parameter estimation methods for single neuron models

Journal of Computational Neuroscience 9, 215–236, 2000c© 2000 Kluwer Academic Publishers. Manufactured in The Netherlands.

Parameter Estimation Methods for Single Neuron Models

JOEL TABAKEquipe de Neurobiologie, CNRS URA 256, Universite de Rennes 1;Laboratory of Neural Control, NINDS/NIH, Bethseda, MD 20892

[email protected]

C. RICHARD MURPHEYDepartment of Physiology and Biophysics, University of Texas Medical Branch at Galveston

L.E. MOORELaboratoire de Neurobiologie des Reseaux sensorimoteurs, UPRESA 7060, Universite de Paris 5

Received October 30, 1998; Revised June 29, 1999; Accepted February 16, 2000

Action Editor: Alexander Borst

Abstract. With the advancement in computer technology, it has become possible to fit complex models to neuronaldata. In this work, we test how two methods can estimate parameters of simple neuron models (passive soma) tomore complex ones (neuron with one dendritic cylinder and two active conductances). The first method usesclassical voltage traces resulting from current pulses injection (time domain), while the second uses measures ofthe neuron’s response to sinusoidal stimuli (frequency domain). Both methods estimate correctly the parametersin all cases studied. However, the time-domain method is slower and more prone to estimation errors in the cableparameters than the frequency-domain method. Because with noisy data the goodness of fit does not distinguishbetween different solutions, we suggest that running the estimation procedure a large number of times might helpfind a good solution and can provide information about the interactions between parameters. Also, because theformulation used for the model’s response in the frequency domain is analytical, one can derive a local sensitivityanalysis for each parameter. This analysis indicates how well a parameter is likely to be estimated and helps choosean optimal stimulation protocol. Finally, the tests suggest a strategy for fitting single-cell models using the twomethods examined.

Keywords: model identification, time domain, frequency domain, sensitivity analysis

1. Introduction

Mathematical models can be useful to answer questionsthat are difficult or impossible to investigate experimen-tally. However, this usefulness is limited if one doesnot have a way to constrain the values of the model’sparameters. The methods for estimating the parame-ters of single neuronal models have traditionally beenseparated in two parts: (1) estimate the passive elec-

trotonic structure with current pulses (Rall, 1969), and(2) estimate the kinetics, conductance, and reversal po-tential of active currents (Hodgkin and Huxley, 1952).This is problematic for several reasons. First, it is dif-ficult to assess how the passive estimates are contami-nated by active conductances. Second, the dendritic treeis sometimes not taken into account when analyzingvoltage-dependent currents in voltage clamp. Further,the different voltage-dependent currents have to be

216 Tabak et al.

pharmacologically separated. In general, this meansthat one cell provides information on only one cur-rent, making the process of analyzing each current verytime consuming. In addition, models built using dataobtained in such a way have to use an average electro-tonic structure, average potassium conductance, aver-age sodium conductance, and so on. Thus, if any inter-actions between the various parameters are present inindividual cells, they might be lost in the model.

Ideally, one would like to reconstruct individual neu-rons rather than use an average neuron model obtainedafter averaging parameters from several recordings.That way, information about the heterogeneity of thepopulation is kept. Some modelers have started to usedirect estimation methods from current clamp record-ings (Foster et al., 1993; Bhalla and Bower, 1993), butmost have limited themselves to estimating maximalchannel conductances using previously reported valuesof passive and kinetic parameters (for more completeestimation, see Weyhing and Borst, 1997). However,kinetic parameters “from the literature” might need tobe corrected for the particular cases studied (Surkiset al., 1998).

Estimating both passive and active parameters usingfrequency-domain (Murphey et al., 1995) or time-domain (Murphey et al., 1996) analysis provides mod-els that are able to satisfactorily fit electrophysiologi-cal data. However, does that mean that the parametersare correctly estimated? This is questionable becauseof the number of parameters that are used and the in-evitable noise in the data, so nonuniqueness of the so-lutions is a problem. Another issue is to know whichmethod provides the best estimates. The frequency-domain method consists of injecting a cell with white-noise signal of low intensity on top of a steady voltagelevel, at various levels (Moore and Christensen, 1985;Murphey et al., 1995). The (fast) fourier transform ofthe response is divided by the fourier transform ofthe stimulation, providing the transfer function at eachvoltage level. This piecewise linear method is thoughtto be superior because of the ability to control the con-tribution of each frequency, while for a step functionthe power decreases rapidly with frequency (Mooreand Christensen, 1985). However, time-domain anal-ysis has been more widely used so far, with very fewexceptions (see Wright et al., 1996, and their biblio-graphy for passive parameters).

To answer these questions, we have generated time-and frequency-domain data for neuronal models rang-ing from the simplest, isopotential, and passive cell

to a neuron with a dendritic cylinder endowed with apotassium and a sodium conductance to test the abilityof both methods to correctly estimate the model pa-rameters. For each case, the methods are first testedwith noise-free computer-generated data as a best-casescenario: if it fails to find the correct estimates, it ob-viously cannot determine them with noisy data. In asecond step, noisy-generated data are used to indicatehow noise alters the estimation procedure. These testssuggested a working strategy for experimental data.Finally, some ways to assess the quality of the estima-tion are also discussed.

2. Methods

2.1. Estimated Parameters

The neuronal model used in this study consists of asoma coupled to a uniform dendritic cylinder. The pa-rameters are summarized in Table 1. For a completelyuniform passive structure only four parameters have tobe considered:gl , the leakage conductance of the soma;C, the capacitance of the soma;L, the electrotoniclength of the cable; andar , the ratio of cable surfaceand somatic surface. The ratio of cable conductance andsomatic conductanceρ is more commonly used thanar . Note thatar is equal toρ L/tanhL, the constantappearing in the transcendental equation that expressesthe relationship betweenL and the so-called separationconstants (Rall, 1969; Durand, 1984; Kawato, 1984).

Table 1. Glossary of model parameters.

Parameter Unit Definition

gl µS Somatic leakage conductance

gsh µS Somatic shunt conductance

C pF Somatic capacitance

L – Electrotonic length of dendritic cylinder

ar – Area ratio (dendrite/soma)

gi µS Maximal somatic conductance for ioni

vx mV Half-activation potential for gating variablex

sx mV−1 Slope ofx∞(V) at half-activation

tx ms Time constant of gating variablex athalf-activation

rx mV−1 Slope ofτx(V) normalized bytx athalf-activation

Vl mV Leak reversal potential

Vi mV Reversal potential for ioni

Parameter Estimation Methods 217

To account for a possible higher conductance on thesoma, due to tonic inhibition or microelectrode penetra-tion, a somatic shunt conductancegsh is added, bring-ing the total to five parameters. Another parameter thatmight need to be estimated is the electrode resistanceRe. This study assumes electrode compensation andtherefore small values forRe. Electrode capacitancewas not taken into account.

The model neuron also possesses voltage-dependentionic conductances (one fast-inactivating sodium andone potassium). The kinetic model used to describethese conductances is a simplification of the Hodgkin-Huxley model that has previously been described(Murphey et al., 1995). Briefly, ifm is a gating variable,thenm∞(V) andτm(V) are described by four param-eters:vm, the half-activation voltage;sm, the slope ofm∞ at half activation;tm, the time constant at half acti-vation (τm(vm)); andrm, the slope of the time constantat half activation (normalized with respect totm). If oneassumes that the reversal potentials are known but themaximal conductances (gK and gNa) are not known,then five parameters have to be estimated for a non-inactivating conductance and nine for an inactivatingconductance.

The values of the different parameters used togenerate the data (referred to as the test values, ortarget values) were chosen so that the models give re-sponses similar to those ofXenopusembryo spinal neu-rons recorded with sharp microelectrodes for stimulisimilar to those applied experimentally.

For the time-domain analysis the parameter val-ues ranged from 0 to 2 times the test values. For thefrequency-domain analysis the range was generallylarger.

