{"title": "Analytical Solution of Spike-timing Dependent Plasticity Based on Synaptic Biophysics", "book": "Advances in Neural Information Processing Systems", "page_first": 1343, "page_last": 1350, "abstract": "", "full_text": "Analytical solution of spike-timing dependent\n\nplasticity based on synaptic biophysics\n\nBernd Porr, Ausra Saudargiene and Florentin W\u00a8org\u00a8otter\n\nComputational Neuroscience\n\nPsychology\n\nUniversity of Stirling\nFK9 4LR Stirling, UK\n\n{Bernd.Porr,ausra,worgott}@cn.stir.ac.uk\n\nAbstract\n\nSpike timing plasticity (STDP) is a special form of synaptic plasticity\nwhere the relative timing of post- and presynaptic activity determines the\nchange of the synaptic weight. On the postsynaptic side, active back-\npropagating spikes in dendrites seem to play a crucial role in the induc-\ntion of spike timing dependent plasticity. We argue that postsynaptically\nthe temporal change of the membrane potential determines the weight\nchange. Coming from the presynaptic side induction of STDP is closely\nrelated to the activation of NMDA channels. Therefore, we will calculate\nanalytically the change of the synaptic weight by correlating the deriva-\ntive of the membrane potential with the activity of the NMDA channel.\nThus, for this calculation we utilise biophysical variables of the physi-\nological cell. The \ufb01nal result shows a weight change curve which con-\nforms with measurements from biology. The positive part of the weight\nchange curve is determined by the NMDA activation. The negative part\nof the weight change curve is determined by the membrane potential\nchange. Therefore, the weight change curve should change its shape de-\npending on the distance from the soma of the postsynaptic cell. We \ufb01nd\ntemporally asymmetric weight change close to the soma and temporally\nsymmetric weight change in the distal dendrite.\n\n1 Introduction\n\nDonald Hebb [1] postulated half a century ago that the change of synaptic strength depends\non the correlation of pre- and postsynaptic activity: cells which \ufb01re together wire together.\nHere we want to concentrate on a special form of correlation based learning, namely, spike\ntiming dependent plasticity (STDP, [2, 3]). STDP is asymmetrical in time: Weights grow\nif the pre-synaptic event precedes the postsynaptic event. This phenomenon is called long-\nterm potentiation (LTP). Weights shrink when the temporal order is reversed. This is called\nlong-term depression (LTD).\n\nCorrelations between pre- and postsynaptic activity can take place at different locations\nof the cell. Here we will focus on the dendrite of the cell (see Fig. 1). The dendrite has\nattracted interest recently because of its ability to propagate spikes back from the soma\n\n\fof the cell into its distal regions. Such spikes are called backpropagating spikes. The\ntransmission is active which guarantees that the spikes can reach even the distal regions of\nthe dendrite [4]. Backpropagating spikes have been suggested to be the driving force for\nSTDP in the dendrite [5]. On the presynaptic side the main contribution to STDP comes\nfrom Ca2+ \ufb02ow through the NMDA channels [6].\nThe goal of this study is to derive an analytical solution for STDP on the basis of the\nbiophysical properties of the NMDA channel and the cell membrane. We will show that\nmainly the timing of the backpropagating spike determines the shape of the learning curve.\nWith fast decaying backpropagating spikes we obtain STDP while with slow decaying\nbackpropagating spikes we approximate temporally symmetric Hebbian learning.\n\nFigure 1: Schematic diagram of the model setup. The inset shows the time course of an\nNMDA response as modelled by Eq. 2.