Metastable states and quasicycles in a stochastic Wilson-Cowan model of neuronal population dynamics

We analyze a stochastic model of neuronal population dynamics with intrinsic noise. In the thermodynamic limit $N\to \infty$, where $N$ determines the size of each population, the dynamics is described by deterministic Wilson-Cowan equations. On the other hand, for finite $N$ the dynamics is described by a master equation that determines the probability of spiking activity within each population. We first consider a single excitatory population that exhibits bistability in the deterministic limit. The steady-state probability distribution of the stochastic network has maxima at points corresponding to the stable fixed points of the deterministic network; the relative weighting of the two maxima depends on the system size. For large but finite $N$, we calculate the exponentially small rate of noise-induced transitions between the resulting metastable states using a Wentzel-Kramers-Brillouin (WKB) approximation and matched asymptotic expansions. We then consider a two-population excitatory or inhibitory network that supports limit cycle oscillations. Using a diffusion approximation, we reduce the dynamics to a neural Langevin equation, and show how the intrinsic noise amplifies subthreshold oscillations (quasicycles).

Figure 1

Bistability in the deterministic network satisfying $\dot{u}=-u+f\left(u\right)$ with $f$ given by the sigmoid [Eq. (2.2)] for $\gamma =4$ and $\theta =1.0$, $f_0=2$. There exist two stable fixed points $u_{\pm }$ separated by an unstable fixed point $u_0$. As the threshold $\theta$ is reduced the network switches to a monostable regime.

Figure 2

Plots of the steady-state distribution $P_S\left(n\right)$ for various thresholds $\theta$ with $\gamma =4.0$ and $f_0=2$. System size is $N=100$.

Figure 3

Plots of steady-state distribution $P_S\left(n\right)$ for various system sizes $N$. Other parameters are $\gamma =4,\theta =0.87,f_0=2$

Figure 4

(Color online) Phase portrait of Hamiltonian equations of motion for ${\Omega }_{+}\left(x\right)=f_0/\left(1+{\text{e}}^{-\gamma \left(x-\theta \right)}\right)$ and ${\Omega }_{-}\left(x\right)=x$ with $\gamma =4$, $\theta =1.0$, and $f_0=2$. The zero energy solutions are shown as thicker curves.

Figure 5

Plot of calculated escape rates $r_{\pm }$ for a bistable neural network as a function of the threshold $\theta$. Other parameters are $\gamma =4.0$, $f_0=2$, and $N=20$. The escape rates are approximately equal for $\theta \approx 0.85$.

Figure 6

Time series showing a single realization of the stochastic process $X\left(t\right)=N\left(t\right)/N$ where $N\left(t\right)$ is the birth-death process corresponding to the neural master equation (2.3). The parameters are $\theta =0.86$, $\gamma =4.0$, $f_0=2$, and $N=20$. The bistable nature of the process is clearly seen.

Figure 7

Same as Fig. 6 except that $\theta =0.83$.

Figure 8

(Color online) Two-population E-I network

Figure 9

Phase diagram of two-population Wilson-Cowan model (4.10) for fixed set of weights $w_{EE}=w_{IE}=w_{EI}=10,,w_{II}=4$. The dots correspond to Takens-Bogdanov bifurcation points

Figure 10

(Color online) Noise-amplification of subthreshold oscillations in an E-I network. (a) Power spectrum of fluctuations (solid curve) obtained by Fourier transforming the Langevin Eq. (4.18). Data points are from stochastic simulations of full master equation with $N=1000$, averaged over 100 samples. (b) Sample trajectories obtained from stochastic simulations of full master equation with $N=100$. Same weight parameter values as Fig. 9 with $h_E=0$ and $h_I=-2$.