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epl draft Self-organising mechanism of neuronal avalanche criticality D. E. Juanico(a) Rm 3111 Complex Systems Theory Group, National Institute of Physics 7 0University of the Philippines - Diliman, Quezon City 1101, Philippines 0 2 n PACS 05.65.+b–Self-organised systems a J PACS 87.18.-h–Multicellular phenomena 1 PACS 89.75.Da–Systemsobeying scaling laws 1 Abstract.- Aself-organisingmodelisproposedtoexplainthecriticalityincorticalnetworksde- ducedfromrecentobservationsofneuronalavalanches. Prevailingunderstandingofself-organised ] O criticality (SOC) dictates that conservation of energy is essential to its emergence. Neuronal A networkshowever are inherently non-conservativeas demonstrated by microelectrode recordings. ThemodelpresentedhereshowsthatSOCcanariseinnon-conservativesystemsaswell,ifdriven . n internally. Evidence suggests that synaptic background activity provides the internal drive for i non-conservative cortical networks to achieve and maintain a critical state. SOC is robust to l n any degree η ∈ (0,1] of background activity when the network size N is large enough such that [ ηN ∼103. Forsmallnetworks,astrongbackgroundleadstoepileptiformactivity,consistentwith neurophysiological knowledge about epilepsy. 1 v 0 2 0 1 0 Neuronal networks have been demonstrated to exhibit action potentials from the pre-synaptic neuron do not de- 7a type of activity dubbed as “neuronal avalanche” [1]. A polarise the post-synaptic neuron [6]. Information loss 0neuronalavalancheis characterisedbya cascadeofbursts duringtransmissionseemstoprecludetheobservedneural / nof local field potentials (LFP), which originate from syn- synchrony in neuronal networks through avalanche activ- ichronised action potentials triggered by a single neuron. ity. Moreover, the critical branching process model con- l nThe intensity of the recorded LFP is assumed to be pro- cludes that any degree of loss or non-conservation frus- : vportional to the number of neurons synchronously firing tratescriticalityandintroducesacharacteristicsizetothe iaction potentials within a short time interval, and thus is avalanche size distribution [7]. Thus until now a gap ex- X a good measure of neuronal avalanche size [1]. The LFP istsbetweenexperimentandtheoryintermsofexplaining r bursts are brief comparedto the observationperiod, typi- howcorticalnetworksmaintaincriticalitydespiteinherent a callylastingtensofmillisecondsandseparatedbyperiods non-conservation in neuronal transmission. of quiescence that last several seconds. When observed Neurons receive and transmit information using one with a multielectrode array, the number of electrodes de- form of medium—the electric potential, which is both re- tecting LFPs,whichis in turnproportionalto LFP inten- ceived as input (synaptic potential) and fired as output sity, is distributed approximately as an inverse power law (action potential or local field potential). Each neuron with exponent 1.5. The observed power law has been storesthe input asmembrane potentialandwhenthis ex- ≈ intuitivelylinkedtotheprinciplesofself-organisedcritical- ceeds a threshold the neuron fires. Usage of one form of ity[1]throughresultsfromthe studyofcriticalbranching medium in receiving, storing and transmitting informa- processes[2],andhasbeenproposedandmodelled[3,4]to tion, as well as the all-or-none response of neurons par- consequently enhance the information processing capabil- allels with that of SOC sandpile models. A “sand grain” ityofcorticalnetworksinvivo. Ithasbeenexperimentally servesasthecurrencyofexchangeandasandpilesiteonly demonstratedhoweverthatpropagationofneuralactivity transfersgrainswhenthe amountofstoredgrainsexceeds does not conserve information content, which is encoded a particular threshold. Thus, neuronal avalanches may in the frequency of spike firing [5]. This experiment sus- parsimoniously, yet sufficiently, be