{"title": "Correlational Strength and Computational Algebra of Synaptic Connections Between Neurons", "book": "Neural Information Processing Systems", "page_first": 270, "page_last": 277, "abstract": null, "full_text": "270 \n\nCorrelational Strength and Computational Algebra \n\nof Synaptic Connections Between Neurons \n\nEberhard E. Fetz \n\nDepartment of Physiology & Biophysics, \n\nUniversity of Washington, Seattle, WA 98195 \n\nABSTRACT \n\nIntracellular recordings in spinal cord motoneurons and cerebral \ncortex neurons have provided new evidence on the correlational strength of \nmonosynaptic connections, and \nthe relation between the shapes of \npostsynaptic potentials and the associated increased firing probability. In \nthese cells, excitatory postsynaptic potentials (EPSPs) produce cross(cid:173)\ncorrelogram peaks which resemble in large part the derivative of the EPSP. \nAdditional synaptic noise broadens the peak, but the peak area -- i.e., the \nnumber of above-chance firings triggered per EPSP -- remains proportional to \nthe EPSP amplitude. A typical EPSP of 100 ~v triggers about .01 firings per \nEPSP. The consequences of these data for information processing by \npolysynaptic connections is discussed. The effects of sequential polysynaptic \nlinks can be calculated by convolving the effects of the underlying \nmonosynaptic connections. The net effect of parallel pathways is the sum of \nthe individual contributions. \n\nINTRODUCTION \n\nInteractions between neurons are determined by the strength and \ndistribution of their synaptic connections. \nThe strength of synaptic \ninteractions has been measured directly in the central nervous system by two \ntechniques. Intracellular recording reveals the magnitude and time course of \npostsynaptic potentials (PSPs) produced by synaptic connections, and cross(cid:173)\ncorrelation of extracellular spike trains measures the effect of the PSP's on the \nfiring probability of the connected cells. The relation between the shape of \nexcitatory postsynaptic potentials (EPSPs) and the shape of the cross(cid:173)\ncorrelogram peak they produce has been empirically investigated in cat \nmotoneurons 2,4,5 and in neocortical cells 10. \n\nRELATION BETWEEN EPSP'S AND CORRELOGRAM PEAKS \n\nSynaptic interactions have been studied most thoroughly in spinal \nFigure 1 illustrates the membrane potential of a \ncord motoneurons. \nrhythmically firing motoneuron, and the effect of EPSPs on its firing. An \nEPSP occurring sufficiently close to threshold (8) will cause the motoneuron \nto fire and will advance an action potential to its rising edge (top). \nMathematical analysis of this threshold-crossing process predicts that an \nEPSP with shape e(t) will produce a firing probability f(t), which resembles \n\n\u00a9 American Institute of Phy~ics 1988 \n\n\f271 \n\n'I \n\nf; \n:: \ni : I ' \n.. ) : \n\n.\",.\"\"..,.\", \n\nr \nI \n'I \nI \n'\n\\ \n\\ \n.... \n.,...,.\"\" \n\n8 --.----r \n\n...\",/ \n\n----.... ~-\"\"\" \n\n.... \n\n.,.,,,, \n\n,; \n\n/' \n\n/..-::--\n, \n.. \n, .... \n, \n.. \n\n.... \n\nI \n\nI \n\nEPSP \n\ne(t) \n\nCROSS(cid:173)\n\nCORRELOGRAM \n\nf(t) \n\nt \n\nt \n\nTIME \n\nFig. 1. The relation between EPSP's and motoneuron firing. Top: membrane trajectory of \nrhythmically firing motoneuron, showing EPSP crossing threshold (8) and shortening the \nnormal interspike interval by advancing a spike. V(t) is difference between membrane \npotential and threshold. Middle: same threshold-crossing process aligned with EPSP, with \nv(t) plotted as falling trajectory. Intercept (at upward arrow) indicates time of the advanced \naction potential. Bottom: Cross-correlation histogram predicted by threshold crossings. The \npeak in the firing rate f(t) above baseline (fo) is produced by spikes advanced from baseline, \nas indicated by the changed counts for the illustrated trajectory. Consequently, the area in \nthe peak equals the area of the subsequent