1st Year PhD Student in Bioengineering (joint program with UCSF).
My research is on Computational Neuroscience; I use Spectral Graph Theory to understand the relationship between the brain's structural and functional connectivity. I use a lot of linear algebra day-to-day, but I've never yet had a formal course, so I'm excited to rebuild my foundation to support my research. I have essentially zero experience writing formal proofs, so I'm a little anxious about proof heavy questions…
So far I'm very happy with the conceptual approach to the class. I let MATLAB do my computing :)
TIL that diagonalizing a matrix is just a projection into a basis space that spans the matrix.
Future HW goes here…
Using Magnetic Resonance Imaging (MRI) we can take many types of images of the brain. Some of these images give structural information, for example where/how neuron bundles travel between regions, and others give functional information, which is represented by the brain using oxygen in certain brain regions (causing magnetic distortions detectable in MRI). We can construct a network (aka graph) of the brain in two ways: structurally, we can perform tractography to trace 'streamlines' through the brain as a proxy for the density of neural fiber bundles connecting regions (graph nodes are regions, edges are number of streamlines); functionally, we can assess the Blood Oxygen Level Dependent (BOLD) signal as a measure of function in each region, then we say regions are 'connected' if their signals are highly correlated (graph nodes are regions, edges are correlation coefficients). These graphs are represented as a matrix where rows/columns are the nodes(i,j,…) and the entries are the connections from e.g. i to j.
We aim to understand how a diffusing signal on the structural connectivity (SC) graph creates the functional connectivity(FC). For this, we use Spectral Graph Theory, which uses the Laplacian operator on the SC, which describes how browning diffusion spreads around the network space. The Laplacian matrix has special constraints regarding its spectrum (i.e. it's eigenvalues) known as Cheeger's Inequality. The eigenvectors of the Laplacian matrix solve an otherwise NP-hard problem in graph theory known as the Min-Cut/Max-Flow problem, and they also represent a fundamental harmonic series on the graph. Our work finds that the SC Laplacian's spectra are fundamentally related to FC, and we fit differential equations with a few parameters that modify the Laplacian eigenmodes to estimate FC.
Why do we care about this problem? By studying how the brain's function emerges, we can understand characteristics about that emergence, and of special note, compare those characteristics between healthy vs diseased populations. One of my specific interests is relating one of our parameters (representing network global-coupling) to the brain's information content (entropy).