Identifying the components of a postsynaptic potential and theiramplitude, latency and shape fluctuations: analysis by means ofautocovariance functions and a stochastic infinite cable model

Octavio Ruiz *, Pablo Rudomın

Departamento de Fisiologıa, Biofısica y Neurociencias, Centro de Investigacion y de Estudios Avanzados del IPN, Av. IPN 2508, Mexico DF 07360,

Mexico

Received 5 April 2002; received in revised form 2 December 2002; accepted 2 December 2002

Abstract

In addition to amplitude fluctuations, physiological mechanisms may introduce latency and shape fluctuations in the components

of a postsynaptic potential (PSP). Latency fluctuations may be originated mainly by presynaptic factors. Shape fluctuations may be

produced by changes in the background synaptic activity received by the postsynaptic neuron, which affect the cell membrane

resistance. This article aims to develop a unified approach for the analysis of amplitude, latency and shape fluctuations in the

components of a PSP. The analysis is based on: (i) the Autocovariance Functions of the PSP (ACOVs); (ii) a mathematical model

able to predict the average and ACOVs of a PSP with specified components and fluctuations (the ‘Stochastic Infinite Cable Model’

(SICM)); and (iii) a procedure to estimate the SICM parameters that best reproduce the average and ACOVs of a given PSP (the

‘SICM-based PSP identification procedure’ (SICM-IP)). The SICM-IP is tested with simulated PSPs. The results obtained support

the feasibility of the approach.

# 2003 Elsevier Science B.V. All rights reserved.

Keywords: Postsynaptic potentials; Fluctuations; Components; Cable model; Autocovariance functions; Taylor series; Latency fluctuations;

Membrane resistance fluctuations

1. Introduction

Amplitude is the most obvious fluctuating attribute of

a synaptic response (Del Castillo and Katz, 1954;

Redman, 1990; Bekkers, 1994; Auger and Marty,

2000). However, there are evidences and considerations

suggesting that evoked postynaptic potentials (PSPs)

may also exhibit latency and/or shape fluctuations. For

example, in cat spinal motoneurons, Munson and Sypert

(1979) observed latency fluctuations in the order of 100

ms in the PSPs evoked by single muscle spindle afferents,

which have been attributed to the mechanism respon-

sible for the presynaptic inhibition. Similar observations

were made by Collatos et al. (1979), Cope and Mendell

(1982a,b) who reported latency standard deviations

ranging from 39 to 101 ms (Cope and Mendell, 1982b;

but see Jack et al., 1981). Experimental and simulation

studies show that presynaptic inhibitory contacts may

accelerate or delay the propagation of the action

potential without blocking conduction (Sypert et al.,

1980; Segev, 1990; Fig. 3 of Graham and Redman, 1994;

Fig. 6 of Walmsley et al., 1995; Cattaert et al., 2001).

Latency fluctuations of the synaptic response in other

central synapses have been reported by Lu and Trussell

(2000) and Schneider (2001).

Abbreviations: ACOV, autocovariance function, in particular,

autocovariance function of a PSP; Amp-Fluct Two-Cpt PSP,

amplitude-fluctuating two-component PSP; Amp/Lat-Fluct Two-Cpt

PSP, amplitude- and latency-fluctuating two-component PSP; Comb-

Fluct Two-Cpt PSP, combined (i.e. amplitude, latency and PMC)-

fluctuating two-component PSP; C-PSP, PSP simulated in a

compartmental model in response to alpha current pulses; non-

SICM PSP, PSP produced by a structure and current pulses different

from that considered by the SICM; PMC, postsynaptic membrane

conductivity or conductance, reciprocal of the postsynaptic membrane

resistivity or resistance; PSP, postsynaptic potential, either excitatory

(EPSP) or inhibitory (IPSP); SICM, stochastic infinite cable model;

SICM-IP, SICM parameter identification procedure.

* Corresponding author.

E-mail address: oruiz@fisio.cinvestav.mx (O. Ruiz).