2.2. Time-Domain Analysis

2.2.1. Structure. The model has a compartmentalstructure. The leakage conductance and capacity ofeach cable compartment arear gl/N andar C/N withN being the number of compartments. The axial resis-tance between each compartment isL2/(N ar gl ). Inthe case of a passive dendritic neuron, the number ofcompartments used for the cable was 30 forar = 5and 10 forar = 0.5.

2.2.2. Procedure. The simulated data (in the follow-ing, referred to as data, test data, or target data) werecreated with a C program that was also incorporatedin the estimation procedure. The differential equations

were numerically solved using Epsode, a variable-step-size, variable-order, backward-integration pack-age (Byrne and Hindmarsh, 1975) to compute volt-age traces. The computed voltage traces were saved astime series with a time interval of .5 ms between con-secutive points. Each trace was computed for at least20 ms. On a few examples, we checked that test datagenerated by a program implemented in Mathematica(Wolfram Research) gave the same results. To mimicthe noise (amplitude on the order of 0.5 mV) observedon recorded voltage transients we generated a randomseries of points between−0.5 and 0.5 mV that wasadded to the simulated data after being filtered with athree-point average operator.

The model parameters are optimized to fit the datausing the simplex algorithm (Nelder and Mead, 1965;Press et al., 1988). A least-squared error criterion isused, as follows: for each point of each level of in-jected current, the square of the difference between theresponse of the model and the data is computed. Thesquared difference between the slopes of the modeledand target signal can also be useful (see Rall, 1969;Burke and ten Bruggencate, 1971, for passive parame-ters). So ifv(i )(k) is the voltage computed by the modelfor trace numberi at instantk andd(i )(k) the actualdata value, the error computed at that point is

s[(v(i )(k)− d(i )(k))2] + (1− s)[((v(i )(k)

− v(i )(k− 1))− (d(i )(k)− d(i )(k− 1)))2], (1)

wheres is chosen to determine the balance between theactual signal and its derivative. Withs= 0.5, the valuegenerally used, the influence of the derivative is weakbecause the difference terms have not been divided bythe duration between instantsk andk−1 (0.5 ms). Theerror is summed over the chosen time interval and overthe number of traces and then divided by the number oftraces; finally the square root is taken, giving the rootmean square error (rms) used as the error criterion inthe optimization method. The time interval is definedby a starting time, generally equal to the beginning ofcurrent injection, and a final time, which value can beadjusted to increase or decrease the relative importanceof the transient phase of the voltage traces. The simplexalgorithm minimizes thermserror until the relativermsdifference between the best and worse sets of param-eter values used by the simplex is below 10−3 or aftera maximum of 600 iterations. If the fit is not satisfy-ing, the algorithm can then be restarted after perturb-ing all but one of the sets of values. The optimization

218 Tabak et al.

program can be restarted automatically with different(random) initial estimates. Thus, a number of differentsolutions (a solution being a set of parameter valuesthat fit the data) can be obtained and later examined us-ing scatter plots (Foster et al., 1993; Tabak and Moore,1998).

2.2.3. Reversal Potential of the Leak Current.Thefitting procedure does not take into account the voltagelevel before current injection. Nevertheless, at the be-ginning of the current pulse, the voltage of the neuronmodel should be at a steady-state value equal to theresting potential of the model neuron used to generatethe data. If this were not the case, there would be ashift between model and data voltage whose reductionwould lead to estimation errors (that is, wrong valuesfor the estimated parameters). For a passive neuron,Vl

is simply set to the desired resting potential.In the presence of active conductances it is necessary

to adjustVl each time new parameters are used, sothat the net current flowing into the soma is null atrest. For an isopotential cell with potassium and sodiumconductances,

Vl = Vr + gK

gln∞(Vr ) (Vr − VK )

+ gNa

glm∞(Vr ) h∞(Vr ) (Vr − VNa). (2)

Note the importance of calculating the correct valueof Vl . For the case of a soma with potassium conduc-tance, the value ofVl used to generate the data was−53.96 mV. IfVl is fixed to−55 mV (instead of beingrecomputed for each new set of parameters), there is ashift initially between the calculated voltage trace andthe target trace if the parameters have their test value.This shift is compensated for in the estimation process,which leads to an important estimation error for someparameters:

gK ,−18%; sn,+36%; |rn|,−60%.

The decrease ofgK and increase ofsn allow to decreasethe potassium current at rest, so that the resting poten-tial is close to−55 mV. Thus, only a 1 mVdiscrepancyon Vl can have a dramatic effect on the quality of theestimation.

If a dendritic structure is coupled to the soma, oneshould incorporate the current flowing between thesoma and the dendrite in Eq. (2) because the resting po-tentials of the different dendritic compartments are notnecessarily equal to the soma resting potential if thereare nonuniformities. However, we assumed that the

resting potential was nearly uniform and used Eq. (2)to computeVl . We found this approximation to be validin the cases we studied. For more complex structures,Eq. (2) might not be accurate enough, and the calcu-lation of Vl could be more time consuming. Using theexample of a soma with potassium conductance, wehave therefore examined two alternative possibilities:(1) fit using the derivative of the voltage traces (that is,use a very low value ofs in Eq. (1)) or (2) compute thevoltage shift at the beginning of current injection andfit so that this shift is maintained over the entire trace.These two methods allow the shift to be ignored, andthe parameters are well estimated.

2.3. Frequency-Domain Analysis

The test data were created with a Mathematica script,according to the model for the admittance describedbelow. Therefore, the white-noise-injection procedure(described in Section 1) was not simulated. To calcu-late the admittance at the soma and cable junction, it ispossible to use an analytical formulation rather than acompartmental one. The admittance is simply the sumof the soma admittance (including the shunt conduc-tance) and the cable admittance:

Y = gsh+ Ysoma+ ar gl

L

√Ydend/gl tanh

(L√

Ydend/gl).

(3)

This differs from Murphey et al. (1995) and previousstudies, where a compartmental model had been used.The termYsoma(soma admittance) is given by (Mauroet al., 1970; Moore et al., 1993)

Ysoma= δ I

δV= ωC+ gl + gK n∞ + gNa m∞h∞

+ Gn

(1+ ωτn)+ Gm

(1+ ωτm)+ Gh

(1+ ωτh),

(4)

where

Gn = gK (V − VK )dn∞dV

Gm = gNa (V − VNa) h∞dm∞dV

(5)

Gh = gNa (V − VNa)m∞dh∞dV

.

The termYdend is similar toYsoma, except that it mightnot contain the same amount of voltage-dependent


conductances. Note that the steady-state voltage in thedendritic cable (which is used to compute the value ofthe voltage-dependent conductances if any are present)is taken as uniform and equal to that in the soma,which is an approximation similar to the one used tocomputeVl for the time-domain method. For smallperturbations from rest and small nonuniformities ofthe membrane this is often a reasonable approxima-tion (L.E. Moore, unpublished observation). In the caseof a nonnull electrode resistanceRe, the admittancebecomes

Y′ = Y

1+ YRe. (6)

Typically, for each holding potential frequenciesranging from 0.5 to 100 Hz were used (in steps of0.5 Hz). To mimic noisy data, random noise of am-plitude 4 MÄ and 0.4 MÄ was added to the real andimaginary parts of the impedance, respectively. Sincethe impedance decreases with frequency, the noise levelgenerated this way is more important at high frequen-cies, as observed in experimental recordings. However,when the cell is depolarized, the impedance decreases,and the noise can affect more at lower frequencies, dif-fering somewhat from experiments. Our goal is not toassess accurately the effect of noise on the estimationbut more modestly to create a situation in which thequality of the fits cannot be assessed solely on thermsvalue.

Similarly to the time-domain method, anrmserroris computed between the model and the data admit-tances, over the different voltage levels. However, inthis case, therms is normalized, and the optimizationmethod is based on a gradient descent (Dennis et al.,1981).

