Introduction

Despite rapid developments in the field since the demonstration by the Hewlett-Packard Lab of a nanoscale titanium dioxide memristor in 2008, there were numerous reports of inexplicable nonlinear dynamical phenomena. In a widely cited paper1, co-authored by Johanes Georg Bednorz, a recipient of the 1987 Nobel Prize in Physics, the authors reported a SrZrO3 memristor that can be switched from a high-impedance state to a low-impedance state by applying a negative voltage pulse of 2 ms. By applying a positive voltage pulse of 2 ms, the information written to the device is erased and the high-impedance state is recovered. They then demonstrated that faster switching speed is possible by applying higher voltage amplitudes. But none of the co-authors, including Bednorz1, could explain the phenomenon. This is just one among many baffling memristor device phenomena2. It took nine years to explain the fascinating switching time phenomenon in titanium dioxide memristor devices3, but the explanation might not be valid for memristors made from other materials. The generalization of the switching time phenomenon to arbitrary materials was achieved only in 2018 by exploiting the newly discovered dynamic route map (DRM)2 to resolve Enigmas 2 and 3 in Table 1. Most of these unexplained memristor phenomena have been solved in a recent work2, which will be covered in the ‘Non-volatile memristors’ section.

Table 1 Five enigmas in non-volatile memristors resolved via dynamic route maps
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Neuromorphic complex architecture of the human brain, requires both volatile and non-volatile memristors. Moreover, neuromorphic computing requires memristors that are not only volatile but are also endowed with a special physical property dubbed the ‘edge of chaos’. This property enables entropy to decrease over time, which is the long-sought complement of the second law of thermodynamics. This remarkable tool will be covered in depth in the ‘Volatile memristors’ section, particularly focusing on resolving four heretofore unsolved fundamental problems, namely the 243-year-old Galvani irritability, the 72-year-old Hodgkin–Huxley spike mystery, the 72-year-old Turing instability and the 50-year-old Smale paradox.

These four historic problems have recently been solved by applying the ‘principle of local activity’4. So powerful is this principle that it can be used to solve a whole gamut of deep unsolved problems exalted at the altar of ‘complexity’. Indeed, when asked about his vision for science in the next century, Stephen Hawking predicted that it would be the ‘century of complexity’. However, there is no universally accepted definition of complexity. In fact, there is a large community of scholars, scientists and deep thinkers, including Schrödinger, Prigogine, Gell-Mann, Turing and Smale, who have conducted deep research on complexity. The term complexity also encompasses various phenomena studied across disciplines from physics to sociology, often described by terms such as ‘emergence’, ‘self-organization’, ‘synergetics’, ‘non-equilibrium phenomena’, ‘collective behaviours’, ‘slaving principle’, ‘dissipative structure’, ‘cooperative phenomenon’, ‘autopoietic phenomenon’, ‘endogenous phenomenon’, ‘negentropy’, ‘instability of the homogeneous’ and ‘amplification of fluctuations’. Researchers from the complexity community tend to use one jargon synonymously with another. One notable example can be found on page 99 of the much-cited book The Quark and the Jaguar by Nobel laureate Murray Gell-Mann, where, in the last paragraph, he cites the word ‘self-organized’ three times, and ‘emergence’ twice, to mean the same thing, namely, ‘complexity’, which also appears in the same paragraph, an apt illustration of the art of tautology.

Fortunately, unlike the above shortcomings, we have discovered a unique scientific definition of complexity5,6 in our research: namely, we define the complexity of a device, or system, as the emergence of a heterogeneous pattern in a medium of identical cells7,8. Not only do we have a definition for all electronic devices modelled by basic electrical circuit elements and characterized by one or more ordinary differential equations, but we can also apply the ‘edge-of-chaos’ criterion, which is the crown jewel of local activity, to calculate, via an explicit formula, using only matrix algebra, whether the device or system is resting on the edge of chaos. In particular, we have solved the 243-year-old Galvani enigma2 and the 72-year-old Hodgkin–Huxley spike mystery2,9.

In this case, the edge-of-chaos domain is a Goldilocks zone bounded by −9.77003 μA < Iext < −7.8293 μA, where the width of the edge-of-chaos domain, ΔIext = 1.94072 μA, is less than 2 μA: a tour de force in scientific prediction. Likewise, we have solved the 72-year-old Turing instability10 and the 50-year-old Smale paradox11.

In this Review, we delve into the edge of chaos, which is the missing principle that many luminaries, including Boltzmann, Schrödinger, Prigogine, Gell-Mann, Hodgkin and Huxley, Turing and Smale, had sought in vain. We discuss its application in solving the four classic unsolved problems and its pivotal role in the design of brain-like machines for neurocomputing. Finally, we end this Review with discussion of a garden variety of complex neuromorphic phenomena exhibited by a homemade ‘Chua corsage memristor circuit’. The US$10 Chua corsage memristor demonstrates that subnanoscale-cum-subfemtowatt computers can mimic the human brain. Concurrently, we envisage that memristors on the edge of chaos will enable distributed information technology via edge computing and Internet of Things devices (see Supplementary Information section 1 for details).

