The present overview is derived from “Summary of Professional Achievements”submitted in support of Habilitation proposal at Institute of Nuclear Physics,

Polish Academy of Science, Kraków, Poland

Electronic networks of chaotic oscillators as physical models

of some emergent phonomena in neural dynamics

1 List of publications

[H1] L. Minati, Experimental dynamical characterization of five autonomouschaotic oscillators with tunable series resistance, Chaos 24, 033110(2014).

[H2] L. Minati, Experimental synchronization of chaos in a large ring ofmutually coupled single-transistor oscillators: phase, amplitude, andclustering effects, Chaos 24, 043108 (2014).

[H3] L. Minati, P. Chiesa, D. Tabarelli, L. D’Incerti, J. Jovicich, Synchro-nization, non-linear dynamics and low-frequency fluctuations: anal-ogy between spontaneous brain activity and networked single-transistorchaotic oscillators, Chaos 25, 033107 (2015).

[H4] L. Minati, Experimental Implementation of Networked Chaotic Oscil-lators Based on Cross-Coupled Inverter Rings in a CMOS IntegratedCircuit, J Circuit Syst Comp 24, 1550144 (2015).

[H5] L. Minati, Remote synchronization of amplitudes across an experimen-tal ring of non-linear oscillators, Chaos 25, 123107 (2015).

[H6] L. Minati A. De Candia, S. Scarpetta, Critical phenomena at a first-order phase transition in a lattice of glow lamps: Experimental findingsand analogy to neural activity, Chaos 26, 073103 (2016).

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2 Introduction

It can be argued that dilemmas around the genesis of mental states and freeentity will never lose their topicality. They are among the most essential andprimary issues, against which stood thinkers of all cultures and eras. Obvi-ously, the way in which such questions are framed has changed significantlythrough history; after discovering and accepting the brain as the ultimatephysical seat of such phenomena, it has essentially become: how can theenormous richness of conscious experience, cognition and emotions emergefrom this small, wet, warm, jelly, fragile and ultimately messy-looking lumpof matter, which is now even trying to study itself [1, 2]? In a historical per-spective, addressing the mind-body problem has remained a matter almostexclusively for philosophy until rather recently [3]. However, over the lastfew decades the situation has changed drastically, in that we now possessan extensive set of physical and computational tools to measure, model andsimulate brain activity, and thereby attempt to address this problem exper-imentally, opening the way to the formulation of falsifiable theories aboutphysical mechanisms underlying higher brain function [4].

One of the most influential paradigms in neuroscience has been (and is)that of localizationism, namely the hypothesis, even the expectation, thatthere are well-defined one-to-one, or at most one-to-many, relationships be-tween cognitive functions and brain structures. While such view has ancientroots, it was given substantial thrust by the studies of Pierre Paul Broca inthe XIX century, which provided unequivocal clinical evidence on the local-ization of some functions, such as speech production [4, 5]. This view stillexerts a powerful influence on contemporary neuroscience, as demonstratedby the fact that many, or indeed the majority of, contemporary neurophys-iological and neuroimaging studies try to localize which areas or networksof areas of the brain generate activity or have structural properties corre-lating with a given variable of interest, such as a behavioural observable.Perhaps remarkably, this is still done primarily by means of more-or-less so-phisticated general linear models [6]. Such approach is certainly fertile, andlinear correlation studies have indeed yielded substantial insights, becausesensorimotor and associative functions, and to a lesser extent also cognitiveprocesses, are significantly localized. However, localizationism and linearmodelling have fundamental limitations in that higher cognitive functionsare much more than a merely summative expression of separable activity in-volving discrete sets of brain areas [4, 7]. This awareness is deeply linked tothe philosophical current of emergentism, which in its various formulationsposits that consciousness, cognition and emotion are emergent phenomena,which cannot be trivially pinned down to the superposition of the propertiesand states of individual constituent elements of the brain [1, 2, 3, 4, 7]. Thisperspective rhymes very closely with modern complexity science, which seeksto discover universal phenomena that underline the emergence of properties

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which cannot trivially be mapped to individual elements across diverse self-organized systems in physics, biology, social science, and indeed neuroscience[8].

Physics and engineering have given fundamental technical contributionsto modern neuroscience, in that they have provided a comprehensive setof techniques to measure the structure and function of the brain, for ex-ample electro-encephalography (EEG), magneto-encephalography (MEG),magnetic resonance imaging (MRI) in all its forms and variants, in-vivoand in-vitro recording techniques such as patch clamping, cortical grids andmulti-electrode arrays (MEA), and two-photon microscopy, among others[4, 9]. However, as highlighted in a recent editorial, physics can and shoulddo much more than that, namely provide robust theoretical paradigms foraddressing core questions about emergence in the brain, and its relationshipto other physical systems [10]. Paraphrasing, the risk otherwise is that ofusing new tools in an old manner, namely to remain bound to a localization-ist approach, which inherently negates the possibility of capturing emergentphenomena; as stated above, the risk is very real, since much of current neu-roscientific research remains hinged around linear models, and only a smallminority of studies explicitly attempt to model higher cognitive processes asemergent phenomena.

The physics of how the brain works has, predictably, attracted muchinterest in itself. On one hand, there is a well-developed body of experi-mental literature on biophysics, attempting to rigorously model micro-scalephenomena, such as the dynamics of synapses, receptors etc. On the other,as regards to understanding macro-scale properties relevant to the mind-brain duality, there has been a sort of “innate revulsion” at modelling braindynamics in classical mechanistic terms, which would seemingly be at oddswith our subjective perception of free will [11]. Disparate theoretical av-enues have consequently been considered, extending all way into positingthat stochastic, even quantum-like, phenomena are essential for emergenceof the mind, or, vastly more credibly in my opinion, asserting the centralityof deterministic chaotic dynamics; the prevailing view at the time of writingis that understanding how the mind emerges from, and eventually shapes,the brain fundamentally requires the contributions of self-organization the-ory, dynamical systems theory and networks theory [7, 12, 13, 14, 15, 16].

The relationship between physics and neuroscience could be deepened byadopting a physical approach to the study of the brain, whereby it is moreexplicitly compared with other natural and artificial physical systems. Whilethis concept is clearly not new, relatively little research has been done alongthe way of explicitly trying to “engineer” emergent phenomena observedfor the brain into other physical systems, and drawing direct comparisonsbased on experimental data [13, 14, 16, 17]. Particularly in computer sci-ence, the notion of building neuromorphic computational architectures iswell-established, and there is an abundance of algorithms (e.g., multi-layer

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perceptron neural networks) and hardware (e.g., retina-like image sensors)which explicitly attempt to replicate select aspects of the structure and func-tion of biological neural systems [18, 19]. However, for the most part suchneuromorphic systems are intended to solve specific problems in a predeter-mined manner, and not to replicate the emergent phenomena observed inthe brain as such. At present, the two requirements are, effectively, antithet-ical since we mostly lack the theoretical tools to “drive” emergence towardssolving prescribed computational operations.

Hence, there is a need for work on developing physical systems that canreplicate at least some select aspects of emergent dynamics in the brain.Such efforts are, in my opinion, motivated to complement more “main-stream” computational neuroscience research, for several reasons. First,in physical systems the emergence of global properties is often greatly in-fluenced by “nuances” such as small parametric mismatches between con-stituent elements, electrothermal noise in the oscillatory variables, noise inthe dynamical parameters, non-ideal behaviours such as presence of “par-asitic” elements, lack of discretization and so on. In practice, such as-pects are not trivial to capture in numerical simulations, hence they areoften neglected in computational models, as including them can lead toover-complicated equation systems that pose troubles to numerical solvers[17, 20, 21]. Second, there is an inherent interest in recapturing aspects ofemergent brain dynamics in artificial physical systems, for the purpose of ob-taining experimental platforms allowing the verification of hypotheses whichmay be difficult to test on biological neural systems, and for the purpose ofeventually engineering practical applications of such systems. Indeed, theprevailing approach to building computing machines is presently the digitalone because of their flexibility and universality; yet, hypothetical alterna-tive paradigms based on computing via emergence in distributed, analogsystems could unlock enormous amounts of computational complexity: thisis readily exemplified by the fact that realistically simulating neural net-works requires digital machines which are many orders of magnitude largerand more power-hungry than their biological counterparts [22, 23, 24, 25].

The present body of work is about eventually accomplishing this goal bymeans of realizing electronic networks of chaotic oscillators. From a physi-cal viewpoint, there is nothing inherently special about electric or electroniccircuits, and one could as well try to recapture aspects of brain dynamicsin non-linear systems realized by mechanical, optical and chemical means.However, at this time electronic circuits have major practical advantages inthat they are undemanding and inexpensive to realize, network and repli-cate; indeed, much more so than in-vitro biological preparations such asneural cultures. The study of chaos in electronic circuits dates back to thefirst incidental observations of “irregular behaviour” by van der Pol, and wassubsequently greatly propelled by the discovery by Leon Chua of the firstelectronic circuit that intentionally behaves in a chaotic manner, generating

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an extraordinarily large repertoire of attractors [26, 27, 28, 29]. While thefield of non-linear electronics has received substantial attention over the re-cent decades, three limitations, discussed more extensively below, are noted.First, much work has focused on “canonical” circuits like Chua’s circuit, or“chaotic adaptations” of well-known oscillators such as Hartley and Colpittsoscillators. Second, only a limited proportion of existing studies has actuallyinvestigated in an experimental manner the behaviour of physically-realizednetworks of chaotic oscillators. Third, not much has yet been done in theway of explicitly establishing relationships between the behaviour of thesenetworks, and emergent phenomena observed for the brain.

The overarching aim of this body of work is to demonstrate that emer-gent features (statistical properties) of brain activity can to some extentbe recaptured in electronic networks of chaotic oscillators, and in doing sopropose new oscillator circuits which hopefully can reinvigorate interest inthis area. The focus is on general emergent phenomena, such as sponta-neous pattern formation via synchronization, and no attempts are made toreplicate specific brain functions, or build models around oscillatory vari-ables that have an immediate biological interpretation; from this viewpoint,the present approach is clearly different and complementary to mainstreamcomputational neuroscience and to the realization of neuromorphic comput-ing architectures. While caution should be exercised to avoid falling in thetrap of over-inferring from superficial similarities, in my opinion the abilityto recapitulate electronically select aspects of emergence in brain networkscan be of value for inspiring the formulation and testing the validity of hy-potheses across different scales (micro-, maso- and macro-scale networks, asdiscussed below) and settings (in-vivo, in-vitro, in-silico).

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3 Discussion of the publications

[H1] L. Minati, Experimental dynamical characterization of five autonomouschaotic oscillators with tunable series resistance, Chaos 24, 033110(2014).

One of the most fundamental aspects of neural dynamics is that neurons,and under certain circumstances even isolated axons, have access to a mul-titude of dynamical behaviours leading to the periodic, quasi-periodic andchaotic activity, which is expressed during the generation of isolated spikes,bursts and irregular sequences of spikes and bursts [30, 31, 32, 33]. The dy-namical properties of individual neurons are knowingly determined by theirphenotype (level of expression of genes related, for example, to receptor for-mation and maintenance), and by electric and biochemical variables (suchas intra- and extra-cellular concentrations of ions) [4]; with reference to thetime-scale of cognitive activity, the former can largely be viewed as fixed pa-rameters, and the latter as control parameters which can vary dynamically.Importantly, such diversity of neural dynamics has been observed acrossmicro-, meso- and macroscopic scales, for example in giant squid axons [34],assemblies of cells [35, 36] and entire brains [31, 37]. There is now consensusthat it is central to the emergence of time-varying partial synchronizationpatterns, which support switching between diverse activity modes based ona largely fixed structural connectivity, ultimately yielding the multiformityand wilfulness of cognitive activity [11, 13, 38].

