{"title": "Nonlinear Pattern Separation in Single Hippocampal Neurons with Active Dendritic Membrane", "book": "Advances in Neural Information Processing Systems", "page_first": 51, "page_last": 58, "abstract": null, "full_text": "Nonlinear Pattern Separation in Single Hippocampal \n\nNeurons with Active Dendritic Membrane \n\nAnthony M. Zador t \nt Depts. of Psychology and Cellular \n&  Molecular Physiology \n\nYale University \nNew Haven, CT 06511 \nzador@yale.edu \n\nBrenda J. Claiborne \u00a7 \n\nt \nThomas H. Brown \n\n\u00a7Division of Life Sciences \nUniversity of Texas \nSan Antonio, TX 78285 \n\nABSTRACT \n\nThe dendritic trees of cortical pyramidal neurons seem ideally suited to \nperfonn local processing on inputs.  To explore some of the implications \nof this complexity for the computational power of neurons, we simulated \na realistic biophysical model of a hippocampal pyramidal cell in which a \n\"cold spot\"-a high density patch of inhibitory Ca-dependent K channels \nand  a  colocalized  patch  of Ca  channels-was  present  at  a  dendritic \nbranch  point.  The cold spot induced a non monotonic relationship  be(cid:173)\ntween the strength of the synaptic input and the probability of neuronal \nfIring.  This effect could also be interpreted as an analog stochastic XOR. \n\nINTRODUCTION \n\n1 \nCortical neurons consist of a highly branched dendritic tree that is electrically coupled to \nthe soma.  In a typical hippocampal pyramidal cell, over 10,000 excitatory synaptic inputs \nare distributed across the tree (Brown and Zador, 1990).  Synaptic activity results in current \nflow  through  a transient conductance increase at the  point of synaptic  contact with  the \nmembrane.  Since the primary means of rapid intraneuronal signalling is electrical, infor(cid:173)\nmation flow can be characterized in tenns of the electrical circuit defIned by the synapses, \nthe dendritic tree, and the soma. \nOver a dozen nonlinear membrane channels have been described in hippocampal  pyrami(cid:173)\ndal neurons (Brown and Zador, 1990).  There is experimental evidence for a heterogeneous \ndistribution of some of these channels in the dendritic tree (e.g. Jones et al .\u2022  1989).  In the \nabsence of these dendritic channels, the input-output function can sometimes be reasonably \napproximated by a modifIed sigmoidal model.  Here we report that introducing a cold spot \n\n51 \n\n\f52 \n\nZador, Claiborne, and Brown \n\nat the junction of two dendritic branches can result in a fundamentally different, nonmono(cid:173)\ntonic input-output function. \n\n2  MODEL \nThe biophysical details of the circuit class defined by dendritic trees have been well char(cid:173)\nacterized (reviewed in RaIl,  1977; Jack et al., 1983).  The fundamental circuit consists of a \nlinear and a nonlinear component  The linear component can be approximated by a set of \nelectrical compartments coupled in series (Fig. 1C), each consisting of a res is tor and capac(cid:173)\nitor in parallel (Fig.  1B).  The nonlinear component consists of a set of nonlinear resistors \nin parallel with the capacitance. \nThe model  is summarized in Fig.  1A.  Briefly,  simulations were  performed on a \n3000-compartment anatomical reconstruction of a region CAl hippocampal neuron (Clai(cid:173)\nborne et aI.,  1992; Brown et al., 1992).  All dendritic membrane was passive, except at the \ncold spot (Fig. 1A).  At the soma, fast K and Na channels (cf. Hodgkin-Huxley, 1952) gen(cid:173)\nerated action potentials in response to stimuli.  The parameters for these channels were \nmodified from Lytton and Sejnowski (1991; cf. Borg-Graham, 1991). \n\nA \n\nSynapti_c ......;+ \ninput-~ \nCold spot \n\nFast somatic \nandNa \n\nc \n\nr-------------------------------- I \n\nI \n\" \nI \n, \n\"  ___________________ .. _____________ ~  I \n\n, \nt \n\n~ \n\n, \n\n) \n< \nRadial and longitudinal Ca+2 diffusion \n\nFig. 1  The model.  (A)  The 3000-compartrnent electrical model used in these simulations was ob(cid:173)\ntained from a 3-dimensional reconstruction of a hippocampal region CAl pyramidal neuron (Clai\u00b7 \nborne et al,  1992).  Each synaptic pathway  (A-D)  consisted of an adjustable number of synapses \narrayed along the single branch indicated (see text).  Random background activity was generated with \na spatially uniform distribution of synapses firing according to Poisson statistics.  The neuronal mem\u00b7 \nbrane was completely passive (linear), except at the indicated cold spot and at the soma.  (B)  In the \nnonlinear circuit associated with a patch a neuronal membrane containing active channels, each chan\u00b7 \nnel is described by a voltage-dependent conductance in series with its an ionic battery (see text).  In \nthe present model the channels were spatially localized, so no single patch contained all of the non\u00b7 \nlinearities depicted in this hypothetical illustration.  (Cl.  A dendritic segment is illustrated in which \nboth electrical and ca2+ dynamics were modelled.  Ca + buffering, and both radial and longitudinal \nCa2+ diffusion were simulated. \n\n\fNonlinear Panern Separation in  Single Hippocampal Neurons \n\n53 \n\nWe distinguished four synaptic pathways A-D (see Fig. lA).  Each pathway consisted of a \npopulation of synapses activated synchronously.  The synapses were of the fast AMP A type \n(see Brown et. al., 1992).  In addition. random background synaptic activity distributed uni(cid:173)\nformly across the dendritic tree fIred according to Poisson statistics. \nThe cold spot consisted of a high density of a  Ca-activated K channel. the AHP current \n(Lancaster and Nicoll. 1987; Lancaster et. aI.,  1991) colocalized with a low density patch \nofN-type Ca channels (Lytton and Sejnowski, 1991; cf. Borg-Graham, 1991).  Upon local(cid:173)\nized depolarization in the region of the cold ~t. influx of Ca2+ through the Ca channel re(cid:173)\nsulted in a transient increase in the local rCa  +].  The model included Ca2+ buffering, and \nboth radial and longitudinal diffusion in the region of the cold spot.  The increased [Ca2+] \nactivated the inhibitory AHP current.  The interplay between the direct excitatory effect of \nsynaptic input, and its inhibitory effect via the AHP channels formed the functional basis \nof the cold spot. \n\n3  RESULTS \n3.1  DYNAMIC BEHAVIOR \n\nRepresentative behavior of the model is illustrated in Fig. 2.  The somatic potential is plot(cid:173)\nted as a function of time in a series of simulations in which the number of activated syn(cid:173)\napses in pathway AlB was increased from  0 to about 100.  For the fIrst  100 msec of each \nsimulation, background synaptic activity generated a noisy baseline.  At t = 100 msec, the \nindicated number of synapses fired synchronously five times at 100 Hz.  Since the back(cid:173)\nground activity was noisy, the outcome of the each simulation was a random process. \n\nThe key effect of the cold spot was to impose a limit on the maximum stimulus amplitude \nthat caused firing, resulting in a window of stimulus strengths that triggered an action po(cid:173)\ntential.  In the absence of the cold spot a greater synaptic stimulus invariably increased the \nlikelihood that a spike fIred.  This limit resulted from  the relative magnitude of the AH P \n\nSample  Soma \n\ntic  Voltage  Tracelll \n\n60 \n\n0 \n\n~ .. \nI 0 >-\n\n-60 \n\n<:> \n\nFig. 2  Sample runs.  The membrane voltage at the soma is plotted as a f'wtction of time and synaptic \nstimulus intensity.  At t = 100 msec, a synaptic stimulus consisting of 5 pulses was activitated.  The \nnoisy baseline resulted from random synaptic input.  A single action potential resulted for input in(cid:173)\ntensities within a range determined by the kinetics of the cold spot \n\n\f54 \n\nZador, Claiborne, and Brown \n\ncurrent \"threshold\" to the threshold for somatic spiking.  