\n\n2 The Model\n\nThe goal is to de\ufb01ne a weight change rule which correlates the dynamics of an NMDA\nchannel with a variable which is linked to the dynamics of a backpropagating spike. The\nprecise biophysical mechanisms of STDP are still to a large degree unresolved. It is, how-\never, known that high levels of Ca2+ concentration resulting from Ca2+ in\ufb02ux mainly\nthrough NMDA-channels will lead to LTP, while lower levels will lead to LTD. Several\nbiophysically more realistic models for STDP were recently designed which rely on this\nmechanisms [7, 8, 9]. Recent physiological results (reviewed in detail in [10]), however\nsuggest that not only the Ca2+ concentration but maybe more importantly the change of\nthe Ca2+ concentration determines if LTP or LTD is observed. This clearly suggests that\na differential term should be included in the learning rule, when trying to model STDP.\nOn theoretical grounds such a suggestion has also been made by several authors [11] who\ndiscussed that the abstract STDP models [12] are related to the much older model class of\ndifferential Hebbian learning rules [13]. In our model we assume that the Ca2+ concen-\ntration and the membrane potential are highly correlated. Consequently, our learning rule\nutilises the derivative of the membrane potential for the postsynaptic activity.\n\nAfter having identi\ufb01ed the postsynaptic part of the weight change rule we have to de\ufb01ne\nthe presynaptic part. This shall be the conductance function of the NMDA channel [6].\n\nPlasticSynapseBP-Spike(cid:229)=iiIdtdVCBPINMDAggpostsyn. event = BP-spikepresyn. event at theplastic NMDAsynapseTrt100ms0\fThe conventional membrane equation reads:\n\nC\n\ndv(t)\n\ndt\n\n= \u03c1 g(t)[E \u2212 v(t)] + iBP (t) + Vrest \u2212 v(t)\n\n,\n\n(1)\n\nR\n\nwhere v is the membrane potential, \u03c1 the synaptic weight of the NMDA-channel and g, E\nare its conductance and equilibrium potential, respectively. The current, which a BP-spike\nelicits, is given by iBP and the last term represents the passive repolarisation property\nof the membrane towards its resting potential Vrest = \u221270 mV . We set the membrane\ncapacitance C = 50 pF and the membrane resistance to R = 100 M\u2126. E is set to zero.\nThe NMDA channel has the following equation:\n\ng(t) = \u00afg\n\ne\u2212b1t \u2212 e\u2212a1t\n\n[a1 \u2212 b1][1 + \u03bae\u2212\u03b3V (t)]\n\nFor simpler notation, in general we use inverse time-constants a1 = \u03c4\u22121\n, etc. In\naddition, the term a1 \u2212 b1 in the denominator is required for later easier integration in the\nLaplace domain. Thus, we adjust for this by de\ufb01ning \u00afg = 12 mS/ms which represents the\npeak conductance (4 nS) multiplied by b1 \u2212 a1. The other parameters were: a1 = 3.0/ms,\nb1 = 0.025/ms, \u03b3 = 0.06/mV . Since we will not vary the M g2+ concentration we have\nalready abbreviated: \u03ba = \u03b7[M g2+], \u03b7 = 0.33/mM, [M g2+] = 1 mM [14].\nThe synaptic weight of the NMDA channel is changed by correlating the conductance of\nthis NMDA channel with the change (derivative) of the membrane potential:\n\na , b1 = \u03c4\u22121\n\nb\n\nTo describe the weight change, we wish to solve:\n\n\u03c1 = g(t)v0(t)\n\nd\ndt\n\nZ \u221e\n\n\u2206\u03c1(T ) =\n\ng(T + \u03c4)v0(\u03c4)d\u03c4,\n\n0\n\nwhere T is the temporal shift between the presynaptic activity and the postsynaptic activ-\nity. The shift T > 0 means that the backpropagating spike follows after the trigger of\nthe NMDA channel. The shift T < 0 means that the temporal sequence of the pre- and\npostsynaptic events is reversed.\n\nTo solve Eq. 4 we have to simplify it, however, without loosing biophysical realism. In\nthis paper we are interested in different shapes of backpropagating spikes. The underly-\ning mechanisms which establish backpropagating spikes will not be addressed here. The\nbackpropagating spike shall be simply modelled as a potential change in the dendrite and\nits shape is determined by its amplitude, its rise time and its decay time.