Connectivity and Dynamics of Neuronal Networks as Defined by the Shape of Individual Neurons
aa r X i v : . [ q - b i o . N C ] M a y Connectivity and Dynamics of Neuronal Networks as Defined by the Shape ofIndividual Neurons
Sebastian Ahnert
Theory of Condensed Matter, Cavendish Laboratory,University of Cambridge, JJ Thomson Avenue, Cambridge CB3 0HE, UK
Luciano da Fontoura Costa ∗ Instituto de F´ısica de S˜ao Carlos,Universidade de S˜ao Paulo,Av. Trabalhador S˜ao Carlense 400,Caixa Postal 369, CEP 13560-970,S˜ao Carlos, S˜ao Paulo, Brazil (Dated: 7 September 2007)Neuronal networks constitute a special class of dynamical systems, as they are formed by in-dividual geometrical components, namely the neurons. In the existing literature, relatively littleattention has been given to the influence of neuron shape on the overall connectivity and dynamicsof the emerging networks. The current work addresses this issue by considering simplified neuronalshapes consisting of circular regions (soma/axons) with spokes (dendrites). Networks are grown byplacing these patterns randomly in the 2D plane and establishing connections whenever a piece ofdendrite falls inside an axon. Several topological and dynamical properties of the resulting graphare measured, including the degree distribution, clustering coefficients, symmetry of connections,size of the largest connected component, as well as three hierarchical measurements of the localtopology. By varying the number of processes of the individual basic patterns, we can quantifyrelationships between the individual neuronal shape and the topological and dynamical features ofthe networks. Integrate-and-fire dynamics on these networks is also investigated with respect totransient activation from a source node, indicating that long-range connections play an importantrole in the propagation of avalanches. ‘Nothing in excess.’ (Delphic proverb)
I. INTRODUCTION
Many discrete systems in nature can be modeled interms of networks which explains the remarkable devel-opment of the field of complex networks research over thelast decade (e.g. [1, 2, 3, 4, 5]). Because such systemsinvolve a large number of elements exchanging mass, en-ergy or information, their representation as a graph ornetwork is intrinsic. A wide range of complex systems,from the Internet to protein-protein interaction, has beensuccessfully mapped and studied in terms of complex net-works (e.g. [6]). The connectivity of a network also influ-ences the properties of dynamical processes which maytake place on it. This relationship between topology anddynamics is of particular interest in many networks [7]including neuronal ones.Despite the many applications of complex networksresearch, little attention has been given to systems inwhich the overall connectivity is affected (or even de-fined) by the geometrical features of the individual con-stituent elements. Indeed, the majority of works address-ing structure and dynamics in complex networks tends ∗ Electronic address: [email protected] to equate the structure with the topology of the net-works, without regarding the geometry of the involvedcomponents. While it is true that many networks do nothave a geometrical basis — such as author collaborationand disease transmission networks, as well as the Inter-net — some important complex systems are formed byelements with a well-defined geometry which influencesor even defines the emerging connectivity and dynam-ics. Such morphological networks include protein-proteininteraction as well as the biological neuronal networksin the central nervous systems (CNS) of animals, whichare called morphological neuronal networks . In such net-works which are the result of spatial interactions betweengeometrical constituent elements, the information aboutthe geometry of the original elements may not be avail-able to the researcher. This is often the case in protein-protein interaction and neuronal networks, where onlythe connectivity between the nodes is provided.A particularly important example of morphologicalnetworks are neuronal systems, which can be representedas directed networks with each neuron being a node andeach synapse a directed edge. The resulting connectivityand dynamics are closely related to the geometry of theinvolved neurons. Indeed, a large number of morpholog-ical neuronal types (e.g. [8]) have been identified in theCNS of vertebrates as well as more primitive animals (e.g.insects). Morphological neuronal types range from cellsas simple as the bipolar neurons of the retina to the in-tricate Purkinje neurons of the cerebellum. Because theneuronal shapes are highly variable, no consensus existson how they should be categorized.The contribution of the individual morphology of neu-rons to the overall network connectivity mostly occursin two ways: (a) the biochemical differentiation of neu-ronal types constrains the respective geometries and con-nections (e.g. a Purkinje cell will never exhibit radialorganization); and (b) the growth of each dendrite oraxon takes into account influences from the