Journal of Neuroscience Methods 124 (2003) 1�/26

www.elsevier.com/locate/jneumeth

0165-0270/03/$ - see front matter # 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0165-0270(02)00368-0

Fluctuations in PSP shape, particularly in the decay

phase, can arise from changes in the postsynaptic

membrane conductivity (PMC), as shown in experi-

ments (Peng and Frank, 1989) and simulations (Holmes

and Rall, 1992). The membrane resistivity of a cell

depends on the conductance and number of ionic

channels open at a given moment. Synaptic actions

occur through the opening and/or closing of transmitter-

dependent ionic channels. Almost every neuron in the

central nervous system receives thousands of excitatory

and inhibitory synapses. It is then reasonable to assume

that the background synaptic activity impinging on a

neuron contributes, at least in part, to determine its

membrane resistance. Changes in PMC due to changes

in background synaptic activity have been actually

observed. The input resistance (Rin, inversely related to

PMC) of pyramidal cells in the cat’s cerebral cortex was

reduced up to 70% during periods of intense sponta-

neous synaptic activity. Also control Rin was increased

by approximately 30�/70% after applications of TTX in

vivo, approaching in vitro values (Pare et al., 1998, see

also Destexhe et al., 2001; Perreault, 2002). All these

considerations suggest an influence of spontaneous

activity in the PSP shape, via PMC fluctuations.

Finally, a single presynaptic axon usually makes

synaptic contacts on the dendrites of the postsynaptic

cell at different distances from the soma, thus producing

responses with several electrotonic components (Burke,

1967; Rall, 1967; Rall et al., 1967). To our knowledge,

there is no systematic approach to characterize the

latency and PMC fluctuations in the components of a

PSP.

This article introduces a novel approach to disclose (i)

the number of components in a PSP; (ii) the average

shape of each component; (iii) the electrotonic site where

each component could have originated; (iv) the ampli-

tude and latency fluctuations of each component; (v) the

shape fluctuations produced in all components by PMC

fluctuations; and (vi) the correlation between amplitude,

latency and PMC fluctuations. The approach is based

on three elements (Fig. 1). The first is the use of

Autocovariance Functions (ACOVs) to analyze the

synaptic response. ACOVs provide information not

only about the variability of a PSP, but also on the

correlation of voltage variations at different times of the

PSP. Second: a mathematical model able to predict the

average and ACOVs of a PSP of specified composition

and fluctuations (the ‘Stochastic Infinite Cable Model’

(SICM)). Third: a method to find the parameters of the

model which best reproduce the average and ACOVs ofa given PSP (the ‘SICM-based PSP identification

procedure’ (SICM-IP)). The paper provides a prelimin-

ary characterization of the SICM-IP, shows the feasi-

bility of the approach and some of its advantages and

limitations.

2. Analysis

2.1. A function to represent the components of a PSP

We sought a mathematical model capable of reprodu-

cing the electrotonic components of a PSP, with simple

means to represent the amplitude, latency and PMC

fluctuations of the components. Comprehensive and

detailed descriptions of the geometry and electrical

characteristics of two interconnected neurons were

discarded because of their complexity and the largenumber of parameters that should be estimated from

experimental data. As a compromise, we adopted the

representation of each component of a PSP by the

voltage response of a homogeneous infinite cable to a

Dirac delta current pulse injected at a distance xk from

the recording site (Fig. 2, and Jack and Redman, 1971a;

Jack et al., 1975):

where V (t ; uk ) is the voltage in the infinite cable, at time

t , recorded at the origin (x�/0), in response to a Dirac

delta current pulse injected in the k th ‘synaptic site’

(mV); uk , the parameters defining the response to the

k th ‘synaptic site’, uk �/(r ,c ,g ,xk ,ak ,bk ); r , the cable’s

axial resistance per unit length (MV/mm); c , the cable’s

capacitance per unit length (nF/mm); g , the cable’s leak

conductance per unit length (mS/mm); xk , the distance,

along the cable, between the recording site and the k th

‘synaptic site’ (mm); ak , the charge injected by the

current pulse at the k th ‘synaptic site’ (pC). With other

parameters constant, this parameter determines the

amplitude of the k th component. bk is the latency of

the Dirac delta current-pulse injected by the k th current

source, representing the latency of the k th component

relative to the presynaptic action potential (ms). t is the

time relative to the presynaptic action potential (ms). t is

V (t; uk)�

0; t�bk50

ak

2ffiffiffip

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir

c(t � bk)

se�(rcx2

k)=4(t�bk)�(g=c)(t�bk); t�bk�0

; t � t

8><>: (1)

O. Ruiz, P. Rudomın / Journal of Neuroscience Methods 124 (2003) 1�/262

the set of times during which the PSP occurs,

[t1,t2,. . .,tl].

A multiple-component PSP is then represented by:

V (t; u)�Xs

k�1

V (t; uk); t � t (2)

where u is the set of parameters defining the multiple

component response, u�/(r , c , g , x1, a1, b1, x2, a2,

b2,. . ., xs , as , bs), and s is the number of components.