analysed using SOC tainspreviousfindingsthatsynapticallyconnectedpairsof sandpile models. neurons exhibit informationtransmissionfailures wherein The large number of neurons and the high density of (a)E-mail: [email protected] non-local synaptic connections (i.e., linking neurons dis- p-1 D. E. Juanico tally located from each other) that comprise the brain [8] tial which is less negative than the resting potential, but allows the approximation of the physics underlying its varies across different types of neurons. When the mem- information transport by mean-field models. A conve- brane potential crosses this threshold, an action potential nient mean-field sandpile model that appropriately cap- is initiated. These three neuronal states are mapped to tures neuronal avalanche dynamics is the self-organised sandpile height z in the following manner: z = 0 (resting critical branching process (SOBP) model introduced by state); z = 1 (critical state); and z = 2 (excited state). Zapperi, Lauritsen and Stanley [2], which represents the Neurons only fire action potentials when they are in the avalanche as a branching process. SOC in this model is excited state. These mappings also agree with the defi- closelyconnectedwiththecriticalityofthebranchingpro- nition of the Manna sandpile [13]. Lastly, grain addition cess for which the theory is well-established [9]. In the andsubtractionoperationsinsandpile models correspond branching process theoretical framework, activity at one to depolarisationand hyperpolarisation,respectively. De- site generates subsequent activity in a number of other polarisationdisplacesthemembranepotentialofaneuron sites. When the number of subsequently activated sites— towardsa lessnegativevalue,while hyperpolarisationdis- the branching parameter σ—is equal to unity on the av- places the membrane potential towards a more negative erage, the branching process is critical. Criticality in the value [8]. In the model, depolarisation corresponds to a SOBP model has been achieved only when grain transfer change ∆z = +1 in the membrane potential whereas hy- duringanavalancheisconservative. Thishasbeenproven perpolarisation to a change ∆z = 1 or 2. Albeit the − − byincorporatingintothemodelaprobabilityǫthatgrains actual values of V are continuous, the discretisation em- m which dislodge from a toppling activated site is absorbed ployedhere is sufficient to emulate the crucial role played (or dissipated) rather than exchanged. For any ǫ>0, the by the threshold potential to neuronal activation. branching parameter σ < 1, and a characteristic size in The model cortical network is assumed to be a fully- the avalanche size distribution, which scales with ǫ−2, is connectedrandomnetworkinorderto simplify the mean- introduced. Avalanches are not sustained when the con- field treatment. Although fully-connected, only two servation law is violated; they easily die out after a few randomly-chosensynapticconnectionsofaneuronarepo- topplings. The characteristicsize divergeswhen ǫ 0,or tentially utilised in every avalanche event. The network → when the SOBP system is conservative [7]. Thus, repre- has a size of N =2n+1 1 neurons, where n is the upper − sentationof neuronalavalanches,which display criticality boundonthenumberofdepolarisations(causedbyaction while occurring in a non-conservative substrate, through potentialstriggeredbyasingleneuron)thatcantakeplace the SOBP model seems to be a contradiction [10]. in a single avalanche. This serves as the boundary condi- A plausible source of sustaining the drive to neuronal tiontoamean-fieldmodelthatneglectsspatialdetails[2]. avalanchesissynapticbackgroundactivity[11],whichhas Thenetworkisina“quiescent”statewhennoexcitedneu- beenfoundto enhanceresponsivenessofneocorticalpyra- ronsarepresent;otherwiseitis“activated.”Thedensities midal neurons to sub-threshold inputs. Synaptic back- of critical neurons and resting neurons in a