The electrical characteristics of many neurons are remarkably robust in the face of changing internal and external conditions. At the same time, neurons can be highly sensitive to neuromodulators. We find correlates of this dual robustness and sensitivity in a global analysis of the structure of a conductance-based model neuron. We vary the maximal conductance parameters of the model neuron and, for each set of parameters tested, characterize the activity pattern generated by the cell as silent, tonically firing, or bursting. Within the parameter space of the five maximal conductances of the model, we find directions, representing concerted changes in multiple conductances, along which the basic pattern of neural activity does not change. In other directions, relatively small concurrent changes in a few conductances can induce transitions between these activity patterns. The global structure of the conductance-space maps implies that neuromodulators that alter a sensitive set of conductances will have powerful, and possibly state-dependent, effects. Other modulators that may have no direct impact on the activity of the neuron may nevertheless change the effects of such direct modulators via this state dependence. Some of the results and predictions arising from the model studies are replicated and verified in recordings of stomatogastric ganglion neurons using the dynamic clamp.