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Eurich *\nInstitut fu\u00a8r Theoretische Physik, Universita \u00a8t Bremen, Otto-Hahn-Allee 1, D-28334 Bremen, Germany\nJ. Michael Herrmann\nMax-Planck-Institut fu \u00a8r Stro\u00a8mungsforschung, Bunsenstrasse 10, D-37073 Go \u00a8ttingen, Germany\nUdo A. Ernst\nInstitut fu\u00a8r Theoretische Physik, Universita \u00a8t Bremen, Otto-Hahn-Allee 1, D-28334 Bremen, Germany\n~Received 14 September 2000; published 31 December 2002 !\nWe study the avalanche dynamics of a system of globally coupled threshold elements receiving random\ninput. The model belongs to the same universality class as the random-neighbor version of the Olami-Feder-Christensen stick-slip model. A closed expression for avalanche size distributions is derived for arbitrarysystem sizes Nusing geometrical arguments in the system\u2019s con\ufb01guration space. For \ufb01nite systems, approxi-\nmate power-law behavior is obtained in the nonconservative regime, whereas for N!\u2018, critical behavior with\nan exponent of 23/2 is found in the conservative case only. We compare these results to the avalanche\nproperties found in networks of integrate-and-\ufb01re neurons, and relate the different dynamical regimes to theemergence of synchronization with and without oscillatory components.\nDOI: 10.1103/PhysRevE.66.066137 PACS number ~s!: 05.65. 1b, 05.70.Ln, 45.70.Ht, 87.18.Sn\nI. INTRODUCTION\nIn the last decade, a considerable number of publications\nhave been dedicated to the occurrence of power-law behaviorin systems involving interacting threshold elements drivenby slow external input. The dynamics accounts for phenom-ena occurring in such diverse systems as piles of granularmatter @1#, earthquakes @2#, the game of life @3#, friction @4#,\nand sound generated in the lung during breathing @5#.A n\navalanche of theoretical investigations was triggered by Bak,Tang, and Wiesenfeld @6#who linked the occurrence of\npower laws to the notion of self-organized criticality ~SOC!.\nIn the so-called sandpile models, locally connected elementsreceiving random input self-organize into a critical statecharacterized by power-law distributions of avalanches with-out the explicit tuning of a model parameter ~e.g., Refs.\n@7\u201318 #!. Analytical results were derived for sandpile models\n@14,15 #, and it was shown that the existence of a conserva-\ntion law is a necessary prerequisite to obtain SOC @16\u201318 #.\nA second class of models inspired by earthquake dynam-\nics employs continuous driving and nonconservative interac-tion between the elements of the system @4,19#. In the Olami-\nFeder-Christensen ~OFC!model @19#, where the amount of\ndissipation is controlled by a parameter\na, power-law behav-\nior of avalanches occurs for a wide range of avalues. Sub-\nsequent investigations emphasized the importance of bound-ary conditions and tied the existence of the observed scalingbehavior to synchronization phenomena induced by spatialinhomogeneities @20\u201324 #. More speci\ufb01cally, Lise and Jensen\n@25#introduced a random-neighbor interaction in the OFC\nmodel to avoid the buildup of spatial correlations. Furtheranalysis indeed revealed that the random-neighbor OFCmodel does not display SOC in the dissipative regime @26\u2013\n28#.\nIn these avalanche models with nonconservative interac-\ntion, analytical results have been obtained only for system\nsizeN!\u2018so far @26,29 #. Here we introduce a model that\nnot only circumvents the problem of system boundaries,", "doc_id": "b3e21c43-f122-4228-800c-1f3b373b3960", "embedding": null, "doc_hash": "d098bb5f2200807de284af0693d8ca5e48f9c980b95b8a1b438583a8dc241e95", "extra_info": null, "node_info": {"start": 0, "end": 3410}, "relationships": {"1": "950acfd7-35b6-40e0-bf25-1f85e4b3951d", "3": "ecd2b3d1-32cd-40e9-a2cb-2a1318531b90"}}, "__type__": "1"}, "ecd2b3d1-32cd-40e9-a2cb-2a1318531b90": {"__data__": {"text": "a wide range of avalues. Sub-\nsequent investigations emphasized the importance of bound-ary conditions and tied the existence of the observed scalingbehavior to synchronization phenomena induced by spatialinhomogeneities @20\u201324 #. More speci\ufb01cally, Lise and Jensen\n@25#introduced a random-neighbor interaction in the OFC\nmodel to avoid the buildup of spatial correlations. Furtheranalysis indeed revealed that the random-neighbor OFCmodel does not display SOC in the dissipative regime @26\u2013\n28#.\nIn these avalanche models with nonconservative interac-\ntion, analytical results have been obtained only for system\nsizeN!\u2018so far @26,29 #. Here we introduce a model that\nnot only circumvents the problem of system boundaries, butyields an analytical access also for \ufb01nite system sizes N. The\nelements are globally connected, which makes the system amean-\ufb01eld model. Randomness is not introduced throughrandom neighbors but by providing a random external input.During an avalanche, the elements become unstable and re-lax in a \ufb01xed order determined by the state of the systemimmediately prior to the avalanche. Therefore, the system is\nstrictly Abelian for dissipation parameters\nasmaller than a\nthreshold value, which can be readily worked out. In thiscase, a geometrical approach in the N-dimensional con\ufb01gu-\nration space yields an exact equation for the distribution ofavalanche sizes.\nIn Sec. II, the model is speci\ufb01ed and compared with other\ndissipative avalanche models, in particular, with the random-neighbor OFC model. In Sec. III, avalanche properties arepresented both numerically and analytically, whereby detailsof the analytical calculation of the avalanche size distribu-tions can be found in Appendixes A\u2013C. Extensions and ap-plications of the model are formulated in the terminology ofneural networks: The model allows for an interpretation interms of a fully connected neural network of nonleakyintegrate-and-\ufb01re neurons. Implications of this view such asthe synchronization behavior of local, densely connectedpopulations of cortical neurons will be discussed in Sec. IV.The paper concludes with a brief summary and discussion.\nII. THE AVALANCHE MODEL\nA. De\ufb01nition\nIn the model, time is measured in discrete steps, t\n50,1,2,....Consider a set of Nidentical threshold ele- *Electronic address: eurich@physik.uni-bremen.dePHYSICAL REVIEW E 66, 066137 ~2002!\n1063-651X/2002/66 ~6!/066137 ~15!/$20.00 \u00a92002 The American Physical Society 66066137-1\nments characterized by a state variable u>0, which will\nhenceforth be called energy. The system is initialized with\narbitrary values uiP@0,U)(i51 ,...,N), where Uis the\nthreshold above which elements become unstable and relax.\nDepending on the state of the system at time t, theith ele-\nment receives external input Iiext(t) or internal input Iiint(t)\nfrom other elements, resulting in an activation u\u02dcat timet\n11,\nu\u02dci~t11!5ui~t!1Iiext~t!1Iiint~t!. ~1!\nFrom the activation u\u02dci(t11), the energy of the ith ele-\nment at time t11 is computed as\nui~t11!5Hu\u02dci~t11!ifu\u02dci~t11!,U,\ne~u\u02dci~t11!2U!ifu\u02dci~t11!>U,~2!\ni.e., if the activation exceeds the threshold U, it is reset but\nretains a fraction e(0<e<1) of the suprathreshold portion\nu\u02dci(t11)2Uof the energy.\nFor the external input Iiext(t), one element is randomly\nchosen from a uniform distribution at each time step, and", "doc_id": "ecd2b3d1-32cd-40e9-a2cb-2a1318531b90", "embedding": null, "doc_hash": "12abe7258d39bcc8c697ce5683d58295831d4bc34ed4bd1ffab9ac698f98e390", "extra_info": null, "node_info": {"start": 2782, "end": 6105}, "relationships": {"1": "950acfd7-35b6-40e0-bf25-1f85e4b3951d", "2": "b3e21c43-f122-4228-800c-1f3b373b3960", "3": "b2006ef0-2c96-48f2-b5a4-9f5ecebab917"}}, "__type__": "1"}, "b2006ef0-2c96-48f2-b5a4-9f5ecebab917": {"__data__": {"text": "timet\n11,\nu\u02dci~t11!5ui~t!1Iiext~t!1Iiint~t!. ~1!\nFrom the activation u\u02dci(t11), the energy of the ith ele-\nment at time t11 is computed as\nui~t11!5Hu\u02dci~t11!ifu\u02dci~t11!,U,\ne~u\u02dci~t11!2U!ifu\u02dci~t11!>U,~2!\ni.e., if the activation exceeds the threshold U, it is reset but\nretains a fraction e(0<e<1) of the suprathreshold portion\nu\u02dci(t11)2Uof the energy.\nFor the external input Iiext(t), one element is randomly\nchosen from a uniform distribution at each time step, and a\nconstant amount of energy DUP(0,U#is added to the el-\nement\u2019s energy. The external input is considered to be deliv-ered slowly compared to the internal relaxation dynamics,i.e., it occurs only if no element has exceeded the thresholdin the previous time step. This corresponds to an in\ufb01niteseparation of the time scales of external driving and ava-lanche dynamics discussed in the SOC literature @11\u201313 #.\nThe external input can formally be written as I\niext(t)\n5dr(t),idM(t21),0DU.r(t) is an integer random variable\ndrawn at time step tfrom a uniform distribution between 1\nandN, indicating the chosen element, M(t21) is the num-\nber of suprathreshold elements in the previous time step, and\ndi,jis the Kronecker delta.\nThe internal input Iiint(t) is given by Iiint(t)5M(t\n21)aU/N, where aU/Nis the coupling strength between\nthe elements. We assume connections to be excitatory, that\nis,a.0.\nAt some time t0an avalanche starts, M(t0)51, provided\nthe element receiving external input becomes unstable. Thesystem is globally coupled, such that during an avalanche allelements receive internal input, including the unstable ele-\nments themselves. The avalanche duration D>0 is de\ufb01ned\nto be the smallest integer for which the stopping condition\nM(t\n01D)50 is satis\ufb01ed. The avalanche size Lis given by\nL5(k50D21M(t01k). The model allows the calculation of the\nprobability P(L,N,a) of an avalanche of size L>0 in the\nregime 0 <L<Nof a system consisting of Nelements with\ncoupling parameter a. Avalanche size distributions can alter-\nnatively be described by a function p(L,N,a) forL>1,\nwhich is related to P(L,N,a) via\np~L,N,a![P~L,N,a!