Abstract
<jats:p>Background and Objectives: Understanding the mechanisms underlying the generation of spike activity in neurons is a central problem of modern theoretical neuroscience. Neurons encode and transmit information through short electrical impulses, and transitions from the resting state to periodic spiking are governed by distinct bifurcation mechanisms. Although several mathematical neuron models have been developed – such as the Hodgkin-Huxley, FitzHugh-Nagumo, Morris-Lecar, and Izhikevich models – most of them reproduce only a limited subset of bifurcation scenarios or are too complex for detailed qualitative analysis. The object of this study is a novel two-dimensional neuron model that, despite its minimal form, reproduces all four classical scenarios of transition from rest to spiking activity known in the theory of dynamical systems. The purpose of this work is to identify and describe these mechanisms analytically and numerically, and to demonstrate the correspondence between the model’s phase-space structures and distinct types of neuronal excitability. Materials and Methods: The model is formulated as a system of two coupled nonlinear differential equations with one fast and one slow variable. Analytical investigation of equilibrium states and their stability was performed by examining the Jacobian matrix. The bifurcation structure was explored using continuation methods and parametric analysis on the (b, k) plane, revealing the boundaries of qualitative transitions between dynamical regimes. Numerical integration of the system and phase-plane visualization were conducted to confirm the theoretical predictions and illustrate the phase portraits, trajectories, and time series corresponding to each scenario. Results: The analysis has revealed four distinct bifurcation mechanisms responsible for the onset of spiking activity: (1) the Andronov-Hopf bifurcation, where a stable equilibrium loses stability and a small-amplitude limit cycle emerges; (2) the saddle-node on invariant circle (SNIC) bifurcation, characterized by the merging of a saddle and a node on a closed trajectory leading to low-frequency oscillations; (3) the homoclinic bifurcation, associated with the reconnection of a saddle separatrix and the generation of large-period oscillations; and (4) the fold bifurcation of two limit cycles, in which a stable and an unstable cycle collide and disappear. Each mechanism corresponds to a specific type of neuronal excitability, determining the threshold and the temporal structure of spike generation. The constructed bifurcation diagram in the parameter plane clearly separates the domains corresponding to steady, periodic, and bistable dynamics. Conclusion: The proposed minimal neuron model successfully unifies four key bifurcation scenarios within a single framework, combining analytical tractability with rich dynamical behavior. Such universality makes it a convenient tool for studying transitions between quiescent and oscillatory activity, as well as for modeling hybrid networks of neurons with diverse excitability types. The results contribute to the theoretical understanding of neuronal dynamics and can serve as a foundation for the development of reduced models of biological neurons and for the analysis of collective activity in neural ensembles.</jats:p>