The Universality and EmergentComputation in Cellular NeuralNetworksby Radu DogaruWorld Scientific Publishing Co., Singapore(ISBN 981-238-102-3)

Homogeneous media represent important and beau-tiful structures in the real world. Research on emer-gence, also called complexity, generated from

homogeneous media has gained much attention in thelast decade. It has been recognized that, unlike the quan-tum theory, no classical mathematical models and theo-ries can provide a basis for accurate prediction ofemergence that arises from homogeneous media.

The cellular neural (or, more generally, nonlinear) net-work (CNN) and the Reaction-Diffusion (R-D) CNN pre-sented by Leon Chua et al. [1, 2] not only have providednew tools for modeling homogeneous medium structures,representing different approximations to nonlinear par-tial differential equations, but also have been developedto CNN universal chips whose theoretical computationalspeed can be at least a thousand times faster than that ofthe current digital processors [3, 4].

Translating the classical positive-real criterion on lin-ear systems synthesis from circuit theory to R-D CNN,Leon Chua has presented, in his exposition of more than200 pages [3] (see also [4]), a local activity principle forCNN. The principle has four conditions to calculate theparameter range necessary for the emergence of nonho-mogeneous (static or dynamic) patterns in a homoge-neous medium [5]. These four conditions are expressedas a set of analytical inequalities to calculate the parame-ter range via computers. Generally speaking, with theincrease of numbers of state variables and ports of CNN,the corresponding analytical inequalities become moreand more complex. A homogeneous medium cannotexhibit complexity unless it is locally active, and the localactivity is the origin of complexity [3, 5].

Most of Dogaru’s book consists of a collection ofrecent papers on local activity and universality for CNN,written by Dogaru and Chua [6–10], with some openingoverviews and short comments for further interpreta-tions. The contents of this book are listed as follows.

1. Introduction2. Cellular Paradigms: Theory and Simulation

3. Universal Cells4. Emergence in Continuous-Time Systems5. Emergence in Discrete-Time Systems6. Unconventional Applications: Bio-Metric Authen-

ticationThe core of the introduction is to address the issue of

emergence and its computations. From different points ofview, emergence is difficult to define analytically, or topredict even from determinate systems. Using modelsand theories for describing emergent phenomena, com-putations in CNN and criteria of local activity for CNN areimportant research topics. This book compares the local-ly active cells or local passive cells for CNN with theexample of students who respond or do not respond to alecturer’s questions in a seminar room. Over reactingnoisy students correspond to active but unstable cells;the students who send the lecturer’s information to theirneighbors correspond to active and stable cells. If oneassumes, furthermore, that active students do not alwaysreact to the lecturer’s information and that passive stu-dents may receive their neighbors’ unformed opinions,then this example can vividly represent the spirit ofChua’s local activity theory.

In the following chapter, whenever possible, Dogarudescribes definitions and representations on CNN in verysimple ways. The reader who is not familiar with CNN issuggested first to read Chua’s excellent papers [3, 5]. Mat-lab simulation programs are provided in the book. Theprograms can be used, after some slight modification pos-sibly, for numerical simulations of dynamic behaviors ofmost CNN equations used in the following chapters of thebook. Practically, an engineer always hopes to design asystem that has both universality and robustness. Thismeans that the system can well undertake universal tasksand work, not only for nominal models but also for a largeset of perturbed models.

A standard uncoupled CNN cell can realize only linear-ly separable Boolean functions. The next chapter dis-cusses the issue of how to design a generalized CNN torealize arbitrary Boolean functions. The principle of thedesign is to guarantee the universality and the robustnessof the CNN. The first design approach is analytic and con-structive. The corresponding generalized (universal)CNN is a combination of polynomials. The secondmethod is based on the theory of canonical piecewise-lin-ear functions [11], where the resulting CNN is a combina-tion of piecewise-linear functions. In order to minimize

18 IEEE CIRCUITS AND SYSTEMS MAGAZINE FOURTH QUARTER 2003

Lequan Min

Book Reviews

the hardware cost, the CNN with uniform multi-nested uni-form cells is presented. To find such a uniform multi-nest-ed realization for all Boolean functions, a universal CNNgene enumeration algorithm is given. Subsection 3.3introduces a simplicial neural cell (SNC) architecture withadaptive nature in order to learn arbitrary input-outputimages whose gray-scales vary in a specified boundeddomain. Although lacking theoretical analysis, numericalsimulations have shown that SNC is suitable for someimage filtering, signal classification, and possibly otherapplications as well.

