{"title": "Circuit Model of Short-Term Synaptic Dynamics", "book": "Advances in Neural Information Processing Systems", "page_first": 1099, "page_last": 1106, "abstract": null, "full_text": "Circuit Model of Short-Term Synaptic Dynamics\n\nShih-Chii Liu, Malte Boegershausen, and Pascal Suter\n\nInstitute of Neuroinformatics\n\nUniversity of Zurich and ETH Zurich\n\nWinterthurerstrasse 190\n\nCH-8057 Zurich, Switzerland\n\nshih@ini.phys.ethz.ch\n\nAbstract\n\nWe describe a model of short-term synaptic depression that is derived\nfrom a silicon circuit implementation. The dynamics of this circuit model\nare similar to the dynamics of some present theoretical models of short-\nterm depression except that the recovery dynamics of the variable de-\nscribing the depression is nonlinear and it also depends on the presynap-\ntic frequency. The equations describing the steady-state and transient re-\nsponses of this synaptic model \ufb01t the experimental results obtained from\na fabricated silicon network consisting of leaky integrate-and-\ufb01re neu-\nrons and different types of synapses. We also show experimental data\ndemonstrating the possible computational roles of depression. One pos-\nsible role of a depressing synapse is that the input can quickly bring the\nneuron up to threshold when the membrane potential is close to the rest-\ning potential.\n\n1 Introduction\n\nShort-term synaptic dynamics have been observed in many parts of the cortical sys-\ntem [Stratford et al., 1998, Varela et al., 1997, Tsodyks et al., 1998]. The functionality\nof the short-term synaptic dynamics have been implicated in various cortical models [Senn\net al., 1998, Chance et al., 1998, Matveev and Wang, 2000]. along with the processing\ncapabilities of a network with dynamic synapses [Tsodyks et al., 1998, Maass and Zador,\n1999]. The introduction of these dynamic synapses into hardware implementations of re-\ncurrent neuronal networks allow a wide range of operating regimes especially in the case\nof time-varying inputs.\n\nIn this work, we describe a model that was derived from a circuit implementation of short-\nterm depression. The circuit implementation was initially described by [Rasche and Hahn-\nloser, 2001] but the dynamics were not analyzed in their work. We also compare the\ndynamics of the circuit model of depression with the equations of one of the theoretical\nmodels frequently used in network simulations [Abbott et al., 1997,Varela et al., 1997] and\nshow examples of transient and steady-state responses of this synaptic circuit to inputs of\ndifferent statistical distributions.\n\nThis circuit has been included in a silicon network of leaky integrate-and-\ufb01re neurons to-\ngether with other short-term dynamic synapses like facilitation synapses. We also show\n\n\fexperimental data from the chip that demonstrate the possible computational roles of de-\npression. We postulate that one possible role of depression is to bring the neuron\u2019s response\nquickly up to threshold if the membrane potential of the neuron was close to the resting\npotential. We also mapped a proposed cortical model of direction-selectivity that uses de-\npressing synapses onto this chip. The results are qualitatively similar to the results obtained\nin the original work [Chance et al., 1998].