\n12P~0,N,a!. ~3!\nAvalanche duration distributions will be denoted by\npd(D,N,a)(D>1).Due to the global coupling of the elements, there are no\nboundary conditions to be speci\ufb01ed in the model.\nB. The case e\u02dc1\nBoth the coupling parameter aand the reset parameter e\ncontrol the amount of dissipation in the system.An analytical\napproach will be possible for e51, that is, if all suprathresh-\nold elements are reset such that they lose an identical amountUof energy @cf. Eq. ~2!#. We will therefore restrict further\nanalysis to this case and only brie\ufb02y return to the generalsituation in Sec. IV.\nFor\ne51, the value a51 corresponds to the conservative\ncase with respect to the internal dynamics: upon resetting ofa suprathreshold element, the energy it loses is completely\ndistributed in the network. For\na>1, an in\ufb01nite avalanche\nmay eventually occur and we will therefore restrict ourselves\nto the case", "doc_id": "b2006ef0-2c96-48f2-b5a4-9f5ecebab917", "embedding": null, "doc_hash": "a440d7b733705c8d893d807288782fab81aef68d9447049c7d078521883697bb", "extra_info": null, "node_info": {"start": 6328, "end": 9320}, "relationships": {"1": "950acfd7-35b6-40e0-bf25-1f85e4b3951d", "2": "ecd2b3d1-32cd-40e9-a2cb-2a1318531b90", "3": "4d4caa9d-39d9-4341-8719-5c55510fc86f"}}, "__type__": "1"}, "4d4caa9d-39d9-4341-8719-5c55510fc86f": {"__data__": {"text": "the coupling parameter aand the reset parameter e\ncontrol the amount of dissipation in the system.An analytical\napproach will be possible for e51, that is, if all suprathresh-\nold elements are reset such that they lose an identical amountUof energy @cf. Eq. ~2!#. We will therefore restrict further\nanalysis to this case and only brie\ufb02y return to the generalsituation in Sec. IV.\nFor\ne51, the value a51 corresponds to the conservative\ncase with respect to the internal dynamics: upon resetting ofa suprathreshold element, the energy it loses is completely\ndistributed in the network. For\na>1, an in\ufb01nite avalanche\nmay eventually occur and we will therefore restrict ourselves\nto the case a,1. In order to avoid side effects resulting\nfrom the null set of rational values of a,U,o rDU,w e\nassume one of the fractions a/UorDU/Uto be irrational.\nAs will be shown below, a variation of aleads to qualita-\ntively different avalanche size distributions.\nC. Comparison with other avalanche models\nAclass of models discussed in the SOC literature employs\na parameter controlling the amount of dissipation @4,19\u201328 #.\nThe numerically observed power-law behavior in such sys-tems, however, could be ascribed to spatial inhomogeneitiesand the employed boundary conditions ~e.g., @21\u201324 #!.I n\norder to study avalanches of activity in the presence of dis-sipation independent of spatial correlations among elements,Lise and Jensen @25#introduced a random-neighbor version\nof the Olami-Feder-Christensen model described in Ref.@19#. In this model, threshold elements receive a constant,\nuniform input and have random nearest neighbors to whichthey are connected during an avalanche. The temporal vari-ability of the network connectivity avoids the buildup of spa-tial correlations, thus ruling out boundary effects in shapingavalanche distributions. Subsequent studies, however, dem-onstrated that the random-neighbor OFC model does nothave scaling behavior in the dissipative regime @26\u201328 #.\nBro\u00a8ker and Grassberger @26#, in their analytical consider-\nations of the random-neighbor OFC model, applied thetheory of branching processes to yield avalanche size distri-butions. For this purpose it was necessary to consider the\nlimitsd!\u2018~wheredis the dimension of the lattice !and\nN!\u2018in order to make the model effectively Abelian and\navoid correlations among elements @26#. This prevents ava-\nlanches from visiting elements more than once and allowssubavalanches to spread independently of each other suchthat each suprathreshold element has a distinctive predeces-sor which triggered it.\nOur model poses an alternative of the random-neighbor\nOFC model: the global coupling of elements prevents spatialcorrelations and the putative dependence of the system be-havior on boundary conditions. Randomness is introducedthrough the external input rather than the random assignmentof nearest neighbors. This approach has the advantage of not\nrequiring the limit N!\u2018: For\ne51, the system is AbelianEURICH, HERRMANN, AND ERNST PHYSICAL REVIEW E 66, 066137 ~2002!\n066137-2\nfor an arbitrary system size Nbecause at each time step t\nduring an avalanche, all elements receive the same input de-\npending only on the number M(t21) of suprathreshold ele-\nments at time t21.\nThe random-neighbor OFC model and the globally\ncoupled model are complementary in the following sense: inthe random-neighbor OFC model, randomness is introducedthrough the random choice of neighbors during the ava-lanche activity, while the interavalanche dynamics is a\nsimple shift of the energy distribution\nr(u) on theuaxis due\nto the uniform input. In our globally coupled", "doc_id": "4d4caa9d-39d9-4341-8719-5c55510fc86f", "embedding": null, "doc_hash": "4997f4928a0d7b04b9bdca09e7d2d2d84735a83b981d4e994af437f4c780374d", "extra_info": null, "node_info": {"start": 9139, "end": 12760}, "relationships": {"1": "950acfd7-35b6-40e0-bf25-1f85e4b3951d", "2": "b2006ef0-2c96-48f2-b5a4-9f5ecebab917", "3": "4753dd74-9764-40be-8baa-81348163e0bd"}}, "__type__": "1"}, "4753dd74-9764-40be-8baa-81348163e0bd": {"__data__": {"text": "the system is AbelianEURICH, HERRMANN, AND ERNST PHYSICAL REVIEW E 66, 066137 ~2002!\n066137-2\nfor an arbitrary system size Nbecause at each time step t\nduring an avalanche, all elements receive the same input de-\npending only on the number M(t21) of suprathreshold ele-\nments at time t21.\nThe random-neighbor OFC model and the globally\ncoupled model are complementary in the following sense: inthe random-neighbor OFC model, randomness is introducedthrough the random choice of neighbors during the ava-lanche activity, while the interavalanche dynamics is a\nsimple shift of the energy distribution\nr(u) on theuaxis due\nto the uniform input. In our globally coupled model, thestochasticity is due to the random external input betweenavalanches, whereas the avalanche activity corresponds to a\nrotation of\nr(u) on a circle @0,U) with periodic boundary\nconditions. The latter property is due to ~i!the fact that all\nelements\u2014including the unstable ones\u2014receive the same in-\nputIiint(t) at each time step, and ~ii!the update rule ~2!which\nreinjects unstable elements according to the suprathreshold\nportionu\u02dci(t11)2Uof their energy. Therefore, the elements\nbecome unstable in a \ufb01xed order depending on the actual\ndistribution r(u). Below it will be shown that for coupling\ncoef\ufb01cients a,max$12DU/U,N/(N11)%, avalanche sizes\nmay not exceed N, which means that each element can be\nactivated only once. In this regime, avalanche distributionsturn out to be very similar for the random-neighbor OFCmodel and the current model, demonstrating that the differ-ences between the models barely change the statistical prop-\nerties of the avalanches. However, in the globally coupledmodel, this regime can be described by a closed expression\nfor avalanche distributions, p(L,N,\na).\nIII. AVALANCHE PROPERTIES\nA. Avalanche sizes\nFigure 1 shows avalanche size distributions for different\nvalues of a.N510000 was chosen as the system size, but\nthe curves look very similar for any other choice of N.\nFour qualitatively different regimes can be distinguished\nwhich will be termed subcritical, critical, supracritical, and\nmultipeaked. For small values of a, subcritical avalanche\nsize distributions exist, which can be approximated by thegeneral expression\np\n~L,N,a!\u2019p\u02c6~L,N,a!5Lgexp~2L/l!, ~4!\nwhere gis an exponent independent of Nto be characterized\nbelow, and l5l(N,a) describes the range of avalanche\nsizes over which power-law behavior is observed @Fig. 1 ~a!#.\nFor \ufb01xed N,l(N,a) is a monotonically increasing function\nofaas long as a,acwhich we refer to as the \u2018\u2018critical\ncase\u2019\u2019 ~Fig. 2 !. For ac, the system has avalanche distribu-\ntions with an approximate power-law behavior with expo-\nnent 23/2 from L51 almost up to the size of the system,\nwhere the usual exponential cutoff is observed @49#@Fig.\n1~b!#. For \ufb01nite N,acis in the dissipative regime.Above the\ncritical value ac, avalanche size distributions become non-\nmonotonic @Fig. 1 ~c!#. Such supracritical curves have a mini-\nmum at some intermediate avalanche size.\nIn order to \ufb01nd the critical coupling coef\ufb01cient acas a\nfunction of system size N, we computed a conveniently de-\n\ufb01ned distance K(a) between the distribution p(L,N,a) and\nFIG. 1. Probability distributions of avalanche sizes, p(x,N,a),\nand avalanche durations,", "doc_id": "4753dd74-9764-40be-8baa-81348163e0bd", "embedding": null, "doc_hash": "dc1d91cfe2474537aca56dea190192ed6d6119b9b20370fc8a9c529d2e6c9aeb", "extra_info": null, "node_info": {"start": 12777, "end": 16049}, "relationships": {"1": "950acfd7-35b6-40e0-bf25-1f85e4b3951d", "2": "4d4caa9d-39d9-4341-8719-5c55510fc86f", "3": "27891564-af8b-4c1b-b331-bae9cd5f0d38"}}, "__type__": "1"}, "27891564-af8b-4c1b-b331-bae9cd5f0d38": {"__data__": {"text": "from L51 almost up to the size of the system,\nwhere the usual exponential cutoff is observed @49#@Fig.\n1~b!#. For \ufb01nite N,acis in the dissipative regime.Above the\ncritical value ac, avalanche size distributions become non-\nmonotonic @Fig. 1 ~c!#. Such supracritical curves have a mini-\nmum at some intermediate avalanche size.