Chapter

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NeurodynamicsFigure13.1Atwo-dimensionaltrajectory(orbit)ofadynamicsystem.Figure13.2Atwo-dimensionalstate(phase)portraitofadynamicsystem.Figure13.3Atwo-dimensionalvectorfieldofadynamicsystem.Table13.1Figure13.4(a)Stablenode.(b)Stablefocus.(c)Unstablenode.(d)Unstablefocus.(e)Saddlepoint.(f)Center.

Figure13.5Illustrationofthenotionofuniformstabilityofastatevector.Figure13.6

Lyapunovsurfacesfordecreasingvalueofconstantc,withc1<c2<c3.Theequilibriumstateisdenotedbythepointˉx.Figure13.7Illustrationofthenotionofabasinofattractionandtheideaofaseparatrix.Figure13.8Additivemodelofaneuron,labeledj.Figure13.9ArchitecturalgraphofaHopfieldnetworkconsistingofN=4neurons.Figure13.10Plotsof(a)thestandardsigmoidalnonlinearity,and(b)itsinverse.Figure13.11PlotoftheintegralFigure13.12Anenergycontourmapforatwo-neuron,two-stable-statesystem.Theordinateandabscissaaretheoutputsofthetwoneurons.Stablestatesarelocatednearthelowerleftandupperrightcorners,andunstableextremaarelocatedattheothertwocorners.Thearrowsshowthemotionofthestate.Thismotionisnotgenerallyperpendiculartotheenergycontours.(FromJ.J.Hopfield,1984,withpermissionoftheNationalAcademyofSciencesoftheU.S.A.)Figure13.13Illustrationoftheencoding–decodingperformedbyarecurrentnetwork.Table13.2Figure13.14(a)ArchitecturalgraphofHopfieldnetworkforN=3neurons.(b)Diagramdepictingthetwostablestatesandflowofthenetwork.Table13.3Figure13.15(a)Blockdiagramofthebrain-state-in-a-box(BSB)model.(b)Signal-flowgraphofthelinearassociatorrepresentedbytheweightmatrixW.Figure13.16Piecewise-linearactivationfunctionusedintheBSBmodel.Table13.4Figure13.17Illustrativeexampleofatwo-neuronBSBmodel,operatingunderfourdifferentinitialconditions:

?thefourshadedareasofthefigurerepresentthemodel’sbasinsofattraction;

?thecorrespondingtrajectoriesofthemodelareplottedinblue;

?thefourcorners,wherethetrajectoriesterminate,areprintedasblackdots.Figure13.18Feedbacksystemusedasaniteratedpredictorforthedynamicreconstructionofachaoticprocess.Figure13.19Regularizediteratedprediction(N=400,m=20)onLorenzdataatSNR=+25dB