Nonextensivity in a memory network access mechanism

Brazilian Journal of Physics, vol. 39, no. 2A, August, 2009

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Nonextensivity in a memory network access mechanism Roseli S. Wedemann∗

Instituto de Matem´atica e Estat´ıstica, Universidade do Estado do Rio de Janeiro, Rua S˜ao Francisco Xavier 524, 20550-900, Rio de Janeiro, Brazil.

Raul Donangelo†

Instituto de F´ısica, Universidade Federal do Rio de Janeiro, Caixa Postal 68528, 21941-972, Rio de Janeiro, Brazil.

Lu´ıs A. V. de Carvalho‡

Programa de Engenharia de Sistemas e Computac¸a˜ o, Universidade Federal do Rio de Janeiro, Caixa Postal 68511, 21945-970, Rio de Janeiro, Brazil. (Received on 26 January, 2009) We have previously described neurotic psychopathology and psychoanalytic working-through by an associative memory mechanism, based on a neural network. Memory was initially modelled by a Boltzmann machine (BM). We simulated known microscopic mechanisms that control synaptic properties and showed that the network self-organizes to a hierarchical, clustered structure. Some properties of the complex networks which result from this self-organization indicate that a generalization of the BM may be necessary to model memory. We have thus generalized the memory model, using Generalized Simulated Annealing, derived from the nonextensive formalism, and show some properties of the resulting memory access mechanism. Keywords: Consciousness-unconsciousness, Neuroses, Self-organized neural networks, Boltzmann machine, Generalized simulated annealing, Hierarchical memory model

1.

INTRODUCTION

Early psychoanalytic research regarding the neuroses found that traumatic and repressed memories are knowledge which is present in the subject, but which is symbolically inaccessible to him. It is therefore considered unconscious knowledge [1, 2]. Freud observed that neurotic patients systematically repeated symptoms in the form of ideas and impulses, and called this tendency a compulsion to repeat [3], which he related to repressed or traumatic memory traces [1]. Neurotic analysands have obtained relief and cure of painful symptoms through a mechanism called working-through. This procedure aims at developing knowledge regarding the causes of symptoms by accessing unconscious memories, and understanding and changing the analysand’s compulsion to repeat [3]. The technique involves mainly free associative talking and the interpretation of dreams during analytic sessions, among other processes. We have proposed a functional and computational model which describes neurotic behavior as an associative memory process [4, 5], where the network returns a stored pattern when it is shown another input pattern sufficiently similar to the stored one [6]. We modelled the compulsion to repeat neurotic symptoms [1], by supposing that such a symptom is acted when the subject is presented with a stimulus which resembles a repressed or traumatic memory trace. The stimulus causes a stabilization of the neural net onto a minimal energy state, corresponding to the trace that synthesizes the original repressed experience, which in turn generates a neurotic re-

sponse (an act). The neurotic act is not a result of the stimulus as a new situation but a response to the repressed memory. We mapped the symbolic associative process involved in psychoanalytic working-through onto a corresponding process of reinforcing synapses among memory traces in the brain. In our model, memory functioning was originally modelled by a Boltzmann machine (BM). However, the power-law and generalized q-exponential behavior we have found, for the node-degree distributions of the network topologies generated by our model, indicate that they may not be well described by Boltzmann-Gibbs (BG) statistical mechanics, but rather by nonextensive statistical mechanics [7–9]. We have therefore modelled memory by a generalization of the BM called Generalized Simulated Annealing (GSA) [8]. In GSA, the probability distribution of the system’s microscopic configurations is not the BG distribution, assumed in the BM, and this should affect the chain of associations of ideas which we are modelling. In this paper, we review in Section 2 our associative memory model for neurosis and the self-organizing mechanism for generating hierarchically clustered memory modules, representing sensorial and symbolic memories. In Section 3, we discuss the use of GSA as a memory access mechanism, as compared to the BM. We then show results from computer simulations with some properties of these complex networks’ memory access mechanisms. In the last section, we draw some conclusions and perspectives for future work.

