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An Invitation to the Study of Brain Networks, with Some Statistical Analysis of Thresholding Techniques

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Discrete and Topological Models in Molecular Biology

Part of the book series: Natural Computing Series ((NCS))

Abstract

We provide a brief introduction to the nascent application of network theory to mesoscale networks in the human brain. Following an overview of the typical data-gathering, processing, and analysis methods employed in this field, we describe the process for inferring a graph from neural time series. A crucial step in the construction of a graph from time series is the thresholding of graph edges to ensure that the graphs represent physiological relationships rather than artifactual noise. We discuss the most popular currently employed methodologies and then introduce one of our own, based on the theory of random matrices. Finally, we provide a comparison of our random-matrix-theory thresholding approach with two dominant approaches on a data set of 1,000 real resting-state functional magnetic resonance imaging scans.

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Notes

  1. 1.

    A voxel is simply the three-dimensional analog of a pixel: a (usually cube-shaped) volume assigned a homogenous scalar or vector value. In the context of fMRI, a typical voxel will represent a volume of size roughly 1–5 mm3, depending on several technical details of the hardware and software used for the imaging.

  2. 2.

    Spin is a quantum property of elementary particles and, as such, may take on only a discrete number of states (or some quantum superposition thereof). In the case of fermions such as the proton, the two possible spin states are \(\{\frac{1} {2},-\frac{1} {2}\}\).

  3. 3.

    Pulse sequences contain integrated information about the application of magnetic gradients, electromagnetic pulses, and the recording of electromagnetic radiation emitted from the subject. For more detail, see [4].

  4. 4.

    This figure comes from an analysis in [20] in which known “ground truth” networks we used to construct simulated fMRI data, to which various network inference techniques we then applied. The accuracy reflects how much of the topology of the ground truth network the inference method managed to capture.

  5. 5.

    The careful reader may find, as does this author, this conclusion somewhat disturbing. The brain is a Turing-complete computational system, and accurate measures of its state should show useful statistics far transcending the first moment; yet, the smoothed, preprocessed BOLD signal does not, it would appear.

  6. 6.

    “Dorsal” refers here to the top of the brain – the part above the eyeline in a human standing upright – and “ventral” refers to the bottom part.

  7. 7.

    The rostro-caudal axis follows a curved path through the head, beginning roughly in the region of the nose, and proceeding straight back towards the midbrain, at which point it bends 90∘ downwards to follow the spine.

  8. 8.

    We derived is r-value threshold by converting the corrected p threshold to a t-score with 298 degrees of freedom, giving \(t = -6.33\), and computing \(6.33/\sqrt{298 + 6.3{3}^{2}}\).

  9. 9.

    One may, of course, fit the curve to an arbitrarily sophisticated function; we chose cubic splines here, as they have been demonstrated to work well in applications ranging from neutron scattering to quantitative finance.

  10. 10.

    It is also worth noting that significant progress has been made in spectral decomposition on commodity GPU hardware; see, e.g., [15].

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Authors and Affiliations

  1. The University of Western Ontario, London, ON, Canada

    Mark Daley

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  1. Mark Daley
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Correspondence to Mark Daley .

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Editors and Affiliations

  1. Dept. Mathematics & Statistics, University of South Florida, Tampa, Florida, USA

    Nataša Jonoska

  2. Dept. Mathematics & Statistics, University of South Florida, Tampa, Florida, USA

    Masahico Saito

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Daley, M. (2014). An Invitation to the Study of Brain Networks, with Some Statistical Analysis of Thresholding Techniques. In: Jonoska, N., Saito, M. (eds) Discrete and Topological Models in Molecular Biology. Natural Computing Series. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40193-0_5

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  • DOI: https://doi.org/10.1007/978-3-642-40193-0_5

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