Analyzed the data: JHD MWH DX MCS BJW SJN. Wrote the paper: JHD. Conceived and designed the analysis of existing data: JHD BJW SJN. Edited the manuscript: JHD MWH DX MCS VMB KW BJW SJN.
The authors have declared that no competing interests exist.
The functional networks of cultured neurons exhibit complex network properties similar to those found
Many social, technological and biological networks exhibit properties that are neither completely random, nor fully regular. They are known as complex networks and statistics exist to characterize their structure. Until recently, such networks have primarily been analyzed as fixed structures, which enable interaction between their components (nodes). The present work is one of the first empirical studies investigating the adaptation of complex networks
The organizational properties of biological, technological and social systems are increasingly being characterized by representing them as abstract networks of interacting components and quantifying nonrandom features of their structure
For neuronal connectivity, the abstract network (graphtheoretic) approach to analysis has allowed common organizational principles to be identified at both the macroscale level of whole brain imaging
The focus of the present study is the development of complex network properties within cultures of neurons, grown
Two aspects of the cultures that are of particular interest are their structural (anatomical) circuitry and the interactions which take place over this circuitry, both determining the computational capacity of the underlying network. Whilst cultures are typically too dense for accurate observation of their structural connectivity, analysis of functional connectivity provides a probabilistic estimation of the relationship between distributed neuronal units
Existing literature indicates that the functional network properties of cortical cultures change during maturation
Reports that rigorously compare culture's complex network properties under different experimental conditions are very sparse. Mature cultures were assessed in
Experiments utilizing cultures can be undertaken across a range of ages, yet little is known about whether developmental changes occur in culture's complex network properties. Questions such as when and which nonrandom properties are present, their stability over time and the variability between cultures and their ages remain largely unanswered. The nature of such spontaneously occurring changes in a culture's functional network are important
The density at which cultures are seeded exerts an important influence on the rate of maturation. Dense cultures mature faster than their sparse equivalents, and they demonstrate bursting activity earlier in development
The present study establishes the baseline network statistics for cultures at specified stages of development and uses them to characterize culture maturation. The topological, spatial and performance properties of functional networks captured every 7 days (7 to 35 days
Since the graphtheoretic approach and use of complex network statistics is a relatively novel method for investigating functional connectivity in cultures, the key methodological decisions are described next.
Although both structural and functional neuronal networks can be explored using graph theory
Steps 1–3 and step 5 are based on recommendations from Bullmore & Sporns (Nature Reviews Neuroscience, 2009). Steps 4, 6 and 7 refer to techniques specific to analysis of culture activity recorded from multielectrode arrays (MEAs, example pictured top right). The 8×8 grid indicates the recording area of the MEA (inset: closeup of two electrodes with visible neurons in their vicinity).
Firstly, at short timescales (hundreds of milliseconds), connectivity was assessed during each networkwide burst, a threshold was then applied to include only the links between highly related nodes
To compare networks from a sequence of experimental ‘conditions’, the development of the
The dual timescale approach to network definition (
A number of topological metrics may be calculated and from these the complex network statistics
Asides from the level of integration and segregation, an important aspect to characterizing a network's ‘class’ is the form of the degree distribution. This may be measured by determining the best fitting model: A fastdecaying (exponential or Gaussian) model provides a good fit for networks with a homogeneous population of nodes, whereby most nodes have a comparable number of connections and few nodes deviate from this number significantly. Such networks are classed as ‘singlescale’ and their scale is equal to their mean node degree. Conversely, networks that have no characteristic scale are termed ‘scalefree’, these are identified by a degree distribution that decays progressively more slowly towards infinity – hence there is no characteristic mean node degree. These are typically represented by a powerlaw model.
