*Corresponding author. Tel.:#972-3-6429305; fax:#972-3-6407932.E-mail address: horn@neuron.tau.ac.il (D. Horn).

Neurocomputing 32}33 (2000) 409}414

Distributed synchrony in an attractor of spiking neurons

David Horn!,*, Nir Levy!, Eytan Ruppin"

!School of Physics and Astronomy, Tel Aviv University, Raymond and Beverly SacklerFaculty of Exact Sciences, Tel Aviv 69978, Israel

"School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel

Accepted 11 January 2000

Abstract

Recently, it has been shown that there exists a temporal synaptic learning curve, composed ofpotentiation for short time delays between pre- and post-synaptic neuronal spiking, anddepression for spiking events occurring in the opposite order. We investigate the e!ect that sucha curve can have on the temporal realization of an attractor in an associative memory networkof spiking neurons. We "nd that an interesting result of this learning paradigm is the formationof distributed synchrony, i.e. spontaneous division of the Hebbian cell assembly into groups ofcells that "re in a cyclic manner. ( 2000 Elsevier Science B.V. All rights reserved.

Keywords: Distributed synchrony; Spiking neurons; Hebbian cell assembly; Temporal synapticlearning curve

1. Introduction

The Hebbian paradigm that serves as the basis for models of associative memoryis often conceived as the statement that a group of excitatory neurons (the Hebbiancell assembly) are coupled synaptically to one another and "re together when asubset of the group is being excited by an external input. Yet the details of thetemporal spiking patterns of neurons in such an assembly are still misunderstood.Theoretically, it seems quite obvious that there are two general types of behavior:synchronous neuronal "ring and asynchrony where no temporal order exists in theassembly and the di!erent neurons "re randomly but with the same overall rate.

0925-2312/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 9 2 5 - 2 3 1 2 ( 0 0 ) 0 0 1 9 3 - 4

Further subclassi"cations were recently suggested by Brunel [3]. Experimentally, thisquestion is far from being settled because evidence for the associative memoryparadigm is quite scarce. On the one hand, one possible realization of associativememories in the brain was demonstrated by Miyashita [7] in the inferotemporalcortex. This area was recently reinvestigated by Yakovlev et al. [8], who comparedtheir experimental results with a model of asynchronized spiking neurons. On theother hand there exists experimental evidence [2] for temporal activity patterns in thefrontal cortex that Abeles called syn"re-chains. Could they correspond to an alterna-tive type of synchronous realization of a memory attractor?

To answer these questions and study the possible realizations of attractors incortical-like networks, we investigate the temporal structure of an attractor assumingthe existence of a synaptic learning curve that is continuously applied to the memorysystem. This learning curve is motivated by the experimental observations [6,9] thatsynaptic potentiation or depression occurs within a critical time window in whichboth pre- and post-synaptic neurons have to "re. If the pre-synaptic neuron "res "rstwithin 30 ms or so, potentiation will take place. Depression is the rule for the reverseorder.

The regulatory e!ects of such a synaptic learning curve on the synapses of a singleneuron that is subjected to external inputs were investigated by Abbott and Song [1]and by Kempter et al. [5]. We investigate here the e!ect of such a rule within anassembly of neurons that are all excited by the same external input throughouta training period, and are then allowed to in#uence one another through theirresulting sustained activity.

2. The model

We use integrate-and-"re excitatory and inhibitory neurons. They all have mem-brane potentials <

*obeying

qad<

*dt

"!<*#I

*, (1)

where a"e, i speci"es the two excitatory/inhibitory types. A spike is generated when<

*reaches the threshold h, upon which a refractory period of 1 ms is set on and the

membrane potential is reset to <3%4%5

where 0(<3%4%5

(h. We choose h"20 mV and<

3%4%5"10 mV.

