Mathematical Methods for Neural Science
In this coursework you will analyse data from the following paper :
C.-H. Yu, J. N. Stirman, Y. Yu, R. Hira and S.L. Smith, Diesel2p mesoscope with dual independent scan engines for
flexible capture of dynamics in distributed neural circuitry, Nature Communications 12:6639, 2021.
We suggest that you familiarise yourself with the paper to understand where the data comes from,
although there is no need to worry too much about the details of the optical technology described
therein. The paper describes two-photon calcium imaging from the cortex of a mouse, with a novel
instrument that allows collection of data from a large number of neurons simultaneously – thousands
of neurons across a 5x5mm2 region of the mouse cortical surface.
1. Load the calcium imaging dataset provided in the Coursework 2 directory. Note: there are two
channels which contain the left and right halves of the image neurons, and in the skeleton code I
show loading both of them and merging them together; however I strongly recommend you just
use one of these for your analysis. (a) Plot the physical locations of the cells in a 2D
representation. (b) Plot the DF/F fluorescence time series for all cells as a matrix (Cells x Time)
with colourscale indicating signal amplitude. (c) For 5 typical cells in which there are obvious
calcium transients, plot the fluorescence time series in a separate plot with the 5 traces overlaid
on a shared x axis but vertically offset.
3. Calculate the matrix of Pearson correlation coefficients between the calcium fluorescence time
series from each cell. Now plot the distribution of correlation coefficients.
4. Form an undirected graph, with nodes representing each neuron, and edges a binary variable
which is 1 if the correlation coefficient between the time series exceeds the xth percentile of the
distribution of correlation coefficients between different neurons (i.e. you are choosing the
threshold to have (100 – x)% sparsity of connections). Choose x to be sufficiently high (i.e. the
network sparsely connected) such that you can handle the number of edges on your computer,
but sufficiently low to allow you to study the properties of the network. For example, if you
were using the 2167 neurons in channel 1, a threshold at 98% would yield 93,000 edges (2167 *
2167 * 0.02). Do not allow self-edges. (a) Plot the adjacency matrix for the graph as an image,
with white indicating no edge and black indicating an edge. (b) Also plot it using another graph
visualisation method of your choice.
5. What is the average degree 𝑘 for the network described by the above adjacency matrix? What is
the average clustering coefficient? What is the global clustering coefficient? Are these high
compared to what you might expect for a random network? What is the network diameter D?
(hint: is the network fully connected?) Would you describe this network as “Small World”?
6. Plot the degree distribution for the network. Would you describe the network as “Scale-free”? Compare it to the degree distribution predicted by an Erdös-Renyi network of the same mean degree.
咨询 Alpha 小助手，获取更多课业帮助