{"title": "A Systematic Study of the Input/Output Properties of a 2 Compartment Model Neuron With Active Membranes", "book": "Advances in Neural Information Processing Systems", "page_first": 149, "page_last": 159, "abstract": null, "full_text": "A Systematic Study or the Input/Output Properties \n\n149 \n\nA Systematic Study of the Input/Output Properties \n\nof a 2 Compartment Model Neuron \n\nWith Active Membranes \n\nPaul Rhodes \n\nUniversity of California, San Diego \n\nABSTRACT \n\nThe  input/output  properties  of a  2  compartment  model  neuron  are  systematically \nexplored.  Taken from  the work of MacGregor (MacGregor,  1987), the model neuron \ncompartments contain several active conductances, including a potassium conductance in \nthe  dendritic  compartment driven  by  the  accumulation  of  intradendritic  calcium. \nDynamics of the conductances and potentials are governed by a set of coupled first order \ndifferential equations which are  integrated numerically.  There are a set of 17 internal \nparameters  to  this  model,  specificying  conductance rate  constants,  time  constants, \nthresholds, etc. \n\nTo study parameter sensitivity,  a set of trials were run in which the input driving the \nneuron is kept fixed while each internal parameter is varied with all others left fixed. \n\nTo study the input/output relation, the input to the dendrite (a square wave) was varied \n(in frequency and magnitude) while all internal parameters of the system were left flXed, \nand the resulting output firing rate and bursting rate was counted. \n\nThe  input/output  relation of the  model  neuron  studied  turns  out  to  be  much  more \nsensitive  to  modulation  of certain  dendritic  potassium  current  parameters  than  to \nplasticity  of synapse  efficacy per se  (the  amount  of current  influx  due  to  synapse \nactivation).  This would in turn suggest, as has been recently observed experimentally, \nthat the potassium current may be as or more important a focus of neural plasticity than \nsynaptic efficacy. \n\nINTRODUCTION \n\nIn order to model biologically realistic neural systems, we will ultimately be seeking to \nconstruct networks  with thousands of neurons and millions of interconnections.  It is \ntherefor desireable to employ basic  units  with  sufficient computational  simplicity to \nmake meaningful simulations tractable, yet with sufficient fidelity to biological neurons \nthat we may retain a hope of gleaning by these simulations something about the activity \ngoing on during biological information processing. \n\n\fISO \n\nRhodes \n\nThe types of neuron models employed in the computational neuroscience literature range \nfrom binary threshold units to sigmoid transfer functions to  1500 compartment neurons \nwith Hodgkin-Huxley kinetics  for  a whole set of active conductances and  spines with \nrich internal  structure.  In principle,  a model neuron's  functional  participation in the \noperation  of a  network  may  be  fully  characterized by a  complete  description of its \ntransfer  function,  or  input-output  relation.  