A glimpse at the local activity principle

As mentioned earlier, studies on complexity will be the main part of twenty-first-century science5,6,7. The intuitive idea is that global patterns and structures emerge from locally interacting elements, such as atoms in laser beams, molecules in chemical reactions, proteins in cells, cells in organs, neurons in brains, and agents in markets, by ‘self-organization’12. Despite extensive reports across disciplines such as biology, chemistry, ecology, physics, sociology and economy, the underlying mechanisms of self-organization remain debated. Historically, theories such as Schrödinger’s ‘order from disorder’13, Prigogine’s ‘dissipative structure’14, Haken’s ‘synergetics’15, and Packard16 and Langton’s ‘edge of chaos’17 have attempted to explain these phenomena. However, these explanations are often based on examples or metaphors only. We argue for a mathematically precise and rigorous definition of local activity4 as the cause of self-organizing complexity which can be tested in an explicit and constructive manner18.

Boltzmann’s struggle in understanding the physical principles distinguishing between living and non-living matter19, Schrödinger’s negative entropy in metabolism13, Turing’s basis of morphogenesis20, Prigogine’s intuition of the instability of the homogeneous14 and Haken’s synergetics15 are in fact all direct manifestations of a fundamental principle of local activity. It can be considered as the complement of the second law of thermodynamics, explaining the emergence of order from disorder instead of disorder from order, in a computable way, at least for reaction–diffusion systems.

The principle of local activity is precisely the missing concept to explain the emergence of complex patterns in a homogeneous medium. Leon Chua articulated this principle in the theory of nonlinear electronics circuits in a mathematically rigorous way18. The local activity principle can be generalized and proven for the class of nonlinear reaction–diffusion systems in physics, chemistry, biology and brain research. The principle of local activity is the cause of symmetry breaking in homogeneous media12.

A glimpse at the edge of chaos

The phrase ‘edge of chaos’ was first used by Norman Packard16 and Chris Langton17 in the context of cellular automata, originally proposed by John von Neumann21 in the early 1950s, as a class of spatially and temporally discrete deterministic mathematical systems characterized by local interaction and inherently parallel form of evolution22. Von Neumann’s brainchild has since been applied and extended by Stanislaw Ulam23 in the context of cellular automata, by John Conway24 in his immensely popular Game of Life, by Stuart Kauffman25 in his search for the laws of self-organization and complexity, by Stephen Wolfram26 in complex pattern evolutions and by Leon Chua27 in his nonlinear dynamics perspective of cellular automata.

In all cases cited above, the edge of chaos is a discrete dynamical system invoked as a jargon for anecdotes and exemplars without a formal definition, let alone an explicit algorithm to identify its parameter domain of existence. In sharp contrast, in the context of this Review, ‘edge of chaos’ is concerned with a system of nonlinear differential equations whose precise parameter domain can be calculated by an explicit formula4,11,28.

The edge-of-chaos domain within a system’s parameter space is where complex phenomena germinate. In the case of the Turing instability and Smale paradox, a hidden excitability allows a cell to be destabilized when interacting with dissipative environments. Although a diffusion process tends to equalize differences, an originally dead or inactive cell can become alive, or active, on coupling with other cells by diffusion29. This phenomenon seems to be counterintuitive but can be mathematically and rigorously proven and confirmed in different applications in reality7,11.

In the parameter spaces of reaction–diffusion systems, the domains of local activity can be visualized by computer simulations. The edges of chaos are domains, typically very small regions, with the ability of creation. For example, in the classic Hodgkin–Huxley axon circuit model, the edge-of-chaos domains of the brain can be determined with spectacular accuracy, namely, less than 1 mV and 2 μA (ref. 28). In the brain, tiny domains are the origin of action potentials and attractors, which are correlated with emerging mental and intellectual abilities. Thus, the edge-of-chaos domains seem to be hidden in the domains of local activity like pearls in shells on the sea floor. In research, the edge of chaos has often been used as a metaphor but not as a mathematically precise concept. Its discovery in the parameter spaces of dynamical systems is completely different in complexity research18. The local activity principle and its pearl, the edge of chaos, are couched in rigorous mathematics. Above all, they are characterized by constructive procedures to compute and visualize their complexity. Therefore, researchers from other disciplines who can describe their dynamical systems via differential equations, such as reaction–diffusion equations, can actually easily calculate, with a computer, the parameter values where complexity and creativity can occur.

The principle of local activity and edge of chaos are fundamental tools of science. In some applications of physics, chemistry, biology, brain research and technology, the domains of local activity and edge of chaos are computed, visualized and exploited for high-tech and scientific applications. For example, in quantum cosmology, the edge of chaos can be identified as symmetry breaking of local gauge symmetries, generating the complexity of matter and forces in our Universe. Supramolecular chemistry and nanoscience are also examples of twenty-first-century science exploring self-organizing complex molecular systems.

The second law of thermodynamics rules out the possibility that complex phenomena may ever appear in isolated physical media. The physics-based principle of the edge of chaos4,5,6,7, establishing the conditions under which a non-isolated system might exhibit complex dynamics, can be interpreted as the extension of the second law of thermodynamics to physical media that might exchange energy with the external environment. In fact, for a system to undergo complex phenomena, it must be constantly supplied with an external supply of energy, to sit on some ‘bias’ equilibrium point (Q), about which it might acquire the capability to amplify infinitesimal fluctuations in energy6. This way, the system enters an operating regime known as local activity4,6. However, for complex phenomena to appear across its physical medium, the operating point Q of the locally active system must additionally be asymptotically stable. Under these circumstances, the system is said to be poised on the edge of chaos. In such conditions, the system hides a high degree of excitability behind an apparently quiet steady state. Tiny changes to the environmental conditions might then be sufficient to induce the destabilization of the asymptotically stable operating point of the system, which would then exhibit some complex dynamical phenomenon, such as limit cycle oscillations, or chaos30,31,32. To mention an example of complexity par excellence, when a biological neuron enters the edge-of-chaos operating regime28, it is endowed with the capability to generate an ‘all-or-none’ spike, also known as an action potential9, which, enabling synaptic adaptation, underlies the development of intelligence in the brain.

