Network algorithmics and the emergence of the cortical synaptic-weight distribution
aa r X i v : . [ q - b i o . N C ] J un Network Algorithmics and the Emergence of theCortical Synaptic-Weight Distribution
Andre NathanValmir C. Barbosa ∗ Universidade Federal do Rio de JaneiroPrograma de Engenharia de Sistemas e Computa¸c˜ao, COPPECaixa Postal 6851121941-972 Rio de Janeiro - RJ, Brazil
Abstract
When a neuron fires and the resulting action potential travels down itsaxon toward other neurons’ dendrites, the effect on each of those neuronsis mediated by the weight of the synapse that separates it from the firingneuron. This weight, in turn, is affected by the postsynaptic neuron’sresponse through a mechanism that is thought to underlie important pro-cesses such as learning and memory. Although of difficult quantification,cortical synaptic weights have been found to obey a long-tailed unimodaldistribution peaking near the lowest values, thus confirming some of thepredictive models built previously. These models are all causally local, inthe sense that they refer to the situation in which a number of neurons allfire directly at the same postsynaptic neuron. Consequently, they neces-sarily embody assumptions regarding the generation of action potentialsby the presynaptic neurons that have little biological interpretability. Inthis letter we introduce a network model of large groups of interconnectedneurons and demonstrate, making none of the assumptions that charac-terize the causally local models, that its long-term behavior gives rise toa distribution of synaptic weights with the same properties that were ex-perimentally observed. In our model the action potentials that create aneuron’s input are, ultimately, the product of network-wide causal chainsrelating what happens at a neuron to the firings of others. Our modelis then of a causally global nature and predicates the emergence of thesynaptic-weight distribution on network structure and function. As such,it has the potential to become instrumental also in the study of otheremergent cortical phenomena.
Keywords:
Cerebral cortex, Synaptic weights, Synaptic-weight distribu-tion, Distributed algorithms, Complex networks. ∗ Corresponding author ([email protected]). D whose nodes correspond to neuronsthat can be either excitatory or inhibitory. For i and j two distinct nodes suchthat at least one of them is excitatory, an edge directed from i to j represents asynapse with associated weight w ij . No edge exists between two inhibitory nodes[2]. The algorithmic component turns each node in D into a simple simulator ofthe corresponding neuron, employing message passing on the edges along theirdirections to simulate the signaling through the corresponding synapses whennodes fire. Collectively, the nodes behave as an asynchronous distributed algo-rithm [6], here referred to as A, each executing a simple procedure P wheneverreceiving a message, possibly sending messages itself while executing P but re-maining idle at all other times. Because nodes only do any processing in thisreactive manner, at least one node is needed that initially executes P once with-out any incoming message to respond to and then starts behaving reactivelylike the others. We call such a node an initiator.At node j , let v j stand for the node’s potential. Let also v and v t bea node’s rest potential and threshold potential, respectively, the same for allnodes. The effect of running P is for j to probabilistically decide whether to2re and, if it does fire, to send messages on all outgoing edges while setting v j to v . If P is run as the initial processing by an initiator, then the firing occurswith probability 1 and P involves no actions other than the ones just described.If not, then let i be the sender of the triggering message. The firing occurswith probability min { , ( v j − v ) / ( v t − v ) } after v j has been updated to either v j + w ij (if i is excitatory) or v j − w ij (if i is inhibitory). Then the weight w ij is considered for an update.The updating of w ij seeks to mimic the commonly accepted generalization ofthe Hebbian rule embodied in the spike-timing-dependent plasticity principles[1, 23], according to which the change incurred by a synapse’s weight depends onthe extent to which there is a causal dependency of what happens at a neuronupon the other’s firing. As a general rule, the synaptic weight is increased(potentiated) if the postsynaptic neuron fires in response to the firing by thepresynaptic neuron, decreased (depressed) otherwise. In either case the amountof change to the synaptic weight depends on how close in time the relevantfirings are, becoming negligible with increasing separation. Procedure P followsthese principles by keeping track of the latest firing by j so that a decision canbe made on whether to increase or decrease w ij . If j does fire in response to themessage received from i , then w ij is increased. If it does not but the previousmessage received from any source did cause j to fire, then w ij is decreased.The weight w ij remains unchanged in all other cases. The actual amount ofchange to w ij depends on whether it is to be increased or decreased, and sodoes the nature of the change (by a fixed amount or by proportion) [9, 10, 18].An increase in w ij is implemented by setting w ij to min { , w ij + δ } with δ > w ij to (1 − α ) w ij with 0 < α <