3. Tests of the Identification Methods

We have studied the following cases: passive soma,passive soma with a passive dendritic cylinder (alsoreferred to as passive dendritic neuron), soma withpotassium conductance, soma with potassium conduc-tance and dendritic cylinder and finally neuron withpotassium and sodium conductances. For each case, aschematic, the voltage transient(s) (left side), and thecomplex admittance(s) (right side) used for the fits arerepresented in Fig. 1 (passive models) or Fig. 3 (modelswith voltage-dependent conductances).

The admittance diagrams have been previously des-cribed (Murphey et al., 1995). They represent the realpart (horizontal axis) and imaginary part (vertical axis)

of the admittance, computed for each frequency ac-cording to Eq. (3). Each trace starts from the horizontalaxis, since at DC the imaginary part of the impedance isnull.

3.1. Passive Soma

The passive and isopotential neuron is modeled as anRC circuit, so two parameters are estimated,gl andC.The traces used for the estimation are shown in Fig. 1A.When no noise is added, both time- and frequency-domain methods give perfect estimates. The additionof noise only results in a small loss of accuracy (esti-mation error around 1% or less). For both methods, theconductance is better estimated than the capacitance.

3.2. Passive Dendritic Neuron

We now consider a passive soma coupled to a pas-sive cable. A shunting conductance is also added onthe soma: The structure, its response to a current stepinjection (−.2 nA), and its complex admittance are rep-resented in Fig. 1B. The tests were conducted for twodifferent values ofar . Note how increasingar increasesthe cell’s conductance and affects the admittancerepresentation.

Also shown on this figure is the response obtained forwrong values of the dendritic parameters (thin curve,indicated by an arrow) in both time and frequency do-mains (forar = 5). Although this solution would be re-jected in both cases, it is obviously harder to distinguishit from the right solution in the time domain than in thefrequency domain. To quantify this, we have calcu-lated the average of normalized square differences forboth representations. This error measure is about oneorder of magnitude higher for the frequency domain.Calculations done for other parameter sets (not shown)give similar results. This suggests that the frequency-domain method might be better for estimating the pas-sive parameters. The results presented below seem toconfirm this observation.

3.2.1. Time Domain. The fit is computed for 45 msfrom the beginning of current injection. The estimatesare often perfectly estimated but not always. However,the best solutions are found when thermsis lowest, sothe method is correct.

The fit does not always find the perfect solution be-cause errors on the parameter estimates might result invery smallrmserrors (see Fig. 1B and examples below).

220 Tabak et al.

Figure 1. Schematic of each passive case studied, with the voltage trace and complex admittance used for the time- and frequency-domain fits.The voltage traces in each case were computed for a current injection of−.2 nA. A: Passive soma, the estimated parameters aregl , 0.01µS;C, 30 pF.B: Passive soma with shunt, coupled to a passive cylinder, parameters aregl , 0.001µS; gsh, 0.009µS; C, 30 pF;L, 1; ar , 5.0 (solidcurves) or 0.5 (dashed curves). The membrane time constantτd equals 30 ms (the somatic time constant isτs= 3 ms). The input conductanceGin is 0.013808µS forar= 5 and 0.0103808µS forar= 0.5. The thin curves correspond to the model output for altered values of the dendriticparameters (gl , 0.0006;L, 1.5;ar, 10.0). The normalized squared difference between the thin and thick curves is 0.000013 for the time domain(correspondingrms is 0.24 mV) and 0.00037 for the frequency domain.

In other words, the addition of even a very small amountof noise can lead to substantial errors in some of theparameter estimates (White et al., 1992). The errors oncable parameters become even larger for small valuesof ar , since their contribution on the model’s responsebecomes smaller.

To illustrate the problem of nonuniqueness of thesolutions, we have taken 50 estimations for each valueof ar and selected the results for which thermserroris lower than 0.01 mV (the resulting voltage curvescannot be distinguished visually). There are 24 of suchsolutions forar= 5 and 47 forar= 0.5. Table 2 givesthe maximal error found out of these solutions for eachparameter. The purely somatic parameters,C andgsh

are much better estimated than the dendritic ones. Thedifference is more marked forar= 0.5 since a smallerdendrite has less effect on the response to somatic cur-rent injection.

Using the estimated parameters to calculate othercharacteristics of the neuron gives very good esti-mates for the input conductance (Gin) and somatictime constant (τs), but not for the dendritic (or mem-brane) time constant (τd), as shown in Table 2. Thesethree characteristics are shaping the response of themodel neuron; therefore, one might expect to deter-mine them accurately. However, if the soma has a muchgreater influence on the total response than the den-drite,τd is less important and might not be determinedaccurately.

The estimation procedure has to choose parametersso that the value ofGin is conserved. This imposes someconstraints on the relationships that some parameterswill have with each other. The relationships betweentwo given parameters can be illustrated by plottingthe points whose coordinates correspond to the valuesof these two parameters for each acceptable solution.


Table 2. Maximal discrepancy (in percentage of thetest value) found out of 50 fits, forrms< 0.01 mV. Inthe casear= 0.5, the numbers in parentheses are thepercentages obtained when the results were restrainedby rms< 0.0035 mV so that 24 solutions are selected.Gin (input conductance),τd (dendritic time constant),andτs (somatic time constant) were calculated fromthe estimated values of the parameters.

Maximum Error (%)

ar = 5 ar = 0.5

gl 18 60 (36)

gsh 15 10 (6)

C 6 2 (0.8)

L 19 50 (30)

ar 27 100 (30)

Gin 0.24 0.7 (0.15)

τd 29 112 (58)

τs 4.9 2 (1)

Figure 2. A, B, C: Scatter plots showing the relationships between pairs of parameters in the casear = 5. The parameter values are standardizedas deviations from the test values—that is, 0 corresponds to the test value, 1 to a value that is one standard deviation from the test value.D:Global sensitivity plot forC.

Examples are shown in Fig. 2. A trivial case is that ifgl is underestimated thengsh has to be overestimatedto keep the same value ofGin, so there is a negativecorrelation betweengl and gsh. Fig. 2C shows that,for the same reason,L and ar have to be positivelycorrelated. In the same way, to preserve the value ofτs = C/(gl + gsh) there is a positive relationship be-tweenC and gsh (Fig. 2B). Because of this positivecorrelation betweenC andgsh and because of the neg-ative correlation betweengsh andgl , there is a negativerelationship betweenC andgl (Fig. 2A). Thus, an er-ror onC will lead to an opposite error ongl and botherrors will add in the calculation of the membrane timeconstantτd=C/gl .

Figure 2D shows therms error as a function ofC.Each value ofC represented on this graph correspondsto a particular solution, so each point corresponds todifferent values of the other parameters as well. Be-cause it uses many different solutions, we have called

222 Tabak et al.

Fig. 2D a global sensitivity plot of the error to this pa-rameter (Tabak and Moore, 1998), in contrast to classi-cal sensitiviy analysis that is local to one solution. Thisplot indicates that the estimation is very accurate, sincethere is a clear minimum at the target value. However,this is because no noise was added to the test data. Be-cause of the strict constraint used to select solutions(rms< 0.01 mV), the scatter plots are concentrated. Ifone parameter is known, the graphs suggest that thevalues of the other parameters will be well determined.For the casear= 0.5, let us assume that the value ofaris given by a morphological examination of the neu-ron and therefore fixed to its target value. Then, outof 10 fits, nine give close to perfect solutions, and onefit does not meet therms error criterion. Therefore,morphological constraints can provide the supplemen-tary information necessary to obtain a more accurateparameter estimation.

In the case of noisy data the scatter plots are morespread (not shown), but the relationships remain. Also,there is no sharp minimum in the global sensitiv-ity plots, and the minimumrms does not in gen-eral correspond to the target value. The estimationerrors are increased about threefold, so the maximalestimation errors can reach 100% for the dendriticparameters.

3.2.2. Frequency Domain. For both values ofar ,the estimations with noise-free data are perfect andfaster than with the time-domain method. The algo-rithm rarely converges to a local minimum. The elec-trode resistance (Re, with a value of 0 MÄ in the targetdata) is then added in the list of parameters to esti-mate. Again, perfect estimation is found, but local min-ima occur more often. Also, these local minima corre-spond to such low values of the quadratic error that theyare hardly distinguishable from the right solution eventhough the associated estimation errors can be quitelarge.

