Detecting Neuronal Communities from Beginning of Activation Patterns
aa r X i v : . [ phy s i c s . s o c - ph ] J a n Detecting Neuronal Communities from Beginning of 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: 27th Jan 2008)The detection of neuronal communities is addressed with basis on two important concepts fromneuroscience: facilitation of neuronal firing and nearly simultaneous beginning of activation of setsof neurons. More specifically, integrate-and-fire complex neuronal networks are activated at each oftheir nodes, and the dissemination of activation is monitored. As the activation received by eachneuron accumulates, its firing gets facilitated. The time it takes for each neuron, other than thesource, to receive the first non-zero input (beginning activation time) and the time for it to producethe first spike (beginning spiking time) are identified through simulations. It is shown, with respectto two synthetic and a real-world (
C. elegans ) neuronal complex networks, that the patterns ofbeginning activation times (and to a lesser extent also of the spiking times) tend to cluster intogroups corresponding to communities of neurons in the original complex neuronal network. Such aneffect is identified to be a direct consequence of the almost simultaneous activation between the nodesinside the same community in which the source of activation is placed, as well as of the respectivetrapping of activation implied by the integration of activiation prior to firing. Interestingly, theaccumulation of activity and thresholds inside each neuron were found to be essential for constrainingthe initial activations within each respective community during the transient activation (no clearclusters were observed when using overall activation or spiking rates). In addition to its intrinsicvalue for neuroscience and structure-dynamics studies, these results confirm the importance of theconsideration of transient dynamics in complex systems investigations.
PACS numbers: 87.18.Sn, 05.40Fb, 89.70.Hj, 89.75.Hc, 89.75.Kd ‘The brain is a wonderful organ. It starts working themoment you get up in the morning and does not stopuntil you get into the office. (R. Frost)
I. INTRODUCTION
Much has been investigated about neuronal systemsfrom both the biological and exact sciences points of view(e.g. [1]). More recently, neuronal networks met complexnetworks (e.g. [2, 3, 4, 5, 6, 7, 8, 9, 10]). Such an inter-face is particularly promising, as it allows the emphasisof dynamical systems characterizing research in neuronalnetworks to be integrated with the structural approachesof complex networks research (e.g. [11, 12, 13, 14]).The intersection of these two major areas, which liesat the heart of the structure and dynamics relation-ship (e.g. [12, 15]), is henceforth called
Complex Neu-ronal Networks research . However, despite the growingattention to this are, few works have considered simpleneuronal models such as the integrate-and-fire. In addi-tion, rather few studies have addressed transient dynam-ics (e.g. [16]) or the accumulation of stimuli responsiblefor the facilitation of firing [1].In a recent study [17], the transient dynamics ofintegrate-and-fire networks underlain by several typesof connectivity was characterized with respect to a se-ries of dynamical properties, including the activationof nodes, spiking, and onset times for activation andspiking. As the activation received by each neuronwas stored inside it as its state (facilitation [1]), it be-came possible to investigate the activation and spiking separately. The neuronal dynamics was found to varymarkedly with respect to the connectivity, with abrupttransitions of initiation of generalized spiking being ob-served for some complex networks models, as well asfor the
C. elegans network. The current work contin-ues such a investigation in order to investigate for pos-sible simultaneous neuronal activation as a consequenceof concentrated interconnectivity between groups of neu-rons. The basic idea is that more intensely intercon-nected groups of neurons, i.e. communities of nodes(e.g. [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29]),would imply concentration of the accumulated activa-tion because of the thresholds, delaying the disseminationof the activation to other parts of the network. Morespecifically, external activation is fed into each of the N neurons, and the unfolding activation is monitored.The beginning activation time of each neuron i (exceptthat used as source) is experimentally determined, corre-sponding to the time from the beginning of the externalactivation to the first arrival of non-zero stimuli at neu-ron i . Therefore, a pattern of N − C. elegans net-work. Similar, though a little less effective, results wereobtained by considering the beginning spiking times. Noclusterings are observed in case the activation of spikingpatterns at specific times (or even averages) are takeninto account. Such results corroborates the importanceof transient dynamics in complex systems investigations.This article starts by presenting the basic concepts incomplex networks, integrate-and-fire complex neuronalnetworks and the statistical method of Principal Compo-nent Analysis. The results, discussion and perspectivesfor future investigations are presented subsequently.
