Memristive Excitable Automata: Structural Dynamics, Phenomenology, Localizations and Conductive Pathways
The memristor (a passive resistor with memory) is a device whose resistance changes depending on the polarity and magnitude of a voltage applied to the device’s terminals and the duration of this voltage’s application. Its existence was theoretically postulated by Leon Chua in 1971 based on symmetry in integral variations of Ohm’s laws (Chua in IEEE Trans. Circuit Theory 18:507–519, 1971; Chua and Kang in Proc. IEEE 64:209–223, 1976; Chua in IEEE Trans. Circuits Syst. 27:1014–1044, 1980). The memristor is characterised by a non-linear relationship between the charge and the flux; this relationship can be generalised to any two-terminal device in which resistance depends on the internal state of the system (Chua and Kang in Proc. IEEE 64:209–223, 1976). The memristor cannot be implemented using the three other passive circuit elements—resistor, capacitor and inductor—therefore the memristor is an atomic element of electronic circuitry (Chua in IEEE Trans. Circuit Theory 18:507–519, 1971; Chua and Kang in Proc. IEEE 64:209–223, 1976; Chua in IEEE Trans. Circuits Syst. 27:1014–1044, 1980). Using memristors one can achieve circuit functionalities that it is not possible to establish with resistors, capacitors and inductors, therefore the memristor is of great pragmatic usefulness. The first experimental prototypes of memristors are reported in Williams (IEEE Spectrum 2008-12-18, 2008), Erokhin and Fontana (arXiv:0807.0333v1, 2008), and Yang et al. (Nature Nano 3(7), 2008). Potential unique applications of memristors are in spintronic devices, ultra-dense information storage, neuromorphic circuits, and programmable electronics (Strukov et al. in Nature 453:80–83, 2008).
Despite explosive growth of results in memristor studies there is still a few (if any) findings on phenomenology of spatially extended non-linear media with hundreds of thousands of locally connected memristors. We attempt to fill the gap and develop a minimalistic model of a discrete memristive medium. Structurally-dynamic (also called topological) cellular automata (Ilachinsky and Halpern in Complex Syst. 1:503–527, 1987; Halpern and Caltagirone in Complex Syst. 4:623–651, 1990) seem to be an ideal substrate to imitate discrete memristive medium. A cellular automaton is structurally-dynamic when links between cells can be removed and reinstated depending on states of cells these links connect. Structurally-dynamic automata are now proven tools to simulate physical and chemical discrete spaces (Rosé et al. in Physica A 206:421–440, 1994; Hasslacher and Meyer in Int. J. Mod. Phys. C 9:1597–1605, 1998; Hillman in Combinatorial spacetimes. Ph.D. dissertation, 1998; Requardt in J. Math. Phys. 44:5588, 2003; Alonso-Sanz in Chaos Solitons Fractals 32:1285–1295, 2006) and graph-rewriting media (Tomita et al. In: Understanding complex systems, pp. 291–309, 2009); see overview in Ilachinsky (Structurally dynamic cellular automata. In: Encyclopedia of complexity and systems science, vol. 19, pp. 8815–8850, 2009).
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