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Distributed Synchrony in a Cell Assembly of Spiking NeuronsNir Levy David HornSchool of Physics and AstronomyTel Aviv University Tel Aviv 69978, IsraelIsaac Meilijson Eytan RuppinSchool of Mathematical Sciences,Tel Aviv University Tel Aviv 69978, IsraelFebruary 18, 2001AbstractWe investigate the formation of a Hebbian cell assembly of spiking neurons, usinga temporal synaptic learning curve that is based on recent experimental �ndings. Itincludes potentiation for short time delays between pre- and post-synaptic neuronalspiking, and depression for spiking events occurring in the reverse order. The couplingbetween the dynamics of synaptic learning and that of neuronal activation leads tointeresting results. One possible mode of activity is distributed synchrony, implyingspontaneous division of the Hebbian cell assembly into groups, or subassemblies, ofcells that �re in a cyclic manner. The behavior of distributed synchrony is investigatedboth by simulations and by analytic calculations of the resulting synaptic distributions.
1 IntroductionConsider the process of formation of a Hebbian cell assembly. Conventional wisdom wouldproceed along the following line of reasoning: start out with a group of neurons that areinterconnected, using both excitatory and inhibitory cells. Feed them with a common inputthat is strong enough to produce action potentials, and let the excitatory synapses growuntil a consistent �ring pattern can be maintained even if the input is turned o�. Usingtheoretical models of neuronal and synaptic dynamics we follow this procedure and studythe resulting �ring modes. Although the model equations may be an oversimpli�cation oftrue biological dynamics, the emerging �ring patterns are intriguing and may connect toexisting experimental observations.Recent studies of �ring patterns by Brunel (1999) have shown in simulations, and inmean-�eld calculations, that large scale sparsely connected neuronal networks can �re indi�erent modes. Whereas strong excitatory couplings lead to full synchrony, weaker cou-plings will usually lead to asynchronous �ring of individual neurons that can exhibit eitheroscillatory or non-oscillatory collective behavior. For fully connected networks there existsevidence of the possibility of cluster formations, where the di�erent neurons within a cluster�re synchronously. This phenomenon was analyzed by Golomb et al: (1992) in a networkof phase-coupled oscillators, and was studied in networks of pulse-coupled spiking neuronsby van Vreeswijk (1996) and by Hansel et al: (1995).In contrast to previous studies, the present investigation concentrates on the study of anetwork storing patterns via Hebbian synapses. We mainly concentrate on a single Hebbiancell-assembly, where full connectivity is assumed between all excitatory neurons. We employsynaptic dynamics that are based on the recent experimental observations of Markramet al: (1997) and Zhang et al: (1998). They have shown that potentiation or depression ofsynapses connecting excitatory neurons occurs only if both pre- and post-synaptic neurons1
�re within a critical time window of approximately 20ms. If the pre-synaptic neuron �res�rst, potentiation will take place. Depression is the rule for the reverse order. The regulatorye�ects of such a synaptic learning curve on the synapses of a single neuron that is subjectedto external inputs were investigated by Song et al: (2000) and by Kempter et al: (1999). Weinvestigate here the e�ect of such a rule within an assembly of neurons that are all excitedby the same external input throughout a training period, and are allowed to in
uenceone another through their resulting sustained activity. We �nd that this synaptic dynamicsfacilitates the formation of clusters of neurons, thus splitting the Hebbian cell-assembly intosubassemblies and producing the �ring pattern that we call distributed synchrony (DS).In the next section we present the details of our model. It is based on excitatoryand inhibitory spiking neurons. The synapses among excitatory neurons undergo learn-ing dynamics that follow an asymmetric temporal rule of the kind observed by Markramet al: (1997) and Zhang et al: (1998). We study the resulting �ring patterns and synapticweights in sections 3 and 4. The phenomenon of distributed synchrony is displayed anddiscussed. To understand it better, we perform in sections 5 and 6 a theoretical analysisof the in
