  Filtered shot noise processes have proven to be very effective in modelling
the evolution of systems exposed to stochastic shot noise sources, and have
been applied to a wide variety of fields ranging from electronics through
biology. In particular, they can model the membrane potential Vm of neurons
driven by stochastic input, where these filtered processes are able to capture
the non-stationary characteristics of Vm fluctuations in response to
pre-synaptic input with variable rate. In this paper, we apply the general
framework of Poisson Point Processes transformations to analyse these systems
in the general case of variable input rate. We obtain exact analytic
expressions, and very accurate approximations, for the joint cumulants of
filtered shot noise processes with multiplicative noise. These general results
are then applied to a model of neuronal membranes subject to conductance shot
noise with continuously variable rate of pre-synaptic spikes. We propose very
effective approximations for the time evolution of Vm distribution and simple
method to estimate the pre-synaptic rate from a small number of Vm traces. This
work opens the perspective of obtaining analytic access to important
statistical properties of conductance-based neuronal models such as the the
first passage time.
