Gating-signal propagation by a feed-forward neural motif

We study the signal propagation in a feed-forward motif consisting of three bistable neurons: Two input neurons receive input signals and the third output neuron generates the output. We find that a weak input signal can be propagated from the input neurons to the output neuron without amplitude attenuation. We further reveal that the initial states of the input neurons and the coupling strength act as signal gates and determine whether the propagation is enhanced or not. We also investigate the effect of the input signal frequency on enhanced signal propagation.

Figure 1

Signal propagation by system (1) at frequency $\Omega =\pi /5$. The response $Q_3$ [Eq. (3)] is shown versus coupling strength $\lambda$. Blue stars denote the numerical values of $Q_3$ obtained for the case when the states of the input neurons 1 and 2 are different, i.e., either $(L,R)$ or $(R,L)$. Green stars show $Q_3$ when the input neurons are in the same state $(R,R)$ or $(L,L)$. Dotted red and black lines denote the corresponding theoretical approximations of $Q_3$ (see text for details). Dashed line shows the value of $Q_{1,2}$ obtained numerically as well as analytically using Eq. (7).

Figure 2

Bifurcation diagram for system (1) with $I=0$. The maximal value of $x_3$ is shown versus $\lambda$. Red (respectively green) lines correspond to the case when the input neurons are both in the same steady state $R$ ($L$ respectively). Blue and black lines correspond to the case when the input neurons are in the different states $R$ and $L$. Solid-color lines denote stable and dashes unstable fixed points. Black line denotes periodic solution, which emerges in Hopf bifurcations $H_1$ and $H_2$. Inset shows the parameter values ($\lambda ,\epsilon$) between two Hopf bifurcations, where stable oscillations exist.

Figure 3

(a) Trajectories of the output neuron of Eq. (1) at $\lambda =0.16$ with $A=0.01$ (green) and $A=0$ (blue) in the phase space. The red dashed lines with arrows represent a simplified trajectory. Dotted lines are the nullclines. (b) Frequency ${\Omega }_3^0$ of the autonomous oscillations for $A=0$ versus coupling strength $\lambda$. The dotted red line denotes an analytic approximation (see text for details) and blue stars denote numerical values. (c) Frequency ${\Omega }_3^A$ of the oscillations for $A\ne 0$ versus coupling strength $\lambda$. Resonance (frequency locking) with ${\Omega }_3^A\approx \Omega$ is observed.

Figure 4

Time series of the original $x_3^A\left(t\right)$ (solid) and rescaled $x_3^0\left(\zeta t\right)$ (dashed): (a) $\lambda =0.11$, (b) $\lambda =0.14$, (c) $\lambda =0.16$, and (d) $\lambda =0.22$.

Figure 5

The response $Q_3$ versus coupling strength $\lambda$ for three different signal frequencies: $\Omega =0.3$, 2, and 20, respectively. Inset: the critical value of $A_T$, at which the input neurons start spiking, versus input frequency $\Omega$.