It is therefore evident that an essential element for any attempt to re-capture the complexity of neural dynamics in other physical systems, is theavailability of oscillators which can generate signals having diverse dynami-cal characteristics as a function of one or more control parameters that areeasily tunable. In this regard, the paradigmatic case of Chua’s circuit readilycomes to mind [26], however other transistor-based oscillator topologies havesome advantages in terms of obtaining complex dynamics with elementarycircuits that can be easily realized practically and coupled in large networks(e.g., [39]). There is a rich literature on transistor-based chaotic oscilla-tors, however much of it represents modifications of pre-existing topologiessuch as the Colpitts oscillator: while such adaptations retain desirable fea-tures for practical applications (such as noise-like signal generation), theyalso constrain exploration away from novel topologies that may have fea-tures more relevant for the present purpose. While a general methodologyfor synthesizing chaotic oscillators is presently lacking [40], in a prelimi-nary study I proposed using genetic algorithms to obtain novel circuits forphysically-realizable transistor-based chaotic oscillators. In that study, thenode connections and component values were encoded in a bit-string, evo-lution was driven by maximization of Approximate Entropy (ApEn), andsome viable circuits were obtained, which do not intentionally represent

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Figure 1: a) Circuit diagram of the simplest chaotic oscillator, where R1

represents the control parameter which was varied between [90, 3500]Ω, andL1 = 8.2µH, L2 = 10µH and C1 = 30pF. b) Example realization on circuitboard.

modifications of pre-existing oscillators [41].In this study, five among those circuits were chosen, realized and a de-

tailed experimental characterization was performed using a dedicated lab-oratory test-bench, in preparation for using them as building blocks fornetworks of coupled oscillators. The circuits are composed by elementarycomponents: one or two bipolar-junction transistors (BJT), inductors, ca-pacitors and a resistor. They are autonomous, in that they oscillate withoutrequiring external excitation. They represent “atypical” topologies that werenot described previously. In these circuits, multiple oscillation modes areavailable at frequencies that are not related by a trivial (e.g., integer) ratio:these are primarily determined by the possible LC combinations of inductorsand capacitors, including transistor junction capacitances. The fundamentalprinciple of these circuits is that because the BJT(s) operate in large-signalmode, there are significant and variable non-linearities (as represented forinstance in the Ebers-Moll equation), which determine and are themselvesinfluenced by the generated signals, ultimately controlling which frequenciesamong the available LC combinations are actually generated. Because of theBJT non-linearity, during oscillation at multiple frequencies Ω1 . . .Ωm, otherfrequency combinations such as niΩi + njΩj can also be generated, whichcan eventually lead to chaos via the so-called “quasi-periodicity” route (with-out excluding other mechanisms such as period-doubling) [42]. One way ofviewing the way in which these circuits transition to chaos, is as having m si-nusoidal voltage sources with frequencies Ω1 . . .Ωm and amplitudes v1 . . . vm,connected in series to a non-linear resistor of value R(v), where Ω1 . . .Ωm

and v1 . . . vm themselves are variable as a function of current i (due to BJTnon-linearity) [43].

The special relevance of these circuits to creating a physical model, or

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Figure 2: a) Representative time-series recorded at the collector node forR1 = 1700Ω, wherein cycle amplitude (maxima) fluctuations are evident.b) Corresponding broad frequency spectrum, confirming chaoticity.

more properly “analogy” of neural activity, is in the fact that their dynamicscan be profoundly influenced by changing the value of a series resistor, whichis always placed in between the connection to the voltage source poweringthe circuit. By determining the voltage/current slope at the supply node,and with it the average bias current too, the series resistor R1 influences thestate trajectory on the BJT’s operating plane (exploring cut-off, saturationand active regions, and in some cases, even reverse saturation and activeregions). It can be argued that the most interesting circuit considered inthis study is the one shown in Figure 1, which comprises an NPN-typetransistor connected in grounded-emitter configuration, shunted by a seriesLC network (L1,C1) and with feedback between collector and base via aninductor (L2). It is reminiscent of a Hartley oscillator [44].

For suitable values of R1, spike-like activity was recorded which had aclear qualitative resemblance to neural activity, as it consisted of ampli-tude fluctuations arranged as isolated events or oscillation “volleys”, withmarkedly greater variance for maxima than minima, representing a com-mon feature across these oscillators; the important difference with respectto neural data, evidently, is that here irregularity was manifest primarily ascycle-amplitude fluctuations, rather than inter-spike interval variations [45].An example time-series is shown in Figure 2, and its very broad spectrumunequivocally confirms chaoticity. While I refrained from making such anexplicit claim in the published manuscript, it is remarkable that this cir-cuit is topologically of similar complexity to the simplest known electronicchaotic oscillator, the Lindberg-Murali-Tamasevicius (LMT), yet its dynam-ics appear probably richer [39].

More generally, to characterize the dynamics of the five oscillators cir-cuits under consideration, in this study I resorted to obtaining the respectivebifurcation diagrams and spectrograms, and plotting the corresponding cor-relation dimension D2, largest Lyapunov exponent λmax and approximate

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entropy e [46, 47]. All measures were considered with respect to the seriesresistor value R1 acting as control parameter, which was made possible bya custom-built electromechanical bench allowing fully automated data ac-quisition. While R1 was swept, for these circuits the following phenomenawere observed: excitation of discrete frequencies corresponding to specificLC combinations, addition of harmonics (corresponding to bifurcation), ap-pearance of frequencies given by other LC paths and their combinations,eventually chaos.

As regards to the spectrograms, most circuits exhibited sharp transitionsbetween periodicity and chaos, and a minority also displayed large-scale bi-furcation structure or intermittency, which is close to the behaviours ob-served in simulated and experimental neural systems [30, 31, 32, 33, 34, 35,36, 37]. This is exemplified by the bifurcation diagram in Figure 3a for thecircuit herein described previously, where multiple sharp transitions coexistwith a fine-grained structure within the chaotic regions. While the largestLyapunov exponent λmax and approximate entropy e were calculated mainlyfor comparability with other studies, the correlation dimension D2 (whichmeasures the change in level of detail as a function of scale accounting for thenumber of times each box is visited) provided the best insight into the dy-namics of these circuits, as it is less vulnerable to limited sampling rate andtime-series length, and noise [46, 47, 48]. Here, the correlation dimension D2

was calculated using the box-assisted search algorithm over time-delay em-bedding as x(t) = [v(t), v(t+ τ) . . . v(t+ τ(m− 1))], making arbitrary choiceof the voltage-measurement node non-critical; the embedding delay τ anddimension m were calculated according to the first minimum of time-lagmutual information function and ratio of false nearest neighbours respec-tively, and temporal autocorrelation bias was attenuated by application ofa Theiler window w [46, 47]. As exemplified in Figure 3b, the correlationdimension D2 clearly demarcated periodic and chaotic regions, confirmingthe capability of this circuit to exhibit multiple phase transitions accordingto the value of the series resistor R1. Analogous results were obtained forthe other four circuits considered, but here the focus is on the simplest one,as it was used as a “building block” for networks in the following studies.

While for brevity I shall not dwell on this aspect in the present summary,these circuits were also extensively simulated using the Simulation Programwith Integrated Circuit Emphasis (SPICE) program, representing separatelysituations with different parasitic elements [49]. The simulations broadly re-captured the chaotic nature of these oscillators, however the agreement withexperimental data was relatively poor, as the overlap between experimentaland simulated bifurcation diagrams was limited. In part, this was probablydown to BJT model parameter uncertainties and other limitations in modelparameter accuracy, however this result was also interpreted as confirmingthe known difficulty in performing accurate numerical simulations of thesecircuits, which was already reported and discussed in a previous study on

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Figure 3: a) Experimental bifurcation diagram, showing multiple transitionsbetween periodic and chaotic oscillation, and fine-grained structure in thechaotic regions. b) Correlation dimension D2 as a function of R1.

the LMT circuit [50]. This aspect is particularly important, because the in-ability to obtain accurate results even with a single oscillator underlines theparamount importance of combining numerical explorations with measure-ments on physical circuits when studying the complexity of entire networks.

In summary, this study revealed five novel single-transistor oscillator cir-cuits, which could generate a rich repertoire of regular and chaotic signals,wherein chaos was primarily evident as cycle amplitude fluctuations, in mul-tiple spiking and bursting patterns. Owing to their simplicity and tunabil-ity, they represent efficient building blocks for experimental networks, whichwere realized in the studies described below. Importantly, the straightfor-ward tunability via a resistor allows exploring dynamical behaviours in var-ious regimes, including close to order-to-chaos transitions; furthermore thistunability in principle allows representing “degenerative” processes, whereinthe effect of altering the dynamical response of individual nodes on whole-network properties is explored.

[H2] L. Minati, Experimental synchronization of chaos in a large ring ofmutually coupled single-transistor oscillators: phase, amplitude, andclustering effects, Chaos 24, 043108 (2014).

Synchronization of non-linear agents receives ever-increasing interest acrossdiverse disciplines: physics, engineering, biology, economics etc. Arguablythe main underlying reason is that in presence of non-linear dynamics, syn-chronization, intended in various forms depending on the context, supportsthe emergence of patterns that are not initially present in the structuralconnections between system elements [20, 51, 52, 53]. For example, thenon-linear dynamics of electrical power distribution networks and meteo-rological systems engender preferential inter-dependences (synchronization)between entities physically not adjacent, which are not straightforward to

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predict a-priori [54, 55]. Even more generally, in nature the formation ofcomplex patterns such as regular or even self-similar shapes of living crea-tures, known as morphogenesis, can often be traced down to synchronizationphenomena [56]. For the case of neural systems, structural connections areembodied by synapses or, at a more coarse-grained scale, by the layout ofaxon bundles in the brain, whereas synchronization (in this specific contextalso referred to as functional connectivity) is observable in terms of statisti-cal inter-dependence of activity between neurons, neural populations or en-tire brain regions. The correspondence between structural connectivity andsynchronization in neural systems is often detectable but incomplete, sincethe latter is highly dynamical and includes features not present in structuralconnectivity: understanding the inter-dependence between the two is crucialto develop theoretically-grounded models of brain function, and comparingthe brain to other systems where the emergent synchronization is decoupledfrom structure is a fundamental part of this process [38, 57].

Much available research on synchronization phenomena has been cen-tred around the Kuramoto oscillator, which has the remarkable feature ofhaving analytical solutions under many circumstances; despite the fact thatthe only free variable is phase, Kuramoto networks have been able to re-capitulate findings from diverse physical systems, including populations ofneurons [58, 59]. Extensive numerical studies have also been performed,on diverse network topologies, using oscillators, such as the Stuart-Landauequation, that have free amplitude too, which yields even superior generality[60, 61]. However, by comparison less has been done in the way of study-ing synchronization experimentally, between agents that are physically builtand measured in the laboratory, such as mechanical, electrical, chemicaland optical systems. Such physical approach appears well motivated, be-cause in nature many emergent phenomena are heavily influenced by smallor large parametric mismatches between oscillators, various non-idealities inthe constituent elements and links, noise in the oscillatory variables and sys-tem parameters and so on [17, 62, 63]. As previously discussed, such factorsunavoidably break the “artificial” symmetry found in the paradigmatic net-works often considered in numerical simulations (such as rings, stars) and,by hindering the onset of global synchronization, make way for regions of in-complete synchronization where complex patterns can emerge. While noise,mismatches and so on can obviously be encapsulated in numerical models,experimental investigation is an important complement because such factorsare often not trivial to capture, and their inclusion typically leads to issueswith numerical solvers such as those related to stiffness and step-size choice,even discretization-related effects [20]. It is from this viewpoint, that study-ing networks of physically-realized coupled electronic oscillators appears tobe a relevant manner to recapitulate the complexity of synchronization asobserved in natural systems.

As indicated above, owing to their availability, ease of implementation,

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Figure 4: a) Coupling scheme: neighbour oscillators were connected in aring via resistors of value R2 at each transistor’s collector node. b) View ofthe circuit board, comprising an “outer ring” with the oscillators themselves,and an “inner ring” with precision amplifiers and multiplexers allowing signalreadout with minimal circuit disturbance.

flexibility and low cost, electronic circuits currently represent the preferredmeans of physically realizing networks of non-linear oscillators. This is par-ticularly true for the simplest circuits, those based on a single or at mostfew transistors, as described in [H1]. There is a well-developed literaturedemonstrating the ability to synchronize chaotic modifications of the Col-pitts and Hartley oscillators, as well as the Lindberg-Murali-Tamasevicius(LMT) oscillator, by means of resistors (bi-directional, diffusive coupling)or operational amplifiers and even diodes (unidirectional, master-slave; e.g.[64, 65, 66, 67]). In the case of these circuits, mismatches making oscillatorsnon-identical are inherently present, in the form of parametric tolerancesin both the passive components (inductors, capacitors, resistors) and in thetransistors themselves; such tolerances usually range 1%-10%. Furthermore,non-idealities are present for example in the form of junction capacitanceswithin the transistors, parasitic parallel capacitances for inductors and se-ries inductances for capacitors, yielding these components a vastly morecomplex behaviour than in an ideal case [44]. However, crucially, while theexperimental synchronization of pairs or at most small groups of oscillatorshas been demonstrated repeatedly, for example while pursuing the aim ofgenerating signals having prescribed spectra, until this study surprisinglylittle work had been done in the way of studying large physical networksof electronic chaotic oscillators. This situation was at odds with numericalsimulation studies, where ensembles of hundreds of units have often beenconsidered [68, 69].