1, thus ensuring thatsynaptic weights remain in the [0 ,
1] interval if so started.Running algorithm A starts with choosing one or more initiators, each ofwhich executes P and then starts behaving like all other nodes. At any time itmay happen that a node has more than one input message to process, in whichcase the order in which they are taken is the order of message reception. Be-cause this order is in principle arbitrary, A is seen to acquire another degree ofindeterminacy, in addition to that which is already present owing to the proba-bilistic decisions. We have conducted extensive computational experimentationwith A on a graph D intended to model a simple cortex, in line with significantrecent work that draws on the theory of graphs to help solve problems in neu-roscience [25, 27, 3, 8, 14, 15, 21, 26, 28]. We regard D as a random graph but,unlike some of the early work on cortical modeling by such graphs [2], wherefully random graphs [13] were used, we let D have a scale-free structure [19],with parameter as suggested by some of the more recent finds [12, 29]. Thus,a randomly chosen node i in D has k outgoing edges with probability propor-tional to k − . . Moreover, inspired by recent work on the modeling of corticalsystems [17, 16], we let each outgoing edge of i lead to another randomly chosennode j with probability proportional to e − d , where d is the Euclidean distancebetween i and j when the nodes of D are placed uniformly at random on aradius-1 sphere (Figure 1), provided i and j are not both inhibitory.3igure 1: Network topology. Restricted to two dimensions for visual clarity, a D instance comprises nodes positioned randomly on a radius-1 circle and edges,drawn as chords of the circle, that tend to be more abundant over lower Eu-clidean distances. Excitatory nodes are represented by filled circles, inhibitorynodes by empty circles.All computational experiments have adhered to the methods described next,which refer to sequences of 10 000 runs of algorithm A. The first run in a sequenceoperates on initial node potentials and synaptic weights chosen randomly fromthe intervals [ v , v t ] and [0 , v = −
15 and v t = 0. Eachsubsequent run operates on the potentials and weights left by the previous run.For n the number of nodes in D , a new set of 0 . n initiators is chosen randomlyat the beginning of each run. A run of A is implemented as a sequential programthat selects the next node to be processed randomly (first out of the group ofinitiators for their first executions of P, then out of those nodes that have atleast one message to be received). A new run in a sequence is only startedafter the previous one has died out (no more messages to be processed remain),which is guaranteed to happen eventually with probability 1. The remainingparameters used by procedure P are δ = 0 .
01 and α = 0 .
05. All our results referto 50 000 independent sequences, of which each 500 sequences correspond to anew D instance. A D instance is constructed by first placing all nodes uniformlyat random on a radius-1 sphere, then selecting the number of outgoing edgesfor each node. Nodes are then chosen to be excitatory or inhibitory randomly,provided a certain proportion is respected, and the destination of each edgeis decided. The graph that is actually used in the run sequences is the giantstrongly connected component of D [11], so a directed path exists from any node4o any other. For the connectivity distribution and construction method in usethis component comprises about 0 . n nodes on average.Our results, here given for n = 1 000 and the well accepted proportion of0 . n inhibitory nodes [2, 4], show that the synaptic-weight distribution becomesanalogous to the distribution unveiled by experimentation along the sequences ofruns of algorithm A described above (Figure 2). The process is gradual, leadingthe weights to become relatively concentrated around a single low-value modewhile still allowing some residual probability to remain at the higher values.The long-term distribution is seen to stabilize even as the weights continue toevolve, thus suggesting the existence of an underlying weight dynamics whoseeffect on the overall distribution is nevertheless practically imperceptible. Theexistence of this persistent dynamics is revealed by the causal history