Communities in Neuronal Complex Networks Revealed by Activation Patterns

0.73 MB·English

by  Luciano da Fontoura Costa

Communities in Neuronal Complex Networks Revealed by Activation Patterns
Luciano da Fontoura Costa
Institute of Physics at S˜ao Carlos, University of S˜ao Paulo,
PO Box 369, S˜ao Carlos, S˜ao Paulo, 13560-970 Brazil
(Dated: 29th Jan 2008)
Recently, it has been shown that the communities in neuronal networks of the integrate-and-
fire type can be identified by considering patterns containing the beginning times for each cell to
receive the first non-zero activation. The received activity was integrated in order to facilitate
the spiking of each neuron and to constrain the activation inside the communities, but no time
decay of such activation was considered. The present article shows that, by taking into account
exponential decays of thestored activation, it is possible to identify the communities also in terms
8 of thepatternsof activation along theinitial stepsof the transient dynamics. The potential of this
0 method is illustrated with respect to complex neuronal networks involving four communities, each
0
of a different type (Erd˝os-R´eny, Barab´asi-Albert, Watts-Strogatz as well as a simple geographical
2
model). Though the consideration of activation decay has been found to enhance the communities

n separation, too intense decaystend to yield less discrimination.
a
J PACSnumbers: 87.18.Sn,05.40Fb,89.70.Hj,89.75.Hc,89.75.Kd

0
3
‘Zora’s secret lies in the way your gaze runs over pat- sentedasanode)havebeenstudiedwithrespecttotheir

] terns following one another as in a musical score...’ (I. transient dynamics. Figure 1 illustrates the type of neu-
C
Calvino, Inivisible Cities) ronal cell adopted in those works. The incoming activa-
N tion, received through the n(i) dendrites, is integrated
. andaccumulatedinthe internalstateS(i)untilits value
o
exceeds the threshold T(i), in which case the cell fires,
i I. INTRODUCTION
b liberating the accumulated activation between the m(i)
- outgoingedges(axons). Inthepreviousworks[32,33],in
q
Neuronal networks (e.g. [1, 2, 3]) and complex net-
[ ordertomaintainthetotalreceivedactivation,whichwas
works (e.g. [4, 5, 6, 7]) can be understood as sister re- fedthroughasingleselectedneuron,theaccumulatedac-
1 search areas. However, as the latter is much younger
tivation S(i) was uniformly distributed among the m(i)
v (especiallyregardingthe developmentsfrom1999),these
outgoing connections, each therefore receiving a share of
4
two sisters have yet to get fully acquainted one an-
S(i)/k (i), where k ut(i) = m(i) is the out-degree of
8 out o
other. Such a natural integration has already begun
6 node i.
(e.g. [8, 9, 10, 11, 12, 13, 14, 15, 16]) and is poised
4
. to continue to the point that these two areas become
1
not only close relatives, but also best friends. This in-
0
tegration is particularly interesting for both neuronal
8
0 networks and complex networks because of the comple-
: mentation of the approaches which have been respec-
v
tively adopted. More specifically, while neuronal net-
i
X works have relied strongly on pattern recognition and
r dynamicalsystems,complexnetworkshavebeenstrongly
a focusing on structure, with a recent surge of interest
on dynamics (e.g. [5, 17]). However, as special em-
phasis has been placed on the important problem of FIG. 1: The integrate-and-fire neuronal cell adopted in the
linear synchronization (e.g. [17]), few works have ad- previousworks[32,33]incorporatesthreestages: (i)integrat-
ing of input activations; (ii) memory of activation S(i); and
dressed non-linear or transient dynamics (e.g. [18, 19]).
(iii) non-linear transfer function involving a threshold T(i)
In complex networks, emphasis has been placed on the
(a hard limitter). While those previous works adopted full
modularity of the connections or community structure conservation of the activation (i.e. decay rate α=0), in the
(e.g.[20,21,22,23, 24,25,26,27,28,29,30,31]),which
presentworkthestoredactivationundergoesexponentialtime
hasimportantimplicationsforboththestructureanddy- decay with rate α.
