Figure 1

Recording avalanches from cortical tissue. (a) Micrograph of cultured cortical slice on a 512 electrode array. Black rectangle ($1\mathrm{mm}\times 2\mathrm{mm}$) added to highlight location of array. (b) Voltage trace from one electrode. Arrow marks a spike from an individual neuron, expanded in (c). Time of spike is marked by black dot. (d) Raster plot of spike times (dots) from many neurons over a 48 s interval. Recordings lasted up to 8 h. (e) Expanded view of network activity reveals an avalanche. Each frame represents the array at one 5 ms bin. Small dots are electrode locations; large dots are spikes on array. An avalanche consists of consecutively active frames, bracketed by inactive frames, as shown here. (f) Avalanche shape is obtained by plotting the number of spikes in each frame versus time.Reuse & Permissions

Figure 2

Contrasting size and duration distributions and average size for fixed duration from a critical data set (left) and a subcritical data set (right). Experimental data are shown by lines with markers and data from the corresponding models are shown by smooth lines. Note that each experimental data set has its own simulation as we use information from the experiment to determine the parameters of the simulation (see SM [14]). The dashed lines on the left (near critical) column correspond to power laws with exponent 1.7, 1.9, and 1.3, corresponding to the critical exponents $\tau$, $\alpha$, and $1/\sigma \nu z$, respectively. These values satisfy the exponent relation $\frac{\alpha -1}{\tau -1}=\frac{1}{\sigma \nu z}$, as is expected for a system near criticality. Statistical error bars are only significant for the largest and longest events for the experimental data and are too small to see in the figure for the simulations.Reuse & Permissions

Figure 3

Avalanche shape collapses. Shapes and attempted collapses from three data sets: two experimental and one simulated. Shapes are produced by averaging the temporal profiles of all avalanches of a particular duration; different colors here represent different durations. The collapses are plotted by rescaling the horizontal and vertical axes. The left- and rightmost data correspond to experimental data close to and far from criticality, respectively (note these are the same data sets used in Fig. 2). Sample 8 clearly shows the roughly parabolic shapes in the raw data and a corresponding very clean collapse, as would be expected from critical data. Sample 6 shows neither. The middle plots are a simulation of sample 8, using transfer entropy data from that set [14]. They clearly show similar shapes and collapse to a universal scaling function.Reuse & Permissions

Figure 4

Avalanche size histograms from three different samples of rat cortex. The horizontal line represents a path in some representative (such as increasing mean connection strength or increasing number of connections) direction in the complex parameter space of network, representing how critical the sample of interest is. As we move along this path towards greater network connectivity, samples change from having early exponential cutoffs, to straightforward power laws, to having humps in their distribution. These humps could be caused by the possibility that the finite system size, rather than the degree of network connection, is constraining the size of the largest events. These histograms show that different samples exist at different points in the phase diagram, such as the mean connection strength with fixed network topology, toward increasing excitability. Factors that determine the phase of a given sample include number of connections and connection strength. Simulations using a mean field (all-to-all connectivity) version of our model can move along the phase diagram by increasing connection strength [19, 20]. Experiments have used drugs to move the network along this line in the parameter space [2, 21, 22].Reuse & Permissions