{"title": "Finding the Key to a Synapse", "book": "Advances in Neural Information Processing Systems", "page_first": 138, "page_last": 144, "abstract": null, "full_text": "Finding the Key to a Synapse \n\nThomas Natschlager & Wolfgang Maass \nInstitute for Theoretical Computer Science \n\nTechnische Universitat Graz, Austria \n{tnatschl, maass}@igi.tu-graz.ac.at \n\nAbstract \n\nExperimental data have shown that synapses are heterogeneous: different \nsynapses respond with different sequences of amplitudes of postsynaptic \nresponses to the same spike train. Neither the role of synaptic dynamics \nitself nor the role of the heterogeneity of synaptic dynamics for com(cid:173)\nputations in neural circuits is well understood. We present in this article \nmethods that make it feasible to compute for a given synapse with known \nsynaptic parameters the spike train that is optimally fitted to the synapse, \nfor example in the sense that it produces the largest sum of postsynap(cid:173)\ntic responses. To our surprise we find that most of these optimally fitted \nspike trains match common firing patterns of specific types of neurons \nthat are discussed in the literature. \n\n1 Introduction \n\nA large number of experimental studies have shown that biological synapses have an in(cid:173)\nherent dynamics, which controls how the pattern of amplitudes of postsynaptic responses \ndepends on the temporal pattern of the incoming spike train. Various quantitative models \nhave been proposed involving a small number of characteristic parameters, that allow us to \npredict the response of a given synapse to a given spike train once proper values for these \ncharacteristic synaptic parameters have been found. The analysis of this article is based \non the model of [1], where three parameters U, F, D control the dynamics of a synapse \nand a fourth parameter A - which corresponds to the synaptic \"weight\" in static synapse \nmodels - scales the absolute sizes of the postsynaptic responses. The resulting model pre(cid:173)\ndicts the amplitude Ak for the kth spike in a spike train with interspike intervals (lSI's) \n.60 1 , .60 2 , \u2022 .. ,.6ok-l through the equations l \n\nAk = A\u00b7 Uk' Rk \nUk = U +Uk-l(1- U)exp(-.6ok-dF) \nRk = 1 + (Rk-l - Uk-1Rk-l - 1) exp( -.6ok-d D) \n\n(1) \n\nwhich involve two hidden dynamic variables U E [0,1] and R E [0,1] with the initial \nconditions Ul = U and Rl = 1 for the first spike. These dynamic variables evolve in de(cid:173)\npendence of the synaptic parameters U, F, D and the interspike intervals of the incoming \n\nITo be precise: the term Uk-1Rk-l in Eq. (1) was erroneously replaced by ukRk-l in the cor(cid:173)\n\nresponding Eq. (2) of [1]. The model that they actually fitted to their data is the model considered in \nthis article. \n\n\fA \n\n0 .75 \n\nu \n! 0.5 \n\nC \n\n0 .25 \n\n0.75 \n\no 0 \n\nB \n\ninput spike train \n\nfA \n\nI IIIIII \nI I fi I \nF2 I lUI I \n\na \n\nIIIIIII IIIII \nIII \nII m! I I ] I II \nI II \n~ lIn! Un \n\n2 \n3 \ntime [sec) \n\n4 \n\n5 \n\nFigure 1: Synaptic heterogeneity. A The parameters U, D, and F can be determined for \nbiological synapses. Shown is the distribution of values for inhibitory synapses investigated \nin [2] which can be grouped into three mayor classes: facilitating (Ft), depressing (F2) \nand recovering (F3). B Synapses produce quite different outputs for the same input for \ndifferent values of the parameters U, D, and F. Shown are the amplitudes Uk \u2022 Rk (height \nof vertical bar) of the postsynaptic response of a FI-type and a F2-type synapse to an \nirregular input spike train. The parameters for synapses fA and F2 are the mean values for \nthe synapse types FI and F2 reported in [2]: (U, D, F) = (0.16,45 msec, 376 msec) for FI , \nand (0.25,706 msec, 21 msec) for F2 \u2022 \n\nspike train. 