of eachterminal message reception (one that does not lead to the firing of the receiver),which can be significantly deep with respect to the relatively short average pathof a scale-free network [20] [Figure 3(a)]. The sending of every message by anon-initiator causes a synaptic weight to be increased, unless it already equals1, but weight-1 synapses are very rare, especially when arranged as a path in D . So the causal histories we have discovered do indeed hint at the existenceof a dynamics of weight evolution in which weights both increase and decreasein complex patterns. Additional confirmation is provided by the average weightof the synapses involved in the causal histories of terminal message receptions,which is consistently less than 1 and also decreases throughout the runs as thesynaptic-weight distribution settles [Figure 3(b)].Every run in the sequences to which Figures 2 and 3 refer involves a newgroup of initiators and as such provides new possibilities regarding the branchingof causal histories and how they affect firings and weight changes throughoutthe network. Monitoring the traffic of messages as they traverse edges andreach nodes is then a means to do some quantification of how the cascadingruns, with their intermingling causal trees rooted at many different initiators,cooperate in promoting the emergence of the synaptic-weight distribution. Wehave found that the long-term distributions of how many runs traverse an edgeor reach a node (Figure 4), allowing as they do for relatively high numberswith significant probabilities, suggest that some sort of information integrationis taking place among portions of the network as the runs unfold. Perhaps suchintegration occurs in a sense similar to that which has been theorized recentlyregarding the emergence of higher functions such as consciousness [5]. If so,then network algorithmics such as we have discussed may come to provide apowerful framework to test the assumptions and eventual predictions of suchtheories. Acknowledgments
We acknowledge partial support from CNPq, CAPES, and a FAPERJ BBPgrant. 5 .0 0.2 0.4 0.6 0.8 1.00.000.020.040.060.08 Initial 0.0 0.2 0.4 0.6 0.8 1.00.000.020.040.060.08 Run 10000.0 0.2 0.4 0.6 0.8 1.00.000.020.040.060.08 Run 3000 0.0 0.2 0.4 0.6 0.8 1.00.000.020.040.060.08 Run 50000.0 0.2 0.4 0.6 0.8 1.00.000.020.040.060.08 Run 9000 0.0 0.2 0.4 0.6 0.8 1.00.000.020.040.060.08 Run 10000 P r obab ili t y Synaptic weight
Figure 2: The synaptic-weight distribution, shown after selected runs of algo-rithm A. Probabilities are binned to a fixed width of 0 . C au s a l dep t h (a) 0 2000 6000 100000.100.150.200.250.30 Run S y nap t i c w e i gh t (b) Figure 3: Causal depth of a message reception and associated synaptic weights.The causal depth of a message reception is the size of its causal history, i.e.,the number of firings that precede it along the chain of firings that begins atsome initiator when it fires for the first time, each preceding the next by directcausation: given any two subsequent firings in this chain, the first entails thesending of a message whose reception triggers the second. (a) Maximum andaverage causal depth of terminal message receptions during the course of eachrun. (b) Average weight (before updates) of the synapses involved in the causalhistories of terminal message receptions.7
00 1500 2500 35000.000.010.020.030.040.05 Edges2000 4000 6000 80000.000.010.020.03 Number of runs Nodes P r obab ili t y Figure 4: Final distributions of the number of runs in which an edge is traversedor a node is reached. An edge is said to be traversed in a run when at leastone message is sent along it during the course of that run. A node is said to bereached in a run when it receives at least one message during the course of thatrun. Probabilities are binned to a fixed width of 50 for edges, 100 for nodes.8 eferences [1] L. F. Abbott and S. B. Nelson. Synaptic plasticity: taming the beast.
Nature Neuroscience , 3:1178–1183, 2000.[2] M. Abeles.
Corticonics: Neural Circuits of the Cerebral Cortex . CambridgeUniversity Press, Cambridge, UK, 1991.[3] S. Achard, R. Salvador, B. Whitcher, J. Suckling, and E. Bullmore. Aresilient, low-frequency, small-world human brain functional network withhighly connected association cortical hubs.
Journal of Neuroscience , 26:63–72, 2006.[4] R. Ananthanarayanan and D. S. Modha. Anatomy of a cortical simula-tor. In
Proceedings of the 2007 ACM/IEEE Conference on Supercomputing ,page 3, 2007.[5] D. Balduzzi and G. Tononi. Integrated information in discrete dynami-cal systems: motivation and theoretical framework.
PLoS ComputationalBiology , 4:e1000091, 2008.[6] V. C. Barbosa.
An Introduction to Distributed Algorithms . The MIT Press,Cambridge, MA, 1996.[7] B. Barbour, N. Brunel, V. Hakim, and J. P. Nadal. What can we learnfrom synaptic weight distributions?