surround-ing environment, such as electric fields and gradients ofconcentration of molecules. Therefore, the shape of aneuron in the CNS is a consequence of both nature (i.e.the biochemical content associated to cell differentiation)and nurture (i.e. the effects of the surrounding environ-ment). The spatial distribution of the bodies of the neu-rons (called somata or perykaria) also plays an importantrole in defining the overall connectivity. It is importantto bear in mind that neuronal systems are highly plas-tic, undergoing structural and dynamical changes duringtheir whole lifetime. Therefore, it is particularly impor-tant to consider the interrelationship between neuronalstructure and dynamics in growing networks.Here we provide the first systematic investigation onthe relationship between the geometry and dynamics ofgrowing morphological networks. Because of the stochas-tic nature of biological neuronal networks, it becomes es-sential to consider a large number of realizations in orderto obtain meaningful measurements of topology and dy-namics. Therefore, we keep the total number of involvedparameters as small as possible, focussing on those whichare clearly related to neuronal shape and connectivity.First, we restrict our investigation to 2D neurons andneuronal networks. Such a simplification is justified bythe fact that several real neurons such as retinal ganglioncells, Purkinje and even the basal dendritic structure ofthe ubiquitous pyramidal cells are mainly planar. Next,we adopt prototypical radial neurons involving a centralregion (axon) from which a given number of dendriteswith the same length (spokes) emerge. This model im-plies as parameters: the number and length of dendrites,angular distribution of the dendrites, and the radius ofthe central region. By assuming a fixed dendritic areafor all neurons, the length of the dendrites can be relatedto their number, so that one of these parameters can beomitted.Once the types of neurons are defined, the networkscan be obtained by progressively placing new neuronalcells and establishing connections whenever a part of adendrite touches an axon. We use a uniform randomdistribution of the position of the cells. Before beingadded to the network, each cell is rotated by a uniformlyrandom angle. Therefore, the only additional parametersimplied by the network growth are the current numberof added cells and the size of the space along which theneurons are distributed.In order to investigate the relationship between struc-ture and dynamics in these networks, we make a series ofdifferent measurements. In order to express the topology of the networks we determine the degrees of the nodes,their clustering coefficient, Garlaschelli’s symmetry in-dex [9], as well as the size of the largest connected com-ponent in the network. The dynamics of the respectivenetworks is obtained in terms of integrate and fire dynam-ics [10, 11]. More specifically, each node is understood asa neuron which integrates the received activation until athreshold is reached, in which case a spike is produced asoutput. It has been verified [12, 13] that integrate-and-fire complex networks can undergo avalanches of acti-vations when stimulated from individual nodes, with thetype of connectivity substantially affecting the dynamics.More specifically, such avalanches have been related tospecific concentric organization of the network connectiv-ity [12, 13]. Given the source node i , the nodes which areat topological distance 1 from i are called the first con-centric level, the neurons which are at distance 2 consti-tute the second hierarchical level, and so on. Avalanchesare related to the existence of concentric levels with alarge number of nodes. More specifically, if a concen-tric level h contains many nodes while the preceding andsubsequent levels are less populated, the firing of neu-rons in the level h tends to induce overall activation ofthe neighboring levels, which then propagates throughthe whole network. In the case of undirected networksit is possible to obtain a simple equivalent model of theoriginal network, formed by a chain of a few equivalentnodes across the hierarchical levels [12, 13]. This modelallows predictions of avalanche characteristics, such asthe required activation ratios [13], even for symmetrizedversions of directed networks. Such an approximation isparticularly justified when the original directed networkinvolves a relatively high degree of symmetric connec-tions, as is the case for our morphological networks withlarge number of spokes n .Possible interdependencies between the structure anddynamics of the networks are investigated by consideringthe Pearson correlation of these measurements.We begin with a brief overview of the basic conceptsof complex networks theory and morphological neuronalnetworks. II. BASIC CONCEPTSA. Complex Networks Topology