\n\nThe similarity of the circuit responses to the responses from Abbott and colleagues\u2019s synap-\ntic model means that we can use these VLSI networks of integrate-and-\ufb01re (I/F) neurons as\nan alternative to computer simulations of dynamical networks composed of large numbers\nof integrate-and-\ufb01re neurons using synapses with different time constants. The outputs of\nsuch networks can also be used to interface with neural wetware. An infrastructure for a re-\nprogrammable, recon\ufb01gurable, multi-chip neuronal system is being developed along with\na user-de\ufb01ned interface so that the system is easily accessible to a naive user.\n\n2 Comparisons between Models of Depression\n\nWe compare the circuit model with the theoretical model from [Abbott et al., 1997] describ-\ning synaptic depression and facilitation. Similar comparisons with [Tsodyks and Markram,\n1997] give the same conclusions. Here, we only describe the circuit model for synaptic\ndepression. The equivalent model for facilitation is described elsewhere [Liu, 2002].\n\n2.1 Theoretical Model for Depression Model\n\nis:\n\nis the maximum synaptic strength. The recovery dynamics of\u0001\n\nIn the model from [Abbott et al., 1997], the synaptic strength is described by\u0002\u0001\u0004\u0003\u0006\u0005\b\u0007 , where\nis a variable between 0 and 1 that describes the amount of depression (\u0001\n\t\f\u000b means no\ndepression) and\nwhere \r\u0015\u000e\nspike at time\u0005\u0016\t\u0017\u0005\u0019\u0018\u001b\u001a\n\nis the recovery time of the depression. The update equation for \u0001\n\n(2)\nwhere\nis\nthe time of the spike. The average steady-state value of depression for a regular spike train\nwith a rate\n\n\t\u0011\u000b\u0013\u0012\u0014\u0001\n\nright after a\n\n\u000f\u000e\u0004\u0010\n\n(1)\n\n\u0007\u0016\t\n\u0001\u0004\u0003\u0006\u0005\u001d\u001c\n\u0018\u001b\u001a\n\u000b ) is the amount by which \u0001\n\u000b\u0013\u0012'&\n\u000b\u0013\u0012\n\n\u0018\u001d\u0018\n\n\u0001\u0004\u0003\u001e\u0005 \u001f\n\u0018!\u001a\nis decreased right after the spike and \u0005$\u0018!\u001a\n\u001f)(\b*,+.-$/\b0 1\n\u001f2(\u0019*,+.-3/\n\n134\n\n2.2 Circuit Model of Depressing Synapse\n\n(\n\n\u0010#\"\n\nis\n\n(3)\n\nis\n\nIn this circuit model of synaptic depression, the equation that describes the recovery dy-\nis nonlinear. This nonlinearity comes about because\nthe exponential dynamics in Eq. 1 was replaced with the dynamics of the current through\n(de-\nrived from the circuit in the region where a transistor operates in the subthreshold region or\nthe current is exponential in the gate voltage of the transistor) can be formulated as\n\nnamics of the depressing variable, \u0001\na single diode-connected transistor. Hence, the equation describing the recovery of\u0001\n\n\t\u00175\n\n67\u000b\u0013\u0012'\u0001\n\n(\b* 879\n\n(4)\n\nwhere \u000b\u0015:;5\nThe maximum value of\u0001\n\nis the equivalent of \r<\u000e\n\nin Eq. 1 and\n\n(a transistor parameter) is less than 1.\n\nis 1. The update equation remains as before:\n\n\u0001\u0004\u0003\u001e\u0005\n\n\u0018!\u001a\n\n\u0007>\t\n\n\u0001?