Trends in Neurosciences , 30:622–629,2007.[8] D. S. Bassett and E. Bullmore. Small-world brain networks.
Neuroscientist ,12:512–523, 2006.[9] G. Q. Bi and M. M. Poo. Synaptic modifications in cultured hippocampalneurons: dependence on spike timing, synaptic strength, and postsynapticcell type.
Journal of Neuroscience , 19:10464–10472, 1998.[10] G. Q. Bi and M. M. Poo. Synaptic modification by correlated activity:Hebb’s postulate revisited.
Annual Review of Neuroscience , 24:139–166,2001.[11] S. N. Dorogovtsev, J. F. F. Mendes, and A. N. Samukhin. Giant stronglyconnected component of directed networks.
Physical Review E , 64:025101,2001.[12] V. M. Egu´ıluz, D. R. Chialvo, G. A. Cecchi, M. Baliki, and A. V. Apkarian.Scale-free brain functional networks.
Physical Review Letters , 94:018102,2005.[13] P. Erd˝os and A. R´enyi. On random graphs.
Publicationes Mathematicae—Debrecen , 6:290–297, 1959. 914] Y. He, Z. J. Chen, and A. C. Evans. Small-world anatomical networks inthe human brain revealed by cortical thickness from MRI.
Cerebral Cortex ,17:2407–2419, 2007.[15] C. J. Honey, R. K¨otter, M. Breakspear, and O. Sporns. Network structureof cerebral cortex shapes functional connectivity on multiple time scales.
Proceedings of the National Academy of Sciences USA , 104:10240–10245,2007.[16] M. Kaiser and C. C. Hilgetag. Modelling the development of cortical sys-tems networks.
Neurocomputing , 58–60:297–302, 2004.[17] M. Kaiser and C. C. Hilgetag. Spatial growth of real-world networks.
Phys-ical Review E , 69:036103, 2004.[18] A. Kepecs and M. C. W. van Rossum. Spike-timing-dependent plasticity:common themes and divergent vistas.
Biological Cybernetics , 87:446–458,2002.[19] M. E. J. Newman. Power laws, Pareto distributions and Zipf’s law.
Con-temporary Physics , 46:323–351, 2005.[20] M. E. J. Newman, S. H. Strogatz, and D. J. Watts. Random graphs witharbitrary degree distributions and their applications.
Physical Review E ,64:026118, 2001.[21] J. C. Reijneveld, S. C. Ponten, H. W. Berendse, and C. J. Stam. Theapplication of graph theoretical analysis to complex networks in the brain.
Clinical Neurophysiology , 118:2317–2331, 2007.[22] J. Rubin, D. D. Lee, and H. Sompolinsky. Equilibrium properties of tempo-rally asymmetric Hebbian plasticity.
Physical Review Letters , 86:364–367,2001.[23] S. Song, K. D. Miller, and L. F. Abbot. Competitive Hebbian learningthrough spike-timing-dependent synaptic plasticity.
Nature Neuroscience ,3:919–926, 2000.[24] S. Song, P. J. Sj¨ostr¨om, M. Reigl, S. Nelson, and D. B. Chklovskii. Highlynonrandom features of synaptic connectivity in local cortical circuits.
PLoSBiology , 3:507–519, 2005.[25] O. Sporns, D. R. Chialvo, M. Kaiser, and C. C. Hilgetag. Organization,development and function of complex brain networks.
Trends in CognitiveSciences , 8:418–425, 2004.[26] O. Sporns, C. J. Honey, and R. K¨otter. Identification and classification ofhubs in brain networks.
PLoS ONE , 2:e1049, 2007.1027] O. Sporns, G. Tononi, and R. K¨otter. The human connectome: a structuraldescription of the human brain.
PLoS Computational Biology , 1:245–251,2005.[28] C. J. Stam and J. C. Reijneveld. Graph theoretical analysis of complexnetworks in the brain.
Nonlinear Biomedical Physics , 1:3, 2007.[29] M. P. van den Heuvel, C. J. Stam, M. Boersma, and H. E. Hulshoff Pol.Small-world and scale-free organization of voxel-based resting-state func-tional connectivity in the human brain.
Neuroimage , 43:528–539, 2008.[30] M. C. W. van Rossum, G. Q. Bi, and G. G. Turrigiano. Stable Hebbianlearning from spike timing-dependent plasticity.