Stochastic synchronization of neural activity waves

We demonstrate that waves in distinct layers of a neuronal network can become phase locked by common spatiotemporal noise. This phenomenon is studied for stationary bumps, traveling waves, and breathers. A weak noise expansion is used to derive an effective equation for the position of the wave in each layer, yielding a stochastic differential equation with multiplicative noise. Stability of the synchronous state is characterized by a Lyapunov exponent, which we can compute analytically from the reduced system. Our results extend previous work on limit-cycle oscillators, showing common noise can synchronize waves in a broad class of models.

Figure 1

Common noise-induced phase locking of bumps evolving in two uncoupled stochastic neural fields, Eq. (1) with $w\left(x\right)=cos\left(x\right)$, and $C\left(x\right)=cos\left(x\right)$. (a) In a single realization, noise eventually drives the bump positions ${\Delta }_1\left(t\right)$ (solid line) and ${\Delta }_2\left(t\right)$ (dashed line) to the absorbing state ${\Delta }_1\left(t\right)={\Delta }_2\left(t\right)$. Here, $f\left(u\right)=H(u-0.5)$ and $\varepsilon =0.01$. (b) Realizations of the log phase difference $ln\left|\varphi \right|=ln|{\Delta }_1-{\Delta }_2|$ (thin lines) compared to theory $ln\left|\varphi \right(t\left)\right|=ln\left|\varphi \right(0\left)\right|+\lambda t$ (thick line) given by Eq. (14). (c) Numerically calculated Lyapunov exponents $\lambda$ (circles) are well approximated by Eq. (14) (line), increasing in amplitude $\left|\lambda \right|$ as $\theta$ moves toward the saddle-node bifurcation of bumps [21].

Figure 2

Independent noise prevents complete phase locking (see the Appendix). (a) Noise correlations drive the bumps close to one another [${\Delta }_1\left(t\right)\approx {\Delta }_2\left(t\right)$], but independent noise prevents their remaining in the phase-locked state. (b) Stationary density $M_0\left(\varphi \right)$ widens for $\chi <1$ in Eq. (16) theory (line) and numerics (circles). Parameters are $\theta =0.5,\varepsilon =0.01,\chi =0.95$, and $C_j\left(x\right)=cos\left(x\right)$ $\left(j=1,2,c\right)$.

Figure 3

Noise-induced phase locking of traveling waves evolving in Eq. (1) with $w\left(x\right)=cos(x-\phi )$. (a) Noise perturbations drive wave phases to the phase-locked state ${\Delta }_1\left(t\right)={\Delta }_2\left(t\right)$. Here $\phi =0.2$, while other parameters are as in Fig. 1. (b) The Lyapunov exponent $\lambda$ calculated from numerical simulations (circles) is approximated by Eq. (17) (line), increasing in amplitude $\left|\lambda \right|$ with the skewness $\phi$ of the weight function [21].

Figure 4

Noise-driven synchronization of breather phases in a pair of adapting stochastic neural fields, Eq. (18) with $w\left(x\right)=cos\left(x\right)$ and $I\left(x\right)=I_{0}cos\left(x\right)$. (a) Noise drives breather phases to the absorbing state where ${\vartheta }_1\left(t\right)={\vartheta }_2\left(t\right)$ (see text); $I_0=0.1$. (b) Numerically computed Lyapunov exponent $\lambda$ describing the stability of phase-locked breathers increases in amplitude $\left|\lambda \right|$ with stimulus strength $I_0$. Parameters are $\alpha =0.1,\beta =0.2,\varepsilon =0.04$. Other parameters are as in Fig. 1.