### Bifurcation analysis of traditional rivalry with fixed inputs

Previously, three main types of dynamical behaviors were found in many models of rivalry with fixed inputs (including the Wilson model): Winner-take-all (WTA), Rivalry oscillations (RIV), Simultaneous activity (SIM). The bifurcation diagram with fixed inhibition strength \(g=1.5\), and varying adaptation strength *h*, is presented in Fig. 3(A). A trivial symmetric equilibrium always exists, which is stable for large values of adaptation strength and corresponds to simultaneous activity (SIM) (Fig. 3(C)). This equilibrium loses its stability with decreasing adaptation strength and a stable limit cycle emerges from a supercritical Hopf bifurcation (H). These relaxation oscillations correspond to rivalry (RIV). For *h* increasing from zero on the symmetric unstable branch, a pair of unstable fixed point branches emerge at a pitchfork bifurcation (PF) with the \(E_{1}=E_{2}\) symmetry broken. The complementary branches undergo fold bifurcations (L) nearby but remain unstable. These two unstable fixed point branches go through two supercritical Hopf bifurcations and become stable. These two stable equilibria create a bistable parameter range known as winner-take-all (WTA), which exists for small *h*. The qualitative transformation of the system from RIV to WTA remains unclear and must involve as yet undetermined bifurcations at changes in stability on the RIV branch (Fig. 3(B), green curve).

In this study, a more complete numerical bifurcation analysis has been carried out to find other stable solutions in the small parameter gap between the WTA and RIV (Fig. 3(B)). As seen in Fig. 4(A), a pair of stable limit cycles emerge from two supercritical Hopf bifurcations (on upper and lower WTA branches) as adaptation strength is increased. These low-amplitude oscillations move around the top-most and bottom-most of the five equilibrium branches existing between PF and L. We call this dynamical behavior low-amplitude-winner-take-all (LAWTA) since there is bistability like WTA behavior; however, the stable states are oscillatory solutions with very small amplitude around asymmetric unstable equilibria (Fig. 4(A)). By further increasing adaptation strength, a cascade of period-doubling bifurcations emerges from the LAWTA branch (Fig. 4(C)). For example, one period of the original limit cycle emerging from a supercritical Hopf bifurcation looks like a sinusoidal signal with one peak and one trough. After the first period-doubling bifurcation (PD), one period of oscillations will have two peaks and two troughs with tiny differences between the peak and trough amplitudes. As seen in Fig. 5, even after three PD bifurcations, the difference in amplitude between the 8 peaks remains small.

On the other side, the limit cycle emerging from the supercritical Hopf bifurcation (green branch labeled RIV in Fig. 3(A–B)) becomes unstable as adaptation strength decreases; however, the continuation software does not detect any bifurcation where the stability changes. This appears to be a global bifurcation which is not detectable by local analysis. We will refer to this putative global bifurcation at the change in stability at the RIV branch as the global bifurcation. Without any specific bifurcation in hand, we cannot follow any emerging stable branch. To tackle this issue we compute the stable periodic solution (assuming one exists, using numerical integration) for a specific value of adaptation strength (in a range between last PD and the change in stability on the RIV branch) and then follow any periodic solution branch using numerical continuation. On the right side of the global bifurcation, time simulations show relaxation oscillations that reach high amplitude rapidly and before relaxing to baseline (Fig. 6(B)). However, on the left side of the change in stability (close to the global bifurcation), we observe that in addition to high amplitude oscillations, one low amplitude oscillation appears (Fig. 6(C)). By further decreasing adaptation strength, there is always one high amplitude oscillation, but the number of low amplitude oscillations increases (Fig. 6(C–F)). The period of these oscillations increase sharply as we move toward a critical parameter value \(h\approx 4.22843\) (Fig. 6(A)). This complex behavior is known as mixed-mode oscillations (MMOs) since it is a mixture of low and high amplitude oscillations. Using the approach described above we could compute stable branches on the left side of global bifurcation (Fig. 4(B)). Interestingly, stable solutions occur through a series of discrete branches. The discontinuous transitions from one branch segment to the next are similar to the spike-adding mechanism from [30]. The bifurcation structure here appears similar to the canard-induced MMOs identified in a spiking neuron model [31]. The sharp increase in amplitude of the limit cycle branch over a short parameter range emerging near the critical value of *h* (Fig. 6 caption) suggests the complex behavior in this region is also associated with canards. Bifurcation analysis of another simple rivalry model with 4 ODEs and different nonlinearity revealed a similar structure for MMOs through the interaction of canards and a singular Hopf point [32]. Whilst, there are some similarities with the MMOs found in the present study, a more rigorous approach would be needed to say whether these MMOs are canard-induced, Hopf-induced, or result from an interaction of both mechanisms. The global bifurcation and the transitions between MMO branches remain to be determined. The behaviors reported here are confined to a narrow region of the \((g,h)\) parameter plane near the left-hand locus of Hopf bifurcation (Fig. 4(D)).