Directed networks are defined by a set of N nodes and E edges which can be represented by an adjacency matrix A , such that its element a ij = 1 indicates the presenceof a directed edge from node i to node j , while a ij = 0expresses its absence.The topological properties of a network can be quan-tified and characterized in terms of a comprehensive setof measurements [5]. In this work we consider the follow-ing features: (i) in- and out-degree; (ii) clustering coeffi-cient; (iii) Garlaschelli’s symmetry index; and (iv) size ofthe largest connected component in the network. Morespecifically, we consider the averages and standard devia-tions of the measurements taken over the whole networks.Each of these measurements, as well as the motivation fortheir adoption, are described in detail as follows: Average Node In- and Out-degree:
The in- and out-degree of an individual node i corresponds to the numberof edges leading to and from that node. These measure-ments can be obtained directly from the adjacency matrixas: a ini = N X j =1 a ji (1) a outi = N X j =1 a ij (2)The average in- and out-degrees, (cid:10) k ini (cid:11) and h k outi i ,taken over the whole network, are identical and providea quantification of the overall degree of connectivity inthe network. Average Clustering Coefficient:
The clustering coeffi-cient of a node i measures how interconnected the neigh-bors of node i are. It does so by counting the numberof connections between neighbors of i and dividing it bythe number of possible connections - in other words, bythe number of pairs of neighbors. In directed networksthis can be written as: c i = nd i ( d i −
1) (3)where d i = P j ( a ij + a ji ) is the number of neighborsof node i and n = P j,k [1 − (1 − a ij )(1 − a ji )][1 − (1 − a ik )(1 − a ki )]( a jk + a kj ) is the number of directed edgesbetween those nodes. Note that 0 ≤ c i ≤
1, with c i = 0indicating a total lack of interconnectivity between theneighbors and c i = 1, i.e. all neighbors are connected toeach other. Another version of the clustering coefficientfor directed networks, which takes the variety of directedtopological environments into account, has been reportedin [14]. Garlaschelli’s Symmetry Index:
Introduced by Gar-laschelli [9], this measurement quantifies the symmetry ofconnections between pairs of nodes relative to an Erd¨os-R´enyi network of the same size and density. Let E be thetotal number of directed edges in the network, and E b bethe total number of bidirectional (symmetric) edges, anddefine the ratios r = E b /E and a = E/ ( N − N ). Thesymmetry index is then given by: ρ = r − a − a (4) Size of the Largest Connected Component:
At anytime during the growth of a network, it is of interestto quantify the overall path connectivity among the ex-isting nodes. This can be done by measuring the size C of the largest connected component in the network.More specifically, a connected component [22] is a sub-graph such that each node can be reached from any othernode through at least one path.Because the dynamics of complex neuronal networks isdefined by the concentric organization (e.g. [12, 15]) oftheir topology, we also consider three hierarchical mea-surements, namely the hierarchical number of nodes, thehierarchical degree and the intra-ring degree. Given anundirected network, the concentric level h of a node i isdefined as the set of nodes which are separated from i by a shortest path of length h . The maximum value of h is H . The hierarchical number of nodes at level h , withrespect to a reference i , is equal to the number of nodesat that level. The hierarchical degree of a node i at level h corresponds to the number of edges between the levels h and h + 1. The intra-ring degree of a node i at level h is the number of edges between nodes of that level. B. Morphological Networks
Morphological networks are networks formed by aspatial distribution of individual geometrical compo-nents [16] with connections between them defined bytheir overlap. For instance, a biological neuronal net-work is composed of neurons with a specific geometrygiven by their dendritic and axonal arborizations. Mor-phological networks can be classified into subcategorieswith respect to: (i) the individual shape of the basic ele-ments, (ii) the degree of homogeneity of such elements inthe network (e.g. networks containing a single or multi-ple type of individual shape), and (iii) the way in whichthese elements are spatially distributed. Each of thesecases is discussed below in detail:
Individual Neurons:
The individual shape of the net-work elements can closely mirror real elements (e.g. byusing images of neurons) or they can take a more abstractform (e.g. by using some pattern generation method).In the case of neural morphological networks, the shapeof the neurons is composed of dendrites and axons. Asneuronal connections (i.e. synapses) extend from axonto dendrite, such networks are directed. The shape ofthe elements can be represented in a discrete form, suchas digital images or a continuous form, e.g. by (possiblypiecewise) continuous curves. The geometrical propertiesof the basic elements in the network can be quantified byusing a series of measurements such as area, perimeter,fractal dimension, symmetry, etc.
