Neural circuits display both input-driven activity that is necessary for the real-time control of behavior and internally generated activity that is necessary for memory, planning, and other cognitive processes. A key mediator between these intrinsic and evoked dynamics is arousal, an internal state variable that determines an animal's level of engagement with its environment. In this work, we show that a continuous Hopfield network embellished with a single additional parameter---recurrent gain---captures some essential effects of arousal state at neural and cognitive levels. Using the model's formal connections to the Boltzmann machine and the Ising model, we offer functional interpretations of arousal state rooted in Bayesian inference and statistical physics. Finally, we liken the dynamics of neuromodulator release to an annealing schedule that facilitates adaptive behavior in ever-changing environments. In summary, we present a minimal neural network model of arousal state that exhibits rich but analytically tractable emergent behavior and reveals conceptually clarifying parallels between arousal state and seemingly unrelated phenomena.