Here, a more complete analysis has revealed MMOs emerge from high amplitude RIV oscillations (Fig. 4(B)) and a cascade of period-doubling bifurcations emerge from LAWTA oscillations (Fig. 4(C)) which have not been reported before in the Wilson model. This analysis describes the mechanism of state transition from WTA to RIV which was not clear before. Whilst MMOs have been reported in another model of rivalry [32, 33], the appearance of a stable PD cascade has revealed richer dynamics in the Wilson model which may also play a part in the mechanisms that lead to appearance and disappearance of limit cycles associated with MMOs. An interesting avenue of investigation will be to understand how the low-amplitude PD cascade (Fig. 4(C)) interacts with periodic forcing. This provides the context to fully understand the periodically forced case.

### Bifurcation analysis of binocular rivalry with periodic forcing

#### Flicker (18 Hz) only

Bifurcation analysis with the flickering stimulus shows that periodic forcing with high frequency (e.g. 18 Hz) modulates the three main types of behaviors that occur with fixed inputs. Instead of WTA and SIM fixed point branches in traditional rivalry with a fixed stimulus (Fig. 3(A)), modulated WTA (WTA-Mod) and modulated SIM (SIM-Mod) periodic solution branches are found with the flickering stimulus (Fig. 7(A)). Subsequently, the SIM-Mod branch undergoes a torus bifurcation (T) giving rise to a torus branch with aperiodic oscillations corresponding to modulated slow rivalry alternations (RIV-Mod). Following the torus bifurcation in the \((g,h)\) parameter plane defines the boundary of rivalry oscillations (Fig. 7(B)). The locus of a pitchfork bifurcation (BP) remains close to the left-hand torus curve (Fig. 7(B), not shown). It appears that the MMO branches and PD cascade identified for fixed inputs disappear with the introduction of flicker. This analysis found evidence that slow rivalry alternations RIV-Mod can exist at parameter values adjacent to regions where stimulus-induced oscillations exist (SIM-Mod). Time histories of these dynamical behaviors are shown in (Fig. 7(C)). These results are consistent with experiments: Flicker stimuli do not differ from the traditional rivalry case [13].

#### Swap (1.5 Hz) only

The dynamical behavior with low frequency periodic forcing (around 1.5 Hz, so-called swap) is different, and in addition to WTA-Mod behavior (for small values of adaptation strength) and SIM-Mod (for large values of adaptation strength), cycle skipping occurs through a PD bifurcation on the SIM-Mod branch (Fig. 8(A)). Cycle skipping refers to the response of each competing population to every other stimulus onset. This means two populations respond in turn to stimulus cycles and while one has high activity during a cycle the other one stays inactive (Fig. 9(A)). We note that the period of the period-doubled solution is 1.333 s and is plausibly the result of a resonance with the fixed-input limit-cycle, which has a period >2 s, for larger values of *h* [11]. SIM-Mod and cycle skipping have been reported in a simpler model of rivalry [34] and via direct simulations of the Wilson model in [25].

As seen in (Fig. 8(B–C)), the cycle skipping branch loses stability at a fold bifurcation (L) for decreasing *h*. For *h* increasing, branching off from the PD point on the WTA-mod branch leads to a cascade of PD bifurcations (Fig. 8(D)). In order to find the possible stable periodic branches between the last PD and the fold bifurcation, we compute the stable periodic solution (assuming it exists) using numerical integration for a specific value of adaptation strength, and then start continuation from this solution. With this approach, we found a family of discontinuous branches, which correspond to multi-cycle skipping (Fig. 8(C)). On these branches, the number of stimulus cycles between switches of activity from one population to the other is variable and increases by one as the bifurcation parameter decreases (Fig. 9(A–E)). Here we found a small region of bistability between the one-cycle skipping and two-cycle skipping behaviors (at around \(h=0.05 \) in Fig. 8(C)). Therefore, in the full hierarchical model it would be possible to find an asymmetric solution where the HL-VR units at the first stage behave differently. However, as explained in the discussion the model normally operates close to the SIM-Mod region, which is far away in parameter space from the cycle skipping region with bistability.

Another interesting behavior is the appearance of a chaotic attractor in a parameter range between the PD cascade and multi-cycle skipping family branches (Fig. 8(C)). Figure 9(F–G) represents chaotic firing activity for each population in a 200 s simulation. The number of cycles between switches does not show any regular or repeating pattern.

#### Flicker (18 Hz) & (1.5 Hz) Swap

The bifurcation structure with both high frequency flicker and low frequency swap appears to be analogous to bifurcation structure with swap only case (Fig. 10(A–B)). However, the right-hand PD bifurcation point in the transition from cycle skipping to SIM-Mod moves down in adaptation strength. This is shown in a direct comparison of two-parameter bifurcation diagrams for the F&S and swap only cases in (Fig. 11(A)). In fact, with the same values of adaptation strength that we might expect cycle skipping from swap only case, with F&S stimulus we can get SIM-mod, which turns out to be critical for obtaining slow alternations, see discussion.

#### Blanks (150 ms) & (1.5 Hz) Swap

Our result shows that the effect of blank insertion before swap times, similar to the effect of adding flicker to swap stimuli, moves the PD bifurcation point down the bifurcation parameter *h* (Fig. 11(B)), but to a much lesser extent than the introduction of flicker; compare Fig. 11(A) and (B).