  The dynamics of living systems often serves the purpose of reaching
functionally important target states. We previously proposed a theory to
analyze stochastic biological dynamics evolving towards target states in
reverse time. However, a large class of systems in biology can only be
adequately described using state-dependent noise, which had not been discussed.
For example, in gene regulatory networks, biochemical signaling networks or
neuronal circuits, count fluctuations are the dominant noise component. We
characterize such dynamics as an ensemble of target state aligned (TSA)
trajectories and characterize its temporal evolution in reverse-time by
generalized Fokker-Planck and stochastic differential equations with
multiplicative noise. We establish the classification of boundary conditions
for target state modeling for a wide range of power law dynamics, and derive a
universal low-noise approximation of the final phase of target state
convergence. Our work expands the range of theoretically tractable systems in
biology and enables novel experimental design strategies for systems that
involve target states.