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Figure 5: Representative waveforms from two adjacent nodes (red, green),under situations of a) near-complete de-synchronization (R2 = 15000Ω),b) near-complete phase synchronization but largely de-synchronized ampli-tudes (R2 = 1000Ω), c) near-complete synchronization of both phases andamplitudes (R2 = 33Ω).

Figure 6: Plots of average synchronization, measured in terms of NormalizedMutual Information (NMI) as a function of R2, with respect to distancebetween nodes d, for a) continuous measured signal, and b) interpolatedfluctuations of cycle amplitudes.

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Here, chaotic oscillators of the novel type introduced in [H1] were hard-wired in a ring network. A ring was chosen for structural connectivity,because rings are the most symmetric type of elementary network, whereineach node “sees” exactly the same configuration of first and more distantneighbours: rings have homogeneous degree and betweenness centrality, andzero modularity and clustering [20, 51, 70]. The dynamics of ring networkshave demonstrated remarkable generality in their ability to recapture diversebiological phenomena, for example via the formation of “communities” (alsoreferred to as “clusters”) of preferentially-synchronized oscillators (clustersynchronization), long-distance synchronization (remote synchronization),and emergence of travelling and standing waves, and other more complexcollective oscillation modes [20, 51, 71, 72]. In this study, as shown in Figure4, 30 oscillators were coupled diffusively by means of resistors placed betweenthe collector nodes of adjacent transistors, and a special circuit board withconcentric layout, named STRANGE-1, was conceived, which allowed quickselection and precise readout of all time-series, without electrically disturb-ing the oscillators. Those resistors, which were set homogeneously (within5% tolerance) to value R2, effectively acted as second “control parameter” ofthe system, together with the supply series resistors R1, which as describedabove controlled the dynamical regime of the individual oscillators, and wereset arbitrarily (non homogeneously, R1 = 1174±94Ω) for chaotic oscillation,close to the point of transition between periodic and chaotic regions. Threeboards were realized with identical specifications: because in this systemit is the small manufacturing tolerances of the components that break thesymmetry of the ring via the non-linear dynamics, patterns having overallsimilar features but different topography were expected for these exemplars.

In virtue of the fact that the realized circuits were not identical, the fol-lowing behaviours were observed: i) for high values ofR2, e.g. R2 ≫ 10000Ω,the oscillators were completely de-synchronized, ii) as R2 was lowered (whichcorresponds to increasing coupling strength), e.g. R2 ≈ 10000Ω, phase syn-chronization gradually ensued, manifesting itself as time-locking of oscilla-tion cycles alongside uncorrelated cycle amplitudes, iii) as R2 was loweredfurther, e.g. R2 ≪ 10000Ω, amplitudes too became synchronized, until even-tually near-complete synchronization of the whole network was attained.This represents a known phenomenon in networks of parametrically mis-matched (or otherwise non-identical) oscillators: it arose because even forweak coupling strengths, the mismatch in dynamical parameters was suffi-ciently small that the available energy transfer rate between units was capa-ble to some extent of “locking” the periodicity of trajectory in phase space,irrespective of the absence of an invariant manifold (no common trajec-tory) [20, 64, 73, 74, 75, 76]. Representative time-series for situations i)-iii)are presented in Figure 5, and the “progression” of synchronization is well-evident in the plots in Figure 6; as discussed in the published manuscript,the precise choice of synchronization measure was not important. A syn-

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chronization gradient with distance was always present, since the energytransfer rate was inversely determined by coupling resistance R2 × d, whered is the number links separating each node pair.

The aspect that is principally of interest here, is that in situations of“weak” phase- or amplitude-synchronization, very clear pattern formationwas observed according to the respective synchronization measures. Thisappeared in the form of so-called cluster synchronization (also known as“partial” synchronization): pairs and groups of oscillators became prefer-entially entrained as “communities”, in a similarly arbitrary manner. Thisphenomenon is a hallmark of modern synchronization theory, and has beenobserved across diverse physical and simulated systems, having lattice-typeor more complex structural connectivity, though frequently with limitedsize. It represents the formation of manifolds that correspond to the syn-chronized behaviour of different oscillators, that preferentially group or split(effectively “ignoring” each other) depending on dynamical properties alone,in other words, largely irrespective of structural connections. Two obser-vations were particularly relevant in this experiment. First, the intensityof cluster synchronization, measured by means of the modularity index Qcalculated via the Louvain method [77], peaked for different values of R2

depending on whether synchronization was considered for the entire signal(weaker coupling required, R2 ≈ 2200Ω), or just for amplitudes (strongercoupling required, R2 ≈ 120Ω). Second, depending on R2 the size of the“communities” that were formed varied, starting from isolated node pairsand eventually engulfing the entire network, for example this was most ev-ident for phase synchronization in the range 470Ω - 4700Ω. Importantly,as predicted, the topography of the “communities” was completely differentacross the three realized boards: detailed study confirmed that these did nottrivially correspond to similarities in component values, frequency etc. butrepresented a vastly more complex phenomenon, inherently related to thenon-linear dynamics of the system [20, 71, 78, 79]. Representative examplesof such pattern formation are shown in Figure 7, from which it is well-evident that the collective dynamics generated inter-dependencies betweenoscillators that are not present in the elementary, symmetrical structuralconnections.

While for brevity I shall not dwell on this aspect here, also in this studySPICE simulations yielded a limited level of agreement with experiment:the main qualitative phenomena were reproduced, but in a distorted man-ner, for example as regards to the difference in onset of phase and amplitudesynchronization. Notably, simulating this system was particularly computa-tionally demanding due to choice of small steps by the solver, representingthe consequence of different dynamics across the nodes. Together with theresults from [H1], this underlines the important role of physical experimen-tation in the study of these complex systems.

In summary, this study revealed the substantial generative potential of

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Figure 7: Examples of pattern emergence. The elementary ring structuralconnectivity (adjacency, blue, left) yielded richer functional connectivity(time-series synchronization, red, right) featuring long-distance links andcommunity formation. The three functional networks correspond to threephysical boards differing only in small component tolerances, which nev-ertheless lead to completely different “symmetry-breaking” configurations.Functional connectivity networks binarized at 25% completeness for viewing.

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Figure 8: Qualitative similarity between cluster synchronization in a) this in-silico system (micro-scale, 30 oscillators), b) networks spontaneously formedby rat neurons cultured in-vitro (meso-scale, ≈ 50 × 103 neurons, multi-electrode array (MEA) data) and c) spontaneous human brain activityrecorded in-vivo during idleness (macro-scale, ≫ 1010 neurons, functionalmagnetic resonance imaging (fMRI) data). Note: b) and c) only shownin the present document and for visual comparison, derived from publicly-available data courtesy of the authors of Refs. [80, 81]. White-to-red colour-map rescaled to data range.

the single-transistor oscillators described in [H1] when coupled in a networkby means of diffusive coupling. The resulting setup is relatively inexpen-sive and undemanding, yet produced a particularly rich set of synchroniza-tion phenomena, which qualitatively closely resembled observations in neuralcultures and in the brain, particularly as regards to the generation of syn-chronized “communities” (or clusters) at multiple scales, as exemplified inFigure 8. Compared to previous experimental studies of synchronization ofelectronic oscillators, here the larger network size enabled observing spatialsynchronization gradients and the formation of such patterns. This setupappears particularly viable for future experiments, also in light of its flexi-bility, since coupling and dynamics can be readily tuned for each node bymeans of R1 and R2. One example application is the matter of the studydescribed next.

[H3] L. Minati, P. Chiesa, D. Tabarelli, L. D’Incerti, J. Jovicich, Synchro-nization, non-linear dynamics and low-frequency fluctuations: anal-ogy between spontaneous brain activity and networked single-transistor

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chaotic oscillators, Chaos 25, 033107 (2015).

The brain is a complex system characterized by non-trivial network topo-logical and dynamical features, which have evolved under evolutionary pres-sure, seeking an optimal trade-off between performance (e.g., cognitive ca-pability, speed of reaction to external stimuli, resilience to damage) and cost(e.g. volume, metabolic requirements) [82]. One such topological feature isthe presence of a modular network organization, visible at multiple scales(that is, nested small and large “communities”), according to which thereis significant segregation of structural connections and synchronization be-tween sets of cortical regions, loosely corresponding to the inter-dependencebetween cognitive functions [57, 83]. In the previously-described study, itwas shown that networks of chaotic oscillators can readily replicate self-organization into such type of architecture [H2]. Another very importanttopological feature is the co-existence of small-world and scale-free orga-nization, in other words brain connectivity at the same time realizes twofeatures: it maximises the information transfer efficiency for a given densityof connections, and it has a power-law type distribution of node degrees.With some differences, this holds true for both structural and functionalconnectivity, and these features are detectable across micro-, meso- andmacro-scales, subject to application of an appropriate rescaling function tocoarse-grain the system [57, 84, 85, 86, 87, 88] The immediate consequenceof having a power-law distribution of node degrees, is that a small fractionof cortical areas have disproportionally stronger connectivity with respect tothe rest of the brain. These so-called “hub” regions are primarily located inthe precuneus, superior-lateral parietal and medial frontal cortex, and areheavily interconnected between themselves as a “rich-club”. Results fromneuroimaging and lesion studies indicate that these regions are a primaryseat of high-order cognitive processes, and are essential for information in-tegration, memory consolidation, self-awareness and consciousness [89].

From the point of view of dynamical features, the principal propertyis that brain networks seem to operate preferentially as a thermodynamicsystem close to the point of criticality, in other words, at the transitionbetween ordered and disordered collective oscillation [15, 90]. As I shallelaborate more in detail later, evidence for this comes primarily from theobservation of activity avalanches, whose distributions of size and durationobey universal scaling exponents; such observations have been establishedusing a variety of neurophysiological techniques, such as electroencephalog-raphy (EEG) and functional magnetic resonance imaging (fMRI) [91, 92].Recent computational investigations have demonstrated that operation closeto criticality may confer specific advantages [93]. During idle wakefulness,spontaneous brain activity clearly shows the establishment of a small numberof distinguishable resting-state networks (RSN), which are groups of regionsthat consistently generate correlated activity [94]. Remarkably, simulations

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Figure 9: Topographical distribution of synchronization and dynamical in-dices over the brain cortex. a) The distribution of node degree k clearly de-lineated highly-connected “hub” regions. b) In those regions, the correlationdimension curves settled on a “plateau” that was lower for the experimentalthan the surrogate time-series (D2 −D2 > 0). c) Correspondingly, there wastighter “saturation” of the correlation dimension curves, indicating greaterevidence of self-similarity (δD2 − δD2 > 0).

of emergent dynamics based on models of brain structural connectivity re-produce with good accuracy the emergence of such RSNs, even in highlysimplified scenarios such as Kuramoto oscillators or discrete excitable units,insofar as the dynamics are tuned close to criticality [38, 95, 96].

Knowledge that the brain collectively operates close to criticality, to-gether with the presence of a small number of “hub” regions having dis-proportionally intense connectivity, naturally leads to asking the questionwhether the temporal dynamics of activity in such regions have differentproperties compared to the rest of the brain. Such question could be an-swered based on blood oxygen level-dependent (BOLD) time-series repre-senting spontaneous activity (awake rest), recorded using fMRI. Unfortu-nately, the temporal resolution of BOLD time-series is several orders of mag-nitude coarser than the underlying neural activity (i.e., ≈ 1s vs. ≈ 1ms),and there is significant contamination from physiological sources of noiseun-related to neural activity (such as heart beat, respiration) [97]. However,particularly in such “hub” regions, brain activity follows a scale-free distri-bution with 1/fβ-like spectrum, and most power is concentrated ≪ 0.1Hz,that is in a range which can be tracked by the haemodynamical processesat the basis of BOLD activity, and which can be sampled effectively viafMRI recordings [38, 98]. This holds true particularly for recordings per-formed with recently-developed “multi-band” techniques, which provide a

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Figure 10: Representative blood oxygen level-dependent (BOLD) activitytime-series for a) precuneus (high synchronization), and b) pre-/post-centralgyri (low synchronization). Greater amplitude of low-frequency fluctuationsand regularity are clearly noticeable in the former.

sample rate > 1Hz and reduced aliasing of physiological noise [99]. Based onthe time-series made publicly available by the Human Connectome Project(HCP), in a previous study I demonstrated the possibility of performing veryhigh-resolution mapping of brain functional connectivity (≈ 230000 nodes),through solving the resulting computational challenge by means of l1-normapproximation of synchronization (linear correlation coefficient) and appli-cation of dedicated computational hardware [100]. Because on the resultingnode degree maps the “hub” regions appear exceptionally well-delineated,it is reasonable to assume that, despite the limitations described above, thecorresponding time-series must contain some relevant information also aboutdynamical properties.