2 It is reported in [2] that the synaptic parameters U, F, D are quite heteroge(cid:173)\nneous, even within a single neural circuit (see Fig. IA). Note that the time constants D and \nF are in the range of a few hundred msec. The synapses investigated in [2] can be grouped \ninto three major classes: facilitating (FI), depressing (F2) and recovering (F3). Fig. IB \ncompares the output of a typical FI-type and a typical F2-type synapse in response to a \ntypical irregular spike train. One can see that the same input spike train yields markedly \ndifferent outputs at these two synapses. \n\nIn this article we address the question which temporal pattern of a spike train is optimally \nfitted to a given synapse characterized by the three parameters U, F, D in a certain sense. \nOne possible choice is to look for the temporal pattern of a spike train which produces the \nlargest integral of synaptic current. Note that in the case where the dendritic integration is \napproximately linear the integral of synaptic current is proportional to the sum 'E~=l A . \nUk . Rk of postsynaptic responses. We would like to stress, that the computational methods \nwe will present are not restricted to any particular choice of the optimality criterion. For \nexample one can use them also to compute the spike train which produces the largest peak \nof the postsynaptic membrane voltage. However, in the following we will focus on the \nquestion which temporal pattern of a spike train produces the largest sum 'E~=l A\u00b7 Uk . Rk \nof postsynaptic responses (or equivalently the largest integral of postsynaptic current). \n\nMore precisely, we fix a time interval T , a minimum value ~min for lSI's, a natural number \nN , and synaptic parameters U, F, D . We then look for that spike train with N spikes during \nT and lSI's 2:: ~min that maximizes 'E~=l A\u00b7 Uk' Rk. Hence we seek for a solution(cid:173)\nthat is a sequence ofISI's ~l' ~2' ... , ~N-I -\n\nto the optimization problem \n\nN \n\nmaximize LA. Uk . Rk under L ~k ~ T and ~min ~ ~k' 1 ~ k < N . \n\nN-I \n\n(2) \n\nk=l \n\nk=l \n\nIn Section 2 of this article we present an algorithmic approach based on dynamic program-\n\n2It should be noted that this deterministic model predicts the cumulative response of a population \n\nof stochastic release sites that make up a synaptic connection. \n\n\fming that is guaranteed to find the optimal solution of this problem (up to discretization \nerrors), and exhibit for major types of synapses temporal patterns of spike trains that are \noptimally fitted to these synapses. In Section 3 we present a faster heuristic method for \ncomputing optimally fitted spike trains, and apply it to analyze how their temporal pattern \ndepends on the number N of allowed spikes during time interval T, i.e., on the firing rate \nf = NIT. Furthermore we analyze in Section 3 how changes in the synaptic parameters \nU, F, D affect the temporal pattern of the optimally fitted spike train. \n\n2 Computing Optimal Spike Trains for Common Types of Synapses \nDynamic Programming For T = 1000 msec and N = 10 there are about 2100 spike \ntrains among which one wants to find the optimally fitted one. We show that a computation(cid:173)\nally feasible solution to this complex optimization problem can be achieved via dynamic \nprogramming. We refer to [3] for the mathematical background of this technique, which \nalso underlies the computation of optimal policies in reinforcement learning. We consider \nthe discrete time dynamic system described by the equation \n\nXl = (U, 1, 0) and Xk+1 = g(Xk, ak) for k = 1, ... , N - 1 \n\n(3) \n\nwhere Xk describes the state of the system at step k, and ak is the \"control\" or \"action\" taken \nat step k. In our case Xk is the triple (Uk, Rk, tk) consisting of the values of the dynamic \nvariables U and R used to calculate the amplitude A . Uk . Rk of the kth postsynaptic \nresponse, and the time tk of