This  relation  would  necessarily  be \nparameterized by a host of internal variables  (which  would  include  conductance  rate \nconstants and parameters defining the neuron's morphology) as well as a very rich space \ncharacterizing possible variations in input (including location of input in dentritic tree). \nIn learning  to  judge  which  structural  elements  of highly  realistic  models  must  be \npreserved and which may be simplified, one approach will be to test the degree to which \nthe  input-output relation of the  simplified neuron  (given  a  physiologically  relevant \nparameter range and input space) is sufficiently close to the input-output properties of \nthe highly realistic model. \n\nTo define 'sufficiently close', we will ultimately refer to the operation of the network as \na whole as follows:  the transfer function of a simplified neuron model will be considered \n'sufficiently close' to a more realistic neuron model if a chosen information processing \ntask  carried  out  by  the  overall  network  is  performed  by  a  network  built  up  of the \nsimplified neurons in a manner close to that observed in a network of the more realistic \nneurons. \n\nWe propose to begin by exploring the input/output properties of a greatly simplified 2 \ncompartment model neuron with active conductances.  Even in this very simple structure \nthere are many (17) internal parameters for things like time constants and activation rates \nof currents.  We wish to understand the parameter sensitivity of this model system and \ncharacterize its input-output relation. \n\n1.0  DESCRIPTION OF THE MODEL NEURON \n\nTHE  MODEL  NEURON  CONSISTS  OF A  SOMA WITH  A  VOLTAGE-GATED \nPOTASSIUM  CONDUCTANCE  AND  A  SINGLE  COMPARTMENT DENDRITE \nWITH  A VOLTAGE-GATED CALCIUM  CONDUCTANCE AND A  [CAl-GATED \nPOTASSIUM CONDUCTANCE \n\nWe will choose for this study a simple model neuron described by MacGregor (I987).  It \npossesses a single compartment dendrite.  This is viewed as a crude approximation to the \nlumped reduction of a dendritic tree.  In this approximation, we are neglecting spatial and \ntemporal summing of individual synaptic EPSP's distributed over a dendritic tree, as well \nas the spatial and temporal dispersion (smearing) due to transmission to the soma.  The \nindividual inputs we will be using are large enough to drive the soma to firing,  and so \nwould represent the  summation of many relatively  simultaneous  individual  EPSPs, \nperhaps  as  from  the  set  of contacts  upon  a  neuron's  dendritic  tree  made  by  the \narborization of one different axon.  The dendritic  membrane  possesses a  potassium \nconductance gated by intradendritic calcium concentration and a voltage gated calcium \nconductance.  The  soma  contains  its  own  voltage-gated  potassium  channels  and \nmembrane time constants.  Electrical connection between soma and dendrite is expressed \nby an input impedance in each direction.  The soma fires  an action potential,  simply \nexpressed by raising its voltage to 50 mv for one msec after its internal voltage has been \n\n\fA Systematic Study or the Input/Output Properties \n\n151 \n\ndriven to firing threshold.  Calcium accumulation in the dendrite is modelled assuming \naccumulation proportional to calcium conductance.  Calcium conductance itself increases \nin proportion to the difference between the dendrite's voltage and a threshold, and calcium \nis removed from the dendrite by means of an exponential decay.  This system is modelled \nby a set of coupled frrst order differential equations as follows: \n\n1.1  THE  SET  OF  EQUATIONS  GOVERNING  THE  DYNAMIC \nVARIABLES  OF THIS  MODEL \n\nThe soma's voltage ES is governed by: \n\ndES/dt={ -ES+SOMAINPUT +GDS *(ED-ES)+GKS * (EK-ES)} IfS \n\nwhere  SOMAINPUT  is  obtained  by  dividing  the  input current by the  total  resting \nconductance of the dendrite (therefor it has units of voltage).  