A limited number of previous studies have queried the presence of non-linear structure in BOLD time-series for specific cortical regions chosen a-priori without reference to connectivity [101, 102]. Here, the more ambitiousgoal was set of performing exhaustive, voxel-level analyses and comparingmeasures of non-linear dynamics to the node degree of activity synchroniza-tion k for each voxel, representing its “level of involvement” in whole-brainsynchronization of spontaneous activity. Besides low sampling frequency(≈ 1.4Hz) and short time-series length (nominally 1,200 points), a challengewas the weak level of determinism expected in the time-series, combinedwith the fact that standard techniques for reduction of physiological noisecannot be applied, as these can introduce spurious non-linear effects. To ad-dress this goal, after careful consideration of the properties of the time-seriesa tailored analysis approach was devised.

First of all, non-linear noise reduction was performed via orthogonalprojection onto a 2-dimensional manifold using a so-called “tricky” met-ric [46, 47, 103]. Second, for each measured time-series a corresponding

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surrogate time-series was calculated, making use of an iterative techniquethat attempts to simultaneously preserve both the temporal autocorrela-tion and the measured value distribution, by means of iterating filteringtowards the desired Fourier amplitudes and rank-ordering the value distri-bution [46, 47, 104]. Third, as already done in [H1] for chaotic signals fromthe transistor oscillators, time-delay embedding was performed and a box-assisted search implementation of the Grassberger-Procaccia algorithm wasused to calculate the correlation dimension curves D2(m, ǫ) (where m and ǫrepresent, respectively, embedding dimension and length scale), separatelyand independently for the measured and surrogate time-series [46, 47, 48].Fourth, we looked for evidence of non-linear structure by means of consider-ing two aspects, namely whether for the measured compared to the surrogatedata i) there was stronger evidence of “saturation” of the correlation dimen-sion curves to a “plateau” in the over-embedding region [m, 2m], and ii) theestimated correlation dimension “plateau” occurred at lower D2 values.

For stochastic dynamics, the correlation dimension curves do not satu-rate to a “plateau” and as a consequence the estimated correlation dimen-sion is only determined by embedding parameters. These analyses werecompleted by means of purpose-developed robust fitting procedures, whichyielded for each voxel two measures, namely D2 −D2 and δD2 − δD2, thatrepresent how well the correlation dimension curves converged to a lowerand tighter plateau for the measured (D2, δD2) than the surrogate data(D2, δD2). In presence of at least some level of deterministic, non-lineardynamics, as opposed to purely stochastic, noise-like activity, one expectsD2 −D2 and δD2 − δD2.

Topographical maps of these parameters were obtained for 10 healthyparticipants, averaged, and compared to the distribution of node degree k.As demonstrated in Figure 9, there was an unequivocal relationship: “hub”regions of the cortex displayed a much stronger signature of non-linear dy-namics compared to the rest of the brain, which was manifest as tighterconvergence of the correlation dimension curves to a lower plateau com-pared to the surrogate data. The corresponding topographical correlationswith node degree k were of moderate intensity and clearly statistically sig-nificant, with ρ = 0.55, p < 0.001 and ρ = 0.52, p < 0.001 for D2 − D2 > 0and δD2 − δD2 > 0 respectively. As previously reported, this finding wasalso accompanied by a spectral shift of activity, which was considerably con-centrated in the < 0.1Hz band for these regions; this is also evident in therepresentative time-series shown in Figure 10. These results have to be con-sidered with caution due to the extensive number of potentially confoundingfactors, however they were corroborated by a range of confirmatory analyses,on which I shall not dwell here. They were in good agreement with existingliterature indicating that temporal fluctuations of brain activity cannot sim-ply be accounted for by linear stochastic processes, suggesting the action oflow-dimensional chaotic dynamics [101, 102]. Importantly, these results go a

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Figure 11: Electronic network of chaotic oscillators. a) Network topology,characterized by intertwined connections among oscillators belonging to seg-ments corresponding to the “hub” nodes (highlighted in red). b) View ofthe realized physical system, comprising three STRANGE-1 boards.

step further and point to a clear topographical relationship between networktopology and dynamics, suggesting that non-linear, possibly chaotic, dynam-ics may be preferentially expressed in the most densely-interconnected areasof the brain cortex.

Predicated on this finding, the question was asked whether such relation-ship could be replicated by networks of single-transistor chaotic oscillators ofthe type considered in the previous study [H2]. To address such question, asshown in Figure 11 the following experimental setup was engineered: threeSTRANGE-1 boards were cascaded, yielding a single ring of 90 oscillators,and additional links were inserted to delineate four hardwired “hub” regions(ring segments). Oscillators in these regions were connected, via an addi-tional resistor per node, to distant oscillators according to an intertwinedpattern, increasing the node degree of each oscillator from 2 to 3, and cre-ating a network that had some level of small-world features. Albeit at anextremely low resolution compared to brain connectivity, this “toy” networkrepresented a similar situation, in that it was characterized by the fact thata minority of nodes had markedly greater connectivity and centrality thanthe rest of the network. To evaluate the effect of oscillator dynamics in re-spect to the collective behaviour of such network, the individual oscillatorswere tuned so that, in absence of any coupling, they would oscillate eitherwell into the chaotic region, or in periodic mode but close to the order-to-chaos transition point. The resulting values of R1 determined empirically(to attain desired oscillation given component tolerances) were, respectively,R1 = 816±156Ω and R1 = 1223±230Ω. Following preliminary experiments,

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Figure 12: Synchronization matrices, determined according to NormalizedMutual Information (NMI) between oscillator time-series, after tuning a)in chaotic region, and b) close to criticality. In both cases, it was evidentthat the additional, long-distance structural connections created “islands”of synchronization engulfing the “hub” regions (nodes 8–10, 25–33, 48–50,and 70–72) and increasing their overall synchronization to the rest of thenetwork.

the resistance value for coupling neighbour nodes was set to R2 = 750Ω andthe resistance for long-distance links was set to R3 = 40Ω; the mismatchwas necessary for experimental reasons, including compensating for lossesin long-distance wiring. The time-series representing the spontaneous activ-ity of this population of oscillators were recorded with the same techniquesdetailed in [H2].

In close similarity to the situation for the brain, the additional struc-tural connections lead to significantly greater synchronization among theoscillators located within the distant “hub” regions that were coupled [38].Effectively these links created “islands” of synchronization that engulfed the“hub” regions and pervasively increased synchronization between them andthe entire network. As shown in Figure 12, this effect was similar regard-less of whether the individual oscillators were tuned in the chaotic region orclose to criticality. Contrariwise, there was a dramatic difference in the rela-tionship between network topology and oscillation of the individual circuits(nodes). When the circuits were individually tuned for chaotic oscillation,this was maintained largely unaltered regardless of network connectivity.However, when they were tuned close to criticality, the following “split”appeared. For the nodes outside the “hub” regions, oscillation remainedlargely periodic, with small cycle amplitude fluctuations probably relatedmainly to frequency mismatches. For the nodes within the “hub” regions,by contrast, the onset of chaotic oscillation was well-evident, in the form oflarge low-frequency fluctuations of cycle amplitude. Remarkably, the gener-ation of strong low-frequency from much faster spike trains is precisely a keyproperty of emergent brain dynamics [38, 95]. Representative time-series are

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Figure 13: Representative time-series from oscillators located a) outside(nodes 17, 40, 60 and 85) and b) within (nodes 9, 29, 49 and 71) the highly-interconnected “hub” regions of the ring. It is evident that in the formercase, oscillation was mostly periodic, whereas in the latter there was onsetof chaotic dynamics, leading to the generation of large, low-frequency cycleamplitude fluctuations.

shown in Figure 13.In the case of the brain, cortical regions outside the “hubs” (low node de-

gree/synchronization) exhibited activity that was primarily stochastic, with-out evidence of non-linear structure; by contrast, such structure appeared foractivity within the “hub” regions (high node degree/synchronization). In thecase of this electronic network, the situation was different because, when theoscillators were tuned close to criticality, outside the “hubs” the activity waslargely periodic (hence low entropy) whereas inside the hubs transition tolow-dimensional chaos occurred, which brought about self-similar dynamicalstructure visible to the correlation dimension analyses. Nevertheless, thereis a commonality because in both cases the evidence for non-linear behaviourwas markedly stronger within than outside the “hubs”, and in both cases thiswas accompanied by the generation of low-frequency, high-amplitude activ-ity. For reasons detailed in the published manuscript and on which I shallnot dwell here for brevity, eventually the effect on D2 −D2 and δD2 − δD2

was remarkably similar, since in the case of the electronic oscillator time-series, the “scrambling” effect of phase randomization was more marked forthe broadband chaotic than the narrow-band periodic time-series.

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In summary, this study revealed a relationship between the regionaltopological and dynamical properties of brain networks, according to whichregions that have higher connectivity generate activity with stronger sig-natures of non-linear dynamics and low-frequency fluctuations. Despiteprofound differences in network size, dynamics, coupling mechanism andspatiotemporal scale, a network obtained from the circuits introduced in[H1,H2] was able to recapture this relationship remarkably closely. Thiswas on the basis of two common properties: i) oscillator tuning close to crit-icality, and ii) presence of “hub” regions, characterized by stronger connec-tivity and greater centrality with respect to the rest of the network. Whilethe results obtained for the brain data require extension and confirmationusing other techniques such as electro- and magneto-encephalography, theclose correspondence with the phenomena observed in the electronic net-work suggests the existence of a relationship that may have general validity.This study thus offers a compelling practical demonstration of the potentialusefulness of electronic networks of chaotic oscillators as models of some as-pects of brain dynamics. Since in this network it is straightforward to tunethe oscillation and coupling parameters, as well as the topology of the ad-ditional links, future studies could attempt to utilize it with the purpose ofmodelling disease-related (e.g., neuro-degenerative) effects, and then com-paring the results with known changes occurring in patients.

[H4] L. Minati, Experimental Implementation of Networked Chaotic Oscil-lators Based on Cross-Coupled Inverter Rings in a CMOS IntegratedCircuit, J Circuit Syst Comp 24, 1550144 (2015).

The studies [H1,H2,H3] discussed above have demonstrated the gen-erative potential of networks of single-transistor chaotic circuits in repli-cating some key features of brain dynamics. Such studies were based onelementary structural connectivity, namely a ring and a simple small-worldnetwork, hence their natural extension would be to evaluate the emergentdynamics based on a more realistic model of structural connectivity in thehuman brain [38, 57, 88, 89, 95, 96]. Even in its simplest embodiment,such model would require around a hundred nodes, and at least few hun-dred links; it is evident that a network this large cannot realistically bebuilt using discrete components, as shown in Figures 4 and 11 for the previ-ous experiments. This problem may be addressed by realizing a dedicatedintegrated circuit, on which a much larger density of oscillators and linkscan be implemented, in more controlled conditions. Yet, an obstacle tosuch approach is represented by the fact that the capacitors and inductorsrequired by the chosen oscillator circuit, shown in Figure 1, are physicallyvery large, hence impractical to realize using standard complementary metal-oxide-silicon (CMOS) processes. This situation is in common with virtually

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Figure 14: Circuit diagram of the proposed “pure” CMOS implementationof the chaotic oscillator, based on three “cross-coupled” inverter rings.

all other chaotic oscillators realized with discrete transistors and operationalamplifiers [64, 65, 67, 76, 105, 106, 107].

In this study, a solution to this problem was sought by means of goingback to the notion of “quasi-periodicity”, which is deemed to possibly un-derlie chaotic oscillation of the circuit in Figure 1, and re-implementing it denovo using a completely different circuit design, better suited for implemen-tation in CMOS technology. While the strict definition of “quasi-periodicity”involves the coexistence of oscillations at frequencies linked by an irrationalratio, in practice, as argued in [H1], given sufficient non-linearity physicaloscillators can become chaotic via the so-called “quasi-periodicity” routestarting from oscillations at frequencies that are linked by a non-trivial,e.g. non-integer, ratio; the requirement for the ratio to be irrational is ef-fectively not relevant, insofar as the ratio is such that oscillation patternswould not repeat excessively closely [43, 108, 109, 110]. The oldest, sim-plest and most compact type of CMOS oscillator is the inverter ring [44],and one can readily realize rings having length n equal to the smallest oddprime numbers: 3, 5, 7 etc. Given two rings having lengths according todifferent odd prime numbers n1, n2, to which we shall refer as n1-ring andn2-ring, the corresponding period ratio n1/n2, while inherently rational, isnot “trivial” in that it cannot embody a fixed phase relationship betweensubsequent oscillation cycles.

