Memristor has been claimed as the passive fourth fundamental circuit element in 1971. About four decades later, a physical realization which presents memristor behavior has resulted in significant interest on memristor and it is continuing to increase at a growing rate. Memristor can provide new possibilities in analog circuit design thanks to its properties which cannot be mimicked by older passive circuit elements. Since no practically available memristor exist on the market yet, obtaining of a practical implementation which behaves like a memristor is important from the point of view real-world circuit design. In this paper, an ultra low-voltage ultra low-power DTMOS-based memristor design is presented. Ultra low-voltage, ultra low-power operational amplifier and ultra low-voltage, ultra low-power multiplier are also designed to use in the implementation. Memristor design is composed of these two type active blocks using CMOS 0.18 A mu m process technology with symmetric +/- 0.25 V supply voltages. Our memristor is used in a second order Sallen-Key band-pass filter topology. Designed memristor-based Sallen-Key band-pass filter is then used for real electroencephalogram data processing. The simulation results show that the design of adjustable memristor memristance/memductance provides tunable filter parameters and without any significant distortion with appropriate parameter set even at ultra-low power levels and at very-low frequencies.