Random and lattice networks both have a singlescale degree distribution, conversely, many realworld networks have been found to possess a powerlaw degree distribution
Since cultured neuronal networks are embedded in physical space, spatial and temporal characteristics of interaction, such as internode distance and signal propagation speed, can also be informative about changes in the activity patterns. Therefore, physical link length (derived from interelectrode distance), and networkwide signal propagation efficiency (via mean burst propagation time) were also assessed. Additionally, the frequency with which individual links are activated can provide information on the influence of a given link in the various network interactions. Therefore, the reliability of link activation was calculated from the analog (weighted) persistence adjacency matrix.
Property type, name and Description  Range & units  Interpretation  


Mean path length: 
1+ (# hops btw nodes)  Integration: Ability for any two nodes to interact via a minimal number of intermediary nodes. A short (low) mean path length reflects high integration (i.e. a low average number of hops between nodes) as found in random networks. 

Mean clustering coefficient: 
0–1  Segregation: Ability for groups of nodes to interact. A high level of segregation (as found in lattice networks) reflects the presence of highly interconnected node subgroups (clusters) within the network.  

‘Smallworldness’: 
0+  Complexity: balance between integration and segregation  

Network broadcast time  Measured as the burst propagation time  ∼100–1000 ms  Performance of the network in terms of the time required for a signal to reach all nodes 
Property type, name and Description  Range, units and Interpretation  

Node degrees  Node degree distribution  Relative influence of nodes in the network (node degree = 1–58) Nodes with a high degree have many connections: A fat tailed degree distribution indicates presence of highly influential nodes, whilst homogeneity indicates lack of network structure. 

Link lengths  Spatial properties: Linklength distribution  Assess form and the proportion of links between nearby 
Link persistence levels  ‘Reliability’ props: Link activation frequency distribution.  Assess form and the contribution of persistent links (link persistence = 0–1). Persistent links represent frequent interactions between neural units. 
Results are split into two sections. The first presents topological, then spatial network statistics from persistent networks. The second presents statistics on the propagation of activity over the network (from the transient networks). Network statistics were obtained for each culture at each age (DIV 7, 14, 21, 28, 35). NOTE, at DIV 7 only one culture was found to have a persistent network, therefore this age was not considered for the significance testing.
The number of nodes and links for a given culture was used to calculate the edge density of its network.
Number of nodes, links and edge density; calculated for 10 cultures at each age (DIV). Left: mean number of nodes and links found in the persistent networks. Note, although the number of nodes is a very different magnitude from the number of links, number of nodes was not found to change significantly (P = 0.272). Results for numbers of links at each age suggested an increase between younger (DIV 14 and DIV 21) and older (DIV 28 and 35) ages, however the increase was not significant (P = 0.074). Right: mean edge density of the persistent networks. Edge density (i.e. link density) quantifies the ‘cost’ of the network in terms of the number of links (
Complex network statistics from each culture's persistent network were used to assess changes in network topology as the cultures matured.
Mean path length, clustering coefficient and conservative smallworldness; averages (
To assess the relative influence of nodes in the network, the form of the node degree distributions were compared between ages. As the cultures matured, the number of nodes with a high degree increased, leading to a fatter tailed node degree distribution (
Node degree distributions, obtained from all the nodes of the persistent networks of all cultures using a bin size of 10%. Panel A: bar graphs represent node degree distributions on a linear scale. Solid lines show the best fitting model at each age, broken lines represent 95^{th} percent confidence interval. Top left: DIV 14, bottom left: DIV 21, top right DIV 28, bottom right: DIV 35. DIVs 14 and 21 show exponential fit on a linear scale, DIVs 28 and 35 show power law fit on a linear scale. Panel B: scatter plots represent node degree distributions on a loglog scale, DIVs 28 and 35 are shown with a linear fit. The fat tailed node degree distribution found at DIVs 28 and 35 is indicative of nodes with a high degree (hubs).