Degree of Homogeneity:
Morphological networks mayinvolve single or multiple basic elements, possibly involv-ing geometrical transformations (e.g. rotation or scal-ing).
Spatial Distribution:
Morphological networks are cre-ated by distributing the element shapes in space and con-sidering their overlap. Several types of distributions canbe considered, e.g. uniformly random or preferential tospecific positions (e.g. a normal distribution centered ata given point).In order to keep the number of involved parameters anddegrees of freedom as small as possible, allowing statisti-cally representative simulations, we choose the followingrepresentation of individual neurons: (a) the dendritesare represented by a star of spokes with uniform anglebetween them; (b) the total length of the dendrites isfixed and therefore independent of the number of spokes;(c) the axon, which corresponds to the soma, is a circleof fixed radius at the center of the star. Identical neu-rons, rotated randomly, are used to build each network.The neurons are placed sequentially at uniformly randompositions within the unit square.Figure 1 illustrates a simple morphological network ob-tained for N = 20 and n = 3. Each neuron correspondsto a circle (the soma and axon) from which n = 3 spokesemanate. The neurons, which are placed with varying ro-tations, are identified by their respective central points.The network obtained is shown in black, superimposedinto the original morphological structure. Observe thatneuron number 4 remains isolated. All other 19 cells be-long the largest strongly connected component. -0.2 0 0.2 0.4 0.6 0.8 1 1.2-0.2 0 0.2 0.4 0.6 0.8 1 1.2-0.2 0 0.2 0.4 0.6 0.8 1 1.2-0.2 0 0.2 0.4 0.6 0.8 1 1.21 2 345 6 789 101112 131415 1617 181920 FIG. 1: An example of morphological network obtained byconsidering N = 20 and n = 3. A directed connection isestablished whenever a dendrite crosses the axon of anotherneuron. C. Integrate-and-Fire Complex Neuronal Networks
In this work, the dynamics of the morphological net-works is simulated by considering the integrate-and-fire neuronal model (e.g. [10]), which incorporates themost important elements characterizing real neuronalnetworks, namely integration of the activation receivedby each cell along time, related to the biological phe-nomenon of facilitation [8], and a non-linear transferfunction. Figure 2 illustrates the main components of the integrate-and-fire neuronal cell. The incoming ax-ons are connected to the dendrites of the neuronal cell i ,their signals being added by the integrator Σ at each timestep and stored in the internal state S ( i ). In this work weassume equal synaptic weights. Once the value of S ( i )reaches a given threshold T ( i ), the cell produces a spikeand its internal state S ( i ) is cleared. The activation con-veyed by the axons is fixed at intensity 1, reflecting theconstant amplitude of the action potential characteristicof real neuronal networks. FIG. 2: The integrate-and-fire model of neuronal cell dynam-ics adopted in this work incorporates v ( i ) incoming connec-tions with respective weights, u ( i ) outgoing links, an integra-tor Σ, the internal memory S ( i ) storing the time-integratedactivation, and the non-linear transfer function (a hard limitis adopted in this work). Every time the internal activation S ( i ) exceeds the threshold T ( i ), the neuron fires, generating aspike of fixed intensity 1 and clearing the internal activation. Though other types of activations can be considered,in the present work we activate the networks through a source node . More specifically, one of the network nodesis chosen and injected with constant activation equal to1. All neurons have the same threshold T ( i ) = T , andall synaptic weights are fixed at 1.The networks studied here are of sizes N = 200 and400, with a varying number of dendrites (i.e. spokes).The radius of the soma was 0.0625 and the total lengthof the dendrites was 0.5. Only the largest connected com-ponents are considered for the analysis of the integrate-and-fire dynamics.The overall integrate-and-fire dynamics unfolding in anetwork can be characterized in several ways (e.g. [10, 12,17]). Here, we consider the internal activation and spikesproduced by each neuron over time. More specifically,we will focus attention on the transient phenomenon of avalanches of spikes (e.g. [12, 17, 18]) which may takeplace during the initial activation of the network. Be-cause we are considering the largest connected compo-nent in each network, the activation arriving at the sourceeventually reaches all the neurons in the networks. Inter-estingly, such an activation may progress either graduallyor involve an abrupt onset of spikes, i.e. an avalanche.The intensity and sharpness of these avalanches are de-fined by the concentric properties of the topology