Breathers in Two-Dimensional Neural Media

In this Letter we show how nontrivial forms of spatially localized oscillations or breathers can occur in two-dimensional excitable neural media with short-range excitation and long-range inhibition. The basic dynamical mechanism involves a Hopf bifurcation of a stationary pulse solution in the presence of a spatially localized input. Such an input could arise from external stimuli or reflect changes in the excitability of local populations of neurons as a precursor for epileptiform activity. The resulting dynamical instability breaks the underlying radial symmetry of the stationary pulse, leading to the formation of a nonradially symmetric breather. The number of breathing lobes is consistent with the order of the dominant unstable Fourier mode associated with perturbations of the stationary pulse boundary

Figure 1

(a) Plot of the Mexican hat weight function $w$ for $a_e=1$, ${\sigma }_e=1$, $a_i=1.4$, ${\sigma }_i=1.8$. (b) Corresponding pulse existence curves with black (gray) indicating stability (instability) of the stationary pulse solution. The labels S and H indicate saddle-node and Hopf bifurcation points, respectively. Other parameters are $\kappa =0.15$, $\beta =2.25$, $ϵ=0.03$, and $\sigma =5.2.$

Figure 2

Plot of the the functions $D(a)$ (dark curves) and $D_c^n(a)$ (light curves) for different values of $n$ with $\sigma =5.2$; see Fig. 1 for other parameters. The stationary pulse is stable if $D(a)>D_c^n(a)$ for all $n$, with a Hopf bifurcation occurring at the first value of $a$ for which this is no longer true. The largest $D_c^n(a)$ determines the mode that dominates the instability for pulse width $a$; e.g., a value of $\mathcal{I}=0.53$ corresponds to $a=2$, in which case the $n=1$ mode should dominate the instability.

Figure 3

Small perturbations in terms of Fourier modes (light curves) associated with general perturbations of the threshold boundary of a stationary pulse (dark curves).

Figure 4

Breathers for the Mexican hat weight function. Light colors denote suprathreshold values, with the number of lobes corresponding to the dominating unstable Fourier mode.

Figure 5

(a) Strictly expanding or contracting fourfold breather on a rectangular grid. (b) Threefold rotor.