The spatial organization of nodes and links also changed as cultures matured. At DIV 14, the proportion of links between distant nodes was significantly higher than the proportion of links between nearby nodes (P = 0.028), whilst at subsequent ages there was no significant difference (P = 0.27, 0.83, 0.5, for DIV 21, 28, and 35, respectively),
Panel A: Each bar represents the median proportion of links between nodes up to (and including) two electrodes apart (classified as ‘nearby’) and links between nodes greater than two electrodes apart (classified as ‘distant’), diagonal neighbors were included; values were calculated from all cultures at each age. Upwards error bars represent the 75^{th} percentile and downwards bars the 25^{th} percentile. Notably, at DIV 14 there was a significantly higher number of connections between distant nodes. Panel B: Normalized histograms of link lengths at each culture age, constructed from the link lengths of all cultures, measured as the proportion of each culture's links at each length. Median values from all cultures were used for each bin in the histogram. Bin size was based on spacing between electrodes of MEA, with one bin for each electrode distance (i.e. bin 1 is all links between neighboring electrodes  including diagonal neighbors, bin 2 is all links between nodes up to two electrodes distance, and so forth until seven electrodes distance which is the maximum between any two nodes on the MEA). Bin edges (Xaxis) specify the start of each bin, measured as the distance between electrodes on the MEA (micrometers). Yaxis is the same for all histograms in panel, only DIV 14 Yaxis is labeled to avoid overcrowding.
Network graphs were generated to depict the spatial arrangement of each culture's network components.
Graphs illustrate the spatial organization of network components at each culture age: the 8 by 8 grid corresponds to positions of the electrodes on the multielectrode array (MEA). Nodes that are part of the network (i.e. for which a link was identified) are numbered according to their MEA hardware numbers, and the lines between electrodes represent undirected links between nodes. Panel A: graphs from the networks thresholded at 25% link persistence. Panel B: graphs from the networks thresholded at 15% link persistence, this lower threshold results in more nodes and links.
Graphs illustrate the location of hubs in the persistent network of a representative culture at two separate ages. The 8 by 8 grid corresponds to positions of the electrodes on the multielectrode array (MEA). Nodes that are part of the network (i.e. for which a link was identified) are numbered according to their MEA hardware numbers, and the lines between electrodes represent undirected links between nodes. At DIV 28 (left hand graph), nodes 4 and 38 were classified as hubs in the network, whilst at DIV 35 (right hand graph), nodes 34, 38, 40, 48, 49 and 53 were hubs. Hubs were classified as nodes having a high degree (degree greater than mean node degree plus one standard deviation) and are highlighted with blue circles.
Results presented thus far have focused on identifying changes in the network infrastructure (via the persistent interactions between different areas [nodes] in the cultures). Here, the results focus upon the activity that takes place over this infrastructure. Each transient network is considered as a ‘snapshot’ of network activity, measured over a short timescale (duration of a networkwide burst) and reflects interactions between different areas of the culture in this period.
As per the persistent networks, the basic properties relating to network size were compared. Additionally, since there were multiple transient networks for each culture, the coefficient of variation was also analyzed (see
Panels A, B: mean number of links and nodes (respectively) in transient networks, averaged over all cultures at a particular age (solid black lines). Error bars represent ± s.e.m. The mean numbers of links at each culture age suggested an increasing trend in the number of links between DIVs 14 and 28, however the trend was not significant (P = 0.087). Likewise the mean numbers of nodes suggested an increasing trend (P = 0.089). The mean numbers of persistent network links and nodes are shown for reference (dotted red lines). Panels C, D: expected coefficient of variation for the numbers of links and nodes (respectively) in each culture's set of transient networks. Error bars represent ± s.e.m. Coefficient of variation for number of links was significantly higher at DIV 21 than DIV 14 (P = 0.021).
