  We investigate how the activation function can be used to describe neural
firing in an abstract way, and in turn, why it works well in artificial neural
networks. We discuss how a spike in a biological neurone belongs to a
particular universality class of phase transitions in statistical physics. We
then show that the artificial neurone is, mathematically, a mean field model of
biological neural membrane dynamics, which arises from modelling spiking as a
phase transition. This allows us to treat selective neural firing in an
abstract way, and formalise the role of the activation function in perceptron
learning. The resultant statistical physical model allows us to recover the
expressions for some known activation functions as various special cases. Along
with deriving this model and specifying the analogous neural case, we analyse
the phase transition to understand the physics of neural network learning.
Together, it is shown that there is not only a biological meaning, but a
physical justification, for the emergence and performance of typical activation
functions; implications for neural learning and inference are also discussed.
