Figure 1.
Neural network activity in experiments and in the cellular automaton model.
A. A snapshot of electrocorticographic (ECoG) data of brain activity, measured by 8×6 subdural array of electrodes. Data is interpolated between nodes, white areas correspond to high activity. B. A snapshot of activity from a cellular automaton model in an 400×400 network. The network is subject to noisy input from spontaneously activating cells (rate ). Active cells are white, refractory and excitable are black (simplified color code). C. Snapshot of activity in a 10×10 sub-network with detailed color code: red for active, blue for refractory, black for excitable nodes. Lines show links between nodes. D. Rules of the CA model: excitable node (black) may become active (red), if activated by a neighbor. After being activated, the node becomes refractory (blue) for a period of time
, after which it becomes excitable again. Data in A is a courtesy of Miles Whittington, recorded in Patient B of [29].
Figure 2.
Initiation of wave in a CA model on a random network.
The first 4 time steps of wave initiation are shown for an 11×11 network. A. t = 0; B. t = 1; C. t = 2; D. t = 3. Colorcode: red for active, blue for refractory, black for excitable cells. Lines show links between cells, red square shows the connectivity footprint of the central cell (shown only in A). Parameters: = 4,
(small for demonstration purposes).
Figure 3.
Traveling waves of activity in random networks.
Traveling waves emerging in the CA model on random networks with A. square, B. quasi-1D connectivity footprint. The cell which initiates the wave is shown by a red asterisk. Active cells are white, refractory and excitable cells are black. Directions of wave propagation are shown by arrows. C. A snapshot of wave () with spatial profiles of all three states: grey for excitable, bold red for active, light blue for refractory cell density. In the center, the wake of excitable cells (grey) grows by recovering from the refractory state (blue). D. Profiles of the active state at four time steps, showing two traveling waves emerged from a single active cell. Once formed, the speed and width of a wave remain constant. Profiles were calculated by averaging active cell counts over 100 bins along X. Parameters
,
(
is shown in the bottom right corners in A,B).
Figure 4.
Wave speed predicted by the parabolic and hyperbolic PDEs compared to simulations of CA on random networks.
The parabolic (Fisher-Kolmogorov) PDE gives wave speed that indefinitely grows with network degree (red line and diamonds). In contrast, the suggested hyperbolic PDE (given in text) provides a reaso`nable wave speed
(given in text, shown by green line and diamonds). The
grows moderately and saturates to the maximum possible speed
, in agreement with CA simulations (blue circles) and intuitive expectations. The solid lines show analytic formulae, the diamonds show simulations of corresponding full PDE systems.
Figure 5.
Wave speed derived from the hyperbolic PDE compared to CA simulations.
The wave speed (red line, high) is derived assuming all links have maximum length. CA simulations are shown in two variants, with maximum-length links (red triangles) and generic random-length links (blue circles). The naive speed scaling
(blue line, low) is derived assuming that link lengths are uniformly distributed. This discrepancy is explained in Results, showing that maximum-length link is a better predictor of wave speed. The dashed lines show the high-order analysis, proving that the hyperbolic PDEs capture the wave behavior sufficiently well, and derivatives of order above 2 are not necessary.
Figure 6.
Role of maximum link lengths in wave propagation.
A. The distribution of link lengths between the cells at the wave front, and the cells which triggered their firing. The front cells (top 1 or 5 %) were selected by their positions in a wave. The mean distances are given in the legend, parameters ,
. B. Estimate of the wave speed by numerical estimate of mean maxima of
i.i.d radii taken from uniform distribution
(broken line). The formula
is the expected value of the mean maxima (solid line). The CA simulations of wave speed are shown by circles. As seen, the mean maxima give a good wave speed estimate, in contrast to naive scaling
derived earlier from the PDE (dotted line).
Figure 7.
Effect of link lengths (radii) distribution on wave speed.
A. The radii distributions between nodes in the random networks: black o - fixed value, cyan x - uniform, red (blue) triangles - exponentially increasing (decreasing), green squares - bell-shaped distribution (see Methods for detailed formulae). B. Wave speeds in the networks with corresponding radii distributions (markers are consistent with panel A). Broken lines are computed mean maxima out of radii samples from each distribution, used as a plausible estimates of the true wave speeds (solid lines). Networks are Erdös-Rényi SCC, so the degree distribution (links per node) is Poissonian.
Figure 8.
Wave speeds in networks with different degree distributions.
Wave speeds in networks of six different degree distributions (explained in legend) are plotted A. against mean degree ; B. against ratio of network moments
. Note the convergence of wave speeds in B. The mean field formulae are shown in both panels by broken lines (Eqn. 7 in A; Eqn. 11 in B). Inset. The
versus
in the simulated networks. Line markers are consistent with legend in panel A. Errorbars are smaller than symbols, due to simulations on many networks of large size (1000×1000).