the arrival of the kth spike at the synapse. The \"action\" ak \nis the length Ilk E [Ilmin , T -\ntkJ of the kth lSI in the spike train that we construct, \nwhere Ilmin is the smallest possible size of an lSI (we have set Ilmin = 5 msec in our \ncomputations). As the function gin Eq. (3) we take the function which maps (Uk, Rk, tk) \nand Ilk via Eq. (1) on (uk+l,Rk+l,tk+1) for tk+1 = tk + Ilk. The \"reward\" for the \nkth spike is A . Uk . Rk, i.e., the amplitude of the postsynaptic response for the kth spike. \nHence maximizing the total reward J(Xl) = 2:~=1 A\u00b7 Uk\u00b7 Rk is equivalent to solving the \nmaximization problem (2). The maximal possible value of J l (Xl) can be computed exactly \nvia the equations \n\nIN(XN) = A\u00b7 UN\u00b7 RN \nJk(Xk) = \n\nmax \n\n~E[~min,T-tkl \n\n(A\u00b7 Uk\u00b7 Rk + Jk+1(g(Xk, Il))) \n\n(4) \n\nbackwards from k = N - 1 to k = 1. Thus the optimal sequence al, ... , aN-l of \n\"actions\" is the sequence Ill, .. . , IlN -1 of lSI's that achieves the maximal possible value \nof 2:~=1 A . Uk . Rk . Note that the evaluation of Jk(Xk) for a single value of Xk requires \nthe evaluation of Jk+1 (Xk+1) for many different values of Xk+1. 3 \n\nThe \"Key\" to a Synapse We have applied the dynamic programming approach to three \nmajor types of synapses reported in [2]. The results are summarized in Fig. 2 to Fig. 5. \nWe refer informally to the temporal pattern of N spikes that maximizes the response of \na particular synapse as the \"key\" to this synapse. It is shown in Fig. 3 that the \"keys\" \nfor the inhibitory synapses Fi and F2 are rather specific in the sense that they exhibit a \nsubstantially smaller postsynaptic response on any other of the major types of inhibitory \nsynapses reported in [2]. The specificity of a \"key\" to a synapse is most pronounced for \nspiking frequencies f below 20 Hz. One may speculate that due to this feature a neuron can \nactivate -\na particular subpopulation of its target \nneurons by generating a series of action potentials with a suitable temporal pattern, see \n\neven without changing its firing rate -\n\n3When one solves Eq. (4) on a computer, one has to replace the continuous state variable Xk by a \ndiscrete variable Xk , and round Xk+l := g(Xk'~) to the nearest value of the corresponding discrete \nvariable Xk+l. For more details about the discretization of the model we refer the reader to [4]. \n\n\f0.75 \n\n~ .!!!.. 0.5 \no \n\n0.25 \n\nFl \n\nIII I I I I I I I I I I I \n\nII \n\nI \n\nI \n\nF3 II \ni.~ 0.5 \no \n\na a \n\n0.25 \n\nu \n\n0.2 \n\n0.4 \n\n0.6 \n\n0.8 \n\ntime [sec] \n\nFigure 2: Spike trains that maximize the sum of postsynaptic responses for three com(cid:173)\nmon types of synapses (T = 0.8 sec, N = 15 spikes). The parameters for synapses \nFi, F2, and F3 are the mean values for the synapse types FI, F2 and F3 reported in \n[2] : (U, D, F} = (0.16,45 msec, 376 msec} for Fl , (0.25,706 msec, 21 msec} for F2 , and \n(0.32, 144 msec, 62 msec} for F3 . \n\nkey to synapse Fl \nII I I I I I I I I I I I I \nkey to synapse F2 \nIIIII \n\nIIII \n\nresponse of a \n\nresponse of a \n\nFl-type synapse F2-type synapse \n\nI \nII \n\n({/ % \n\n81% I \nI \n\nthe \"keys\" to the synapses Fl and F2 -\n\nFigure 3: Specificity of optimal spike trains. The optimal spike trains for synapses Fl and \nobtained for T = 0.8 sec and N = 15 \nF2 -\nspikes are tested on the synapses Fl and F2 . If the \"key\" to synapse Fl (F2 ) is tested on \nthe synapse Fl (F2 ) this synapse produces the maximal (l00 %) postsynaptic response. If \non the other hand the \"key\" to synapse Fl (F2) is tested on synapse F2 (Fl) this synapse \nproduces significantly less postsynaptic response. \n\nFig. 4. Recent experiments [5, 6] show that neuromodulators can control the firing mode \nof cortical neurons. In [5] it is shown that bursting neurons may switch to regular firing \nif