Modalidades de la Cooperación Sur-Sur

The two noise amplification mechanisms discussed before are enhanced by the presence of metric correlations in the network, and are highly sensitive to the local wiring. The growth of avalanches at a given point in the system will be statistically biased towards the directions of stronger amplification. These ’prefered’ directions however, are only revealed after time–averaging over many avalanches, resulting in a clear picture in the form of effective networks and activity flows. These two representations allows us to characterize the noise focusing mechanism as we will now see.

2.6.1 Effective networks

The complex internal structure of the activity avalanches is revealed after a correct time–averaging. Instead of focusing on the activity of neurons during avalanches, we focus directly on the connections. The information obtained from time–averaging neuronal activity is equivalent to calculating the individual firing rate of the neurons, and we have already seen inFigure 2.11cthat they are not a good indicator of the internal dynamics. Instead, we weight each connection by the number of times it participates in an avalanche, i.e., it is involved in two causally–connected firings. This operation creates a new network which is a weighted version of the original (structural) one. If we now apply a certain threshold to only keep the most active connections we obtain an effective network that shows the structure of the most active connections and the subnetwork they might form.

InFigure 2.25awe show the effective network for a particular case which contains

the top 1% of most active connections during BAs. These connections only involve about 25% of the neurons. The structure of such a ‘dressed’ network is remarkably different from the original ‘bare’ network. Not only is it strongly inhomogeneous in space, but also exhibits a more hierarchical structure, with a fundamentally different degree distribution consistent with a power-law (seeFigure 2.25d). A community structure, also appears, as indicated by the different colors inFigure 2.25a. Many of the modules concentrate around the nucleation sites (presented inFigure 2.25c), however, a substantial amount also appear far from the nucleation sites. The con- struction of this effective network prunes the less active links, effectively filtering the small avalanches, those that are self-similar and essentially homogeneous and isotropic, while keeping the contribution of large avalanches, which grow selec- tively in more specific locations and directions as a result of the noise focusing mechanism.

An equivalent effective network can be constructed with only the IAs, as shown

2.6. Noise focusing 79 0.08 0.30 0.51 0.73 Nucleation PDF (mm-2₎ 100 101 102 10−4 10−3 10−2 10−1 100 connectivity k p(k) IA BA Random Structural 1 mm a b c d

Figure 2.25 Effective networks. a, Spatial representation of the background avalanche activity

in a circular culture 5 mm–wide and density ρ =300 neurons/mm2_{, with an average connectivity}

of hki ∼ 70. Only the top 1% of the most active connections that participate in the avalanches are shown. Different colors identify communities according to standard community detection

algorithms [Blondel 2008]. The background activity forms a subnetwork clustered in specific

regions of the culture containing only 25% of the total population. b, Spatial representation of the ignition avalanche activity from the same network in a,. c, Nucleation probability density, coarse–grained over the connectivity correlation length scale of 0.26 mm. Each nucleation point is defined by the geometrical center of the neurons that are the first to fire in a burst. Their distribution is highly localized in specific regions of the system which define the nucleation sites (3 in this case). The nucleation sites are more focused than the IAs activity itself. d, Degree probability distribution p(k) of the different networks and subnetworks studied. The distribution of the IA (yellow) and BA (blue) subnetworks is completely different from the structural one (black), and shifts from a Gaussian-like profile to one that is consistent with a scale-free network. The distribution of a randomized version of the original connectivity, with only 1% of the links taken at random, is also shown for comparison (purple).

80 2. Noise focusing: the emergence of coherent activity in neuronal cultures BA one, more concentrated in fewer modules. All the modules in the IAs effective network are located around the nucleation sites, clearly showing how the activity of IAs concentrate around the nucleation sites. The smallest nucleation site from the left however, is missed out, partially because the activity that concentrates there comes from a wide area and is not as concentrated. This reflects the importance of spatio-temporal correlations in the avalanche dynamics that have been removed by the time-averaging. Remarkably, not even the characterization of these subnetworks is sufficient enough to fully identify the nucleation sites.

Figure 2.25dshows how the effective network structure is shaped by the dynamics.

The original network connectivity distribution has a Gaussian profile, while both the BA and IA effective networks resemble a power–law distribution, with a few nodes acting as hubs. These hub neurons are not particularly more active than the other neurons, but they are able to amplify its activity much more effectively. As a side note, let us point out that the presence of these effective networks also reveals the complexity of the problem of network inference in neuroscience. Any network reconstruction method that is based on activity recordings will be heavily influenced by the presence of these networks, and the resulting reconstruction might be more closely related to this network than to the structural one (even in a system as ’simple’ as a neuronal culture).

2.6.2 Activity flow

A new picture emerges if, instead of focusing on the network structure, we identify where the activity flows to during ignition avalanches. Every IA ends up nucleating a wave at a specific point in the system, and we can assign a unit vector to every neuron that participates in the avalanche that points towards the nucleation point. This method overcomes many of the problems from the other analyses, namely heterogeneity and variability. By directly correlating every participating neuron with the final nucleation point we are actually taking into account the long range spatio–temporal correlations and averaging over all possible paths. After iterating over many avalanches, we can finally coarse–grain the resulting vectors to create a flow map (as we did with the other observables and the network maps). Instead of plotting the flow map itself we plot its streamlines, which better describe the flow.

InFigure 2.26we see the streamlines for the same network presented inFigure 2.11.

The two nucleation sites now clearly appear as sinks of activity. Activity originating in regions away from the nucleation sites flow, on average, towards them following the corresponding streamlines. This picture also clearly shows the effective basin of attraction of each nucleation site, i.e, its area of effect, which in this particular case appears to be around 2 mm–wide. The small nucleation site in the bottom left creates a perturbation to the streamlines that go to the nucleation site on the right,

2.7. Discussion and conclusions 81