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Picosecond pulsed electric fields have been shown to have stimulatory effects, such as calcium influx, activation of action potential, and membrane depolarization, on biological cells. Because the pulse duration is so short, it has been hypothesized that the pulses permeate a cell and can directly affect intracellular cell structures by bypassing the shielding of the membrane. This provides an opportunity for studying new biophysics. Furthermore, radiating picosecond pulses can be efficiently done by a compact antenna because the antenna size is comparable to the pulse width. However, all of the previous bioelectric studies regarding picosecond pulses have been conducted in vitro, using electrodes. There is not yet a device which can non-invasively deliver picosecond-pulsed electric fields to neurological tissue for therapeutic applications. It is unclear whether a radiated electric field at a given penetration depth can reach the threshold to cause biological effects. In this dissertation, a picosecond- pulsed electric field system designed for the electrosimulation of neural cells is presented. This begins with the design of an ultra-wideband biconical dielectric rod antenna. It consists of a dielectrically loaded V-conical launcher which feeds a cylindrical waveguide. The waveguide then transitions into a taper, which acts like a lens to focus the energy in the tissue target. To describe the antenna delivery of picosecond pulses to tissues, the initial performance was simulated using a 3-layer tissue model and then a human head model. The final model was shown to effectively deliver pulses of 11.5 V/m to the brain for a 1 V input. The spot size of the stimulation is on the order of 1 cm. The electric field was able to penetrate to a depth of 2 cm, which is equal to the pulse width of a 200 ps pulse. The antenna was constructed and characterized in free space in time domain and in frequency domain. The experimental results have a good agreement with the simulation. The ultimate biological application relies on adequate electric field. To reach a threshold electric field for effective stimulation, the antenna should be driven by a high voltage, picosecond-pulsed power supply, which, in our case, consists of a nanosecond charging transformer, a parallel-plate transmission line, and a picosecond discharging switch. This transformer was used to charge a parallel-plate transmission line, with the antenna as the load. To generate pulses with a rise time of hundreds of picoseconds, an oil switch with a millimeter gap was used. For the charging, a dual resonance pulse transformer was designed and constructed. The novel aspect of this transformer is has a fast charge time. It was shown to be capable of producing over 100 kV voltages in less than 100 ns. After the closing of the peaking switch and the picosecond rise time generation, the antenna was able to create an electric field of 600 V/cm in the air at a distance of 3 cm. This field was comparable to the simulation. Higher voltage operation was met with dielectric breakdown across the insulation layer that separates the high voltage side and the ground side. Before the designed antenna is used in vivo, it is critical to determine the biological effect of picosecond pulses. This is especially important if we focus on stimulatory effects, which require that the electric field intensity be close to the range that the antenna system can deliver. Toward that end, neural stem cells were chosen to study for the proliferation, metabolism, and gene expression. Instead of using the antenna, the electrodes were used to deliver the pulses to the cells. In order to treat enough cells for downstream analyses, the electrodes were mounted on a 3-D printer head, which could be moved freely and could be controlled accurately by programming. The results show that pulses on the order of 20 kV/cm affect the proliferation, metabolism, and gene expression of both neural and mesenchymal stem cells, without reducing viability. In general, we came to the conclusion that picosecond pulses can be a useful stimulus for a variety of applications, but the possibility of using antennas to directly stimulate tissue functions relies on the development of a pulsed power system, high voltage insulation, and antenna material.