Reduction of colored noise in excitable systems to white noise and dynamic boundary conditions

In this study we derive an analytical expression for the transfer function of the leaky integrateand-fire (LIF) neuron model exposed to synaptic filtering. To this end we first develop a general framework that reduces a first order stochastic differential equation driven by fast colored noise to an effective system driven by white noise. A perturbation expansion in the ratio of the time scale of the noise to the time scale of the system, combined with boundary layer theory reduces the twodimensional Fokker-Planck equation to a one-dimensional effective system with dynamic boundary conditions. Finally, we solve the effective system of the LIF neuron model using analogies to the quantum harmonic oscillator.