Figure 2. Multi-stable dynamics and anisotropic information routing.
(a) Two identical and symmetrically coupled neuronal circuits of Wilson–Cowan-type (dark and light green, modular sub-network in Fig. 1e). The noise free (that is, input free) network displays two different stable oscillatory dynamical states α and β. (b) Phase difference Δφ1,2(t):=φ1(t)−φ2(t) between the extracted phases of the two neuronal populations is fluctuating around a locked value 
of a the stable collective state α of the deterministic system (orange); a strong external perturbation (purple arrow) induces a switch to stochastic dynamics around the second stable deterministic reference state β (brown) with phase difference 
. (c) Delayed mutual information dMI1,2 and (d) transfer entropy dTE1→2 curves between the phase signals in states α (orange) and β (brown) for numerical data (dots) and theory (lines) as a function of the time delay d between the stochastic phase signals φ1(t) and φ2(t+d). The change in peak latencies form 
to 
in the dMI1,2 curves and the asymmetry of the dTE1→2 curves show anisotropic information routing for the two different states. Switching between the two dynamical states reverses the effective information routing pattern (IRP) (graphs, bottom). (e) Phase coupling function γ(Δφ)=γ1,2(Δφ)=γ2,1(Δφ) (blue) between the two neuronal oscillators and its anti-symmetric part 
(red). The two zeros of 
with negative slope indicate the stable deterministic equilibrium phase differences 
and 
of the dynamical states α and β, receptively. The directionality in the IRP arises due to symmetry breaking in the dynamics reflected in the different slopes of γ(Δφ) (dashed lines): In the state α, oscillator 1 receives inputs from oscillator 2 proportional to 
, while oscillator 2 is coupled to 1 proportional to 
in linear small-noise approximation (cf. equation (5)). As 
is large deviations from the phase-locked state of oscillator 2 due to the noise inputs are strongly propagated to oscillator 1 to restore the phase-locking. Information injected to oscillator 2 is thus transmitted to oscillator 1. In contrast, inputs to oscillator 1 only weakly impact oscillator 2 as 
is small. In total, the information is thus dominantly routed from 2 to 1. Switching to the dynamical state β reverses the roles of the oscillators and thus also the directionality of the information routing motive.