We consider the direct electrical interfacing of semiconductor chips with individual nerve cells and brain tissue. At first, the structure of the cell-chip contact is studied. Then we characterize the electrical coupling of ion channels--the electrical elements of nerve cells--with transistors and capacitors in silicon chips. On that basis it is possible to implement signal transmission between microelectronics and the microionics of nerve cells in both directions. Simple hybrid neuroelectronic systems are assembled with neuron pairs and with small neuronal networks. Finally, the interfacing with capacitors and transistors is extended to brain tissue cultured on silicon chips. The application of highly integrated silicon chips allows an imaging of neuronal activity with high spatiotemporal resolution. The goal of the work is an integration of neuronal network dynamics with digital electronics on a microscopic level with respect to experiments in brain research, medical prosthetics, and information technology.