Chapter 4 is perhaps the most interesting one. First,using the FitzHugh-Nagumo (FN) CNN model as an exam-ple, it presents some analytical inequalities for the localactivity of a CNN with two state variables and one or twoports (called also coupling input). It might be better tokeep the strict definition of the local activity principlefor CNN introduced by Chua in [3, 4], from which itmight be easier for the reader to understand the mean-ing of the corresponding parameter inequalities from alocal activity dogma.

The definition of edge of chaos follows next. The edge-of-chaos domain is a cell-parameter domain such thatthere exists at least one equilibrium point, Qi, which isboth locally active and stable, that is, the real parts ofthe eigenvalues λ of the Jacobian matrix (the linearizedequation without considering the port(s)) of the CNNequation at Qi are all negative. Recent research revealsthat this definition could be slightly improved for the fol-lowing reasons.

(a) Practically, different equilibriums may representdifferent meanings. For instance, a tumor growth CNN[12] has three kinds of equilibriums, representing com-plete recovery, complete recovery with immunity, com-plete cancerization coexistence of cancer cells, deadcells, and free and bound cytotoxic cells. Therefore, thedefinition of edge of chaos given with respect to a spe-cific equilibrium point may be useful for examining thedynamic behaviors of CNN at different equilibriumpoints [12].

(b) The requirement for stability can also be extend-ed. In addition, there exists some eigenvalue λ such thatReal(λ) = 0 and the multiplicity of λ equals 1 (see Defini-tion 2.1 in [13]). In fact, one interesting phenomenon isthat the tumor growth CNN may exhibit a periodic oscil-latory pattern with coexistence of cancer cells, deadcells, and free and bound cytotoxic cells, if the cell param-eters are located nearby or in the edge of chaos withrespect to this kind of equilibrium point [12].

After addressing an edge of chaos algorithm, 44 bifur-cation diagrams for FN CNN with one or two ports and different cell parameters are given. The following simu-lations display homogeneous static patterns or Turing-

like patterns, and spiral wave patterns, if the cell parame-ters are selected in locally passive domains or locallyactive domains.

Emergent behaviors of the Brusselator (B) CNN are dis-cussed next, with only one equilibrium for the restrictedlocal activity domain. First, the bifurcation diagrams of BCNN with one and two ports are calculated. Simulationresults show that non-homogeneous static patterns can be observed if the corresponding cell parameters are chosen in the edge of chaos domains. The B CNN,whose cell parameters are located in locally active unsta-ble domains and near the edge of chaos, may also gener-ate non-homogeneous static patterns. This means thatthose non-homogeneous static patterns correspond to thestable equilibrium points of the B CNN equations,although they are unstable equilibrium points when theports are set to zero. If the cell parameters of a B CNN areselected in a locally active unstable domain, then chaoticand periodic patterns with periodic pulses can appear.

An application to Gierer-Meinhardt (GM) CNN followsnext. Dynamic simulations of sixteen GN CNN are provid-ed based on the bifurcation diagrams of the GM CNN withone or two ports. The first thirteen simulate patterns con-verging to non-homogeneous patterns. The other pat-terns display periodic synchronization, chaos, andconvergence to homogeneous patterns, respectively.