II. BASIC CONCEPTS
A directed, unweighted network Γ can be fully repre-sented in terms of its adjacency matrix K . Each edgeextending from node i to node j implies K ( j, i ) = 1. Theabsence of connection between nodes i and j is repre-sented as K ( j, i ) = 0. The out-degree of a node i , hence-forth expressed as k , corresponds to the number of out-going edges of that node. An analogue definition holdsfor the in-degree. The immediate neighbors of a node i are those nodes which can be reached from i through anoutgoing edge. Two nodes are adjacent if they share anedge; two edges are adjacent if they share a node. A walk is a sequence of adjacent edges, with possible repetitionsof nodes and edges. A path is a walk without repetitionof nodes or edges. The length of a walk (or path) is equalto the number of the edges it contains. The shortest pathlength between two nodes is the length of the shortestpath between them.An integrate-and-fire neuron involves two successivestages: (i) summation of the inputs x i , i.e. S = P i x i ;and (ii) thresholding, i.e. a spike is produced whenever S is larger than a threshold T . In this work, each node rep-resents an integrate-and-fire neuron. The incoming acti-vation (from dendrites) is accumulated as its state until aspike occurs, in which case the accumulated activation iscompletely flushed out through the outgoing connections(axons). In order to ensure conservation of the total ac-tivation, ach outgoing axon conveys a fraction S/k ( i ) ofthe respective previously accumulated activation, where k ( i ) is the out-degree of neuron i . The activation andspiking of all neurons in the network can be representedthrough the respective activogram and spikegram , namelymatrices storing the activation or occurrence of spikes forevery node along all considered times. In this article, allneurons have the same threshold T = 1, and the externalactivation of the network is always performed by inject-ing activation of intensity 1 at each of the neurons. Foreach of these activations (with the source of activationplaced at node v ), the time one neuron i takes, from thebeginning of the external initiation, to receive the firstnon-zero input is henceforth called its respective begin-ning activation time T a ( i, v ). The time it takes for thatneuron to produce the first spike is the beginning spikingtime T s ( i, v ). The dimensionality of the measurement space definedby the onset activation times for each node in a networkinvolves N measurements [35] (the beginning times). So,we have a total of N measurements for each of the N nodes, which is a high dimensional space involving sev-eral correlations between the times. The dimensionalityof such measurements spaces can be convenient and op-timally reduced (decorrelation) by using the PrincipalComponent Analysis (PCA) methodology (e.g. [14, 30]).Let each of the N observations v = { , , . . . , N } , charac-terized by the set of beginning times at each neuron i as aconsequence of activation placed at node v , be organizedas respective feature vectors ~f v , with respective elements f v ( i ) = T a ( i, v ), i ∈ { , , . . . , N } . Let the covariancematrix between each pair of measurements i and j bedefined in terms of its elements C ( i, j ) = 1 N − N X v =1 ( f v ( i ) − µ i )( f v ( j ) − µ j ) (1)where µ i is the average of f v ( i ) over the N observa-tions. The eigenvalues of C , sorted in decreasing order,are represented as λ i , i = 1 , , . . . , M , with respectiveeigenvectors ~v i . The following matrix, obtained from theeigenvectors of the covariance matrix, defines the stochas-tic linear transformation known as the Karhunen-Lo`eveTransform [14, 30]. G = ←− ~v −→←− ~v −→ . . . . . . . . . ←− ~v m −→ (2)where m = N . Because such a transformation concen-trates the variance of the observations along the first axes(the so-called principal axes), it is frequently possible toreduce the dimensionality of the measurements withoutlosing much information (the variances along the otheraxes tend to be small as a consequence of correlations be-tween the original measurements) by considering in theabove matrix only the m < N eigenvectors associated tothe larges eigenvalues. The new measurements ~g , withdimension m , can now be straightforwardly obtained as ~g = G ~f . (3)
III. RESULTS AND DISCUSSION
The potential of the neuronal community detectionapproach reported in this article is illustrated with re-spect to the three following directed networks: (a) a syn-thetic network (
Net1 ) containing 3 small communities(Figure 1); (b) a medium-sized synthetic network (
Net2 )containing 4 communities (Figure 1); and (c) the networkof
C. elegans ( NetCe ) [22]. The two synthetic commu-nities were obtained by randomly assigning direct edgesamong each of the communities, extracting the connectedcomponent, and interconnecting the communities accord-ing to a fixed probability. Each of the three communitiesin
Net1 contains 5, 7 and 7 nodes, respectively. Eachof the four communities in
Net2 contains 20, 37, 22 and24 nodes. The largest strongly connected component inthe
C. elegans network, used in this work, contained 239nodes.
FIG. 1: A simple network (
Net1 ) involving 3 communitieswith 5, 7 and 7 nodes considered in this work for illustrativepurposes.