of therespective networks [12, 17]. After an avalanche, the to-tal number of spikes in the network tends to exhibit well-defined oscillations [17].The type of activation transition can be predicted [19]by considering the three following ratios derived from thehierarchical number of nodes n h ( i ), hierarchical degrees k h ( i ), and intra-ring degrees a h ( i ) s h ( i ) = k h ( i ) n h +1 ( i ) 1 T (5) s h ( i ) = a h ( i ) n h ( i ) 1 T (6) s h ( i ) = k h ( i − n h − ( i ) 1 T (7)(8)The quantity s h ( i ), called forward activation ratio ,quantifies the intensity of the transfer of activation fromthe neurons in level h into level h + 1 after most of theneurons in level h have fired (because of degree regular-ity, the nodes within each concentric level tend to fire atsimilar times). The reflexive activation ratio s h ( i ) ex-presses how much of the activation of the neurons at level h remains at that same level. The transfer of activationfrom level h to level h − backward activation ratio s h ( i ). All ratios are normal-ized with respect to the threshold T of every cell so thatratio values of 1 indicate a full transfer of the receivedactivation.It is important to note that the propagation of theavalanches is influenced by the finite size and local topol-ogy of the networks. Consider a perfectly regular butfinite orthogonal lattice. For any given node, the con-centric layers of neighbours surrounding it will be char-acterized by levels with progressively increasing numberof nodes, up to a point where this number starts to de-cline again as a consequence of the finite size of the net-work. This maximum (see [13] for the complete analyt-ical results), plays an important role in the formationof avalanches. Several other types of complex networkshave been shown to undergo avalanches under specificparameter configurations [12, 13]. It is also possible todesign networks which will never produce avalanches, e.g.by having more uniform distributions of nodes along theconcentric levels. Real-world neuronal systems are oftenorganized into modules with sometimes only a few dozensof neurons, which is in line with the size of networks inour models. III. CHARACTERIZATION OF THETOPOLOGY
Figure 3 shows the average out-degree h k outi i (a) , theaverage clustering coefficient h c i i (b), the symmetry co-efficient ρ (c) as well as the size of the largest connectedcomponent divided by the number of neurons (d) in termsof the time t (which is equal to the number of placed neurons) for 50 morphological networks with n = 1 to 20spokes (identified by different colors).It is clear from Figure 3(a) that the average in-degreeincreases roughly quadratically as additional neurons areincorporated into the network. In addition, the averagein-degree also increases with the number of spokes up to n = 5, decreasing thereafter. As shown in Figure 3(b),the average clustering coefficient tends to increase lin-early with the number of added neurons t . It also in-creases with the number of spokes n up to n = 10, beforeleveling off at higher values of n . The symmetry index,shown in Figure 3(c), remains nearly constant with t butincreases steadily with the number of spokes n . This re-flects the fact that for large numbers - and thus a highangular density - of spokes, there is a high probabilitythat if the axon of neuron A is reached by the dendriteof neuron B, the opposite will also be true. The size ofthe largest connected component C always converges to1, with faster convergence being observed for larger val-ues of n . Except for n = 1, most of the neurons will bepart of the largest connected component for t > N = 200), known to play an importantrole in the integrate-and-fire dynamics, is illustrated inFigure 4 with respect to the average standard deviationof the hierarchical number of nodes n h ( i ), hierarchicaldegrees k h ( i ), and intra-ring degrees a h ( i ).Figure 3 makes it clear that all three measurementsbehave similarly, exhibiting a peak near the middle con-centric level. In addition, larger number of spokes tendto increase the total number H of concentric levels in thenetworks, which goes from about 6 for n = 1 to 13 for n = 20. Similar results were obtained for N = 400.Figure 5 shows the mean values of the three consid-ered concentric measurements for n = 1 , , ,
