Two-Parameter Bifurcation Structure and Multistability in a Piecewise-Smooth Neural Decision-Making Model

In this work, we perform a bifurcation analysis of a planar piecewise-smooth continuous system with eight parameters, previously introduced to model neural decision-making in social behaviors. While previous studies of related models have relied mainly on numerical simulations, there is a lack of rigorous analytical results describing their global organization in parameter space. We provide a structured two-parameter bifurcation characterization of this system, showing how a suitable parameter selection allows to uncover the global dynamics. We prove that the model is topologically equivalent to a reduced formulation that preserves the qualitative dynamics while simplifying its bifurcation structure. Within this reduced framework, we characterize equilibria and derive analytical conditions for saddle-node and boundary equilibrium bifurcations in the two-parameter [Formula: see text]-plane. Using numerical continuation and targeted search strategies, we compute Hopf, grazing, and homoclinic bifurcation curves. These curves organize the [Formula: see text]-plane into regions with distinct dynamics, revealing in particular small parameter regions where multistability occurs. In particular, the diagram identifies regions where equilibria and limit cycles coexist. The resulting bifurcation diagram provides a unified and rigorous description of the global dynamics, complementing previous numerical studies and offering a consistent interpretation of the multistability observed in the original model.