In summary, the local activity principle provides a newtool for studying the dynamic behaviors arising from R-FCNN, which gives a necessary but not sufficient conditionfor emergence. Some new phenomena have beenobserved with the help of the bifurcation diagrams ofthree R-F CNNs, generated based on the local activity cri-teria. For instance, the existence of two nonzero diffusioncoefficients is not a necessary condition for generatingstatic non-homogeneous Turing-like patterns. In view ofthe 3 D-F CNNs originated from models of nerve mem-branes, to explain the self-organization phenomenon andliving systems, the above research results may providehelpful interpretations of the underlying mechanisms forstudying corresponding technical problems.

Some analytical criteria for local activity have beendeveloped for CNN with 3, 4, or 5 state variables [14,15, 13], [16]. Most numerical simulations on CNN haveconfirmed Chua’s assertion [3, 5] that a wide spectrumof complex behaviors may exist if the correspondingcell parameters of CNN are chosen on or nearby theedge of chaos.

It is generally not easy to determine the robustnessand universality of a generalized cellular automata withpiecewise linear function that can represent someBoolean functions. Chapter 5 provides a novel approachfor analytically and precisely partitioning the cell param-eter space into subdomaims via the so-called failure

19FOURTH QUARTER 2003 IEEE CIRCUITS AND SYSTEMS MAGAZINE

boundaries, in which every subdomain corresponds to aspecific Boolean function and each parameter grouplocated in the center of a subdomain supports the CNNwith the maximal possible robustness. This approach canbe achieved via simply introducing two parameters in thepiecewise-linear function of the CNN. A well known “gameof life” CNN, whose prototype is the classic Conway logicfunction, is presented as an example for interpretation ofthe approach.

In order to classify emergent behaviors, a cellular dis-order measure is introduced. As a result, cell parameterspaces can be classified into an unstable-like region, anedge of chaos-like region, and a passive-like region,respectively. The emergent phenomena in continuousstate and coupled generalized cellular automata are dis-cussed. As applications, some examples “game of life”and image reconstructions are provided.

Local activity criteria for difference-equation CNN hasbeen developed recently. “The results highlight a funda-mental difference between the qualitative properties ofsystems of nonlinear differential—and difference—equations” [17]; “discrete time maps may be used tomodel spatial physical systems because matter is discon-tinuous on microscopic scales and may be fractal in com-position”. The local activity principle of discrete-timeCNN systems has also been presented and used to studythe logistic map CNN, the magnetic vortex pining mapCNN, and the spiral wave reproducing map CNN [18].

The last chapter [19] attempts to stimulate the likeand dislike reactions of human brains via CNN with piecewise-linear functions—the gene functions. It seemsthat the five given gene functions are too simple to beused for stimulating reactions of human brains. The lackof strict definitions and representation of CNN modelsmay prevent the reader from understanding the exactintention and meaning of the author. However, the char-acteristics of (generalized) CNN imply that they are prom-ising candidates for simulating brain functions.

Dogaru’s book is a timely addition to the literaturepublished in recent years, which is strongly orientated tomulti-disciplinary nonlinear science areas. This book ishighly recommendable to any one who wants to studynonlinear dynamics, circuits, devices and systems, and tothe individuals who have interest in expanding their cur-rent expertise of mathematics to understand more aboutthe complex nonlinear world. With this introductorybook, some uncharted mathematical and physical terri-tories may be further explored for novel applications ofvarious CNN in the near future.

Acknowledgment: The reviewer would like to thank Pro-fessor Guanrong Chen of the City University of Hong Kongfor helpful suggestions, and thank the support of theNational Natural Science Foundations of China (GrantNos. 60074034, 70271068), and the Research Fund for theDoctoral Program of Higher Education (No.200200080004) by the Ministry of Education of China.