The shape of each component is determined by the

cable parameters r , c , g and the location of the ‘synaptic

sites’ x1, x2,. . ., xs . If one considers {xk} to be fixed

throughout an experiment, the shape of each component

will be affected by changes in any of the parameters r , c

or g . There are no data or considerations suggesting that

the specific resistance of the cytoplasm or the specific

Fig. 1. Diagram of the SICM based PSP identification procedure (SICM-IP). The SICM-IP aims to characterize a PSP as a sum of electrotonic

components, and to estimate the amplitude, latency and shape fluctuations of each component. The SICM-IP consists of two parts. The SICM

parameter fitting algorithm (central white area) improves an initial solution until the SICM outcome maximally resembles the average and ACOVs of

a PSP. The remainder of the SICM-IP (gray area) compares the results of different configurations and initial solutions, searching for the best global

fitting. Dashed blocks represent the simulations performed in this work to test the performance of the SICM-IP. Further explanation in text.

O. Ruiz, P. Rudomın / Journal of Neuroscience Methods 124 (2003) 1�/26 3

capacitance of the cell’s membrane may change between

successive PSPs. On the contrary, g , the parameter

representing the membrane conductivity of the post-

synaptic cell, would change during variations of the

background synaptic activity received by the postsynap-

tic neuron. Consequently g will be the parameter

dedicated to account for the shape fluctuations of thePSP components. This model will be referred to as the

SICM.

2.2. Average and ACOVs of the SICM

We calculated next the average and ACOVs of the

SICM, as a function of the means, variances andcovariances of the stochastic parameters. The calcula-

tion cannot be carried out directly because parameters

{bk} and g occur nonlinearly in Eq. (1). Yet, the average

and ACOVs of a nonlinear function of several stochastic

parameters can be approximated as follows (see Papou-

lis, 1991): (i) replace the function (in our case, Eq. (1)) by

its Taylor-series expansion around the mean values of

the stochastic parameters; (ii) truncate the series toinclude first- and second-order terms only; (iii) intro-

duce this approximation into the formulas defining the

mean and ACOVs of the function, in our case:

V (t; u)�EV (t; u)

COVV (h; t; u)�E(V (h; u)�V (h; u))(V (t; u)�V (t; u));(3)

(iv) the expectations of the approximations yield poly-

nomials whose terms contain either the first- or second-

order derivatives of V with respect to the stochasticparameters, and the variances or covariances of the

stochastic parameters. The polynomials can be conve-

niently expressed using Linear-Algebra operations.

Hence, the Taylor series approximation to the SICM

average becomes:

V (t; u):V (t; u;J)�V (t; u)�1

2tr(JWt;u); t � t (4)

while the SICM autocovariances are approximated by:

COVV (h; t; u):COVV (h; t; u;J)�VT

h;uJVt;u;

h � h; t � t(5)

where V (t; u) is the average of the cable-model responseat time t (mV); V (t; u;J) is the Taylor-series approx-

imation to the average of the cable-model response at

time t (mV); u/�/(r ,c ,/g;/x1,/a1;//b1;/x2,/a2;//b2;/. . .,xs ,/as;//bs) is the

vector of parameters, containing the mean values of the

stochastic parameters; J is the covariance matrix of the

stochastic parameters (Eq. (6)); V (t ; u)/ is the cable-

model response, evaluated for the mean values of the

stochastic parameters (mV); tr() specifies the trace of amatrix, i.e. the sum of its main-diagonal entries; Wt;u is

defined in Eq. (7); COVV (h ,t ; u ) is the autocovariance

of the cable-model response (value at time t) relative to

the reference time h (mV2); COVV (h ,t ;/u;/J) is the SICM

autocovariance, i.e. the Taylor-series approximation to

the ACOV of the cable-model response relative to the

reference time h (mV2); ()T represents the transpose of a

vector or a matrix; Vh;u amd Vt;u are defined in Eqs. (8)and (9); and h is the set of ‘reference’ times, h�/[h1, h2,

. . ., hm ], a subset of t.