10 and 20.It is clear from this figure that the increase of the num-ber of spokes implies a larger number of concentric levelsas well as smoother distributions of measurements alongthe h − axis. Furthermore it is evident that the disper-sions of each configuration decrease with n . Because theintegrate-and-fire dynamics is defined by the concentricorganization of the network, the distribution of the con-figurations shown in Figure 5 implies that the adoptednetworks are poised to yield distinct types of avalanchedynamics.In order to obtain additional insights about the latterissue, namely the type of activation transitions duringthe transient regime, we calculated also the three acti-vation ratios (Eqs. 5– 6) for each of the configurationsof morphological networks. Table I shows the mean andstandard deviation values of the three activation ratiosobtained for the two concentric levels associated to thelargest number of original nodes assuming T = 7. Itshould be recalled that these three ratios are defined forundirected networks, so that the morphological struc-tures were symmetrized before the respective calcula-tions. It has been verified that the ratios obtained for thesymmetrized versions of a directed network still capture FIG. 3: The average out-degree h ok i (a), average clustering coefficient h cc i (b), symmetry coefficient ρ (c) as well as the size ofthe largest connected component divided by the number of neurons (d) in terms of the time obtained along the growth of 50morphological networks with n = 1 to 20 spokes ( n = 1 in black, n = 3 in blue, n = 5 in green, n = 10 in cyan, n = 20 in red). to a good deal the integrate-and-fire important dynami-cal features.Most networks involve at least one ratio with valuessmaller or near 1, suggesting that the activation in thesenetworks proceed in a relatively gradual manner. Indeed,the activations tend to be more gradual for the configu-rations with larger number of spokes. IV. CHARACTERIZATION OF THEDYNAMICS
In this section we investigate the transient activationdynamics of the morphological networks with respect toseveral configurations. Figure 6 illustrates the spikegram(a) and total number of spikes (b) obtained for a networkwith N = 200, n = 3 and T = 3.The spikegram in Figure 6(a) shows the occurrenceof a spike (shown in white) over time (vertical axis) foreach of the N = 200 neurons in the network (horizontalaxis). As the network is fed from node 1, a few neurons(the most immediate neighbors of node 1) start sporadicfiring, up to nearly t = 50, when most of the neuronsbegin producing spikes, signaling the occurrence of anavalanche. Figure 6(b) shows the total number of spikesalong time. In the case of this specific example, this sig-nal involves three preliminary peaks of spiking, followed by the avalanche. The total number of spikes tends toundergo regular oscillations after the avalanche.As predicted by the respective three activation ratioscalculated for each of the networks configuration (see pre-vious Section), gradual activations and almost completelack of avalanches were observed for T = 7, as illustratedin Figure 7. In other words, the configurations of mor-phological networks considered in the present work in-corporate concentric organizations which imply the tran-sient activation to be distributed in a relatively gradualmanner.So far we have considered identical model neuronswith uniformly distributed spokes of equal length. How-ever, real-world biological neuronal systems are charac-terized by the coexistence of short and long range connec-tions. The latter interconnect diverse cortical modulesand regions [8, 20]. In order to investigate the effect oflong-range connections in our morphological networks,we added a number of random edges. Figure 4 showsthe three main hierarchical measurements obtained forthe same network used to produce Figure 6, but with20% E new randomly chosen directed edges. It is evi-dent from this figure that the incorporation of additionaledges completely changes the concentric organization ofmorphological networks, by reducing the total number ofconcentric levels H (compare with Fig. 4) as well as im-plying sharper peaks in the hierarchical number of nodes FIG. 4: The hierarchical number of nodes n h ( i ), the hierarchical degrees k h ( i ) and the intra-ring degrees a h ( i ) obtained for themorphological networks with N = 200 and n = 1 , , ,
10 and 20 spokes.(a) (b) (c)FIG. 5: Superposition of the mean values of the three considered concentric measurements for n = 1 , , ,
10 and 20. and intra-ring degrees signatures.Table II gives the mean and standard deviation valuesof the three activation ratios obtained for the 50 morpho-logical networks with 20% E additional edges and T = 7respectively to the two concentric levels containing thelargest number of nodes. It should be observed that, as aconsequence of the smaller diameter of the networks with20% E additional edges, the level with the second highestnumber of nodes contains only a few nodes. Relativelyto Table I, we can see that the addition of edges impliedin substantially larger values for all the three ratios, pre-dicting sharper and more intense avalanches (confirmedexperimentally). Observe that both Tables I and II referto T = 7. Such an major change in the activation ra-tios is a direct consequence