Furthermore, through an in-depth analysis of a simple cellular neural network7, leveraging the capability of solid-state volatile threshold switches8,33,34,35,36,37 to exhibit, similar to the sodium and potassium ion channels in a biological neuron38, a negative differential resistance39, which is a signature for local activity, it has recently been demonstrated that the edge of chaos lies behind the emergence of Turing instabilities10 in reaction–diffusion systems, providing an answer to an open question that troubled the beautiful mind of the father of artificial intelligence (AI), Alan Turing20. Last but not least, choosing a bio-inspired circuit, based upon a locally active memristor from NaMLab40, as the object of the investigations11, this physics principle has also been invoked to resolve the Smale paradox29, explaining why two identical biological cells, sitting in a silent state on their own, may come alive, pulsing together, when allowed to interact by means of a diffusion process, which would be typically expected to equalize the concentrations of chemical species of the two cells rather than to induce sustained oscillations41 across the homogeneous cellular medium. The Russian luminary Ilya Prigogine coined a phrase to refer to symmetry-breaking phenomena in homogeneous cellular media, namely the ‘instability of the homogeneous’14,42. So far, to the best of our knowledge, no cellular network has ever been reported to undergo dissipation-induced higher-order dynamics, such as a route to chaos or an uncontrollable upsurge in the respective physical variables43. After a brief discussion on the pillars of local activity and edge-of-chaos theory, we show a sixth-order array in which two identical circuit-based reaction cells, silent on their own, are found to support higher-order complex phenomena of this kind, when allowed to interact by means of a diffusion process, which, in general, would be expected to equalize the cells’ dynamical states in pairs. Powerful methods from nonlinear dynamics30,31,32 and local activity theory4 are used to carry out a comprehensive investigation of the nonlinear dynamics of the circuit, to identify all the possible regions of the cell parameter space, within which the cell itself is poised on the edge of chaos, which is the sine qua non14,42 for Prigogine’s instability of the homogeneous. For some of these parameter sets, endowing the cell with a high degree of excitability hidden behind a seemingly quiet state, numerical simulations reveal emergence of high-order dynamical phenomena.

All in all, Nature obeys physics laws. One of these, the edge of chaos, lies behind the appearance of complexity in a non-isolated physical medium. The robust foundations of the local activity theorem on page 671 of lecture 10 of the Chua Lectures6 enable us to determine under which conditions on the respective physical parameters a medium of this kind may support complex phenomena, which is a crucial preliminary step for design and control applications (see Supplementary Section 2 for details).

Multiple local fading memories

Fading memory44 is the capability of a physical system to approach a unique asymptotic behaviour, irrespective of the initial conditions, when stimulated by an input from a certain class. Standard stimuli from the a.c. periodic class typically induce fading memory effects in non-volatile memristors, as uncovered back in 201645,46,47. Since spring 2024, a deep investigation of resistance switching phenomena in a TaOx resistive random-access memory cell revealed the ability of the non-volatile nanodevice to exhibit one of two possible oscillatory behaviours, depending on the initial condition, when subject to a particular periodic excitation. This interesting finding was, however, left unexplained. Bistability is the simplest form of local fading memory. In a system endowed with local fading memory under a given stimulus, the initial condition does not affect the long-term behaviour of the state as long as it is drawn from the basin of attraction of either of the distinct coexisting state-space attractors (two limit cycles for the periodically forced memristor acting as a bistable oscillator). Here the history of the system, encoded in the initial condition, is thus erasable only locally through ad hoc stimulation. Motivated by the discovery of local history-erasing effects in our resistive random-access memory cell, our research applies a powerful systematic theoretic tool (enabling analysis of the response of first-order systems to square-pulse-train periodic stimuli, and known as time-average state dynamic route2) to an accurate physics-based mathematical model, earlier fitted to the nanodevice, to determine a strategy for specifying the parameters of an excitation signal consisting of a sequence of two square pulses of opposite polarity per period, so as to induce various forms of monostability, or multistability, in the non-volatile memristor48,49,50,51. In particular, as an absolute novelty in the literature, experimental measurements validate the theoretical prediction of the capability of the device to operate as one of two distinct oscillators, depending on the initial condition, under a specific pulse-train excitation signal. The coexistence of multiple oscillatory operating modes in the periodically forced resistive random-access memory cell, an example par excellence of their unique nonlinear dynamics, might inspire the development and circuit implementation of new sensing and mem-computing paradigms.

Non-volatile memristors

There are two kinds of memristors: volatile or non-volatile. A non-volatile memristor is a two-terminal device whose resistance depends on the magnitude and polarity of the voltage applied to it and the length of time that voltage has been applied. When you turn off the voltage, the memristor remembers its most recent resistance until the next time you turn it on, whether that happens a day later or a year later.