Figure 3 shows the activograms and the spikegrams,as well as the respective beginning activation and begin-ning spiking times diagrams for activation placed at node3 (a), 9 (b) an 16 (c). It is clear from the respective be-ginning activation time diagram in Figure 3(a) that theactivation being received by node 3 implied in early andsimultaneous conveyance of non-zero activation to theother nodes in the community to which node 3 belongs(i.e. the community including nodes 1 to 5). Observethat the beginning spiking times, shown in the respectivediagram, are larger than the activation time, because therespective neuronal firing requires accumulation of theactivation received by the dendrites of the neurons inthat community. Similar nearly simultaneous activationsof the other communities can be identified in Figure 3(b)and (c). However, a less uniform initiation of activationin the this community is observed in Figure 3(c).Figure 4 shows the distribution of the beginning activa-tion times after PCA projection onto a two-dimensional space defined by the principal variables pca pca
N et
2. Again, each of the communitieswas clearly mapped into respective clusters in the two-dimensional PCA projected space. Again, the borderingnodes in each cluster can be found to correspond to theinterface nodes between communities in the original net-work. It is interesting to observe that, compared to theprevious example, the larger number of communities andnodes in this network tended to imply a more cluttereddistribution, especially at the interface between the redand magenta communities. Substantially more separatedclusters have been observed in three-dimensional PCAprojected spaces.The distribution of nodes obtained for the
C. elegans network by two-dimensional PCA projection is shown inFigure 6. A concentration of nodes can be observed atthe left-hand side of the space, containing the nodes withhigher numbers (the numbers follow the original assigne-ment as in [22]). A community can also be discernedat the lower right-hand side of the transformed mea-surement space. Because of the large number of nodesin this network, it is interesting to consider additionaldimensions in the PCA projection. The measurementspace defined by the principal variables pca pca pca
3, witha more compact cluster of nodes appearing at the upperleft-hand side of Figure 7. In addition, a small clusterinvolving nodes 136, 146, 148, 149, 234, 235 and 236 isnow identifiable at the lower left-hand side of Figure 7.Less definite results were obtained in all cases by con-sidering the beginning spiking times, and no clear clusterstructure was observed when other activation of spikingmeasurements (at given instants or averaged) were used.
IV. CONCLUDING REMARKS
This work has addressed an important issue related tothe structure-dynamics paradigm in neuronal and com-plex networks research. More specifically, we have inves-tigated how communities of neurons in directed complexneuronal networks can be identified by considering thetransient dynamics of beginning activation of nodes. As aconsequence of the integration period required for reach-ing the firing threshold in each integrate-and-fire neuron,the activation incoming from the source node tends tobe trapped inside the respective community, unfoldingto other portions of the network only after most of the
FIG. 2: A medium-sized complex network (
Net2 ) containing 4 communities with 20, 37, 22 and 24 nodes each. neurons in that community have started spiking. Thedistribution of the activation flushed outside each neuronat the spikings, required for the conservation of the acti-vation, was also critical for the compartmentalization ofthe activation inside communities. The distinct patternsof beginning activation times obtained by placing the ac-tivation source at each of the neurons of each communitywere clearly revealed by the optimal statistical methodof Principal Component Analysis. More specifically, thenodes tended to cluster into respective groups, with thenodes at the borders of such groups corresponding tothose nodes implementing the intercommunity connec-tion in the original network. In addition to its intrinsicvalue for biological neuronscience, these results also pro-vide effective and simple practical means for obtaining neuronal communities.Several interesting future works are possible. First,it would be important to perform a more systematicand comprehensive study of the separability of the com-munities by considering other types of networks, withdistinct interconnectivity between communities, amongother possibilities. Also interesting is to use hierarchicalclustering methods (e.g. [14, 19, 30]) in order to obtaina hierarchical organization of the neuronal communities,as well as investigating how such hierarchies (e.g. [29])are organized with respect to time. As the suggestedmethod can be immediately extended to identificationin other types of networks, including non-directed struc-tures, it would be interesting to compare this methodwith other more traditional approaches not involving (a) (b) (c)FIG. 3: The activograms and spikegrams, as well as the respective beginning activation and beginning spiking times, are shownwith respect to the situations where the activation source corresponded to nodes 3 (a), 9 (b) and 16 (c). The beginning timediagrams show in black the time instants preceding the first activation or spiking of each neuron. For instance, in the beginningspiking times diagram in (a), neuron 5 started spiking at the 8th time step from the initiation of the external activation arrivingat node 3. thresholds. Because abrupt beginning of spiking has beenobserved [17] for several types of complex networks, itwould be also interesting to search for possible phasetransitions of activation inside each community, whichcould be ultimately responsible for the activation trap-ping inside each neuronal community during the tran-sient activation period.All in all, the findings and perspectives reported inthis article have supported the fact that investigations of transient non-linear dynamics are specially promisingand useful in the study of complex systems (see also [17,31, 32, 33, 34]).
Acknowledgments
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Net2 . Each of the four originalcommunities can be clearly identified from the respective clusters in this scatterplot, with the nodes at the borders of theclusters corresponding to the nodes at the borders of the original communities.
FIG. 6: The distribution of nodes obtained for the
C. elegans network considering the two principal PCA variables pca pca FIG. 7: The distribution of nodes obtained for the
C. elegans network considering the first and third principal PCA variables pca pcapca