GDS  is  proportional  to \ninput  resistance  from  dendrite  to  soma,  and  multiplies  the  difference  between  the \ndendrite's  voltage  ED  and  the  soma's  voltage  ES;  GKS  is  the  soma's  aggregate \npotassium conductance  (modelled below);  EK is  the voltage of the potassium battery \n(assumed constant at -1 Omv);  and TS is  the  soma's time constant.  All potentials  are \nrelative to resting potential, and all conductances are dimensionless. \n\nThe dendrite's voltage ED is govened by: \n\ndED/dt={-ED+DENDINPUT+GSD*(ES-ED)+GCA*(ECA-ED)+ GKD*(EK-ED)}IID \n\nwhere  DENDINPUT  is  obtained by dividing the  input  current by the  total  resting \nconductance of the dendrite and  so  has units of voltage.  GSD is  proportional to  the \ninput resistance from soma to dendrite, and hence multiplies the difference between ES \nand  ED;  GCA is  the dendrite's calcium conductance  (modelled  below),  ECA  is  the \ncalcium battery (assumed constant at 50mv), and GKD is proportional to the dendrite's \npotassium conductance (modelled below).  All potentials are relative to resting potential. \n\nThe  soma's voltage is  raised artificially to  50mv for  I  msec after the  soma's  voltage \nexceeds a (fixed) threshold, thus simplifying the action potential. \n\nThe potassium conductance in the soma, GKS,  is governed by: \n\ndGKS/dt={ -GKS+S*B}lfGK \n\nwhere S is  1 if an action potential has just fired and 0 otherwise, B is an activation rate \nconstant governing the rate of increase of potassium conductance, and TGK is the time \nconstant  of the  potassium  conductance  decay.  This  rather  simplified  picture  of \npotassium conductance will be replaced by a more realistic version with a Markov state \nmodel  of the potassium channel in  a  subsequent publication in preparation.  For the \npresent investigation then we  are  modelling the voltage dependence  of the potassium \nconductance by the  following:  potassium conductance builds up by a  fixed  amount \n(proportional to BlfGK) during each action potential, and thereafter decays exponentially \nwith time constant TGK. \n\n\f152 \n\nRhodes \n\nThe dendrite's calcium conductance is governed by: \n\ndGCNdt={ -GCA +D*(ED-CSPlKETHRESH)} IfGCA \ndGCNdt={ -GCNlGCA} \n\nED>CSPIKETHRESH \n\nED<CSPlKETHRESH \n\nwhere  CSPIKETHRESH  is  the  minimum  dendritic  voltage  above  which  calcium \nconducting channels begin to be opened, D is an activation rate governing the rate of \nincrease in calcium conductance, and TGCA is  the time  constant assumed  to  govern \nconductance decay when voltage is below threshold \n\nThe dendrite's internal calcium concentration [CA] is governed by: \n\nd[ CAYdt={ -[ CA]+ A *GCA} IfCA \n\nwhere TCA is the time constant for the removal of internal CA, and A is  a parameter \ngoverning the accumulation rate of increase of internal CA for a given conductance and \ntime constant.  A is inversely proportional  to the effective relevant volume in  which \ncalcium is  accumulating.  An increase in internal  calcium buffer would decrease the \nparameter A. \n\nFinally, the dendrite's potassium conductance is governed by: \n\ndGKD/dt={ -GKD+ BD} /TGKD \ndGKDldt={ -GKD} IfGKD \n\n[CA]>CALCTHRESH \n[CA]<CALCTHRESH \n\nwhere CALCTHRESH is the internal calcium concentration threshold above which the \ncalcium gated potassium channel begins to open, BD is the parameter governing the rate \nof increase  of  dendritic  potassium  conductance,  and  TGKD  is  the  time  constant \ngoverning the exponential decay of potassium conductance. \n\nThis  entire  system  of  equations  is  taken  from  the  work  of  MacGregor \n(MacGregor,  1987). \n\nThe system of coupled fIrst order differential equations is integrated using the exponential \nmethod,  also  discussed  in MacGregor.  