The five enigmas

In the 1960s, researchers who worked on non-volatile solid-state memory devices observed pinched hysteresis loops through the origin of the V–I plane. However, they believed that these loops were caused by outdated instrumentation and parasitics. As a result, they often massaged the device parameters to make them disappear. When they failed, most were too ashamed to report them in their publications (Box 1).

The preceding reported observations can all be explained by one of the five well-articulated sentences identified in Table 1 as Enigmas 1, 2, 3, 4 and 5. They are dubbed enigmas because they were based on observations reported but not proved to be true in general. Indeed, they shall remain enigmas until rigorously proved to be theorems when a nonlinear graphical tool, dubbed the dynamic route map (henceforth abbreviated DRM), sees the light of day2. The DRM has been found to be an indispensable tool for understanding and predicting complex nonlinear and bifurcation dynamics of first-order memristor circuits.

Brief history of the dynamic route map

The DRM is a simple and powerful graphical tool for explaining and predicting the highly complex nonlinear dynamical phenomena exhibited by non-volatile first-order memristors. The concept of DRM was proposed in figure 14 on page 372 of ref. 52. It provides a geometric perspective of nonlinear dynamical trajectories in a 2D phase space, dubbed the phase plane. The original concept of phase space was introduced by Paul and Tatiana Ehrenfest in their seminal work on the kinetic theory of gases in 1911. Paul Ehrenfest was a student of Boltzmann and is known for his formulation of the Ehrenfest theorem, Bose–Einstein statistics, Plank’s energy quanta and the initial concept of spin. For an in-depth exposition on DRM, the reader is referred to refs. 2,52.

Applying DRM to real memristors

The DRM technique has been used to analyse and ascertain the potential applications of the two devices shown in Fig. 1. The calculated DRM for a HfO2 device (grey surface) and the experimental results recorded from multiple measurements of the device53 are shown in Fig. 1a. This figure shows the relation between the time (t) derivative of the conductance (dG/dt) as a function of both the conductance (G) and the applied voltage (V). As the surface contains all the possible states of the memristor, any dynamical change of conductance related to a given waveform applied to the memristor can be represented as a trajectory on this surface, providing a powerful visual tool to analyse the dynamics of the system.

Fig. 1: Exploiting dynamic route maps.
figure 1

a, Example of the experimental and calculated dynamic route map (DRM) of a HfO2 memristor (the inset image), showing the conductance (G) and its temporal derivative (dG/dt), using the voltage V as a parameter. The grey surface represents the fitting of dG/dt as a function of V(t) and G(t) for the measured points. The coloured scattered points represent experimentally measured points from the HfO2 memristor. b, A phase-change memory (PCM) device55, along with its I–V pinched hysteresis loop. The measured DRM can be used to predict the final value of the conductance, which is invariant for d.c. and a.c. measurements. Part a is adapted with permission from ref. 53, Elsevier. Part b is reprinted from ref. 55, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/).

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This insightful idea is explored in ref. 53 to construct the DRM from experimental measurements at constant voltage, using it also to predict the results of measurements made with sinusoidal waveforms. As expected by the DRM technique, all of the different measurements fall over the same surface, as they correspond to the evolution of the same system starting from different initial conditions and under different forcing functions, as shown in Fig. 1. This was also used in ref. 54 to fit an empirical relation to the experimental DRM and then applied to predict the evolution of the system under different conditions.

In this figure, the coloured scattered points represent experimentally measured points from HfO2 devices as described in ref. 53. These devices were forced with a given waveform V(t), and each point corresponds to their state (voltage V, conductance G, derivative of G) at a given time. That is, these points represent [V(t), G(t), dG/dt] for a given time t. Owing to the intrinsic cycle-to-cycle variability, which causes different initial conditions, the points are scattered. On the other hand, the grey surface represents the fitting of dG/dt as a function of V(t) and G(t) for the measured points. Notice that this function provides the dynamic behaviour of the conductance, which can be interpreted as the internal state variable of the system; any new possible state will be on this surface, and the behaviour of the system for any stimulus will be a trajectory on this surface. For instance, we can write, up to first order, that the conductance at time (t + dt) is \(G(t+{\rm{d}}t)=G(t)+{\rm{d}}t\times \frac{{\rm{d}}G(V(t),G(t))}{{\rm{d}}t}\), and the new position on the surface will be [V(t + dt), G(t + dt), \(\frac{{\rm{d}}G\left(V(t+{\rm{d}}t,G(t+{\rm{d}}t))\right.}{{\rm{d}}t}\)], where the only ‘free’ parameter is the voltage V, which is an external signal, and all the information related to the specific device is encoded in dG/dt.

A similar approach is found in ref. 55. Their phase-change memory device (Fig. 1b) is measured and modelled using a differential equation and other relations between structural, thermal and electrical variables. It is also shown in ref. 55 that the DRM is invariant for d.c. and a.c., without considering small parasitic effects. This means that the slow speed measurements are enough to determine the shape of the DRM, thus unequivocally determining the evolution of a given system, where we know its initial state and the forcing waveform. Therefore, it indicates that the DRM is a powerful tool to determine the optimal waveform to reach a desired state, allowing an easy way to optimize power consumption and thermal budget.