The variances and covariances of the stochastic

parameters are included in the covariance matrix of

the stochastic parameters:

J�

ja1a1ja1b1

ja1a2ja1b2

ja1g

jb1a1jb1b1

jb1a2jb1b2

� � � jb1g

ja2a1ja2b1

ja2a2ja2b2

ja2g

jb2a1jb2b1

jb2a2jb2b2

jb2g::: njasas

jasbsjasg

n jbsasjbsbs

jbsg

jga1jgb1

jga2jgb2

� � � jgasjgbs

jgg

266666666664

377777777775(6)

The entries in the main diagonal of J are the variances

of the stochastic parameters, ja1a1�/VAR(a1), . . ., jgg �/

VAR(g ). The off-diagonal entries represent either cov-

ariances between fluctuations of the same component or

between fluctuations of different components. For

example, ja1b2�/jb2a1

�/COV(a1, b2)�/COV(b2, a1) is

the covariance between the amplitude fluctuations of

the first component and the latency fluctuations of the

second component, and so on.

The matrix Wt;u (in Eq. (4)) includes the second-orderderivatives of the cable response with respect to each

stochastic parameter, evaluated for the mean values of

the parameters, in the following arrangement:

Fig. 2. Homogeneous infinite cable model (SICM) used to represent

composite PSPs. (A) i1 to is are current sources representing synaptic

contacts at distances x1 to xs from the ‘soma’, located at x�/0. (B)

Electrical parameters of the homogeneous infinite cable model.

O. Ruiz, P. Rudomın / Journal of Neuroscience Methods 124 (2003) 1�/264

The last entry in Wt;u; as

k�1 Vgg(t; uk); is the sum of

the second-order derivatives of V (t , uk ), valued for each

‘synaptic’ site uk; k�/1, . . ., s . This entry differs from the

others because g , the membrane conductivity of the

postsynaptic structure, affects every component.

Finally, vectors Vh;u and Vt;u; in Eq. (5) contain the

first-order derivatives of the model response, with

respect to the stochastic parameters, arranged as:

Vt;u�

Va1(t; u1)

Vb1(t; u1)

nVas

(t; us)

Vbs(t; us)Xs

k�1

Vg(t; uk)

266666666664

377777777775

(9)

The calculation of the SICM average (Eq. (4)), for

each time t , or the calculation of the SICM ACOVs (Eq.

(5)) for a pair of times h and t , requires the evaluation of

the derivatives of V appearing in Eqs. (7)�/(9). The

analytical expressions for these derivatives may be found

in Appendix A.

Summarizing, if a PSP can be represented by the

voltage response of a homogeneous infinite cable to a

Dirac delta current pulse (Eqs. (1) and (2)), then an

approximation to the average and ACOVs of the PSP is

provided by Eqs. (4) and (5). This pair of equations will

be referred to as the Outcome of the SICM.

2.3. Identification of a PSP in terms of the SICM

2.3.1. SICM-based PSP identification procedure

This section addresses the following problem: given

the average and ACOVs of a PSP, estimate the number

of components in the PSP and the SICM parameterswhich best account for the data. The problem has not a

direct solution because (i) some SICM parameters occur

nonlinearly in the expressions for the SICM outcome;

(ii) the average and ACOVs of the PSP consist of a

number of points larger than the number of SICM

parameters to be estimated and, most importantly; (iii)

the number of components in the PSP is not known a

priori. We have devised the following procedure to

estimate the number of components in a PSP, together

with the SICM parameters. Step 1: Start by considering

a single-component PSP configuration. Step 2: Propose

an initial guess of SICM parameters making the SICM

outcome to resemble the average and ACOVs of the

PSP. Step 3: Optimize the SICM parameters to minimize

the differences between the average of the PSP and the

SICM average by means of the SICM parameter-fitting

algorithm (Section 2.3.2). Step 4: Consider other con-

figurations, with increasing number of components and

different combinations of amplitudes, latencies and

locations. Step 5: Optimize the proposed initial solutions

by repeating step 3. Step 6: Stop trying larger-number-

of-component configurations when ERRAverage (Eq.

(10)) cease to diminish or when the algorithm fails to

converge for every initial solution tested. Step 7: Asses

the goodness-of-fit of all the solutions through the x2-

test described in Appendix D. Step 8: Discard all the

solutions fulfilling the criteria described in Appendix B.