\nIn order to \ufb01nd the critical coupling coef\ufb01cient acas a\nfunction of system size N, we computed a conveniently de-\n\ufb01ned distance K(a) between the distribution p(L,N,a) and\nFIG. 1. Probability distributions of avalanche sizes, p(x,N,a),\nand avalanche durations, pd(x,N,a), in the subcritical @~a!,a\n50.8], critical @~b!,a50.99], supracritical @~c!,a50.999], and\nmultipeaked @~d!,a50.99997] regime. ~a!\u2013~c!Solid lines and\nsymbols denote the analytical and the numerical results for the ava-lanche size distributions, respectively. In ~d!, the solid line shows\nthe numerically calculated avalanche size distribution. The dashedlines in ~a!\u2013~d!show the numerically evaluated avalanche duration\ndistributions. In all cases, the presented curves are temporal aver-ages over 10\n7avalanches with N510000, DU50.022, and U\n51.\nFIG.2. Range l(N,a) ofavalanchesizesoverwhichpower-law\nbehavior is observed in the subcritical regime. l(N,a) has been\nplotted for four different system sizes, namely, for N5102~solid\nline!,N5103~dashed line !,N5104~dashed-dotted line !, andN\n5105~dotted line !.To obtain l,p\u02c6(L,N,a) as de\ufb01ned in Eq. ~4!has\nbeen \ufb01tted to the analytically calculated avalanche size distribution\np(L,N,a) by maximizing the symmetric version of the Kullback-\nLeibler distance K(l) as de\ufb01ned by K(l)5(L(p2p\u02c6)@ln(p)\n2ln(p\u02c6)#.FINITE-SIZE EFFECTS OF AVALANCHE DYNAMICS PHYSICAL REVIEW E 66, 066137 ~2002!\n066137-3\nan \u2018\u2018ideal\u2019\u2019 power-law distribution p\u02dc(L,N)5L23/2/(LL23/2.\nThenK(a) was numerically minimized to yield the param-\neteracfor which the distribution is closest to a power law.\nWe chose the symmetric version of the Kullback-Leibler dis-\ntance as de\ufb01ned by K(a)5(L(p2p\u02dc)@ln(p)2ln(p\u02dc)#, which\nrevealed a critical coupling constant\nac~N!\u201912N2mwith m50.560.01 ~5!\n~obtained for system sizes ranging from N5102up toN\n5107). An alternative approach to obtain the exponent mis\nto compute the slope of the avalanche size distribution\np(L,N,a) for avalanche sizes L5N/2 using the analytical\nexpression to be derived below. The result is m50.5, in\nagreement with the numerics.\nAbove the supracritical case, a fourth regime exists for\nvalues of aclose to 1, where the distributions show multiple\npeaks located at L5N,2N11,3N11 ,....These peaks arise\nfrom the high coupling strength because elements can be-come suprathreshold more than only once during an ava-lanche. This is not possible in the subcritical, critical, andsupracritical regimes. Figure 1 ~d!shows an example with\nthree peaks ~note that the last maximum is not referred to as\na peak !.\nConditions for the occurrence of", "doc_id": "27891564-af8b-4c1b-b331-bae9cd5f0d38", "embedding": null, "doc_hash": "3defb0441bbacc24d7af3eec133fae96aad0894cf8f2d61aa5d4b36ea8f539e9", "extra_info": null, "node_info": {"start": 16116, "end": 19019}, "relationships": {"1": "950acfd7-35b6-40e0-bf25-1f85e4b3951d", "2": "4753dd74-9764-40be-8baa-81348163e0bd", "3": "67231d57-e066-4cda-84fe-b3df6c8e09a8"}}, "__type__": "1"}, "67231d57-e066-4cda-84fe-b3df6c8e09a8": {"__data__": {"text": "compute the slope of the avalanche size distribution\np(L,N,a) for avalanche sizes L5N/2 using the analytical\nexpression to be derived below. The result is m50.5, in\nagreement with the numerics.\nAbove the supracritical case, a fourth regime exists for\nvalues of aclose to 1, where the distributions show multiple\npeaks located at L5N,2N11,3N11 ,....These peaks arise\nfrom the high coupling strength because elements can be-come suprathreshold more than only once during an ava-lanche. This is not possible in the subcritical, critical, andsupracritical regimes. Figure 1 ~d!shows an example with\nthree peaks ~note that the last maximum is not referred to as\na peak !.\nConditions for the occurrence of kpeaks in the avalanche\nsize distributions can be readily worked out. Consider the\ncasek51 corresponding to the situation that neurons may\n\ufb01re twice at most during an avalanche. First, an avalanche\nsizeL5N11 must be possible. Since all elements receive\nthe same internal input and \ufb01re in a \ufb01xed order as describedabove, this is equivalent to the condition that the elementwhich originally triggered the avalanche may \ufb01re twice. Af-terN\ufb01ring events, this element has received the total input\nDU1\naU. A second \ufb01ring can thus occur if this input ex-\nceeds the threshold, or a.12DU/U. Second, after N11\n\ufb01ring events, the total internal input to each element must\nexceed the threshold to allow for further \ufb01ring, ( N\n11)aU/N.Uora.N/(N11). Similar arguments hold\nfor the general case of kpeaks. The above conditions must\nthen be replaced by\na.amin~k!5maxH12DU\nkU,kN\nkN11J. ~6!\namin(k),a<amin(k11) then gives the range of coupling con-\nstants for which kpeaks can be observed.\nExamples for avalanche distributions in the multipeaked\nregime are shown in Fig. 3. The distribution functions be-\ntween two peaks at L5kNandL5(k11)Nare always non-\nmonotonic. This can be seen as follows: In an avalanche of\nsize larger than kN, the energies uimust have been in an\nappropriate order to allow for this size. Because the interava-lanche dynamics corresponds to a simple shift of\nr~U!on the\ncircle ~0,U!, the ordering after kNevents is nearly similar to\nthe ordering prior to the start of the avalanche, except for theelement which received external input. This element hasbeen responsible for triggering the avalanche, and only thiselementhaschangeditspositionrelativetotheothers.There-fore, it is highly probable that again all Nelements will takepart in the continuing avalanche, which explains the increase\nof the distribution towards L5(k11)N. As can be seen\nfrom Fig. 3, all distributions have minima at avalanche sizes\nL5N/2,3N/ 2 ,...,(k11/2)N,....\nB. Avalanche durations\nIn comparison to the avalanche size distributions de-\nscribed before, Fig. 1 also shows examples of avalanche du-\nrationdistributions in the four different regimes. Qualita-\ntively, the duration distributions have similar shapes. In thesubcritical regime, the distributions are described by mono-tonically decreasing functions as in Eq. ~4!, and above the\ncritical regime, the functions show one or more maxima as\nthe coupling\naincreases, going from the supracritical to the\nmultipeaked regime.\nThe critical case occurs for the same value acfor which\nthe size distribution is also critical @Fig. 1 ~b!#, and the critical\nexponent is the same. This holds for all system sizes Nwe\nhave tested ~data not shown !. That is, the dependence of the\ncritical aon the system size Nis given by the", "doc_id": "67231d57-e066-4cda-84fe-b3df6c8e09a8", "embedding": null, "doc_hash": "f6774b2ad52f55383766520d0c887c8a752e8f1d48e31584245ff8738fdecb79", "extra_info": null, "node_info": {"start": 18926, "end": 22375}, "relationships": {"1": "950acfd7-35b6-40e0-bf25-1f85e4b3951d", "2": "27891564-af8b-4c1b-b331-bae9cd5f0d38", "3": "84db127e-7780-4291-987b-f77a9b7f9dc4"}}, "__type__": "1"}, "84db127e-7780-4291-987b-f77a9b7f9dc4": {"__data__": {"text": "before, Fig. 1 also shows examples of avalanche du-\nrationdistributions in the four different regimes. Qualita-\ntively, the duration distributions have similar shapes. In thesubcritical regime, the distributions are described by mono-tonically decreasing functions as in Eq. ~4!, and above the\ncritical regime, the functions show one or more maxima as\nthe coupling\naincreases, going from the supracritical to the\nmultipeaked regime.\nThe critical case occurs for the same value acfor which\nthe size distribution is also critical @Fig. 1 ~b!#, and the critical\nexponent is the same. This holds for all system sizes Nwe\nhave tested ~data not shown !. That is, the dependence of the\ncritical aon the system size Nis given by the same expres-\nsion~5!for the avalanche sizes and the avalanche durations.\nThe main difference to size distributions lies in the fact\nthat duration distributions start to differ from an \u2018\u2018ideal\u2019\u2019power-law distribution at lower values of L. This behavior\ncan be explained by an intuitive argument. For avalanche\nsizes ofL5N, it is unimportant how many elements are\ntriggered in each step of the avalanche as long as the totalnumber of toppling elements is N. For an avalanche duration\nofN, it is not only required that the avalanche composed of\nNelements is being triggered, but it is also necessary that in\neach step of the avalanche, exactly oneelement is triggered.\nHence large avalanche durations have a far lower probabilitythan large avalanche sizes.\nC. Analytical considerations\nWe use combinatorial arguments in the system\u2019s\nN-dimensional con\ufb01guration space to derive expressions for\navalanche size distributions in the subcritical, critical, and\nsupracritical regimes. The con\ufb01guration space PN(0,U)~or\nsimplyPiN) is de\ufb01ned to be the Cartesian product\nFIG. 3. Different avalanche size distributions p(L,N,a) with\na50.996 for N550~thin solid line, four peaks !,N5100~thin\ndashed line, two peaks !,N5200~thick solid line, one peak !, and\nN5250~thick dashed line !.The curves show maxima at L5kNand\nminima at L5(k10.5)N~both marked with dotted lines !.I na l l\ncases, the presented curves are temporal averages over 2 3108ava-\nlanches with DU50.022 and U51.EURICH, HERRMANN, AND ERNST PHYSICAL REVIEW E 66, 066137 ~2002!\n066137-4\nPN(0,U)5@0,U)Nwith periodic boundary conditions, i.e., it\nhas the topology of an N-torus ~see Appendix A !.