The AHP current required a rela(cid:173)\ntively high level of activity for its activation.  This AHP current \"threshold\" reflected the \nsigmoidal voltage dependence of N-type Ca current activation (V1I2 = -28 mV), since only \nas the dendritic voltage approached V1I2  did dendritic  [Ca2+]  rise enough to activate the \nAHP current.  Because V1I2 was much higher than the threshold for somatic spiking (about \n-55 mV under current clamp), there was a window of stimulus strengths sufficient to trigger \na somatic action potential but insufficient to activate the AHP current  Only within this \nwindow of between about 20 and 60 synapses (Fig. 2) did an action potential occur. \n3.2  LOCAL NON-MONOTONIC RESPONSE FUNCTION \nBecause the background activity was random, the outcome of each simulation (e.g. Fig. 2) \nrepresented a sample of a random process.  This random process can be used to defme many \ndifferent random variables.  One variable of interest is whether a spike fired in response to \na stimulus.  Although this measure ignores the dynamic nature of neuronal activity, it was \nstill relatively informative because in these simulations no more than  one spike fired per \nexperiment \nFig. 3A shows the dependence of firing probability on stimulus strength.  It was obtained \nby averaging over a population of simulations of the type illustrated in Fig. 2.  In the ab(cid:173)\nsence of AHP current (dotted line), the fIring probability was a sigmoidal function of activ(cid:173)\nity.  In its presence, the firing probability was a smooth nonmonotonic function of the \nactivity (solid line).  The firing probability was maximum at about 35 synapses, and oc(cid:173)\ncurred only in the range between about 10 and 80 synapses.  The statistics illustrated in Fig. \n3A quantify the nonmonotonicity that is implied by the single sample shown in Fig. 2. \n\nSpikes required the somatic Hodgkin-Huxley-like Na and K channels.  To a first approxi(cid:173)\nmation, the effect of these channels was to convert a continuous variable-the somatic volt(cid:173)\nage-into a  discrete variable-the presence or absence of a  spike.  Although  this \napproximation ignores the complex interactions between the soma and the cold spot, it is \nuseful for a qualitative analysis.  The nonmonotonic dependence of somatic activity on syn-\n\nA \n..... -;0 \n\n1.0 \n>.. \n.....  0 .6 \n\nIII \n,ll  0.6 \n0 \n'\"' \nPo.  0.4 \nDO \nC \n.~ \nr;:  0.2 \n\nCold  Spol-\nCold  Spol+ \n\nI \nI \nI \nI \n~ \n\n, \n\u2022 \n\n> \n\nB \n-56 - -56 \nE -cu  -60 \n.. 0 p \n\ntIO \nCI  -62 \n\n-64 \nJI: \nCI cu  -66 \nPo. \n\n.. -.-.-\n\n-\" \n\nCold  Spot-\nCold  Spot+ \n.-\n\" \n\n\"  .. '-\" \n\n.-\n\n0.0 '--__ '--_ ........ _ \n\n........ --.i::o ......... ~\"__._.......J \n60 \n\n80 \n\n120 \n\n40 \n\n20 \n100 \nNumber  of  active  synpases \n\no \n\no  20  40  60  60  100  120 \nNumber  of  active  synpases \n\nFig. 3  Nonmonotonic input-output relation.  (A) Each point represents the probability that at least \none spike was fIred at the indicated activity level.  In the absence of a cold spot, the fIring probabil(cid:173)\nity increased sharply and monotonically as the number of synapses in pathway C/ D increased (dot(cid:173)\nted Une).  In contrast, the fIring probability reached amaximumforpathwayA/B and then decreased \n(solid line).  (B) Each point represents the peak somatic voltage for a single simulation at the indi(cid:173)\ncated activity level in the presence (pathway AlB; solid line)  and absence (pathway C/D,\u00b7  dotted \nUne) of a cold spot  Because each point represents the outcome of a single simulation, in contrast \nto  the average used in (A), the points reflect the variance due to the random background activity. \n\n\fNonlinear Pattern Separation in  Single Hippocampal Neurons \n\n55 \n\naptic activity was preserved even when active channels at the soma were eliminated (Fig. \n3B).  