quiescent net- groundactivity occurs in the form of membrane potential work are ρ and 1 ρ, respectively. The network is slowly − fluctuations. Because of these voltage fluctuations, small stimulated by external stimuli. The probability that the excitatoryinputs are able to generate action potentials in stimulus excites a critical neuron simply corresponds to neurons. Theenhancementofweaksignaldetectioncapa- the density ρ. At this point, the network is activated and bility in the presence of background activity is analogous avalanche ensues. The excited neuron fires an action po- to a well-studied nonlinearphenomenon inphysics known tential that is transmitted to two post-synaptic neurons, as stochastic resonance. It has been suggested that the chosenatrandom. Threedifferentthingsmayhappen: (i) level of background activity within actual in vivo corti- with probability α the excited neuron hyperpolarises to cal networks is optimal and possibly keeps neurons in a resting state (z : 2 0) and both post-synaptic neurons → highly responsive state [11]. In this Letter, a phenomeno- depolarise; (ii) with probability β the excited neuron hy- logical model of synaptic background activity is incorpo- perpolarises to critical state (z : 2 1) and only one of → ratedintotheSOBPmodeltorealiseSOCinthepresence the two post-synaptic neurons depolarises; and (iii) with of a violation of the transmission conservation law. The probabilityǫ=1 α βtheexcitedneuronhyperpolarises − − non-conservative SOC model proposed here parallels the torestingstate(z :2 0)buttheactionpotentialfailsto → theoreticalworkofPruessnerandJensenonasandpile-like transmitthereby none of the post-synaptic neurons depo- model defined to approach the random-neighbour forest- larise. Theserulesareappliediterativelyforeveryexcited fire model in the non-conservative limit [12]. neuron that emerges until the whole network recovers its A crucial starting point for describing the model is quiescent state or when the action potential propagates translating the sandpile language into terms that define a total of n steps, after which no subsequent depolarisa- neuronal phenomena, done in the following. A site cor- tion takes place. The n-th neuron, which serves as the responds to a neuron, and its height is analogous to the boundary of avalanche propagation,may loosely be inter- membrane potential. At “rest,” the membrane potential preted as a peripheral or a motor neuron. In the wake is typically 90mV V 40mV [8]. A neuron can of an avalanche, the density ρ of the quiescent network m − ≤ ≤ − undergo a critical state determined by a threshold poten- changes. Avalanches take place instantaneously with re- p-2 SOC in neuronal networks spect to the period that the network lingers in the qui- The phenomenological model for synaptic background escent state. This timescale separation is also observed activity is conceived by assuming the presence of sub- experimentally [1], and is evident in the fast processing threshold membrane potential fluctuations in neurons time of the brain when presented with a sensory stimu- when the network is in the quiescent state. The back- lus [8]. ground activity level is quantified by a parameter η ∈ Treatingthe neuronalavalancheas a branching process (0,1]. A resting neuron depolarises with probability η allows us to define a branching probability to become critical (z : 0 1); otherwise it remains in → the resting state. Parallel update of all neurons through πk =αρδk,2+βρδk,1+[1 (1 ǫ)ρ]δk,0, (1) this process contributes a change η(1 ρ) to the density − − − ρ of critical neurons. This process represents the stochas- wherekdenotesthenumberofpost-synapticneuronsthat tic fluctuations arising from excitatory post-synaptic po- get depolarised by the action potential released by an ex- tentials (EPSPs) on the membrane potential [8]. On the cited neuron. The first term of Eq. (1) represents case other hand, a