Volatile memristors

Recall that there are two types of memristor: volatile or non-volatile. In general, a two-terminal device that exhibits pinched hysteresis loop loci passing through the origin (V,I) = (0,0) in the plane of voltage V(t) versus current I(t) when driven by any zero-mean periodic voltage V(t) or current I(t) of frequency ω ≥ 0 is called a memristor. The periodic input signal includes sinusoidal signals with zero time average and arbitrary frequency ω. When ω = 0, the input is said to be d.c., and the corresponding loci in the V–I plane are called the d.c. V–I loci. In this case, the d.c. input voltage source (battery) or d.c. input current source must be held constant until all transients disappear. Enigma 5 in ref. 2 asserts that non-volatile memristors do not have d.c. V–I loci, because the transient response never subsides to zero. Hence all memristors that have d.c. V–I loci are volatile memristors. Typical examples include d.c. V–I curves that have a negative slope, and the exotic multivalued self-intersecting d.c. V–I loci exhibited by the Chua corsage memristor56.

Galvani irritability and Hodgkin–Huxley spike mystery

This subsection chronicles Galvani’s serendipitous discovery that a frog leg could be induced to contract by using a Leyden jar so nearly completely discharged as to be undetectable with the most sensitive electroscope. He aptly questioned how such a tiny electrical force could produce muscle contraction if it were not setting into motion some internal force and triggering some extremely mobile principle existing in nerves that then excites the action of the nerve-muscular force.

Galvani’s home laboratory (shown in Fig. 2) is of special historic importance in this Review because it was used to conduct the first reproducible scientific experiment to prove that animal cells can not only be stimulated by static electricity, such as that generated by manmade machines or by lightning, but can also be stimulated internally. After many years of more delicate experiments to reconfirm his rigorous analysis, in 1791 Galvani published his monumental masterpiece57 “A commentary on the effects of electricity on the motion of muscles” (translated from the Latin) in which he asserted that ‘animal electricity’ was fundamentally different in principle from that generated by electrostatic generator or by lightning, and insisted that the animal electricity was inherent in the animal itself. In particular, Galvani’s concerted experiments demonstrated that the frog’s muscle and nerve acted like the two sides of a Leyden jar, allowing the discharge of this animal electricity. But how could it be that electrical forces could produce muscle contraction without some extremely mobile principle? How could the spark of life — also known as ‘all or none’, or action potential — be binary in nature: why can’t there exist an action potential with half, or a tenth, or any fraction in between, of its normal amplitude? Galvani’s seminal research was based on many years of rigorous scientific research, publishable even in current high-ranking journals. It was in fact Galvani’s Commentarius57 that inspired both Alan Hodgkin and Andrew Huxley (1963 Nobel Prize in Physiology) and Roderick Mackinnon (2003 Nobel Prize in Chemistry) to continue Galvani’s quest for learning the ultimate biological, chemical, physical and mathematical mechanisms behind the spark of life.

Fig. 2: Galvani’s lab inspires Hodgkin–Huxley and Mackinnon.
figure 2

The left panel is a hand-cranked friction machine for charging a Leyden jar (at the right) through an overhead wire. Three frog leg dissections are shown on the table. The small lace-cuffed hands pointing to the instruments were a common way to mark a point during the Renaissance57. The right panel illustrates two frog legs which Galvani carefully prepared to carry out a critical experiment that proved that contacts between nerves and muscle alone could produce muscle twitches. He used the electrostatic machine on the left end of the table to generate a spark that caused the frog leg to twitch. Galvani carefully positioned the sciatic nerve of the first leg so that its cut end touched the sciatic nerve of the second leg, with another contact between two nerves being made elsewhere. Galvani found that the contacts between the two nerves were sufficient to elicit a muscle contraction in the first leg. Since no metal had been involved in the circuit, the electricity could only have come from the frog and the nerves themselves. This decisive experiment proved the existence of ‘animal electricity’ and led to the birth of electrophysiology. His momentous experiment inspired Alan Hodgkin and Andrew Huxley to derive a circuit model reproducing the action potential via ion channels, and they were awarded the Nobel Prize in 1963. Fast forward 40 years, and, inspired by Galvani’s laboratory and Hodgkin and Huxley’s ion channels, Roderick Mackinnon was awarded the Nobel Prize in 2003 for discovering the molecular structure of the Hodgkin–Huxley potassium ion channel, a tour de force in experimental research. Figure 2 is reprinted from ref. 57, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/).

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Galvani’s quest for his elusive ‘extremely mobile principle’ led him to exploit, unsuccessfully, the doctrine of irritability, one of the most important conceptual elaborations of eighteenth-century physiology. Fast forward 243 years, we can now prove rigorously via nonlinear circuit theory and advanced theory of complex variables that Galvani’s quest for an extremely mobile principle is the local activity principle and its crown jewel, the edge of chaos, as conceived and presented in ref. 4.

Galvani’s serendipitous discovery of the phenomenon of frog leg muscle contraction in 1781 (depicted in Fig. 2) and his irritability doctrine were largely ignored for 171 years until Hodgkin and Huxley published their 1952 paper, which was recognized by their 1963 Nobel Prize. But a mathematically rigorous explanation of the nonlinear mechanism that generates the action potential spikes (Fig. 3) remained unsolved until the discovery of the subcritical Hopf bifurcation emerging from the edge of chaos, which spawns an unstable periodic orbit that grows to coalesce with a large-amplitude stable orbit via a saddle-node bifurcation, as depicted in ref. 9.