2. ∗ Electronic

address: [email protected] address: [email protected] ‡ Electronic address: [email protected]

† Electronic

ASSOCIATIVE MEMORY MODEL FOR NEUROSIS

We proposed a memory organization, where neurons belong to two hierarchically structured modules corresponding to sensorial and symbolic memories. Traces stored in sen-

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sorial memory represent mental images of stimuli received by sensory receptors. Symbolic memory stores higher level representations of traces in sensorial memory, i.e. symbols, and represents brain structures associated with symbolic processing, language and consciousness. Sensorial and symbolic memories interact, producing unconscious and conscious mental activity [4]. Memory functioning was initially modelled by a BM with N neurons, where node states take binary values and connections have symmetrical weights wi j = w ji . The states Si of the units take binary output values in {0, 1}. Because of the symmetry of the connections, there is an energy functional 1 H({Si }) = − ∑ wi j Si S j , 2 ij

(1)

which allows us to define the BG distribution function for network states     H({Si }) H({Si }) / ∑ exp − , (2) PBG ({Si }) = exp − T T {S } i

where T is the network temperature parameter. In the BM, pattern retrieval on the net is achieved by a standard simulated annealing process, in which the network temperature T is gradually lowered by a factor α, according to the BG distribution given by Eq. 2. A detailed treatment of the BM may be found in [6, 10]. Brain neural topology is structured by cooperative and competitive mechanisms, controlled by neurosubstances, where neurons interact mainly with spatially close neighbors, having fewer long-range synaptic connections to more distant neurons [11, 12]. This is started and controlled by environmental stimulation and is the process whereby the environment represents itself in the brain. We summarize the clustering algorithm developed based on these biological mechanisms [4] to model the selforganizing process which results in a structured, clustered topology of each memory as follows. In Step 1, neurons are uniformly distributed in a bi-dimensional square sheet. In Step 2, we assume a Gaussian approximation for the numerical solution of the diffusion equation of neural substances that control neural competition and cooperation. A synapse is thus allocated to connect two neurons ni and n j , according to a Gaussian probability distribution Pi j of the distance that separates the pair, with standard deviation σ. A synapse connecting ni to n j has strength proportional to Pi j and weights are symmetrical. Step 3 clusterizes neurons in the memory sheets, based on mechanisms for forming cortical maps [4, 6, 13], where a group of neurons spatially close to each other represents a sensorial stimulus or an idea. Step 4 regulates synaptic intensities by strengthening synapses within a cluster and reducing synaptic strength between clusters, disconnecting them. Neurons, which have received stronger sensorial stimulation and are more strongly connected, stimulate their neighborhoods and promote even stronger connections. Long-range synaptic growth is energetically more costly than short-range, and consequently the former is less frequent in the brain than the latter. We represent association of ideas or symbols (such as in culture and language) by long-range

synapses, which should connect clusters by considering the basic Hebbian learning mechanism [6, 11, 12], where synaptic growth between two neurons is promoted by simultaneous stimulation of the pair. Since we are still unaware of synaptic distributions which result in such topologies, as a first approximation, we allocated synapses randomly among clusters. A full description of the algorithm, with memory storage and retrieval and working-through simulation, can be found in [4]. 3.

NETWORK TOPOLOGY AND ASSOCIATIVITY

We have generated 10000 different initial topologies with the clustering algorithm, for different values of N, such that Nsens = Nsymb = N/2 of them belong to sensorial and symbolic memories respectively and σ = 0.58. In [4], we explain how we represent and interpret these as neurotic topologies, by linking neurons between sensorial and symbolic memories more weakly. Other model parameters are also described in [4]. We then measured the average node degree (k) distributions of these topologies. Figure 1 [14] shows an asymptotic power law behavior with exponent γ ≈ −3.2, which indicates scale independence [9]. For N = 4000, the deviation from the fit by the Poisson distribution, Pλ (k) = λk exp(−λ)/k!, for k > 10 is quite evident. The deviation from Poisson for higher values of k may be attributed to the cooperative-competitive biological mechanisms mentioned earlier, which introduce structure. Smaller values of k correspond to neurons that did not significantly participate in the competition-cooperation process and, hence, distributions for small k are approximately fitted by Poisson forms. Figure 1 also shows a fit by a generalization of the qexponential function [9, 17] given by 1 Pq (k) = p0 kδ h i 1 , τ τ (q−1)µk q−1 1− µ + µe