critical neuron hyperpolarises to resting (i), the secondtermrepresentscase(ii), andthe lastterm state (z :1 0) with probability η multiplied by the de- is the sum of the probability described in case (iii) and → viationofthebranchingparameterpercriticalneuronσ/ρ the probability (1 ρ) that the neuron subsequently de- from unity. Parallel update over all neurons through this − polarised is a resting neuron, which also yields no action process contributes a change η(σ/ρ 1)ρ to the density potential. Havingǫ>0(orequivalentlyα+β <1)makes − − ρ, and represents the stochastic fluctuations arising from the sandpile model violate the transmission conservation inhibitory post-synaptic potentials (IPSPs) on the mem- law. A branching parameter σ = (2α+β)ρ is calculated brane potential [8] and the feedback mechanism to sup- from Eq. (1) according to the definition σ = kπ [9]. k k press runaway excitation [15]. Both effects of EPSPs and The branchingprocessis sub-criticalifσ <1,Pwherebyan IPSPs operate on the same timescale and can be conve- avalanche typically dies out after a few (< n) transmis- nientlyaddedtogethertogiveanetchangeΓ=η(1 σ)to sion steps. The branching process, on the other hand, is − the density ρ of critical neurons in the quiescent network. supra-criticalifσ >1,describinganavalanchethatperco- Γ is dominantly excitatory to the network when σ < 1, lates the entire network with nonzero probability. At the whicheffectivelyincreasesρ;ontheotherhand,itisdom- critical value σ = 1, the avalanche size distribution P(s) inantly inhibitory when σ >1, whicheffectively decreases iscalculatedusingageneratingfunctiondirectlyderivable ρ. Experimental evidence shows that this type of see- from Eq. (1): sawbetweencorticalexcitationandinhibitionisaformof synaptic plasticity that contributes in stabilising cortical 1 bω √1 2bω+aω2 F(ω)= − − − . (2) networks, keeping them on the border between inactivity 2αρω and epileptiform activity [6]. The parameter η takes on a more physical meaning because experiments reveal an in- where a = β2ρ2 4αρ[1 (1 ǫ)ρ] and b = βρ, and is − − − ternal mechanismfor differential synaptic depressionthat related to P(s) as a complex-variable power series dynamically adjusts the balance between cortical excita- tionandinhibitiontofostercorticalnetworkstability[15]. N F(ω)= P(s)ωs. (3) Summing the changes in ρ brought about by synaptic Xs=1 backgroundactivityΓandtheneuronalavalanchelaysout the following dynamical equation Taking N , expansion of Eq. (2) about the singu- → ∞ dρ ξ larity ω = 0 followed by the comparison of terms to the =η[1 σ(ρ)]+A(p)+ . (6) dt − N coefficients of Eq. (3) results to a recurrence relation of P(s) valid for s 2: The first term is Γ and the second term represents the ≥ change in ρ in the wake of a neuronal avalanche, P(s)= bP(s−1)(2s−1s)+−1aP(s−2)(s−2), (4) A(ρ)= 1 1 σn ǫρ 1+ 1−σn+1 2σn , N (cid:26) − − 1 (1 ǫ)ρ(cid:20) 1 σ − (cid:21)(cid:27) − − − (7) and subject to the end conditions: P(0) = 0 and P(1) = (b2 a)/4α. Atthecriticalstate,a=2b 1orequivalently, which is derived by following closely the branching pro- − − cess arguments of the analysis in [7]. The last term in (6) the function represents the fluctuations, which properly vanish in the Q(ρ)=β2ρ2 4αρ[1 (1 ǫ)ρ] 2βρ+1 (5) limit of large N, around average quantities assumed in − − − − the calculation of Γ and A. This noise term is therefore isequaltozero,whichfollowsfromσ =1. Eq.