\n1. An example with two elements\nThe case N52 demonstrates the basic mechanisms for\nevaluating the avalanche dynamics. The avalanche distribu-tion is calculated by determining the equilibrium density of\nstates in P\n2,r(u1,u2), and subsequently considering the\nregions which lead to avalanches of sizes 0,1, and 2. Figure4 shows the con\ufb01guration space P\n2and the shifts resulting\nfrom external input DU, internal input aU/2, and ava-\nlanches of size L51. In the latter case, the system leaves P2\nand is reinjected on the opposite side. We consider r(u1,u2)\nonly at times between avalanches. Then, the total internalinput distributed during an avalanche leads to a shift vectorwhich guarantees that systems will never be reinjected into\nthe region denoted by L\n2(a,U), i.e., r(u1,u2)50 for\n(u1,u2)PL2. The density in P2\\L2is solely determined by\nthe randomly distributed external input. This input can be\ndecomposed into deterministic shifts of size DU/A2 along\nthe diagonal u15u2and a random walk", "doc_id": "84db127e-7780-4291-987b-f77a9b7f9dc4", "embedding": null, "doc_hash": "ae075fdcd61b5d6e17b02fd331837cffda11840c53f9317ca4325f5e01ca3d66", "extra_info": null, "node_info": {"start": 22357, "end": 25717}, "relationships": {"1": "950acfd7-35b6-40e0-bf25-1f85e4b3951d", "2": "67231d57-e066-4cda-84fe-b3df6c8e09a8", "3": "7e380a6d-4fc5-4186-b584-08ef2cf17f80"}}, "__type__": "1"}, "7e380a6d-4fc5-4186-b584-08ef2cf17f80": {"__data__": {"text": "the shifts resulting\nfrom external input DU, internal input aU/2, and ava-\nlanches of size L51. In the latter case, the system leaves P2\nand is reinjected on the opposite side. We consider r(u1,u2)\nonly at times between avalanches. Then, the total internalinput distributed during an avalanche leads to a shift vectorwhich guarantees that systems will never be reinjected into\nthe region denoted by L\n2(a,U), i.e., r(u1,u2)50 for\n(u1,u2)PL2. The density in P2\\L2is solely determined by\nthe randomly distributed external input. This input can be\ndecomposed into deterministic shifts of size DU/A2 along\nthe diagonal u15u2and a random walk orthogonally to u1\n5u2. As a consequence of this stochasticity in combination\nwith the reinjection after avalanches, the density in P2\\L2\nbecomes constant for large times t, and a normalization\nyields the value r(u1,u2)5@U2(12a)#21. Figure 4 also\nidenti\ufb01es those regions in P2which lead to avalanches of\nsizes 0 (B), 1 (C), and 2 ~D!following external input to\nelement 1. Avalanche probabilities P(L,2,a) are obtained by\nintegrating r(u1,u2) over the respective region. Using Eq.\n~3!, the result is\np~1,2,a!52~12a!\n22aandp~2,2,a!5a\n22a.~7!2. The general case of Nelements\nSimilar arguments hold for the general situation of Nel-\nements. The topology of region LN\u2014the region which is not\ninhabited between avalanches after transients havedecayed\u2014and the regions leading to avalanches of certainsizes, however, are more complicated. We will outline thederivation of the distribution functions in the following; thedetailed, rather tedious calculations can be found in Appen-dixesA\u2013C.The \ufb01rst step is to obtain a general expression for\nthe volume of region L\nN,V\u0084LN(a,U)\u0085,i nPN. For this\npurpose, a rule can be derived showing how LNis composed\nof direct products of N-dimensional and lower-dimensional\nhypercubes of varying side lengths; V(LN) is then given by\nthe sum of the products of the volumes of these hypercubes.\nAs a result, the particularly simple expression V\u0084LN(a,U)\u0085\n5aUNis obtained for arbitrary N~seeAppendixesAand B !.\nFor the regions in PN(0,U) leading to different avalanche\nsizes, we suppose without loss of generality that the external\ninput DUis given to element 1. Upon receiving input, ele-\nment 1 \ufb01res if u1.U2DU. In the second step, the corre-\nsponding phase space region, whose volume is given by\nUN21DU, has to be partitioned into regions where L21\n50,1,2,...,N21 further elements will \ufb01re in the respective\navalanche. The volumes of these regions will be denoted as\nZ(L,N,a). The regions and their volumes are constructed\niteratively as shown in Appendix C. In the last step, ava-\nlanche probabilities p(L,N,a) are obtained by subtracting\nthe volumes of the intersections of the regions Z(L,N,a)\nwith region LN, and subsequently normalizing by the vol-\nume of PN\\LN~see Appendix C !. Using Eq. ~3!, the ava-\nlanche distributions become independent of DUandU,\np~L,N,a!5LL22SN21\nL21DSa\nNDL21S12La\nNDN2L21\n3N~12a!\nN2~N21!afor 1 <L<N. ~8!\nAs an example, Figs. 1", "doc_id": "7e380a6d-4fc5-4186-b584-08ef2cf17f80", "embedding": null, "doc_hash": "183c0390ce98320f1efc3a25d3ed26af27b203a053cc2c6360864631b90c8cb3", "extra_info": null, "node_info": {"start": 25790, "end": 28811}, "relationships": {"1": "950acfd7-35b6-40e0-bf25-1f85e4b3951d", "2": "84db127e-7780-4291-987b-f77a9b7f9dc4", "3": "e282d256-9f11-4515-b813-b90e589f831e"}}, "__type__": "1"}, "e282d256-9f11-4515-b813-b90e589f831e": {"__data__": {"text": "denoted as\nZ(L,N,a). The regions and their volumes are constructed\niteratively as shown in Appendix C. In the last step, ava-\nlanche probabilities p(L,N,a) are obtained by subtracting\nthe volumes of the intersections of the regions Z(L,N,a)\nwith region LN, and subsequently normalizing by the vol-\nume of PN\\LN~see Appendix C !. Using Eq. ~3!, the ava-\nlanche distributions become independent of DUandU,\np~L,N,a!5LL22SN21\nL21DSa\nNDL21S12La\nNDN2L21\n3N~12a!\nN2~N21!afor 1 <L<N. ~8!\nAs an example, Figs. 1 ~a!\u20131~c!demonstrate the perfect\nagreement between the analytical result ~8!and the numeri-\ncal avalanche size distributions for N5104.\nEquation ~8!resembles the avalanche size distribution\nwhich Bro \u00a8ker and Grassberger @@26#, Eq. ~36!#have found\nfor the random-neighbor OFC model using branching theory.The results differ, in that the result in Ref. @26#yields an\nexpression for avalanche sizes in systems of an arbitrary sizeN, but is valid only in the in\ufb01nite-size limit where simulta-\nneous avalanches are nonoverlapping. In contrast, Eq. ~8!\nholds for arbitrary system sizes Nin our model. Formally,\nEq.~8!contains a correction factor which is calculated by\nconsidering the region L\nNwhere the density of states even-\ntually vanishes, instead of assuming a uniform density over\nthe whole con\ufb01guration space PN(0,U) divided into regions\nleading to different avalanche sizes.\n3. The thermodynamic limit\nAvalanche behavior in the thermodynamic limit N!\u2018\ncan directly be assessed from Eq. ~8!. Numerical results and\nFIG. 4. The dynamics in the con\ufb01guration space for N52 ele-\nments. Effects of an external input to element 1 ~line marked as\nDU/U) followed by an avalanche of size 1 resulting in an input to\nboth elements ~arrow pointing along the diagonal !.L2denotes the\nregion where the density of states eventually vanishes; A,B, andC\ndenote regions leading to avalanches of size L50,1,2, respectively,\nif triggered by an external input to element 1. The hatched arealeads to an avalanche of size 1 but lies within L\n2.FINITE-SIZE EFFECTS OF AVALANCHE DYNAMICS PHYSICAL REVIEW E 66, 066137 ~2002!\n066137-5\nanalytical considerations @26#suggest a critical coupling pa-\nrameter ac51 forN!\u2018. Indeed, an evaluation of Eq. ~8!\nshows that the local exponent\ng~L!5lim\na!1lim\nN!\u2018lnp~L,N,a!\np~L11,N,a!YlnL\nL11~9!\nbecomes constant for L!\u2018: limL!\u2018g(L)523/2. Thus, in\nthe conservative system, power-law behavior with an expo-\nnent of 23/2 is reached in the regime of large avalanche\nsizes.The critical exponent is identical to that of the random-neighbor OFC model @26#and, for example, for mean-\ufb01eld\npercolation @30#.\nFor \ufb01nite L, the distribution is actually very close to\npower-law behavior. Since the critical case corresponds to\nthe conservative system\na51, the supracritical regime be-\ncomes smaller and smaller as N!\u2018: the occurrence of non-\nmonotonic avalanches is a \ufb01nite-size effect.\n4. Avalanche durations\nFor avalanche durations pd(D,N,a), an iterative equation\nfor the corresponding regions and their volumes in the", "doc_id": "e282d256-9f11-4515-b813-b90e589f831e", "embedding": null, "doc_hash": "52a88528898827b36ca9fd1337b9b45eeb03c04dc56e9942b8786cdb13a86d86", "extra_info": null, "node_info": {"start": 28920, "end": 31943}, "relationships": {"1": "950acfd7-35b6-40e0-bf25-1f85e4b3951d", "2": "7e380a6d-4fc5-4186-b584-08ef2cf17f80", "3": "bba7a515-7ef3-4ba6-9572-9438b8bc5ebb"}}, "__type__": "1"}, "bba7a515-7ef3-4ba6-9572-9438b8bc5ebb": {"__data__": {"text": "system, power-law behavior with an expo-\nnent of 23/2 is reached in the regime of large avalanche\nsizes.The critical exponent is identical to that of the random-neighbor OFC model @26#and, for example, for mean-\ufb01eld\npercolation @30#.\nFor \ufb01nite L, the distribution is actually very close to\npower-law behavior. Since the critical case corresponds to\nthe conservative system\na51, the supracritical regime be-\ncomes smaller and smaller as N!\u2018: the occurrence of non-\nmonotonic avalanches is a \ufb01nite-size effect.\n4. Avalanche durations\nFor avalanche durations pd(D,N,a), an iterative equation\nfor the corresponding regions and their volumes in the con-\ufb01guration space can be derived; cf. Eq. ~C23!. A closed ex-\npression corresponding to the avalanche size distribution ~8!,\nhowever, is not available.\nIV. EXTENSIONS AND APPLICATIONS OF THE MODEL\nIN THE CONTEXT OF NEURAL NETWORKS\nModels of SOC can usually be interpreted in terms of\nneural networks ~e.g., Refs. @23,31\u201334 #!. Single elements are\nidenti\ufb01ed with model neurons that receive both external andinternal input. The energy variable corresponds to some in-ternal state of a neuron, usually interpreted as its excitationor membrane potential. Upon reaching a threshold, the neu-ron is reset and subsequently sends an input to other neuronsin the network. In the following, we will study extensionsand applications of the avalanche model using neural net-work terminology.