To assess whether network properties influenced the transfer of information across the culture, burst propagation time was compared at each age (
Bar chart shows the median burst propagation time (from all transient networks of all cultures at each age), values outside the 5^{th} to 95^{th} percentiles were removed as outliers, giving
To investigate whether links became more reliable (persistent) as the cultures matured a histogram of link persistence values was generated. The fat tailed link persistence distributions at DIVs 28 and 35 reflected the fact that persistent links became more numerous and were activated more frequently (
Each histogram shows the percentage of links found at each link persistence level for all cultures at each age (normalized count of links found at the persistence value, expressed as the percentage of transient networks (bursts) in which the link was found, bin size 5%, bin edges specify the end of each bin). Top left: DIV 14, bottom left: DIV 21, top right: DIV 28, bottom right: DIV 35. Red (solid) line is the link persistence threshold (link presence in at least 25% of the transient networks). The histograms are cropped to show the detailed distribution, inset histograms show the full scale. At DIV 21, many links were found infrequently (i.e. below the link persistence threshold). The more pronounced tail of the distribution as the cultures matured, reflected a significantly higher contribution of persistent links in the network's of mature cultures.
The present study characterizes the evolution of functional networks observed in cortical cultures and extends previous work where network properties of cultures were investigated at a single developmental stage
Immature cultures (DIV 14) exhibited limited interactions between neuronal units, resulting in a network of few nodes and links. The observation that at DIV 14 activity could spread rapidly between any two neuronal units (short mean path length in
Whilst interactions at DIV 14 were clearly unstructured, the subsequent 14 days of development represented a critical window, during which functional complexity increased (
We consider the possible driving forces behind this topology change to include the level of synchronization, the ratio of excitationinhibition and the mechanism of Hebbian learning.
Synchronization of culture activity can be defined over a range of timescales – from ‘synchronous busting’
In a previous study of functional connectivity during development
Crucially, the combination of varied burst properties and transient synchronization at DIV 21 indicates a mixture of regular and irregular activity. Modeling studies have suggested that such mixed activity constitutes optimal conditions for the emergence of a smallworld topology via Hebbian learning rules and activity driven plasticity
Our results demonstrate that functional clustering increases from DIV 14. Moreover, this increased clustering (rather than a reduced mean path length) was the cause of increased smallworldness, which continued until cultures reached a state of semimaturity at DIV 28. We note that the increased clustering was accompanied by a change in the distribution of link lengths, from a clear dominance of longrange links at DIV 14 to an increased proportion of short links thereafter, which suggests an increase in localized latticelike clustering. The change of the link length distributions from Gaussian to bimodal, to longtailed at DIVs 14, 21 and 28–35 respectively (
During culture maturation, the distribution of node connections changed from a rapidly decaying and homogeneous degree distribution to one with a longer tail, indicating a small but nonnegligible proportion of highly connected nodes (hubs). The smallworld topology does not require hubs
Networks at DIVs 28 and 35 were classified as smallworld, exhibiting several highly connected areas (clusters of highly interconnected neural units), alongside the ability for any two areas to interact via few intermediary connections (short mean path length). Interestingly, when the network properties at DIVs 28 and 35 were compared, smaller differences were found than between earlier ages, suggesting a state of maturity
Smallworld networks have an architecture which supports efficient information transfer
The increasing prevalence of highly connected nodes in older cultures suggests that such hubs play a greater role in network activity as the cultures mature, perhaps indicating sources
The present study has demonstrated that networks derived from the spontaneous activity of cultures develop nonrandom properties despite a lack of external input. Based on these results, we draw four main conclusions. Firstly, to mitigate fluctuations in spontaneous activity, multiple network bursts should be assessed to obtain the persistent network. Secondly, the functional network of a cortical culture evolves from an initial random topology to a smallworld topology; we propose this is due to a change in the culture's spontaneous activity patterns that is driven by the maturing excitatoryinhibitory balance and an increase in networkwide synchronization. Thirdly, the reduction in burst propagation time with culture maturation that accompanies the evolution of a smallworld topology supports the efficient networkwide flow of information afforded by a smallworld network. Lastly, the presence of hubs and increasing contribution of links with high persistence suggests a proportion of highly influential nodes and links.