Fig. 3: Neurons function near the edge of chaos.
figure 3

a, The original Hodgkin–Huxley circuit for generating action potential. The original Hodgkin–Huxley model consists of an erroneous time-dependent sodium ion-channel resistor (RNa) and potassium ion-channel resistor (RK), respectively, in addition to a standard linear leakage resistor (Rl) and a membrane capacitor (CM)38. b, The standard textbook schematic of a membrane, showing the main internal structure, including a potassium and a sodium channel. c, The first experimentally measured action potential between the inside and outside of a squid axon70,71. The upper part of a periodic sinusoidal waveform shown at the bottom is used to provide the timescale. This waveform was recorded by Hodgkin and Huxley in 1939, when commercial oscilloscopes were not yet available. To identify the timescale, they applied simultaneously a 500-Hz sinusoidal signal as the time marker. d, The Hodgkin–Huxley memristor circuit model. By replacing the two conceptually ill-defined time-dependent resistors from part a by a sodium ion-channel memristor and a potassium ion-channel memristor, respectively, we obtain the immortal memristive Hodgkin–Huxley circuit model9. Parts a,b and d are reprinted from ref. 9, CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). Part c is reprinted from ref. 70, Springer Nature Limited.

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Turing instability and Smale paradox

Alan Turing, the father of computing and AI, is also the creator of the chemical basis for morphology. He posed this in a paper in 1952, where he asked: how can a homologous assemblage of identical cells exhibit non-homogeneous structures? We have recently resolved the Turing instability problem10, where we proved, via an explicit example, that the edge of chaos is a sine qua non for Turing instability. An illuminating exposition of the calculation by the edge-of-chaos criterion is presented in Fig. 4.

Fig. 4: Turing instability emerges from the edge of chaos.
figure 4

The top panel shows two identical chemical cells with identical initial conditions. Turing proved, by brute force calculations, that they tend to different concentrations in equilibrium. This demonstrates that skin cells from animals such as leopards that are identical over broad areas could exhibit patterns. Turing’s two-cell example is emulated by two locally active volatile memristors made from niobium oxide in T. Mikolajick’s laboratory34 in Dresden, Germany. The two identical memristive 1-ports10 are coupled by a positive resistance Rc. Unlike Turing’s two-cell example, where the diffusion coefficients are found by time-consuming trial and error, the coupling resistance Rc is calculated by the explicit edge-of-chaos formula. By straightforward application of this formula, we were able to derive the critical bifurcation parameter Rc = 49.7 Ω. The bottom four graphs show a symmetry breaking when the coupling resistance Rc is tuned to the bifurcation parameter value Rc = 25 Ω, where the data are extracted from ref. 10. The photo of the leopard is reprinted with permission from ref. 5, World Scientific.

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Inspired by the beauty and depth of the Turing instability, Stephen Smale, a 1966 Fields medallist, upped the ante by asking a deeper related mathematical problem: “How can two mathematically dead cells in higher-dimensional space, coupled via diffusion, become alive?”29 Even such a highly revered mathematician as Smale had a pessimistic prognosis when he warned, in 1974, that “any sort of systematic understanding or analysis seems far away.” Smale was right, for it took almost half a century for the Smale paradox to be solved, in March 202211. An even more inspiring testimony of the profound power of edge of chaos is unveiled in Fig. 5.

Fig. 5: Smale paradox springs from the edge of chaos.
figure 5

Inspired by the eminent mathematician Smale’s quest for a higher-dimensional analogue of the Turing instability, we have successfully invented a simple higher-dimensional locally active volatile memristor circuit that could manifest exotic complex nonlinear dynamics29. The niobium oxide memristor cited above is shown on the left of the top panel. The two identical cells of the Smale paradox circuit are shown in the top right, arranged in a mirror-symmetric configuration. The two identical Turing cells referred to in Fig. 4 are in fact identical to the above Smale twin cells, upon deleting the two capacitors. By coupling the twin Smale cells with the resistor Rc, we obtain the Smale paradox circuit, where the waveforms exhibited in the bottom two panels are measured. Remarkably, on just adding a capacitor (C) across the same niobium oxide memristor from the preceding Turing instability example, as shown in the top panel, applying the same edge-of-chaos formula, and using the same critical bifurcation parameter Rc = 49.7 Ω, we were dumbfounded to observe higher-dimensional nonlinear dynamical bifurcation phenomena11. The examples in the bottom panel are extracted from these two 3D locally active cells, coupled by the positive resistor Rc. Borrowing Smale’s inspiring language, we have found ‘nirvana’, when the memristor cells become alive when we tune Rc to the critical bifurcation value Rc = 49.7 Ω. The bottom panel is reprinted with permission from ref. 11, IEEE.

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The Chua riddle

A riddle is shown in Fig. 6 that I routinely posed in the opening lecture of a graduate course on nonlinear circuits and nonlinear dynamics that I taught at the University of California, Berkeley, and at the Chua hp lecture series, as well as at numerous keynote lectures5,6. It has attracted immense recent scientific attention and research activities in view of its deep nonlinear dynamical relevance in the electrophysiology of ion channels, in complexity theory, and above all in demystifying the time-honoured Turing instability conundrum20 (unsolved since 1952) and Smale’s paradox29 (unsolved since 1974).

