Scale-free and economical features of functional connectivity in neuronal networks

A form of activity that is highly studied in cultured cortical networks is the neuronal avalanche, characterized by bursts whose distribution follows a power law. While the statistics of neuronal avalanches are well characterized, much less is known about the neuronal interactions from which they arise. We examined statistical dependencies between pairs of cells in spontaneously active cultures of cortical neurons using an information measure of transfer entropy. We show that the distribution of transfer entropy follows a power law with a slope near 3/2. Using graph-theoretic approaches of weighted networks, we demonstrate that this power law maximizes a measure of global economy that accounts for both the efficiency of neuronal interactions as well as the overall traffic in the network. Finally, we describe a pairwise Poisson model that captures the statistics of information transfer in a population of spiking neurons. Using this model, we show that avalanches can occur in systems with weak pairwise interactions, and that strong pairwise interactions can arise without avalanches, suggesting that these two measures capture distinct properties of brain dynamics.

Figure 1

Neuronal activity in cultures of neocortical neurons. (a) Culture of cortical neurons plated on microelectrode array (only a subset of array shown). Electrodes are spaced 200 μm apart. (b). Spike raster showing the activity of a population of neurons. (c) Distribution of the number of cells participating in avalanches, with time windows of 10 ms (black), 15 ms (blue), and 20 ms (green). Red, shuffled spikes (window of 10 ms). Dashed lines show best-fitting power law obtained with maximum likelihood estimation.

Figure 2

Information transfer between pairs of neurons. (a) Distribution of transfer entropy obtained from all pairs of neurons, using windows of 1 ms (black), 2 ms (dark gray), and 4 ms (light gray). Dashed lines show best-fitting power law. (b). Matrix of transfer entropy for all pairs of neurons in a recording. (c) Log likelihood of the λ parameter estimated for the exponential function. Solid black, dashed, and gray lines are obtained from 100%, 50%, and 25% of the data, respectively. (d). Same as panel (c), but obtained for the α parameter of the power law function. (e). Mean square error between values of transfer entropy obtained for consecutive time windows of 1,2,…, 20 min in duration. (f) Relationship between mean transfer entropy ($T$) and physical distance between microelectrodes. Panels (a) and (c)–(f) are obtained after pooling data from all recordings.

Figure 3

Relation between power law distribution of information transfer and network economy. (a) Strength of information transfer ($w$) with different slopes (α). These distributions were obtained from synthetic data [Eq. (14)]. (b) Network efficiency decreases as the slope (α) of the power distribution of strength increases. Different colored lines correspond to networks where each node had $k$ degree by forming random connections. (c) Economy is a measure that trades off connection strength and efficiency [Eq. (15)] and is highest when strength follows a power law with slope near 3/2 (dashed vertical line). (d) Network economy is diminished when network weights are increased or decreased by a constant value $c$. Data are averaged over all recordings. (e) Network economy improves when adjusting individual weights of a network with a search algorithm. (f) Power law distribution is largely unaltered after adjusting individual network weights. Black circles, initial distribution; gray circles, distribution following weight adjustments for 5000 time steps.

Figure 4

Dissociation between avalanches and functional connectivity. (a) Matrix of transfer entropy for all pairs of neurons after constrained pairwise shuffling. (b) Distribution of transfer entropy for original (solid black line) and shuffled (solid gray line) activity. Dashed line, best-fitting slope of power law of original data obtained by maximum likelihood estimation. (c) Distribution of the number of cells active during avalanches obtained after shuffling. (d) Raster of spike data obtained from the pairwise Poisson model. (e). Transfer entropy ($T$) obtained with a pairwise Poisson model is strongly correlated with transfer entropy obtained with the original data. (f) Avalanche distribution obtained from the pairwise Poisson model. Panels (b), (c), (e), and (f) are obtained after pooling data from all recordings.