This result emphasizes that the critical nonlinearity was the cold spot itself. \n3.3  NONLINEAR PATTERN SEPARATION \nSo far. we have treated the output as a function of a scalar-the total activity in pathway \nAlB (or CID).  In Fig. 3  for example. the total activity was defmed as the sum of the activ(cid:173)\nities in pathway A and B.  The spatial organization of the afferents onto 2 pairs of branch(cid:173)\nes-A & B and C & D  (Fig.  I)-suggested considering the output as  a function  of the \nactivity in the separate elements of each pair. \nThe effect of the cold spot can be viewed in terms of the dependence of fIring as a function \nof separate activity in pathways A and B (Fig. 4).  Each fIlled circle indicates that the neuron \nfIred for the indicated input intensity of pathways A and B. while a small dot indicates that \nit did not fire.  As suggested by  (Fig. 3). the fIring probability was highest when the total \nactivity in the two pathways was at some intennediate level.  The neuron did not fIre when \nthe total activity in the two pathways was too large or too small.  In the absence of the cold \nspot, only a minimum activity level was required. \nIn our model the probability of fIring was a continuous function of the inputs.  In the pres(cid:173)\nence of the dendritic cold spot, the corners of this function suggested the logical operation \nXOR.  The probability of fIring was high if only one input was activated and low if both or \nneither was activated. \n\n4  DISCUSSION \n4.1  ASSUMPTIONS \nNeuronal morphology in the present model was based on a precise reconstruction of a re(cid:173)\ngion CAl pyramidal neuron.  The main additional assumptions involved the kinetics and \ndistribution of the four membrane channels. and the dynamics of Ca2+in the neighborhood \nof influx.  The forms assumed for these mechanisms were biophysically plausible. and the \nkinetic parameters were based on estimates from a collection of experimental studies (listed \nin Lytton and Sejnowski. 1991; Zador et aI .\u2022  1990).  Variation within the range of uncer(cid:173)\ntainty of these parameters did not alter the main conclusions.  The chief untested assump(cid:173)\ntion of this model was the existence of cold spots.  Although there is experimental evidence \n\n...... ....... .. \nt  --..... .. .. . \n\n. . . . .  \n\n- . e  .\u2022\u2022. .\n\n~ _ ...... . _ ..... \n~ _ ..... \n~  :~ .. ~~~:.: \n=='  =::::~ \n::::: ::::: :. \np...  ~::~:~ \n:::::::;Z!Zi \n...... \niiiilli Iii i I \n\n\u2022 \u2022\u2022\u2022\u2022 \n\u2022 \u2022\u2022\u2022\u2022 \n\n\u00b7 ... . \niq:I:I!'! : \niii \nfifl \n\nFig.4  Nonlinear pattern separation  Neuronal fIring is represented as  a joint nmction of two input \npathways (AlB).  Filled circles indicate that the neuron fIred for the indicated stimulus parameters. \nSome indication of the stochastic nature of this function. resulting fonn the noisy background, is giv(cid:173)\nen by the density of interdigitation of points and circles. \n\nInput  A  -+ \n\n\f56 \n\nZador, Claiborne, and Brown \n\nsupporting the presence of both Ca and AHP channels in the dendrites. there is at present \nno direct evidence regarding their colocalization. \n4.2  COMPUTATIONS IN SINGLE NEURONS \n4.2.1  Neurons and Processing Elements \nThe limitations of the McCulloch and Pitts (1943) PE as a neuron model have long been \nrecognized.  Their threshold PEt in  which  the output is the weighted  sum of the inputs \npassed through a threshold, is static, deterministic and treats all inputs equivalently.  This \nmodel ignores at least three key complexities of neurons: temporal, spatial and stochastic. \nIn  subsequent years, augmented models have attempted to  capture aspects of these com(cid:173)\nplexities.  For example, the leaky integrator (Caianiello,  1961; Hopfield,  1984) incorpo(cid:173)\nrates the  temporal dynamics implied by  the  linear RC component of the circuit element \npictured in Fig. IB.  