trough. \n\n\f272 \n\nthe derivative of the EPSP 4,8. Specifically, for smooth membrane potential \ntrajectories approaching threshold (the case of no additional synaptic noise): \n\nf(t) = fo + (fo/v) del dt \n\n(1) \nwhere fo is the baseline firing rate of the motoneuron and v is the rate of \nclosure between motoneuron membrane potential and threshold. This \nrelation can be derived analytically by tranforming the process to a \ncoordinate system aligned with the EPSP (Fig. 1, middle) and calculating the \nrelative timing of spikes advanced by intercepts of the threshold trajectories \nwith the EPSP 4. The above relation (1) is also valid for the correlogram \ntrough during the falling phase of the EPSP, as long as del dt > -v; if the EPSP \nfalls more rapidly than -v, the trough is limited at zero firing rate (as \nillustrated for the correlogram at bottom). The fact that the shape of the \ncorrelogram peak above baseline matches the EPSP derivative has been \nempirically confirmed for large EPSPs in cat motoneurons 4. This relation \nimplies that the height of the correlogram peak above baseline is proportional \nto the EPSP rate of rise. The integral of this relationship predicts that the area \nbetween the correlogram peak and baseline is proportional to the EPSP \nthat the effects of \namplitude. \nsimultaneously arriving EPSPs will add linearly. \n\nlinear relation further \n\nimplies \n\nThis \n\nThe presence of additional background synaptic \"noise\", which is \nnormally produced by randomly occurring synaptic inputs, tends to make the \ncorrelogram peak broader than the duration of the EPSP risetime. This \nbroadening is produced by membrane potential fluctuations which cause \nadditional threshold crossings during the decay of the EPSP by trajectories \nthat would have missed the EPSP (e.g., the dashed trajectory in Fig. 1, \nmiddle). On the basis of indirect empirical comparisons it has been proposed \n6,7 that the broader correlogram peaks can be described by the sum of two \nlinear functions of e(t): \n\nf(t) = fo + a e(t) + b deldt \n\n(2) \n\nThis relation provides a reasonable match when the coefficients (a and b) can \nbe optimized for each case 5,7, but direct empirical comparisons 2,4 indicate \nthat the difference between the correlogram peak and the derivative is \ntypically briefer than the EPSP. \n\nThe effect of synaptic noise on the transform -between EPSP and \ncorrelogram peak has not yet been analytically derived (except for the case of \nGaussian noise1). However the threshold-crossing process has been \nsimulated by a computer model which adds synaptic noise to the trajectories \nintercepting the EPSP 1. The correlograms generated by the simulation match \nthe correlograms measured empirically for small EPSP's in motoneurons 2, \nconfirming the validity of the model. \n\nAlthough synaptic noise distributes the triggered firings over a wider \npeak, the area of the correlogram peak, i.e., the number of motoneuron firings \nproduced by an EPSP, is essentially preserved and remains proportional to \nEPSP amplitude for moderate noise levels. For unitary EPSP's (produced by \n\n\fa single afferent fiber) in cat motoneurons, the number of firings triggered per \nEPSP (Np) was linearly related to the amplitude (h) of the EPSP 2: \n\nNp = (O.l/mv)\u00b7 h (mv) + .003 \n\n(3) \n\n273 \n\nThe fact that the number of triggered spikes increases in proportion to EPSP \namplitude has also been confirmed for neocortical neurons 10; for cells \nrecorded in sensorimotor cortex slices (probably pyramidal cells) the \ncoefficient of h was very similar: 0.07/mv. This means that a typical unitary \nEPSP with amplitude of 100 Ilv, raises the probability that the postsynaptic \ncell fires by less than .01. Moreover, this increase occurs during a specific \ntime interval corresponding to the rise time of the EPSP - on the order of 1 - 2 \nmsec. The net increase in firing rate of the postsynaptic