\u0003\u001e\u0005\n\n\u0018\u001b\u001a\n\n(5)\n\n\u0001\n\u0001\n\u0010\n\u0005\n\u0010\n\u0007\n\u0010\n%\n\u0001\n\t\n\u0010\n&\n0\n\u0010\n\u0001\n\u0010\n\u0005\n=\n\u001c\n\u0010\n\u001f\n\u0007\n4\n\fVa\n\nM1\n\nIr\n\nM2\n\nVd\n\nVpre\n\nM3\n\nVgain\n\nM6\n\nVx\n\nC\n\nM5\n\nIsyn\n\nVpre\n\nM4\n\nM7\nId\n\nC2\n\n0.35\n\n0.3\n\n0.25\n\n)\n\nV\n\n(\n \n\nx\n\nV\n\n0.2\n\n0.15\n\nUpdate \n\nV\n=0.3 V\nd\n\nSlow recovery\n\nV\n=0.26 V\nd\n\nV\n=0.28 V\nd\n\nFast recovery\n\n(a)\n\n0.1\n0.04\n\n0.06\n\n0.08\n\n0.14\n\n0.16\n\n0.12\n\n0.1\nTime (s)\n(b)\n\n\u000f\b\n\nor\n\n\u0002\u0001\n\nis around\n\nweight current\n\nrecovers to the voltage,\n\nis exponential in the voltage,\n\n while the synaptic term \u0002\u0001\nconsisting of transistors, 5\ninput goes to the gate terminal of5\n\nFigure 1: Schematic for a depressing synapse circuit and responses to a regular input spike\ntrain. (a) Depressing synapse circuit. The voltage\ndetermines the synaptic conductance\n. The subcircuit\n\u0018\u0005\u0004\u0007\u0006\n. The presynaptic\n\f which acts like a switch. When there is a presynaptic\n. In between\n\u000e\b\nthrough the diode-connected transistor, 5\n. When\n\u0010\u0001\n. When the presynaptic input comes from a regular spike\ndecreases with each spike and recovers in between spikes. It reaches a steady-\nturns on and the synaptic\n\nspike, a quantity of charge (determined by\nspikes,\nthere is no spike,\ntrain,\n\n\t\b\n\u0018\u0005\u0004\u0007\u0006\n\u000e ) is removed from the node\n\n, 5\u000b\n , and 5\r\f , control the dynamics of\n\nstate value as shown in (b). During the spike, transistor 5\u0012\u0011\nmirror circuit consisting of5\u0015\u0014 , 5\u000b\u0016 , and the capacitor\n\n\u0018\u0013\u0004\u0007\u0006 charges up the membrane potential of the neuron through the current-\n\u0018\u0005\u0004\u0007\u0006 current\n\u000e with some gain and a \u201ctime constant\u201d by adjusting the\n1\u0006*\u001aB\n\n\u001b\u0005\u001c\u001e\u001d \u001f\n(\u001d\u001c('*)\n. In a normal synapse circuit (that is, without short-\nis controlled by an external bias voltage. (b) Input spike train at a\n(top curve) of the circuit\nC\b\nhas nonlinear dynamics. The\ndepends on the distance of the present value of\n.\n\n . We can convert the\n\u0003\u001e\u0005\b\u0007\u0002\t\n\n\u00109\nterm dynamics),\nfrequency of 20 Hz (bottom curve) and corresponding response\nfor\n\n\t 0.26,0.28, 0.3 V. The diode-connected transistor5\nD\u0001\n\nrecovery time of the depressing variable \u0001\n\u000f\b\n2.2.1 Circuit\n\n. The recovery rate of\u0001\n\nsource into a synaptic current\nvoltage\n\nincreases for a larger difference between\n\n. The decay dynamics of\n\n\u001c$&\n\n+\"!$#%!\n\u001c+\u001d\u0013\u001f*,.-0/*-0\u001c$&21\n354\n\n\u001f=;?>\u0005@ A\n\nis given by\n\n8<+<;\n\n0\u001d0\n\n\u000f\u0018\u0007\u0001\u001a\u0019\nand\n\nand\n\nD\u0001\n\nE\b\n\nwhere\n\nfrom\n\n687\n\n=\b\n\nEquations 4 and 5 are derived from the circuit in Fig. 1. The operation of this circuit is\n\nis described in [Liu, 2002]. The voltage\nis set by both\n\ndescribed in the caption. The detailed analysis leading to the differential equations for \u0001\nwhile the dynamics of \u0001\n\u000f\b\nThe synaptic weight is described by the current,\n\n. The time taken for the present value of\nis determined by the current dynamics of the diode-connected transistor\n\n. The conductance\n\nto return to\nand\n\nin Fig. 1(a):\n\nis set by\n\nis set by\n\n%\u0001\n\n.