We have demonstrated that the input-output function of a realistic neu(cid:173)\nron model  can have qualitatively different behavior from  that of a single processing ele(cid:173)\nment(pE). \n4.2.2  Interactions Within The Dendritic Tree \nThe early work ofRall (1964) stressed the spatial complexity of even linear dendritic mod(cid:173)\nels.  He noted that input from different synapses cannot be considered to arrive at a single \npoint, the soma.  Koch et al.  (1982) extended this observation by exploring the nonlinear \ninteractions between synaptic inputs to different regions of the dendritic tree.  They empha(cid:173)\nsized that these interactions can be local in the sense that they effect subpopulations of syn(cid:173)\napses and suggested that the entire dendritic tree can be considered in terms of electrically \nisolated subunits.  They proposed a specific role for these subunits in computing a veto(cid:173)\nan analog  AND-NOT ---that might underlie directional selectivity in retinal ganglion cells. \nThe veto was achieved through inhibitory inputs. \nThe underlying neuron models of Koch et al. (1982) and Rall (1964) were time-varying but \nlinear, so it is not surprising that the resulting nonlinearities were monotonic.  Much steeper \nnonlinearities were achieved by Shepherd and Brayton (1987) in a model that assumed ex(cid:173)\ncitable spines with fast Hodgkin-Huxley K and Na channels.  These channels alone could \nimplement the digital logic operations AND and OR.  With the addition of extrinsic inhibi(cid:173)\ntory inputs, they showed that a neuron could implement a full complement of digital logic \noperations, and concluded that a dendritic  tree could in principle implement arbitrarily \ncomplex logic operations. \nThe emphasis of the present model differs from that of both the purely linear and of the dig(cid:173)\nital approaches, although it shares their emphasis on the locality of dendritic computation. \nBecause the cold spot involved strongly nonlinear channels, it implemented a non mono ton(cid:173)\nic response function, in contrast to strictly linear dendritic models.  At the same time, the \npresent model retained the essentially analog nature of intraneuronal signalling, in contrast \nto the digital dendritic models.  This analog mode seems better suited to processing large \nnumbers of noisy inputs because it preserves the uncertainties rather than making an imme(cid:173)\ndiate decision.  Focussing on the analog nature of the response eliminated the requirement \nfor operating within the digital range of channel dynamics. \nThe NMDA receptor-gated channel can give rise to an analog AND with a weaker voltage(cid:173)\ndependence than that induced by fast Na and K channels.  Mel (1992) described a model in \nwhich synapses mediating increases to both the NMDA and AMP A conductances were dis(cid:173)\ntributed across the dendritic tree of a cortical neuron.  When the synaptic activity was dis-\n\n\fNonlinear Pattern Separation  in  Single  Hippocampal Neurons \n\n57 \n\ntributed in appropriately  sized clusters,  the resulting neuronal response function  was \nreminiscent of that of a sigma-pi unit  With suitable preprocessing of inputs. the neuron \ncould perform complex pattern discrimination. \n\nA unique feature of the present model is that functional inhibition arose from purely exci(cid:173)\ntatory inputs.  This mechanism underlying this inhibition -the AHP current-was intrinsic \nto the membrane.  In both the Koch et ale  (1982) and Brayton and Shepherd (1987) models. \nthe veto or NOT operation was achieved through extrinsic synaptic inhibition.  This requires \nadditional neuronal circuitry.  In the case of a dedicated sensory system like the direction(cid:173)\nally selective retinal granule cell. it is not unreasonable to imagine that the requisite neu(cid:173)\nronal circuitry is hardwired.  