(4)asymp- neglected in the mean-field approximation for large N. totically approaches power-law behaviour with exponent The first term inside of (7) represents the depolarisa- {·} 3/2for s>>1, whichcan be showngraphically. A closed tioncausedbyexternalstimulithatconsequentlybringsa form solution for Eq. (4) can also be easily solved by set- critical neuron to excited state and initiate an action po- ting β =0 [14]. tential. Thesecondtermrepresentstheaverageamountof p-3 D. E. Juanico action potential dissipated by the n-th neuron. The third large ( 1012 neurons), which is typical of actual cortical ∼ term is present when ǫ > 0 and represents the average networks[8],thenawiderangeofηenablesthenetworkto amount of action potential absorbed due to transmission maintainitscriticalphase,aslongasηN 103. However, ≥ failure. Hence, ǫ is the degree of non-conservation in the since the fluctuations ξ/N are neglected in analysing the cortical network. stationary behaviour of Eq. (6), this conclusion does not ThestationarybehaviourofEq.(6)isanalysedinphase necessarilyholdtrue fornetworkswithintermediatesizes, space. Fig. 1 plots the phase portraits of ρ for different for which the fluctuations may not be negligible. Thus, network sizes N. All networks have the same degree of simulation results are essential in probing the stationary non-conservation and background intensity, ǫ = 0.25 and behaviour of the network for this case. ∗ η =0.0625,respectively. Thefixedpoints ρ arethe roots of the phase portraits. Also shown is the function Q(ρ) defined in Eq. (5) with a unique root at ρ = 2/3. As c A B ∗ 0.9 N increases, ρ ρ . However, even for a network as n = 12 c → ∗ n = 16 small as N = 131071 neurons, ρ ρ <<1. The rapid ∗ | − c| n = 20 convergence of ρ to ρc from N = 31 to N = 131071 and 0.8 n = 36 thediminishinglyperceptibledifferencebetweenthephase n = 52 portraits from N = 131071 to N indicate a phase * → ∞ 0.7 transition. 0.03 0.6 SUB-CRITICAL CRITICAL 0.02 0.5 Q( ) -13 -9 -5 -1 3 7 11 10 10 10 10 10 10 10 0.01 . Fig. 2: Fixed point ρ∗ versus ηN exhibiting data collapse for 0.00 networks of varying sizes with two different degrees of non- conservation: (A) ǫ = 0.45, ρ = 0.909, filled polygons; and c A (B) ǫ = 0.25, ρc = 0.667, open polygons. Transition occurs -0.01 B between sub-critical phase (ρ∗ < ρ ) and critical phase (ρ∗ = C c D ρ ). Phases are separated by a solid line footed at ηN ∼103. c -0.02 0.0 0.2 0.4 0.6 0.8 1.0 A simulation of the model is performed for a network with size N = 131071 driven by a synaptic background Fig.1: Phase portrait ofthedensityρofcritical neuronsfora with η = 0.025 such that ηN 3670. The degree of ≈ network with degree of non-conservation ǫ=0.25 driven by a non-conservation in the simulated network is ǫ = 0.25 backgroundwithη=0.03125atdifferentsizesN: (A)N =31, such that the critical density ρ =0.8, in accord with the c dotted curve; (B) N = 511, chain curve; (C) N = 131071, findings of Vogels and Abbott that depolarisationsdue to dashed curve; and (D) N →∞, full curve. Also plotted is the synchronous action potentials evoke post-synaptic spikes function Q(ρ) with root at ρ = 2. c 3 only 80% of the times [5]. After rescaling the simulated avalanche size with a factor deduced from data in [1], the Indeed,asshowninFig.2,atransitionfromsub-critical simulated network successfully fits the experimental data ∗ ∗ phase (ρ < ρc) to critical phase (ρ = ρc) occurs. Data for the neuronal avalanche distribution adopted from [3], collapse reveals that the profile of the phase transition as shown in Fig. 3 (Top panel). A power-law with ex- does not depend on η and N separately, but rather on ponent 3/2 mainly characterises the distribution. How- the product ηN. In the sub-critical phase, the quiescent ever,anexponential cutoff appearsbecause ofthe limited network has ρ= 1 of critical neurons, in accordance with number of microelectrodes used to resolve LFP intensity 2 theresultsof[7]. ButstartingatηN 10−1,the network during the experiments [1]. The cutoff shifts to the right ∼ driven by a background activity abruptly transforms to (i.e., towards larger avalanche sizes) when the number of a critical phase up to ηN < 103. The critical phase is microelectrodesisincreased. Thesimulationdataalsofits characterisedbyρ=ρ =(2∼α+β)−1 (orρ =[2(1 ǫ)]−1 this exponential tail, which also arises from the boundary c c − for β = 0) for which the branching parameter σ = 