This study was based on an oscillator which consisted of a ring of 3 in-verters, to which longer rings were gradually attached, via a specific “cross-coupling” mechanism based on MOSFET diodes and pass-gates. Followingpreliminary work on which I shall not dwell for brevity, each possible pairof inverter rings was interconnected via three diodes placed between i) theinputs of two chosen inverters, ii) their outputs and iii) the input of theinverter in the shorter ring to the output of the inverter in the longer ring.

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Figure 15: a) Microphotograph depicting abutted instances of the oscillator.b) View of the test circuit board, at the centre of which the custom integratedcircuit is visible.

Coupled mismatched inverter rings had already been considered for chaosgeneration, but in the form of an implementation which, in contrast to thepresent one, involved capacitors and resistors, and as such was significantlyless area-efficient [107]. In this study, a “pure” CMOS implementation wasproposed, wherein chaos ensured due to the “competition” between the nat-ural frequencies associated with the mismatched inverter rings, via the non-linearity inherent in the operation of the inverters themselves [42, 43]. Arepresentative realization of the proposed circuit included inverter rings oflengths 3, 5 and 7 and is shown in Figure 14.

A highly simplified model could be formulated by representing each in-verter as a capacitor of value C charged by a transconductance stage withio = vaGi, wherein io represents the output current, va input voltage andGi < 0 transconductance. Because of the finite available supply voltage Vs,practically one has io = Gi[R(va)H(Vs + vo) −R(−va)H(Vs − vo)], whereinR() and H() are, respectively, the ramp and Heaviside functions, which arethe only form of non-linearity present in this model of the circuit. Diodescould be represented as piece-wise resistors between nodes at voltage va andvb having id = GcR(va − vb − Vt). For suitable values of C, Gi and Gc,numerical simulation of this model revealed the generation of time-serieshaving irregular cycle amplitude, which was evident primarily as fluctua-tions of the maxima. The corresponding correlation dimension and largestLyapunov exponent were, respectively, D2 = 2.1±0.1 and λ1 = 0.032±0.001,confirming chaoticity and close overlap with the dynamical features of thesingle-transistor oscillator considered in [H1,H2,H3].

Predicated on this result and further SPICE simulations with a morecomplex model of MOSFET non-linearity, the circuit was physically pro-totyped on a standard 0.7 µm, 1-poly 2-metal CMOS process (C07-D; ONSemi. Inc., Phoenix AZ, USA). As shown in Figure 15, in an attempt toreplicate the results obtained in [H2], a ring network of 24 oscillators was

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Figure 16: Representative signals from an uncoupled oscillator, as a functionof progressive coupling of the inverter rings with length 5, 7 and 9 to the3-ring: a) time-series, b) spectra and c) zero temporal-derivative Poincarésections. Increasing “complexity” of the time-series and underlying attractorgeometry is well-evident.

realized on a single-chip integrated circuit device, named CHARM-1. Diffu-sive coupling was implemented by means of MOSFETs linking neighbours ata chosen circuit node (equivalent to resistor R2 in Figure 4a), and couplingstrength was controlled by a global coupling voltage Vc. A dedicated testboard and fixture allowing read-out of the time-series was also realized. Forlayout-related reasons discussed in the study, for this prototype each oscil-lator comprised 4 inverter rings having lengths 3, 5, 7 and 9, which couldbe individually connected to the ring having length 3 under the control ofdigital signals.

Example waveforms, spectra and Poincaré sections for an uncoupled os-cillator Vc = 0 are shown in Figure 16. As predicted, when the 3-ringwas operating in isolation, a periodic waveform was generated. Couplingthe 5-ring elicited cycle amplitude fluctuations, but the pattern was ratherregular and the spectrum had a comb-like appearance. By contrast, addi-tionally coupling the 7- and 9-rings lead to the appearance of highly irregu-lar cycle amplitude fluctuations, and correspondingly broader spectra. ThePoincaré sections clearly demonstrated increasing geometrical complexity ofthe underlying attractor, which was also apparent as increasing correlationdimension values, namely D2 = 1.1 ± 0.0, 1.8 ± 0.1, 2.2 ± 0.3, 2.5 ± 0.6. Thisresult confirmed the controllability of oscillation dynamics in this circuit, inclose similarity to the consequences of varying the value of the supply series

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Figure 17: Synchronization matrices from five integrated circuit specimens(α, β, γ, δ, ǫ), demonstrating the formation of small “communities” of am-plitude synchronization, whose location depended on manufacturing toler-ances.

resistor R1 visible in Figure 1; straightforward digital control of dynami-cal “complexity” also represents a relevant feature for possible engineeringapplications of this generator [111, 112].

As observed in [H2] for the network of single-transistor oscillators, grad-ually increasing the coupling strength (voltage Vc) initially lead to the ap-pearance of phase synchronization, followed by amplitude synchronization.Namely, for Vc < 1.5V the network was almost completely de-synchronized.At Vc ≈ 2.5V , phase synchronization was strong, but amplitudes remainedlargely de-synchronized. Increasing Vc further, the behaviour depended onwhether the 9-rings were coupled or not: if so, amplitude synchronizationensued above a higher Vc threshold, but then increased more rapidly aboveit. Without the 9-rings, the formation of “communities” of preferentially-synchronized oscillators was also observed, with a topographical distributionthat depended on the manufacturing tolerances (mainly gate oxide thick-ness), and thus was different across integrated circuit specimens as shownin Figure 17.

In summary, this study demonstrated that the oscillator considered in[H1,H2,H3] can be successfully re-implemented in a form that is suitablefor efficient implementation on a CMOS integrated circuit. This is an essen-tial element to allow the future realization of larger networks, for examplehaving the aim of representing more realistically the structural connectiv-ity of the brain. Experiments with such networks are necessary to fullyrealize the approach of studying the relationship between structural con-nectivity and synchronization in the human brain, by means of comparisonto another physical system. Albeit with some unavoidable differences, thepossibility to influence oscillator dynamics, the main qualitative and quanti-tative features of the generated time-series, and the emergence of phase- andcluster-synchronization phenomena depending on coupling strength were allreplicated. More generally, in this study a novel, simple and area-efficientchaos generator was proposed, which may find application also in industrialscenarios such as the generation of random numbers or broadband modula-tion of radio signals [113].

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[H5] L. Minati, Remote synchronization of amplitudes across an experimen-tal ring of non-linear oscillators, Chaos 25, 123107 (2015).

As discussed above, a fundamental aspect of modern synchronizationtheory is that the coupling of non-linear oscillators leads to formation ofsynchronization patterns which do not trivially reflect the structural connec-tivity (adjacency) [20, 51, 52, 53]. Accordingly, in the brain the correspon-dence between structural connectivity and synchronization, albeit statisti-cally significant, is relatively weak because synchronization networks pos-sess topological features not present in structural connectivity [38, 95, 96].Importantly, this decoupling appears to be closely related to the integrityand state of the brain: the correspondence between structural and synchro-nization networks is relatively weak during wakefulness, becomes strongerduring sedation and general anaesthesia inversely correlating with the levelof consciousness, and is also inversely related to the degree of consciousnesspreservation in patients with brain lesions [114, 115, 116]. One emergent fea-ture of brain synchronization networks is the presence of a strongly modulararchitecture, which plausibly represents a form of cluster synchronization(also known as partial synchronization) [83]; in [H2,H3,H4], it was demon-strated that elementary chaotic oscillators wired in a ring topology couldsupport the emergence of cluster synchronization, which was manifest inthe seemingly arbitrary formation of “communities”, intended as subsets ofoscillators that preferentially synchronized as a consequence of small para-metric heterogeneities. Another emergent feature of brain synchronizationnetworks, which by comparison has been less thoroughly investigated but isplausibly at least as important, is the fact that some distant cortical areas,which do not have direct anatomical connections between them, generate ac-tivity that is preferentially synchronized, seemingly without entraining othercortical or subcortical structures which could act as “intermediate stations”for information transfer. This phenomenon has received some attention inreference to addressing the “binding problem” in cognitive psychology, andrecent research has linked it to a form of “remote synchronization”, whichcan emerge as a consequence of symmetry patterns present in the architec-ture of the structural network [117, 118].

In physical terms, remote synchronization is defined with reference toat least three oscillators, two of which are structurally coupled exclusivelythrough the other(s), and with reference to a chosen synchronization met-ric. Remote synchronization is deemed to occur when the two oscillatorsattain a high degree of synchronization, without seemingly entraining theintermediate node(s). As mentioned previously, this phenomenon seems tooccur spontaneously in a variety of distributed physical systems, includingelectrical power distribution networks, meteorological phenomena and brain

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function [54, 55, 118]. Remote synchronization was first studied system-atically in a small star-like network of Stuart-Landau oscillators, whereinsufficient frequency mismatching of the “hub” node allowed the “leaves” tobecome selectively synchronized without involving it [72]. While in thatsystem remote synchronization required the availability of free amplitude,other studies have shown that remote synchronization can also emerge innetworks of phase (Kuramoto) oscillators, contingent for example on de-layed or phase-frustrated coupling [117, 119]. It has also been demonstratedthat remote synchronization can readily ensue in complex network topolo-gies, leading to the emergence of significant decoupling between structuralconnectivity and synchronization, in line with experimental observations forthe brain [120].

In this study, the purpose was to conduct a comprehensive experimentalinvestigation of remote synchronization in a physical network of coupled os-cillators. The motivation is that thus far, remote synchronization had onlybeen demonstrated in the laboratory in the context of the small star-likenetwork mentioned above (5 circuits), implemented by means of operationalamplifiers and analog multipliers, and limited network size severely con-strained the opportunities of observing emergent synchronization patternswith complex topological features [72, 120]. In the course of preliminaryexperiments on which I shall not dwell here, it was concluded that the net-works considered in [H2,H3,H4] probably cannot yield remote synchroniza-tion effects. Hence, a novel implementation approach was devised to allowthe realization of a suitable ring of 32 non-linear oscillators, with completedigital (software) control of all circuit parameters. This feature was par-ticularly important to enable flexible selection of the control parameters,and exhaustive exploration of the corresponding space, searching for re-gions where remote synchronization would occur. To realize this device, theField Programmable Analog Array (FPAA) technology was chosen. In brief,FPAAs are the analog equivalent of the well-known programmable logic de-vices: they contain a wide range of building blocks (amplifiers, filters etc.),which are constructed with switched-capacitor circuits, and interconnectedvia a programmable array of switches. As a consequence, one can realizenear-arbitrary circuits which operate in discrete time but continuous ampli-tude, and change the circuit topology and parameters on-the-fly simply bydownloading a new bit-stream [121, 122].