To the authors' knowledge, this is the first demonstration of smallworld properties evolving in the functional networks of cortical neurons grown
An important area for future work is to investigate the role of frequently activated nodes (hubs) in cultured neurons; including whether the presence of networksynchrony controlling hubs in the underlying substrate could mediate the timing and extent of functional interactions between otherwise segregated clusters, perhaps coordinating synchronous networkwide bursting. Additionally, the use of staining to identify the location and proportion of the different neuron types and subtypes, and the use of pharmacological manipulation to verify their effect on activity may help elucidate mechanisms behind the different network properties.
Data used for the present study was collected for
Culture's electrical activity was recorded daily during their first 5 weeks of development. For the present study, a sample population was selected from the large number of cultures recorded, specifically, 10 cultures from 4 preparations (plating batches). Cultures were arbitrarily selected from those that had recordings every 7 DIV, i.e. those which survived for the full 5 weeks and for whom none of the weekly recordings were missed. The use of multiple preparations is important as bursting patterns across preparations vary considerably
Data were recorded from cultures for 30 minutes daily in the incubator used for culture's maintenance. Unit and multiunit spontaneous spike firing was recorded from the MEA (8×8 array of 59 planar electrodes, each 30 µm diameter with 200 µm interelectrode spacing [centre to centre]). The preamplifier was from Multi Channel Systems (MCS), excess heat was removed using a custom Peltiercooled platform. Data acquisition and online spike detection was performed using MEABench
Spikes were detected online (using MEABench), positive or negative excursions beyond a threshold of 4.5× estimated RMS noise, were classed as spikes. Their peak amplitude timestamp (µs), plus electrode number were stored. For the present study, all positive amplitude spikes were removed to avoid counting spikes on both upwards and downwards phases.
In cortical cultures, global bursts (population bursts), characterized by an increase in culture activity across the entire MEA, are typically present from DIV 4–6 onwards
All activity occurring from the first spike in the nwburst to the last spike in the nwburst (including tonic activity from electrodes not included in the nwburst) was used for assessing the relationships between channel pairs. Spike occurrences were counted in 1 ms bins, this allowed a certain amount of jitter in the spike arrival times (which could otherwise decrease the likelihood of identifying correlated activity). Bin size was selected based on experimentation with 1, 5 and 10 ms bins. The 1 ms bins provided a greater separation between correlated and uncorrelated channels, data not shown.
Functional connectivity was assessed by correlating spike times recorded on pairs of electrodes during a networkwide burst (as per
Channels that had fewer than 8 spikes recorded during the burst were excluded from the crosscovariance analysis, as results from synthetic data testing showed that performing crosscovariance on vectors with fewer than 8 spikes was poor at distinguishing related vectors from independent ones (data not shown).
The crosscovariance function calculates the covariance of two random vectors:
Calculation of the crosscovariance at each lag resulted in a crosscovariance plot for each channel pair. The maximum crosscovariance value (peak of the plot) was used to determine whether a link between nodes was present by comparing it to a threshold as detailed next.
Under the assumption that a peak in the crosscovariance (XCov) plot indicates a relationship between the channel pairs
The transient networks obtained over the duration of a recording were found to be highly variable (see
To compute the persistent network a weighted adjacency matrix comprising the count of each link's occurrence over all transient networks was obtained by summing the binary adjacency matrices of all transient networks. A link persistence threshold was applied to this ‘adjacency frequency matrix’ to obtain a binary adjacency matrix representing the persistent network. At a threshold of 1, the persistent network is simply the superset of all transient networks (and thus, not strictly speaking, ‘persistent’); conversely a threshold set equal to the total number of transient networks, requires link presence in every transient network. Setting the threshold equal to link presence in 25% of transient networks provided a good balance between minimizing the number of overly dense and overly sparse networks (see complex network analysis).