In the limiting case of the digital model, the requisite circuitry \nwould involve a separate inhibitory interneuron for each NOT-gate. \n4.2.3  Adaptive Dendritic Computation \nWhat algorithms can harness the computational potential of the dendritic  tree?  Adaptive \ndendritic computation is a very new subject. Brown et ale (1991, 1992) developed a model \nin which synapses distributed across the dendritic tree showed interesting forms of spatial \nself-organization.  Synaptic plasticity was governed by a local biophysically-motivated \nHebb rule (Zador et al' J  1990).  When temporally correlated but spatially uncorrelated in(cid:173)\nputs were presented to the neuron, spatial clusters of strengthened synapses emerged within \nthe dendritic tree.  The neuron converted a temporal correlation into a spatial correlation. \n\nThe computational role of clusters of strengthened synapses within the dendritic tree be(cid:173)\ncomes important in the presence of nonlinear membrane.  If the dendrites are purely pas(cid:173)\nsive. then saturation ensures that the current injected per synapse actually decreases as the \nclustering increases.  If purely regenerative nonlinearities are present (Brayton and Shep(cid:173)\nherd.  1987; Mel.  1992), then the response increases.  The cold spot extends the range of \nlocal dendritic computations. \n\nWhat might control the formation and distribution of the cold spot itself?  Cold spots might \narise from  the fortuitous colocalization of Ca and KAHP channels.  Another possibility is \nthat some specific biophysical mechanism creates cold spots in a use-dependent manner. \nCandidate mechanisms might involve local changes in second messengers such as [Ca2+] \nor longitudinal potential gradients (if. Poo, 1985).  Bell (1992) has shown that this second \nmechanism can induce computationally interesting distributions of membrane channels. \n4.3  WHY STUDY SINGLE NEURONS? \nWe have illustrated an important functional difference between a single neuron and aPE. \nA neuron with cold spots can perform extensive local processing in the dendritic tree, giv(cid:173)\ning rise to a complex mapping between input and output.  A neuron may perhaps be likened \nto a \"micronet\" of simpler PEs. since any mapping can be approximated by a sufficiently \ncomplex network of sigmoidal units (Cybenko, 1989).  This raises the objection that since \nmicronets represent just a subset of all neural networks, there may be little to be gained by \nstudying the properties of the special case of neurons. \n\nThe intuitive justification for studying single neurons is that they represent a large but high(cid:173)\nly constrained subset that may have very special properties.  Knowledge of the properties \ngeneral to all sufficiently complex PE networks may provide little insight into the proper(cid:173)\nties specific to single neurons.  These properties may have implications for the behavior of \ncircuits of neurons.  It is not unreasonable to suppose that adaptive mechanisms in biolog(cid:173)\nical circuits will utilize the specific strengths of single neurons. \n\n\f58 \n\nZador, Claiborne, and Brown \n\nAcknowledgments \n\nWe thank Michael Hines for providing NEURON-MODL assisting with new membrane \nmechanisms.  This research was supported by grants from the Office of Naval Research, \nthe Defense Advanced Research Projects Agency, and the Air Force Office of Scientific \nResearch. \n\nReferences \nBell, T. (1992) Neural in/ormation processing systems 4 (in press). \nBorg-Graham, L.J. (1991)  In H. Wheal and J. Chad (Eds.) Cellular and Molecular Neurobiology: A \n\nPractical Approach. New York: Oxford University Press. \n\nBrown, T.H. and Zador, AM. 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Acad. Sci. USA 87:  6718-6722. \n\n\f", "award": [], "sourceid": 539, "authors": [{"given_name": "Anthony", "family_name": "Zador", "institution": null}, {"given_name": "Brenda", "family_name": "Claiborne", "institution": null}, {"given_name": "Thomas", "family_name": "Brown", "institution": null}]}