While a large literature on chaotic oscillators implemented in FPAAsis available, the known circuits primarily implement multi-scroll systems orare too large to physically realize a network with a large enough number ofnodes [29, 123, 124, 125]. Hence, in this study a novel circuit was devised,pursuing the following aims: i) it needed to be small enough to fit in a singleFPAA, so as to allow realizing a large network, ii) it needed to maintain someof the features of the oscillators introduced in [H1,H2,H3,H4], particularlyexpression of chaoticity in the form of continuously variable cycle amplitude,

31

Figure 18: a) Architecture of the non-linear oscillator circuit implemented inan FPAA, demonstrating the ring of negative-gain stages, the mixing stageand the integrators. Each oscillator received as input the signal from theprevious one on the ring. b) Time-series from all internal nodes of an oscil-lator obtained for a representative set of parameter values, demonstratingthe occurrence of saturation, particularly at the output nodes of the twointegrators, v3 and v5.

without discretization imposed by a landscape of separate attractors, iii) itneeded to oscillate for a wide, continuous range of parameter settings, thusallowing exploration of network dynamics over multiple parameter regions,iv) it needed to be easily synchronizable. The oscillator that was designedcan be viewed as a type of ring oscillator, conceptually similar to the onesshown in Figure 14. At its heart, a loop of three negative-gain stages wasinstanced. This loop was closed by means of a mixing stage, which allowedcontrolling the gain of the loop, and mixing its signal with that of anotheroscillator (another node in the network). Since implementing longer loopswould not have been viable due to limited resources inside a single FPAA,to introduce chaos two integrators were instanced instead, and “overlapped”to two of the negative-gain stages in the loop. Their integration constantswere related by the fixed ratio 3.67, which hypothetically “promoted” theonset of chaos via quasi-periodicity, as previously discussed with referenceto [H4]. The resulting circuit is shown in Figure 18, wherein the waveforms

32

corresponding to a representative oscillator are also visible: it is evidentthat in this circuit, non-linearity was introduced primarily by saturationat the outputs of the two integrators (nodes v3 and v5). Accordingly, anacceptable piece-wise linear approximation of the dynamics could be ob-tained considering the following system of ordinary differential equations,wherein n ∈ [1, 32] represents oscillator position on the ring, v1 . . . v6 theinternal voltages, G1 . . . G7 the internal gains, F1 . . . F4 the low-pass cut-offfrequencies, and K1,K2 the integration constants:

dv1,n

dt= Γ (2πF1,n(G4,nv4,n +G5,nv5,n − v1,n), v1,n, Sn)

dv2,n

dt= Γ (2πF2,n(G1,nv6,n − v2,n), v2,n, Sn)

dv3,n

dt= Γ (K1,nv6,n, v3,n, Sn)

dv4,n

dt= Γ (2πF3,n(G2,nv2,n +G3,nv3,n − v4,n), v4,n, Sn)

dv5,n

dt= Γ (K2,nv2,n, v5,n, Sn)

dv6,n

dt= Γ (2πF4,n(G6,nv1,n − 0.4G7,nv6,n − v6,n), v6,n, Sn)

and where saturation to the supply voltage Sn is represented by the function

Γ

(

dv

dt, v, S

)

= R

(

dv

dt

)

H(S − v) −R

(

−dv

dt

)

H(S + v) .

Without dwelling on the detailed description of the equations and param-eters, which is provided in the published manuscript, here it is highlightedthat the gains G6 and G7 represented, respectively, the internal loop gainand the strength of coupling of each oscillator to its preceding neighbouron the ring via v6; in this sense, the network was different from thosein [H2,H3,H4], because here coupling was not diffusive but master-slave.These two parameters, alongside K1, acted as control parameters for thenetwork, and the others were treated as constants.

Depending on G6, G7 and K1, numerical simulations according to thepiece-wise linear approximation revealed distinct collective behaviours ofthe oscillators. For suitable values, the following scenario ensued: phaseswere near-completely synchronized, but cycle amplitude fluctuations con-comitantly delineated two forms of remote synchronization, one wherebydistant ring segments became entrained without involving the intermediatenodes, and another whereby oscillators at approximately fixed, non-unitarytopological distance became entrained. It was found that such effect re-quires parametric mismatch between the oscillators, however this could beas small as 10−8 in relative terms. As demonstrated in Figure 19, while the

33

Figure 19: Representative synchronization matrices, calculated according tothe maximum cross-correlation coefficient, demonstrating the emergence ofremote synchronization given G1 = −3.6, G2 = −3.12, G3 = −0.5, G4 =−3.08, G5 = −0.71, G6 = 0.188, G7 = −1.14, F1 = F2 = F3 = 2kHz,F4 = 100kHz, S = 4V, K1 = 0.11 µs−1. The matrices represent 10 separateruns with 0.5% random parametric variation, which determined the observeddistribution of remotely-synchronized node subsets.

average settings of the control parameters determined whether remote syn-chronization could ensue or not, the precise pattern depended on such smallparametric mismatches. This situation recalls previous results on clustersynchronization [20, 78, 79], as well as the experimental observations re-ported in [H2,H4]; as regards remote synchronization, the observed effectwas novel because the previous studies did not demonstrate in such explicitterms the possibility of formation of extended patterns. Moreover, while in[72, 120] free amplitude was posited as a requirement for “hidden” informa-tion transfer but remote synchronization was actually observed for phases,by contrast here phases were globally synchronized and it was amplitudefluctuations that exhibited the remote synchronization effect.

To confirm these results experimentally and extend them, in this studythe largest experimental network of FPAAs known to date was designed andbuilt in the form of a dedicated circuit board, named LYAPUNOV-1 andvisible in Figure 20, which comprised a 4-by-8 array of FPAAs connectedin a ring, alongside the supporting data acquisition, signal conditioning andinterfacing circuitry. A large set of time-series was acquired, for a range ofsettings of G6, G7 and K1; these were made freely available, alongside theboard design, to foster future research in this area. In this experimentalsystem, the parametric mismatches were on the order of 0.1-1% and causedprimarily by integrated circuit manufacturing tolerances, furthermore thenon-linearity was more complex than a piece-wise linear approximation, ow-ing to the behaviour of the physical circuits as saturation is approached;notably, performing data acquisition by means of this board also yielded

34

Figure 20: a) Architecture of the realized electronic system enabling ex-periment control and data acquisition, effectively an analog plug-in com-puter in the form of a PCI expansion board. b) View of the correspondingLYAPUNOV-1 circuit board.

considerable acceleration with respect to numerical simulation of the net-work on CPU [29, 122].

The average correlation dimension of the time-series 〈D2〉, calculatedacross all oscillators as in [H1,H3,H4], revealed that the dynamics of thenetwork delineated a “chaotic band” as a function of G6, G7 and K1, whereincollectively chaotic oscillation ensued with various levels of “complexity”,represented by 〈D2〉 reaching ≈ 3.5. Consideration of the degree of phasecoherence (also referred to as phase locking, Kuramoto order parameter),separately for the measured signal and for its amplitude fluctuations deter-mined via the analytic signal, referred to respectively as r and r, revealedthat the network almost invariably maintained strong phase synchroniza-

35

Figure 21: Parametric maps of the average correlation dimension 〈D2〉 ofthe recorded time-series as a function of the control parameters: oscillatorloop gain G6, coupling strength G7 and integrator constant K1. The ap-pearance of a “chaotic band” is evident, and the attained level of dynamicalcomplexity within it increased with K1.

tion, while undergoing a number of transitions as regards to the synchro-nization of amplitudes. The emergent behaviour, summarized in Figure 21,was therefore vastly more complex compared to that of the networks con-sidered in [H2,H4]. For the case K1 = 0.11µs−1, the network operated inan intermediate phase between regular and fully chaotic oscillation, whichcould have represented a phase close to criticality: it is in this phase thatremote synchronization was observed most markedly.

Owing to the master-slave coupling scheme, the system also exhibited lagsynchronization, which could be artefactually interpreted as remote synchro-nization [126]. Consideration of the maximum cross-correlation coefficient,or time-lag variants of the phase coherence index etc. nevertheless confirmedthat the system also exhibited “true” remote synchronization, with qualita-tive features closely similar to observations in the numerical simulations. Inthis sense, the agreement was better in comparison to the SPICE simulationsreported in [H1,H2]. An example of remote synchronization, alongside theunderlying time-series, is shown in Figure 22. Two aspects were particularlynoteworthy.

First, remote synchronization was highly robust to the choice of synchro-nization measure, appearing irrespective of whether phase coherence, linearcorrelation, or normalized mutual information were considered: in this sense,all these measures were equivalently “blind” to the “hidden” informationtransfer taking place between the remotely-synchronized nodes. However,generalized synchronization, which involves considering not just single mea-

36

Figure 22: Representative matrix demonstrating a case of remote synchro-nization (maximum cross-correlation coefficient) and corresponding under-lying time-series for varying node separation distance. Synchronization wasinitially high between neighbouring nodes, then decreased down to near-complete de-synchronization, then increased again inside an “island” of re-mote synchronization. To aid visual comparison, the time-series were re-aligned according to lag of maximum cross-correlation.

37

Figure 23: Representative networks with significant small-world features, ob-tained by thresholding the emergent remote synchronization patterns (max-imum cross-correlation coefficient), a) G6 = 0.132, G7 = −1.44, K1 = 0.11yielding 〈SWS〉 = 1.81 with k = 0.31, b) G6 = 0.12, G7 = −1.56, K1 = 0.121yielding 〈SWS〉 = 1.73 with k = 0.39, c) G6 = 0.136, G7 = −1.53,K1 = 0.132 yielding 〈SWS〉 = 1.97 with k = 0.30.

surement times but immersed time series, was capable of detecting a grad-ual gradient of synchronization otherwise not visible. This suggests thatthe underlying inter-dependence may have been according to a more generalfunctional relationship of the form y(t) = ψ(x(t)), that is concealed whenonly considering pair-wise correspondence between measurements [127, 128].

Second, for reasonable threshold values, the networks resulting from bi-narizing the remote synchronization patterns had strong small-world fea-tures according to the formula

SWS = γWSg /λg =

(

CWSg Lrand

)

/(

CWSrandLg

)

,

where CWS represents average clustering and L harmonic mean path length,for the measured network g and a for corresponding random network. Namely,for the three representative examples shown in Figure 23, 〈SWS〉 ≥ 1.7 (av-erage over n = 100 random networks), indicating that in comparison toErdős–Rényi random networks, there was significantly higher clustering rel-ative to path length [129, 130]. While for brevity I shall not dwell on thisaspect here, it was also found that the synchronization patterns were highlynon-stationary, posing a challenge to the validity of scalar, “grand average”measures of synchronization and closely recalling observations of similarphenomena in brain activity [20, 131].

In summary, this study demonstrated that physically-realized rings ofelectronic chaotic oscillators can support the emergence not only of clustersynchronization as shown in [H2,H4] but also of remote synchronization.To my knowledge, the system that was built and investigated representsthe largest experimental network in which this phenomenon has been ob-served thus far, and the network size enabled observing in a clear mannerthe formation of extended synchronization patterns with aspects of remote

38

synchronization. This phenomenon emerged specifically for the synchro-nization of amplitude fluctuations, representing a difference with respect tosystems considered previously, and appeared related to collective oscillationin an intermediate phase between periodicity and maximum complexity. Thefact that the resulting synchronization networks possessed significant small-world features yields a powerful demonstration of the generative potentialof these circuits, and provides further motivation for considering this phe-nomenon as one of the mechanisms by means of which complex topology mayhave emerged in neural networks [57, 82], and even more generally as oneof the possible mechanisms underlying morphogenesis in nature [56]. Theobservation that a measure of generalized synchronization could, at least tosome extent, detect the information transfer between nodes motivates futurestudy focused on the dynamics of this system, represented not just as scalartime-series but in a more general (higher dimensional) embedding space.

Indeed, while the results are intriguing and there was good overall agree-ment between numerical simulations and experiment, the underlying mech-anism by which remote synchronization appeared in this system remainsunclear. Unlike previous studies wherein the “hub” nodes were a-priori,intentionally mismatched to prevent their synchronization, here remote syn-chronization emerged completely spontaneously, potentially representing amore “ecologically valid” metaphor of dynamics in self-organized neural sys-tems. The synchronization patterns that emerged, plausibly, represent both“local” effects, understandable in terms of given oscillators “coding” synchro-nization information in a manner that is eventually “reversed” after suffi-cient hops, and “global” effects, that are the reflection of collective oscillationmodes related to the dimension and symmetry of the underlying ring net-work. The present results motivate investigating these aspects more deeply,numerically and experimentally, and searching for corresponding phenomenain time-series derived from neurophysiological recordings acquired in-vitroand in-vivo.

[H6] L. Minati A. De Candia, S. Scarpetta, Critical phenomena at a first-order phase transition in a lattice of glow lamps: Experimental findingsand analogy to neural activity, Chaos 26, 073103 (2016).

The studies discussed above [H2,H3,H4,H5] have shown that networksof chaotic oscillators can give rise to complex synchronization phenomena,leading to the formation of patterns not present in the structural connec-tivity. However, it is widely agreed upon that non-linear dynamics andpotential chaoticity are not, in themselves, sufficient to support emergenceof the diverse complex phenomena observed in the brain, and in otherself-organized dynamical systems. The other essential element is opera-tion close to criticality, namely, close to the point of transition between

39

ordered (periodic, laminar) and disordered (chaotic, turbulent) dynamicalphases [15, 90, 132, 133]. This experiment represented an initial attempt toreplicate in a simple, non-linear electronic circuit some phenomena directlyrelated to this property.