Since some of the complex network statistics are defined only for certain ranges of network size, it was important to ensure that each persistent network was within the size range suitable for complex network analysis. Specifically, an assumption made when assessing smallworld properties is that the networks are sparse:
The expected persistent network statistics for each age (DIV) were obtained from all 10 cultures. For the numbers of nodes, links and edge density, data outside the 5^{th} to 95^{th} percentile were removed as outliers (maximum removed = data from 4 cultures, leaving minimum
For each persistent network, the networkwide statistics (mean path length [average shortest path length] (
Smallworldness of the network
To check that the smallworld metric was not influenced by network disconnectedness (as mean path length is only defined for connected graphs and not all graphs were connected), the ratio of global efficiency to the clustering coefficient
Node degree distribution was calculated using all nodes of the network by counting the number of nodes with each degree in bins of size 2. Bin size was selected to provide a sufficient number of data points, whilst minimizing the number of empty bins (sizes 1, 2 and 3 were tested, data not shown). Hubs were identified as nodes with a degree greater than mean node degree plus one standard deviation
Network size and density may influence the magnitude of complex network statistics
In addition to the networks' topological properties, the spatial and temporal features of the networks were also assessed; link distance was calculated as the Euclidean distance between the electrodes on the MEA, based on 200 µm centretocentre spacing of the electrodes. For the present study, connections between nodes up to 566 µm (2 electrodes) apart were considered as ‘nearby’ and those greater than 566 µm as ‘distant’. Link persistence was calculated using the weighted persistent network adjacency matrix (i.e. prior to thresholding), normalized so that the persistence value was the percentage of transient networks in which the link was found.
For both link length (derived from the distance between connected nodes) and link persistence, histograms were obtained over all links from all cultures at each age. Thus, for link length, a count of the number of links in each bin (bin size = 1 electrode spacing) was calculated for each network, this was normalized to the total number of links in the network. For link persistence, a count of the links at each persistence level (bin size 5%) was calculated for each network. In both cases, median bin values were obtained over all 10 cultures, therefore the histogram proportions may not always sum to 1.
To quantify the changes in link length and persistence, two further measures were assessed: for link length, the proportion of links between spatially nearby
The efficiency of information broadcast was measured as burst propagation time (time to recruit all channels in a networkwide burst). This was calculated in milliseconds from the time of the first spike in the burst, until the time at which all channels participating in the burst had been recruited. Channels could be recruited to the burst whilst the burst was in progress (i.e. sufficient channels displayed the required activity) but once the number of channels bursting dropped below the threshold, channels could no longer be recruited. For each channel included in the burst, recruitment time was the timestamp of the first spike in the burst activity sequence. Burst propagation times were calculated for all bursts of a culture at each age and the median of these was calculated for each age. Outliers (values <5^{th} and >95^{th} percentile) were removed from the data.
Network graphs were visualized using a freely available script
All statistics were obtained using SPSS version 17.0 (SPSS Inc., Chicago, USA). Unless otherwise specified P<0.05 was set as the significance level. Statistical tests for each network property were selected based on the experiment design and form of the resultant data; Checks were performed to ensure that the assumptions of each test were met. Following test selection, statistical power was verified at the 80% level (checking that the proposed test statistic had sufficient power to detect a genuine effect
To check for a significant increasing or decreasing linear trend of the network properties as a function of the culture age, results for each network property were compared using a oneway ANOVA. Culture age (DIV) was the factor, and the network property was the dependent variable. The following properties were assessed in this manner: number of nodes, number of links, edge density, normalized mean path length, normalized clustering coefficient, smallworldness, goodness of fit ratio. In cases where a significant trend was found, Bonferroni and Tukey posthoc tests were performed to check for significant differences between each pair of conditions, where found, the homogeneous subsets are mentioned in the results. Homogeneity of variances was tested using the Levene test.
Normality was tested using the ShapiroWilk normality test. In cases where the sample means were not normally distributed, nonparametric tests were used. For the burst propagation times a KruskalWallis test was performed on the median burst propagation times for each culture at each age, with culture age as the grouping factor and median burst propagation time as the dependent variable. For the proportion of links to nearby
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