Generally a  1 msec timestep is  used,  with a \nsmaller timestep of .1  msec used for the relaxation between the dendritic voltage ED and \nthe somatic voltage ES. \n\n2.0  THE  EFFECT  OF  CHANGES  IN PARAMETERS  (TIME \nCONSTANTS,  CONDUCTANCE  RATES,  ETC.)  ON  THE \nMODEL  NEURON'S  INPUT-OUTPUT  PROPERTIES  WILL \nBE EXPLORED \n\nAs  is clear from  a review of the  above  set  of interrelated  equations  governing  the \ndynamics of the state variables of the model neuron, there are quite a  few externally \nspecified parameters (I7) even in such a simple model.  Presumably the thresholds are \nfairly well measureable, and the rate constants and time constants may be specified by \nmeasurement of time courses in patch clamp experiments.  We are nevertheless dealing \nwith  parameters of which  some  are  thought to  be variable  and  which  are  probably \n\n\fA Systematic Study or the Input/Output Properties \n\n153 \n\nmodulated explicitly by normal mechanisms in neurons.  Therefor we wish to explore \nthe effect that variation of any of these parameters has on the input-output properties of \nthe  model  neuron.  In  fact,  we  will  find  indication  that  the  modulation  of \nthese  parameters,  in  particular  the  rate  constants  governing  the \ndendritic  potassium  current  and  internal  calcium  accumulation,  may  be \nvery  effective  targets  of neural  plasticity.  We  find  that  the  neuron's \ninput-output  properties  are  more  sensitive  to  these  parameters  than  to \nmodulation  of the  efficacy of the  synapse  strength  per see \n\n2.1  PROTOCOL  FOR  SYSTEMATIC  EXPLORATION  OF  THE \nEFFECT  OF  VARIATION  IN  THE  MODEL'S  PARAMETERS  ON  THE \nINPUT-OUTPUT  PROPERTIES  OF THE  MODEL  NEURON \n\nWe started with the parameters all set to a set of benchmarks and drove the neuron with a \nconstant input to the dendrite.  (We  could have driven the soma instead, or both soma \nand dendrite,  and we  could have chosen more complex  input streams.  See below for \ntrials where we systematically vary the input but the parameter values are held steady.) \nThe input was  a  steady command input of 35mv.  The values of all  the  benchmark \nparameters are given in Table 1. \n\nWe then systematically halved and doubled each of the  17 parameters in turn, while \nleaving all other parameters fixed.  Note that in all cases and in fact with any  driving \ninput this  model  neuron  fires  in bursts.  This  is  due  to the long time  course  of the \npotassium current in the dendrite, which enforces a long refractory period (about 40-\n80msec) even during continuous stimulation. \n\n2.2  RESULTS  OF  SYSTEMATIC  VARIATION  OF  PARAMETERS \nOF MODEL  NEURON \n\nThe  results  are  summarized  in the  notes  to Table  1.  Following are several \nobservations about the different parameters' varying degree of efficacy in modulation of \nthe input-output function. \n\n1)  The most  striking finding  is that variation of the activation rate  of the potassium \ncurrent, particularly the potassium current in the dendrite, is the most effective means of \nmodulating the input-output properties of the model neuron.  