\n\nwhere \u0011\t\n+<;\n0\u001d0\n\u0003GF\n\n\u001f=;?H,1\u0006*\u001aB\n\n\u0003\u001e\u0005\b\u0007\n\n\u0018\u0005\u0004\u0007\u0006\n>\u0001\u0004\u0003\u001e\u0005\b\u0007\nis the synaptic strength, \u0001\u0004\u0003\u0006\u0005\b\u0007\n8=;2@,*\u001aB\n. The recovery time constant (\u000b\n:;5\n\nis\n\n,\"L\nJ\u0013K\n) of \u0001\n\n(6)\n\n10O\u0013P\n0\u001d02MNL\n\u001b\u0005Q\u001e&\n+\"!\u001e1\n\u001bSR\u0013T\nis set by\n\n, and\n\nV\u0001\n\n-IU\n\n(5\n\n. The recovery time constant (\u000b\n\u0003GF\n\ncodes for>\u0001\n\u000e and\nD\u0001\n) of\u0001\n:;5\n\u0018\u0005\u0004\u0007\u0006\n&\u00158=;2H\u000f*IB\n\n\u0003\n(\n\u0003\n\n(\n\n\b\n\n\u0001\n\n\b\n\u0003\n\u0017\n\u0003\n\u0003\n\u0006\n\u0003\n\u000e\n\u0003\n\u000e\n1\n\t\n\u0017\n7\n:\n\t\n&\n\u001f\n4\n\n\u000e\n(\n\n\b\n\n\u0001\n\n5\n(\n\n\u0001\n\n\u0001\n\u0003\n\u0003\n\t\n\u0006\n4\n\t\n\u0003\nF\n\u0006\n&\n4\n@\n4\n\u0003\n\t\n\u001a\n&\n8\n4\n\t\n\f+<;\n\n0\u001d0,\u001f%;\n\n( \u001f28\u000f1\n\n4 ). The synaptic current,\n\n\u001b\u0005Q\u001e&\n\u0018\u0005\u0004G\u0006 which lasts for the duration of the pulse width of the presynaptic spike. However, we\ncan set a longer time constant for the synaptic current through\n. The equation describ-\ning this dependence (that is, the current equation for a current-mirror circuit) is given in the\ncaption of Fig. 1.\n\nto the neuron, is then a current source\n\n1\u0006*\u001aB\n\n\u0018\u0007\u0001\u001a\u0019\n\na)\n\n1\n\ns\ne\nk\np\nS\n\ni\n\n0.5\n\nAbbott\u2019s model\nCircuit model\n\n500\n\n1000\n\n1500\n\n2000\n\n2500\n\n3000\n\n3500\n\n4000\n\n0\n0\n\u221260\n\n\u221265\n\n\u221270\n\nb)\n\n)\n\nV\nm\n\n(\n \n\nm\n\nV\n\n\u221275\n0\n1\n\nc)\n\nD\n\n0.5\n\n0\n0\n\n500\n\n1000\n\n1500\n\n2000\n\n2500\n\n3000\n\n3500\n\n4000\n\n500\n\n1000\n\n1500\n\n2000\n\n2500\n\n3000\n\n3500\n\n4000\n\nTime (msec)\n\nFigure 2: Comparison between the outputs of the two models of depression. An optimiza-\ntion algorithm was used to determine the parameters of the models so that the least square\nerror in the difference between the EPSPs from the two models was at a minimum. The\ndistribution is shown in (c). (a) Poisson-distributed input with an initial\nfrequency of 40 Hz and an end frequency of 1 Hz. (b) The EPSP responses of both models\nis close\n\ncorresponding \u0001\nwere identical. (c) The\u0001\nto 1. Parameters used in the simulations: \r\u0002\u0001\n\nvalues were almost identical except in the region when\u0001\n\t\u000b\u0003\n\n\t\u0004\u0003\u0006\u0005\b\u0007\n\t ,\n\n\u000e ,5\n\n\t\u000b\u0005\n\n\t\r\u0005\n\n\f ,\n\n\t\u000b\u0005\n\ncase of\n\n(a transistor pa-\nIt is dif\ufb01cult to compute a closed-form solution for Eq. 4 for any value of\nrameter which is less than 1). This value also changes under different operating conditions\nin the\n\nand between transistors fabricated in different processes. Hence, we solve for \u0001\u0004\u0003\u001e\u0005\b\u0007\n\nis far from its recovered value of 1, we can approximate its recovery dynamics by\n\n\u0010 given that the last spike occurred at\u0005\u0016\t\u0017\u0005\u0012\u0011 :\n\u0005\b\u0007\u001e\u001d\u001f\u0019\u001c \"!#\u001b\n\u0007\u0016\u0015\u0018\u0017\u001a\u0019\u001c\u001b\n\u0003!5\n\u0007\u0014\u0013\n\u0001\u0004\u0003\u0006\u0005'\u0011\n\u0003!5\n\u0005\b\u0007&\u001d\n\u0007\u0016\u0019\u001c \"!