On Networks, Neurons and the Brain

When we think of our brain what comes in mind? Well, typically, words as memory and neurons are some keywords associated with the word brain. Moreover, other keywords can be schizophrenia, depression, Alzheimer’s, Parkinson’s. All these characterize different functional aspects of the brain. But, lets take a more spherical look at this structure which holds the most valuable process in humans: cognition. The brain can be considered as the world, consisting of many countries and smaller communities. Countries will be brain regions, such as the prefrontal cortex, the parietal cortex or the hippocampus. Communities would then be smaller neuronal networks within the individual brain areas. All these functional units work together to produce an output, may it be a single thought or better an array of thoughts, or, a complex grasping movement. At the heart of this output, lie neuronal units.  This post provides some introduction on the examination of the functional and architectural hierarchy of the brain, under the scope of Graph Theory.  The information is taken by the excellent review of Bullmore and Sporns [2009]. I urge you to read this review for more details and references on the topic.

Graph Theory, which is the study of graphs, has been steadily gaining interest in brain research. Regardless of time and scale and species [Bullmore and Sporns, 2009], the brain has been shown to have the characteristics of a small-world network. Before going into the specifics of brain networks, lets go over some definitions.

What is a network? A network is characterized by a set of nodes (may it be anatomical brain areas, functional brain areas as measured by electroencephalography or neurons) and a set of edges that connect these nodes. The connection is determined on the basis of the actual experimental question e.g. an edge can represent anatomical connections between brain areas e.g. the corpus callosum between the two cerebral hemispheres (a very simple 2 node graph/network with one edge) or the individual 200–250 million (!!!) contralateral axonal projections of the corpus callosum ( more complex graph; the nodes in this case may be the individual neurons receiving these connections and their topology). Key attributes of a network are: Node Degree, the number of connections of the specific node; the Clustering Coefficient, which is the ratio of the number of connections of a node to the maximum possible number connections a node may have. Moreover other important network attributes are: Hubs, which are nodes with high degree or else, high centrality, which is simply the number of shortest paths of other nodes, that cross the node under question. Short path, denotes the minimum number of edges, one may cross between a specific pair of nodes.

What then is a Small-World Network? A small-world network is a type of graph/network which is characterized by high node degree, hence, most nodes are neighbors between them. Also, a small-world network is characterized by small short path lengths, thus each node can be reached from another within a small number of lengths. This type of graph has received a lot of attention, when studying social networks. A famous example is the Six-Degrees of Separation which refers to the theory that a person is approximately 6 steps (edges of connection) away from another person on Earth. The brain satisfies the criteria of a small-world network, and this holds true on various scales in time and space and along the evolutionary ladder.

Caenorhabdidis elegans is the first organism to have its  nervous system described at a cellular level and shown to be a small-world network in terms of connectivity [Watts and Strogatz, 1998]. The organism has merely 302 neurons, a fact that made mapping of its connectome very easy in comparison to higher organisms such as human. In the human case, data produced by many methods of brain imaging have been used to build structural brain networks. For example, application of graph theory on data from Diffusion Tensor Imaging, revealed that a few areas such as the superior frontal cortex, superior parietal cortex, the precuneus and the insula have high correlation between them, in terms of centrality [Sporns and Kötter, 2004]. Moreover, functional brain networks have been constructed with the use of fMRI, MEG, EEG. Combination of data from the BOLD signal of fMRI which provides better spatial resolution and data from electrophysiological methods such as EEG, which provide better temporal resolution, and application of interpolation methods made it possible to generate more accurate networks of brain activation. Moreover, science has gone further down the line, and scientists are able to record neuronal network activity in a dish (in vitro) by MEA (multielectrode array). From these in vitro studies and animal studies, graph theory has shown that neuronal nodes with  similar connection patterns, tend to have similar functions [White et al., 1986]. Graph theory has and is proving very useful in understanding the changes that take place in various neurological and neuropsychiatric syndromes. For example, network analysis of MEG and fMRI data from Alzheimer’s and schizophrenic patients showed that small-world organization is lost and this held true both on the functional and on the structural scale. Moreover network analysis, offers the ability to ”visualize” changes that happen in the molecular level, and allows better understanding of processes such as synaptic plasticity, which is a process fine-tuned on the scale of milliseconds, and it is considered (LTP), to be the molecular underpinning of memory formation.  Comparison of the functional networks on the molecular scale, to functional networks collected from humans while performing a memory task , may help us understand the functional and even the structural changes taking place in the brain during memory formation and shed light in the pathophysiology of memory-related conditions.

This post only provides a brief introduction on the subject of Graph Analysis and brain function and structure. Nevertheless, it is clear from the little described here, that Graph Analysis along with other mathematical and statistical methods, may provide the key to understand better this eloquently-build structure called brain.

References

Bullmore, E. & Sporns, O. Complex brain networks: graph theoretical analysis of structural and functional systems.         Nat.Rev.Neurosc.10, 186-198 (2009)

Watts, D. J. & Strogatz, S. H. Collective dynamics of “small-world” networks. Nature 393, 440–442 (1998).

Sporns, O. & Kötter, R. Motifs in brain networks. PLoSBiol. 2, 1910–1918 (2004)

White, J. G., Southgate, E., Thomson, J. N. & Brenner, S. The structure of the nervous system of the nematode Caenorhabditis elegans. Philos. Trans. R. Soc. Lond. B Biol. Sci. 314, 1–340 (1986).

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Filed under Biology, Graph Theory, Neurons, Neuroscience

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