Fig. 6: Chua riddle as example par excellence of the edge of chaos.
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The black box contains only two linear circuit elements (chosen from linear time-invariant resistor, inductor or capacitor). In the top circuit, the external current i(t) entering the black box is found to be always a stable exponential waveform with a positive time constant52 τ > 0 for any initial current (if an internal element is an inductor), or initial voltage (if an internal element is a capacitor). On adding a positive resistor in series with the black box, the corresponding external current i(t) is found to be an unstable exponential waveform with a negative time constant τ < 0, for any initial internal inductor current, or initial capacitor voltage. The Chua riddle is: ‘what are the two circuit elements inside the black box?’ The Chua riddle circuit is the simplest electrical circuit endowed with the edge of chaos.

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As a scintillating riddle with an evangelical message, the reader is urged to pause and contemplate the profound scientific principle ensconced within the modest and unimpressive Chua riddle circuit. So what is inside the black box dubbed the Chua riddle circuit in Fig. 6? The answer could not be simpler: the circuit contains a −1 Ω resistor in series with a −1 H inductor. The R = −1 Ω resistor (or L = −1 H inductor) can be built by connecting an R = 1 Ω resistor (or L = 1 H inductor) across port 2 of a negative impedance converter58. The negative impedance converter can be built by using a transistor or op-amp.

Any sophomore from electrical engineering can trivially derive and show the exact analytical solution i(t) of the circuit in Fig. 6 with initial current i(0) = −2 A is the boring damped exponential function shown on the right of the circuit, with a positive time constant τ = 1 s (ref. 52). Inserting the 2-Ω resistor in series with the −1-Ω resistor gives the corresponding exact solution (with the same initial current i(0) = −2 A) shown on the right of the ‘dissipated’ Chua riddle circuit. The adjective ‘dissipated’ is added to emphasize that the inserted 2-Ω resistor adds dissipation to the Chua riddle circuit. Our intuition would suggest that the solution should look like the preceding boring damped exponential waveform but moving a little more slowly towards a slightly different d.c. steady state. Yet even the smartest kid would be shocked to see that the actual solution blows up towards infinity! The immediate reaction would range from “This is insane” to “This is impossible!”

A rigorous and comprehensive analysis of the dynamical phenomena exhibited by the Chua riddle circuit for arbitrary values of the internal negative resistor and negative inductor is given in a recent paper9. The doubting Thomases are encouraged to peruse it. Indeed, it is reassuring that the small-signal equivalent circuit about a d.c. equilibrium point of both the potassium ion-channel memristor, and the sodium ion-channel memristor, in the immortal Hodgkin–Huxley circuit model shown in Fig. 3d, contains the Chua riddle circuit as its two crucial components, as illustrated in ref. 9, along with the corresponding exact value of the negative resistor and negative inductor. But even more amazing is that Turing and Smale were both dumbfounded that the two identical cells became unstable when coupled by diffusion, which is equivalent to coupling the two identical cells in Figs. 4 and 5 by a positive resistor, as in the Chua riddle circuit in Fig. 6.

The four classic unsolved problems presented earlier are all resolved using volatile memristors operating in a typically minuscule edge-of-chaos subset of the huge device parameter space. Finding this tiny sweet spot is more difficult than searching for a needle in a haystack. In the case of the Turing instability example in Fig. 5, and the Smale paradox exhibited in Fig. 6, the sweet spot is calculated, and not searched, from an explicit formula extracted from the edge-of-chaos theorem derived and presented in ref. 4. Indeed, the critical resistance value is Rc = 49.7 Ω, which produces a bifurcation from an informationless, dull, homogeneous parameter region to the sought-for information-rich sweet spot. The volatile memristors chosen in Figs. 4 and 5 are NbOx memristors40 made in the NaMLab in Dresden, Germany, and an NbO2-Mott memristor made in a high-tech lab in Palo Alto, California. We end this section with the astounding revelation that the two-element Chua riddle circuit is stable but potentially unstable, the defining characterization of edge of chaos.

Neuromorphic memristors

Encouraged by the above success stories, we now pose our ultimate challenge — build a neuromorphic computer that mimics the human brain. This challenge calls for the development of subnanoscale and subfemtowatt memristors. Once such volatile memristors are developed, it would be a cinch to emulate most high-level brain functions and observe the garden variety of well-known phenomena reported in the literature59,60.

