Fig. 4.

Bifurcations in the 2-unit model. Each plot shows the parameter sets $(\mathrm{\Delta }\hat{b}={\hat{b}}_1-{\hat{b}}_2,q)$ leading to each type of qualitative dynamics described in Sect. 4.2. The 2-unit model shows hysteresis (Type III dynamics) when $q<(w^0-1)+g(-|\mathrm{\Delta }\hat{b}|)$, fully represents the location with the stronger input while suppressing the response to the weaker input (Type I or Type II dynamics) when $(w^0-1)+g(-|\mathrm{\Delta }\hat{b}|)<q<(w^0-1)+g(|\mathrm{\Delta }\hat{b}|)$, and linearly combines the two embedded activity patterns (Type IV dynamics) when $q>(w^0-1)+g(|\mathrm{\Delta }\hat{b}|)$. These bifurcations are shown by the solid black lines. The points specify the type of dynamics found in numerical simulations of the 2-unit model, where light blue (left region) indicates Type I, dark blue (right region) indicates Type II, dark red (bottom region) indicates Type III, and light red (top region) indicates Type IV. In all cases, the numerical simulations agree with the analytical predictions given by Eqs. (14)–(16). The initial state is set to the desired activity pattern such that the active unit is the unit driven by the weaker input. We classified the dynamics as Type I or Type II when the only active unit in the equilibrium state is the unit receiving the stronger input, as Type III when the initially active unit remains the only active unit in the equilibrium state, and as Type IV when both units are active in the equilibrium state. (a) The parameters of the 2-unit model approximate the reduced parameters from the megamap model (Eq. (7)), as used in Fig. 3(b) and (d). The four regions predict the response of the corresponding megamap as q and Δb̂ vary. (b) and (c) Bifurcations given a smaller reduced inhibitory weight. Reducing ${\hat{w}}^{\mathrm{I}}$ reduces the range of permissible values for q, shrinking the relative size of the parameter space with Type IV dynamics compared to that with Type III dynamics. The transition between operational modes ($q=0.2$) is not affected by ${\hat{w}}^{\mathrm{I}}$. (d) Bifurcations given a smaller inhibitory threshold, which makes the nonlinearity in $g(x)$ more apparent. The full ranges of permissible q and Δb̂ are shown for each plot