\nA. The case e\u00b81\nThe results described in Sec. III are valid for the Abelian\ncasee51. In terms of neural networks, this corresponds to a\nfast neural relaxation such that the excess energy u\u02dci2Uis\naccumulated afterthe reset. For e,1 in Eq. ~2!, the reset of\na neuron is slower, such that a fraction 1 2eof the excess\nenergy is lost @34#.\nWe show examples of avalanche size distributions in Fig.\n5~a!, and examples of duration distributions in Fig. 5 ~b!, for\ne50.1.\nA conspicuous feature is the appearance of additional\npeaks also in the regime where avalanches are restricted to\nsizesL<N. The distributions thus deviate from a power law\nwith a single exponent as in the conservative case e51.\nWhen some neurons cross the threshold, the differences be-\ntween their membrane potentials before the avalanche, ui\n2uj, will become smaller after the avalanche stopped,\ne(ui2uj). Therefore e,1 induces peaks in r(u), whichintroduce length scales in the distributions pandpd, when\nr(u) is rotated in uduring an avalanche. The differences\nbetween distributions for e51 and e,1 are most pro-\nnounced above a5ac, as can easily be seen in Fig. 5. Small\necan also prevent avalanches larger than Nin the multi-\npeaked regime\u2014the dissipation during the reset of the mem-brane potentials eats up the excess energy which otherwisewould make the same neuron \ufb01re twice during an avalanche.\nSimilar avalanche size distributions were described by\nCorralet al. @23#for locally connected networks of integrate-\nand-\ufb01re neurons receiving uniform input to which somenoise was added. As in our model, the dissipation of energywas responsible for the occurrence of the peaks whereas inthe conservative case, approximate power-law behavior wasobserved.\nB. Avalanches in networks of leaky integrate-and-\ufb01re neurons\nIn the context of biologically motivated neural networks,\nadditional parameters such as time delays of interaction or\nFIG. 5. Distributions of ~a!avalanche sizes and ~b!avalanche\ndurations for a subcritical coupling strength a50.8~dashed line", "doc_id": "bba7a515-7ef3-4ba6-9572-9438b8bc5ebb", "embedding": null, "doc_hash": "d312cb2862d10d41648bdbba37d3d7fb89d3d51ea236838c0f8b9a846385c5aa", "extra_info": null, "node_info": {"start": 31825, "end": 35278}, "relationships": {"1": "950acfd7-35b6-40e0-bf25-1f85e4b3951d", "2": "e282d256-9f11-4515-b813-b90e589f831e", "3": "851bfea7-7ea6-48e4-a4ae-3ed590d606c2"}}, "__type__": "1"}, "851bfea7-7ea6-48e4-a4ae-3ed590d606c2": {"__data__": {"text": "same neuron \ufb01re twice during an avalanche.\nSimilar avalanche size distributions were described by\nCorralet al. @23#for locally connected networks of integrate-\nand-\ufb01re neurons receiving uniform input to which somenoise was added. As in our model, the dissipation of energywas responsible for the occurrence of the peaks whereas inthe conservative case, approximate power-law behavior wasobserved.\nB. Avalanches in networks of leaky integrate-and-\ufb01re neurons\nIn the context of biologically motivated neural networks,\nadditional parameters such as time delays of interaction or\nFIG. 5. Distributions of ~a!avalanche sizes and ~b!avalanche\ndurations for a subcritical coupling strength a50.8~dashed line !\nfor a critical coupling a50.99~solid line !, a supracritical coupling\na50.999 ~dashed-dotted line !, and a coupling strength of a\n50.9998 ~dotted line !. Compare also the distributions shown in Fig.\n1 using identical a\u2019s. In all cases, the presented curves are temporal\naverages over 106avalanches with N510000, DU50.022, e\n50.1, andU51. For comparison, the thick solid line in ~a!shows\nthe critical size distribution for e51.EURICH, HERRMANN, AND ERNST PHYSICAL REVIEW E 66, 066137 ~2002!\n066137-6\ndecay time constants for the elements\u2019 dynamical variable u\nare usually employed ~see, e.g., Refs. @35\u201341 #!. Here we\nbrie\ufb02y show how the avalanche statistics changes by the in-troduction of a leak term into the dynamical equation ~1!.\nWithout input to element i, this leak term yields an exponen-\ntial decay of u\nito zero with time constant t. For our simu-\nlations with leaky threshold neurons, we used a discretizedversion of the continuous dynamical system\ntu\u02d9i~t!52ui~t!1Iiext~t!1Iiint~t!~i51 ,...,N!,~10!\nwith external input Iiext(t)5d(t2kDt)dr(k),iDU,kPZ.W e\nde\ufb01ne DUto be\nDU5u0@12exp~2Dt/t!N#, ~11!\nwhere 1/ Dtis the rate of the external input and u0the\nasymptotic energy to which an uncoupled neuron would bedrivenintheabsenceofa\ufb01ringthreshold.Ifneuron ireaches\nits threshold U, the energy is reset to u\ni50.\nIn the previous, Abelian case we had only one parameter\nDUcontrolling the input, which had apparently no in\ufb02uence\non the shape of the avalanche distributions @see Eq. ~8!#.\nNow, there are two parameters controlling the neuron\u2019s\ninput-output characteristics, u0andDt. In the following, we\ndemonstrate the phenomena resulting from varying these in-put parameters.\nIn Fig. 6, we choose the critical case\na5acin a system of\nN51000 neurons, while varying u0. The respective time in-\nterval Dtis chosen such that the input DUis constant. Ef-\nfectively, the case u05\u2018@Fig. 6 ~a!#corresponds to neurons\nwithout leakage, and decreasing u0yields the network be-\nhavior for increasing leakage. Such a decrease imposes twochanges: \ufb01rst, large avalanches get more and more improb-able, and second, oscillations are induced into the size dis-tributions. Both effects can be understood by observing the\nenergy densities\nr(u)~small insets in Fig. 6 !. While in the\nnonleaky case @Fig. 6 ~a!#,ris nearly uniform, a leaky inte-\ngration causes more neurons to have energies", "doc_id": "851bfea7-7ea6-48e4-a4ae-3ed590d606c2", "embedding": null, "doc_hash": "141dab415abe67e410cc6f369155ff1a1a62e80b09276d348e64705d9eda762b", "extra_info": null, "node_info": {"start": 35225, "end": 38299}, "relationships": {"1": "950acfd7-35b6-40e0-bf25-1f85e4b3951d", "2": "bba7a515-7ef3-4ba6-9572-9438b8bc5ebb", "3": "0631eda8-3675-4993-bc6f-915690b5017e"}}, "__type__": "1"}, "0631eda8-3675-4993-bc6f-915690b5017e": {"__data__": {"text": "u0. The respective time in-\nterval Dtis chosen such that the input DUis constant. Ef-\nfectively, the case u05\u2018@Fig. 6 ~a!#corresponds to neurons\nwithout leakage, and decreasing u0yields the network be-\nhavior for increasing leakage. Such a decrease imposes twochanges: \ufb01rst, large avalanches get more and more improb-able, and second, oscillations are induced into the size dis-tributions. Both effects can be understood by observing the\nenergy densities\nr(u)~small insets in Fig. 6 !. While in the\nnonleaky case @Fig. 6 ~a!#,ris nearly uniform, a leaky inte-\ngration causes more neurons to have energies near the \ufb01ringthreshold Uthan energies near the resting potential, \ufb01nally\nintroducing oscillations and peaks in\nr@Figs. 6 ~b\u2013d!#. These\ndensity oscillations lead to the observed oscillations in thesize distributions due to the deterministic readout mechanismof the avalanches: during an avalanche, the neural energiesare uniformly shifted on the uaxis. We observe that the\nnumber of oscillatory peaks decreases as u\n0decreases, while\nthe oscillation amplitude increases.\nIn a second numerical experiment, we held the leakiness\nconstant, while we varied both the rate 1/ Dtat which exter-\nnal input DUwas delivered, and the coupling constant a\nsuch that a transition from subcritical to supracritical oc-\ncurred. Our results are summarized in Fig. 7. With highlyvariable external driving, and subcritical coupling ~upper left\nplot in Fig. 7 !, the neurons do not show any sign of synchro-\nnization. When the external driving gets more frequent~lower left plot in Fig. 7 !, even a small coupling leads to\nsynchronization, accompanied by a strong oscillation.Thingsdo not change signi\ufb01cantly when the coupling gets stronger~lower right plot in Fig. 7 !, only the oscillation period getsshorter and the noise appears to have a stronger in\ufb02uence on\nthe dynamics. When both the variability of the external inputand the coupling is high, the system synchronizes without\noscillating. Here, one element is likely to trigger a large por-tion of the other elements in the network ~synchronization !,\nbut the input variability ensures that the membrane potentialsof the elements get desynchronized before another avalancheis triggered, preventing an oscillatory component to build upin the cross-correlation functions.\nV. SUMMARYAND CONCLUSION\nIn summary, we presented an avalanche model involving\nrandom input and global coupling between its elements. Ava-lanche size distributions can be calculated exactly for an ar-bitrary system size through combinatorial arguments in thesystem\u2019s con\ufb01guration space. The model therefore accountsfor phenomena in \ufb01nite systems and elucidates the transitionto the thermodynamic limit.\nThe model belongs to the same universality class as the\nrandom-neighbor OFC model, showing similar distributionsin the subcritical and critical regimes, and the same critical\nexponent 23/2 in the conservative case\na51a sN!\u2018.\nThe analytical access to avalanche size and duration dis-\ntributions in \ufb01nite systems is especially important when mod-\neling systems that in reality have some 100 to 10000 ele-\nments. For example, cortical columns are examples neural\nnetworks with an order of 1000 to 10000 elements which are\nFIG. 6. Distributions of avalanche sizes, p(L,N,a), of a fully\nconnected network of leaky threshold elements receiving randominput for different leakiness constants u\n0, namely, for ~a!u05\u2018,\n~b!u055,~c!u051.5, and ~d!u051.01. The insets show the cor-\nresponding mean energy", "doc_id": "0631eda8-3675-4993-bc6f-915690b5017e", "embedding": null, "doc_hash": "ca73dab4289caa0018ad9b5d032ad89f64c179e14e4cd3cf82c49afefb7c3e74", "extra_info": null, "node_info": {"start": 38393, "end": 41900}, "relationships": {"1": "950acfd7-35b6-40e0-bf25-1f85e4b3951d", "2": "851bfea7-7ea6-48e4-a4ae-3ed590d606c2", "3": "3487ebc9-ffaa-48a7-aa8e-872ddaa89c8c"}}, "__type__": "1"}, "3487ebc9-ffaa-48a7-aa8e-872ddaa89c8c": {"__data__": {"text": "and the same critical\nexponent 23/2 in the conservative case\na51a sN!