(3)

where p0 , δ, τ, µ and q are additional adjustable parameters. The curves indicate that, asymptotically, the power-law and generalized q-exponential fits are appropriate, with q ≈ 1.113. This is a common feature of many biological systems and indicates that they may not be well described by BG statistical mechanics. There is no theoretical indication of the exact relation between network topology and memory dynamics. There have been some indications that complex systems which present a power-law behavior, i.e. which are asymptotically scale invariant, may be better described by the Nonextensive Statistical Mechanics formalism [8, 9, 15, 16]. Since the neural systems we are studying do not have only local interactions and present the scale-free topology characteristic, we have begun to investigate memory dynamics with a generalized acceptance probability distribution function [8] for a transition from state S ≡ {Si } to S0 , if H(S0 ) ≥ H(S), given by PT (S → S0 ) =

1

[1 + (qA − 1)(H(S0 ) − H(S))/T ]1/(qA −1)

, (4)

where qA is a model parameter and other variables and parameters are the same as defined in Section 2.

Brazilian Journal of Physics, vol. 39, no. 2A, August, 2009 Brazilian Journal of Physics, vol. ??, no. ??, (mes), 200? 10000 Eq.(3) Poisson 10**6.5/k**3.2 N = 10000 N = 4000 N = 2000 N = 1000 N = 282 N = 100 N = 50 N = 32

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FIG. nodetodegree distributions forδvarious by Eq.1:3Average corresponds q = 1.113, p0 = 610, = 4.82, N. τ =The 2.34fitand Eq.µ3=corresponds to q = 1.113, p = 610, δ = 4.82, τ = 2.34 and 0 0.014. The Poisson fit corresponds to a = 2950 and λ = 6.4. For µ =large 0.014. TheisPoisson fit corresponds N = 2950 and λ = 6.4. k, there an exponential finite sizetoeffect. For large k, there is an exponential finite size effect.

from state S ≡ {Si } to S0 , if H(S0 ) ≥ H(S), given by

The memory access mechanism is1thus modelled by a genPT (S →of S0 )the = BM, called Generalized Simulated , (4) eralization Anneal0 − H(S))/T ]1/(qA −1) [1 + (qA from − 1)(H(S ing (GSA) [8], derived the )nonextensive formalism, based on qtheisgeneralized acceptance probability, given by Eq. where a model parameter and other variables and paramA 4. eters In GSA, the probability distribution of the system’s miare the same as defined in Section 2. croscopic configurations is not the BG distribution given by The memory access mechanism is thus modelled by a genEq.eralization 2, assumed BM, and Generalized this should affect the chain of of in thethe BM, called Simulated Annealassociations of ideas which we are modelling. In the BM, ing (GSA) [8], derived from the nonextensive formalism, depending there is a probability tendency of based onon thetemperature, generalized acceptance 4. reaching In GSA, onetheofprobability the energydistribution minima closer the initial configuration of thetosystem’s microscopic conpresented to the network. The probability figurations is not the BG distribution givendistribution by equation given 2, asin the BM, this should the minima chain ofofassoby sumed Eq. 4 should allowand reaching moreaffect distant the ciations of ideasH, which modelling. In the BM, deenergy functional fromwe anare initial configuration, than the pending on temperature, there is a tendency of reaching one BM for the same temperature values, which corresponds to of the energy minima closer to thetraces. initial configuration more associativity among memory This implies prethat to the network. Thetoprobability distribution given by thesented GSA machine will tend make many local associations equation 4 should allow the and, more often than the reaching BM, willmore alsodistant make minima looser, of more energy functional H, from an initial configuration, than the distant associations. This would correspond to a more flexithe same temperature values, bleBM and for creative memory dynamics in thewhich brain.corresponds to more associativity among memory traces. This implies that To this, will we tend compare themany energies of the patthe illustrate GSA machine to make local associations terns accessed by the BM and GSA for an initial temperaand, more often than the BM, will also make looser, more disturetant T= 0.2. Since we are searching for local minima, we associations. This would correspond to a more flexible useand lower initial temperature values and higher values of the creative memory dynamics in the brain. annealing schedule α. we In the following simulations, have To illustrate this, compare the energies of thewe patterns analyzed smaller with for N= Nsenstemperature = Nsymb =T16, accessed by thenetworks BM and GSA an32, initial = 0.2.simulation Since we are searchingaccess for local minima, use lower since of memory is very timewe consuming. initial temperature of theThe annealing Memory sheets have values size 1.5and X higher 1.5 andvalues σ = 0.58. simuschedule α. In the following we to have analyzed lation experiment followed wassimulations, to perform up 10000 minsmaller procedures, networks with N =one 32, starting Nsens = Nfrom 16, since simsymb = imization each a different ranulation of memory access is When very time consuming. dom network configuration. a new pattern isMemory found, have X 1.5 and = 0.58.from The simulation exit issheets stored andsize the1.5 procedure is σ repeated other random periment followed was to perform up to 10000 minimization starting configurations, otherwise the search stops. We note procedures, starting each one from a different random network in Fig. 2 that, for T = 0.2, there are many patterns found by configuration. When a new pattern is found, it is stored and GSA that are not found using the BM. In these simulations, GSA appears to visit state space more loosely, while the traditional BM visits state space more uniformly. For very low temperatures, the BM functions more like a gradient descent method and, when presented with randomly generated patterns, the network will stabilize at the closest local minima.