of the drastic influence of theadditional edges over the concentric organization of the respective morphological networks. Even more intensechanges have been obtained for larger number of addededges.The above results imply that morphological networksinvolving short and long-range connections are substan-tially more likely to exhibit avalanches of activation thanmorphological networks involving only short-range con-nections. Such a finding presents important potentialimplications for neuroscience, especially because it showsthat the activation of morphological networks, includingtheir susceptibility to avalanches, can be controlled bythe density of short and long-range links. n s first ) s second ) s first ) s second ) s first ) s second ) n = 1 2.27 ± ± ± ± ± ± n = 3 2.46 ± ± ± ± ± ± n = 5 2.16 ± ± ± ± ± ± n = 10 1.71 ± ± ± ± ± ± n = 20 1.30 ± ± ± ± ± ± s s s n = 1 , , ,
10 and 20. T = 7. n s first ) s second ) s first ) s second ) s first ) s second ) n = 1 3.53 ± ± ± ± ± ± n = 3 5.45 ± ± ± ± ± ± n = 5 4.60 ± ± ± ± ± ± n = 10 2.82 ± ± ± ± ± ± n = 20 2.55 ± ± ± ± ± ± s s s E additional edges and n = 1 , , ,
10 and 20. T = 7.FIG. 6: The spikegram (a) and total number of spikes alongtime (b) obtained for a morphological network with N = 200, n = 3 and T = 3. V. RELATING STRUCTURE AND DYNAMICS
Having investigated the integrate-and-fire dynamicsfor several configurations of morphological networks, ourattention is now focused on the particularly relevant is-sue of relating the structural and dynamical aspects ofthe considered networks.Three measurements of each avalanche were extractedautomatically: (i) its onset time t i ; (ii) the mean of thetotal number of spikes after the avalanche takes place h N s i , and (iii) the respective standard deviation σ N s .The avalanches had to be detected before such measure- FIG. 7: The spikegram (a) and total number of spikes alongtime (b) obtained for a morphological network with N = 200, n = 3 and T = 7. ments could be calculated. This was achieved by thresh-olding the total number of spikes with signals at one fifthof the maximum height. A total of 500 time steps wasconsidered.Figure 9 shows the scatterplots of h N s i against s s s s s s N = 200, and T = 3 for all values of n . Thisfigure shows that the mean intensity of the avalanchestends to increase steeply with the activation ratios andthen saturate at a value of about 65. This results shows FIG. 8: The hierarchical number of nodes n h ( i ), the hierarchical degrees k h ( i ) and the intra-ring degrees a h ( i ) obtained for themorphological networks with N = 200 and n = 1 , , ,
10 and 20 spokes and 20% additional edges. that important features of the integrate-and-fire dynam-ics, such as the average intensity of the avalanches, are in-trinsically related to the respective structural propertiesof the network, here represented by the three activationratios, which are ultimately derived from the concentricorganization. No relationships have been identified be-tween the activation ratios and the other two avalanchemeasurements.
VI. CONCLUDING REMARKS
The intersection between complex networks and neu-ronal networks, which has been called complex neuronalnetworks , represents one of the most challenging andpromising research areas because it allows neuroscienceto be revisited while considering the relationship betweenstructured connectivities such as small-world and scale-free and the respectively obtained dynamics. The currentarticle has addressed this important paradigm with re-spect to more biologically realistic networks, called mor-phological networks , whose connectivity is defined as aconsequence of the shape and distribution of geometricalneuronal cells. By assuming a uniformly random spatial distribution of geometrically simple neurons consisting ofa circular axon and a set of straight dendrites radiatingfrom the soma with uniform angles, it was possible tokeep the number of involved parameters as small as pos-sible, allowing the systematic investigation of the effectof the shape of the neurons on the overall topologicalproperties of the resulting networks as well as on the re-spective integrate-and-fire dynamics. Several interestingresults were obtained:
Characterization of the Topology of Morphological Net-works in Terms of Traditional Measurements:
The topol-ogy of the considered morphological networks was char-acterized in terms of several measurements, includingaverage out-degree, clustering coefficient, symmetry in-dex, and size of the largest connected component. Wefound that networks obtained by considering neuronswith larger number of dendrites tended to exhibit greatersymmetry and larger connected components. The aver-age out-degree and clustering coefficient tended to in-crease with the number of dendrites up to a maximum,decreasing thereafter.