For data with added random noise, only a few trialshave been done (forar = 5). The fits are good whenthe signal to noise ratio can be increased by using mul-tiple (target) data sets. With four traces, the estimationerror is lower than 10% for each parameter. When onlyone trace is used, the average estimation error on cableparameters is about 20%, but the maximal error can beup to 95% (8 trials). Furthermore, all fits yield similarquadratic error (0.22 to 0.24), so this criterion cannotbe used to select the best estimates.

3.3. Soma with Potassium Conductance

This case is illustrated in Fig. 3A. It is clear from thevoltage traces that an outward current activates atdepolarized potentials. Note, on the admittance dia-gram, how increasing the voltage level increases the ab-solute value of the admittance. Also, at−44 mV, thereis a resonance, the absolute admittance goes through aminimum (for a more conventional representation of aresonance, see Fig. 4).

3.3.1. Time Domain. Estimations are conducted for15 ms after current injection, over 10 traces obtainedfor I = −0.2 nA to 0.8 nA. The reversal potential forpotassium,VK , is assumed to be known and fixed to itsvalue. Thus, seven parameters are estimated. Withoutnoise, perfect fit is found most of the times, even if theparameters range is increased to up to four times thetest values.

Fits with Noisy Data. The results from 23 fits aresummarized in Table 3. Thermserror for each of thefits is between 0.23 and 0.24 mV, so no solution can bedistinguished from each other on this basis.

Table 3 shows that the average of a parameter valueover all the fits is a good estimate. In general, the me-dian, which is less sensitive to outliers, provides evenbetter estimates than the average (not shown). An in-dication of the quality of the method is given by theaverage error or by the maximal error. The maximalerror allows to evaluate the risk one takes by using theestimates given by a single fit of the data but dependson the number of fits conducted. The average error isa better criterion if one wants to compare two estima-tion methods with a limited number of runs because if

Table 3. Summary of results from 23 fits for a soma withpotassium conductance. The passive parameters andvn arewell estimated.

Average Mean Error (%) Maximum Error (%)

gl 0.010 1.2 5.3

C 30 0.42 1.2

gK 0.022 13 49

vn −34 3.3 12

sn 0.048 8.2 28

tn 9.7 5.2 22

rn − 0.050 13 42


Figure 3. Schematic for each case studied that included voltage-dependent conductance(s), with the voltage traces and complex admittancesused for the time- and frequency-domain fits. The voltage traces in each case were computed for current injections ranging from−.2 to .8 nA.The admittance traces were computed for the following voltage levels (in mV):−84,−74,−64,−54,−44 (and−34,−24 in C). The tracesfor −84 and−74 mV are superimposed. Note the resonance for traces at−44 mV and higher.A: Soma with potassium conductance.B: Somawith shunt and potassium conductance, coupled to a dendritic cylinder (without potassium conductance for the traces represented).C: Somawith potassium and sodium conductances. For the frequency domain, there is also a shunt and a passive cylinder. Values of passive parametersare the same as described in Fig. 1. For the potassium conductance,gK , 0.02µS;vn,−35 mV;sn, 0.05 mV−1; tn, 10 ms;rn,−0.05 mV−1; VK ,−84 mV. For the sodium conductance,gNa, 0.04µS;vm,−15 mV;sm, 0.06 mV−1; tm, 0.5 ms;rm,−0.1 mV−1, vh,−25 mV;sh,−0.06 mV−1;th, 5 ms;rh,−0.1 mV−1; VNa, 50 mV.

one very bad solution is found, the average will be lessaffected than the maximal error.

The potassium reversal potential might not be knownwith great accuracy. If we use an inaccurate value ofVK during the fit, the main effect is on the estimateof gK to maintain an equivalent amount of potassiumcurrent without affecting the kinetics. To know if the

reversal potential could also be estimated, we have donea series of fits for whichVK is also a parameter tobe estimated (range:−100 to−55 mV). The resultsindicate that the inaccuracy ongK can be much greaterthan whenVK is known. However, the estimation ofthe other parameters are not affected very much. Theparametersvn andVK are well estimated.

224 Tabak et al.

Figure 4. Two different sets of parameters give nearly identical frequency response. For each of the two equivalent representations (modulusand phase of the impedance, or complex admittance) and for different voltage levels, the generated data (dots) and the two candidate fits aredrawn. The traces corresponding to a voltage level of−84 mV are not shown (they are identical to the ones for−74 mV). Parameter values forboth solutions are as follow, with the values for the worse solution in parentheses:Re, 0.19 (1.8);gl , 0.0010 (0.00012);gsh, 0.0095 (0.014);C,33 (45);L, 0.94 (0.23);ar , 4.2 (1.8);gK , 0.022 (0.030);vn, −35 (−34); sn, 0.049 (0.048);tn, 11 (12);rn, −0.039 (−0.036);VK , −82 (−77);fraK, 0 (0).

3.3.2. Frequency Domain. Five admittance traces,obtained for potential levels ranging from−84 to−44 mV, are used for the fits (Fig. 3A, right panel).All parameters are perfectly estimated when noise isabsent, even when the fits also includeVK . It is also

possible to estimateRe with the other parameters (VK

fixed in that case), but local minima are encountered.As in the time domain, using noisy data results in

only a small loss of accuracy since multiple traces areused.C andgl are estimated with an accuracy better


than 0.5%, while the maximal error on the other pa-rameters is about 10% (forgK andrn).

3.4. Dendritic Neuron with Potassium Conductance

The addition of a dendritic cylinder to the previous caseleads to an increased input conductance and affects theadmittance traces in the same way as in the passive case.This is illustrated in Fig. 3B (compare with Fig. 3A andFig. 1). Ten parameters have to be estimated (whenVK

andRe are fixed).

3.4.1. Time Domain. The potassium conductance isrestricted to the soma for the cell used to generate thetime-domain data. As in the previous case,Vl is ad-justed according to Eq. (2); hence the current betweensoma and dendrite is not taken into account. The result-ing shift in resting potential is negligible in most cases.No fits are done with noisy data.

Out of nine trials (ar= 5), only one gives anrmserror lower than 0.01 mV. The values obtained showthe following discrepancies with the test values:gl ,15%;gsh, 8%;C, 0.7%;L, 16%;ar , 14%;gK , 5%;vn,1%; sn, 5%; tn, 0.4%;rn, 10%. The active parametersare better estimated than the passive cable parametergl , L, andar .

Since an inaccuracy on the cable parameters doesnot prevent a good estimation of the active parameters,we have also tried to fit a model without cable to thesame data. In four trials, the maximal estimation errorson the potassium parameters are 20 to 40% (exceptrn,100%). Therefore, this is a low-cost method to get agross estimate of the potassium parameters.

If one tries to fit the data with a model for whichthe potassium conductance has a uniform distributionalong soma and cable,gK is always underestimated(about 50%) to compensate for the extra potassiumconductance on the dendrite, and there is a lot of vari-ability in the solutions from one fit to another. Thedifference with the correct model is large enough tobe detectable. Therefore, the method can distinguishwhether the potassium conductance is present only onthe soma or uniformly distributed.

3.4.2. Frequency Domain. We have studied cases fordata generated with and without dendritic potassiumchannels. We callfraK the ratio of dendritic potas-sium channel density over somatic potassium channeldensity. For a uniform distribution,fraK = 1, if no

potassium channels are present on the dendritefraK =0. As for the case of an isopotential cell (Section 3.3.2),five admittance traces are used for voltage levels rang-ing from−84 to−44 mV (Fig. 3B, right).

In most cases perfect estimations are obtained, evenif Re, VK and fraK are estimated, showing that themethod is sensitive enough to distinguish betweendifferent distributions of the potassium conductance.However, the high number of parameters leads to morelocal minima problems, so finding the best minimumdepends critically on the initial guess. To avoid localminima, it is possible to fit only the passive parametersusing the most hyperpolarized trace(s) and then to usethese values as initial guess for the complete fit. Thedifferent cases studied are described in Appendix.