1. condition that effectively puts an upper bound n to the Starting at ηN 103, the network is in the critical phase number of transmission steps during an avalanche. The ∼ for any degree of non-conservationǫ (0,0.5] (at ǫ>0.5, cutoffalsoshiftstothe rightbyincreasingn. Hence,both ∈ the critical density ρ > 1, hence impossible to achieve). modelandexperimentaldataagreenotonlyinthepower- c The phase transition implies that if the network is very law behaviour of the avalanche size distribution, but also p-4 SOC in neuronal networks in the mechanism that gives rise to the exponential tail. 0 10 The inset graph illustrates the evolution of the density ρ inthequiescentnetworkwithtimet,approachingthecrit- 1.0 ical value ρ = 0.8 in the steady state. This steady-state c behaviour is supported by a phase plot of ρ (Fig. 3, Bot- -1 0.8 10 tom panel). The fixed point (B) corresponds to ρ = 0.8. c 0.6 Also shown are two other dynamical attractors (A, C), P(s) t which correspond to background activity fluctuations ne- 0 5000 10000 15000 glectedinthefixed-pointanalysisofEq.6. Thesymmetry 10-2 = 3/2 between(A)and(C)evidentlysuggeststhebalanceofcor- tical excitation and inhibition, imparting stability to the networkatthe criticalstate. Thus, the backgroundactivity is essential in maintaining the network at the critical -3 10 state such that a power-law avalanche size distribution is generated. 1 10 100 Actual neuronal networks are found to be redundant— s several identical neurons that perform similar roles are present [8]. Redundancy thereby enlarges the network, and may be vital to its robustness. Through the model, robustness of the critical behaviour due to network size is explored by driving a small (N = 131071) and a large (N = 4194303) network with a strong background (η =1.0). Fig.4(Leftpanel)showsasmallnetworkdriven by strongbackgroundexhibiting an asymmetryin the excitatory and inhibitory dynamical attractors—there is a larger region of excitation than inhibition. This mechanism is believed to be the precursor of epileptic seizures, which manifests in the synchronous firing of a large number of neurons. The inset graph illustrates the avalanche sizedistributionforthisnetwork. Amarkedpeakappears (indicated by arrow) for large avalanche sizes of the distribution. Hence, there is a high probability for a large Fig. 3: Top panel.—Distribution of neuronal avalanche size number of neurons to synchronously depolarise and fire data adopted from [3] (circles), and of a simulated net- action potentials. This feature indicates that small network (full-curve) with ηN ≈ 3670 having a degree of non- works show epileptiform activity when driven by a strong conservation ǫ = 0.25. Inset graph shows ρ versus iterations background. This is consistent with neurophysiological t approaching a critical value ρ = 0.8 in the steady state. c knowledge that seizure development is triggered by dam- Bottom panel.—Phase plot of the simulated network, starting agestobraincellscausedbyinjury,drugabuse,degenera- at ρ= 0, converging towards three dynamical attractors: (A) tive neurodiseases, brain tumors, and brain infections [8]. excitatory background fluctuations; (B) fixed point (ρ∗ =ρ ); c On the other hand, the large network is robust to strong and (C) inhibitory background fluctuations. The solid line is background activity (Fig. 4, Right panel). The avalanche the mean-field prediction. sizedistributionofthis network(insetgraph)exhibits the expectedpower-lawbehaviourandthereisnopronounced peak for large avalanche sizes. between corticalexcitationandinhibition when drivenby Summary and Conclusion.