cell is calculated by \nthe proportional decrease in interspike intervals produced by the triggered \n(While the above values are typical, unitary EPSP's range in size \nspikes 4. \nfrom several hundred IlV down to undetectable levels of severalllv., and \nhave risetimes of.2 - 4 msec.) \n\nInhibitory connections between cells, mediated by \n\ninhibitory \npostsynaptic potentials (IPSPs), produce a trough in the cross-correlogram. \nThis reduction of firing probability below baseline is followed by a \nsubsequent broad, shallow peak, representing the spikes that have been \ndelayed during the IPSP. Although the effects of inhibitory connections \nremain to be analyzed more quantitatively, preliminary results indicate that \nsmall IPSP's in synaptic noise produce decreases in firing probability that are \nsimilar to the increases produced by EPSP's 4,5. \n\nDISYNAPTIC LINKS \n\nthe underlying monosynaptic connections. \n\nThe effects of polysynaptic links between neurons can be understood \nas combinations of \nA \nmonosynaptic connection from cell A to cell B would produce a first-order \ncross-correlation peak P1(BIA,t), representing the conditional probability that \nneuron B fires above chance at time t, given a spike in cell A at time t = O. As \nnoted above, the shape of this first-order correlogram peak is largely \nproportional to the EPSP derivative (for cells whose interspike interval \nexceeds the duration of the EPSP). The latency of the peak is the conduction \ntime from A to B (Fig. 2 top left). \n\nIn contrast, several types of disynaptic linkages betw.een A and B, \nmediated by a third neuron C, will produce a second-order correlation peak \nbetween A and B. A disynaptic link may be produced by two serial \nmonosynaptic connections, from A to C and from C to B (Fig. 2, bottom left), \nor by a common synaptic input from C ending on both A and B (Fig. 2, \nbottom right). In both cases, the second-order correlation between A and B \nproduced by the disynaptic link would be the convolution of the two first(cid:173)\norder correlations between the monosynaptically connected cells: \n\n(4) \n\n\f274 \n\nAs indicated by the diagram, the cross-correlogram peak P2(BIA,t) would be \nsmaller and more dispersed than the peaks of the underlying first-order \ncorrelation peaks. For serial connections the peak would appear to the right \nof the origin, at a latency that is the sum of the two monosynaptic latencies. \nThe peak produced by a common input typically straddles the origin, since its \ntiming reflects the difference between the underlying latencies. \n\nMonosynaptic connection => First-order correlation \n\nI \\ \nt \n\\ \n\nLJA,,-_~_(_B_I_A_' t_)_ \n\n@ \n\n-----..'t-\n1 \n\n~(AIB,t) = ~(~IA,-t) \n\nDisynaptic connection ~ Second-order correlation \n\nSerial connection \n\n\\ : \\. A ~ (C I A) \nH\\ \n\n\\ ~ \n\\ \n\\ \n\\ \n\\ \n\nt \\ \nI \n\nt \nt \nt \nt \nI \n\n: \n\n@ \n\n\\~(BIC) \n\nP(BIA) \n_________ ~,_2 __ ----\n\nCommon input \n\nt \nt \nI \n\n: \nj \n:\" \n\nll(AIC) \n\nr--A----\nV I'\" \n\nt \\ \nt \n\n\\. \n\nP(BIC) \n\n@ \n\n1 \\/\\ ' \n\nJ t\"-_------\n\n___ /\\.~(BIA) \n\n-L \n\nFig. 2. Correlational effects of monosynaptic and disynaptic links between two neurons. \nTop: monosynaptic excitatory link from A to B produces an increase in firing probability of B \nafter A (left). As with all correlograms this is the time-inverted probability of increased firing \nin A relative to B (right). Bottom: Two common disynaptic links between A and B are a \nserial connection via C (left) and a common input from C. In both cases the effect of the \ndisynaptic link is the convolution of the underlying monosynaptic links. \n\n\f275 \n\nThis relation means that the probability that a spike in cell A will \nproduce a correlated spike in cell B would be the product of the two \nprobabilities for the intervening monosynaptic connections. Given a typical \nNp of .Ol/EPSP, this would reduce