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

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

*Correspondence:

This is an open-access article subject to a non-exclusive license between the authors and Frontiers Media SA, which permits use, distribution and reproduction in other forums, provided the original authors and source are credited and other Frontiers conditions are complied with.

Sensory and cognitive processing relies on the concerted activity of large populations of neurons. The advent of modern experimental techniques like two-photon population calcium imaging makes it possible to monitor the spiking activity of multiple neurons as they are participating in specific cognitive tasks. The development of appropriate theoretical tools to quantify and interpret the spiking activity of multiple neurons, however, is still in its infancy. One of the simplest and widely used measures of correlated activity is the pairwise correlation coefficient. While spike correlation coefficients are easy to compute using the available numerical toolboxes, it has remained largely an open question whether they are indeed a reliable measure of synchrony. Surprisingly, despite the intense use of correlation coefficients in the design of synthetic spike trains, the construction of population models and the assessment of the synchrony level in live neuronal networks very little was known about their computational properties. We showed that many features of pairwise spike correlations can be studied analytically in a tractable threshold model. Importantly, we demonstrated that under some circumstances the correlation coefficients can vanish, even though input and also pairwise spike cross correlations are present. This finding suggests that the most popular and frequently used measures can, by design, fail to capture the neuronal synchrony.

The first observations of spiking activity in groups of neurons in the 1960s and 1970s established that the spiking activity is correlated across neurons (Gerstein and Clark,

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What is the origin of ^{4} other neurons and sends out signals via its synapses to about 10^{4} others (Abeles,

The first step toward deciphering the information encoded in the spike trains of neurons is to simultaneously record the spike times of multiple neurons. Multiple electrodes implanted into the cortical tissue (Gerstein and Clark, ^{ij}_{1}(_{2}(_{cond,12}(τ) (Binder and Powers,

Here ν_{1} and ν_{2} are the mean firing rates of neurons 1 and 2, respectively. 〈·〉 denotes the average over all spikes of the reference neuron 1 and subsequently averaged over multiple realizations of the same experimental condition. Because the temporal spike resolution can be limited, it is often beneficial to consider instead the number of spikes emitted in a given time period _{12}

where _{1}_{2}

The leaky integrate and fire (LIF) model has long served as the preferred neuron model for addressing the computational properties of spike cross correlations (Moreno-Bote and Parga, _{C}

The mixing ratio is determined by the correlation strength _{C}_{1}_{2}_{I}

Upon reaching the threshold value ψ_{0} a spike is emitted and the voltage is reset to a subthreshold value. To obtain the firing rate, spike auto and cross correlations coupled Fokker–Planck differential equations for the probability densities of _{1}(_{2}(_{1}(_{2}(

Over the last two decades numerous authors have shed light on the spike statistics of integrate and fire neurons. Burkitt (

A key challenge for a viable theoretical framework is to quantify the impact of the input current correlations on pairwise spike correlations. A subtle change in the amount of

Do correlation coefficients depend linearly on the input correlation strength? For example, are experimentally recorded weak spike correlation coefficients truly indicative of weak spike and input correlations?

Do vanishing correlation coefficients reliably indicate independent or uncorrelated spike trains and justify correlation-free decoding strategies?

How does one best compare correlation coefficients obtained under different experimental conditions?

In the following, we address these questions in a novel threshold framework.

As discussed above, even the most simple and popular integrate and fire models have severe restrictions on the type of accessible input correlations. However, neuronal dynamics might not always unfold within these artificial limits. Therefore, alternative methods with a broader analytically accessible range of correlation strengths and temporal forms of

In this model framework, the somatic voltage fluctuations of cortical neurons are approximated by a stationary, temporally correlated Gaussian process. Starting from a stationary Gaussian potential

where _{cond}(τ) in Eq.

Figure

_{1} − _{3} and the corresponding spike trains _{1}(_{2}(_{1}(_{2}(_{0} (bottom left), correlation coefficient ρ_{12} as a function of correlation strength _{cond,12}(0) − ν as a function of firing rate ν (bottom right).

