Background avalanches

InFigure 2.20we show the distribution of BAs sizes and durations. InFigure 2.20a

we observe a clear scaling relation, over 2 decades in size and 6 in frequency, independent of network properties, with an universal exponent close to the one from classical percolation in a Cayley tree, ∼ 5/2 [Albert 2002]. For larger avalanches however, this scaling relation starts to break down and becomes more sensitive to the specific details of the network. Does that mean that our system is in a critical state? Not quite. To understand what is going on we need to understand how our system evolves during bursts. Right after a burst, synapses are completely depressed, and although the neurons can spontaneously fire, they are unable to transmit any signal to their neighbors. In this regime, the firings are completely uncorrelated,

70 2. Noise focusing: the emergence of coherent activity in neuronal cultures

Avalanche size

Avalanche frequency per neuron (Hz)

100 101 102 103 104 10−9 10−7 10−5 10−3 10−1 = 1/3 = 2/3 = 1 α α α ~x-2.5 100 101 102 103 104 10−4 10−3 10−2 10−1 100 Avalanche size Avalanche duration T (s) T / α = 1/3 = 2/3 = 1 α α α a b

Figure 2.20 Statistics of Background Avalanches (BAs). a, Statistics of BAs for networks with different connection probability α and different density ρ. The mean connectivity is fixed at hki ∼ 70. The avalanche size distribution shows power-law statistics for almost three decades. b, Relationship between avalanche duration and avalanche size for the same networks as in a,. Inset: the curves collapse into a single one when rescaled with the connection probability α, although deviations start to appear at larger sizes. In all cases the calculated exponent is below 1. Each curve from a, and b, is averaged over 5 different network realizations and over 3 hours of simulated activity.

and if we tried to detect avalanches we would observe a curve consistent with a Poisson process (since the firings are uncorrelated). While the system is recovering, however, the connections are being strengthened, and the neurons are able to induce firings in their neighbors so the system is effectively subcritical. If the connections keep being strengthened, there will be a point when a single spike is able to induce more than one firing in average (characteristic of supercritical behavior), when this happens however, a burst can also develop (which is different from a large avalanche), triggering the nucleation process, wave propagation and complete activity saturation. So our system is continuously evolving from subcritical to supercritical, thus crossing the critical state in between. Averaging all over these phases results in the curve observed inFigure 2.20a, the system is always moving around the critical regime12but never being exactly there.

Why is the exponent so similar to that of percolation in a Cayley tree? The defining feature of a Cayley tree is the absence of loops (zero clustering coefficient), and our networks are far from that, as they are highly clustered. As we will see the answer relies in the fact that our process is dynamical, and neurons can fire multiple times in a single avalanche.

2.4. Microscopic dynamics and activity avalanches 71 A Cayley tree (also known as a Bethe lattice) is a loopless network where each node (neuron) has the same number of connections z (coordination number). In a Cayley , the cluster (avalanche) size distribution nsis calculated as follows. The

first node is chosen randomly (since the tree is loopless and infinite they are all equivalent) and ignited with probability p. If it is ignited, the same is performed with its z neighbors. For those neighbors that were ignited, the process is repeated with its z − 1 neighbors (since one of the nodes is already ignited, the precursor), until no nodes can be ignited. The total number of nodes that were ignited n define the cluster size. It can be shown [Albert 2002] that around the percolation threshold ns∝ s−5/2e−cs with c ∝ (p − pc)2 (2.32)

where pcis the percolation threshold. It is also easy to see that pc= 1/(z − 1). On

average, a node will ignite p(z − 1) new nodes. So only when p(z − 1) ≥ 1 we can obtain an infinite cluster. Below that the cluster will eventually stop. Above that, however, the cluster size will increase exponentially. Note that a Cayley tree is essentially a random graph with fixed degree, since both show zero clustering in the limit N → ∞.

If we look closely to the way the clusters are computed in a Cayley tree we can see the parallelism with our activity avalanches. If we consider that each iteration in the cluster formation takes a time ∼ τ, this becomes equivalent to the time it takes for a given spike to induce another in any of its neighbors. In highly clustered networks, as is our case (usually hCCi ' 0.3) the percolation transition differs because the number of available neighbors at every iteration is reduced because of the loops. Since our process is dynamical however, given that our degree is large hki ≥ 70 and we are close to the percolation threshold, after a few iterations the neurons have returned to their resting state and can be activated again, essentially making the system infinite and loopless.

This parallelism helps explain the universal exponent −5/2 found for BAs, as shown inFigure 2.20a. The region of interest for wave nucleation, however, is that of large avalanches, where the presence of loops is statistically significant and breaks that scaling. It is also important to compute the distribution of avalanche times (seeFigure 2.20b), the fact that the exponent is close to 1 further indicates that the system is close to the percolation transition, and that the internal avalanche structure is self–similar (an avalanche twice as large, lasts twice as long and so on). The exponent, however, is not strictly one (it is slightly smaller) and depends on the clustering of the network, the deviations being more apparent at large avalanches. This indicates that larger avalanches have a smaller duration when the clustering is high, and this is caused by the fact that when the avalanches are large, loops enhance the probability of induced firings, as we will later see.

A typical BA is shown inFigure 2.21. Its spatio–temporal structure is quite complex

(seeFigure 2.21a) and is better observed when mapped to the unit circle (seeFig-

ure 2.21b). In this representation the internal structure of the avalanche is revealed.

72 2. Noise focusing: the emergence of coherent activity in neuronal cultures time 45 90 0 time (ms) a b 1 mm

Figure 2.21 Structure of Background Avalanches. a, Spatio–temporal structure of a back- ground avalanche composed of 91 spikes in a 5 mm–wide circular network with hki ∼ 70,

ρ = 300 neurons/mm2and α = 2/3. Big circles correspond to the neurons that fired during

the avalanche (color coded by the first time they fired) and the involved connections. This particu- lar avalanche starts in a small area, spreads out and dies. In gray, all the other neurons that did not participate. b, The avalanche structure is mapped to a circular graph where the angle represents firing time. Each neuron is positioned across the circle by the first time they fire (denoted by a small line). Causal interactions are represented by curved paths whose curvature depends on the time difference between the two firings and color–coded by the first.

a backward connection in the graph (a dark line connecting to a light area), and the temporal profile fluctuates greatly. There are many spikes in the first 10 ms, but only a few are able to propagate past that point. Around the 40 ms mark the activity increases again to finally die out 90 ms after the first spike.