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.