An intriguing line of research is to use memristors at the edge of chaos to emulate the Hodgkin–Huxley model of the neuron, following the discovery that neurons are made of memristors38. This approach was first demonstrated using two NbO2 memristors and two capacitors to represent the four state variables of the Hodgkin–Huxley equation61. The concept was then extended to demonstrate a wide variety of biological behaviours, in this case using two VO2 memristors35. The common factor was that both memristor materials displayed first-order Mott insulator-to-metal transitions with increasing temperature that dramatically increased their conductivity. This was shown to produce negative differential resistance in devices made from such materials, caused by the dynamic coupling of the Joule heating in the device to the thermal conductance that removes heat from the system, and resulted in the temperature being the state variable for such ‘electrothermal’ memristors62. This understanding resulted in the experimental demonstration of various types of regular and chaotic oscillations in structures with a single physical device that incorporated one memristor and one capacitor to demonstrate biological behaviour and computation63,64. These materials were limited in terms of the temperatures in which they could operate, and by an incomplete understanding of how their properties were described by local activity theory, with the latter issue resolved in ref. 36. However, this left the problem of how to identify or predict new materials and device structures that would be strong candidates for edge-of-chaos memristors. This is challenging because the typically extremely small volume within the parameter space that exhibits the edge of chaos makes finding new materials by simple trial-and-error methods nearly impossible (NbO2 and VO2 were discovered by serendipity). Thus, local activity theory was reinterpreted in terms of static electronic and thermal properties that have been measured in various laboratories for decades and are easily collected today with simple instruments and procedures39. This analysis showed that although negative differential electrical resistance is not necessary for the existence of local activity and edge of chaos (for example, negative differential thermal resistance can also yield edge of chaos), it is the most easily observed property of real materials measured in a lab that can act as a beacon for experimentalists to find new candidates for edge-of-chaos memristors. This was followed by a demonstration of the experimental procedure for measuring and identifying the edge of chaos and the region in which it exists, applied to the known system VO2 (ref. 8). Using these ideas has led to searches of the literature for new candidate materials, for example a list of over 30 known oxides with insulator–metal transitions, as well as chemical methods to modify these materials to engineer their electrical and thermal properties for desired applications65.

We end this short but visionary section with a sample of 20 commonly observed waveforms recorded from a US$10 homemade Chua corsage memristor56. Each of the 20 waveforms shown in Fig. 7 is identified either by well-known terminology, such as ‘action potential’ in Fig. 7b and ‘chaotic waveform’66,67 in Fig. 7i, or by somewhat obscure jargon introduced in ref. 59. As these terminologies are not defined in a precise scientific manner, we refer their characterization to the above highly regarded and widely cited reference by Izhikevich59. Our goal here is to impress on readers that our brain can generate a wide variety of complicated waveforms, most of which originate from poorly understood higher brain regions. Most of the waveforms displayed in Fig. 7 are rather common and reproducible, as evidenced by the ease of observing them from the poor-man’s Chua corsage memristor.

Fig. 7: A variety of neuromorphic phenomena from the Chua corsage memristor.
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a–t, The Chua corsage memristor can exhibit a broad variety of garden-variety neuromorphic phenomena, including the 20 experimentally measured samples (a–t) shown here. The memristor can be built with off-the-shelf electronic components for less than US$10 (ref. 56). It is dubbed a corsage memristor because it has an unusual d.c. V–I curve that resembles a corsage and has been traced in one contiguous piece without gaps56,72,73, a tour de force to behold. The global nonlinear dynamical bifurcation mechanisms that give birth to the neuromorphic phenomena are complicated and require a deep mathematical background. An in-depth rigorous mathematical explanation of several neuromorphic phenomena can be found in refs. 30,31,59. Figure reprinted with permission from ref. 56, IEEE.

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Visions of memristors on edge of chaos

Edge computing is a distributed information technology architecture in which data are processed in proximity to the sources, for example sensors or Internet-of-Things devices68,69, that generate them. In the traditional computing approach, data are preliminarily sent to a remote centre before elaboration. This unavoidably slows down the extraction of information, as the transfer procedure is susceptible to bandwidth limitations, latency issues and potential disruptions over the Internet. Another major issue is the huge amount of associated energy consumption. The energy consumption for sending data to a data centre is more than ten times that for processing it locally on the edge using memristive systems, because sending data involves converting analog data into digital, storing them and transferring them over long distance.

A distributed signal processing paradigm is of vital necessity for a time-efficient and energy-efficient solution of demanding computational challenges, emerging frequently in the Big Data era we are currently living in. Technical systems, operating within an edge-computing information technology environment, are endowed with data storage and processing blocks, which, together with power and transceiver units, allow local real-time control over critical processes in various domains, including the military sector, the medical field and industry. Only the main results of the local computations may then be transferred, upon request, across the Internet to some remote centre for further elaboration, which resolves the data communication issues affecting the traditional computing approach.

In the past three years, there has been an increased emphasis on solving practical design challenges for implementing neuro-memristive solutions for energy-efficient computing on the edge. The complexity of problems at the edge keeps on increasing, in problems such as autonomous driving, surveillance and tracking of multiple objects, and real-time tracking of humans, all of which demand increased computational density and at the same time need to meet low energy requirements. The dot-product computations that are at the heart of neural computations in a crossbar architecture align well with the energy-efficient requirements of edge devices. However, the increasing challenge in such systems has been cycle-to-cycle and device-to-device variability that requires careful circuit design considerations to be followed. Although there are several ways to mitigate the variability at device to systems level, there is no one solution that fits all. Selecting the right architecture and cells to be robust against variability becomes a multiobjective optimization problem.

Design optimization and tools that can help to automate the design are an emerging area of interest in designing edge-computing hardware. More recently, there have been efforts to make use of large language models to generate the chip designs and automate the time-consuming verification tasks. The use of generative AI and automating chip design for edge AI computing is futuristic, yet quickly becoming a reality due to emergence of foundry supported open-source chip design tools. Open chip movements such as RISC-V initiatives will further speed up the design-to-test times for neuro-memristive chips for edge computing, as the designers have a well-developed hardware and software stack that can be readily integrated to implement a complete ecosystem for building applications with edge AI chips.