Fig. 2

A simulation of spatially periodic non-travelling patterns in a two-dimensional spiking neural field model with an \(I_{\mathrm{h}}\) current, solved on a spatial grid of 1000 × 1000 points. Displayed is the voltage component across the entire network at \(t=7000\mbox{ ms}\) (left) and \(t=10\mbox{,}000\mbox{ ms}\) (right). The model supports periodic patterns of localised activity. Note that these patterns are not static, but oscillate in time. Parameters: \(C = 1\mbox{ }\upmu \mbox{Fcm}^{-2}\), \(\tau_{\text{h}}=400\mbox{ ms}\), \(V_{\text{h}}=40 \mbox{ mV}\), \(g_{l} = 0.25\mbox{ mS/cm}^{-2}\), \(g_{h} = 1\mbox{ mS/cm}^{-2}\), \(\tau_{\mathrm{R}} = 200\mbox{ ms}\), \(V_{\text{th}} = 14.5\mbox{ mV}\), \(V_{\text{r}} = 0\mbox{ mV}\), \(V_{1/2} = -10\mbox{ mV}\), \(k=10\), \(g_{\text{syn}} = 15\mbox{ mS/cm}^{-2}\), \(w_{0} = -10\), \(\sigma= 25\), \(\beta ^{-1}=0\), and \(\alpha^{-1} = 20\mbox{ ms}\). The choice of a long refractory time-scale in the model is useful for eliciting a single (rather than multiple) spike rebound response. Spatial domain \(\Omega= [-L,L] \times[-L,L]\) where \(L=10 \sigma\). See also the video in Additional file 2, showing the emergence of more exotic spatio-temporal structures, including hexagons