Can anybody think of a real(ish) world example of a problem that can be solved by a single neuron neural network? I'm trying to think of a trivial example to help introduce the concepts.
Using a single neuron to classification is basically logistic regression, as Gordon pointed out.
Logistic regression is the appropriate regression analysis to conduct when the dependent variable is dichotomous (binary). Like all regression analyses, the logistic regression is a predictive analysis. Logistic regression is used to describe data and to explain the relationship between one dependent binary variable and one or more metric (interval or ratio scale) independent variables. (statisticssolutions)
This is a good case to apply logistic regression: Suppose that we are interested in the factors that influence whether a political candidate wins an election. The outcome (response) variable is binary (0/1); win or lose. The predictor variables of interest are the amount of money spent on the campaign, the amount of time spent campaigning negatively and whether or not the candidate is an incumbent. (ats)