Entrainment in up and down states of neural populations: non-smooth and stochastic models.

Slow oscillations in firing rate of neural populations are commonly observed during slow wave sleep. These oscillations are partitioned into up and down states, where the population switches between high and low firing rates (Sanchez-Vives and McCormick in Nat Neurosci 3:1027-1034, 2000). Transitions between up and down states can be synchronized at considerably long ranges (Volgushev et al. in J Neurosci 26:5665-5672, 2006). To explore how external perturbations shape the phase of slow oscillations, we analyze a reduced model of up and down state transitions involving a population neural activity variable and a global adaptation variable. The adaptation variable represents the average of all the slow hyperpolarizing currents received by neurons in a large population. Recurrent connectivity leads to a bistable neural population, where a low firing rate state coexists with a high firing rate state, where persistent activity is maintained via excitatory connections. Adaptation eventually inactivates the high activity state, and the low activity state then persists until adaptation has significantly decayed. We analyze the phase response of the rate model by taking advantage of the separation of timescales between the fast activity and slow adaptation variables. This analysis reveals that perturbations to the neural activity variable have a considerably weaker effect on the oscillation phase than adaptation perturbations. When noise is not incorporated into the rate model, the period of the slow oscillation is determined by the timescale of the slow adaptation variable. In the presence of noise, times at which the population transitions between the low and high activity states become variable. This is because the rise and decay of the adaptation variable is now stochastically-driven, leading to a distribution of transition times. Interestingly, common noise in the adaptation variable can lead to a correlation of two distinct slow oscillating populations. This effect is still significant in the event that each population contains its own local sources of noise. We also show this phenomenon can occur a spiking network. Our results demonstrate the relative contributions of excitatory input and hyperpolarizing current fluctuations on the phase of slow oscillations.