Hopf bifurcation and stochastic spiking in an antiferromagnetic FitzHugh--Nagumo normal form

Abstract

Antiferromagnets offer ultrafast, stray-field-free dynamics that are attractive for neuromorphic spintronic devices. Here we analyze an antiferromagnetic spin-Hall nano-oscillator in the overdamped regime and derive a reduced set of equations for the N\'eel-vector dynamics constrained to the unit sphere. For spin polarization along the easy axis, the model reduces to an asymmetric rotator, for which analytic solutions and the associated spin-pumping signal are obtained in selected limits. We further show that near a suitable operating point, the projected dynamics can be transformed into a local FitzHugh-Nagumo normal form. The resulting mapping identifies the effective fast variable, recovery variable, bias current, and Hopf condition in terms of magnetic material parameters. We finally extend the reduced model to an It\^o stochastic FitzHugh-Nagumo equation driven by spin-pumping input and additive thermal or electronic fluctuations. The stochastic phase portrait shows that the deterministic nullcline geometry organizes noisy spike cycles and produces controlled spike-time variability. These results provide a minimal analytic framework for AFM-based spiking-neuron elements and suggest design criteria for future neuromorphic spintronic devices.