namics of networks. The integration between neuronal
networks and complex networks is henceforth referred
to as complex neuronal networks, which has special im-
portance for non-linear dynamical systems underlain by Several interesting dynamic features are implied by
structured and complex connectivity. such a simple neuronal model. First, the accumulation
Recently[32,33],complexneuronalnetworksinvolving of the received activity is related to the important phe-
simple integrate-and-fire neurons (each neuron is repre- nomenon of facilitation of firing. Roughly speaking, the
2
income of a spike into a cell enhances the probability of tical method known as Principal Component Analysis
its future spiking by occasion of subsequent activations. (PCA) is applied to the activation patterns in order to
Second, the non-linear element implies the activation to reduce their dimensionality, which is optimally obtained
remain stored until the threshold is reached, which con- by decorrelation of the activation. The original commu-
tributes stronglytoconstrainingthe activationlocallyin nities could be properly detected in most cases, even for
the networkalongtopologyandtime. As allneuronsare theBaraba´si-Albertandgeographicalmodels. Combined
henceforthassumedto havethe samethresholdT =1 (a with the investigations reported previously [33], the re-
biologicallyreasonablechoice),thedistributionofoutgo- sults obtained in the current work substantiate further
ing activation implied by each spiking becomes impera- the importance of the transient regime for characteriza-
tive in order not to yield one spike at every time step. tion of modularity regardingboth structure and dynam-
Similar effects can be obtained by associating weights icsincomplexsystems. Withrespectto the specific area
smaller or equal to one to each edge (synaptic weight). of neuronal networks, the relationships between struc-
The combination of such non-linear effects has been ob- tured connectivity, in the form of communities, and the
served[33]tocontributedecisivelyforconstraining,along activationandspikingdynamics provideseveralimplica-
a transientperiodoftime, the activationinside the com- tions for synchronization, pattern recognition and mem-
munity which contains the source of activation. Such an ory. The proposed methodologies may also prove useful
effect allows the identification of neuronal communities as practicalmethods for identification of communities in
by considering the transient non-linear dynamics in the more general types of networks.
whole network while it is stimulated by sources of acti- The current article starts by presenting the basic con-
vations placed at each of its neurons. It has been exper- cepts in complex neuronal networks, the four adopted
imentally verified that the time it takes for each cell to theoretical models of complex networks, and the statis-
receive non-zero activation in any of its dendrites, called tical method of Principal Component Analysis. The re-
the beginning activation time of each cell, seems to be sults, discussion, and perspectives for future works are
particularlyrelevantforthe identificationofthe commu- presented subsequently.
nities. Promising results were obtained with respect to
two synthetic (networks including 3 and 4 communities
with uniform connectivity) as well as a real-world net- II. BASIC CONCEPTS
work (C. elegans [20]).
However, the previous investigations reported in [33]
A directed, unweighted network Γ can be completely
considered no time decay of the stored activation S(i),
specified in terms of its adjacency matrix K. Each edge
whichseemsto havebeenresponsiblefor makingthe be-
extending from node i to node j is representedK(j,i)=
ginningactivationtimesdecisivefortheproperidentifica-
1. The absence of connection between nodes i and j
tionofthecommunities. Inthepresentworkweconsider
implies K(j,i) = 0. The nodes which receive a direct
the more biologically realistic situation involving expo-
edgefroma node i arecalledthe immediate neighbors of
nential decays of the activations. More specifically, at
i. The out-degree of a node i is equal to the number of
each time step each stored activation is decreased at a
its immediate neighbors.
constant rate α, i.e.
Four theoretical models of complex networks (e.g. [4,
5, 6, 7]) have been used in order to construct the hy-
St+1(i)=St(i)−αSt(i) (1) bridcommunitynetworksconsideredinthiswork: Erdo˝s-
R´enyi(ER),Baraba´si-Albert(BA),Watts-Strogatz(WS)
where t is the time step and 0≤α<1. as well as a simple geographical type of network (GG).