Firing rate as a function of mean current in Figure

Pairwise spike cross correlations as a function of input strength in Figure _{s}

Weak pairwise spike cross correlations as a function of rate dependence in _{s}

As a reminder ν is the firing rate, ψ_{0} the spiking threshold, τ_{I}_{s}_{M}_{I}_{V}

This framework also allows us to go one step further and explore a previously inaccessible territory: the influence of a broad range of correlation strengths and temporal form of spike correlations on various measures of spike synchrony. Studying the effect of the temporal form of input correlations on the correlation coefficient in (Tchumatchenko et al.,

_{12} can vanish in pairs of cross correlated neurons. This effect can be expected if ρ_{12} is computed for large time bins in neuronal pairs where the integral over the spike cross correlation function vanishes

the firing rate, bin size, and temporal form of the voltage cross correlations play a crucial role for spike cross correlations. Importantly, weak spike correlation coefficients do not necessarily imply weak input correlations, because the conversion from inputs to spikes is in general substantially sub-linear.

spike output correlations can vanish despite finite input cross correlations. In particular, spike count correlations on long time scales can vanish even if count correlations on shorter time scales do not.

Relating a change in correlation coefficients to a change of input synchrony is best achieved in two pairs of equal firing rates and intrinsic time constants, where the coefficients are computed for small bin sizes.

With all the benefits and insights provided by the threshold model, what are its limits? One of its main limits is its restriction to the fluctuation-driven regime and Gaussian voltage statistics. Yet, these specific regimes are supported by growing experimental evidence showing that the fluctuation rather than mean depolarization driven regime is the primary operation scheme in cortical neurons. The first lines of evidence are the exceptionally low firing rates <1 Hz (Margrie et al.,

Correlated activity has been shown to play an important role in sensory encoding and cognitive functions. Yet, understanding its properties has been difficult with the prevalent theoretical methods. In particular many questions regarding the implications of measured spike correlations remained open. Are weak correlation coefficients indicative of weak input correlations? Does a vanishing spike correlation coefficient in a pair of neurons imply that they are uncorrelated? Can spike synchrony across experimental conditions be compared simply by comparing correlation coefficients? Many of these basic questions were neglected in the quest for an easy-to-use measure of spike synchrony. To address such fundamental questions, there are two possible approaches to choose from. One approach is to resort to advanced simulations of specific neuronal morphology and ion channel composition, studying numerically the properties of spike correlation measures in various parameter regimes. Another approach is to take a minimal model, which can replicate the essential properties of spike correlations in real neurons, yet which is simple enough to deal with analytically. We took the second route and found that a minimal model based on the threshold crossings of correlated Gaussian potentials can do just that – replicate essential features of cortical spike correlations and offer an unprecedented analytical transparency to address the above questions. Using this model, we found that weak spike correlations, as characterized by a weak correlation coefficient, do not necessarily imply weak input correlations. In fact, substantial spike correlations and input correlations can be present in pairs with vanishing spike correlation coefficients. This implies that particular care needs to be given to the way correlation coefficients are computed. In particular they should be calculated with time scales shorter than the intrinsic time constants, and large integration times should be avoided. Let us stress again that many important features of neuronal activity can be emulated even if the biophysical basis of spiking mechanisms is not modeled in exquisite detail.

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 the referees for numerous helpful comments and Yoram Burak, Mirko Lukovic and Maximilian Puelma Touzel for fruitful discussions, Michelle Monteforte for English corrections and the Bundesministerium für Bildung und Forschung (#01GQ0430, #01GQ1005B, #01GQ07113), German-Israeli Foundation (#I-906-17.1/2006), Deutsche Forschungsgemeinschaft (SFB889), University of Connecticut and the Max Planck Society for financial support.

Interdependencies in the spiking of multiple neurons. Pairwise spike correlations are correlations between the spike trains of two neurons, quantified by the spike (cross) correlation function. Higher order correlations include three-neuron, four-neuron, and the general

Is a measure of interdependence between two signals. A pairwise (cross) correlation function quantifies their similarity at any two time points. It reaches its lowest negative values if the two points are each other's opposites, is positive if the two points are on average similar, and it is 0 if they are uncorrelated. It measures only linear interdependencies and therefore a lack of pairwise cross correlations does not imply statistical independence.

The spike cross or auto correlation functions in a pair of neurons. Spike auto correlations quantify the temporal structure within a spike train, and cross correlations quantify the temporal coordination of spikes across the two spike trains.

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If the spike trains are identical then this commonly used figure of merit is equal to one, if they are independent of each other then the coefficient is 0. The inverse route is less straight forward; the interpretation of a correlation coefficient in terms of input correlation strength is not possible without prior knowledge of relevant time scales, firing rates and bin size.

Coincident events in the activity of multiple neurons. Synchronous spiking activity across neurons is characterized by a high degree of temporal fidelity in the emission of spikes. Figure

Pairwise input correlations are the temporal auto and cross correlation functions of the net somatic input currents in two neurons. They determine the strength and temporal structure of pairwise voltage auto and cross correlations and ultimately give rise to pairwise cross correlations in two spike trains.