Using methods from nonlinear dynamics and interpolation techniques from applied mathematics, we show how to use data alone to construct discrete time dynamical rules that forecast observed neuron properties. These data may come from simulations of a Hodgkin-Huxley (HH) neuron model or from laboratory current clamp experiments. In each case, the reduced-dimension, data-driven forecasting (DDF) models are shown to predict accurately for times after the training period. When the available observations for neuron preparations are, for example, membrane voltage V(t) only, we use the technique of time delay embedding from nonlinear dynamics to generate an appropriate space in which the full dynamics can be realized. The DDF constructions are reduced-dimension models relative to HH models as they are built on and forecast only observables such as V(t). They do not require detailed specification of ion channels, their gating variables, and the many parameters that accompany an HH model for laboratory measurements, yet all of this important information is encoded in the DDF model. As the DDF models use and forecast only voltage data, they can be used in building networks with biophysical connections. Both gap junction connections and ligand gated synaptic connections among neurons involve presynaptic voltages and induce postsynaptic voltage response. Biophysically based DDF neuron models can replace other reduced-dimension neuron models, say, of the integrate-and-fire type, in developing and analyzing large networks of neurons. When one does have detailed HH model neurons for network components, a reduced-dimension DDF realization of the HH voltage dynamics may be used in network computations to achieve computational efficiency and the exploration of larger biological networks.