The net effect of the decay is to generally delay the An Erdo˝-R´enyi network (see also [34]) can be obtained
firing of cells. Interestingly, such an effect seems to al- by establishing connections between pairs of nodes with
lowproperidentificationofthecommunitiesalsobycon- constant probability. The BA networks were obtained
sidering the average activation of the network along an by starting with m0 nodes and progressivelyincorporat-
intervalofthetransientdynamics,insteadofonlythebe- ing new nodes with m edges, which are attached to the
ginningactivationtime. Thisispossiblyaconsequenceof remainder nodes with probability proportional to their
thefactoftheenlargedperiodoftimerequiredtoconvey respective degrees. The WS structures were obtained
the activation from one community to another, which is by starting with a linear regular network of suitable de-
enhanced by the decays. This possibility is experimen- gree and subsequently rewiring 10% of its edges. The
tally investigated in the current article by considering geographical structures are obtained by distributing N
hybrid networks containing four communities of differ- nodes along a two-dimensional space and then connect-
ent types (Erd˝o-R´enyi, Baraba´si-Albert, Watts-Strogatz ing each pair of nodes whose distance does not exceed
as well as a simple geographical model). Several combi- a given threshold. Though all these networks are undi-
nations of inter and intra-community intensities of con- rected,weobtainedtherespectivedirectedneuronalcom-
nections are considered. The activation is averagedfrom plex networks by considering the incoming on outgoing
the beginning of the source operation for a total of H directions of each edge as dendrites and axons, respec-
steps along the transient dynamics. Then, the statis- tively. Therefore, the so-obtained networks are directed
3
and have in-degree identical to the out-degree. III. RESULTS AND DISCUSSION
Theintegrate-and-fireneuron adoptedinthisworkhas
been described and discussed in the Introduction. The
Figure 2 illustrates the 9 networks adopted in the
activation and spiking of all neurons in the network can
presentinvestigation. Eachoftheminvolves4 communi-
berepresentedintermsofdiagramswhicharehenceforth
ties,ofrespectiveER,BA,WSandGGtypes(seelegend
called activogram and spikegram, respectively. These di-
atthe bottomof the figure)andapproximately50nodes
agrams are matrices storing the transient activation or
each. The intra-community degrees, expressed in terms
occurrence of spikes for every node. In this article, the
ofthe BA parameterm,increase alongthe columns (top
activationofthenetworkisalwaysperformedbyinjecting
to bottom), and the inter-community degrees k increase
external activation of intensity 1 at each of the neurons.
along the rows (left to right). The considered values of
Thetimeittakesforeachneuroni,fromtheonsetofthe
intra- and inter-connectivity are shown in Figure 2. The
external initiation, to receive the first non-zero input is
same intra-connectivity degree, defined with respect to
henceforthcalleditsrespective beginningactivation time
the parameter m of the BA model, was adopted for all
T (i,v). Thetimeittakesforthatneurontoproducethe
a the 4 communities in each case. The consideration of
first spike is the beginning spiking time T (i,v).
s hybrid communities involving several network models is
Because the activation patterns obtained with the
particularly useful for investigating the community de-
sourceineachoftheN neuronsinvolveN measurements,
tectionmethodologywithrespecttovaryingconnectivity
a highly dimensional space is implied. As a consequence
patterns.
of the intrinsic correlations between the activation pat-
Each network was searched for community structure
terns, it is possible to apply the PCA method to opti-
by placing the activation source (with intensity 1) at
mally decorrelated those patterns and yield meaningful
each of its neurons and simulating the respective acti-
2D and 3D projections. Let each of the N observations
vation and spiking along the initial H = 200 steps of
v = {1,2,...,N} be characterized by the average acti-
the transient dynamics. Three whole set of simulations
vations of all nodes as a consequence of the activation
whereperformedby consideringrespective decayratesα
sourceplacedatnode v. These measurementscanbe or-
equalto0.02and0.5. Figure3showstheactivogramand
ganized into respective feature vectors f~ , with elements
v spikegram, as well as the diagrams of beginning activa-
f (i), i ∈ {1,2,...,N}. Let the covariance matrix be-
v tion times and beginning spiking times for the network
tween each pair of measurements i and j be defined as
with m=3, k =0.2 and α=0.02.
The constant activation fed through neuron 25 is
1 N clearly identified as the white column in the activogram
C(i,j)= N −1X(fv(i)−µi)(fv(j)−µj) (2) and spikegram. Because of the more intense intercon-
v=1 nectivity between the nodes in the community to which
where µ is the average of f (i) considering all the N thisneuronbelongs(ER,inthiscase)thepropagationof
i v
observations (i.e. activations). The eigenvalues of C, activation and spikes tend to occur first inside this com-
sorted in decreasing order, are henceforth represented munity, being propagatedto the other communities only
as λ , i = 1,2,...,M, with respective eigenvectors v~. later. One exception are the neurons around node 125,
i i
The matrix G given in Equation 3, obtained from the which belong to the WS community. Because neuron
eigenvectorsofthecovariancematrix,definesthestochas- 125 is connected to the ER community with particular
tic linear transformation known as the Karhunen-Lo`eve intensity (more than one edge), it receives considerable
Transform [7, 35]. activation sooner, leading to a progressive spreading of
activation within the WS community. However, because
this effect is not verified for most of the other nodes of
←− v~1 −→ theERcommunity,ittendstobecomelessrelevantinthe
G=←− v~2 −→ (3) subsequent decorrelation projection implemented by the
... ... ...