Wt;u�

Va1a1(t; u1) Va1b1

(t; u1) 0 0 Va1g(t; u1)

Vb1a1(t; u1) Vb1b1

(t; u1) 0 0 � � � Vb1g(t; u1)

0 0 Va2a2(t; u2) Va2b2

(t; u2) Va2g(t; u2)

0 0 Vb2a2(t; u2) Vb2b2

(t; u2) Vb2g(t; u2)::: n

n Vasas(t; us) Vasbs

(t; us) Vasg(t; us)

Vbsas(t; us) Vbsbs

(t; us) Vbsg(t; us)

Vga1(t; u1) Vgb1

(t; u1) Vga2(t; u2) Vgb2

(t; u2) � � � Vgas(t; us) Vgbs

(t; us)Xs

k�1

Vgg(t; uk)

2666666666666664

3777777777777775

(7)

VT

h;u� Va1

(h; u1) Vb1(h; u1) � � � Vas

(h; us) Vbs(h; us)

Xs

k�1

Vg(h; uk)

" #(8)

O. Ruiz, P. Rudomın / Journal of Neuroscience Methods 124 (2003) 1�/26 5

From the retained valid solutions, that producing the

minimal ERRAverage will be the SICM characterization

of the PSP.

2.3.2. SICM parameter fitting algorithm

The problem is to find a set of parameters (/u; J)

which minimize, at the same time, the difference

between the SICM average and the PSP average,

kV (t; u;J)�V (t)�k; and the differences between the

SICM ACOVs and the ACOVs of the PSP,

al

j�1½½COVV (hj; t; u;J)�COVV (hj; t)�½½: We might op-

timize all the SICM parameters in u and J by means of anon-linear optimization algorithm. However, a more

economical calculation can be performed after noticing

that the elements of J occur linearly in Eqs. (4) and (5)

and do not intervene in the matrix and vectors with the

derivatives of V (Eqs. (7)�/(9)). This kind of problem

belongs to a class known as Separable Nonlinear Least

Squares Optimization, which can be solved by optimiz-

ing a functional constructed exclusively with the non-linear parameters and calculating the linear parameters

by Least Squares (Golub and Pereyra, 1973). We have

tried several variants of this idea and devised the

algorithm described in what follows.

Define the error functional:

ERRAverage(u;J; V (t)�)

�1000½½V (t; u;J)�V (t)�½½g(u;J) (10)

where V (t)� is the average of the PSP, calculated fromexperimental data: V (t)�� [V (t1) V (t2) � � � V (tl)];

V (t; u;J) is the SICM average, as defined by Eq. (4);

1000 is a factor included to increase the numerical

magnitude of the error and improve the fitting; jja�/bjjis the Euclidean distance between vectors a and b; and

g (/u;/J) is a penalty function as described below.

Minimize Eq. (10) by means of the Simplex Method

(Nelder and Mead, 1965) estimating, within each itera-tion of the method, the SICM parameter covariance

matrix J as described in Appendix C. If the estimated

amplitude, latency and PMC fluctuations are less than

the limits specified by the second criterion of Appendix

B, set g to 1. Otherwise, if the variance of one parameter

exceeds its limit, set g to 10, if two parameters exhibit

excessive variances, set g to 20, and so on. If the

algorithm converges, one has optimized the SICMparameters u and J corresponding to a given config-

uration.

3. Methods

3.1. Simulation of the PSPs

Two models were used in the PSP simulations; either

(i) the same model embedded in the SICM, i.e. a

homogeneous infinite cable, excited by infinitesimal-

width ‘synaptic’ currents (Eqs. (1) and (2)), or (ii) a

compartmental model of a tapering finite structure

receiving alpha pulses as ‘synaptic’ inputs. The PSPssimulated with the infinite cable were produced by

setting the SICM arbitrary parameters (see Section

3.4) equal to:

c�0:628 nF=mm; r�2:23 MV=mm (11)

The compartmental model was made of eight r �/c �/g

compartments (Fig. 2B) with r ranging from 10 to 300MV, c from 10 to 2 pF, and g from 470�1 to 7000�1 mS.

During a PSP simulation with either model, the

stochastic parameters were modified from record to

record in order to obtain specified amplitude, latency

and PMC fluctuations in the PSP components. The

mean, variances and covariances of these simulated

parameters were calculated in the usual way, to be

compared with the estimations of the SICM-IP (Fig. 1,dashed arrows).

Noisy-PSP records were obtained by adding filtered

white noise to the PSPs. The noise was calculated by

means of:

N(tj)�aN(tj�1)�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib(1�a2)

pw(tj);

j�2; 3; . . . ; l(12)

where a is a factor determining the spectral character-

istics of the noise (for the examples in the present work,

a�/0.81); b�/VARN is the noise variance (mV2); w(tj)

is a normally distributed pseudorandom variable with

mean 0 and variance 1; and l is the number of points

forming a record.