Noisy Data(Done only for fraK= 0, ar = 5). Thefit was done for 11 parameters (fraK and Re not in-cluded). The two estimations found are reasonable:gl ,5%; gsh, 0.7%;C, 6%; L, 1%;ar , 8%; gK , 10 to 60%;vn, 1 to 9%;sn, 3 to 7%;tn, 11 to 18%;rn, 26%;VK , 4%.As observed without a dendritic cylinder (for both timeand frequency domain), the passive parameters are bet-ter estimated in the presence of noise. Because of thefew trials performed, we cannot exclude the possibilityto find a better fit; however, it will not be possible todistinguish the estimations by comparing therms. If asixth admittance trace is added, obtained at a voltagelevel of−49 mV, the estimation of the potassium pa-rameters is improved:gK , 3 to 14%;vn, 0.3 to 2%;sn, 2to 4%;tn, 7 to 8%;rn, 20%;VK , 3%. If we try to also es-timateRe andfraK, the effect of noise combined to thehigh number of parameters to estimate leads to fits withvery different parameter values. Fig. 4 shows the exam-ple of two different sets of parameters that fit the dataequally well. This is illustrated with both the complexadmittance representation and the more classical repre-sentation of absolute value and phase of the impedance.The main difference between the two solutions used ison the passive parameters, which are reasonably esti-mated in one case. Is it possible to distinguish thesetwo solutions using the time-domain data? The voltagetransients obtained for the two sets of parameters areshown in Fig. 5. They suggest that the two solutionscan be distinguished over many voltage traces, and thebetter solution clearly gives a response closer to thedata. However, for a large enough amount of noise inthe data, it might not be possible to distinguish the twosolutions.

226 Tabak et al.

Figure 5. The two sets of parameters that produce indistinguishable responses in the frequency domain give slightly distinct traces in responseto current injections. Arrows point to the worse solution. Data are shown dotted. The simulations were done with a five-compartment dendriticcable.

3.5. Neuron with Potassiumand Sodium Conductances

This case is illustrated in Fig. 3C. The amount of in-ward current is too small to generate an action potential.However, the effect of the negative slope conductancedue to the activation of sodium channels can be noticedfor potentials as low as−44 mV, slightly shifting theadmittance trace to the left (compare with the one justabove in Fig. 3B) and is strong for the highest voltagelevel (−24 mV; for this trace there is also a decreaseof the termdn∞/dV from Eq. (5), further decreas-ing the conductance). The sodium reversal potentialVNa is not estimated. The case of a cell with a den-dritic cylinder is addressed only for the admittances(frequency domain), and all tests are done withoutnoise.

3.5.1. Time Domain. The tests are restricted to anisopotential cell; hence 11 parameters have to be esti-mated. The fit is conducted for 20 ms following currentinjection.

Because no noise is present the better estimation cor-responds to the lowerrmserror. In the best case (out offour trials), passive and potassium parameters are verywell estimated (maximal error onrn, 0.4%), and theinaccuracy on sodium parameters is reasonably small(3 to 16%). However, if we do not constrain all the pa-rameters to stay between 0 and 2 times the test value,large inaccuracies can occur. Also, the high numberof parameters induces a slow convergence of theoptimization.

3.5.2. Frequency Domain. The tests are conductedwith a dendritic neuron (ar = 5), but the active


Table 4. Summary of the tests conducted.

Test Time Domain Frequency Domain

Passive soma Perfect estimations

Only small loss of accuracy with noise

Passive dendritic neuron Good, but nonuniqueness problems Perfect estimations

Large inaccuracies for small signal/noise

Soma withgK Perfect estimations

Reasonable estimations with noise

Dendritic neuron withgK Nonuniqueness (passive params) Perfect estimations

Nonuniqueness problems with noise

Neuron withgK andgNa Inaccuracies mostly on sodium parameters

conductances are restricted to the soma. Seven admit-tance traces were used as target data, for potential lev-els ranging from−84 to−24 mV (Fig. 3C, right). Asfor the time domain, passive and potassium parameterswere perfectly estimated but not the sodium parame-ters, the worst estimated parameter beinggNa (15%).If the sodium conductance is not taken into account,the passive parameters are well estimated but not thepotassium parameters.

Finally, a summary of all the tests is presented inTable 4.

4. Sensitivity Analysis

As demonstrated with the tests, it might be difficultto distinguish between several solutions or simply tobe sure estimates are accurate. We have shown (usingfits in the time domain) the usefulness of doing mul-tiple fits and that scatter plots of the solution couldprovide interesting information on the interactions be-tween the different parameters. Here, we present a sen-sitivity analysis for the frequency domain fits to gaininformation on the quality of the parameter estimates.

Because the admittance is given by an analytical for-mula, it is easy to calculate a sensitivity function foreach parameter. The sensitivity of the admittanceY,with respect to parametera, is defined by

Sa =∣∣∣∣Relative change ofY

Relative change ofa

∣∣∣∣ = ∣∣∣∣ aY ∂Y

∂a

∣∣∣∣ . (7)

It is a function of frequency. Note that it is the sameas the sensitivity of the impedance. If the admittance ismore sensitive to parametera than to parameterb, thenwe might expect thata will be better estimated thanb.

4.1. Passive Dendritic Neuron

Fig. 6 shows the admittance sensitivities of the passivedendritic neuron model, whose parameter values arethose of Fig. 1B. The first thing to note is the variationof the sensitivities with frequencies. The shapes forgsh

andC are easily understandable:gsh is the main pa-rameter controlling the DC input conductance, so thesensitivity to this parameter is maximal at DC and thendrops with frequency because the capacitative termof the admittance takes over, which in turn explainswhy the sensitivity toC increases with frequency. Theshapes of the sensitivities to dendritic parameters areless intuitive. Forar and speciallyL, there is an initialrise with frequency followed by a plateau. Note thatif the range of frequencies used was 0 to 30 Hz in-stead of 0 to 100 Hz, the estimation ofL could havebeen much less accurate. So if instead of white noise apulse is applied, it is possible that the frequency con-tent would be too limited since the power spectrum ofsuch signal decreases like a sinus cardinal function ofthe frequency (squared). The shape of the sensitivityto gl is a little more complex. It first decreases, as theconductance role in the admittance decreases in favorof the capacitance, and then increases back like thetwo other cable parameters, as propagation within thestructure becomes important.

The second thing to note is that the sensitivity islarger forgsh andC over most of the frequency rangewhenar is small, suggesting that the dendritic param-eters can be less well estimated, as observed. How-ever, for largerar , the dendritic admittance becomesmore important; therefore, the sensitivities togl , ar ,and L increase while they decrease forC and gsh.This study is limited toar = 5 andar = 0.5, but wealso show in Fig. 6 that the sensitivity toar further

228 Tabak et al.

Figure 6. Admittance sensitivity as a function of frequency for each parameter of the passive dendritic neuron and for both values ofar . Inaddition, the sensitivity toar was shown forar = 50 to show the effect of an increase in dendritic surface relative to somatic surface.

increases forar = 50 (it also increases forL andgl , notshown).

4.2. Soma with Potassium Conductance

For simplicity, we limited this sensitivity study to oneactive conductance on a soma. Because of the volt-age dependency of the potassium conductance, thesensitivities are dependent not only on the frequencybut also on the voltage level. The variation of the

sensitivity with respect to each parameter is thereforerepresented as a surface or multiple profiles in Fig. 7.Note that for purpose of illustration we exaggerated thevariations of sensitivity with voltage by changing thevalues ofgl (0.001µS) andgK (0.2µS).

The variations of the sensitivity to passive parame-ters with frequency is the same as in the previous case(except that heregl is the analog ofgsh). In addition,it should be noted that the sensitivity togl is signifi-cant only at hyperpolarized levels, for which the K+


conductance is shut down. Note also that the higherthe level of depolarization, the higher the maximal fre-quency of the signal necessary to reach a high sensitiv-ity with respect toC.