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compare, in Fig. 3from the frequency with whichconfigdifferent theWe procedure is repeated other random starting patterns, corresponding to minimum energy states, are found urations, otherwise the search stops. We note in Fig. 2 that with the BM and with the GSA for q = 1.4, for the same simA for T = 0.2, there are many patterns found by GSA that are ulations Fig.the 2. BM. We observe is degeneracy in network not foundof using In thesethere simulations, GSA appears to states. Wespace notice that,loosely, in the case GSA, the frequency with visit state more whileofthe traditional BM visits whichspace the hardest to detect patterns is much the larger state more uniformly. For very are lowfound temperatures, BM more like a ones gradient descent and, when thanfunctions the corresponding in the BM.method In particular, many presented withare randomly generated patterns, the network will patterns that not found by the BM are detected employing stabilize at the closest local GSA. This corresponds tominima. the gaps encountered in the specWeshown compare, in Fig. the frequency with which different trum in Fig. 2(a).3 GSA tends to prefer the lower energy patterns, corresponding to minimum energy states,higher are found states, but will also find, with low probability, energy with the BM the GSA qA = 1.4, for the same states. Oneand canwith observe an for exponential upper limit simfor the ulations of Fig. 2. We observe there is degeneracy in network frequency of visits, as a function of energy for GSA. The BM states. Wevisit notice that,with in thea case GSA, thedistribution frequency with tends to states moreofuniform of frewhich the hardest to detect patterns are found is much larger quencies, as is expected from the characteristic of the locality than the corresponding ones in the BM. In particular, many of visits of state space, which we mentioned before. patterns that are not found by the BM are detected employing GSA. This corresponds to the gaps encountered in the spectrum shown in Fig. 2(a). GSA tends to prefer the lower energy 4. with CONCLUSIONS states, but will also find, low probability, higher energy states. One can observe a power-law upper limit for the frequency of visits, as a afunction energy for GSA. The BM We have proposed memoryoforganization, where two hiertends to visit states with a morecorresponding uniform distribution of fre-and archically structured modules to sensorial quencies, is expected from theproducing characteristic of the locality symbolicasmemories interact, sensorial and symof visits of staterepresenting space, which unconscious we mentionedand before. bolic activity, conscious men-