norepinephine is applied. Together with the specificity of synapses to certain temporal \npatterns these findings point to one possible mechanism how neuromodulators can change \nthe effective connectivity of a neural circuit. \n\nRelation to discharge patterns A noteworthy aspect of the \"keys\" shown in Fig. 2 (and \nin Fig. 6 and Fig. 7) is that they correspond to common firing patterns (\"accommodat(cid:173)\ning\", \"non-accommodating\", \"stuttering\", \"bursting\" and \"regular firing\") of neocortical \ninterneurons reported under controlled conditions in vitro [2, 5] and in vivo [7]. For ex(cid:173)\nample the temporal patterns of the \"keys\" to the synapses Fl , F2, and F3 are similar to \nthe discharge patterns of \"accommodating\" [2], \"bursting\" [5, 7], and \"stuttering\" [2] cells \nrespectively. \n\nWhat is the role of the parameter A? Another interesting effect arises if one compares \nthe optimal values of the sum Ek=l Uk . Rk (i.e. A = 1) for synapses H, F 2 , and F3 (see \nFig. 5A) with the maximal values of E~=l A . Uk \u2022 Rk (see Fig. 5B), where we have set \n\nN \n\n-\n\n-\n\n-\n\n\fsynaptic response \n\nkey to synapse Fl \n\n11111111 I I I I I I I \n\nFl lo--\n\nsynaptic response \n\nFl i 0--\n\nkey to synapse F2 \n\n-te--=-III -----....:11=---------=.11_-=-1 ~< I \n\nFigure 4: Preferential addressing of postsynaptic targets. Due to the specificity of a \"key\" to \na synapse a presynaptic neuron may address (i.e. evoke stronger response at) either neuron \nA or B, depending on the temporal pattern of the spike train (with the same frequency \nf = NIT) it produces (T = 0.8 sec and N = 15 in this example). \n\nF2 \n\n(1-\n\nA 4 ,-----\n\n-\n\n-\n\nB 15 \n(3 \n~10 \n\n,-----\n\nr - - - - -\n\n-\n\nA::1 \n\nA::1 \n\nA::1 \n\nA::3.24 \n\nA=7.76 \n\nA:3.44 \n\no \n\nFl \n\nF2 \n\nF3 \n\no \n\nFl \n\nF2 \n\nF3 \n\nFigure 5: A Absolute values of the sums 2::=1 Uk . Rk if the key to synapse Pi is applied \nto synapse Pi, i = 1,2,3. B Same as panel A except that the value of 2::=1 A . Uk . Rk is \nplotted. For A we used the value of G max (in nS) reported in [2]. The quotient max I min \nis 1.3 compared to 2.13 in panel A. \n\nA equal to the value of Gmax reported in [2]. Whereas the values of Gmax vary strongly \namong different synapse types (see Fig. 5B), the resulting maximal response of a synapse \nto its proper \"key\" is almost the same for each synapse. Hence, one may speculate that the \nsystem is designed in such a way that each synapse should have an equal influence on the \npostsynaptic neuron when it receives its optimal spike train. However, this effect is most \nevident for a spiking frequency f = NIT of 10 Hz and vanishes for higher frequencies. \n\n3 Exploring the Parameter Space \n\nSequential Quadratic Programming The numerical approach for approximately com(cid:173)\nputing optimal spike trains that was used in section 2 is sufficiently fast so that an average \nPC can carry out any of the computations whose results were reported in Fig. 2 within a few \nhours. To be able to address computationally more expensive issues we used a a nonlinear \noptimization algorithm known as \"sequential quadratic programming\" (SQP)4 which is the \nstate of the art approach for heuristically solving constrained optimization problems such \nas (2). We refer the reader to [8] for the mathematical background of this technique and \nto [4] for more details about the application of SQP for approximately computing optimal \nspike trains. \n\nOptimal Spike Trains for Different Firing Rates First we used SQP to explore the \neffect of the spike frequency f = N IT on the temporal pattern of the optimal spike train. \nFor the synapses PI, P2 , and Pa we computed the optimal spike trains for frequencies \n\n4We used the implementation (function constr) which is contained in the MATLAB Optimiza(cid:173)\n\ntion Toolbox (see http : //www . ma thworks . com/products/ optimiz a tion/). \n\n\fkeys to Fl synapse \n\n111111 11111 11 \n\nh 40 11 111111 111 1111111111111111111111111 \n\n--~ 35 111 11 1111 11 \n!. 30 11 111 11111 11111 111111 \ng 25 111 11111 1 11 111111 1 II I \n~ 20 11 1111 1 1 1 1 111 1 1 \n~ 15 1111 11 1 111 1 I I \n\nIII \n\nI\n\nI \n\nh 40 ~ \n\n11 11 --~ 35 1 \n\nII \n\"-. 