From Propagator to Oscillator: The Dual Role of Symmetric Differential Equations in Neural Systems

In our previous work, we proposed a novel neuron model based on symmetric differential equations and demonstrated its potential as an efficient signal propagator. Building upon that foundation, the present study delves deeper into the intrinsic dynamics and functional diversity of this model. By systematically exploring the parameter space and employing a range of mathematical analysis tools, we theoretically reveal the system 's core property of functional duality. Specifically, the model exhibits two distinct trajectory behaviors: one is asymptotically stable, corresponding to a reliable signal propagator; the other is Lyapunov stable, characterized by sustained self-excited oscillations, functioning as a signal generator. To enable effective monitoring and prediction of system states during simulations, we introduce a novel intermediate-state metric termed on-road energy. Simulation results confirm that transitions between the two functional modes can be induced through parameter adjustments or modifications to the connection structure. Moreover, we show that oscillations can be effectively suppressed by introducing external signals. These findings draw a compelling parallel to the dual roles of biological neurons in both information transmission and rhythm generation, thereby establishing a solid theoretical basis and a clear functional roadmap for the broader application of this model in neuromorphic engineering.