, ${c}_{0}(a)=1\Phi (a)=\overline{\Phi}(a)$, ${c}_{0}(b)=\overline{\Phi}(b)$,

for $n=1$, ${c}_{1}(a)=\varphi (a)$, ${c}_{1}(b)=\varphi (b)$,

for $n\ge 2$, ${c}_{n}(a)=\frac{\varphi (a)}{n!}{H}_{n1}(a)$, ${c}_{n}(b)=\frac{\varphi (b)}{n!}{H}_{n1}(b)$.

for $n=0$, ${c}_{0}(a)=\overline{\Phi}(a)$, ${c}_{0}(b)=\varphi (b)$,

for $n=1$, ${c}_{1}(a)=\varphi (a)$, ${c}_{1}(b)=b\varphi (b)\Phi (b)+1$,

for $n\ge 2$, ${c}_{n}(a)=\frac{\varphi (a)}{n!}{H}_{n1}(a)$, ${c}_{n}(b)=\frac{\varphi (b)}{n!}(b{H}_{n1}(b)+{H}_{n2}(b))$.

for $n=0$, ${c}_{0}(a)=\varphi (a)$, ${c}_{0}(b)=\overline{\Phi}(b)$,

for $n=1$, ${c}_{1}(a)=a\varphi (a)\Phi (a)+1$, ${c}_{1}(b)=\varphi (b)$,

for $n\ge 2$, ${c}_{n}(a)=\frac{\varphi (a)}{n!}(a{H}_{n1}(a)+{H}_{n2}(a))$, ${c}_{n}(b)=\frac{\varphi (b)}{n!}{H}_{n1}(b)$.
Note that the first two terms in Eq. (42) correspond to orders $n=0$ and $n=1$, respectively. Denoting $\mathbb{E}[{Y}_{1}{1}_{\{{Y}_{1}\in [0,\mathrm{\infty})\}}{Y}_{2}{1}_{\{{Y}_{2}\in [0,\mathrm{\infty})\}}{X}_{1}=\psi ,{X}_{2}=\psi ]={\mathcal{C}}_{(a,b)}(\tau )$ we find $\u3008{s}_{1}(t){s}_{2}(t+\tau )\u3009={\mathcal{C}}_{(a,b)}(\tau ){p}_{\tau}(\psi ,\psi )$. Here, ${\mathcal{C}}_{(a,b)}(\tau )$ is a uniformly convergent series. □
6.2 Zero Time Lag Correlations
Solving this integral we obtain Eq. (23).
6.3 Proof of Theorem 4.1
where ${G}_{q}^{i}({x}_{1},{x}_{2})={\sum}_{k+j=q}{d}_{j}^{(i)}({\psi}_{i}){a}_{k}{H}_{j}({x}_{1}){H}_{k}({x}_{2})$. A Gaussian distribution is a stable limit distribution for a sum of independent finite variance variables. Therefore, all that is left to prove is that contributions $q\ne {q}^{\prime}$ are independent and have finite variance. From Mehler’s Formula we recognize that the contributions for $q\ne {q}^{\prime}$ are independent. The finite variance follows from the observation that for all q the variance of ${G}_{q}^{i}({X}_{i}(s),{X}_{i}^{\prime}(s))$ is proportional to the expectation of a product of four Hermite polynomials, which has been proven to be finite (Theorem 10.10 in [2]) if the conditions of Theorem 4.1 are satisfied.
This is the result reported in Theorem 4.1. □
6.4 Modified Breuer–Major Theorem
Here, we adapt the Breuer–Major Theorem [34] to show that the bivariate vector $({J}_{q}^{1}(T,{X}_{1},{X}_{1}^{\prime}),{J}_{q}^{2}(T,{X}_{2},{X}_{2}^{\prime}))$ is Gaussian.
This follows from the Central Limit Theorem for $\frac{1}{\epsilon}$dependent random vectors and concludes the proof. □
We provide the MATHEMATICA 8 (Wolfram Research) code to iteratively calculate ${\nu}_{\mathrm{cond}}(\tau )$. The code can be found at: http://www.tchumatchenko.de/CodeNuCond_Fig2.nb.
Declarations
Acknowledgements
The authors thank the two anonymous referees, and Sabrina Münzberg, Amadeus Dettner, and Laurent Badel for constructive comments on a previous version of the paper, Sabrina Münzberg for help with Hermite polynomial calculations and associated problem solving and Sara Gil Mast for English corrections. EDB and JRL are supported by a ECOSNord project under the reference V12M01. TT is funded by the Volkswagen foundation and the Max Planck Society and is thankful for the support of the Center for Theoretical Neuroscience at Columbia University during her stay there.
Authors’ Affiliations
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