Dear students,
welcome to today's lecture! Today we will discuss how to analyze large networks of neurons. We consider not only fully connected networks where every neuron is connected to everybody else but also randomly connected network. A particularly interesting random connectivity is the one where each neuron receives input from K (randomly selected) other neurons.
The method we use for the network analysis is called mean-field theory. It is at the heart of network analysis and has been in the background (without naming it) during the last two weeks when we analyzed the Hopfield model. In fact, we can write down overlap equations for the Hopfield model, because the network structure is such that mean-field methods work.
Importantly in the current context of the COVID epidemics, the mean-field methods today are also those that are used to analyze epidemics in networks of people. So if you want to know how to go from interacting people to the spread of one big wave of infections, watch carefully the lecture today.
The lectures today are limited to homogeneous networks. We analyze a single population of neurons that are all somehow equivalent. But neither the brain nor human populations are completely homogeneous. However, in the next weeks we will also extend the approach to structured networks. In the context of the brain, we will be interested in how cognitive effects can arise from structured networks of neurons in our brains.
I hope you will enjoy the lectures today. Have a good start into the week!
Wulfram Gerstner
