Spike correlations - what can they tell about synchrony?

Abstract

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.

Keywords: correlation coefficient; count correlations; spike correlations; synchrony.

Figure 1

Sensory stimuli are represented in the spiking activity of cortical neurons. A sensory stimulus (left) is encoded, albeit with potential information loss, in the activity of an interconnected cortical population (middle). Colored circles and black arrows depict neurons and their connections. The spike times of each neuron (right, same color code as in the middle scheme) are depicted as dots. A black square highlights the spike train t^ij of neuron j and the red squares highlight examples of synchronous spikes. Sensory information is contained in the total number of spikes, their timing and the spike cross correlations between two or more neurons. Potential decoding strategies can benefit from any of these sources.

Figure 2

The correlation transfer from input currents to spikes. Cross correlations in the net somatic currents of neurons 1 and 2 can originate from a multitude of synaptic interactions: direct or recurrent synaptic connections, common inputs or their combination. The presence of current cross correlations results in pairwise correlations in the spikes of neurons 1 and 2. Currently few neuronal models can offer tractable solutions describing how the pairwise input correlations translate to spike cross correlations.

Figure 3

Pairwise spike correlations in the novel threshold model framework. The spike times neurons t₁ − t₃ and the corresponding spike trains s₁(t) and s₂(t) represented by rectangles in red (top) or blue (center) are modeled in the threshold framework by positive threshold crossings (black dashed line) of a fluctuating potential V₁(t) or V₂(t), respectively. The correlations between neurons are incorporated as symmetric voltage cross correlations. Tractable analytical results can be obtained for the dependence of firing rate on the mean driving current I₀ (bottom left), correlation coefficient ρ₁₂ as a function of correlation strength r (bottom center) and the weak pairwise spike correlation ν_cond,12(0) − ν as a function of firing rate ν (bottom right).

Figure 4

Vanishing spike count correlations do not imply uncorrelated spike trains. While a lack of spike cross correlations implies 0 correlation coefficients, the opposite does not always hold. We have shown that count correlation coefficient ρ₁₂ can vanish in pairs of cross correlated neurons. This effect can be expected if ρ₁₂ is computed for large time bins in neuronal pairs where the integral over the spike cross correlation function vanishes $\underset{-\infty }{\overset{\infty }{\int }}{\nu }_{\text{cond}}(\tau )d\tau =0$ (Tchumatchenko et al., 2010a).