  PCA.In addition, exceptfor a few other cells, the nodes
←− v~ −→
m whichareinside the same community tend to receiveac-
with m = N. Because such a transformation opti- tivation relatively soon, as illustrated in the respective
mallydecorrelatesthe activationpatterns,concentrating diagram of beginning activation times. A less regularly
the variance of the observations along the first axes (the simultaneous activation is obtained for the spikes in the
so-calledprincipalaxesorvariables),itisfrequentlypos- respective beginning spiking times diagram. The incor-
sible to reduce the dimensionality of the measurements porationofsuitable(nottoolarge)valuesofdecayseems
withoutsubstantiallossofinformationbyconsideringthe topromotemorestableactivationpatternsforthesource
abovematrixwithm≪N. Thenew,projectedmeasure- placedatneuronsofasamecommunityasaconsequence
ments~g,withdimensionm,cannowbestraightforwardly offurtherconstraintsonthe dispersionofthe activation.
obtained in terms of the following linear transformation In this work we consider for the community identifica-
tion the patterns of activation obtained by integrating
the activationfromtime 0to H =200. Observethatthe
~g =Gf~. (4) parameter H has important implications for the compu-
4
tational cost, in the sense that the larger its value, the identification of the topological communities as clusters
larger the number of computations. appearing in scatterplots obtained by optimally decorre-
Figure 4 depicts the clusters obtained in the two- lated PCA projections.
dimensionalspacedefinedbythefirsttwoPCAvariables Such a phenomenon has been experimentally investi-
considering the average activation patterns and decay gated by considering severalhybrid networks, with com-
α = 0.02. Figure 5 shows the respective scatterplots munitiesofdifferenttypes,andvaryingratiosofintra-to
obtained for the first and third PCA variables. There- intercommunity connectivity. In most cases the original
fore, it is possible to have a clear idea of the 3D PCA communities were mapped onto adjacent sets of points
space by considering these two images. In these figures, (clusters)which were often well-definedand delimitated.
aswellasalltheothersubsequentones,theoriginalcom- As in [33], the nodes at the interfaces between the ob-
munities are identified by respective colors: ER in blue; tained clusters tended to correspond to those nodes im-
BA in green; WS in red and GG in magenta. In most plementing the intercommunity connectionsl. The ER
cases,especially for low ratiosk/m, the originalcommu- and BA modules tended to produce more concentrated
nities were mapped into well-defined respective clusters clusters, with the WS and GG communities often yield-
in the PCA space. For instance, in the case m = 2 and ing scattered, but still separated, distributions in the
k =0.1, we have dense clusters obtained for the ER and PCAdiagrams. Thediscriminationbetweenthecommu-
BA communities. Because of their intrinsic nature, the nities was undermined when a substantially large decay
WS andGG models tended to yield largerdispersionsin was considered. As observed previously [33], the tran-
most of the cases considered in this work. Yet, they are sient confinement of the activation inside each commu-
well-separated, as it can be verified by considering both nity seems to be related to an abrupt pattern of activa-
the pca1×pca2 (Figure 4) and pca1×pca3 (Figure 5) tionobservedforseveralmodelsofcomplexnetworks[32].
diagrams. Except for the WS case, the other 3 com- As a matter of fact, the WS and GG models had indeed
munities still tended to map to reasonably well-defined been observed [32] to produce less abrupt activations.
local regions in the PCA projections for larger values of The situations involving higher values of α would also
intercommunity connection (i.e. k = 0.5 and 1). The imply substantially higher computational cost required
separationbetweenthecommunityclusterstendedtoin- for the simulation of the activation along longer tran-
creasesubstantiallyfromtoptodownalongeachcolumn sient periods. Nevertheless, it seems to follow from the
in Figures 4 and 5 as a consequence of the increase of currently reported results that relatively small non-zero
the intra-community connectivity relatively to the inter- decay of the accumulatedactivation, to a certainextent,
community density of connections. emphasizesthetransientconfinementofactivationinside
The PCA scatterplots obtained by considering more the communities.
