User:Boldingd/sandbox/SmallWorldNetworkInNeuroscience

Note: I am perhaps abusing wikipedia's resources to write an article and use it for a class project.

Recently, there has been interest in applying the techniques and tools provided by graph theory to the problem of understanding the organization of the brain. The motivation for doing this is in part that the structural connection patterns within the brain are significant to how it operates - and this is perhaps true on many scales, from populations of single neurons up to the connection pattern between different functional centers in the brain - and because those connection patterns can be well-described by graphs.

Graph Theory
Formally, we describe a graph $$G = (V, E)$$ as a set of vertices $$V$$ (or, equivalently, nodes) and a set of edges $$E$$; edges connect two nodes, and can themselves be represented as a pair of nodes that are their end-points $$E \subseteq V \times V$$.

It is worth noting that there are significantly more complex formulations for graphs, which might include assigning directions or weights to the edges; these constructions are typically not used in the context of Small-World or Scale-Free networks. That is, we small-world and scale-free networks are typically undirected, and their edges are unweighted.

There are two properties of graphs that are significant to scale-free and small-world networks, the degree of a vertex and the shortest-path between two vertices. The degree of a vertex $$v \in V$$ is the number of edges that have $$v$$ as an end-point; in the case of an undirected graph, we do not distinguish the two ends of an edge. A path between any two vertices $$v_{start}, v_{end} \in V$$ is an ordered set of edges $$e \in E$$, such that each edge shares an and-point with the edge that preceded it, that the first edge has $$v_{start}$$ as an end-point and the last edge has $$v_{end}$$ as an end-point; the shortest path between two vertices is then the shortest of all such sequences.

We will also consider the longest shortest path between any pair of nodes. We consider this property because, loosely, it is a measure of the network's ability to communicate information between pairs of "distant" nodes efficiently. An interesting property of both small-world and scale-free networks is that they have low longest shortest paths (and thus can quickly send messages across great distances) while still ensuring that the degrees of the individual nodes and the number of edges in the graph are low (and thus the network would be cheap to physically construct).

Small-World and Scale-Free Networks
Two major families of graphs have been useful in attempting to describe how the brain is organized: small-world networks and scale-free networks. Both are families of graphs that attempt to satisfy a single property: that both the number of edges in the graph be kept small, and that the paths between any pair of nodes must be kept small. Intuitively, this is a reasonable pair of goals for the brain: internally, it is extensively connected, with many of those connections being between different neurons, potentially in completely different parts of the brain; however, at the same time, maintaining those connections is expensive in terms of nourishment, which would imply that they should be kept to the minimum that is necessary to support the operation of the brain - and in particular, that long-range connections should be minimized.

Small-world networks encode these conditions fairly directly. They were originally developed to model social connections, and are the principle that underlies the classical 7 degrees of Kevin Bacon concept. The idea is to create a network where each node has very many short-range connections, but a few nodes also have long-range connections; these nodes can then act as "bridges" between different parts of the network, which keeps the shortest-path-lengths down while also minimizing the length of the edges.

There is a simple method to create an example small-world network - although note a network can have the "small-world property" without being generated in the following manner. Consider a ring network, consisting of N neurons; first, connect each neuron to its k nearest neighbors; then, for each edge in the network, break that edge with (small) probability P, and wire reconnect that edge to a random neuron. The family of networks constructed in this way are referred to as "Wattz-Strogatz toy networks." The resulting network will have a very high local connectivity (and a small number of long range connections, proportional to P), but it will also have a very low shortest-path-length between any two pairs of neurons.

Scale-free networks are a more recent development; they tackle the problem more from the perspective of the nodes of the graph than from its edges. They attempt to construct graphs where, in loose language, "most nodes have very few edges, while a few nodes have very many edges." Nodes with high degree tend to serve as hubs, in that they connect groups or chains of nodes of very low degree. These networks also tend to have the property that both the longest shortest path in the network remains low, while the total number of edges remains low.

On significant caveat to scale-free networks is that they can be vulnerable to attack; the hubs in a scale-free network are significant weak-points, and removing them can disconnect sections of the network.

Applications to Neuroscience
Perhaps the major application of small-world and scale-free networks has been in the study of neurological disorders. If we assume that the connection-structure of the brain is significant in determining its behavior as a system, then it is perhaps reasonable to assume that individuals demonstrating neurological dysfunction might also possess abnormalities in the structural organization of the brain. Neuroscientists can exploit this by experimentally measuring the connective patterns in the brains of healthy patients, repeating those measurements in affected patients and comparing the results. While it is obvious that we do not yet have the capability to correct such a structural abnormality, this technique at least allows us to learn what might be driving neurological dysfunction.

Connectivity is generally approximated by using some measure of activity in regions of the brain, such as an EEG, and recording a time-series; generally, patients are recorded in an "eyes-closed, no task" state and told to think about nothing in particular, in an attempt to control for task-specific activations in the brain. Once a time series is acquired, correlations are computed between the activities of each pair of regions; loosely, for any two sensors, the degree to which the activity recorded at one sensor matches the activity of the other sensor is computed. This lattice of correlation coefficients is then thresholded; if the correlation coefficient rises above a certain threshold, we assume that the regions monitored by that sensor are strongly-connected, and otherwise we assume that they are not. We then create a graph representing these correlations, with nodes representing sensors and edges connecting nodes that represent correlated sensors.

Early studies simply computed the connection properties of these graphs in order to determine whether or not the brain might be described as a scale-free network; after promising early results, later studies have attempted to compare the organizational properties of healthy rains to the organizational properties of the brains of those with neurological conditions.

It is worth noting that this experimental set-up is not without caveat. One significant issue is that it does not provide direct observation of a small-world or scale-free arrangement. While many experimenters describe this procedure as measuring functional connectivity (rather than physical connectivity), it perhaps properly does not even measure that, as the areas beneath the sensors of for example an EEG are not necessarily correlated with single functional regions of the brain.

Some studies have used interesting processing approaches; one study in particular used frequency filtering to attempt to detect whether the brain demonstrated small-world organization across different frequency regimes. This was motivated by the theory that the mechanisms that mediate different processes in the brain operate at different frequencies, and thus different functional connectivity patterns might be evidenced with the different mechanisms in operation.

Significant work has been done in modelling alzheimer's, schizophrenia and seizures as disorders primarily of connectedness.

One study found a correlation between reduced functional correlation between distant brain regions with schizophrenia, leading that group to assert that schizophrenia is caused by a reduced ability to integrate the functioning of different regions of the brain, as a result of a breakdown of the long-range wiring in the brain.

Another significant study using a similar methodology identified the degradation of small-world network properties (through the loss of long-range connectivity between certain brain regions) with Alzheimer's disease. That group did frequency-filter their raw data; their results were constrained primarily to the Beta-band of brain activity, which is associated with active, attentive cognition.

Another study connected small-world properties with seizure disorders, albeit they did so by simulating a network of spiking neurons, controlling their connectivity properties and detecting regions of connectivity where seizure-like spiking patterns could and could not arrive.

Boldingd (talk) 04:00, 3 April 2014 (UTC)