30 I \nC 25 1 \nt:: \ng. 20 1 \nQ) \n~ 15 1 \n\nkeys to F2 synapse \nI I 1 I I I I \nI I I 1 1 1 I \nI I I 1 1 1 \n\nI \n\nI \n\nkeys to F3 synapse \n\nh 40 111111111 11111 1111 111 1111 11111 11111 11 \n\n11 111111 11111 11111 111111 111111 \n\n11 11 1111 111 111111 111 1111 \n\n111 111 111 1111 111 1111 \n\nII \n\nII \n\nII \n\nII \n\nI I \n\nII \n\nI I I \n\n--~ 35 \n\nII \n\"-. 30 \n\nC 25 \nt:: \ng. 20 \nQ) \n<./:: 15 \n\nQ) \n\n0 \n\n0.2 \n\n0.4 \n0.6 \ntime [sec] \n\n0.8 \n\n0.2 \n\n0.4 \n0.6 \ntime [sec] \n\n0.8 \n\n0 \n\n0.2 \n\n0.4 \n0.6 \ntime [sec] \n\n0.8 \n\nFigure 6: Dependence of the optimal spike train of the synapses FL F2 , and F3 on the \nspike frequency f = NIT (T = 1 sec, N = 15, ... ,40). \n\n0 .60 \n0.50 \n0 .45 \n0.40 \n0 .35 \n::) 0 .30 \n0 .25 \n0.20 \n0.15 \n0 .10 \n\nI \nI \nI \n\nI \n\nI \nI \nI \n\nI \nI \nI \nII \nII \n\nI \nI \nI \n\nI \nI \nI \n\nII \nII \n\n1111 \n\nIII \n\nI \nI \nI \n\nII \nII \n\nI \nI \nI \n\nIII \n\n11111 \n\ni \n0 \n\n11111 \n\ni \n\n0.25 \n\nI \nI \nI \n\nI \nI \nI \n\nII \nII \n\n1111 \n\ni \n\n0 .5 \n\nI \nI \nI \n\nI \nI \nI \n\nI \nI \nI \n\nIII \n\nII \nII \n\n11111 \n\nI \nI \nI \n\nIII \n\nII \nII \n\n1111 \n\ni \n\n0.75 \n\ntime [sec] \n\nI \nI \nI \n\nI \nI \nI \nII \n\nII \n\nI \nI \nI \n\nII \nII \nII \nI II \nIII \n1111 \n\nFigure 7: Dependence of the optimal spike train on the synaptic parameter U. It is shown \nhow the optimal spike train changes if the parameter U is varied. The other two parameters \nare set to the value corresponding to synapse F3: D = 144 msec and F = 62 msec. \nThe black bar to the left marks the range of values (mean \u00b1 std) reported in [2] for the \nparameter U. To the right of each spike train we have plotted the corresponding value of \nJ = Ef=i ukRk (gray bars). \n\nranging from 15 Hz to 40 Hz. The results are summarized in Fig. 6. For synapses Fi and \nF2 the characteristic spike pattern (Fi ... accommodating, F2 ... stuttering) is the same for \nall frequencies. In contrast, the optimal spike train for synapse F3 has a phase transition \nfrom \"stuttering\" to \"non-accommodating\" at about 20 Hz. \n\nThe Impact of Individual Synaptic Parameters We will now address the question how \nthe optimal spike train depends on the individual synaptic parameters U, F, and D. The \nresults for the case of F3-type synapses and the parameter U are summarized in Fig. 7. For \nresults with regard to other parameters and synapse types we refer to [4]. We have marked \nin Fig. 7 with a black bar the range of U for F3-type synapses reported in [2]. It can be \nseen that within this parameter range we find \"regu]ar\" and \"bursting\" spike patterns. Note \nthat the sum of postsynaptic responses J (gray horizontal bars in Fig. 7) is not proportional \nto U. While U increases from 0.1 to 0.6 (6 fold change) J only increases by a factor of 2. \nThis seems to be interesting since the parameter U is closely related to the initial release \nprobability of a synapse, and it is a common assumption that the \"strength\" of a synapse is \nproportional to its initial release probability. \n\n\f4 Discussion \n\nWe have presented two complementary computational approaches for computing spike \ntrains that optimize a given response criterion for a given synapse. One of these meth(cid:173)\nods is based on dynamic programming (similar as in reinforcement learning), the other one \non sequential quadratic programming. These computational methods are not restricted to \nany particular choice of