#\u001b\n) and solving for\u0001\u0004\u0003\u0006\u0005\b\u0007 , we get\n\u0001\u0004\u0003\u001e\u0005\b\u0007\n\nWhen\u0001\nIn this regime, \u0001\u0004\u0003\u0006\u0005\b\u0007 follows a linear trajectory. Note that the same is true of Eq. 1 when\n\n(irrespective of\n\n\u0001\u0004\u0003\u001e\u0005\b\u0007>\t\n\n\u0003\u001d\u000b\u0013\u0012\u0014\u0001\n\n\u0001\u0004\u0003\u0006\u0005'\u0011\n\n\u0001\u0004\u0003\u0006\u0005\n\n\u0005\u001e\u001d\n\n\u0015\u0012\u0017$\u0019%\u001b\n\n\u0005\b\u0007\n\u0005\b\u0007\n\n\u0003\u001b5\n\u0003\u001b5\n\n\u001f2( .\n\n\u000e .\n\n\u0003\u0002\u0005\u000f\t\n\n(7)\n\n8\n\n4\n&\n\u001f\n+\n@\n\u0003\n\u000e\n\u0003\n\n\u0006\n\u0010\n4\n=\n4\n4\n=\n=\n4\n\u0010\n\u0001\n:\n\u0010\n\u0005\n\t\n5\n\n\u0011\n4\n\u0010\n\u0001\n:\n\u0010\n\u0005\n\t\n5\n=\n\t\n5\n\u0007\n4\n\u0005\n\"\n\"\n\n\f)\n\nV\n\n(\n \ne\ns\nn\no\np\ns\ne\nr\n \n\nn\no\nr\nu\ne\nN\n\n0.8\n\n0.6\n\n0.4\n\n0.2\n\n0\n\n\u22120.2\n\n\u22120.4\n\n\u22120.6\n0\n\n1.4\n\n1.2\n\n1\n\n0.8\n\n0.6\n\n0.4\n\n0.2\n\n)\n\nV\n\n(\n \n\ne\nd\nu\n\nt\ni\nl\n\n \n\np\nm\na\nP\nS\nP\nE\n\nPoisson spike train\n\nVd = 0.2 V, Va = 1.01 V\n\nVd = 0.4 V, Va = 1.03 V\n\nNon\u2212depressing synapse \n\nVd = 0.6 V, Va = 1.15 V\n\n1\n\n2\nTime (s)\n(a)\n\n3\n\n0\n0\n\n10\n\n20\nFrequency (Hz)\n\n30\n\n40\n\n50\n\n(b)\n\nFigure 3: Transient EPSP responses to a 10 Hz Poisson-distributed train (a) and dependence\nof steady-state EPSP responses on the input frequency for different values of depression (b).\nThe data was measured from the fabricated circuit. In (a), the amplitude of the EPSP de-\ncreases with each incoming input spike clearly showing the effect of synaptic depression.\nIn (a), the EPSP amplitude depends on the occurrence of the previous spike. The asterisks\nare the \ufb01ts of the circuit model to the peak value of each EPSP. The \ufb01ts give a\nvalue\nof 0.79. The input is the bottom curve of the plot. (b) Steady-state EPSP amplitude ver-\nsus frequency for a Poisson-distributed input. The solid lines are \ufb01ts from the theoretical\nequation.\n\n3 Comparison between Models\n\nWe compare the two models by looking at how \u0001\n\nchanges in response to a Poisson-\ndistributed input whose frequency varied from 40 Hz to 1 Hz as shown in Fig. 2. We used a\n:\nsimple linear differential equation to describe the dynamics of the membrane potential\n\n\u0003\u001e\u0005\b\u0007\n\n\t\u0001\u0003\u0002 \u0003\u0006\u0005\b\u0007\nis the membrane time constant and \u0002 \u0003\u0006\u0005\b\u0007\n\n\u0003\u001e\u0005\b\u0007\n\nis the synaptic current. We ran an op-\ntimization algorithm on the parameters in the two models so that the least square error\nbetween the EPSP outputs of both models was at a minimum. In this case, the EPSP re-\nvalues (Fig. 2(c)) were almost\n\nwhere \r\nsponses were identical (Fig. 2(b)) and the corresponding \u0001\nidentical except in the region where\u0001 was close to the maximum value. We performed the\n\nsame comparison with Tsodyks and Markram\u2019s model and the results were similar. Hence,\nthe circuit model can be used to describe short-term synaptic depression in a network sim-\nulation. However, the nonlinear recovery dynamics of the circuit model leads a different\nfunctional dependence of the average steady-state EPSP on the frequency of a regular input\nspike train.