Memristor for Neuromorphic Applications: Models and Circuit Implementations
Since the first experimental evidence for the existence of the memristor in nature, a large number of memristor mathematical models have been proposed in the literature. Among them the generalized Boundary Condition Memristor model sticks out for the adap
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Memristor for Neuromorphic Applications: Models and Circuit Implementations Alon Ascoli, Fernando Corinto, Marco Gilli, and Ronald Tetzlaff
13.1 Introduction The current-controlled ideal memristor is a passive bipole linking charge q(t) and flux ϕ (t) through a nonlinear relation, i.e. ϕ (t) = ϕ (q(t)). From application of Faraday’s Law and of the chain rule it follows that voltage v(t) depends upon current i(t) through v(t) =
d ϕ (t) = M(q(t)) i(t), dt
(13.1)
where M(q) = d ϕdq(q) is the memristance (i.e. memory-resistance) of the bipole. Since t t i(t )dt , then M(q) = M( −∞ i(t )dt ). In other words the resistance of q(t) = −∞ the memristor depends upon the time history of the current flowed through it. This explains the memory capability of the memristor, theoretically envisioned by Chua in 1971 [1] and later classified by Chua and Kang in 1976 as the simplest element from a large class of nonlinear dynamical systems endowed with memristance, the so-called memristive systems [2]. In [2] a memristive system (or memristor system1 ) is a nonlinear dynamical circuit element defined by the following differential-algebraic system of equations: 1 In
the following memristive systems are referred to as memristor systems, whereas the term ideal memristor is used for systems described by (13.1).
A. Ascoli () • R. Tetzlaff Technische Universitat ¨ Dresden, Mommsenstraße 12, 01062 Dresden, Germany e-mail: alon.ascoli@tu-dresden; ronald.tetzlaff@tu-dresden F. Corinto • M. Gilli Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy e-mail: [email protected]; [email protected] R. Tetzlaff (ed.), Memristors and Memristive Systems, DOI 10.1007/978-1-4614-9068-5__13, © Springer Science+Business Media New York 2014
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dx(t) = f(x(t), u(t)), dt y(t) = g(x(t), u(t))u(t),
(13.2) (13.3)
where2 x ∈ Rn is the state, u ∈ R refers to the input, y ∈ R describes the output, f(x, u) : Rn × R → Rn stands for the state evolution function, while g(x, u) : Rn × R → R denotes the memductance (memristance) if input u is in voltage (current) form. Since 2008, when its existence at the nano-scale was certified at Hewlett-Packard (HP) Labs [3], the memristor has attracted a strong interest from both industry and academia for its central role in the setup of novel integrated circuit (IC) architectures, especially in the design of high-density nonvolatile memories [4], programmable analog circuitry [5], neuromorphic systems [6], and logic gates [7,8]. The development of innovative strategies for the design of memristor-based electronic systems requires the availability of mathematical models [3, 9–14, to name but a few] for the memristor nano-structures under study. A good model should be as general as possible, i.e. it should be able to capture the memristor dynamics of a large number of nano-films. In this respect the Boundary Condition Memristor (BCM) model, recently introduced in [12], was developed so as to meet this generality requirement. In fact the distinctive feature
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