By Thomas Goodwin, University of Technology Sydney
The brain is made of billions of neurons. Neurons transfer tiny electrical signals to one another which describes how the brain is activated when doing different tasks. One approximation to the number of neurons in the brain is 85 billion ! One recent open question in biology is explaining the mechanism of neural structures between neuroscience and behavioural therapy. How does the brain structure change and adapt as we learn something new?
One important insight shown in the late 20th century was neuroplasticity. This describes how the brain changes overtime, even in the adult brain. As we learn something new, neuron connections can strengthen or weaken overtime, and new neural circuits can be formed with the repetition of a movement or behaviour.
Directed graph, i.e. a graph with vertices and edges connecting these vertices with arrows on each edge to give a direction of information flow, we begin to get an idea about the structure and flow of signals in the brain. A Bratteli Diagram is a directed graph with vertices at each level n and edges connected at each level. We say that each vertex represents a neuron and the edges between each vertex is the synaptic connections between each neuron.
Using Measure theory, a mathematical way of measuring the size of sets (distance or volume of strange sets), we can assign probabilities on each edge, we can see how different paths on the Bratteli Diagram are analogous is neural circuits being activated in the brain.
By changing these probabilities over time, we can see how these represents neuroplasticity in the brain. By analysing how this measure of neural paths changes overtime, we discuss how future research can relate this to modern techniques of fMRI (imaging brain activity) and how, with some further work we can simulate brain activity by simulating random walks down Bratteli Diagrams.
 Robert W Williams and Karl Herrup. The control of neuron number. Annual review of neuro-science, 11(1):423–453, 1988.
Thomas Goodwin was a recipient of a 2018/19 AMSI Vacation Research Scholarship.