The sensitivities togK , VK , vn, andsn show the samevariations with frequencies as the sensitivity togl : lowpass for−74 and−64 mV, band pass for−54 mV,and almost large band for−44 mV. But of course their

Figure 7. Admittance sensitivity as a function of frequency and voltage for each parameter of the model soma with K+ conductance. Theparameters values are given in captions of Figs. 1 and 3, except forgl (0.001µS) andgK (0.2µS). Due to this change, the sensitivities togl andC are not high compared to the sensitivities to active parameters, and the variations with voltage are sharpened.

(Continued on next page.)

variations with voltage are opposite to the variation ofthe sensitivity togl . They are negligible before−64 mV,except at low frequencies as just noted. The sensitivitiesto tn andrn are always null at 0 Hz. This is becausethese two parameters only control the conductance timeconstant and therefore play no role at DC.

The variations with potential of the sensitivities toall potassium parameters are similar, except the one for

230 Tabak et al.

Figure 7. (Continued).

gK . This parameter will have its maximal impact whenall potassium channels are open; therefore, the sensi-tivity to gK increases with voltage andplateausas soonasn∞(V) equals 1. On the other hand, if the sensitivi-ties to the other K+ parameters begin to increase withvoltage, theydecreasewhen activation becomes largebecause the kinetics do not play any role once all chan-nels are opened. ForVK , the reason for this decreaseafter a given level is that a change in reversal potential

has a small effect when the voltage is far from it. Eventhough the variations of the sensitivities with voltagewere artificially sharpened, we want to emphasize thatit might be possible to find a rather sharp maximum.An example is given by the sensitivity torn, showingthat if a trace obtained around−54 mV was not used, itmight be difficult to accurately estimate this parameter.

Finally, the vertical scales show that the best esti-mated parameters should bevn, VK , andsn. Although


we have no mathematical proof that this result appliesto the time-domain data, we observed that the parame-ters best estimated using voltage transients werevn andVK .

5. Discussion

The primary goal of this work was to investigate towhat extent one can estimate the passive and activeproperties of single neurons when multiple active con-ductances are present. The parameters used in this studywere chosen so that the neuron properties mimic thoseof Xenopusspinal neurons. Considering these partic-ular values, the tests showed that for a dendritic neu-ron with potassium and sodium conductances, accept-able estimates could be obtained, provided the signalto noise ratio was high enough and assuming that pa-rameters like reversal potentials were known (which istrue for whole cell recordings). Because of the possiblyhigh number of parameters to estimate and the noise in-herent to physiological data, the principal problem en-countered is the nonuniqueness of the solutions. This isan issue when estimating passive parameters with time-domain data (see below) and was aggravated in our casebecause a long, not short, hyperpolarizing pulse wasused. This problem had previously been reported fordendritic neurons with somatic shunt (Stratford et al.,1989; White et al., 1992); estimation errors on parame-ters can compensate each other, and the resulting modelresponse cannot be distinguished from the data. It canbe reduced by increasing the signal to noise ratio (Majoret al., 1990, 1994) or by including some morpholog-ical constraints (White et al., 1992). In the case ofa passive dendritic neuron, we showed that knowingthe value ofar allowed unique estimation of the otherparameters.

The second major problem was to estimate the pa-rameters of the sodium current because errors on activa-tion and inactivation can compensate. The interactionbetween potassium and sodium current does not seemto lead to large estimation errors. Therefore, we be-lieve that to better estimate the sodium parameters, onewould need to obtain more voltage (or current) levelsor to focus more on the very transient part of the time-domain response, without the need to use a pharmaco-logical blocker to separate the two currents. However, ifmultiple inward currents or multiple outward currentsare present on the neuron, the errors resulting from in-teractions between similar currents will preclude find-ing a unique solution. Therefore, one would have to first

gather data without any blocker, then with one specificblocker, then with a second blocker if possible, andso on, as long as all recordings can be made on thesame cell. The fits would then be done with the dataobtained when most currents are blocked. The follow-ing fits would incorporate the additional currents untilthe final fit, which would include all parameters.

A better but costly method would be to first ana-lyze each voltage-dependent conductance with voltage-clamp protocols to have (preliminary) estimates of thekinetic parameters of each current. Then fitting datafrom individual cells would allow to determine the rela-tive quantity of these currents on each individual. In thissecond step, the kinetic parameters would not have tobe estimated or would be estimated within a restrictedrange, decreasing the complexity of the task.

5.1. Comparison of the Methods

With some limitations, a comparison can be done be-tween the results of the time- and frequency-domainmethods. The first limitation is that the optimizationschemes were different: a simplex algorithm was usedfor the time domain while a gradient descent was usedin the frequency domain. However, a previous studyshowed little difference between simplex- and gradient-based schemes when fitting noisy time-domain data(Murphey et al., 1996). The major limitation is thatthe noise used to contaminate the generated data is notidentical for each method. Therefore, the present studyis limited to a preliminary account of the effects ofnoise on the quality of the estimations.

The first notable difference between the two meth-ods is the speed. The admittance fits are much fasterbecause there are no differential equations to be solved.Also, for the first time, we used an analytical model tocompute the admittances instead of a compartmentalmodel that requires recurrent iterations of the sameformula (Murphey et al., 1995). This speed differ-ence was quite important: for the dendritic neuronwith a potassium conductance, using a 10-compartmentmodel, it took more than an hour to fit the voltage tran-sients, while the admittance fit required just a coupleof minutes (but note that the gradient-method descentused to fit the admittance data generally requires lessiterations than the simplex algorithm, which partly ac-counts for the speed difference). These calculationswere done on Sun Sparcstations IPC and LX. Theanalytical formulation of the admittance also allowseasy derivation of analytical formulas for sensitivity

232 Tabak et al.

functions. Another advantage is that calculation ofVl

is not necessary, since onlygl appears in the admittanceformula, while the computation of voltage transients re-quires evaluatinggl (V − Vl ). Also, we have assumedthat the reversal potential associated withgshwas equaltoVl , which is probably an approximation (Staley et al.,1992; Surkis et al., 1998). This is not an issue whencomputing the admittance.

As noted above, the second point is that the white-noise approach seems to better estimate the passivestructure. Although the simplex algorithm might bepartly responsible for not converging reliably towardthe best minimum, the time-domain method is less sen-sitive to errors in the parameter estimates. This waseven worse for smaller value ofar , and significant esti-mation errors for the cable parameters were also foundwhen potassium channels were present. Conversely,when multiple noisy traces were used (increasing thesignal to noise ratio), the frequency-domain methodwas able to find good estimates of the cable parame-ters. Only when estimating an additional parameterRe

did nonunicity problems become an issue. Why is thefrequency-domain superior? The voltage traces can bedecomposed as a sum of exponential components withdiffering amplitudes and time constants; therefore, thesensitivity of the time-domain method to parameterchanges depends on the ability to distinguish signalswith small differences in those components, which isdifficult. Another advantage of the white-noise methodis that the measurements are made at steady state, sothat passive properties cannot be contaminated by tran-sient active responses. Superiority of white-noise stim-ulation over current pulses was shown previously, witha different methodology, by other authors (Wright etal., 1996).

Still, there are some drawbacks of the admittancemethod. First, the analytical model has assumed uni-form membrane potential in the calculation of thevoltage-dependent conductances present on the den-dritic cable. This approximation can create little error,but when the voltage profile has to be taken into ac-count, one has to resort to compartment models andcalculate the steady-state potential in each compart-ment. This will limit the speed advantage over fittingvoltage transients and require knowledge of the rever-sal potential associated with the leak conductance.

The main disadvantage of the frequency-domainmethod is the difficulty to obtain measurements forlarge depolarizations because activation of sodium orcalcium channels will tend to destabilize membrane

potential. Therefore, analysis of voltage transients isnecessary when dealing with strong outward currents.

