# The Role of Sensory Cues in Collective Dynamics: A Study of Three-Dimensional Vicsek Models
## 3 Results  
We conduct numerical simulations to examine how combined visual and auditory cues influence group-level behavior, comparing these effects to those of purely visual and auditory cues. In the composite model, each particle has two distinct sensing cones: one for the field of vision and the other for the acoustic beam. In this study, we ensure that the sensing cones for three different modalities are geometrically similar when comparing group behaviors across these modalities. This approach enables us to attribute any observed differences in group behavior to the nature of the sensing mechanisms themselves rather than to variations in the size of the sensing regions. In other words, for any given simulation, the values of the sensing radius \(R\) and the sensing angle \(\phi\) for auditory sensing are matched with those for visual sensing, and thus for the composite sensing, which combines auditory and visual sensing regions. Note that although the sensing cones for auditory and visual modalities are geometrically identical, the method of defining neighbors in each modality differs, leading to distinct sets of neighbors for each particle. In our simulations, we set the side length of the cubic domain to \(L=5\), the constant particle speed to \(v_{0}=0.03\), and the radius of the sensing cones to \(R=0.5\), so the arena is relatively large compared to the interaction region (\(L=10R\)). The initial positions and heading vectors of the particles are randomly assigned within the cubic domain and the unit sphere, respectively, using uniform distributions. Sample simulation snapshots for auditory, visual, and composite sensing modalities at time step \(t=1500\), with parameters \(\rho=10\), \(\eta=0.2\) and \(\phi=6\pi/15\), are presented in Figure 2.  
We observe distinct group-level behavior emerge as a result of the differences in sensing modalities used. For example, the auditory mode in Figure 2a and the composite mode in Figure 2c lead to higher polarization, while the visual mode in Figure 2b leads to higher cohesion. To further explore these differences, we examine the evolution of group alignment over time. Considering the presence of random noise in the system, we perform a Monte Carlo simulation for 1000 time-steps with 20 repetitions and \(\rho=2\). Keeping the initial conditions fixed, we vary five specific pairs of parameters with increasing noise levels and different sensing angles as \((\eta,\phi):(0.1,2\pi/15)\), \((0.1,5\pi/15)\), \((0.3,5\pi/15)\), \((0.7,5\pi/15)\), and \((0.8,6\pi/15)\). The results are presented in Figure 3. Each sub-figure shows the polarization as a function of time for the auditory, visual, and composite modes represented as red, green, and blue lines, respectively. The dotted lines show the results of each iteration, while the solid lines represent the averages computed over the 20 iterations at each time step. The results demonstrate the dependence of the group alignment on the noise intensity and the sensing region. We observe that composite sensing enables the group to achieve higher polarization (alignment) compared to auditory and visual modalities, specifically when there is low noise (Figure 3a-c). Moreover, when comparing the sub-figures (Figure 3a,b), it is observed that the polarization increases with an increasing sensing region (larger \(\phi\)) for all three modalities with noise intensity kept constant at 0.1. Furthermore, with the use of composite sensing, this high polarization is achieved at a fast rate, within 600 simulation time steps (Figure 3b). In addition to the sensing region, the group alignment is also influenced by noise intensity. For example, comparing Figure 3b,c, we observe that increasing noise intensity from \(\eta=0.1\) to \(\eta=0.3\) (keeping the sensing region fixed at \(\phi=5\pi/15\)) results in a decrease in polarization; however, composite sensing still maintains a higher polarization compared to the auditory and visual modes.