Implanted antennas for implant-to-air data communications must be composed of material compatible with biological tissues. We design single and dual-polarization antennas for wireless ultra-wideband neural recording systems using an inhomogeneous multi-layer model of the human head. Antennas made from flexible materials are more easily adapted to implantation; we investigate both flexible and rigid materials and examine performance trade-offs. The proposed antennas are designed to operate in a frequency range of 2-11 GHz (having S11 below -10 dB) covering both the 2.45 GHz (ISM) band and the 3.1-10.6 GHz UWB band. Measurements confirm simulation results showing flexible antennas have little performance degradation due to bending effects (in terms of impedance matching). Our miniaturized flexible antennas are 12 mm×12 mm and 10 mm×9 mm for single- and dual-polarizations, respectively. Finally, a comparison is made of four implantable antennas covering the 2-11 GHz range: 1) rigid, single polarization, 2) rigid, dual polarization, 3) flexible, single polarization and 4) flexible, dual polarization. In all cases a rigid antenna is used outside the body, with an appropriate polarization. Several advantages were confirmed for dual polarization antennas: 1) smaller size, 2) lower sensitivity to angular misalignments, and 3) higher fidelity.