The transfer function is \n250%  more  sensitive  to  an  increase  in  the  [CA]-gated  dendritic  potassium current \nactivation rate than it is to an increase in synaptic efficacy ~~. \n\n2)  Changing the time constant of the [CA]-gated potassium current in the dendrite is \nthe  only parameter change  which  effectively  modulates  the  number of bursts  per \nsecond (see  Figure  I).  Changing the  time constant of the voltage-gated potassium \ncurrent in the soma, does not have any effect on the number of bursts per second. \n\n\f154 \n\nRhodes \n\n3.0  MEASUREMENT OF THE INPUT/OUTPUT RELATION \nOF THE MODEL NEURON \n\nThe input/output relation was detennined  by the  following  protocol:  The input was \nsupplied in the fonn of a square wave of current injected into the dendritic compartment, \nand the frequency of the pulses and their magnitude was systematically varied. \n\nThe output of the soma, in the form of action potentials fIred  per second, was  plotted \nagainst the  input rate,  defined as  the product  of the square wave frequency  and the \nmagnitude of the injected current.  The duration of pulses was kept fixed at 20 msec (but \nsee below), all internal parameters were fIXed  at their benchmark levels. \n\n3.1  THE  SHAPE  OF THE  INPUT/OUTPUT  RELATION \n\nFigure 2 depicts the above described plot in the case where all the internal parameters \nwere fixed  at  purported  \"benchmark\"  values  except  for  the  parameters  governing \nintradendritic calcium accumulation..  It is  clearly not  strictly  monotonic  (there  are \nresonance points) though  a  smoothed version is  monotonic,  and it does not faithfully \nrender a sigmoid. \n\n3.2  THE  INPUT/OUTPUT  RELATION  IS  UNCHANGED  IF  THE \nSQUARE  SHAPE  OF  THE  EPSP  DRIVING  THE  DENDRITE  IS \nREPLACED  BY  AN  ALPHA  FUNCTION \n\nThe trials in this study were largely conducted using a square wave as the input driving \nthe dendritic compartment.  In order to check whether the unphysical square shape of the \nenvelope of this current injection was coloring the results, the input/output relation was \nmeasured in a set of trials wherein the alpha function commonly used to model the time \ncourse of EPSP's replaced the square pulse.  The total current injected per pulse was kept \nuniform.  The  results,  shown  in  Figure  3,  are  surprising:  The \ninput/output  relation  was  almost  completely  unaltered  by  the \nsubstitution.  This  suggests  that  the  detailed  shape  and  fourier \nspectrum  of  the  time  course  of synaptic  input  has  nearly  no  effect  of \nthe  neuron's  output.  Thus it is suggested that very adequate models  can be built \nwithout the need for a strict modelling of the synaptic EPSP.  I expect this effect is due \nto  the temporal  integration ongoing in the  summation of input to this system,  which \nblurs the exact shape of any input envelope. \n\n3.3  MODULATION  OF  THE  INPUT/OUTPUT  RELATION  BY \nVARIATION  OF  INTERNAL  MODEL  PARAMTERS \n\nFigure 1 portrays the input/output relation measured in three cases in which all internal \nparameters are  identical except the rate of accumulation of intradendric calcium.  The \nlower curve is  the case where  the calcium accumulation rate  is  highest.  Since  [Ca] \naccumulation drives  the  dendritic  potassium current,  the  activation of which  in tum \nhyperpolarizes the dendrite and thus indirectly suppresses firing in the soma, we expect \noutput in this case to be lower for a given input as is  indeed the result observed.  Note \nthat the parameter being varied would be expected to be inversely proportional to  the \namount  of  available  intradendritic  calcium  buffer.  