The frequency-domain method is less often usedcompared with the traditional methods of estimatingneuronal model parameters, even when direct optimiza-tion methods are used. We do not suggest that conven-tional methods should be replaced by the frequency-domain method but rather that both should be usedwhenever possible. If only passive parameters are tobe estimated, then we think that admittance mea-surements are preferable. Other admittance measure-ment methods have been described that are probablyeven more accurate: Ali-Hassan et al. (1992) avoidedfourier-transforming noisy data and instead inverse-transformed the model transfer function; Wright et al.(1996) computed the first-order Wiener kernel beforeswitching to the frequency domain and favorably com-pared this method to short-pulse measurement.

5.2. Possible Strategy

It follows from the tests and previous discussion that afitting strategy for a dendritic neuronal model includingboth potassium and sodium conductances could be asfollow:

1. Estimate passive parameters using the most hyper-polarized admittance trace(s),

2. Use these values as initial guesses for a fit of bothpassive and potassium parameters, on a limited setof admittance traces (moderate depolarization),

3. Use the previous estimates as initial guess and fitpotassium and sodium parameters on all traces ob-tained with current clamp protocol, and

4. Check that the parameters found also fit the ad-mittance traces (otherwise go back to step 2 or fitsimultaneously both sets of data).

In the case of multiple inward or outward currents, thisstrategy has to be modified because of the likely use ofblockers, but the overall strategy of fitting first passive,then outward, and then inward currents still applies.

Note that the estimates of passive parameters ob-tained in step 1 are not fixed in step 2, since we haveshown that they could be misleading. This is the samefor the potassium parameters obtained in step 2; theyserve only as initial guesses for step 3 to accelerate con-vergence. However, at this point the passive parametersmight be kept fixed to simplify step 3.


A point that was not illustrated here but results fromthe use of the two data sets (time and frequency domain)is the ability to detect that a critical conductance was leftout of the model. For example, a potassium A current,which inactivates at depolarized levels, might not havea dramatic effect on the admittance that is obtainedat steady-state but would have a definite influence onvoltage transients. A slow calcium-activated potassiumcurrent, on the other hand, could have no effect on thefirst 100 ms of a voltage transient but a huge effect onthe admittance. Therefore, if one of these currents ispresent on the neurons but left out of the model, therewill probably be a discrepancy between the results ofeach method (Tabak et al., 1996).

5.3. Choice of the Optimization Method

We have used a simplex and a gradient descent methodin this study. It is well known that while descent meth-ods can converge quickly, they are prone to local min-ima. The simplex method also suffers from local min-ima problems, only to a lesser extent. There are morepowerful (and computationally expensive) minimiza-tion methods that could be used, like sophisticatedrandom search techniques (Foster et al., 1993) or ge-netic algorithms (Weyhing and Borst, 1997). One mightwonder if the use of such a technique would have im-proved our results.

Most probably, some of these optimization methodswill always provide perfect fits of the (noise-free) volt-age responses obtained from dendritic neuron models,where the simplex might fail. However, the failures tofind the perfect fit are due, at least in part, to the ill-posedness nature of the problem: even a very smallamount of noise can produce large errors on some pa-rameter estimates. Furthermore, the presence of noisein the data generally precludes the existence of a sharpminimum, so the search for the best minimum is point-less. Therefore, instead of spending a lot of computa-tion time trying to find the best minimum, one shouldrather find a lot of these solutions to gain knowledgeof the interactions that exist between the different pa-rameters and determine their confidence intervals (seebelow).

As pointed out by Foster et al. (1993), depending onthe shape of the error surface, low-cost optimizationtechniques (like gradient-descent methods) might beunable to converge toward the right region in parameterspace. Only then should utilization of high-cost tech-niques (like simulated annealing) be recommended.

From our experience with various parameter estima-tion problems, we feel that the simplex is often a goodcompromise between speed and accuracy. In problemswith only three parameters, one can even successfullyuse a grid search method (Borst and Haag, 1996).

In addition, the simple grid search allows the plot-ting of error surfaces as a function of two parameters.Combined with a criterion for acceptable match be-tween model and data, this in turn defines confidenceregions—that is, the ranges of parameter values givingan acceptable fit. The criterion defined by Borst andHagg (1996) is a least-square error, but it is normalizedby the standard error from the measurements. There-fore, for any value lower than 1 on the error surface themodel is within the measurement noise.

The values lower than 1 thus define a confidencearea, which can be shown as a contour plot (Borst andHaag, 1996). Not only does this indicate how well eachparameter is estimated, but this plot could also be usedto determine the relationships between each parameter.The only drawback of this method is that these two-dimensional areas are dependent on the values chosenfor the other parameters. The usefulness of this methodis therefore limited to a small number of parameters.When at least five parameters are estimated, we favorfinding a large number of solutions using, for instance,the simplex method and then plotting two-dimensionalscatter plots (Foster et al., 1993) or global sensitivityplots, as discussed below.

5.4. Necessity of Sensitivity Analysis or MultipleRepeats of the Estimation Procedure

Because of the large number of parameters or the pres-ence of noise in the data, there are possibly large esti-mation errors and nonuniqueness of the solutions. Howdoes one assess the quality of the estimation?

The sensitivity analysis that we have shown in thecase of the frequency domain gives information aboutthe importance of the contribution of each parameterto the admittance, for a given solution. In the two ex-amples we have shown, a poor sensitivity measure fora parameter indicated a poor estimation. Further, thisnot only allows assessment of how likely it is that eachparameter is accurately estimated but also provides in-formation about what voltage level or what frequencyshould be used to better estimate those parameters forwhich the sensitivity was low. If the sensitivity curvesand surfaces are not too dependent on the parametersvalues, it will be possible to determine the best set of

234 Tabak et al.

frequencies and voltage levels to use from preliminaryexperiments.

The other way to assess the quality of the estimation,which we want to emphasize, is to repeat the estimationprocedure a number of times. This was used in thepresent work mostly for the time domain method. Itsimportance is demonstrated by the following facts.

• A great deal of variability between solutions mightindicate a problem in the model.• A single fit often results in estimation errors. With

multiple fits, one might choose the best fit accordingto therms if the signal to noise ratio is high enough,but in most cases the level of noise will prevent asharp minimum to be found, and the minimum of thermsdoes not indicate the best estimation. However,our results tend to indicate that taking the average ormedian of all estimations provides better estimatesthan the minimum. Because most estimations weredone on ranges comprising 0 to 2 times the targetvalues (for the time domain data), it is possible thatthis is just an artifact. We do not think so becauseof the nonuniform density of the solutions withinthe range (not shown), but a definitive proof willrequire redoing some of these tests with a range ofparameters values not centered on the target values.• From a large number of solutions one can also draw

“global sensitivity plots”—that is, thermserror foreach value of a given parameter taken from eachsolution. Provided that a criterion for acceptable fitis defined (that is, a cutoff value of therms), this plotgives a “confidence range” for each parameter.• Within these ranges, two-dimensional scatter plots

can be plotted to visualize the relationships thatmust exist between the various parameters (Fosteret al., 1993). Knowing these relationships might beimportant to choose the parameters of model neu-rons in a network, if one wants to model a het-erogenous population. Note also that from these“global” relationships we can deduce the relation-ships that exist between two given parameters if onlythese two parameters were varying (assuming linearrelationships).

These remarks are also valid for the estimationof synaptic parameters used in network simulations,which belongs to the same kind of problems as treatedin this article (Zipser, 1992; Tabak and Moore, 1998).Indeed, our approach in constructing realistic neuralnetwork models is to obtain measurements from thecells during network activity and during stimulation

protocols. Thus, the same cells can be used for estimat-ing intrinsic and network properties. Although biasedby the particular cases and parameter values chosenhere, we hope the observations we made will help otherconstrain their neuronal model parameters.

