  Diffusion processes have been applied with great success to model the
dynamics of large populations throughout science, in particular biology. One
advantage is that they bridge two different scales: the microscopic and the
macroscopic one. Diffusion is a mathematical idealisation, however: it assumes
vanishingly small state changes at the microscopic level. In real biological
systems this is often not the case. The differential Chapman-Kolmogorov
equation is more appropriate to model population dynamics that is not well
described by drift and diffusion alone.
  Here, the method of characteristics is used to transform deterministic
dynamics away and find a coordinate frame where this equation reduces to a
Master equation. There is no longer a drift term, and solution methods there
are insensitive to density gradients, making the method suitable for jump
processes with arbitrary jump sizes. Moreover, its solution is universal: it no
longer depends explicitly on the deterministic system.
  We demonstrate the technique on simple models of neuronal populations.
Surprisingly, it is suitable for fast neural dynamics, even though in the new
coordinate frame state space may expand rapidly towards infinity. We
demonstrate universality: the method is applicable to any one dimensional
neural model and show this on populations of leaky- and
quadratic-integrate-and-fire neurons. In the diffusive limit, excellent
approximations of Fokker-Planck equations are achieved. Nothing in the approach
is particular to neuroscience, and the neural models are simple enough to serve
as an example of dynamical systems that demonstrates the method.