\u2018.\nThe analytical access to avalanche size and duration dis-\ntributions in \ufb01nite systems is especially important when mod-\neling systems that in reality have some 100 to 10000 ele-\nments. For example, cortical columns are examples neural\nnetworks with an order of 1000 to 10000 elements which are\nFIG. 6. Distributions of avalanche sizes, p(L,N,a), of a fully\nconnected network of leaky threshold elements receiving randominput for different leakiness constants u\n0, namely, for ~a!u05\u2018,\n~b!u055,~c!u051.5, and ~d!u051.01. The insets show the cor-\nresponding mean energy densities r(u). In all cases, the presented\ncurves are temporal averages over 106avalanches with N51000,\nU51,DU50.17, and t51. With these parameters, the discretiza-\ntion time step Dtwas chosen to satisfy Eq. ~11!.FINITE-SIZE EFFECTS OF AVALANCHE DYNAMICS PHYSICAL REVIEW E 66, 066137 ~2002!\n066137-7\ndensely connected to each other, but sparsely connected to\nother columns. Our approach may help to understand thesynchronization properties of these local networks receivingapparently stochastic input. Even when the analytically solv-able Abelian model may abstract from a real neuron, theproperties of the avalanche distributions are stable with re-spect to changes in the underlying model itself\u2014we alreadypointed out its similarity with the distributions seen in therandom-neighbor OFC model. In general, it is not easy tomotivate the random-neighbor OFC model, because it em-ploys a coupling changing randomly in each step of an ava-\nlanche. In the neuronal context, however, the model may bean example of a constantly driven, densely connected net-work of elements subjected to synaptic failures that occurrelatively often in reality.\nAmong other dynamical properties, we also observe syn-\nchronization without oscillations. While this phenomenonhas already been observed in biology @42#and modeling\nstudies ~see, e.g., Refs. @43,44 #!, we link its occurrence to the\ntransition from the critical to the supracritical regime. Thefact that the latter disappears for large networks goes to-gether with the synchronized dephasing due to \ufb01nite size \ufb01rstmentioned in Ref. @45#~cf. also Ref. @43#!.\nWith the advance of experimental technologies such as\ne.g., stable long-time multielectrode recordings, the questionof whether one can \ufb01nd similar phenomena in our \u2018\u2018toy\u2019\u2019model as well as in reality arises\u2014our analysis could then\nprovide a tool to understand the mechanisms behind the dy-namics. While there are hints that in some cases, power lawscan be found in the brain\u2019s dynamics @46\u201348 #, it remains to\nelucidate which functional advantage a critical state mayhave for the information processing going on in the brain.\nACKNOWLEDGMENTS\nWe would like to thank Professor Theo Geisel for most\nfruitful discussions at the Max-Planck-Institute for Fluid Dy-namics in Go \u00a8ttingen. This work has been supported by the\nDFG, Sonderforschungsbereich 517 \u2018\u2018Neurokognition\u2019\u2019 ~U.E.\nand C.W.E. !, and by theVolkswagen Foundation, Project No.\n5425 ~U.E.!.\nAPPENDIX A: PRELIMINARIES\nIn the appendixes, we derive the exact avalanche distribu-\ntionsp(L,N,a) for arbitrary system sizes N. Appendix A\nwill introduce a suitable notation for partitioning the con-\n\ufb01guration space Pinto products of lower-dimensional sub-\nsets. InAppendix B, we calculate the volume", "doc_id": "3487ebc9-ffaa-48a7-aa8e-872ddaa89c8c", "embedding": null, "doc_hash": "19efe4c97b68584f1dea866c5bb0f2c6baf0688a80caed2dfd3369de55d770a8", "extra_info": null, "node_info": {"start": 41867, "end": 45252}, "relationships": {"1": "950acfd7-35b6-40e0-bf25-1f85e4b3951d", "2": "0631eda8-3675-4993-bc6f-915690b5017e", "3": "30fa272d-b0ff-4b7f-b7cb-17f3e098b8bd"}}, "__type__": "1"}, "30fa272d-b0ff-4b7f-b7cb-17f3e098b8bd": {"__data__": {"text": "for most\nfruitful discussions at the Max-Planck-Institute for Fluid Dy-namics in Go \u00a8ttingen. This work has been supported by the\nDFG, Sonderforschungsbereich 517 \u2018\u2018Neurokognition\u2019\u2019 ~U.E.\nand C.W.E. !, and by theVolkswagen Foundation, Project No.\n5425 ~U.E.!.\nAPPENDIX A: PRELIMINARIES\nIn the appendixes, we derive the exact avalanche distribu-\ntionsp(L,N,a) for arbitrary system sizes N. Appendix A\nwill introduce a suitable notation for partitioning the con-\n\ufb01guration space Pinto products of lower-dimensional sub-\nsets. InAppendix B, we calculate the volume of the region L\nin phase space which is not inhabited between avalanches,by using a partitioning of the con\ufb01guration space leading toa recursion formula for subregions. In Appendix C, a recur-\nFIG. 7. Raster plots showing\nthe \ufb01ring dynamics of a networkofN5100 neurons. Each spike is\ndrawn as a small black tick in de-pendence of the time t, and the\nnumber of the neuron which emit-ted that spike. The coupling pa-rameter\nawas chosen to be a\n50.6~top left !, 0.92 ~top right !,\n0.6~bottom left !, and 0.92 ~bot-\ntom right !, while the time interval\nbetween two external inputs wasgiven by Dt520/N,20/N,1/Nand\n1/Nm, respectively. The insets\nshow the mean over the cross-correlation functions from 300pairs selected out of the N5100\nneurons. The cross-correlationfunctions have been scaled arbi-trarily, but identically for all fourinsets.u\n0was chosen to be u0\n51.05 and t532.84 ms, yielding\nan output rate of 10 Hz for an un-coupled neuron. The critical cou-pling strength for N5100 neurons\nis\nacrit50.9.EURICH, HERRMANN, AND ERNST PHYSICAL REVIEW E 66, 066137 ~2002!\n066137-8\nsion formula for regions leading to avalanches of a speci\ufb01c\nsizeLwill be derived, and subsequently be modi\ufb01ed by sub-\ntracting the noninhabited regions. This modi\ufb01cation \ufb01nally\nleads to the exact avalanche distributions p(L,N,a).\nBefore starting the analysis, we will shortly summarize\nthe terminology used in the appendix: x,y,zPR, andx,y,z\nPRm;i,j,k,l,m,n,p,q,rPNdenote indices; I,J,Kdenote\nsets of indices. I,J,Kdenote second-order sets of indices;\nPm,Vm,Gm,Qm,Fm#Rmdenote subsets in Rm;D,S,Vde-\nnote volumes of subsets. Overlined symbols will denote re-gions and volumes excluding the subset of the non-inhabitedvolume in con\ufb01guration space.\n1. Subsets and sets of indices\nLetIk,mdenote an arbitrary k-element subset of Im\n\u201c$1 ,...,m%. The superset of all different Ik,mis denoted by\nIk,m.Ik,mcontains thus (km)k-element subsets of Imas its\nelements.\nFor the following analysis, it is convenient to de\ufb01ne\nm-dimensional subsets Pm,\nPm~xmin,xmax!\u201c$xP@xmin,xmax!m#Rm%. ~A1!\nThe con\ufb01guration space of Nunits can then be denoted by\nPN(0,U).\nWe also de\ufb01ne subsets Gkm(l,u,Ik,m) for 0 ,k<mand", "doc_id": "30fa272d-b0ff-4b7f-b7cb-17f3e098b8bd", "embedding": null, "doc_hash": "3d40cbd3dfcc5b054dea5fb7a0f02dab5dc95e4b06bed2cabce48d634840ab3c", "extra_info": null, "node_info": {"start": 45314, "end": 48042}, "relationships": {"1": "950acfd7-35b6-40e0-bf25-1f85e4b3951d", "2": "3487ebc9-ffaa-48a7-aa8e-872ddaa89c8c", "3": "701d0aff-6665-4b22-98b5-4bd41470f1db"}}, "__type__": "1"}, "701d0aff-6665-4b22-98b5-4bd41470f1db": {"__data__": {"text": "in con\ufb01guration space.\n1. Subsets and sets of indices\nLetIk,mdenote an arbitrary k-element subset of Im\n\u201c$1 ,...,m%. The superset of all different Ik,mis denoted by\nIk,m.Ik,mcontains thus (km)k-element subsets of Imas its\nelements.\nFor the following analysis, it is convenient to de\ufb01ne\nm-dimensional subsets Pm,\nPm~xmin,xmax!\u201c$xP@xmin,xmax!m#Rm%. ~A1!\nThe con\ufb01guration space of Nunits can then be denoted by\nPN(0,U).\nWe also de\ufb01ne subsets Gkm(l,u,Ik,m) for 0 ,k<mand 0\n,l<1,\nGkm~l,u,Ik,m!\u201cHxPPm~0,u!Uxi,lk\nmu,iPIk,mJ.\n~A2!\nLetLmdenote a union of Gkm\u2019s,\nLm~l,u!\u201c\u0142\nk51m\n\u0142\nIk,mPIk,mGkm~l,u,Ik,m!. ~A3!\nFor the special case of l51,Gmm(1,u)5Pm(0,u) and, there-\nfore,\nLm~1,u!5Pm~0,u!. ~A4!\nIn order to be able to combine lower-dimensional subsets,\nwe \ufb01nally de\ufb01ne the direct product Qk^Fm2kuIk,mbetween\ntwo subsets. Ik,mdetermines the indices of the components\nof the elements yin the resulting volume assigned to com-\nponents belonging to elements xinQk,\nQk\u201c$xPA#Rk%,\nFm2k\u201c$xPB#Rm2k%,\nQk^Fm2kuIk,m5$yPRmu$yi%iPIk,mPA,$yi%iPImnIk,mPB%.\n~A5!\nNote that this de\ufb01nition is well de\ufb01ned only for sets Abeing\ninvariant under a permutation of the components of xPA.\nThe operator ^is assumed to have higher precedence than\n\u0142,\u00f8, and \\.2. Lemmas\nThe following three lemmas will help to shorten the deri-\nvation of the recursion formulas in the following sections.\nLemma 1. ;k,l<m;;u.0;;l:0,l<1,\nA\u201cPl~0,lu!^Pm2l~lu,u!uJl,m\u00f8Gkm~l,u,Ik,m!\n\u00deB,Ik,m#Jl,m. ~A6!\nProof.Let us choose a suitable disjoint decomposition of\nthe index set Imas\nIm5~Ik,m\u00f8Jl,m!\n\u0142~Ik,m\\Jl,m!\n\u0142~Jl,m\\Ik,m!\n\u0142~Im\\\u0084Ik,m\u0142Jl,m!\u0085. ~A7!\nUsing Eqs. ~A1!,~A3!, and ~A5!, we can then explicitly\nwriteAas\nA5$xPRmu0<xi,luk/m:iPIk,m\u00f8Jl,m,\nlu<xi,luk/m:iPIk,m\\Jl,m,\n0<xi,lu:iPJl,m\\Ik,m,\nlu<xi,u:iPIm\\~Ik,m\u0142Jl,m!%. ~A8!\nBecause of lu>luk/m,Ais nonempty if and only if\nIk,m\\Jl,m5B; and this implies that Ik,m#Jl,m,A\u00deB. Note\nthat ifk.l, condition Ik,m#Jl,mis never ful\ufb01lled.