\n\n4 Circuit Response\n\nThe data in the \ufb01gures in the remainder of this paper are obtained from a fabricated\nsilicon network of aVLSI integrate-and-\ufb01re neurons of the type described in [Boahen,\n1997, Van Schaik, 2001, Indiveri, 2000, Liu et al., 2001] with different types of synapses.\n\n\u0010\n\n\u0001\n\n\u0001\n\u0010\n\n\u0001\n\u0010\n\u0005\n\u0012\n\n\u0001\n\u0001\n\f4.1 Transient Response\n\nWe \ufb01rst measured the transient response of the neuron when stimulated by a 10 Hz Poisson-\ndistributed input through the depressing synapse. We tuned the parameters of the synapse\nand the leak current so that the membrane potential did not build up to threshold. This data\nis shown in Fig. 3(a). The \ufb01t (marked with asterisks with in the \ufb01gure) using Eq. 6 along\n\ncomputed from Eq. 7, describes the experimental data well.\n\nwith\u0001\n\n4.2 Steady-State Response\n\non the presynaptic\nfrequency can easily be determined in the case of a regular spiking input of rate\nby using\nEqs. 5 and 7. The resulting expression is somewhat complicated but by using the reduced\ndynamics expression (\n\nThe equation describing the dependence of the steady-state values of\u0001\n), we obtain a simpler expression for\u0001\n\u0018\u001d\u0018\n\n\u0018\u001d\u0018 :\n\nThis equation shows that the steady-state\u0001\n\ninversely dependent on the presynaptic rate\nobtained in the work of [Abbott et al., 1997] where the data can be \ufb01tted with Eq. 3.\n\nand hence, the steady-state EPSP amplitude is\n. The form of the curve is similar to the results\n\n\u0003\u001d\u000b\n\n(8)\n\nFrom the chip, we measured the steady-state EPSP amplitudes using a Poisson-distributed\ntrain whose frequency varied over a range of 3 Hz to 50 Hz in steps of 1 Hz. Each frequency\ninterval lasted 15 s and the EPSP amplitude was averaged in the last 5 s to obtain the steady-\nstate value. Four separate trials were performed and the resulting mean and the variance of\nthe measurements are shown in Fig. 3(b). The parameters from the \ufb01ts using the response\ndata to a regular spiking input were used to generate the \ufb01tted curve to the data in Fig. 3(b).\n\nThe values from the \ufb01ts give recovery time constants from 1\u20133 s and \u0001\n\nbetween 0.02-0.04.\n\n\u0018\u001d\u0018 values varying\n\n5 Role of Synaptic Depression\n\nDifferent computational roles have been proposed for networks which incorporate synaptic\ndepression. In this section, we describe some measurements which illustrate the postulated\nroles of depression. The direction-selective model of [Chance et al., 1998] which makes\nuse of the phase advance property from depressing synapses have been attempted on a\nneuron on our chip and the direction-selective results were qualitatively similar.\n\nDepressing synapses have also been implicated in cortical gain control [Abbott et al., 1997].