\n\nFirst we observe that the in\ufb02uence of a single (or even a few) NMDA-channels on the\nmembrane potential can be neglected in comparison to a BP-spike1, which, due to active\nprocesses, leads to a depolarisation of often more than 50 mV even at distal dendrites\nbecause of active processes [15]. Thus, we can assume that the dynamics of the membrane\npotential is established by the backpropagating spike and the resting potential Vrest:\n\n(2)\n\n(3)\n\n(4)\n\n(5)\n\n(6)\n\nThis equation can be further simpli\ufb01ed. Next we assume that the second passive repolarisa-\ntion term can also be absorbed into iBP , thus resulting to itotal(t) = iBP (t) + Vrest\u2212v(t)\n.\nTo this end we model itotal as a derivative of a band-pass \ufb01lter function:\n\nR\n\nC\n\ndv(t)\n\ndt\n\n= iBP (t) + Vrest \u2212 v(t)\n\nR\n\nitotal(t) = \u00afitotal\n\na2e\u2212a2t \u2212 b2e\u2212b2t\n\na2 \u2212 b2\n\n1Note that in spines, however, synaptic input can lead to large changes in the postsynaptic poten-\n\ntial. In such cases g(t) contributes substantially to v(t).\n\n\fwhere \u00afitotal is the current amplitude. This \ufb01lter function causes \ufb01rst an in\ufb02ux of charges\ninto the dendrite and then again an out\ufb02ux of charges. The time constants a2 and b2 deter-\nmine the timing of the current \ufb02ow and therefore the rise and decay time. The total charge\n\ufb02ux is zero so that the resting potential is reestablished after a backpropagating spike.\n\nIn this way the active de- and repolarising properties of a BP-spike can be combined with\nthe passive properties of the membrane, in practise by a curve \ufb01tting procedure which yields\na2, b2. As a result we \ufb01nd that the membrane equation in our case reduces to:\n\nWe receive the resulting membrane potential simply by integrating Eq. 6:\n\nC\n\ndv(t)\n\ndt\n\n= itotal(t)\n\nv(t) =\n\n\u00afitotal\nC\n\ne\u2212b2t \u2212 e\u2212a2t\n\na2 \u2212 b2\n\n(7)\n\n(8)\n\n(9)\n\n(12)\n\n(13)\n\n(14)\n\nNote the sign inversion between v (Eq. 8) and i (Eq. 6, the one being the derivative of the\nother.\n\nThe NMDA conductance g is more complex, because the membrane potential enters the\ndenominator in Eq. 2. To simplify we perform a Taylor expansion around v = 0 mV .\nWe expand around 0 mV and not around the resting potential. There are two reasons.\nFirst, we are interested in the open NMDA channel. This is the case for voltages towards\n0 mV . Second, the NMDA channel has a strong non-linearity around the resting potential.\nTowards 0 mV , however, the NMDA channel has a linear voltage/current curve. Therefore\nit makes sense to expand around 0 mV .\nThe NMDA conductance can now be written as:\n1\n\ne\u2212b1t \u2212 e\u2212a1t\n\n\u03ba + 1\nand \ufb01nally the potential v(t) (Eq. 8) can be inserted:\n\na1 \u2212 b1\n\ng(t) = \u00afg\n\n\u00b7 (\n\n+ \u03b3\u03bav(t)\n\n(\u03ba + 1)2 + . . .)\n\ng(t) = \u00afg\n\ne\u2212b1t \u2212 e\u2212a1t\n\n(cid:18) 1\n\na1 \u2212 b1\n+\n\n\u03ba + 1\n\n\u00b7\n\u00afitotal\u03b3\u03bae\u2212b2t\n\nC(\u03ba + 1)2(a2 \u2212 b2)\n\n\u2212\n\n\u00afitotal\u03b3\u03bae\u2212a2t\n\nC(\u03ba + 1)2(a2 \u2212 b2)\n\n(cid:19)\n\n+ . . .\n\n(10)\n\n(11)\n\nterminating the Taylor series after the second term this leads to three contributions to the\nconductance:\n\ng(t) =\n\ne\u2212b1t \u2212 e\u2212a1t\n\na1 \u2212 b1\n\n\u00afg\n\ng(0)\n\n\u03ba + 1\n\n{z\n|\n\u2212 \u00afg\u00afitotal\u03b3\u03ba\n|\n(\u03ba + 1)2C\n|\n\n\u00afg\u00afitotal\u03b3\u03ba\n(\u03ba + 1)2C\n\n+\n\n}\n{z\n{z\n\ng(1a)\n\ng(1b)\n\ne\u2212(b1+a2)t \u2212 e\u2212(a1+a2)t\n\n(a1 \u2212 b1)(a2 \u2212 b2)\n\ne\u2212(b1+b2)t \u2212 e\u2212(a1+b2)t\n(a1 \u2212 b1)(a2 \u2212 b2)\n\n}\n}\n\nTo perform the correlation in Eq. 4 we transform the required terms into the Laplace domain\ngetting:\n\ng(0,1a,1b)(t) = k\n\nitotal(t) = \u00afitotal\n\ne\u2212\u03b2t \u2212 e\u2212\u03b1t\n\n\u03b1 \u2212 \u03b2\n\na2e\u2212a2t \u2212 b2e\u2212b2t\n\na2 \u2212 b2\n\n\u2194 G(0,1a,1b)(s) = k\n\n1\n\n(s + \u03b1)(s + \u03b2)\n\n\u2194 Itotal(s) = \u00afitotal\n\ns\n\n(s + a2)(s + b2)\n\n(15)\n\n(16)\n\n\fwhere \u03b1 and \u03b2 take the coef\ufb01cient values from the exponential terms in g(0), g(1a), g(1b),\nrespectively and k are the corresponding multiplicative factors2.