tal processes. This memory structure and functioning, along with an adaptive learning process, is used to explain a pos4. neurotic CONCLUSIONS sible mechanism for behavior and psychoanalytical working-through. We proposed a memory organization, twofrom hier-the Thehave complex network topologies whichwhere result archically structured modulesbased corresponding to sensorial and self-organizing processes, on biological mechanisms, declarative interact, producing sensorial andcommon symwhich we memories have proposed, have properties that are bolic activity, representing unconscious and conscious men-they features of many biological systems and indicate that tal processes. This memory structure and functioning along may not be well described by BG statistical mechanics, but with learningStatistical process isMechanics. used to explain posratheranbyadaptive Nonextensive Thesea mechasible for neurotic behavior psychoanalytical nismsmechanism are very characteristic of much and of the brain’s functionworking-through. ing and suggest the use of a GSA algorithm to model memory The complex network topologies which result from the selffunctioning and the way we associate ideas in thought. The organizing processes, based on biological mechanisms, which study of network quantities such as node degree distributions we have proposed have properties that are common features and clustering coefficients may indicate possible experiments, of many biological systems and indicate that they may not be which would validate models suchstatistical as the one presentedbut here. well described by Boltzmann-Gibbs mechanics, We by areNonextensive proceeding instatistical further model refinement analyrather mechanics. These and mechasis. It are is necessary to verifyof more thoroughly the dependence nisms very characteristic much of the brain’s functionof model behavior on its various parameters suchmemory as T and ing and suggest the use of a GSA algorithm to model σ. These parameters represent, at least partially, the effects functioning and the way we associate ideas in thought. The of neurosubstances such assuch neural growth factors, neurotransstudy of network quantities as node degree distributions mitters and neuromodulators. Although we do not have exand clustering coefficients may indicate possible experiments, that would validate models the one values, presented perimental indications of such their as absolute thehere. tuning of We relative are proceeding model refinement analy-and their values in is further fundamental for model and stability sis. It is necessary to verify the dependence functioning. This may givemore somethoroughly insight to basic mechanisms of behavior on its and various parameters such as T and asin model real neural networks the emergence of behavioral σ. These represent,inattrying least to partially, the effects pects. Weparameters are also interested map language strucof neurosubstances neural growth factors, ture and processingsuch intoasnetwork topology and neurotransdynamics, almitters Although weisdopossible. not have experthoughand we neuromodulators. are not yet sure whether this imental indications of their absolute values, the ofpropose their Our main contribution in recent work has tuning been to relative values is fundamental for model stability and funca neuronal model, based on known microscopical, biologitioning. This may give some insight to basic mechanisms in cal brain mechanisms, which describes conscious and unconscious memory activity involved in neurotic behavior, as described by Freud. The model emphasizes that symbolic processing, language and meaning are important for consciousness. However, we do not sustain or prove that this is the actual mechanism that occurs in the human brain. Although

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FIG. 2: The numbers taken as abcissas in (a) and in (b) identify the same patterns. In both figures T = 0.2. (a) Left, energy of stored patterns FIG.visited 2: Theby numbers as abcissas andpatterns in (b) identify thethe same patterns. both T= 0.2. configurations (a) Left, energy of (a). stored patterns the BM.taken (b) Right, energy in of (a) stored visited by GSA for qA =In1.4, forfigures the same initial as in FIG. numbers takenenergy as abcissas in (a)patterns and in (b) identify bothfor figures T = 0.2. (a)configurations Left, energy of as stored patterns visited by2:theThe BM. (b) Right, of stored visited bythe thesame GSApatterns. for qA =In1.4, the same initial in (a). visited by the BM. (b) Right, energy of stored patterns visited by the GSA for qA = 1.4, for the same initial configurations as in (a). 0.1

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FIG. 3: (a) On the left, visiting frequency to stored patterns by the BM for T = 0.2. (b) On the right, similar to (a) for the GSA at qA = 1.4 and temperature. FIG.the 3: same (a) On the left, visiting frequency to stored patterns by the BM for T = 0.2. (b) On the right, similar to (a) for the GSA at qA = 1.4 FIG. and 3: (a) thetemperature. left, visiting frequency to stored patterns by the BM for T = 0.2. (b) On the right, similar to (a) for the GSA at qA = 1.4 theOn same

and the same temperature.