\nA depressing synapse acts like a transient detector to changes in frequency (or a \ufb01rst deriva-\ntive \ufb01lter). A synapse with short-term depression responds equally to equal percentage rate\nchanges in its input on different \ufb01ring rates. We demonstrate the gain-control mechanism\nof short-term depression by measuring the neuron\u2019s response to step changes in input fre-\nquency from 10 Hz to 20 Hz to 40 Hz. Each step change represents the same rate change\nin input frequency. These results are shown in Fig. 4(a) for a regular train and in (b) for a\nPoisson-distributed train. Each frequency epoch lasted 3 s so the synaptic strength should\nhave reached steady-state before the next increase in input frequency.\n\nFor both \ufb01gures in Fig. 4, the top curve shows the response of the neuron when stimulated\nby the input (bottom curve) through a depressing synapse (top curve) and a non-depressing\nsynapse (middle curve). Figure 4(a) shows clearly that the transient increase in the \ufb01ring\nrate of a neuron when stimulated through a depressing synapse right after each step increase\nin input frequency and the subsequent adaptation of its \ufb01ring rate to a steady-state value.\nThe steady-state \ufb01ring rate of the neuron with a depressing synapse is less dependent on the\n\n%\n\u0010\n\u0001\n:\n\u0010\n\u0005\n\t\n5\n\u0001\n\t\n5\n\u0012\n\u0010\n\u0007\n%\n4\n%\n\fRegular spike train\n\n5\n\n4.5\n\n4\n\n3.5\n\n3\n\nm\n\nV\n\n2.5\n\n)\n\nV\n\n(\n \n\n2\n\n1.5\n\n1\n\n0.5\n\nPoisson spike train\n\n5\n\n4\n\n3\n\n2\n\n1\n\n0\n\n)\n\nV\n\n(\n \n\nm\n\nV\n\n3\n\n4\n\n5\n\nTime (s)\n(a)\n\n6\n\n7\n\n0\n\n2\n\n4\n\nTime (s)\n\n6\n\n(b)\n\n8\n\nFigure 4: Response of neuron to changes in input frequency (bottom curve) when stim-\nulated through a depressing synapse (top curve) and a non-depressing synapse (middle\ncurve). The neuron was stimulated for three frequency intervals (10 Hz to 20 Hz to 40 Hz)\nlasting 3 s each. (a) Response of neuron using a regular spiking input. The steady-state \ufb01r-\ning rate of the neuron increased almost linearly with the input frequency when stimulated\nthrough the non-depressing synapse. In the depressing-synapse curve, there is a transient\nincrease in the neuron\u2019s \ufb01ring rate before the rate adapted to steady-state. (b) Response of\nneuron using a Poisson-distributed input. The parameters for both types of synapses were\ntuned so that the steady-state \ufb01ring rates were about the same at the end of each frequency\ninterval for both synapses. Notice that during the 10 Hz interval, the neuron quickly built\nup to threshold if it was stimulated through the depressing synapse.\n\nabsolute input frequency when compared to the \ufb01ring rate of the neuron when stimulated\nthrough the non-depressing synapse.\nIn the latter case, the \ufb01ring rate of the neuron is\napproximately linear in the input rate.\n\nThe data in Fig. 4(b) obtained from a Poisson-distributed train shows an obvious difference\nin the responses between the depressing and non-depressing synapse. In the depressing-\nsynapse case, the neuron quickly reached threshold for a 10 Hz input, while it remained sub-\nthreshold in the non-depressing case until the input has increased to 20 Hz. This suggests\nthat a potential role of a depressing synapse is to drive a neuron quickly to threshold when\nits membrane potential is far away from its threshold.