\nA correlation in the Laplace domain is expressed by Plancherel\u2019s theorem [16]:\n\n\u2206\u03c1 =\n\n1\n2\u03c0\n\n(cid:18)Z +\u221e\nZ +\u221e\nZ +\u221e\n\n\u2212\u221e\n\n\u2212\u221e\n\n\u2212\n\n+\n\n\u2212\u221e\n\nG(0)(\u2212\u0131\u03c9)e\u2212\u0131\u03c9T It(\u0131\u03c9)d\u03c9\n\nG(1a)(\u2212\u0131\u03c9)e\u2212\u0131\u03c9T It(\u0131\u03c9)d\u03c9\n\nG(1b)(\u2212\u0131\u03c9)e\u2212\u0131\u03c9T It(\u0131\u03c9)d\u03c9\n\n(cid:19)\n\n(17)\n\n(18)\n\n(19)\n\n(cid:19)\n(cid:19)(cid:21)\n\n(21)\n\n(22)\n\n+ =\n\nThe solution is calculated with the method of residuals which leads to a split of the result\ninto T \u2265 0 and T < 0 and we get:\nFor T \u2265 0:\n\u2206\u03c1(T ) =\n\n\u2212 a1e\u2212a1T\n\nb1e\u2212b1T\n\n(cid:20)\n\n(20)\n\n\u00afg\u00afitotal\n(\u03ba + 1)C\n\n(cid:18)\n(cid:18)\n\n\u2212 \u03b3\u03ba\u00afitotal\n\n(\u03ba+1)(a2\u2212b2)C\n\n+ \u03b3\u03ba\u00afitotal\n\n(\u03ba+1)(a2\u2212b2)C\n\n(0)\n+\n\nB\n\nA\n(b1+a2)e\u2212(b1+a2)T\n\n(0)\n+\n\nB\n\n(1)\n+\n\n\u2212 (a1+a2)e\u2212(a1+a2)T\n\nA\n\n(1)\n+\n\n(b1+b2)e\u2212(b1+b2)T\n\nB\n\n(1)\n+\n\n\u2212 (a1+b2)e\u2212(a1+b2)T\n\nA\n\n(1)\n+\n\n+ = (a1\u2212b1)(a1 +a2)(a1 +b2), A(1)\n\nwith A(0)\n(a1 \u2212 b1)(b1 + b2)(a2 + b1), B(1)\nFor T < 0:\n\n\u2206\u03c1(T ) =\n\n\u00afg\u00afitotal\n(\u03ba + 1)C\n\n+ = (a1\u2212b1)(a1 +2a2)(a1 +a2 +b2), B(0)\n(cid:20)\n\n+ = (a1 \u2212 b1)(2a2 + b1)(a2 + b1 + b2).\n(cid:19)\n(cid:19)(cid:21)\n\n\u2212 \u03b3\u03ba\u00afitotal\n\n(\u03ba+1)(a2\u2212b2)C\n\n\u2212 b2eb2T\n\n\u2212 b2eb2T\n\n(cid:18)\n(cid:18)\n\na2ea2T\n(1a)\nA\n\u2212\n\na2ea2T\n(0)\n\u2212\n\n(1a)\n\u2212\n\n(0)\n\u2212\n\nB\n\nB\n\nA\n\n+ \u03b3\u03ba\u00afitotal\n\n(\u03ba+1)(a2\u2212b2)C\n\na2ea2T\n(1b)\nA\n\u2212\n\n\u2212 b2eb2T\n\nB\n\n(1b)\n\u2212\n\n(23)\n\n(24)\n\n(25)\n\nwith A(0)\u2212 = (a2 \u2212 b2)(a1 + a2)(a2 + b1), A(1a)\u2212 = (a2 \u2212 b2)(a1 + 2a2)(2a2 + b1), A(1b)\u2212 =\n(a2 \u2212 b2)(a1 + b2 + a2)(a2 + b1 + b2), B(0)\u2212 = (a2 \u2212 b2)(a1 + b2)(b1 + b2), B(1a)\u2212 =\n(a2 \u2212 b2)(a1 + a2 + b2)(b1 + a2 + b2), B(1b)\u2212 = (a2 \u2212 b2)(a1 + 2b2)(b1 + 2b2).\nThe resulting equations contain interesting symmetries which makes the interpretation easy.\nWe observe that they split into three terms. For T > 0 the \ufb01rst term captures the NMDA\nin\ufb02uence only, while for T < 0 it captures the in\ufb02uence of only the BP-spike (apart from\nscaling factors). Mixed in\ufb02uences arise from the second and third terms which scale with\nthe peak current amplitude \u00afitotal of the BP-spike.\n\n3 Results\n\nWhile the properties of mature NMDA channels are captured by the parameters given for\nEq. 2 and remain fairly constant, BP-spikes change their shapes along the dendrite. Thus,\n\n2We use lower-case letters for functions in the time-domain and upper-case letters for their equiv-\n\nalent in the Laplace domain.