Hence  the  amount  of \n\n\fA Systematic Study or the Input/Output Properties \n\nISS \n\nintradendritic  buffer  has  a  profound  ability  to  modulate  the  transfer \nfunction  of the  system. \n\n4.0  CONCLUSIONS \n\nAs  regards  the  shape  of the  transfer  function  itself,  we  have  found  it  to  be  non(cid:173)\nmonotonic (there are resonance points) unless it is smoothed.  The shape of the transfer \nfunction  appears  little effected by the envelope of the  EPSP (Le.  square pulse  input \nproduces  nearly  the  same  transfer  function  as  the case  where  alpha  functions  are \nsubstituted for the square pulses in  modelling the EPSP). \n\nA parameter  sensitivity  analysis  of  a  2  compartment  model  neuron  with  active \nmembranes reveals some unexpected results.  For example, the input/output (transfer) \nfunction of the neuron is 250% more sensitive to the activation rate of the  [CA]-gated \ndendritic potassium current than it is to synaptic efficacy per se.  This in turn suggests \nthat, as has indeed been observed (Alkon et~ 1988; Hawkins, 1989; Olds etal, 1989), \nnature might employ mechanisms other than  simply increasing synaptic conductance \nduring the EPSP to enhance the efficacy of the transfer function. \n\nAlkon, D.L. et at, J.  Neurochemistry, Volume 51, 903, (1988). \n\nHawkins,  R.  D.  in  Computational  Models  of Learning  in  Simple  Neural  Systems, \nHawkins and Bower, Eds., Academic Press, (1989). \n\nMacGregor, R., Neural and Brain Modelling, Academic Press, (1988). \n\nOlds, J.  L. et ai, Science, Volume 245,  866, (1989). \n\nTABLE 1 \n\nRESULTS OF PARAMETER SENSITIVITY ANALYSIS \n\nPROTOCOL:  EACH OF THE 17 INTERNAL PARAMETERS OF THE MODEL \nNEURON WAS VARIED IN TURN, WHILE ALL THE OTHERS WERE KEPT \nFIXED AT BENCHMARK VALUES.  THE DENDRITE WAS DRIVEN IN \nEACH CASE WITH A STEADY FIXED INPUT AND THE RESULTING \nBURSTING RATE AND FIRING RATE WAS COUNTED.  IN THE FINAL \nTRIAL, ALL THE PARAMETERS WERE LEFT FIXED AND THE INPUT \nMAGNITUDE WAS VARIED, TO SIMULATE FOR COMPARISON THE \nEFFECT OF MODULATION OF SYNAPTIC EFFICACY. \n\nPARAMETER \n\nSYMBOL \n\nVALUE \n\nSEC \n\nBURST \n\nFREQ.  BE~CHMARK \n\nBURSTS  SPIKES/  FIRING \n\nFIRING FREQ. \n\nAS  % OF \n\nSOMATIC MEMBRANE \nTIME CONSTANT \n\nDENDRITIC MEMBRANE \nTIME CONSTANT \n\nTS \n\nTD \n\nBENCHMARK \nlOW \nHIGH \n\nBENCHMARK \nlOW \nHIGH \n\n5.0 \n2.5 \n10.0 \n\nS.O \n2.5 \n10.0 \n\n13.51 \n13.70 \n12.82 \n\nl3.S1 \n13.51 \n12.66 \n\n2 \n2 \n2 \n\n2 \n2 \n2 \n\n27.03 \n27.40 \n2S.64 \n\n27.03 \n27.03 \n2S.32 \n\n100.0% \n101.4% \n94.9% \n\n100.0% \n100.0% \n93.7% \n\n\f156 \n\nRhodes \n\nPARAMETER \n\nSYMBOL \n\nVALUE \n\nBURSTS  SPIKES!  FIRING \n\nFIRING FREQ. \n\nAS%OF \n\nSEC \n\nBURST \n\nFREO.  BENCHMARK \n\nTHRESHOLD  FOR \n(CAJ-GA TED POTASSIUM \nCURRENT IN DENDRITE (I) \n\nCALCTHRESH  BENCHMARK \n\nLOW \nHIGH \n\nBENCHMARK \nLOW \nmGH \n\n20.0 \n10.0 \n40.0 \n\n33.0 \n16.5 \n66.0 \n\nACTIVATION RATE OF \nSOMA TIC POTASSIUM \nCURRENT (2) \n\nACTIVATION RATE OF \nDENDRITIC (CAJ-GA TED \nPOTASSIUM CURRENT \n\nTIME CONSTANT OF \nSOMATIC POTASSIUM \nCURRENT (2) \n\nTIME CONSTANT OF \nDENDRITIC POTASSIUM \nCURRENT (3) \n\nB \n\nBD \n\nTGK \n\nTGKD \n\nACTIVATION RATE OF \nCALCIUM CONDUCTANCE \n\nD \n\nTIME CONSTANT \nOF DENDRITIC CALCIUM \nCONDUCTANCE \n\nTGC \n\nACCUMULATION RATE OF \nCALCIUM FOR A GIVEN \nCALOUM CONDUCTANCE (4) \n\nA \n\nTIME CONSTANT FOR \nCALCIUM ACCUMULATION \n\nTCA \n\nINPUT CONDUCTANCE FROM  GDS \nDENDRITE TO SOMA (5) \n\nINPUT CONDUCTANCE FROM  