While for brevity I shall not dwell on this aspect, it is a remarkable factthat, upon suitable choice of the physical variables and observation scale,much of the thermodynamical theories about phase transitions can be ap-plied to diverse dynamical systems, yielding a robust theoretical frameworkwithin which to formulate predictions about the statistical properties thatare expected [133, 134]. In physical systems, such as the paradigmatic caseof the Ising model, the correlation length and other observables diverge atthe critical point: as T → Tc, ξ → ∞. In proximity of the critical point,such observables follow the relationship A(T ) ∝ (T − Tc)

α, where α is acritical exponent [135]. It is found that rather diverse physical systems fea-ture identical critical exponents, denoting the presence of a small numberof “universality classes”, which are determined by the nature of the inter-actions, symmetry etc. The study of collective dynamics in systems vastlydifferent from those classically dealt with in thermodynamics has revealedthe pervasive occurrence of power-law relationships according to universalscaling exponents, with examples found, for example, in geophysics and me-teorology, flocking and swarming behaviours, economics, and neuroscience[15, 90, 136, 137, 138]. At present, the most compelling account for whysuch diverse systems universally show critical-like phenomena is providedby the theory of self-organized criticality (SOC), which posits the emer-gence of an attractor drawing dynamics towards criticality irrespective ofthe system-specific features [132, 139, 140].

The view that the brain operates preferentially close to criticality issubstantiated by experimental observations, mainly of the generation ofactivity according to “avalanches” whose size and duration distributionsfollow the critical exponents αS = 3/2 and αD = 2, yielding power spec-tral density which decays as 1/fβ noise; scale-freeness is also evident inthe distribution of node degrees of structural and synchronization networks[15, 84, 90, 98, 141, 41]. While the field is still evolving rapidly, com-pelling evidence of these properties has been acquired across the micro-,meso- and macro-scales, using in-vivo techniques such as electroencephalog-raphy (EEG), magnetoencephalography (MEG) and functional MRI (fMRI),and in-vitro techniques such as recording the activity of spontaneously-developing neural cultures or brain slices on multi-electrode arrays (MEA)[91, 92, 93, 142]. There is also a growing body of computational studieswhich have demonstrated that operation at, or close to, the point of criti-cality may confer specific advantages in terms of maximizing the availabledynamic range to encode stimuli and internal variables, and the ability torapidly switch between activity patterns [93, 143, 144, 145].

Despite the fact that the view of the brain operating as a thermodynam-

40

Figure 24: a) Circuit diagram showing four units of the lattice, each oneconsisting of a glow lamp, connected to a global, static control voltage Vs

via a resistor of value R and coupled to its neighbours via capacitors of valueC. b) View of the corresponding VAN DER POL-1 circuit board, realizinga 34 × 34 array of such units (photograph with long exposure time).

ical system close to criticality is having a profound impact on our under-standing of how it supports such a large repertoire of complex behaviours,little work has been done in the way of attempting to replicate critical braindynamics in other physical systems, such as electronic circuits. While inthe previously-discussed studies [H1,H2,H3,H4,H5] some phenomena weredemonstrated which seemed to occur preferentially when the individual os-cillators were tuned close to a transition point, no attempts were made toexplicitly replicate critical phenomena such as the emergence of scale-freerelationships according to critical exponents [139, 140, 141]. In preliminaryexperiments, not described for brevity, it was concluded that the dynamicalproperties of the circuits considered in [H1,H2,H3,H4,H5] are not idealfor demonstrating critical phenomena in the form of avalanching. Address-ing this issue, in this study a specific, elementary circuit was developed,predicated on the type of integrate-and-fire dynamics which are known togive rise to critical phenomena in, for example, neural systems, earthquakesand nuclear reactions [93, 146, 147, 148].

As shown in Figure 24a the circuit consisted of a two-dimensional latticeof units, each of which comprised a glow lamp (neon bulb) coupled to itsfour neighbours via capacitors of fixed value C, and polarized by means ofa resistor of fixed value R towards a static voltage Vs, which was appliedglobally and represented the only control parameter of the system. Whileglow lamps are nowadays largely relegated to the function of low-cost linevoltage indicators, their physical properties are complex and they can sup-

41

Figure 25: Cycle plots for a) average event rate f , b) degree of spatial order− log10(Fs) and c) degree of temporal order − log10(Ft), where F denotesthe Fano factor, i.e. variance divided by the average. As the voltage Vs wasincreased from 72.4V (near-complete quiescence) to ≈74.2V, the event rategradually increased and activity became more disordered, until transition toPhase II occurred; hystheresis was thereafter well-evident, in that the PhaseI → Phase II transition occurred at ≈73.3V.

port the emergence of very rich dynamics; they were once used to realizeactive circuits, and electronic chaos was indeed discovered by Van der Polin a glow lamp-based oscillator [27, 149, 150, 151]. As a function of theapplied voltage vi, a glow lamp transitions hysteretically between an “on”state wherein it behaves like a non-linear resistor and an “off” state whereinit behaves largely like an open circuit; the transitions are stochastic and toa good approximation governed by rates given by e(vi−Vb)/α and e(Vr−vi)/β ,where the breakdown and recovery voltages in this circuit were Vb ≈ 76.2Vand Vr ≈ 61.3V, determined by physical parameters that are subject tosignificant manufacturing tolerances, and α ≈ β ≈ 0.4 [149, 150]. In thisstudy, a 34 × 34 array was realized on a circuit board visible in Figure 24band named VAN DER POL-1, and its dynamics were studied as a functionof applied control voltage Vs, gathering data by means of a custom setupinvolving two synchronized CCD cameras and a high-speed photodiode.

The fundamental operating principle of this system was that followinga breakdown event (generation of a brief flash) at a given site, a transientvoltage ≫ Vs was generated, with a distribution across the lattice such thatits peak amplitude occurred at the farthest sites: this meant that, despitethe short-range structural coupling and the presence of some short-rangeeffects, interactions were effectively long-range and therefore approximatelymean-field behaviour was expected, as also observed in some other phys-ical systems with short-range coupling [152, 153]. To begin investigatingthe behaviour of the network, separate measurements (50 s-long record-ings) were initially performed, ramping up the control voltage Vs to eachtarget value, evenly distributed between 73V and 75V in steps of 0.1V (ap-proximately corresponding to the distribution of breakdown voltages acrosslamps). These measurements revealed the presence of two clearly-distinct

42

dynamical phases, referred to as Phase I and Phase II, the transition be-tween which occurred at ≈ 74.2V. Phase I was characterized by low rateof breakdown events ≈ 0.01Hz and low level of spatio-temporal order, asrepresented by the spatial and temporal Fano factors (variance divided byaverage, logarithm-transformed); the distributions of inter-event intervalsand number of events per 20 ms-long frame were, respectively, power law-like and exponential-like. By contrast, Phase II was characterized by aconsiderably higher event rate > 0.4Hz, and had normal-like distributionsof inter-event intervals and number of events per frame, with average re-spectively ≈ 3ms and ≈ 10 at Vs = 74.2V. Phases I and II were, therefore,respectively “glass-like” and “crystal-like” dynamical phases; they did notcorrespond to specific states of the isolated glow lamps, but emerged fromthe collective dynamics of the coupled units.

Closer consideration of the results revealed that Phase I was metastable,and that the system transitioned spontaneously to Phase II after a timethat rapidly diminished with Vs → 74.3V. To gain further insight, continu-ous sweeps were thereafter performed, cycling Vs from 72.4V to 74.2V thenback to 73.1V. As visible in Figure 25, these revealed that the transitionPhase I ↔ Phase II was also associated with significant hysteresis: withinthe chosen temporal observation window, the Phase I → Phase II transitionoccurred at Vs ≈ 74.2V, whereas the Phase II → Phase I transition oc-curred at Vs ≈ 73.3V. This observation of metastability and hysteresis wasunexpected and intriguing for multiple reasons. First, these properties didnot trivially map onto the distributions of the “on” ↔ “off” (i.e., breakdownand recovery) transition voltages previously measured for the individual glowlamps. Second, metastability and hysteresis are frequently observed on in-vivo and in-vitro neurophysiological recordings and numerical simulations,and in particular metastability is deemed to be directly related to the brain’sessential ability to “lock in” activity patterns that have long but finite life-times [154, 155]. Third, most importantly the observation of metastabilityand hysteresis unequivocally signalled that the transition between PhasesI and II was a first-order (discontinuous) one. As critical phenomena suchas avalanching and the emergence of power-law relationship are usually as-sociated with second-order (continuous) phase transitions, upon superficialconsideration one might then not have expected to observe such phenomenain this system [135].

Instead, in contrast to such expectation, it was observed that for break-down events occurring in Phase I (that is, prior to spontaneous transition toPhase II), activity (generation of flashes) occurred in the form of avalanches,exemplified in Figure 26. Moreover, for Vs → 74.3V, the branching param-eter σ → 1 [146]. The avalanche size and duration distributions divergedfor Vs → 74.3V, approximating increasingly closely power-laws having ex-ponents αS = 3/2 and αD = 2, as shown in Figure 27. Moreover, power-law scaling of the temporal Fano factor was present, according to exponent

43

Figure 26: Representative avalanche, comprising 16 events and observedduring Phase I for Vs = 74.2V (t = 0 was chosen arbitrarily).

δ ≈ 0.55 [141]. Activity propagation according to a near-unitary branch-ing parameter, divergence of avalanche size and duration, and presence offractal temporal structure altogether strongly hallmarked critical behaviour,and replicated qualitatively and quantitatively the previous observations inneural systems, particularly in-vitro neural cultures, whose dynamics arewell-accounted for by a critical branching process model [91, 93, 156]. Never-theless a relevant difference was that whereas mature neural cultures possessa highly-structured “repertoire” of activity, for this circuit the topographicaldistribution of avalanches appeared to be purely stochastic [80, 157].

The apparent incongruence between the observation of metastability andhysteresis on one hand, and critical behaviour on the other, was resolved byconsidering that, even though this appears to be a less frequent occurrence,critical phenomena can also be found in first-order phase transitions as oneenters the metastability region and approaches the spinodal curve (in thiscase, a spinodal voltage). This situation is exemplified by recent work ongeophysical phenomena, fracture propagation and network recovery, whereinclose to the spinodal, transition precursors follow power-law scaling diverg-ing to infinity on the spinodal itself [152, 153, 158, 159, 160]. For largeenough Vs, Phase I effectively corresponded to the “superheated” state in

the liquid → gas transition of water. Two spinodal voltages V(1)

s < V(2)

s

were predicted to exist: V(1)

s , at which the lifetime of Phase II → 0, and

V(2)

s , at which the lifetime of Phase I → 0. In these experiments, the voltage

Vs = 74.3V corresponded to an approximation of V(2)

s . Additionally, a volt-

age V(m)

s such that V(1)

s < V(m)

s < V(2)

s was predicted to exist, at which thelifetimes of Phases I and II are equal; this voltage was not observed directlydue to the limited length of the recording window compared to the lifetimeof the phases.

The observed critical phenomena can be practically difficult to distin-

44

Figure 27: Distribution of a) avalanche size (number of sites involved in anavalanche) and b) avalanche duration (normalized time-span from first tolast event in an avalanche). As the voltage Vs was increased, the distribu-tions diverged towards power-laws with critical exponents αS = 3/2 andαD = 2 (grey: lower voltages, red: Vs = 74.2V; Vs = 74.3V not shown dueto insufficient duration of Phase I).

guish from those expected in a second-order phase transition. However, inthis study a relevant confirmation of the order of the transition came fromthe fact that σ > 1 was never observed, in other words the system was neversuper-critical; instead, it altogether transitioned to another phase, namelyPhase II, as expected due to the vanishing lifetime of Phase I approachingthe spinodal curve [146, 159]. These results are in agreement with a tran-sition belonging to the same universality class of breakdown in disorderedmedia, or the transition in the democratic fibre bundle model (DFBM), orin the long-range Ising model on the spinodal lines [152, 153, 161, 162, 163].

In summary, this study introduced an elementary electronic network,which was capable of simultaneously reproducing two fundamental prop-erties of neural dynamics: metastability and hysteresis on one hand, andcritical behaviour on the other. While at a phenomenological level the sim-ilarity to biological recordings was remarkable, there is a fundamental dif-ference in that this system was externally tuned via Vs, whereas biologicalneural systems are thought to reach criticality spontaneously, via a self-organized internal process. Moreover, in this system the phase in whichcritical phenomena were observed had a finite life-time. Nevertheless, theco-existence of metastability and hysteresis on one hand and critical phe-nomena on the other, according to dynamics near the spinodal curve of afirst-order phase transition, raised the intriguing question of whether a sim-ilar situation might actually hold true also for biological neural networks:future theoretical and experimental work will need to address this questionin detail. Further research is also necessary to consider networks with aricher connectivity structure (e.g., small-world, scale-free, as opposed to a

45

regular lattice), which I posit could yield more complex dynamics predicatedon the fact that the interactions also exerted short-range influences.

