Definition

The brain is composed of many neurons, functional areas, and layers. Together these components work as a network to produce behavior. At minimum, the network behavior is determined by four things: (1) the network inputs, (2) the dynamics of the individual nodes, (3) the coupling functions between the nodes, and (4) the topology. This encyclopedia section will provide a brief overview on characterizing brain-scale networks. Often the coupled system will have emergent behaviors, behaviors that could not be predicted from analysis of the individual components alone. Understanding how the brain functions requires an understanding of how components work together in a network. In many diseases the cause cannot be pinpointed to dysfunction or failure of a single component, such as an ion channel mutation. Instead, subtle changes in cellular behavior may lead homeostatic mechanisms to alter the coupling between the neurons and brain areas, resulting in pathological activity such as synchronous population oscillations or unchecked excitability. The ultimate goal in applying the network theory to understanding connections within the brain is to develop a measure that can explain the emergence of pathological behaviors and to perhaps develop approaches to treating diseases.

But first, we will introduce a few common terms. A network is a collection of coupled components. Generally, when the components coupled together are different elements, it is referred to as a system. If the components of the system are similar and interchangeable, it is instead called a network. The components in the network are referred to as nodes, which can be individual neurons or brain regions. The coupling between the nodes is referred to as edges, which can be synapses or fiber paths. Coupling in the nervous system is generally through chemical synapses which are directional, where the coupling of neuron or region A onto B will be different than B onto A. However, for networks of neurons coupled through electrical synapses, the coupling can be undirected. Networks are considered weighted if the strengths of coupling between nodes have a distribution and unweighted if they are all the same.

Generally, the dynamics of the nodes and coupling function are highly nonlinear. A linear response is defined as given twice the input, the output will be twice as strong. However, because neurons have thresholds and synapses are plastic, the responses are very nonlinear. Furthermore, neurons are a mixture of deterministic behavior, where its behavior can be determined from its past and its inputs, and stochastic, where the activity is also due to some noise in the system which cannot be accounted for.

The statistics of the coupling within a network is called the topology. A list of all the connections within a network is called the graph. If nodes have a physical location in space, such as brain regions, and coupling is dependent on the distance, the network is considered to have a geometry.

In summary, neuronal networks are nonlinear, directional weighted graphs with a geometry. These networks are called complex networks, and few tools have been developed to analyze them. The development of these network analysis tools is at the cutting edge.

Detailed Description

The goal of these encyclopedia entries is to provide an introduction into the network theory. In the first entry (“Network Theory in Neuroscience”), there is an overview of the network theory and its applications to diseases. In “Functional Network Observations of Diseased Brain States,” there is an introduction to functional networks in neuroscience. In “Determining Network Structure from Data: Nonlinear Modeling Methods,” we will introduce methods for reconstructing networks from the data using nonlinear measures. In “Master Stability Function for Globally Synchronized Systems,” we introduce a universal approach to determining if a network will synchronize given the dynamics of the individual components and the network topology, through the analysis of the master stability function. In “Connectionist Models of CPG Networks,” we will provide an introduction to nonsynchronous network behaviors, such as seen in central pattern generators. In “Neuropathologies and Networks,” we will introduce pathological network function to characterize diseases.

Cross-References