GSD \nSOMA TO DENDRITE \n\nSOMATIC FIRING \nTHRESHOLD \n\nTHRESHOLD \n\nCA SPIKE THRESHOLD  CSPKTHRESH \nIN DENDRITE (6) \n\nSYNAPTIC INPUT TO \nDENDRITE (8) \n\nINPUT \n\nBENCHMARK \nLOW \nmGH \n\n75.0 \n37.5 \n150.0 \n\nBENCHMARK \nLOW \nHIGH \n\nBENCHMARK \nLOW \nHIGH \n\nBENCHMARK \nLOW \nmGH \n\nBENCHMARK \nLOW \nmGH \n\nBENCHMARK \nLOW \nHIGH \n\nBENCHMARK \nLOW \nHIGH \n\nBENCHMARK \nlDW \nHIGH \n\nBENCHMARK \nlDW \nHIGH \n\nBENCHMARK \nlDW \nHIGH \n\nBENCHMARK \nLOW \nmGH \n\nBENCHMARK \nlDW(7) \nHIGH \n\n3.5 \n1.8 \n7.0 \n\n10.0 \n5.0 \n20.0 \n\n2.2 \n1.1 \n4.4 \n\n5.0 \n2.5 \n10.0 \n\n2.0 \n1.0 \n4.0 \n\n5.0 \n2.5 \n10.0 \n\n5.0 \n2.5 \n10.0 \n\n5.0 \n2.5 \n10.0 \n\n12.0 \n6.0 \n24.0 \n\n12.0 \n6.0 \n24.0 \n\n35.0 \n27.0 \n70.0 \n\n13.51 \n12.82 \n13.51 \n\n13.51 \n12.99 \n13.51 \n\n13.51 \n12.35 \n13.16 \n\n13.51 \n13.51 \n13.33 \n\n13.51 \n21.74 \n8.00 \n\n13.51 \n14.71 \n11.11 \n\n13.51 \n14.29 \n12.82 \n\n13.51 \n13.51 \n12.99 \n\n13.51 \n14.71 \nII. 76 \n\n13.51 \n11.90 \n14.29 \n\n13.51 \n13.89 \n10.75 \n\n13.51 \n15.38 \n13. 16 \n\n13.51 \n14.08 \n13.70 \n\n13.51 \n11.63 \n16.95 \n\n2 \nI \n3 \n\n2 \n3 \n1 \n\n2 \n4 \n2 \n\n2 \n2 \n2 \n\n2 \n2 \n3 \n\n2 \n2 \n4 \n\n2 \n2 \n2 \n\n2 \n3 \nI \n\n2 \nI \n3 \n\n2 \nI \n4 \n\n2 \n2 \n2 \n\n2 \n4 \n1 \n\n2 \n2 \n2 \n\n2 \n2 \n2 \n\n27.03 \n12.82 \n40.54 \n\n27.03 \n38.96 \n13.51 \n\n27.03 \n49.38 \n26.32 \n\n27.03 \n27.03 \n26.67 \n\n27.03 \n43.48 \n24.00 \n\n27.03 \n29.41 \n44.44 \n\n27.03 \n28.57 \n25.64 \n\n27.03 \n40.54 \n12.99 \n\n27.03 \n14.71 \n35.29 \n\n27.03 \n11.90 \n57.14 \n\n27.03 \n27.78 \n2U1 \n\n27.03 \n61.54 \n13.16 \n\n27.03 \n28.17 \n27.40 \n\n27.03 \n23.26 \n33.90 \n\n100.0% \n47.4% \n150.0% \n\n100.0% \n144.2% \n50.0% \n\n100.0% \n182.7% \n97.4% \n\n100.0% \n100.0% \n98.7% \n\n100.0% \n160.9% \n88.8% \n\n100.0% \n108.8% \n164.4% \n\n100.0% \n105.7% \n94.9% \n\n100.0% \n150.0% \n48.1% \n\n100.0% \n54.4% \n130.6% \n\n100.0% \n44.0% \n211.4% \n\n100.0% \n102.8% \n79.6% \n\n100.00/0 \n227.7% \n48.7% \n\n100.0% \n104.2% \n101.4% \n\n100.0% \n86.0% \n125.4% \n\n\fA Systematic Study or the Input/Output Properties \n\n157 \n\nNOTES TO PARAMETER SENSITIVITY ANALYSIS \n\n(1)  The number of spikes  per burst  is  altered  by  modulating  the  internal  calcium \nconcentration required to  trigger  the dendritic potassium  current.  In an observation \nrepeated  several  times  herein,  it  seems  clear  that  modulating  the  hyperpolarizing \npotassium current has a marked effectiveness in modulating the neuron's output. \n\n(2)  Modulating the activation rate (B) of the somatic potassium current strongly effects \nruing, but changing the  time constant  of this  current  has almost no  effect either on \nbursts/second or spikeslburst. \n\n(3)  However, note that, among all  17 parameters of this model  neuron, it is only the \ntime  constant  of the  [CAl-gated  dendritic  potassium  current  which  is  effective  in \nmodulating the rate of bursting  (whereas th e somatic  potassium current time constant \ndoes not seem to effect the model neuron's output at alI). \n\n(4)  This quantity, the accumulation rate of calcium in the dendrite per unit calcium \nconductance, would increase as the effectiveness of calcium buffers within the dendrite \ndecreased. \n\n(5)  Despite its efficacy in modulating the neuron's output, this parameter is presumably \nnot a likely candidate for plasticity,  because it depends  on the axial resistance of the \ncytoplasm, the cross section of the base of the dendrite, and the volume of the soma, all \nof which seem unlikely to be the subject to modulation. \n\n(6)  Surprisingly, the overall input-output relation for the neuron is not much effected by \nchanging the threshold for the voltage gated calcium spike activity in the dendrite. \n\n(7)  The minimum dendritic  input required  to  produce any spike activity  (that  is,  to \nincrease the voltage  in the soma above firing threshold) may be calculated to  be  26.4 \nwith all the other parameters at benchmark values.  