Excitation of coherent oscillations in a noisy medium

We numerically study the influence of neuronal threshold modulation on the properties of cortical traveling waves. For that reason we simplify a Wilson-Cowan-type integrodifferential equation model of propagating neocortical activity to a spatially discrete version. Further we introduce a noisy threshold. Depending on the noise level we find different states of the network activity, ranging from asynchronous oscillations, traveling waves, to synchronous oscillations. Finally, we induce the transition between these different states by an external modulation.

Figure 1

(Color online) (a) Traveling wave in a 1D network with periodic boundaries. The wave propagates in both directions. Because of the periodic boundaries the wave fronts meet each other and eliminate themselves. (b) The threshold dependences of the velocity of wave propagation in a 1D network for the Pinto-Ermentrout model (blue/dotted) Eq. (1) for $ϵ=0.2$, compared to the spatial discrete model, Eq. (2), with a nearest neighbor coupling, Eq. (3), for $N=100$ elements with $A=0.4$, $ϵ=0.2$, and $C=0.3$ (red/solid), respectively, $C=0.4$ (green/dashed). The characteristic shape of $c\left(\vartheta \right)$ stays qualitatively the same.

Figure 2

(Color online) The squared fluctuations of the mean field of a 1D ring consisting of 100 units. (a) The simplified model with $ϵ=0.2$, $A=0.4$, $C=0.3$. (b) The Pinto-Ermentrout traveling wave model with $ϵ=0.2$. Both for $\vartheta =0.1$ (red/solid), $\vartheta =0.15$ (green/dashed), and $\vartheta =0.2$ (blue/dotted). The mean-field fluctuations of both show a strong a maximum at $\vartheta \approx D$, indicating a coherence resonance of the collective network dynamics.

Figure 3

(Color online) (a) The phase portrait for an $100*100$ network with $ϵ=0.2$, $A=0.4$, $C=0.3$, and periodic boundary conditions. The color indicates the size of the squared fluctuation: below the red (solid) line traveling waves are possible (traveling waves possible state), and below the blue (dashed) line traveling waves are induced by noise automatically (traveling waves state). (b) The oscillation of the mean field for $\vartheta =0.2$ and $D=0.2$. (c) Noise induces traveling waves for $\vartheta =0.38$ and $D=0.18$.

Figure 4

(Color online) $c\left(\vartheta \right)$ for a $50*50$ network with $A=0.4$, $C=0.3$, $ϵ=0.2$, and periodic boundary conditions. The wave propagation speed and the maximal threshold where wave propagation is possible increase with the noise intensity $D$.

Figure 5

(Color online) (a) The correlation time ${\tau }_c$ for the mean field (red/solid), a single element (green/dashed) in the $50*50$ network $A=0.4$, $C=0.3$, and for a single element (blue/dotted) $A=1.6$, $C=0$ with $\vartheta =0.2$ and $ϵ=0.2$. Note that ${\tau }_c$ is only a measure for temporal coherence, not for the amplitude of the mean-field oscillations. (b) The associated squared fluctuation as a measure for spatial coherence in the network and a measure of activity for a single element.

Figure 6

(Color online) The squared fluctuation over the amplitude of the modulation for a $50*50$ network with $\vartheta =0.3$, $ϵ=0.2$, $A=0.4$, $C=0.3$ and periodic boundary conditions for (a) $D=0.1$ and (b) $D=0.6$. Both compare modulation with a square wave (red/solid), modulation with only the positive part of the square wave (green/dashed), modulation with only the negative part of the square wave (blue/dotted), and constant threshold modulation (purple).