In this paper, fractional-order memristor (fracmemristor), a novel basic element of brain-like computing, is discussed.The concept of memristor is extended from the conventional integer order to the fractional order, i.e. the fracmemristor. Fracmemristor is a compound word of fractional-order memristor, whose fractional impedance is fracmemristance. Accordingly, it is natural to think of a range of theoretical challenges: What is the relationship of fracmemristor to the conventional fractor and the famous memristor? What are the interpolation properties between the memristor and the capacitor or inductor? Where is the location of fracmemristor in the Chua's circuit periodic table? What are the general expressions for fracmemristances of arbitrary-order ideal capacitive and inductive fracmemristors? What are the measuring unit and physical dimensionality of fracmemristor? What are the fingerprint features for identifying fracmemristor? How to implement arbitrary fractional-order memristor effectively in the form of analog circuit with ordinary memristor, capacitor and inductor? This paper makes preliminary discussions on the above challenging theoretical problems based on abundant prior exploratory findings. The fracmemristor solves the problem that fractor is difficult to realize the function of memory charge or magnetic flux. As a basic circuit element, the fracmemristor can be applied to the design of chaotic system, neural circuit and neural network circuit.