\nLemma 2.", "doc_id": "701d0aff-6665-4b22-98b5-4bd41470f1db", "embedding": null, "doc_hash": "5664f702c503e98901ae466b2f5b76b027003ec81c92883f03bce55f71e01d3c", "extra_info": null, "node_info": {"start": 48125, "end": 50039}, "relationships": {"1": "950acfd7-35b6-40e0-bf25-1f85e4b3951d", "2": "30fa272d-b0ff-4b7f-b7cb-17f3e098b8bd", "3": "f5f3457d-3d38-45f9-bea0-60efa8549999"}}, "__type__": "1"}, "f5f3457d-3d38-45f9-bea0-60efa8549999": {"__data__": {"text": "~A1!,~A3!, and ~A5!, we can then explicitly\nwriteAas\nA5$xPRmu0<xi,luk/m:iPIk,m\u00f8Jl,m,\nlu<xi,luk/m:iPIk,m\\Jl,m,\n0<xi,lu:iPJl,m\\Ik,m,\nlu<xi,u:iPIm\\~Ik,m\u0142Jl,m!%. ~A8!\nBecause of lu>luk/m,Ais nonempty if and only if\nIk,m\\Jl,m5B; and this implies that Ik,m#Jl,m,A\u00deB. Note\nthat ifk.l, condition Ik,m#Jl,mis never ful\ufb01lled.\nLemma 2. ;l<m;;u.0;;l:0,l<1,\n\u0142\nk51l\n\u0142\nIk,m#Jl,mGkm~l,u,Ik,m!\n5Ll~ll/m,u!^Pm2l~0,u!uJl,m. ~A9!\nProof.Inserting de\ufb01nition ~A2!into the innermost union\nin Eq. ~A9!yields\n\u0142\nIk,m#Jl,mGkm~l,u,Ik,m!\n5\u0142\nIk,m#Jl,mHxPPm~0,u!Uxj,lk\nmu,jPIk,mJ.\nIn this union, exactly m2lcomponents of xcover the whole\ninterval @0,u). By separating these components forming a-FINITE-SIZE EFFECTS OF AVALANCHE DYNAMICS PHYSICAL REVIEW E 66, 066137 ~2002!\n066137-9\nsubset Pm2l(0,u), the union can be written as a direct prod-\nuct of Pm2lwith a union of dimension l, using suitably\nchosen index sets Kk,l;\nS\u0142\nKk,lPKk,lHxPPl~0,u!UxjPKk,l,ll\nmk\nluJD\n^Pm2l~0,u!uJl,m\n5S\u0142\nKk,lPKl,kGkl~ll/m,u!D^Pm2l~0,u!U\nJl,m.\n~A10!\nThen Eq. ~A9!follows immediately, using the de\ufb01nition\n~A3!forLl.\nLemma 3. ;z<y,\nLm~x,y!\u00f8Pm~0,z!5Lm~xy/z,z!. ~A11!\nProof.This can be achieved by rescaling the parameters u\nandlin de\ufb01nition ~A2!to the smaller subset Pm(0,z), and\ninserting the rescaled de\ufb01nition into Eq. ~A3!.\nAPPENDIX B: CALCULATION OF THE NONINHABITED\nVOLUME\nIn a con\ufb01guration space of dimension Nand volume UN,\nthe volume not inhabited between avalanches mediated by a\ncoupling of strength aU/Nis denoted by LN(a,U). The\npurpose of this section will be to calculate its volume V,\nwhich is done iteratively. The reason for using this strategycan be illustrated by comparing the phase spaces and their\npartitionings for N52~Fig. 4 !andN53~Fig. 8 !. The par-\ntitioning for N52 is similar to the partitioning of the u\n1-u2\nplane in Fig. 8, except for a change in the side lengths of the\nvolumes. This \u2018\u2018self-similiarity\u2019\u2019 continues when proceedingto higher Nand enables the iterative calculation of the vol-\numes L\nN. Note that already L3has a relatively complex\nstructure.\nTheorem. ;l,uand;m.0,V\u0084Lm(l,u)\u0085is given by\nthe particularly simple expression\nV\u0084Lm~l,u!\u00855lum. ~B1!\nThe proof will", "doc_id": "f5f3457d-3d38-45f9-bea0-60efa8549999", "embedding": null, "doc_hash": "d07dbc2768ea7cad29c7adc66e3ea6d99466633df038de89036ff4f836541741", "extra_info": null, "node_info": {"start": 50153, "end": 52301}, "relationships": {"1": "950acfd7-35b6-40e0-bf25-1f85e4b3951d", "2": "701d0aff-6665-4b22-98b5-4bd41470f1db", "3": "4ab5ea03-d5fa-4de3-9a71-cd0118f66b5e"}}, "__type__": "1"}, "4ab5ea03-d5fa-4de3-9a71-cd0118f66b5e": {"__data__": {"text": "using this strategycan be illustrated by comparing the phase spaces and their\npartitionings for N52~Fig. 4 !andN53~Fig. 8 !. The par-\ntitioning for N52 is similar to the partitioning of the u\n1-u2\nplane in Fig. 8, except for a change in the side lengths of the\nvolumes. This \u2018\u2018self-similiarity\u2019\u2019 continues when proceedingto higher Nand enables the iterative calculation of the vol-\numes L\nN. Note that already L3has a relatively complex\nstructure.\nTheorem. ;l,uand;m.0,V\u0084Lm(l,u)\u0085is given by\nthe particularly simple expression\nV\u0084Lm~l,u!\u00855lum. ~B1!\nThe proof will be given by induction over m.\nBasis.From de\ufb01nitions ~A3!and~A2!it is obvious that\nform51,\nV\u0084L1~l,u!\u00855V\u0084G11~l,u,I1,1!\u00855lu. ~B2!\nInduction. For the induction we assume that Eq. ~B2!has\nbeen proven for m5n21. Thus we have to prove that Eq.\n~B2!holds also for m5n.The phase space Pn(0,u) can be expressed as a union of\ndisjoint subsets,\nPn~0,u!5\u0142\nl50n\n\u0142\nJl,nPIl,nPl~0,lu!^Pn2l~lu,u!uJl,n,\n~B3!\nwhose volumes are related to a binomial expansion of\nV(Pn),\nV\u0084Pn~0,u!\u00855un@l1~12l!#n5un(\nl50nSn\nlDll~12l!n2l.\n~B4!\nUsing the de\ufb01nitions ~A1!~A3!, it is clear that\nLn(l,u)#Pn(0,u)nPn(lu,u). Inserting Eqs. ~A3!~B3!\ninto this expression,\nLn~l,u!5@Pn~0,u!nPn~lu,u!#\u00f8Ln~l,u!\n5S\u0142\nl51n\n\u0142\nJl,nPIl,nPl~0,lu!^Pn2l~lu,u!uJl,nD\n\u00f8S\u0142\nk51n\n\u0142\nIk,nPIk,nGkn~l,u,Ik,n!D.\nWe subsequently use Lemmas ~A6!,~A9!, and ~A11!and\nobtain\nFIG. 8. Example of the con\ufb01guration space P3and its partition-\ning. The noninhabited volume L3is highlighted in shades of gray.\nThe volumes V\u00af(L,3,a) leading to avalanches of sizes L50,L\n51,L52, andL53 are outlined with thick black lines at their\nedges.EURICH, HERRMANN, AND ERNST PHYSICAL REVIEW E 66, 066137", "doc_id": "4ab5ea03-d5fa-4de3-9a71-cd0118f66b5e", "embedding": null, "doc_hash": "b9fbec865dfee7f2dfc4365eeb7997e9eaa5f50a5c7d4a80c484714866c129d1", "extra_info": null, "node_info": {"start": 52112, "end": 53782}, "relationships": {"1": "950acfd7-35b6-40e0-bf25-1f85e4b3951d", "2": "f5f3457d-3d38-45f9-bea0-60efa8549999", "3": "e8595aab-231a-4568-b65b-36a42d7ef43d"}}, "__type__": "1"}, "e8595aab-231a-4568-b65b-36a42d7ef43d": {"__data__": {"text": "subsequently use Lemmas ~A6!,~A9!, and ~A11!and\nobtain\nFIG. 8. Example of the con\ufb01guration space P3and its partition-\ning. The noninhabited volume L3is highlighted in shades of gray.\nThe volumes V\u00af(L,3,a) leading to avalanches of sizes L50,L\n51,L52, andL53 are outlined with thick black lines at their\nedges.EURICH, HERRMANN, AND ERNST PHYSICAL REVIEW E 66, 066137 ~2002!\n066137-10\n\u0142\nl51n\n\u0142\nJl,n\nPIl,nSPl~0,lu!^Pn2l~lu,u!uJl,n\u00f8\u0142\nk51l\n\u0142\nIk,n#Jl,nGkn~l,u,Ik,n!D@Eq.~A6!#\n5\u0142\nl51n\n\u0142\nJl,n\nPIl,nSPl~0,lu!^Pn2l~lu,u!uJl,n\u00f8Ll~ll/n,u!^Pn2l~0,u!uJl,nD@Eq.~A9!#\n5\u0142\nl51n\n\u0142\nJl,nPIl,n\u0084Pl~0,lu!\u00f8Ll~ll/n,u!\u0085^\u0084Pn2l~lu,u!\u00f8Pn2l~0,u!\u0085uJl,n,\n5\u0142\nl51n\n\u0142\nJl,n\nPIl,nLlSl\nn,luD^Pn2l~lu,u!U\nJl,n@Eq.~A11!#. ~B5!\nBy construction @see Eq. ~B3!#, the subsets are disjoint and\nthe volume Vof their union can be written as a sum over the\nsubvolumes. In addition, volumes of subsets for different in-\ndex sets Jl,nfor \ufb01xed nandlare identical. Thus we can\ninsert Eq. ~B1!forl,n, and Eq. ~A4!forl5n. Through this\nprocedure we close the induction\nV\u0084Ln~l,u!\u00855(\nl51nSn\nlDl\nn~lu!lun2l~12l!n2l\n5lun(\nl51nSn21\nl21Dll21~12l!n2l\n5lun(\nk50n21Sn21\nkDlk~12l!(n21)2k5lun.j\n~B6!\nBy choosing n5N,u5U, and l5a, we obtain the vol-\numeVfor the noninhabited region as\nV\u0084LN~a,U!\u00855aUN. ~B7!\nAPPENDIX C: AVALANCHE DISTRIBUTIONS\nIn this section, we will prove the following theorem for\nthe avalanche probabilities p(L,N,a).\nTheorem.\np~L,N,a!5LL22SN21\nL21DSa\nNDL21\n3S12La\nNDN2L21N~12a!\nN2~N21!a.~C1!\nProof.It is convenient to divide the proof into three steps.\nThe \ufb01rst step will be to identify the regions in con\ufb01gurationspace leading to avalanches of a certain size L. The second\nstep will be to subtract the noninhabited subset L\nNfrom\nthese regions. By calculating their volume, one \ufb01nally ob-\ntains the correct avalanche probabilities p(L,N,a). As in thepreceding appendix, we will use an iterative procedure, as\nsuggested by comparing Figs. 4 and 8.\n1. Regions representing different avalanche sizes\nTo convey the idea behind the analysis, we \ufb01rst recall the\ndynamics during one event in an avalanche. Typically, m\nunits have still not been active yet, lunits are just \ufb01ring, k\nelements have already \ufb01red and will not be activated", "doc_id": "e8595aab-231a-4568-b65b-36a42d7ef43d", "embedding": null, "doc_hash": "3f8147e4834fbd7098232bc749c91ec7a5472bc45df3a09e03da7fe0b1d40601", "extra_info": null, "node_info": {"start": 53955, "end": 56133}, "relationships": {"1": "950acfd7-35b6-40e0-bf25-1f85e4b3951d", "2": "4ab5ea03-d5fa-4de3-9a71-cd0118f66b5e", "3": "d2ba30fc-0e5f-4891-895d-6d148e261c7d"}}, "__type__": "1"}, "d2ba30fc-0e5f-4891-895d-6d148e261c7d": {"__data__": {"text": "step will be to identify the regions in con\ufb01gurationspace leading to avalanches of a certain size L. The second\nstep will be to subtract the noninhabited subset L\nNfrom\nthese regions. By calculating their volume, one \ufb01nally ob-\ntains the correct avalanche probabilities p(L,N,a). As in thepreceding appendix, we will use an iterative procedure, as\nsuggested by comparing Figs. 4 and 8.\n1. Regions representing different avalanche sizes\nTo convey the idea behind the analysis, we \ufb01rst recall the\ndynamics during one event in an avalanche. Typically, m\nunits have still not been active yet, lunits are just \ufb01ring, k\nelements have already \ufb01red and will not be activated again,andjof themremaining units will be activated until the\navalanche stops. If the coupling strength is\nb5aU/N,n o\nstate variable uof the remaining munits could have been\ninitially larger than U2kb. We will denote the\nm-dimensional subsets of the con\ufb01guration space, which will\nevolve into the situation described above, with Vk,lm(j). The\nfollowing considerations will lead to a recursion formula for\nVk,lm(j) over the variable j.\nLet us start with the subspace Pm(0,U2kb), which can\nbe written as a union over all V\u2019s with \ufb01xed k,l, andm,\nPm~0,U2kb!5\u0142\nj50m\nVk,lm~j!. ~C2!\nIn other words, Eq. ~C2!expresses that an m-dimensional\ncon\ufb01guration space of side length U2kb, onto which an\ninput oflbis given, can be decomposed into subsets where\njunits will \ufb01re. It is obvious that for the case j50 in which\nan avalanche stops, the subset Vk,lm(0) is given by\nVk,lm~0!5Pm\u00840,U2~k1l!b\u0085. ~C3!