Disentangling the critical signatures of neural activity

Abstract

The critical brain hypothesis has emerged as an attractive framework to understand neuronal activity, but it is still widely debated. In this work, we analyze data from a multi-electrodes array in the rat's cortex and we find that power-law neuronal avalanches satisfying the crackling-noise relation coexist with spatial correlations that display typical features of critical systems. In order to shed a light on the underlying mechanisms at the origin of these signatures of criticality, we introduce a paradigmatic framework with a common stochastic modulation and pairwise linear interactions inferred from our data. We show that in such models power-law avalanches that satisfy the crackling-noise relation emerge as a consequence of the extrinsic modulation, whereas scale-free correlations are solely determined by internal interactions. Moreover, this disentangling is fully captured by the mutual information in the system. Finally, we show that analogous power-law avalanches are found in more realistic models of neural activity as well, suggesting that extrinsic modulation might be a broad mechanism for their generation.

Figure 1

(a) Left: scheme of the array used to obtain the LFPs data from all the cortical layers of the barrel cortex (adapted from); right an example of the LFPs signals for different layers and the corresponding discretization. An array of 256 channels organized in a $64\times 4$ matrix is inserted in a barrel column and the signals from the cortical layers are collected by $55\times 4$ electrodes. (b–d) Avalanche statistics obtained from the analysis of LFPs data in a rat. Both the distribution of the avalanches (b) sizes and (c) durations are power-laws, and (d) the crackling-noise relation is satisfied. (e) Scaling of the correlation length with the system size in LFPs data, averaging over four different rats. The error bars are shown as 5 standard deviations from the mean for visual ease. The correlation length scales linearly with the system size with no plateau in sight, a hallmark of criticality.

Figure 2

Avalanche statistics generated by the model at ${\mathcal{D}}^{\ast }=0.3$ (a–b, e–g) and at ${\mathcal{D}}^{\ast }=5$ (c-d,h-j), with ${\gamma }_D=15$ and $\theta =1$ and ${\gamma }_i=\gamma =0.05$ for the extrinsic model. (a–b) Comparison between the trajectories of $\mathcal{D}(t)$, $v_i$ and the corresponding discretization in the low-${\mathcal{D}}^{\ast }$ regime for (a) the extrinsic model and (b) the interacting one. (c–d) Same, but in the high-${\mathcal{D}}^{\ast }$ regime. (e–g) If ${\mathcal{D}}^{\ast }$ is low, avalanches are power-law distributed with almost identical exponents in the extrinsic and interacting model, ${\tau }^{\mathrm{ext}}=1.60\pm 0.01$, ${\tau }^{\mathrm{int}}=1.55\pm 0.01$ and ${\tau }_t^{\mathrm{ext}}=1.77\pm 0.01$, ${\tau }_t^{\mathrm{int}}=1.74\pm 0.01$. The crackling-noise relation is verified in both cases. (h–j) Same plots, now in the high-${\mathcal{D}}^{\ast }$ regime. Avalanches are now fitted with an exponential distribution. Notice that larger events, corresponding to periods in which $\mathcal{D}(t)>{\mathcal{D}}^{\ast }$, show up in the distributions’ tails, suggesting that the shift between exponentials and power-laws is smooth. (j) The average avalanche size as a function of the duration scales with an exponent that, as $D^{\ast }$ increases, becomes closer to the trivial one ${\delta }_{\mathrm{fit}}^{\mathrm{ext}}\approx {\delta }_{\mathrm{fit}}^{\mathrm{int}}\approx 1$.

Figure 3

(a) The correlation length of the interacting model scales linearly with the system size, as in the data. In the extrinsic model, as expected, the correlation length of the fluctuations is constant and equal to 1, i.e., the correlation function drops to zero for adjacent electrodes. (b) Comparison between the mutual information in the extrinsic model ($\theta =1$, ${\gamma }_D=10$, ${\gamma }_1=0.1$, ${\gamma }_2=0.5$) and, as an example, in the interacting model with two units. Notice that the onset of a non-vanishing mutual information induced by $\mathcal{D}(t)$ is also the onset of power-law distributed avalanches, whereas the mutual information arising from interactions is independent of ${\mathcal{D}}^{\ast }$.

Figure 4

Avalanche statistics generated by the Wilson Cowan units. The Wilson Cowan units are always in an inhibition dominated phase, i. e. ${\omega }_I=7$ and ${\omega }_E=6.8$, and $\alpha =1$. Their external input h is instead always in a balanced state, in particular ${\omega }_E^{(h)}=50.5$, ${\omega }_I^{(h)}=49.5$. Its other parameters are $h^{(h)}={10}^{-3}$ and ${\alpha }^{(h)}=0.1$. In Figures (a–d) however, ${\sigma }^{(h)}$, the amplitude of the noise, is increased to $2.5\times {10}^{-2}$ so that the up state can be destabilized by the noise. In Figures (e–h) instead the noise is reduced to $5\times {10}^{-3}$ so that the up state is stable. (a, e) Comparison between the trajectories of h, $\frac{E_i+I_i}{2}$ and the corresponding trains of events in the high (a) and low (e) ${\sigma }^{(h)}$ regime. (b–d) If ${\sigma }^{(h)}$ is high avalanches are power-law distributed and the crackling-noise relation is verified. (f–g) Same plots, now in the low ${\sigma }^{(h)}$ regime. Avalanches are now fitted with an exponential distribution. (h) The average avalanche size as a function of the duration scales with an exponent that, as ${\sigma }^{(h)}$ decreases, becomes closer to the trivial one ${\delta }_{\mathrm{fit}}\approx 1$.