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1. Repetitive firing in space-clamped squid axons bathed in low Ca and stimulated by a just suprathreshold step of current can be annihilated by a brief depolarizing or hyperpolarizing pulse of the proper magnitude applied at the proper phase. 2. In response to such perturbations, membrane potential and ionic currents show damped oscillations toward a steady state. 3. For other, non-annihilating, perturbations repetitive firing resumes with unaltered frequency but with phase resetting. 4. Experimental findings are compared with calculations for the space-and current-clamped Hodgkin-Huxley equations. Annihilation of repetitive firing to a steady state corresponds to a solution trajectory perturbed off a stable limit cycle and into the domain of attraction of a coexistent stable singular point. 5. Experimentally and theoretically the nerve exhibits hysteresis with two different stable modes of operation for a just suprathreshold range of bias current: the oscillatory repetitive firing state and the time-independent steady state. 6. Analogy is made to a brief synaptic input (excitatory or inhibitory) which may start or stop a biological pace-maker.

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