—A self-organising mecha- a strong background (η = 1.0), consequently leading to nism for neuronal avalanche activity is proposed. Criti- epileptiform activity. This finding agreeswith neurophys- cality manifests in the power-law behaviour of neuronal iological knowledge that brain cell damage is a chief con- avalanche sizes. Despite an inherent transmission non- tributor to the onset of seizure development. conservationinneuronalnetworks,thiscriticalityismain- Themodelalsoprovesthatself-organisedcriticalitycan tained. Alargeneuronalnetworkthatis internallydriven be achieved even when a conservation law of dynamical by synaptic background activity self-organises towards transferisviolated. Atransitionoccursfromasub-critical and robustly maintains a critical state for any level of to a critical phase, demonstrating a vast regime for non- background activity η (0,1] and for any degree of non- conservative systems to display SOC behaviour. Thus, it ∈ conservation ǫ (0,0.5]. This finding advocates the role addressesalong-standingissuethatisfundamentaltothe ∈ of redundancy of neuronal networks in fostering robust- theory of self-organised criticality—whether conservation ness against any fluctuations of internal activity. A small is necessary for its emergence. neuronal network, on the other hand, loses the balance Recommendations.—A key assumption in the model is p-5 D. E. Juanico [6] Galarreta M. and Hestrin S., Nature Neurosci., 1 (1998) 587. [7] Lauritsen K. B., Zapperi S.andStanley H. E.,Phys. Rev. E, 54 (1996) 2483. [8] Purves D. et al. (Editor), Neuroscience, Third Edition (SinauerAssociates, SunderlandMA) 2004. [9] HarrisT.E.,Thetheoryofbranchingprocesses(Springer- Verlag, Berlin) 1963 [10] Juanico D. E., Monterola C. and Saloma C., Criti- cality of a dissipative self-organised branching process in a dynamic population, nlin.AO/0604058 preprint,2006. [11] Hoˆ N. and Destexhe A., J. Neurophysiol., 84 (2000) 1488. [12] Pruessner G. and Jensen H. J., Europhys. Lett., 58 (2002) 250. [13] Manna S. S.,J. Phys. A: Math. Gen., 24 (1991) L363. Fig. 4: Phase plots for two networks of different sizes N but [14] Pinho S. T. R.andPrado C. P. C.,Braz. J. Phys.,33 similar degrees of non-conservation ǫ = 0.25 and driven by a (2003) 476. strong background η = 1.0. Critical density is at ρ = 0.8. [15] Nelson S. B.andTurrigianoG. G.,Nature Neurosci., Left panel.—N = 131071 neurons, showing a promincent pro- 1(1998) 539. trusion at ρ > ρ and ρ˙ > 0. Solid line is the mean-field pre- [16] Goh K.-I., Lee D. S.,and Kim D., Phys. Rev. Lett., 91 c diction. Inset graph displays the logarithmically-binned dis- (2003) 148701. [17] RoxinA.,RieckeH.,andSollaS.A.,Phys.Rev.Lett., tribution of avalanche sizes with a marked peak (pointed by arrow) for large sizes, indicating epileptiform activity. Right 92 (2004) 198101. panel.—N = 4194303 neurons, showing stability of the fixed [18] Lee K. E. and Lee J. W., Eur. Phys. J. B, 50 (2006) point at ρ consistent with mean-field prediction (solid line). 271. c Insetgraphdisplaysthelogarithmically-binned distribution of avalanchesizes exhibiting considerable power-law behaviour. network randomness. However, the morphology of actual neuronal networks may actually be more accurately characterised in terms of small-world connectivity pat- terns [16–18]. Thus, an investigation into the behaviour of the model in small-world networks is recommended as a possible extension of this study. Nevertheless, the main conclusions,whichdonotstronglydependonthenetwork morphology as long as the density of connections and the size of the network remain large, would still hold. ∗∗∗ TheauthorwishestoacknowledgetheOfficeoftheVice- Chancellor for Research and Development (OVCRD) of the University of the Philippines, Diliman for Research Grant No. 050501 DNSE and the Philippine Council for AdvancedScience andTechnologyResearchandDevelop- ment (PCASTRD) for funding. REFERENCES [1] BeggsJ.M.andPlenzD.,J.Neurosci,23(2003)11167. [2] Zapperi S., Lauritsen K. B.andStanley H. E.,Phys. Rev. Lett., 75 (1995) 4071. [3] Haldeman C. and Beggs J. M., Phys. Rev. Lett., 94 (2005) 058101. [4] Hsu D. andBeggs J. M.,Neurocomput., 69(2006) 1134. [5] Vogels T. P.andAbbott L. F.,J.Neurosci.,25(2005) 10786. p-6