the effectiveness of a given disynaptic \nlinkage by two orders of magnitude relative to a monosynaptic connection. \nHowever, the net strength of all the disynaptic linkages between two given \ncells is proportional to the number of mediating intemeurons (C}, since the \neffects of parallel pathways add. Thus, the net potency of all the disynaptic \nlinkages between two cells could approach that of a monosynaptic linkage if \nthe number of mediating interneurons were sufficiently large. It should also \nbe noted that some intemeurons may fire more than once per EPSP and have \na higher probability of being triggered to fire than motoneurons 11. \n\nFor completeness, two other possible disynaptic links between A and B \ninvolving a third cell C may be considered. One is a serial connection from B \nto C to A, which is the reverse of the serial connection from A to B. This \nwould produce a P2(BIA) with peak to the left of the origin. The fourth \ncircuit involves convergent connections from both A and B to C; this is the \nonly combination that would not produce any causal link between A and B. \n\nThe effects of still higher-order polysynaptic linkages can be computed \nsimilarly, by convolving the effects produced by the sequential connections. \nFor example, trisynaptic linkages between four neurons are equivalent to \ncombinations of disynaptic and monosynaptic connections. \n\nThe cross-correlograms between two cells have a certain symmetry, \ndepending on which is the reference cell. The cross-correlation histogram of \ncell B referenced to A is identical to the time-inverted correlogram of A \nreferenced to B. This is illustrated for the monosynaptic connection in Fig.2, \ntop right, but is true for all correlograms. This symmetry represents the fact \nthat the above-chance probability of B firing after A is the same as the \nprobability of A firing before B: \n\nP(BIA, t) = P(AIB, -t) \n\n(5) \n\nAs a consequence, polysynaptic correlational links can be computed as the \nsame convolution integral (Eq. 4), independent of the direction of impulse \npropagation. \n\nP ARALLEL PATHS AND FEEDBACK LOOPS \n\nIn addition to the simple combinations of pair-wise connections \nbetween neurons illustrated above, additional connections between the same \ncells may form circuits with various kinds of loops. Recurrent connections \ncan produce feedback loops, whose correlational effects are also calculated by \nconvolving effects of the underlying synaptic links. Parallel feed-forward \npaths can form multiple pathways between the same cells. These produce \ncorrelational effects that are the sum of the effects of the individual \nunderlying connections. \n\nThe simplest feedback loop is formed by reciprocal connections \nbetween a pair of cells. The effects of excitatory feedback can be computed by \n\n\f276 \n\nif the connections were sufficiently potent \n\nsuccessive cO?1volutions of the underlying monosynaptic connections (Fig. 3 \ntop). Note that such a positive feedback loop would be capable of sustaining \nactivity only \nto ensure \npostsynaptic firing. Since the probabilities of triggered firings at a single \nsynapse are considerably less than one, \nreverberating activity can be \nsustained only if the number of interacting cells is correspondingly increased. \nThus, if the probability for a single link is on the order of .01, reverberating \nactivity can be sustained if A and B are similarly interconnected with at least \na hundred cells in parallel. \n\nFeedforward parallel pathways are formed when cell A \n\nConnections between three neurons may produce various kinds of \nis \nloops. \nmonosynaptically connected to B and in addition has a serial disynaptic \nconnection through C, as illustrated in Fig. 3 (bottom left); the correlational \neffects of the two linkages from A to B would sum linearly, as shown for \nexcitatory connections. Again, the effect of a larger set of cells {C} would be \nadditive. Feedback loops could be formed with three cells by recurrent \nconnections between any pair; the correlational consequences of the loop \nagain are the convolution of the underlying links. Three cells can form \nanother type loop if both A and B are monosynaptically connected, and \nsimultaneously influenced by a common interneuron C (Fig. 3 bottom right). \nIn this case the expected correlogram between A and B would be the sum of \nthe individual components -- a common input peak around the origin plus a \ndelayed peak produced by the serial connection. \n\nFeedback loop \n\n1-----' .. l;---~ \n\n. ... / \n... \n..\u2022. :.... \n-', \n.... .... \n.... \n.... \n.. ' \n-', \n\nI: \n\n\" \n\n\\.... \n\n. ...... \n. ......... . \n;'\" \n. ...... . \n.... \n\u2022\u2022\u2022\u2022\u2022 \n\n'\" \n\n\u2022\u2022\u2022. \n\nParallel \n\njeedfOrward path \n\nCommon input loop \n\nPI (BIA) +P 2 (BIA) \n\nI \nI \n: \n\nt \nt \nt \n\nI \nt \n\nPI (BIA)+P 2 (BIA) \n___ .;.J l~ ___ _ \n\n:/\\ \n\nFig. 3. Correlational effects of parallel connections between two neurons. Top: feedback \nloop between two neurons A and B produces higher-order effects equivalent to convolution \nof mono~aptic effects. Bottom: Loops formed by parallel feed forward paths (left) and by a \ncommon mput concurrent with a monosynaptic link (right) produce additive effects. \n\n\f277 \n\nCONCLUSIONS \n\nThus, a simple computational algebra can be used to derive the \ncorrelational effects of a given network structure. Effects of sequential \nconnections can be computed by convolution and effects of parallel paths by \nsummation. The inverse problem, of deducing the circuitry from the \ncorrelational data is more difficult, since similar correlogram features may be \nproduced by different circuits 9. \n\nThe fact that monosynaptic links produce small correlational effects on \nthe order of .01 represents a significant constraint in the mechanisms of \ninformation processing in real neural nets. For example, secure propagation \nof activity through serial polysynaptic linkages requires that the small \nprobability of triggered firing via a given link is compensated by a \nproportional increase in the number of parallel links. Thus, reliable serial \nconduction would require hundreds of neurons at each level, with \nappropriate divergent and convergent connections. It should also be noted \nthat the effect of intemeurons can be modulated by changing their activity. \nThe intervening cells need to be active to mediate the correlational effects. As \nindicated by eq. I, the size of the correlogram peak is proportional to the \nfiring rate (fo) of the postsynaptic cell. This allows dynamic modulation of \npolysynaptic linkages. The greater the number of links, the more susceptible \nthey are to modulation. \n\nAcknowledgements: The author thanks Mr. Garrett Kenyon for stimulating \ndiscussions and the cited colleagues for collaborative efforts. This work was \nsupported in part by Nll-I grants NS 12542 and RR00166. \n\nREFERENCES \n\n1. Bishop, B., Reyes, A.D., and Fetz E.E., Soc. for Neurosci Abst. 11:157 (1985). \n2. Cope, T.C., Fetz, E.E., and Matsumura, M., J. Physiol. 390:161-18 (1987). \n3. Fetz, E.E. and Cheney, P.D., J. Neurophysiol. 44:751-772 (1980). \n4. Fetz, E.E. and Gustafsson, B., J. Physiol. 341:387-410 (1983). \n5. Gustafsson, B., and McCrea, D., J. Physiol. 347:431-451 (1984). \n6. Kirkwood, P.A., J. Neurosci. Meth. 1:107-132 (1979). \n7. Kirkwood, P.A., and Sears, T._ J. Physiol. 275:103-134 (1978). \n8. Knox, C.K., Biophys. J. 14: 567-582 (1974). \n9. Moore, G.P., Segundo, J.P., Perkel, D.H. and Levitan, H., Biophys. J. 10:876-\n900 (1970). \n10. Reyes, A.D., Fetz E.E. and Schwindt, P.C., Soc. for Neurosci Abst. 13:157 \n(1987). \n11. Surmeier, D.J. and Weinberg, R.J., Brain Res. 331:180-184 (1985). \n\n\f", "award": [], "sourceid": 15, "authors": [{"given_name": "Eberhard", "family_name": "Fetz", "institution": null}]}