\nWhile decomposition ~C2!partitions Pmconsidering the\nwhole remaining part of an avalanche with junits \ufb01ring, one\ncan equally well partition Pmconsidering only the next step\nin an avalanche, where the input of lbcan trigger up to m\nunits to \ufb01re immediately. With idenoting the number of\nthese units, the disjoint decomposition then readsFINITE-SIZE EFFECTS OF AVALANCHE DYNAMICS PHYSICAL REVIEW E 66, 066137 ~2002!\n066137-11\nPm~0,U2kb!5\u0142\ni50m\n\u0142\nIi,m\nPIi,mPm2i\u00840,U2~k1l!b\u0085\n^Pi\u0084U2~k1l!b,U2kb\u0085U\nIm2i,m.\n~C4!\nUsing an appropriately scaled ~C2!as a decomposition of\nPm2i, the common input ibdue to the iunits \ufb01ring will\nsubsequently trigger j82ielements until the avalanche\nstops,\nPm2i\u00840,U2~k1l!b\u00855\u0142\nj85im\nVk1l,im2i~j82i!. ~C5!\nInserting Eq. ~C5!into Eq. ~C4!, and comparing Eqs. ~C2!\n~C4!, one obtains after changing the precedence of the\nunions over", "doc_id": "d2ba30fc-0e5f-4891-895d-6d148e261c7d", "embedding": null, "doc_hash": "80b40250f066146c9829a508255ba8e80a84b970f65564c0377d4d3d3e05c06b", "extra_info": null, "node_info": {"start": 55882, "end": 58281}, "relationships": {"1": "950acfd7-35b6-40e0-bf25-1f85e4b3951d", "2": "e8595aab-231a-4568-b65b-36a42d7ef43d", "3": "d1be3b9a-914d-42cd-92c5-efe7ebb6f33e"}}, "__type__": "1"}, "d1be3b9a-914d-42cd-92c5-efe7ebb6f33e": {"__data__": {"text": "an appropriately scaled ~C2!as a decomposition of\nPm2i, the common input ibdue to the iunits \ufb01ring will\nsubsequently trigger j82ielements until the avalanche\nstops,\nPm2i\u00840,U2~k1l!b\u00855\u0142\nj85im\nVk1l,im2i~j82i!. ~C5!\nInserting Eq. ~C5!into Eq. ~C4!, and comparing Eqs. ~C2!\n~C4!, one obtains after changing the precedence of the\nunions over iandj8,\n\u0142\nj51m\nVk,lm~j!5\u0142\ni51m\n\u0142\nIi,mPIi,mS\u0142\nj85im\nVk1l,im2i~j82i!D\n^Pi\u0084U2~k1l!b,U2kb\u0085U\nIm2i,m\n5\u0142\nj851m\n\u0142\ni51j8\n\u0142\nIi,mPIi,mVk1l,im2i~j82i!\n^Pi\u0084U2~k1l!b,U2kb\u0085U\nIm2i,m.~C6!In Eq. ~C6!, we excluded subsets where the input lbtriggers\nnone of the units, because we already know the result fromEq.~C3!.\nIf we require Eq. ~C6!to represent a recursive description\nof the avalanche dynamics, then one speci\ufb01c V\nk,lm(j) should\nbe composed of terms with j8satisfying ( j82i)1i5j,\nVk,lm~j!5\u0142\ni51j\n\u0142Ii,mPIi,mVk1l,im2i~j2i!\n^Pi\u0084U2~k1l!^b,U2kb\u0085uIm2i,m.~C7!\nThis expression is the required recursion formula.\n2. Subtraction of the noninhabited region\nFor the following considerations, we introduce the abbre-\nviation Fn\u201cPn\u0084U2(k1l)b,U2kb\u0085.\nWe de\ufb01ne V\u00afby subtracting LNfrom V,\nV\u00afk,lm~j!^FN2muIm,N\u201cVk,lm~j!^FN2muIm,NnLN~a,U!.\n~C8!\nIfk1l<N, using Eqs. ~A2!and ~A3!reveals that\nFn\u00f8LN(a,U)5B. Through this property, Eq. ~C7!re-\nmains valid if one replaces the V\u2019s by the V\u00af\u2019s.\nThus it suf\ufb01ces to explicitly compute V\u00afk,lm(j) forj50.\nInserting Eq. ~A3!into Eq. ~C8!, and using Lemmas ~A6!,\n~A9!, and ~A11!yields\nV\u00afk,lm~0!^FN2muIm,N\n5Vk,lm~0!^FN2muIm,Nn\u0142\ni51m\n\u0142\nJi,N#Im,NGiN~a,U,Ji,N!@Eq.~A6!#\n5Vk,lm~0!^FN2muIm,NnLm~am/N,U!^PN2m~0,U!uIm,N@Eq.~A9!#\n5FVk,lm~0!nLmSmaU/N\nU2~k1l!b,U2~k1l!bDG^FN2muIm,N@Eq.~A11!#. ~C9!\nFrom this expression, V\u00afk,lm(0) can be extracted as\nV\u00afk,lm~0!5Vk,lm~0!nLmSmaU/N\nU2~k1l!b,U2~k1l!bD.\n~C10!3. Calculation of the volumes of the", "doc_id": "d1be3b9a-914d-42cd-92c5-efe7ebb6f33e", "embedding": null, "doc_hash": "20f220fa1f7ae1983261aba8c2144f6155bd484dd423820af04d97c7a3a353d7", "extra_info": null, "node_info": {"start": 58553, "end": 60321}, "relationships": {"1": "950acfd7-35b6-40e0-bf25-1f85e4b3951d", "2": "d2ba30fc-0e5f-4891-895d-6d148e261c7d", "3": "6995bb4c-d4f0-412b-8b28-2350734610c8"}}, "__type__": "1"}, "6995bb4c-d4f0-412b-8b28-2350734610c8": {"__data__": {"text": "~C9!\nFrom this expression, V\u00afk,lm(0) can be extracted as\nV\u00afk,lm~0!5Vk,lm~0!nLmSmaU/N\nU2~k1l!b,U2~k1l!bD.\n~C10!3. Calculation of the volumes of the regions\nWithSk,lm(j)\u201cV\u0084Vk,lm(j)\u0085andS\u00afk,lm(j)\u201cV\u0084V\u00afk,lm(j)\u0085, Eqs.\n~C3!and~C7!de\ufb01ne recursions for con\ufb01guration space vol-\numesEURICH, HERRMANN, AND ERNST PHYSICAL REVIEW E 66, 066137 ~2002!\n066137-12\nSk,lm~j!5H\u0084U2~k1l!b\u0085m, j50\n(\ni51jSm\niD~lb!iSk1l,im2i~j2i!,j<m,~C11!\nS\u00afk,lm~j!5H\u0084U2~k1l!b\u0085m21\u0084U2~k1l1m!b\u0085,j50\n(\ni51jSm\niD~lb!iS\u00afk1l,im2i~j2i!, j<m,\n~C12!\nwhereS\u00afk,lm(0) was possible to calculate from Sk,lm(0) by sim-\nply subtracting the volume of Lm, because its size in Eq.\n~C10!has been scaled not to extend over Vk,lm(0),\nS\u00afk,lm~0!5Sk,lm~0!2aUm\nN\u0084U2~k1l!b\u0085m21.~C13!\nUsing similar arguments, one also obtains a recursion for\nthe volumes Dk,lm(j) corresponding to regions of avalanche\ndurations j,\nDk,lm~j!5(\ni51m2j11Sm\niD~lb!iDk1l,im2i~j21!, ~C14!\nfor 0 ,j<NandDk,lm(0)5Sk,lm(0).Correcting for the nonin-\nhabited volume leads to the same recursion for the volumes\nD\u00afk,lm(j) for 0 ,j<NwithD\u00afk,lm(0)5S\u00afk,lm(0).To obtain a closed expression for the volumes SandS\u00af,w e\nwill now prove the following proposition.\nProposition. ForU.0,k1l1j,N, andj<m,\nSk,lm~j!5Sm\njDbjl~j1l!j21\u0084U2~k1l1j!b\u0085m2j,\nS\u00afk,lm~j!5Sm\njDbjl~j1l!j21\u0084U2~k1l1j!b\u0085m2j21\n3\u0084U2~m1k1l!b\u0085. ~C15!\nThe proof is possible by induction over n, and it is very\nsimilar for SandS\u00af. We will therefore only give the proof for\nS\u00afin order to shorten this appendix.\nBasis.Form51,jcan either be 0 or 1, and using Eq.\n~C13!leads to\nS\u00afk,l1~0!5\u0084U2~11k1l!b\u00851, ~C16!\nS\u00afk,l1~1!5~lb!1. ~C17!\nInduction. For the induction we assume that Eq. ~C15!has\nbeen proven for m<n21. Thus we have to prove that Eq.\n~C15!holds also for", "doc_id": "6995bb4c-d4f0-412b-8b28-2350734610c8", "embedding": null, "doc_hash": "5c53c1cb8eebb1b4489fea60eea22a3af5e047a1b2a07df3fa50f7f8f11cc1c6", "extra_info": null, "node_info": {"start": 60486, "end": 62188}, "relationships": {"1": "950acfd7-35b6-40e0-bf25-1f85e4b3951d", "2": "d1be3b9a-914d-42cd-92c5-efe7ebb6f33e", "3": "cb1939e2-a3e0-4684-9c87-7cbf0c6f5314"}}, "__type__": "1"}, "cb1939e2-a3e0-4684-9c87-7cbf0c6f5314": {"__data__": {"text": "~C15!\nThe proof is possible by induction over n, and it is very\nsimilar for SandS\u00af. We will therefore only give the proof for\nS\u00afin order to shorten this appendix.\nBasis.Form51,jcan either be 0 or 1, and using Eq.\n~C13!leads to\nS\u00afk,l1~0!5\u0084U2~11k1l!b\u00851, ~C16!\nS\u00afk,l1~1!5~lb!1. ~C17!\nInduction. For the induction we assume that Eq. ~C15!has\nbeen proven for m<n21. Thus we have to prove that Eq.\n~C15!holds also for m5n,\nS\u00afk,lm~j!5(\ni51jSm\niD~lb!iS\u00afk1l,im2i~j2i!\n5(\ni51jSm\niD~lb!iSm2i\nj2iDbj2i\u0084U2~m1k1l!b\u0085\u0084U2~k1l1j!b\u0085m2j21i~j2i1i!j2i21\n5Sm\njDbj\u0084U2~k1l1j!b\u0085m2j21\u0084U2~m1k1l!b\u0085H(\ni51j\nliSj\niDijj2i21J\n5Sm\njDbj\u0084U2~k1l1j!b\u0085m2j21\u0084U2~m1k1l!b\u0085Hl(\ni850j21Sj21\ni8Dli8j(j21)2i8J\n5Sm\njDbjl~l1j!r21\u0084U2~k1l1j!b\u0085m2j21\u0084U2~m1k1l!b\u0085. j\n~C18!\nWith this closed expression, it will be possible to \ufb01nally\ncalculate an expression of the avalanche probabilities.\n4. Avalanche probabilities\nAn avalanche starts if one unit is triggered by an input of\nstrength DUto \ufb01re. Thus the phase space volumes\nV\u00af(L,N,a) andV(L,N,a) for avalanches of size L.0 areobtained by multiplying DUwithS\u00af0,1N21(L21) andS0,1N21(L\n21), respectively. These speci\ufb01c S\u2019s are the volumes of the\nsubsets of dimension N21 containing states for which j\n5L21 neurons will subsequently \ufb01re, triggered by an input\noflbwithl51.V\u00af(0,N,a) andV(0,N,a) can be compute-\ndas the remaining part of the whole phase space, and weobtainFINITE-SIZE EFFECTS OF AVALANCHE DYNAMICS PHYSICAL REVIEW E 66, 066137 ~2002!\n066137-13\nV~L,N,a!5HUN21~U2DU!,L50\nDUS0,1N21~L21!,L.0,\n5HUN21~U2DU!, L50\nDU\nLUN21SN21\nL21DSLa\nNDL21S12La\nNDN2L\n,L.0~C19!\nand\nV\u00af~L,N,a!5HUN21~12a!FU2DUS12a2a\nNDG, L50\nDU\nLUN21SN21\nL21DSLa\nNDL21S12La\nNDN2L21\n~12a!,L.0.~C20!\nThe probability of an avalanche P(L,N,a) is then given\nbyP(L,N,a)5V\u00af(L,N,a)/V\u0084PN(0,a)nLN(0,a)\u0085. With Eq.\n~B1!,V\u0084PN(0,a)nLN(0,a)\u00855UN(12a); then using Eq. ~3!\nleads to the \ufb01nal result @see also Eq.", "doc_id": "cb1939e2-a3e0-4684-9c87-7cbf0c6f5314", "embedding": null, "doc_hash": "c6f55f67f8b25ae149d4795dc1605505565e16fb62fb9efcea3732b0b8bd1e45", "extra_info": null, "node_info": {"start": 61968, "end": 63825}, "relationships": {"1": "950acfd7-35b6-40e0-bf25-1f85e4b3951d", "2": "6995bb4c-d4f0-412b-8b28-2350734610c8", "3": "12fb5d28-5df9-418b-95ed-2632fdfc8513"}}, "__type__": "1"}, "12fb5d28-5df9-418b-95ed-2632fdfc8513": {"__data__": {"text": "L50\nDU\nLUN21SN21\nL21DSLa\nNDL21S12La\nNDN2L21\n~12a!,L.0.~C20!\nThe probability of an avalanche P(L,N,a) is then given\nbyP(L,N,a)5V\u00af(L,N,a)/V\u0084PN(0,a)nLN(0,a)\u0085. 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Cybern. 62,\n407~1990!.\n@49#The term \u2018\u2018critical\u2019\u2019 is applied to \ufb01nite systems here meaning\nan approximate power-law behavior; true criticality requiresthe thermodynamic limit N!\u2018, which will be discussed be-\nlow.\n@50#By mapping the variable names L\u00b0s,L\na/N\u00b0p, and (N\n21)/L\u00b0n, expression ~C22!becomes identical to Eq. ~36!in\nRef.@26#.FINITE-SIZE EFFECTS OF AVALANCHE DYNAMICS PHYSICAL REVIEW E 66, 066137 ~2002!\n066137-15", "doc_id": "569a331a-329c-4d7b-9508-a7b361c3cac0", "embedding": null, "doc_hash": "0c112ca8347fc8ad43aad6378eb6d93c4f34244486345e35b6a4ef6d31955097", "extra_info": null, "node_info": {"start": 67828, "end": 69239}, "relationships": {"1": "950acfd7-35b6-40e0-bf25-1f85e4b3951d", "2": "9a1f2bb5-a1d8-42b2-bcf0-4af29dd3142d"}}, "__type__": "1"}}}