\n\n\fFigure 2: (A-F) STDP-curves obtained from Eqs. 22, 25 and corresponding normalised\nBP-spikes (G-I, \u00afitotal = 1, left y-axis: current, right y-axis: integrated potential). Panels\nA-C were obtained with different peak currents \u00afitotal = 0.5 nA, 0.1nA and 25pA. These\ncurrents cause peak voltages of 40mV, 50mV and 40mV respectively. Panels D-F were\nall simulated with a peak current of \u00afitotal = 5.0 nA. This current is unrealistic, however, it\nis chosen for illustrative purposes to show the different contributions to the learning curve\n(the dashed lines for G(0) and the dotted lines for G(1a,b) and the solid lines for the sum\nof the two contributions). Time constants for the BP-spikes were: (A,D,G) a\u22121\n2 = \u03c4a =\n0.0095 ms, b\u22121\n2 = \u03c4b = 0.01 ms (B,E,H) \u03c4a = 0.05 ms, \u03c4b = 0.1 ms (C,F,I) \u03c4a = 0.1 ms,\n\u03c4b = 1.0 ms.\n\nwe kept the NMDA properties unchanged and varied the time constants of the BP-spikes\nas well as the current amplitude to simulate this effect. Fig. 2 shows STDP curves (solid\nlines, A-F) and the corresponding BP-spikes (G-I). The contributions of the different terms\nto the STDP curves are also shown (\ufb01rst term, dashed, as well as second and third term\nscaled with their fore-factor, dotted). All curves have arbitrary units. As expected we \ufb01nd\nthat the \ufb01rst term dominates for small (realistic) currents (top panels), while the second and\nthird terms dominate for higher currents (middle panels). Furthermore, we \ufb01nd that long\nBP-spikes will lead to plain Hebbian learning, where only LTP but no LTD is observed\n(B,C,E,F).\n\n4 Discussion\n\nWe believe that two of our \ufb01ndings could be of longer lasting relevance for the under-\nstanding of synaptic learning, provided they withstand physiological scrutinising: 1) The\nshape of the weight change curves heavily relies on the shape of the backpropagating spike.\n2) STDP can turn into plain Hebbian learning if the postsynaptic depolarisation (i.e., the\nBP-spike) rises shallow.\n\nPhysiological studies suggest that weight change curves can indeed have a widely varying\nshape (reviewed in [17]). In this study we argue that in particular the shape of the back-\n\n\fpropagating spike in\ufb02uences the shape of the weight change curve. In fact the dendrites\ncan be seen as active \ufb01lters which change the shape of backpropagating spikes during their\njourney to the distal parts of the dendrite [18]. In particular, the decay time of the BP spike\nis increased in the distal parts of the dendrite [15]. The different decay times determine if\nwe get pure symmetric Hebbian learning or STDP (see Fig. 2). Thus, the theoretical result\nwould suggest temporal symmetric Hebbian learning in the distal dendrites and STDP in\nthe proximal dendrites. From a computational perspective this would mean that the distal\ndendrites perform principle component analysis [19] and the proximal dendrites temporal\nsequence learning [20].\n\nNow, our model has to be compared to other models of STDP. We can count our model\nto the \u201cstate variable models\u201d. Such models can either adopt a rather descriptive approach\n[21], where appropriate functions are being \ufb01t to the measured weight change curves. Oth-\ners are closer to the kinetic models in trying to \ufb01t phenomenological kinetic equations\n[7, 22, 23, 9]. Those models establish a more realistic relation between calcium concen-\ntration and membrane potential. The calcium concentration seems to be a low-pass \ufb01ltered\nversion of the membrane potential [24]. Such a low pass \ufb01lter hlow could be added to the\nlearning rule Eq. 3 resulting in: d\u03c1/dt = g(t)hlow(t) \u2217 v0(t).\nThe approaches of [9] as well as of Karmarkar and co-workers [23] are closely related to\nour model. Both models investigate the effects of different calcium concentration levels by\nassuming certain (e.g. exponential) functional characteristics to govern its changes. This\nallows them to address the question of how different calcium levels will lead to LTD or\nLTP [25]. Both model-types [9, 23, 8] were designed to produce a zero-crossing (transition\nbetween LTD and LTP) at T = 0. 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