intense decay(i.e. α=0.5)areshowninFigures6and7
Thesefindingssubstantiatetheimportanceoftransient
respectively to the pca1×pca2 and pca1×pca3 projec-
dynamics for the characterization and analysis of non-
tions. Lessseparatedclustershavebeenobtainedinmost
linear complex systems. Furthermore, the relationship
cases, with intense overlap between communities. How-
betweenmodularinterconnectivityandnearlysimultane-
ever,thenodesbelongingtotheoriginalcommunitiesstill
ousactivationofcommunitieshasseveralimplicationsfor
tended to be mapped to nearby positions in the scatter-
biologicalandcomputationalneuroscience. Inparticular,
plots. Sucha decreaseinthe community identificationis
such a relationship can be intrinsically related to recog-
adirectconsequenceofthefactthatmoreintensedecays
nition of patterns and associativememory. For instance,
tended to produce less stable activation patterns for the the nearly simultaneous activation of the communities
activation source placed at different nodes. In addition,
could play an important role in reconstructing larger
the consideration of more intense decay also would im-
patterns (communities) from incomplete presentations.
ply in averaging the activations along a longer period of
Because of the temporal dynamics of nervous systems,
time, demanding additional computations.
where problems have to be solvedby functional modules
of neurons along a given period of time (e.g. [36, 37]), it
is possible that the phenomenon of simultaneous activa-
IV. CONCLUDING REMARKS tion within communities plays an important generalrole
inneuronalorganizationandfunctionality. Theneuronal
In continuation to recent previous investigations [33], community identification methodology is also promising
therelationshipbetweentopologicalanddynamicalmod- for the identification of functional modules in the cortex
ularity during the transient period of activation of non- or neuronal subsystems because of its intrinsic compat-
linear integrate-and-fire complex neuronal networks has ibility with the non-linear dynamics performed in those
been explored further, with respect to the consideration systems.
ofaverageactivationpatternsasresourcesforcommunity Several are the future works implied by the results
identification. By increasing the latent period in which and methods reported in the current article. First, it
the activation has to increase until firing is reached in wouldbe importanttoperformmoreobjectiveinvestiga-
theneuronalcells,relativelysmallnon-zerodecaysofthe tions of the discriminability between the PCA clusters,
accumulated activation tended to allow the subsequent forinstancebyusingtheintra-andinter-classscatterings
5
(e.g.[35])andcomparingtheresultsobtainedforthesyn- activations verified for several complex network models.
thetic hybrid communities with those yielded by canon- This phenomenon, which is possibly associated to phase
ical analysis (e.g. [7, 38]). It would also be necessary to transition and/or self-organized criticality, seems to lie
consider larger ensemble of networks in order to reach at the heart of the confinement of the activation inside
more generaland definitive conclusions regardingthe ef- the communities during the transient activation.
fect of the few parameters involved (i.e. α and H), as
wellasconceriningpossiblefinite-sizeandscalingeffects.
Valuable insights about the influence of the connectiv-
ity on the transient non-linear dynamics of the complex
Acknowledgments
neuronal networks considered in this work can be po-
tentially achieved by applying the systematic approach
of superedges [18]. Of particular interest are further in- Luciano da F. Costa thanks CNPq (308231/03-1)and
vestigations aimed at the characterization of the abrupt FAPESP (05/00587-5)for sponsorship.
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6
k=0.1 k=0.5 k=1
m=2
(a) (b) (c)
m=3
(d) (e) (f)
m=4
(g) (h) (i)
legend:
FIG. 2: The 9 hybrid networks considered in this work incorporate 4 communities each, of respective ER, BA, WS and GG
types(see legend at thebottom).
7
(a)
(b)
(c)
(d)
FIG. 3: The activogram and spikegram, as well as the diagrams of beginning activation times and beginning spiking times
obtainedforthe200initialtimestepswithactivationsourceatnode25forthecomplexnetworkwithm=3,k=0.5(Figure2e)
and α=0.02.
8
FIG. 4: The clusters obtained by considering the first and second PCA variables for α=0.02.
9
FIG. 5: The clusters obtained by considering the first and third PCA variables for α=0.02.
10
FIG. 6: The clusters obtained by considering thefirst and second PCA variables for α=0.5.