To determine a functional network, a population measure of cell activity is required. In large organs, such as the brain, functional magnetic resonance imaging ( fMRI) signals tend to be used to map interactions between distant areas (Bullmore and Sporns, 2009), whereas calcium (Ca²⁺)-imaging is used to monitor the contribution of individual neurons to the function of discrete neural circuits (e.g., hippocampus) (Peterlin et al., 2000). As the pituitary is a small organ – the size of a pea in humans – large swathes can easily be mapped using the latter approach. Moreover, basal- and secretagogue-stimulated hormone release from most neuroendocrine cells depends on action potential-driven extracellular Ca²⁺-entry through voltage-gated ion channels (a notable exception being mammalian gonadotropes) (Mollard and Schlegel, 1996; Schlegel et al., 1987; Stojilkovic et al., 2005). Thus, simultaneous assessment of cytosolic free Ca²⁺ from hundreds of individual endocrine cells allows the activity dynamics underlying hormone secretion to be reasonably estimated. This technique has been widely used in assessing network organization in discrete brain regions.

Decoding the complex network dynamics underlying hormone release is challenging because of our limited knowledge about the nature, strength, extent, and rapidity of cell–cell communication. For example, paracrine couplings mediated by readily diffusible molecules will be subject to different temporal/spatial limitations compared to those that rely on next-neighbor gap junction linkages. Indeed, both exist inside the pituitary, but their exact details remain largely obscure (reviewed in (Hodson et al., 2012aa)). Are the coupling mechanisms of short- or long range? Do they necessitate precise orchestration of cell activity? Are cell–cell communications predominantly homo- or hetero-typic? Consequently, the investigation of structural and functional endocrine cell connectivity is necessarily statistical and descriptive.

The following sections will describe how the mathematical language of graph theory can be used to analyze correlations in position and activity, providing information about the physiologically relevant cell–cell interactions which may contribute to hormone secretion from the pituitary. These tools will then be used in a worked example to analyze spontaneous activity of a pituitary endocrine cell population in situ.

A branch of statistical physics has emerged in recent years to probe the characteristics of complex systems. Relying principally on graph theory, algorithms have been developed to identify the various components of networks, and derive a better understanding of the interactions that occur across multiple scales (μm; seconds to years) to generate complex behavior. Such analyses have deepened our understanding of how network topology permeates seemingly complex scenarios ranging from scientific publishing (e.g., a few influential authors are responsible for the majority of citations) (Price, 1965) to the world wide web (e.g., just a few websites are referred to in the majority of hyperlinks) (Barabási and Albert, 1999) to biology (e.g., protein–protein networks where some key targets such as p53 have wide-ranging effects by virtue of their multiple interactions) (Barabási and Oltvai, 2004). A critical aspect of understanding networks is borne from the observation that, although the constituents can be different (people versus cells) the resultant behaviors/topologies are highly conserved (e.g., random, scale-free and small world).

For a system to behave as a complex biological system, a set of criteria, as adapted from Alon (Alon, 2003), must be fulfilled (see Text Box 6.1).

1. Nodes and edges: A network is constructed of nodes connected by edges. In biology, the nature of nodes can vary from individual cells to distinct tissue regions responsible for a particular function.
   
2. Node degree (k): The number of edges emanating from each node. The population degree distribution probability largely determines network topology.
   
3. Node association: A measure of similarity between nodes which determines node degree. This is normally based upon statistical observations of nonrandom behavior such as cell–cell correlation measures.
   
4. Adjacency matrix: Usually a binary table (i.e., “0” or “1”) specifying whether cell 1 is statistically connected/associated to cell 2,…,x. It contains all possible pairwise permutations and allows the calculation of the node degree.
   
5. Network topology: A characteristic description of the network structure drawn from the degree distribution in combination with various other network measures (see network jargon heading).
   
6. Undirected network: No inference is made about the line origins, that is, both nodes are considered to potentially give rise to the interaction.
   
7. Directed network: Lines originate from a specified node. Requires statistical measures of association which can take into account directionality, for example, Granger causality.

Quantitative analysis of network topology is critical for a better understanding of how functional connectivity influences network performance. For example, do just a few endocrine cells possess most of the edges, thus acting as hubs, or do all cells contribute equally to network architecture? What are the consequences of this for cell–cell communications, robustness in the face of insults, and downstream outputs, such as hormone secretion and transcription? To aid this, some common metrics are listed in the following with explanations of what their values mean. The necessary input data are derived from the correlation matrix and Euclidean distances (see later for how to derive this information).

1. Degree distribution (P(k)): The proportion of nodes in the network which possess degree (k). It is otherwise known as the probability distribution of degrees across the population P(k). The value can be derived from summing the columns of the correlation matrix. Cells with the maximum k-value are considered to possess 100% of the connections. The degree distribution is an important indicator of network topology and geometry. For example, if the distribution obeys a power law, then the network is scalefree, i.e., a hub and spoke architecture. Conversely, if P(k) follows a Poisson or Gaussian distribution, then the network is random, that is, all nodes have identical degrees (or edge number).
   