A spike emitted by neuron j at time t0

adds to the input of neuron i Ja1 ,a2ij

mV at thetime t"t

0#q

ijwhere q

ijis the (axonic and dendritic) delay between the two neurons

in question.A single memory is modeled on a group of N

%"100 excitatory neurons that are

connected to N*"50 inhibitory neurons. Some of the connections are preset in

a random fashion, so that with a probability of 0.2 connections are formed and areassigned the values J%,*"0.3 J*,%"0.4 and J*,*"0.4. The excitatory synaptic connec-tions, where learning is going to take place, are preset to a random value in the range

410 D. Horn et al. / Neurocomputing 32}33 (2000) 409}414

of [0, 0.1]. All are allowed to undergo changes but are subject to an upper boundJ%,%(0.6.

During memory training an external Poissonian input with average of 1 mV/ms(the signal) is given to all excitatory neurons for a period of 500 ms. After training allneurons are subjected to a low Poissonian input (average 0.4 mV/ms) that simulatesexternal noise. Successful learning of the memory should lead to sustained activity ofthe system in the absence of the memory signal.

The synaptic learning curve is de"ned via

*J%,%ij

"F(ti, t

j), (2)

such that

F"f1e~j(ti~tj ) if t

i!t

j'e (3)

and

F"!f2e~j(tj~ti ) if t

j!t

i'e, (4)

f1

represents potentiation, taking place when the postsynaptic neuron "res after thepresynaptic one. The f

2term represents depression that occurs for the opposite

temporal order.

3. Results

We have run our system with synaptic delays distributed randomly betweenqij"1,2,3 ms, and temporal parameters of q

%"40 ms and q

*"20 ms, and obtained

sustained "ring activities in the 150 Hz range. We have found, in addition to syn-chronous and asynchronous realizations of this attractor, a mode of distributedsynchrony shown in Fig. 1. The 100 excitatory neurons split into groups such that eachgroup "res at the same frequency and at a "xed phase di!erence from any other group.This comes about through a self-organization of the synaptic matrix, as displayed inFigs. 2a. The block form of this matrix represents the ordered couplings that areresponsible for the fact that each coherent group of neurons feeds the activity ofgroups that follow it. The self-organized groups form spontaneously. When thesynapses are a!ected by some external noise, as can come about from Hebbianlearning in which these neurons are being coupled with other pools of neurons, thegroups will change and regroup, as seen in Fig. 2b and c. After such synaptic noise theattractor may also be realized in a di!erent fashion, e.g. asynchronous "ring.

When a single delay parameter is employed, a distributed synchronous moderemains stable if the sustained "ring is not been interfered with. When several delayparameters exist, a situation that probably more accurately represents the a-functioncharacter of synaptic transmission, distributed synchrony may still be obtained, as isevident from Fig. 1. After a long time it can destabilize and regrouping may occur byitself, without external interference. This comes about because di!erent synapticconnections that have di!erent delays can interfere with one another. Nonetheless,over time scales of the type shown in Fig. 1, grouping is stable.

D. Horn et al. / Neurocomputing 32}33 (2000) 409}414 411

Fig. 1. Distributed synchronized "ring mode. The "ring patterns of six cell assemblies of excitatory neuronsare displayed versus time (in ms). These six groups of neurons formed in a self-organized manner when thesystem underwent synaptic training with f

1"f

2"0.05 and j"1.2 ms~1. The delays were chosen random-

ly from three values, 1 2 or 3 ms. e"0.5 ms and the system is monitored every 0.5 ms.

Fig. 2. The synaptic matrix for distributed synchrony. The synaptic matrix between the 100 excitatoryneurons of our system is displayed in a grey-level code with black meaning 0 and white standing for thesynaptic upper-bound. (a) The matrix that exists during the distributed synchronous mode of Fig. 1. Itsbasis is ordered such that neurons that "re together are grouped together. (b) Using the same basis as in (a)a new synaptic matrix is shown, one that is formed after stopping the sustained activity of Fig. 1,introducing small noise in the synaptic matrix, and reinstituting the original memory training. (c) The samematrix as (b) is shown in a new basis that exhibits connections that lead to a new and di!erent realization ofdistributed synchrony.