46

4 Summary

These studies constitute a cohesive body of work which had the overarch-ing purpose of realizing physical models capturing some emergent propertiesof biological neural systems, in the form of electronic networks of chaoticoscillators. The realization of such oscillators was demonstrated by meansof a heterogeneous set of technologies, including discrete bipolar-junctiontransistors (BJTs) [H1,H2,H3], a custom CMOS integrated circuit [H4],field-programmable analog arrays (FPAAs) [H5], and glow-lamps [H6]. Themain results about the oscillator circuits and their applications are summa-rized below:

• Five novel BJT-based chaotic oscillators, not representing adaptationsof pre-existing oscillators, were built and extensively characterized ex-perimentally. These autonomous oscillators have two relevant features.First, their dynamics are easily and continuously controllable by meansof a resistor connected in series to the supply voltage, acting as controlparameter as a function of which a complex structure of transitionsand bifurcations is observed. Second, they generate signals that are“neural-like” in the sense that as a function of the value of such resis-tor, one observes the generation of periodic activity, volleys or “bursts”of activity, and oscillations characterized by large cycle-to-cycle am-plitude variation. This recalls observations in neural preparations asa function of biochemical variables [4, 30, 31, 32, 33, 34, 35]. [H1]

• One route-to-chaos mechanism in such circuits was postulated to bequasi-periodicity, embodied by means of the large numbers of availableLC combinations resonating at frequencies not linked by trivial ratios,and determined by the chosen components as well as the associatedparasitic elements [42, 43, 45, 108, 109, 110]. Such combinations weremade to resonate depending on the circuit dynamics controlled bythe series resistor. This mechanism was thereafter re-implemented intwo rather different technologies, namely CMOS- and FPAA-basedcircuits, realizing the overlap and interaction between activity at non-trivial frequency ratios by means of structures optimally suited forthese technologies, namely cross-coupled inverter rings for the former,and overlapped ring oscillators and integrators for the latter. As aresult, oscillators with features similar to the BJT-based circuits wereobtained, but with substantial advantages in terms of physical size anddigital controllability. [H1,H4,H5]

• Similar partial synchronization effects were demonstrated experimen-tally across ring networks built with heterogeneous oscillators, namelyBJT-based circuits (30 nodes, diffusively-coupled), CMOS-based cir-cuits (24 nodes, diffusively-coupled) and FPAA-based circuits (32 nodes,

47

master-slave coupled). These effects primarily consisted of the gradualonset of phase-locking, followed by entraining of cycle-to-cycle am-plitude variations, eventually complete synchronization, as the cou-pling strength was increased [20, 43, 51, 73, 74, 75, 76]. Exclusivelyfor FPAA-based circuits, lag synchronization was also observed [126].[H2,H4,H5]

• In the same ring networks, the spontaneous occurrence of pattern for-mation was also demonstrated, and was driven by small parametricvariations due to manufacturing tolerances in the oscillator elementswhich broke the symmetry of the ring. Cluster synchronization, in-tended as the formation of pairs or subsets of structurally-adjacentoscillators that preferentially became synchronized, was observed forintermediate levels of coupling strength in BJT-, CMOS- and FPAA-based circuits [71, 76, 78, 79]. Remote synchronization, intendedas the preferential synchronization of structurally-distant oscillatorswithout entraining the intermediate ones on the ring, also occurred;this phenomenon, however, was only found in FPAA-based circuits,for synchronization of amplitude fluctuations, and for parameter set-tings yielding dynamics with an “intermediate” level of complexity[72, 117, 118, 119, 120]. These results confirm experimentally thatthe dynamics of these networks can support considerable morphogen-esis, and in particular they recall the emergence of multi-scale mod-ular architecture and the decoupling between structural connectivityand synchronization observed in the brain [20, 51, 52, 53, 56, 57, 83].[H2,H4,H5]

• Based on blood oxygen level-dependent (BOLD) time-series represent-ing spontaneous brain activity, a relationship between network con-nectivity and dynamics was identified: activity in the most intensely-synchronized cortical areas was associated with a shift towards large-amplitude low-frequency oscillations, and with emergence of non-linearstructure, visible in the form of self-similarity according to the cor-relation dimension D2(m, ǫ) curves. A “toy” network of BJT-basedoscillators, consisting of a ring onto which four “hub” regions werehard-wired, was built and successfully replicated such relationship,insofar as the individual oscillators were tuned close to criticality[15, 57, 89, 90, 93, 95, 114]. This result is particularly important asit outlines a relationship between connectivity and dynamics which isseemingly generalizable across very different systems, and it confirmsthe concrete usefulness of electronic networks of chaotic oscillationsfor testing hypotheses in this area. [H3]

• A novel approach to realizing chaotic oscillators in CMOS technologywas proposed, which consisted of “cross-coupling” inverter rings having

48

mismatched, odd lengths not related by trivial ratios, for example asspecified by the smallest odd prime numbers. This approach was pred-icated on the hypothesis of quasi-periodicity route to chaos, and en-ables straightforward digital control of chaos generation via enabling ordisabling the coupling of specific rings. It has important practical ad-vantages with respect to existing technology in that it does not requireimplementing any resistor, capacitor or inductor [29, 107, 111, 112].Since such components are physically larger than inverter rings whenrealized on an integrated circuit, the proposed approach maximisesarea efficiency: as such, besides its main intended application in thefuture realization of large neuromorphic networks, it may be worthy ofgeneral consideration as the preferred circuit architecture for realizingcompact chaos generators in various engineering applications (e.g., asentropy source in embedded systems). [H4]

• Similarly, a novel chaotic oscillator circuit suitable for implementa-tion in a single FPAAs was proposed, with advantages over knowncircuits in terms of minimal occupancy, continuous controllability ofchaos generation according to a “chaotic band” and synchronizability[29, 124, 125]. It consisted of a ring of three negative-gain stages, sim-ilar to the CMOS-based realization, but in this case chaos ensued dueto mixing the signal with the outputs of two integrators having mis-matched integration constants, rather than due to coupling with ringshaving different length. Based on a ring network of such oscillators, anovel form of remote synchronization was demonstrated, whereby am-plitude fluctuations become selectively entrained across distant nodes,with underlying global phase synchronization; this result extendedprevious findings, in which free amplitude was necessary for remotesynchronization to occur, but amplitude fluctuations only acted as a“vehicle” for the remote synchronization of phases [61, 72, 120]. Thisresult has relevance for modelling brain dynamics at two levels. First,it provides a simple scenario in which topologically complex (small-world) synchronization networks emerge spontaneously from elemen-tary connectivity via dynamics; this demonstration could have generalrelevance for understanding the emergence of small-world networks inbiological neural systems [57, 82, 129]. Second, it provides a warningabout the potential issues related to drawing inferences about informa-tion flow in networks and underlines the importance of synchronizationmeasure choice, since the “hidden information transfer” between nodeswas only visible to generalized synchronization, a measure which wascalculated on embedded time-series; as such, it prompts careful con-sideration of suitable synchronization measures when analyzing neu-rophysiological signals [127]. [H5]

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• Across these experiments, emergent properties reminiscent of those ofneural dynamics were found mainly when the oscillators were tunedclose to criticality, or when the degree of complexity of collective dy-namics was otherwise intermediate. First, in the BJT-based networkwired according to a ring with four “hub” regions overlaid to it, se-lective transition to non-linear (chaotic) dynamics by the oscillatorsin the “hub” regions was observed only when all oscillators were in-dividually tuned for periodic oscillation close to criticality. Second,in the FPAA-based network, remote synchronization emerged only forspecific combinations of the control parameters (gains and integrationconstants), which yielded intermediate values of correlation dimensionof the individual time-series. Third, in the glow lamp-based network,avalanching according to the same critical exponents observed in neu-ral systems occurred when the supply voltage was carefully tuned asclose as possible to the transition point between the two dynamicalphases of the system. Even though “canonical” critical phenomenawere explicitly observed only for the third case, altogether these re-sults are concordant in suggesting that the capability of these circuitsto recapture neural-like phenomena is probably related not just tochaotic oscillation, but more specifically to oscillation close to or at anorder-to-chaos transition [15, 90, 92, 93, 96, 116, 133]. [H3,H5,H6]

• A novel elementary integrate-and-fire lattice network, based on capacitively-coupled glow lamps, was introduced. Depending on the value of aglobally-applied static control voltage, two well-distinct dynamicalphases emerged, which were characterized by different rates of gen-eration of stochastic lamp break-down events (emission of brief lightflashes) and different degrees of spatial and temporal order. The tran-sition between the two phases was associated with significant metasta-bility and hysteresis, hence it was a first-order one. Yet, setting thevoltage close to the transition point yielded critical avalanching andgeneration of fractal time-series, which are critical phenomena usuallyassociated with second-order transitions. This result is deemed inter-esting as it provides an example of a simple physical system which cangenerate critical phenomena in the context of a first-order transitiondue to spinodal instability, and more generally, as it spurs questionsrelated to the dynamics of neural systems [152, 153, 158, 159, 160]. Inin-vivo and in-vitro neural systems, metastability and hysteresis arecommon observations, as are critical phenomena such as avalanching,but the current level of understanding regarding how the two couldbe reconciled remains limited; indeed, self-organized criticality andsecond-order phase transition are almost assumed a-priori, and thismight not be universally correct [15, 90, 93, 132, 133, 139]. The re-sult obtained with this circuit should instigate further investigation

50

into the possibility that, at least under some circumstances, also thecritical phenomena observed in neural systems could perhaps repre-sent transition precursors arising near the spinodal curve in first-ordertransition. [H6]

Taken together, the findings of these studies demonstrate that manyemergent properties of brain dynamics can be successfully recaptured in ex-tremely simple non-linear electronic networks. As such, the results havemultiple levels of relevance for physics, electronic engineering and neuro-science, which at a general level may be finally conceptualized as follows:

• First, they provide compelling evidence that the dynamical phenomenataking place in the brain are, at least in part, not specific to it, but canbe engendered, and in fact occur spontaneously, also in other physicalsystems, including small electronic networks. This awareness shouldstrongly enhance the motivation for adopting a physical approach tothe study of brain complexity [10, 17, 15, 90, 164].

• Second, they underline the remarkable generative power of non-linearelectronic circuits, beyond the existing experimental results that havebeen largely focused on small networks, based primarily on “canonical”circuits like Chua’s circuit or chaotic adaptations of Colpitts and Hart-ley oscillators. They provide a fresh view on this area by introducinga range of novel circuits and results [26, 29, 44].

• Third, they provide practically-viable building blocks, which are suit-able for building neuromorphic systems of substantial size and com-plexity, as advocated for example in [18, 165]; this is particularly truefor the CMOS- and FPAA-based circuits [H4,H5] that are inherentlywell-suited for realizing larger networks than those considered in theseinitial experiments.

• Fourth, they raise theoretical questions that are directly pertinent tohow neurophysiological time-series are analysed, in particular as re-gards to how information transfer in synchronization networks (e.g.between cortical regions) is measured and studied [H5], and as re-gards to how experimental findings of metastability and hysteresis arereconciled with evidence of critical phenomena (i.e. spinodal instabil-ity in first-order transition vs. criticality in second-order transition)[H6].

• Fifth, owing to their elementary nature and undemanding, inexpen-sive realization these circuits are also of interest from an engineer-ing viewpoint. Once suitable ways are found to “manage” the self-organization into prescribed functions, they could serve as a sub-strate to implement new computational approaches, yielding highly

51

power- and area-efficient realizations of specific computational opera-tions [166, 167]. For example, building on the paradigm of cellular neu-ral networks (CNN), the observed pattern formation phenomena (e.g.,cluster synchronization in [H2,H4,H5] and remote synchronization in[H5]) plausibly could be leveraged upon for enhancing pattern recog-nition and reconstruction operations [28]. On the other hand, chaoticoscillators are already receiving increasing consideration as a substratefor novel, secure modulation schemes in critical telecommunications,and the “hidden information transfer” phenomenon observed in [H5]could have particular relevance within such framework [168]. Not last,a high level of signal complexity combined with digital controllabilityof chaos generation were attained in the CMOS integrated circuit [H4]as well as in the network of FPAAs [H5]; these have direct relevance forhigh-bandwidth generation of “true” random numbers, with potentialapplications in the fields of data encryption and authentication, whichrequire ever-increasing rates of random number generation [169].

52

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