Hence 27  is an input level that is \nonly  2%  above the  minimum level  to get any firing  at all.  Note  that it appears  a  2 \nspike burst is always produced (with the internal parameters set at the benchmark levels) \nif any firing  at all is elicited.  The number of spikes per burst, then,  is  modulated  by \nconductance activation rates and calcium accumulation rates but not by input.  Tables 2 \nand 3 demonstrate this over a wide range of inputs. \n\n(8)  Note  that doubling the  synaptic  input to  the dendrite only increases  the  model \nneuron's firing rate by 25.4%, but that, for example, doubling the activiation rate of the \ndendritic calcium current increases the firing  rate by 64.4%.  Hence  we  suggest that \nmodulation of synaptic efficacy is not the only choice or even the most effective choice \nfor the  mechanism underlying plasticity.  Alkon (1988,1989) and others have in fact \nrecently reported that an increase in protein kinase C, leading to a reduction in calcium(cid:173)\nactivated  potassium  current,  is  observed  to  be  associated  with  conditioning  in \nHermissenda and rabbit.  Thus, plasticity in the nervous system may indeed operate via a \nwhole set of internal dynamic parameters, of which synapse efficacy is only one. \n\n\f158 \n\nRhodes \n\nDENDRITIC  K-CLTRRENT  TIl\\fE  CO NST:  5  !vISEC \n\nFIRING  RATE=43.48  BURST  RATE=21.74 \n\n-\n\nro  ..... . \n\nSOMA  VOLT. \n-.---A---\n\nOCN)  VOLT . \u2022 \n\nI<  roN) OCN \n\n-10 'mnllll1l1lDD1lDD1l111DlllRlmnmnllllllllllRllllmmmmlllll' \n\n1 \n\n20  40 \n\nro  00  100  120  1 40  1 ro  100  200  220  240 \n\nTlAE:(M5\u00a3C) \n\nDENDRITIC  K-CURRENT  TIME  CONST:  20  MSEC \n\nFIRING  RATE=24.00  BURST  RATE=8.00 \n\n~~----------------.-,---------~ \nro  ..... .................................. ....................... ....... ...... . \n\n-\n\n-\n\u2022 \n.. \n\nOCN)  VOLT. \u2022 \n\nK OJN) OCN \n\n20  40 \n\nro  00  1 00  1 20  1 40  1 ro  1 00  200  220  240 \n\nTh1E:  ().15\u00a3C) \n\nFigure  1 \n\n\fA Systematic Study or the Input/Output Properties \n\n159 \n\nTHE  INPUT/OUTPUT  RELATION \n\nCA  ACClThfUIATION  RATE  SET  AT  3  lEVElS \n\nM.-----------------------~ \n70  .............. . \n\n0~-.r-~--~--~--~--~--4 \no \n700 \n\nroo \n\n500 \n\n300 \n\n200 \n\n400 \n\n100 \n\ntRJf RAT[ \nFigure  2 \n\nCOMP ARISON  OF  INPlIT,IOlITPUT  REIATION \n\nEPSP  SQlTARE  PlJIEE  VS  AlPHA  FUNCTION \n\nM~----------------------~ \n70  ............................................................................ . \nro  ............................................................................ . \n\n...... -- ...... ~ ......................................................... -............. - ........................... -\n\n10  ............................................................................ . \n\no~~--~--~--~--~--~~ \no \n700 \n\nroo \n\n500 \n\n200 \n\n100 \n\n400 \n\n3CO \n\ntfllJr  RAT[ \nFigure  3 \n\n\f", "award": [], "sourceid": 219, "authors": [{"given_name": "Paul", "family_name": "Rhodes", "institution": null}]}