real neural networks and the emergence of behavioral aspects. realare neural emergence of behavioral aspects. We alsonetworks interestedand in the trying to map language structure and biologically plausible, ininaccordance with many structure aspects We are also interested trying to map language and processing into network topology and dynamics, althoughdewe scribed by psychodynamic and ispsychoanalytic clinical expeprocessing into network and dynamics, although we are not yet sure whether topology this possible. areOur notmain yet sure whether based this is on possible. rience, and experimentally simulations, the to model is contribution in recent work has been propose Our mainmodel, contribution work beenmetaphorto propose neuronal based inonrecent known biologiveryaschematic. It nevertheless seems tomicroscopical, behas a good a neuronal model, based on known microscopical, biologibrain mechanisms, that describes conscious unconical cal view of facets of mental phenomena, for whichand we seek cal brain mechanisms, that describes conscious and unconscious memory activity neurotic behavior, as dea neuronal substratum, and involved suggestsindirections of search. scious memory activity involved in neurotic behavior, as prodescribed by Freud. The model emphasizes that symbolic scribed by Freud. and The meaning model emphasizes that symbolic processing, language are important for consciouscessing, languagewe anddomeaning are important for consciousness. However, not sustain or prove that this is the Acknowledgements ness. mechanism However, we dooccurs not sustain provebrain. that this is the actual that in theorhuman Although actual mechanism that occurs in the human brain.aspects Although biologically plausible, in accordance with many deWe are gratefulplausible, for fruitful discussionswith regarding basic biologically in accordance manyclinical aspectscondescribed by psychodynamic and psychoanalytic expeceptsrience, of psychoanalytic theory, with psychoanalysts pro-is scribed by psychodynamic and psychoanalytic clinical expeand experimentally based on simulations, theand model fessors Romildo do Rˆ e go Barros, Paulo Vidal and Angela rience, and experimentally based on simulations, the model is very schematic. It nevertheless seems to be a good metaphorvery schematic. It nevertheless seems to be a good metaphorical view of facets of mental phenomena, for which we seek a ical view substratum, of facets of mental phenomena, for which we seek a neuronal and suggests directions of search. neuronal substratum, and suggests directions of search. Acknowledgements

[1] Acknowledgements S. Freud, Introductory Lectures on Psycho-Analysis, Standard We areNew grateful regarding basic conEdition, Yorkfor- fruitful London:discussions W. W. Norton and Company We are grateful for fruitful discussions regarding basic concepts of First psychoanalytic theory, with psychoanalysts and pro(1966). German edition in 1917. cepts of psychoanalytic theory, with psychoanalysts and pro[2] S. Freud, The Unconscious, Standard Edition of The Complete Psychological Works of Sigmund Freud, Volume XIV, London: The Hogarth Press (1957). First German edition in 1915. [3] S. Freud, Beyond the Pleasure Principle, Standard Edition of The Complete Psychological Works of Sigmund Freud, Volume XI, London: The Hogarth Press (1974). First German edition

fessors Romildo do Rˆego Barros, Paulo Vidal and Angela fessors Romildo Rˆego Brasileira Barros, Paulo Vidalalise and of Angela Bernardes of thedoEscola de Psican´ Rio de Bernardes ofthe theDepartment EscolaBrasileira Brasileira Psican´ athe lise of Rio Bernardes of Escola dede Psican´ aof lise of UniversiRio de de Janeiro and of Psychology Janeiro and the the Department Psychology of UniversiJaneiro and Department ofofPsychology of members the the Universidade Federal Fluminense. We also thank the of the Federal Fluminense. We also thank members of the dade Federal Fluminense. We also thank members of the Escola Brasileira de Psican´ alise of Riothe dethe Janeiro for their Escola Brasileira Psican´ alise of Wedemann RioRio de de Janeiro for of their warm reception the author S. in some their Escola Brasileiraofdede Psican´ aR. lise of Janeiro for their warm reception ofof thethe author R. S.R. Wedemann somein of some their seminars. Suggestions and discussions withinSumiyoshi Abe,of warm reception author S. Wedemann seminars. and Constantino discussions with Sumiyoshi Abe, Evaldo M.Suggestions F. Curado and Tsallis regarding comtheir seminars. Suggestions and discussions with Sumiyoshi Evaldo M. F. Curado and Constantino Tsallis regarding complex networks and nonextensive statistical mechanics were Abe, Evaldo M. F. Curado and Constantino Tsallis regardplex networks andenlightening. nonextensive We statistical mechanics were also helpful and acknowledge the Braziling complex networks and nonextensive statistical mechanalso helpful and enlightening. We acknowledge thedeBrazilianwere National Council (CNPq) and Rio Janeiro ics alsoResearch helpful and enlightening. Wetheacknowledge the ian National Research Council (CNPq) and the Riofinancial de Janeiro State Research Foundation (FAPERJ) for partial sup-de Brazilian National Research Council (CNPq) and the Rio State Foundation (FAPERJ) for partial financial support.Research Janeiro State Research Foundation (FAPERJ) for partial fiport.

nancial support.

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