Functional structure of cortical neuronal networks grown in vitro

We apply an information-theoretic treatment of action potential time series measured with microelectrode arrays to estimate the connectivity of mammalian neuronal cell assemblies grown in vitro. We infer connectivity between two neurons via the measurement of the mutual information between their spike trains. In addition we measure higher-point multi-information between any two spike trains, conditional on the activity of a third cell, as a means to identify and distinguish classes of functional connectivity among three neurons. The use of a conditional three-cell measure removes some interpretational shortcomings of the pairwise mutual information and sheds light on the functional connectivity arrangements of any three cells. We analyze the resultant connectivity graphs in light of other complex networks and demonstrate that, despite their ex vivo development, the connectivity maps derived from cultured neural assemblies are similar to other biological networks and display nontrivial structure in clustering coefficient, network diameter, and assortative mixing. Specifically we show that these networks are weakly disassortative small-world graphs, which differ significantly in their structure from randomized graphs with the same degree. We expect our analysis to be useful in identifying the computational motifs of a wide variety of complex networks, derived from time series data.

Figure 1

A neuronal culture (right) over a microelectrode array. The uniform grid of points shows electrodes where neuronal activity is measured. Left panels show individual neurons. White bars are $50\mathrm{\mu }\mathrm{m}$ (center-to-center distances between electrodes is $40\mathrm{\mu }\mathrm{m}$).

Figure 2

Spike time series for the spontaneous (native) activity of a cortical culture with 62 recorded neurons. Much of the activity takes place as part of network-wide collective events [(a), vertical features], known as network bursts. The structure of each neuron’s spike train observed over short time scales [(b), network burst detail] shows strong stochasticity.

Figure 3

Different connectivity arrangements of neurons have distinct roles in short-term memory or in information processing. These can be identified via the information-theoretic analysis of multicell spike time series, with the former associated with $R>0$ and the latter $R<0$; see text.

Figure 4

(Color) Weighted graphs for network of (a) 62 and (c) 33 recorded neurons. Panels (b) and (d) show the corresponding networks selecting only the links with strength above 0.4. The graphs are characterized by a large number of weak links that connect almost every pair of cells and of fewer stronger links that tend to interconnect subsets of neurons. Node size is drawn proportionally to its total connectivity weigh.

Figure 5

The probability distribution for weights of the inferred network of 62 recorded cells (left). This distribution is well described by an exponential (dashed line), with average connectivity weight $0.294\pm 0.019$. The distribution of normalized $r$ for the same network (right). Most trios show information redundancy $(r>0)$, while a smaller number display synergy $(r<0)$. The central peak at $r=0$, with probability $p=0.091$, consists of a fraction of $70\%$ of truly independent trios $(R=0)$, $22\%$ with negative $r\ge -0.05$, and $8\%$ with positive $r\le 0.05$. The other two networks exhibit quantitatively consistent results, but with poorer statistics. These results are suggestive that multineuron connectivity structures associated with both information processing $(R<0)$ and short-term memory $(R>0)$ are present in cultured neuronal networks in vitro.