Abstract
From a pedagogical point of view, the memristor is defined in this tutorial as any 2-terminal device obeying a state-dependent Ohm’s law. This tutorial also shows that from an experimental point of view, the memristor can be defined as any 2-terminal device that exhibits the fingerprints of “pinched” hysteresis loops in the v–i plane. It also shows that memristors endowed with a continuum of equilibrium states can be used as non-volatile analog memories. This tutorial shows that memristors span a much broader vista of complex phenomena and potential applications in many fields, including neurobiology. In particular, this tutorial presents toy memristors that can mimic the classic habituation and LTP learning phenomena. It also shows that sodium and potassium ion-channel memristors are the key to generating the action potential in the Hodgkin-Huxley equations, and that they are the key to resolving several unresolved anomalies associated with the Hodgkin-Huxley equations. This tutorial ends with an amazing new result derived from the new principle of local activity, which uncovers a minuscule life-enabling Goldilocks zone, dubbed the edge of chaos, where complex phenomena, including creativity and intelligence, may emerge. From an information processing perspective, this tutorial shows that synapses are locally-passive memristors, and that neurons are made of locally-active memristors.
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Notes
- 1.
For a generalized memristor defined by v=R(x,i)i, or i=G(x,v)v, the v–i loci tends to a single-valued nonlinear function as ω→∞ [13].
- 2.
We will use the term generalized memristor to refer to the most general case defined in [10], where the memristance (resp., memductance) R(x,i,t) (resp., G(x,v,t)) may depend also on the input current i (resp., voltage v) and the time t.
- 3.
Here we assume “zero” initial state, i.e. φ(0)=0. Hence \(\varphi(t)=\int_{-\infty}^{t}v(\tau )d\tau =\int_{0}^{t}v(\tau )d\tau\).
- 4.
The original Bliss-Lomo experiment was carried out at the perforant path synapses on the dentate gyrus.
- 5.
Hodgkin and Huxley were awarded the 1961 Nobel Prize in Physiology for this seminal contribution.
- 6.
We caution the readers that these pinched hysteresis loops are different from those shown in Fig. 4 of [4], which were calculated with E K =0, and E Na =0, respectively, in the state equations given in Figs. 11b and 11c. We take this opportunity to alert the readers of [21] that the pinched hysteresis loops in Figs. 11, 12, 17 and 18 are calculated with E K =0 and E Na =0.
- 7.
For certain ideal 2-terminal circuit elements [23, 24], only a DC voltage source (resp., current source) restricted to a limited range of terminal voltages (resp., currents) is admissible. For example, for an ideal diode [23, 24], only a non-positive voltage source (resp., non-negative current source) is allowed (by the definition v=0, i≥0, and i=0, v≤0 of an ideal diode) to be connected across the ideal diode, in order to avoid the pathological situation where the circuit does not have a solution!
- 8.
Our reference voltage polarity for V and current direction for I follow Hodgkin-Huxley’s 1952 paper [20], which are opposite to the prevailing reference convention. The corresponding DC V–I curve in conventional reference polarity and direction is obtained by rotating the V–I curve in Fig. 15 by 180∘ degrees.
- 9.
In view of the parallel network topology of Fig. 16b, it is more convenient to analyze the admittance \(Y(s)\triangleq\frac{1}{Z(s)}\) instead of the impedance Z(s). The independent small-signal variable in this case is δv(t).
- 10.
We take this opportunity to alert the readers of [22] of consistently repeated errata in Figs. 14–26, and Fig. 35, where the unit of ω should be rad/ms, and not kHz. This error also occurs in the text on pp. 26–30.
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Chua, L. (2014). Memristor, Hodgkin-Huxley, and Edge of Chaos. In: Adamatzky, A., Chua, L. (eds) Memristor Networks. Springer, Cham. https://doi.org/10.1007/978-3-319-02630-5_5
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