References[1] L.O. Chua and L. Yang, “Cellular neural networks: Theory and appli-cations,” IEEE Trans. Circuits Syst., vol. 35, pp. 1257–1290, 1988.[2] L.O. Chua, M. Hasler, G.S. Moschytz, et al., “Autonomous cellularneural networks: A unified paradigm for pattern formation and activewave propagation,” IEEE Trans. Circuits Syst. I, vol. 42, pp. 559–577, 1995.[3] L.O. Chua, “A version of complexity,” Int. J. Bifurcation and Chaos,vol. 7, no. 10, pp. 2219–2425, 1997.[4] L.O. Chua, CNN: A Paradigm for Complexity. Singapore: World Scien-tific, 1998.[5] L.O. Chua, “Passivity and complexity”, IEEE Trans. Circuit Syst. I, vol.46, no. 1, pp. 71–82, 1999.[6] R. Dogaru and L.O. Chua, “Edge of chaos and local activity domain ofFitzHugh-nagumo”, Int. J. Bifur. Chaos, vol. 8, no. 2, pp. 211–257, 1998.[7] R. Dogaru and L.O. Chua, “Edge of chaos and local activity domain of the brusselator CNN,” Int. J. Bifur. Chaos, vol. 8, no. 6, pp. 1107–1130, 1998.[8] R. Dogaru and L.O. Chua, “Edge of chaos and local activity domain ofGierer-Meinhart CNN,” Int. J. Bifur. Chaos, vol. 8, no. 12, pp. 2321–2340, 1998.[9] R. Dogaru and L.O. Chua, “Universal CNN cells,” Int. J. Bifur. Chaos,vol. 9, no. 1, pp. 1–48, 1998.[10] R. Dogaru and L.O. Chua, “Mutations of the ‘game of life’: A gener-alized cellular automata perspective of complex adaptive systems,” Int.J. Bifur. Chaos, vol. 10, no. 8, pp. 1821–1866, 2000.[11] L.O. Chua and S.M. Kang, “Section-wise piecewise-linear functions:canonical representation, properties, and applications,” IEEE Proceed-ings, vol. 65, no. 6, pp. 915–929, 1977.[12] L. Min, J. Wang, X. Dong, and G. Chen, “Some analytical criteria forlocal activity of three-port CNN with four state variables: Analysis andapplications,” Int. J. Bifur. Chaos, vol. 13, no. 8, 2003, in press.[13] L. Min and N. Yu, “Some analytical criteria for local activity of two-port CNN with three or four state variables: analysis and applications,”Int. J. Bifur. Chaos, vol. 12, no. 5, pp. 931–963, 2002.[14] L. Min, K.R. Crounse, and L.O. Chua, “Analytical criteria for localactivity and applications to the oregonator CNN,” Int. J. Bifur. Chaos, vol. 10, no. 1, pp. 25–71, 2000.[15] L. Min, K.R. Crounse, and L.O. Chua, “Analytical criteria for localactivity of reaction-diffusion CNN with four state variables and applica-tions to the Hodgkin-Huxley equation,” Int. J. Bifur. Chaos, vol. 10, no. 6,pp. 1295–1343, 2000.[16] L. Min and X. Dong, “Analytical criteria for local activity of CNNwith five state variables and applications to Hyper-Chaos synchroniza-tion Chua’s circuit,” J. Univ. Sci. Technol. Beijing, vol. 9, no. 4, pp. 311–-318, 2002.[17] V. Sbitnev, T. Yang, and L.O. Chua, “The local activity criteria for ‘dif-ference-equation’ CNN,” Int. J. Bifur. Chaos, vol. 11, no. 2, pp. 311–419, 2001.[18] V. Sbitnev and L.O. Chua, “Local activity criteria for discrete-mapCNN,” Int. J. Bifur. Chaos, vol. 12, no. 6, pp. 1227–1272, 2002.[19] R. Dogaru and Ioana Dogaru, “Biometric authentication based onperceptual resonance between CNN emergent patterns and humans,” inCellular Neural Networks and Their Applications: Proceedings of the 7thIEEE International Workshop, R. Tetzlaff, Ed. Singapore: World Scientific,2002, pp. 267–274.

Lequan MinUniversity of Science and Technology, Beijing

20 IEEE CIRCUITS AND SYSTEMS MAGAZINE FOURTH QUARTER 2003