the optimality criterion and the synaptic model. In [4] applications \nof these methods to other optimality criteria, e.g. maximizing the specificity, are discussed. \n\nIt turns out that the spike trains that maximize the response of Fl-, F2- and F3-type synapses \n(see Fig. 1) are well known firing patterns like \"accommodating\", \"bursting\" and \"regular \nfiring\" of specific neuron types. Furthermore for Fl- and F3-type synapses the optimal \nspike train agrees with the most often found firing pattern of presynaptic neurons reported \nin [2], whereas for F2-type synapses there is no such agreement; see [4]. This observa(cid:173)\ntion provides the first glimpse at a possible functional role of the specific combinations of \nsynapse types and neuron types that was recently found in [2]. \n\nAnother noteworthy aspect of the optimal spike trains is their specificity for a given synapse \n(see Fig. 3).: suitable temporal firing patterns activate preferentially specific types of \nsynapses. One potential functional role of such specificity to temporal firing patterns is \nthe possibility of preferential addressing of postsynaptic target neurons (see Fig. 4). Note \nthat there is experimental evidence that cortical neurons can switch their intrinsic firing be(cid:173)\nhavior from \"bursting\" to \"regular\" depending on neuromodulator mediated inputs [5, 6]. \nThis findings provide support for the idea of preferential addressing of postsynaptic targets \nimplemented by the interplay of dynamic synapses and the intrinsic firing behavior of the \npresynaptic neuron. \n\nFurthermore our analysis provides the platform for a deeper understanding of the specific \nrole of different synaptic parameters, because with the help of the computational techniques \nthat we have introduced one can now see directly how the temporal structure of the optimal \nspike train for a synapse depends on the individual synaptic parameters. We believe that \nthis inverse analysis is essential for understanding the computational role of neural circuits. \n\nReferences \n[1] H. Markram, Y. Wang, and M. Tsodyks. Differential signaling via the same axon of neocortical \n\npyramidal neurons. Proc. Natl. Acad. Sci. , 95:5323- 5328, 1998. \n\n[2] A. Gupta, Y. Wang, and H. Markram. Organizing principles for a diversity of GABAergic in(cid:173)\n\nterneurons and synapses in the neocortex. Science, 287:273- 278, 2000. \n\n[3] D. P. Bertsekas. Dynamic Programming and Optimal Control, Volume 1. Athena Scientific, \n\nBelmont, Massachusetts, 1995. \n\n[4] T. Natschlager and W. Maass. Computing the optimally fitted spike train for a synapse. sub(cid:173)\n\nmitted for publication, electronically available via http : //www. igi . TUGr a z .at/igi/ \ntn a tschl/psfiles/synkey-journal. ps . gz, 2000. \n\n[5] Z. Wang and D. A. McCormick. Control of firing mode of corticotectal and corticopontine layer \nV burst generating neurons by norepinephrine. Journal of Neuroscience, 13(5):2199-2216, 1993. \n\n[6] J. C. Brumberg, L. G. Nowak, and D. A. McCormick. Ionic mechanisms underlying repet(cid:173)\n\nitive high frequency burst firing in supragranular cortical neurons. Journal of Neuroscience, \n20(1):4829-4843, 2000. \n\n[7] M. Steriade, I. Timofeev, N. Diirmiiller, and F. Grenier. Dynamic properties of corticothalamic \nneurons and local cortical interneurons generating fast rhytmic (30-40 hz) spike bursts. Journal \nof Neurophysiology, 79:483-490, 1998. \n\n[8] M. J. D. Powell. Variable metric methods for constrained optimization. \n\nIn A. Bachem, \nM Grotschel , and B. Korte, editors, Mathematical Programming: The State of the Art, pages \n288- 311. Springer Verlag, 1983. \n\n\f", "award": [], "sourceid": 1813, "authors": [{"given_name": "Thomas", "family_name": "Natschl\u00e4ger", "institution": null}, {"given_name": "Wolfgang", "family_name": "Maass", "institution": null}]}