We describe the random response of a neuron with the random variable \(R\). All the information that we have about the neuron is given by the probability density function \(f(r)\), which for the parametric case would depend on some parameters \(\theta\). For example:

\[f(r) = \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(r-\mu)^2}{2\sigma^2}}\] If we have a sensory neuron, as the response depends on the stimulus, we conditioned the probability density function on the stimulus \(f(r|s)\). For example:

\[f(r) = \frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(r-\beta_0-\beta_1s)^2}{2\sigma^2}}\] Given the stimulus \(s\) and the parameters \(\beta_0\), \(\beta_1\) and \(\sigma\), the response of the neuron is completely specified. The inferential problem is given the pairs of numbers

\[(s_1, r_1), \dots , (s_N, r_N)\] to infer \(\hat{\beta_0}\), \(\hat{\beta_1}\) and \(\hat{\sigma}\). The typical method is MLE.

The regression function

\[r(s)=E[R|S=s]\] is called the tuning function. For the previous example is:

\[r(s)=E[R|S=s] = \int_{-\infty}^{-\infty}rf(r)\,dr = \int_{-\infty}^{-\infty}r\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(r-\beta_0-\beta_1s)^2}{2\sigma^2}}\,dr = \beta_0 + \beta_1s\] This is a linear tuning function.