uence of an ordered �ring pattern on the development of the synaptic couplings.This is derivable in a two-neuron model, and is compared with the results of simulationson a network of neurons. In section 7 we proceed to demonstrate that similar types ofdynamics may appear also in the presence of multiple memory states. A �rst version of ourmodel was presented at Horn et al: (2000).2 The ModelWe study a network composed of NE excitatory and NI inhibitory integrate-and-�re neu-rons. Each neuron in the network is described by its subthreshold membrane potential Vi(t)
obeying _Vi(t) = � 1�nVi(t) +RIi(t) ; (1)where �n is the neuronal membrane decay time constant. A spike is generated when Vi(t)reaches the threshold �, upon which a refractory period of �R sets in and the membranepotential is reset to Vreset where 0 < Vreset < �. For simplicity we set the level of therest-potential to 0. Ii(t) is the sum of recurrent and external synaptic current inputs. Thenet synaptic input charging the membrane of excitatory neuron i at time t is:RIi(t) = NEXj wij(t)Xl Æ �t� tlj � �dij�� NIXk JEIik Xm Æ (t� tmk � �dik) + IE ; (2)summing over the di�erent synapses of j = 1; : : : ; NE excitatory neurons and of k =1; : : : ; NI inhibitory neurons, with synaptic eÆcacies wij(t) and JEIik respectively. The sumover l (m) represents a sum on di�erent spikes generated at times tlj (tmk ) by the respectiveneurons j(k). IE, the external current, is assumed to be random and independent at eachneuron and each time step, drawn from a Poisson distribution.Similarly, the synaptic input to the inhibitory neuron i at time t isRIi(t) = NEXj JIEij Xl Æ �t� tlj � �dij�� NIXk JIIik Xm Æ (t� tmk � �dik) + II ; (3)where II is the external current.We assume full connectivity among the excitatory neurons, but only partial connectivityfor all other three types of possible connections, with connection probabilities denoted byCEI , CIE and CII . In the following we will report simulation results in which the synapticdelays �d were assigned to each synapse, or pair of neurons, randomly, chosen from some�nite set of values. Our analytic calculation will be done for one single value of the synapticdelay parameter.Synaptic eÆcacies between two excitatory neurons, wij , are potentiated or depressedaccording to the temporal �ring patterns of the pre- and post-synaptic neurons. Other3
synaptic eÆcacies, namely those involving at least one inhibitory neuron, JEI , JIE andJII , are assumed to be constant. Each excitatory synapse obeys_wij(t) = � 1�swij(t) + Fij(t) ; (4)where we allowed for a synaptic decay constant �s. We will discuss situations where it isin�nite, but consider also cases when it is �nite but larger than the membrane time constant�n. wij(t) are constrained to vary in the range [0; wmax]. The change in synaptic eÆcacy isde�ned by Fij(t) =Xk;l hÆ(t � tki )KP (tlj � tki ) + Æ(t� tlj)KD(tlj � tki )i ; (5)where KP and KD are the potentiation and depression branches of a kernel function.Following Markram et al: (1997) and Zhang et al: (1998) we distinguish between thesituation where the postsynaptic spike, at tki , appears after or before the presynaptic spike,at tlj . This distinction is made by the use of asymmetric kernel functions that capture theessence of the experimental observations.
−5 0 5
∆ = t
l − t
−15 −10 −5 0 5 10 15
∆ = t
l − t
Figure 1: Two choices of the kernel function whose left part, KP , leads to potentiation ofthe synapse, and whose right branch, KD, causes synaptic depression.4
Figure 1 displays the two kernel functions that are used for analysis and simulations: acontinuous function K1(�) = �c�exp h� (a�+ b)2i (6)as plotted in Figure 1(a), and a discontinuous functionK2(�) = 8>>>>>>>>>: KP (�) = a exp [c�] if � < ��KD(�) = �b exp [�c�] if � > �0 otherwise (7)plotted in Figure 1(b). For both kernels the constants a; b; c change the span, asymmetryand strength of the kernels. In the discontinuous kernel � sets the minimal phase shift thatthe kernel is sensitive to. The shapes of the kernels were determined so that their timewindows match the typical inter-spike-intervals of the excitatory neurons, characteristicallybetween 10ms to 30ms in the simulations that we will report below. As a result the spanof the kernel is somewhat smaller than the experimentally observed ones. In future morerealistic neuronal dynamics one should aim for both larger time-span of the kernel and lowersustained �ring rates of excitatory neurons, thus getting closer to experimental observations.It should be noted that the synaptic dynamics of Eq. (4) do not include short-termsynaptic depression due to high frequency of presynaptic neuronal �ring, such as in Markramand Tsodyks (1996) and Abbott et al: (1997). Neither do the neuronal dynamics includeneuronal regulation, a mechanism proposed by Horn et al: (1998) and termed synapticscaling by Turr