\n\n6 Conclusion\n\nWe described a model of synaptic depression that was derived from a circuit implementa-\ntion. This circuit model has nonlinear recovery dynamics in contrast to current theoretical\nmodels of dynamic synapses. It gives qualitatively similar results when compared to the\nmodel of Abbott and colleagues. Measured data from a chip with aVLSI integrate-and-\ufb01re\nneurons and dynamic synapses show that this network can be used to simulate the responses\nof dynamic networks with short-term dynamic synapses. Experimental results suggest that\ndepressing synapses can be used to drive a neuron quickly up to threshold if its membrane\npotential is at the resting potential. The silicon networks provide an alternative to computer\nsimulation of spike-based processing models with different time constant synapses because\nthey run in real-time and the computational time does not scale with the size of the neuronal\nnetwork.\n\n\fAcknowledgments\n\nThis work was supported in part by the Swiss National Foundation Research SPP grant.\nWe acknowledge Kevan Martin, Pamela Baker, and Ora Ohana for many discussions on\ndynamic synapses.\n\nReferences\n\n[Abbott et al., 1997] Abbott, L., Sen, K., Varela, J., and Nelson, S. (1997). Synaptic de-\n\npression and cortical gain control. Science, 275(5297):220\u2013223.\n\n[Boahen, 1997] Boahen, K. A. (1997). Retinomorphic Vision Systems: Reverse Engineer-\ning the Vertebrate Retina. PhD thesis, California Institute of Technology, Pasadena CA.\n[Chance et al., 1998] Chance, F., Nelson, S., and Abbott, L. (1998). Synaptic depres-\nsion and the temporal response characteristics of V1 cells. 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Differential short-term\nsynaptic plasticity and transmission of complex spike trains: to depress or to facilitate?\nCerebral Cortex, 10(11):1143\u20131153.\n\n[Rasche and Hahnloser, 2001] Rasche, C. and Hahnloser, R. (2001). Silicon synaptic de-\n\npression. Biological Cybernetics, 84(1):57\u201362.\n\n[Senn et al., 1998] Senn, W., Segev, I., and Tsodyks, M. (1998). Reading neuronal syn-\n\nchrony with depressing synapses. Neural Computation, 10(4):815\u2013819.\n\n[Stratford et al., 1998] Stratford, K., Tarczy-Hornoch, K., Martin, K., Bannister, N., and\nJack, J. (1998). Excitatory synaptic inputs to spiny stellate cells in cat visual cortex.\nNature, 382:258\u2013261.\n\n[Tsodyks and Markram, 1997] Tsodyks, M. and Markram, H. (1997). The neural code\nbetween neocortical pyramidal neurons depends on neurotransmitter release probability.\nProc. Natl. Acad. Sci. USA, 94(2).\n\n[Tsodyks et al., 1998] Tsodyks, M., Pawelzik, K., and Markram, H. (1998). Neural net-\n\nworks with dynamic synapses. Neural Computation, 10(4):821\u2013835.\n\n[Van Schaik, 2001] Van Schaik, A. (2001). Building blocks for electronic spiking neural\nnetworks. Neural Networks, 14(6/7):617\u2013628. Special Issue on Spiking Neurons in\nNeuroscience and Technology.\n\n[Varela et al., 1997] Varela, J., Sen, K., Gibson, J., Fost, J., Abbott, L., and Nelson, S.\n(1997). A quantitative description of short-term plasticity at excitatory synapses in layer\n2/3 of rat primary visual cortex. Journal of Neuroscience, 17(20):7926\u20137940.\n\n\f", "award": [], "sourceid": 2233, "authors": [{"given_name": "Shih-Chii", "family_name": "Liu", "institution": null}, {"given_name": "Malte", "family_name": "Boegershausen", "institution": null}, {"given_name": "Pascal", "family_name": "Suter", "institution": null}]}