Appendix: Frequency-Domain Fits for DendriticNeuron with Different Distributionsof Potassium Parameters

Here we present the results of the admittance fits for adendritic neuron with potassium conductance for dif-ferent values offraK, the ratio of the density of den-dritic potassium conductance over somatic potassiumconductance.

fraK = 0 (ar= 0.5)

The estimates are perfect. If we try to fit with a modelhaving a uniform distribution of K+ channels, the mainresult, as for the time domain, is to underestimategK

(by a factor of 3). All the passive parameters are stillwell estimated (estimation error lower than 3%).

If VK is also estimated, the perfect solution is stillfound. In addition, ifRe is also to be estimated, theright solution is rarely found. Active parameters arewell estimated, but the values of the passive dendriticparameters (gl , L andar ) can be seriously wrong andchange a lot from one fit to another.

An alternative strategy is to first estimate the pas-sive parameters using the most hyperpolarized trace(−84 mV). The estimation is perfect, even withRe

(with more difficulties), but withar restricted to theinterval [0, 1]. If we take the trace obtained at−74 mV,for which IK is still tiny, the estimation is still good (at2%) and is slightly altered ifRe is included (<4%). Thepotassium current becomes significant at−64 mV, sothe estimation error increases if this trace is selected tofit the passive parameters (but no more than 20%). Inthat case, addingRe leads to larger errors (up to 45%),the worse estimated parameters being, as before,gl ,ar ,andL (generally in that order). Once estimates of thepassive parameters are obtained, they are used as theinitial guess for a fit on all parameters from all admit-tance traces. Whether we choose the passive estimatesobtained from−84,−74, or−64 mV (includingRe),the perfect solution is always obtained. Local minimawere therefore avoided with this two-round strategy.

Using this two-round strategy we have also tried toestimatefraK—that is, a thirteenth parameter. The right


solution can be achieved, but there are other fits thatare as good visually for which the estimates of passivedendritic parameters are clearly wrong. The only hintthat these fits are wrong is that some of the parametersattain one limit of the allowed range. If the range isincreased and the parameters still reach the limit, thenthe quality of the fit can be suspected. This remark isnot limited to this case.

fraK = 1 (ar= 0.5)

Perfect fits are found even withVK , fraK, andRe. Hereagain, local minima are characterized by one of theparameters reaching a limit. In one case,fraK reachedits limit, 1, which is also the test value, but the fit waswrong. Increasing the upper limit forfraK was neededto obtain the right solution.

If the fit is done with a model that does not incor-porate K+ channels on the dendrite (Re not estimated),the effect on the only trial conducted is not to overesti-mategK (which was even slightly underestimated) butto overestimate|VK |, which also increasesIK . WhenVK is not estimated, the expected overestimation ofgK

is found.

fraK = 0.5 (ar= 5)

For this case, too, all 13 parameters can be perfectlyestimated.

Acknowledgments

We would like to thank J.-M. Bellanger and J.-L.Coatrieux for stimulating discussions. Comments fromR.E. Burke and the two anonymous reviewers allowedus to improve the manuscript. We also would like to ac-knowledge Y. Pichon for his support during this work.

References

Ali-Hassan WA, Saidel GM, Durand D (1992) Estimation of elec-trotonic parameters of neurons using an inverse fourier trans-form technique.IEEE Transactions on Biomedical Engineering39:493–501.

Bhalla US, Bower JM (1993) Exploring parameter space in detailedsingle neuron models: Simulations of the mitral and granule cellof the olfactory bulb.Journal of Neurophysiology69:1948–1965.

Borst A, Haag J (1996) The intrinsic electrophysiological charac-teristics of fly lobula plate tangential cells: I. Passive membraneproperties.Journal of Computational Neuroscience3:313–336.

Burke RE, ten Bruggencate G (1971) Electrotonic characteristics ofalpha motoneurones of varying size.Journal of Physiology212:1–20.

Byrne GD, Hindmarsh AC (1975) A polyalgorithm for the numericalsolution of ordinary differential equations.ACM Transactions onMathematical Software1:71–96.

Dennis JE, Gay DM, Welsch RE (1981) An adaptive nonlinear least-squares algorithm.ACM Transactions on Mathematical Software7:348–368.

Durand D (1984) The somatic shunt cable model for neurons.Bio-physical Journal46:645–653.

Foster WR, Ungar LH, Schwaber JS (1993) Significance of conduc-tances in Hodgkin-Huxley models.Journal of Neurophysiology70:2502–2518.

Hodgkin AL, Huxley AF (1952) A quantitative description of mem-brane currents and its application to conduction and excitation innerve.Journal of Physiology117:500–544.

Kawato M (1984) Cable properties of a neuron model with non-uniform resistivity.Journal of Theoretical Biology111:149–169.

Major G, Larkman AU, Jack JJB (1990) Constraining non-uniqueness in passive electrical models of cortical pyramidal neu-rones.Journal of Physiology430:13P.

Major G, Larkman AU, Jonas P, Sakmann B, Jack JJB (1994)Detailed passive models of whole-cell recorded CA3 pyrami-dal neurons in rat hippocampal slices.Journal of Neuroscience14:4613–4638.

Mauro A, Conti F, Dodge F, Schor R (1970) Subthreshold behaviorand phenomenological impedance of the squid giant axon.Journalof General Physiology55:497–523.

Moore LE, Christensen BN (1985) White noise analysis of cableproperties of neuroblastoma cells and lamprey central neurons.Journal of Neurophysiology53:636–651.

Moore LE, Hill RH, Grillner S (1993) Voltage-clamp frequency do-main analysis of NMDA-activated neurons.Journal of Experimen-tal Biology175:59–87.

Murphey CR, Moore LE, Buchanan JT (1995) Quantitative analy-sis of electrotonic structure and membrane properties of NMDA-activated lamprey spinal neurones.Neural Computation7:486–506.

Murphey CR, Tabak J, Moore LE, Buchanan J (1996) Estimation ofmembrane properties from step current measurement of Xenopusneurons. In: Bower J, ed.Computational Neuroscience. AcademicPress, New York, pp. 107–112.

Nelder JA, Mead J (1965) A simplex algorithm for function mini-mization.Computer Journal7:308–313.

Press WH, Flannery BP, Teukolsky SA, Vetterling WT (1988)Numerical Recipes in C: The Art of Scientific Computing. Cam-bridge University Press, Cambridge.

Rall W (1969) Time constants and electronic length of membranecylinders and neurons.Biophysical Journal9:1483–1508.

Staley KJ, Otis TS, Mody I (1992) Membrane properties of dentategyrus cells: Comparison of sharp microelectrode and whole-cellrecording.Journal of Neurophysiology67:1346–1358.

Stratford K, Mason A, Larkman A, Major G, Jack J (1989) Themodelling of pyramidal neurones in the visual cortex. In: DurbinR, Miall C, Mitchison G, eds.The Computing Neuron. Addison-Wesley, Reading, MA. pp. 296–321.

Surkis A, Peskin CS, Tranchina D, Leonard CS (1998) Recov-ery of cable properties through active and passive modeling ofsubthreshold membrane responses from laterodorsal tegmental

236 Tabak et al.

neurons.Journal of Neurophysiology80:2593–2607.Tabak J, Moore LE (1998) Simulation and parameter estimation

study of a simple neuronal model for rhythm generation: Roleof NMDA and non-NMDA receptors.Journal of ComputationalNeuroscience5:209–235.

Tabak J, Murphey CR, Moore LE (1996) Estimation methods inlocomotion network activity.Society for Neuroscience Abstract22:1372.

Weyhing H, Borst A (1997) Parameter search in compartmental mod-els using genetic algorithms.Society for Neuroscience Abstract

23:655.White JA, Manis PB, Young ED (1992) The parameter estima-

tion problem for the somatic shunt model.Biological Cybernetics66:307–318.

Wright WN, Bardakjian BL, Valiante TA, Perez-Velazquez JL,Carlen P (1996) White noise approach for estimating the pas-sive electrical properties of neurons.Journal of Neurophysiology76:3442–3450.

Zipser D (1992) Identification models of the nervous system.Neuroscience47:853–862.

parameter estimation methods for single neuron models

Documents