   Average path length (L): The shortest path between two nodes, averaged for all pairwise combinations, is determined by the following formula:

   
   
2. 
   
   
   
   

   where N is the total number of nodes, and is the shortest distance between nodes i and j (summed over all possible node pairs).

   
   

   In general, the shorter the average path length, the more efficient the information transfer through the network.

   
   

   Clustering coefficient (C): The number of neighboring edges hosted by a node i as a function of total possible edges (ranging from 0 to 1) and can be calculated using:

   
   
3. 
   
   
   
   

   where j is the number of neighboring nodes and k is the number of edges with all neighbors (i.e., degree). The network clustering coefficient is simply the average for all nodes in the population.

   
   

   High values indicate increased local connectedness with neighbors, streamlining communications due to a tendency toward decreased path length.

   
   

   Degree centrality (D): A measure of how connected a node’s neighbors are. Degree D centrality of a given node i can be summarized by the following:

   
   
4. 
   
   
   
   

   where if an edge exists between node i and j.

   
   

   Nodes with high centrality have neighbors with low degree and, thus, act as relays for information flow, minimizing average path length.

   
   
5. Closeness centrality (CC): The sum of the inverses of the distances between a node and all other nodes:
   
   
   
   

   where i is the node under examination and is the shortest distance between nodes i and j. Information transfer is most efficient if routed through a central node that is close to all other nodes.

The use of transgenic animals allows the identification of signals arising from specific pituitary cell populations/lineages. A variety of strains exist which contain fluorophores driven by specific promoters (e.g., prolactin-Discosoma red fluorescent protein (DsRed), luteinizing hormone-Cerulean, growth hormone-enhanced green fluorescent protein (eGFP)). It is important to consider any excitation/emission overlap between the fluorescent tag and Ca²⁺ indicator when designing imaging protocols.

• Mix 50 g fura-2AM with 5 l DMSO + 10 l 20% pluronic acid (w/v in DMSO) and sonicate for 3 min. Add 500 l bicarbonate bufferand sonicate for a further 3 min to disperse micelles. Make up to 2 ml and incubate slice in fura-2-containing solution for 1–2 h at 37°C/5% CO. Rinse sections twice with bicarbonate buffer and reincubate for 30 min without fura-2 to allow cleavage and activation of fura-2 intracellular esterases (! critical: do not over-incubate slice to avoid buffering of cytosolic free calcium by fura-2).
  
• Mount slices on/in microscope chamber and perfuse with bicarbonate buffer at 34–36°C (! critical: ion channel kinetics are temperature-dependent).
  
• Use a 10–40 water immersion objective depending on desired size of imaged field. Capture a snapshot of the fluo-tagged cell population under investigation. A NA 0.95 water immersion objective allows pituitary slices to be imaged with single-cell resolution whilst retaining a large field of view.
  
• Proceed to functional multicellular imaging (fMCI).

• Perform excitation at 780–800 nm and record emission at 525/50 nm for nonratiometric two-photon imaging of fura-2. Multiphoton microscopy has many advantages over single photon modalities including better depth penetration, less photoxicity and improved axial resolution (see Glossary) (! critical: due to fura-2 excitation above the isosbestic point, increased [Ca²⁺] will present as decreased signal intensity).
  
• Acquire a three-dimensional reconstruction of the cell population under investigation. This will allow functional connectivity (based on association measures) to be directly correlated with structural connectivity (based on physical linkages) (see later).
  
• Image fura-2 in the second cell layer. This avoids artifacts due to recording at the cut surface and ensures optimal overlap with the cell fluo-tag. Generally, Ca²⁺ indicators will penetrate three to four cell layers deep.
  
• According to the Nyquist sampling theorem, the frame rate should be twice the highest frequency which needs to be resolved for analysis purposes. For example, 1 Hz will be required to detect events at 0.5 Hz. For long-term time lapses (>15 min), reduce laser power, and frame rate at the expense of spatial resolution/image quality (! critical: phototoxicity introduces artefacts into signal analysis).
  
• Use binning to average intensity for pixels if using the an EM-CCD-based system.

Note: Fura-2 is ideal for both one- and two-photon imaging modalities as it possesses nonoverlapping excitation/emissions spectra with commonly used eGFP cell-tags, loads pituitary cells better than equivalent Ca²⁺ indicators, and possesses a dissociation constant (145 nM) compatible with resolution of both spontaneous and secretagogue-driven Ca²⁺ events in pituitary cells. Since fura-2 can exit cells via chloride channels, “leak resistant” forms are recommended for long time-lapse experiments.