  We examine the dynamical evolution of the state of a neurone, with particular
care to the non-equilibrium nature of the forces influencing its movement in
state space. We combine non-equilibrium statistical mechanics and dynamical
systems theory to characterise the nature of the neural resting state, and its
relationship to firing. The stereotypical shape of the action potential arises
from this model, as well as bursting dynamics, and the non-equilibrium phase
transition from resting to spiking. Geometric properties of the system are
discussed, such as the birth and shape of the neural limit cycle, which provide
a complementary understanding of these dynamics. This provides a multiscale
model of the neural cell, from molecules to spikes, and explains various
phenomena in a unified manner. Some more general notions for damped
oscillators, birth-death processes, and stationary non-equilibrium systems are
included.