Characterization of the Topology of Morphological Net-works in Terms of Hierarchical Measurements:
The con-centric organization of the morphological networks, as re-0
FIG. 9: The scatterplots of the average total number of spikes along time ( N = 200 and T = 3) and respective activationratios. vealed by the hierarchical number of nodes, hierarchicaldegree and intra-ring degree, exhibited several concentriclevels, yielding respective signatures characterized by apeak nearly the intermediate level. The gradual distri-bution of these measurements along the concentric levelsuggests more gradual activation of the respective net-works. Prediction of the Type of Activation of the Networks:
The concentric organization of a network has been foundto define important features of the respective integrate-and-fire dynamics [12]. We considered the three activa-tion indices introduced in [19] in order to predict the typeof activation of the network. The higher the value of suchratios — which are obtained from the hierarchical num-ber of nodes, hierarchical degree and intra-ring degree,the higher the probability of getting abrupt activation ofthe network and onset of avalanches. Because the ac-tivation ratios for the concentric levels with the largestnumber of nodes are particularly small, the morphologi-cal networks considered in this work are expected to un-dergo relatively smooth dissemination of the activationreceived from a source node. Such a type of dynamicswas experimentally confirmed.
Investigation of Morphological Networks InvolvingShort- and Long-Range Connections:
Because biologi-cal neuronal networks are known to incorporate short-and long-range connections, we repeated our investiga-tions of structure and dynamics considering morpholog-ical networks involving also long-range dynamics. Itwas found that the incorporation of additional connec-tions can change drastically the concentric organizationof morphological networks, implying substantial changesalso in the respective integrate-and-fire dynamics. In par-ticular, it was found that the addition of 20% random di-rected edges reduced substantially the number of concen-tric levels while increasing the activation ratios. The so-obtained morphological networks were found to undergosharp activation, with onset of avalanches. Such a find-ing implies that the way in which biological networks areactivated can be controlled by the ratio between short-and long-range connections, with morphological networksinvolving only short-range connections undergoing moregradual activation.
Characterization of the Activation Dynamics in Termsof the Average Intensity of Avalanches and Structure-Dynamics Relationships:
A specific methodology was de-1veloped in order to automatically identify the presenceof avalanches from the total number of spikes signaturesin terms of time. This also allowed the identification ofthe initiation time and average and standard deviationof the total number of spikes after the occurrence of theavalanches. In order to relate this dynamical behaviorto the topology of the networks we investigated possi-ble relationships between the activation ratios (derivedfrom the concentric organization) and initiation time ofavalanches, and the average and standard deviation ofthe total number of spikes after their respective onset.It was verified that the average of the total number ofspikes after the avalanches tend to increase steeply withany of the three activation ratios, saturating at a maxi-mum value. No relationships were observed between theratios and the other two avalanche features.Possible future directions of research include the con-sideration of neurons with non-uniformly distributedspokes, neurons with higher branching orders, as well asdifferent types of neurons in the same network and non-uniformly distributed neurons. It would also be interest-ing to consider integrate-and-fire dynamics with internalactivation decay [21].
List of Symbols
Γ = a graph or complex network; N = total number of nodes in a network; E = total number of edges in a network; K = the adjacency matrix of a complex network; k ( i ) = degree of a network node i ; ik ( i ) = in-degree of a network node i ; ok ( i ) = out-degree of a network node i ; cc ( i ) = clustering coefficient of a network node i ; C = the size of the largest strongly connected componentin the network; L = the size of the workspace used to build the morpho-logical networks (i.e. L × L ); n ( i ) = the number of spokes (dendrites) of a neuron i ; r ( i ) = the radius of the soma (also axon); S ( i ) = the current internal activation of neuron i ; T ( i ) = the threshold of neuron i ; u ( i ) = the number of incoming connections at neuron i ; v ( i ) = the number of outgoing connections of neuron i ; H = the total number of concentric levels in a network; n h ( i ) = the number of hierarchical nodes at level h ofneuron i ; k h ( i ) = the hierarchical degree at level h of neuron i ; a h ( i ) = the intra-ring degree at level h of neuron i ; s h ( i ) = the forward activation ratio at level h of neuron i ; s h ( i ) = the reflexive activation ratio at level h of neuron i ; s h ( i ) = the backward activation ratio at level h of neu-ron i . Acknowledgments
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