  We study the impact of noise on a neural population rate model of up and down
states. Up and down states are typically observed in neuronal networks as a
slow oscillation, where the population switches between high and low firing
rates (Sanchez-Vivez and McCormick, 2000). A neural population model with spike
rate adaptation is used to model such slow oscillations, and the timescale of
adaptation determines the oscillation period. Furthermore, the period depends
non-monotonically on the background tonic input driving the population, having
long periods for very weak and very strong stimuli. Using both linearization
and fast-slow timescale separation methods, we can compute the phase
sensitivity function of the slow oscillation. We find that the phase response
is most strongly impacted by perturbations to the adaptation variable. Phase
sensitivity functions can then be utilized to quantify the impact of noise on
oscillating populations. Noise alters the period of oscillations by speeding up
the rate of transition between the up and down states. When common noise is
presented to two distinct populations, their transitions will eventually become
entrained to one another through stochastic synchrony.
