Noise-induced transitions in slow wave neuronal dynamics

Abstract

Many neuronal systems exhibit slow random alternations and sudden switches in activity states. Models with noisy relaxation dynamics (oscillatory, excitable or bistable) account for these temporal, slow wave, patterns and the fluctuations within states. The noise-induced transitions in a relaxation dynamics are analogous to escape by a particle in a slowly changing double-well potential. In this formalism, we obtain semi-analytically the first and second order statistical properties: the distributions of the slow process at the transitions and the temporal correlations of successive switching events. We find that the temporal correlations can be used to help distinguish among biophysical mechanisms for the slow negative feedback, such as divisive or subtractive. We develop our results in the context of models for cellular pacemaker neurons; they also apply to mean-field models for spontaneously active networks with slow wave dynamics.

Appendix

1.1 Modification of the duration

In Section 3.2, the durations of states are computed under the assumption that the system moves along the V-nullcline. However, as seen in Fig. 3(b), the trajectory during each state is widely spread along the V-nullcline due to noise. The deviation of the trajectory from the V-nullcline affects the speed of the slow variable in Eq. (3c) and thus, may lead to the discrepancy between the analysis and the simulation (Fig. 3(d)).

The statistics of this deviation can be obtained analytically by the stationary distribution of V for each h. The distribution of V at each h is the stationary solution for the Fokker-Planck equation (Gardiner 1985). When the system has a potential function, U(V,h), then the stationary solution is given as follows:

$$ {p_s}\left( {V,h} \right) = C\exp \left( {{{ - 2U\left( {V,h} \right)} \mathord{\left/{\vphantom {{ - 2U\left( {V,h} \right)} {\sigma {\prime^2}}}} \right.} {\sigma {\prime^2}}}} \right) $$

(10a)

In a double-well potential, we can calculate the conditional probability and the mean given that the system stays in the left well by normalizing Eq. (10a). Here, V _sep (h) is the separatrix between the left and right wells for given h.

$$ {p_{cond}}\left( {V,h} \right) = \frac{{{p_s}\left( {V,h} \right)}}{{\int_{ - \infty }^{{V_{sep}}(h)} {{p_s}\left( {V\prime, h} \right)dV\prime } }} $$

(10b)

$$ \mu (h) = \int_{ - \infty }^{{V_{sep}}(h)} {{p_{cond}}\left( {V\prime, h} \right)V\prime dV\prime } $$

(10c)

With the modified V obtained from Eq. (10c), we can compute the duration. Figure 8 shows the mean deviation of the trajectory from the V-nullcline and comparison of the durations computed using the V-values on the V-nullcline and the modified V. The distributions of h at the entry and leaving points are obtained from the simulations so that the discrepancy between the simulation and analysis does not originate from the transition PDFs. With the modified V, there is a modest change in the distribution of durations. It agrees better with the simulation even though the distribution with the modified V still doesn’t agree with the simulation perfectly. The mean of the deviation is not enough to correct the distribution. However, we can see the influence of the deviation of V on the duration in Fig. 8(b).

Fig. 8

Deviation of a trajectory from the slow manifold and its effect on the duration of the state. (a) Mean V of the noisy trajectory (dotted line) and the V-nullcline in the SP (solid line) are plotted in the V-h phase plane. Due to noise, the mean of the trajectory deviates from the V-nullcline. (b) Distribution of SP durations obtained from a simulation and an analysis. In an analysis, the distributions of starting and leaving locations in the SP are given as the transition PDFs in h obtained from the simulation. The solid line is computed with the V values on the V-nullcline and the dotted line is computed using the mean of the deviated V