Fig. 1 displays a 6-cycle of distributed synchrony. Running the same system forf1"0.075, f

2"0.05 we obtain a 4-cycle. Higher f

1values lead to lower numbers of

cycles and higher frequency of the "ring rate. All these formations are quite stable inspite of the continuous noisy input and continuous synaptic learning that this system

412 D. Horn et al. / Neurocomputing 32}33 (2000) 409}414

Fig. 3. Distributed synchronized "ring for f1"0.075, f

2"0.05. Other parameters are the same as in Fig. 1.

The di!erent frames describe activity of neurons over two time periods separated by 100 ms. (a) The groupsare de"ned according to the synchrony observed at the beginning of this time window. (b) This groupingstructure is being modi"ed after 100 ms or so, as seen here where the di!erent frames describe the samegroups as in (a). (c) The data of (b) are redisplayed in a new basis showing that regrouping of distributedsynchrony took place.

is subjected to. We "nd distributed synchrony also when potentiation is weaker thandepression, e.g. f

1"0.05 f

2"0.075. Corresponding results are shown in Fig. 3. Note

that the group structure is no longer stable. After a relatively short period one "ndsregrouping of neurons that "re together.

4. Summary

We have studied the e!ects of a synaptic learning curve on the formation ofattractors in an associative memory model. We have found the appearance ofdistributed synchrony, a synchronous mode in which the attractor is being realized in

D. Horn et al. / Neurocomputing 32}33 (2000) 409}414 413

an n-cycle. Remarkably, in spite of continuous synaptic learning, the resulting synap-tic matrix and the observed n-cycle are stable over long periods of time.

We have found n-cycles with rather limited values of n and relatively high frequencyof sustained activity. It seems therefore premature to try and identify them with syn"rechains [2] that show recurrence of "ring patterns of groups of neurons with periods ofhundreds of ms. Nonetheless it is interesting to see that n-cycle formation is a prefer-red mode of the asymmetric synaptic learning rule. It is important to note that thedi!erent groups that are elements of the n-cycle do not have semantic meaning of theirown. As such they di!er from realizations of syn"re chains in models [4] thatconstruct the chain out of individual patterns. As demonstrated in Figs. 2 and 3, thesame Hebbian cell assembly can regroup into di!erent realizations of distributedsynchrony.

References

[1] L.F. Abbott, S. Song, Temporally asymmetric hebbian learning, spike timing and neuronal responsevariability, in: M.S. Kearns, S.A. Solla, D.A. Cohn (Eds.), Advances in Neural Information ProcessingSystems 11: Proceedings of the 1998 Conference, MIT Press, Cambridge, 1999, pp. 69}75.

[2] M. Abeles, Local Cortical Circuits, Springer, Berlin, 1982.[3] N. Brunel, Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons, J.

Comp. Neurosci. 8 (2000) 183}208.[4] M. Herrmann, J. Hertz, A. PruK gel-Bennet, Analysis of syn"re chains, Network: Comp. Neural Syst.

6 (1995) 403}414.[5] R. Kempter, W. Gerstner J. Leo van Hemmen, Spike-Based Compared to Rate-Based Hebbian

Learning. in: M.S. Kearns, S.A. Solla, D.A. Cohn (Eds.), Advances in Neural Information ProcessingSystems 11, MIT Press, Cambridge, 1999, pp. 125}131.

[6] H. Markram, J. LuK bke, M. Frotscher, B. Sakmann, Regulation of synaptic e$cacy by coincidence ofpostsynaptic aps and epsps, Science 275 (5297) (1997) 213}215.

[7] Y. Miyashita, Neuronal correlate of visual associative long-term memory in the primate temporalcortex, Nature 335 (1988) 817}820.

[8] V. Yakovlev, S. Fusi, E. Berman, E. Zohary, Inter-trial neuronal activity in inferior temporal cortex:a putative vehicle to generate long-term visual associations, Nature Neurosci. 1 (1998) 310}317.

[9] L.I. Zhang, H.W. Tao, C.E. Holt, W.A. Harris, M. Poo, A critical window for cooperation andcompetition among developing retinotectal synapses, Nature 395 (1998) 37}44.

414 D. Horn et al. / Neurocomputing 32}33 (2000) 409}414