3.2. Average and ACOVs of a PSP

The average of a PSP, obtained from noisy PSP

records was calculated as:

V (t)��1

n

Xn

i�1

Y (t)(i); t � t (13)

where V (t)� is the estimated PSP average (value at time

t ); n , the number of available records; and Y (t)(i )

represents the ith record of the noisy PSP (value at

time t).

The autocovariances of a noisy PSP are given by:

COVY (h; t)�1

n � 1

Xn

i�1

(Y (h)(i)�Y (h))(Y (t)(i)�Y (t));

h�h1; h2; . . . ; hm; t � t

(14)

where COVY(h ,t) is the autocovariance of the noisy

PSP, relative to time h , value at time t ; Y (h)(i ) is the ithPSP record, value at time h ; Y (t)(i ) is the ith PSP record,

value at time t ; Y (h) is the PSP average at time h ; Y (t) is

the PSP average at time t .

O. Ruiz, P. Rudomın / Journal of Neuroscience Methods 124 (2003) 1�/266

The ACOVs of a noisy PSP result from two contribu-

tions: the variability of the PSP itself and the structure

of the background noise. If (i) the PSP and noise

combine additively; (ii) the noise is stationary, i.e. the

statistical characteristics of the noise are the same

regardless of the specific time for which they are

calculated; and (iii) the noise and the PSP fluctuations

are uncorrelated*/i.e. the cross-correlation between

PSP fluctuations and noise is 0 for every pair of times

(h ,t )*/then the best estimation of the PSP autocovar-

iances, COVV (h ,t)*, is obtained by subtracting the noise

autocovariance from the noisy-PSP ACOVs:

COVV (h; t)��COVY (h; t)�COVN(t�h);

h�h1; h2; . . . ; hm; t � t(15)

where the noise autocovariance, COVN(t�/h ), is calcu-

Fig. 3. Estimation of the average and ACOVs of a PSP from noisy-PSP records and records of the background noise. (A) Four noise records (of 2000

used for the calculations) simulated by means of Eq. (12) with a�/0.81 and b�/0.001 mV2. (B) Nine reference times (vertical dotted lines) and

corresponding noise ACOVs (solid traces), as calculated with Eq. (14) for each reference time. (C) Time-shifted noise ACOVs. Every curve in B was

horizontally displaced until its reference time is located at t�/0 ms. (D) Final estimation of the noise ACOV: mean of the nine time-shifted noise

ACOVs shown in C. (E) Four noisy-PSP records (of 2000 simulated and used for the calculations). (F) PSP average, estimated as the mean of the

entire set of noisy-PSP records (Eq. (13)). (G) Noisy-PSP ACOVs, calculated for the three reference times indicated in F (Eq. (14)). The solid trace

represents the h3-relative noisy-PSP ACOV. The dashed trace is the h7-relative noisy-PSP ACOV and the dotted trace is the h43-relative noisy-PSP

ACOV. (H) Estimated PSP ACOVs, obtained by horizontally shifting the noise ACOV in panel D until its peak coincide with the reference time of a

noisy-PSP ACOV (panel G), and subtracting the shifted noise ACOV from the noisy-PSP ACOV (Eq. (15)).

O. Ruiz, P. Rudomın / Journal of Neuroscience Methods 124 (2003) 1�/26 7

Fig. 4. Lat-Fluct Single-Cpt PSP. (A�/C) Individual records of simulated ‘Fast’ (x�/0.5l ), ‘Medium’ (x�/1l ) and ‘Slow’ (x�/2l ) PSPs exhibiting

normally-distributed latency fluctuations. The vertical dotted lines in B represent three reference times used to calculate the autocovariances depicted

in panels E�/G. (D) Average of the medium-PSP (circles) and average of the SICM when set with the same parameters utilized in the PSP simulation

(continuous trace). (E�/G) Autocovariances (ACOVs) of the medium-PSP records (circles) and SICM ACOVs (continuous traces) corresponding to

the reference times indicated by the vertical dotted lines in panel B. (H) Errors introduced by the SICM when reproducing the average (dotted traces)

and ACOVs (continuous traces) of the fast (F), medium (M) and slow (S) PSPs.

O. Ruiz, P. Rudomın / Journal of Neuroscience Methods 124 (2003) 1�/268

lated using Eq. (14) on records where the PSP has not

been evoked. If several individual estimations of the

noise ACOV are available (each calculated for a

different reference time) the estimation of COVN(t�/h )is improved by shifting the curves to align their reference

times and averaging the curves (see Fig. 3 for a step-by-

step explanation).

3.3. Comparison of the outcome of the SICM with the

actual average and ACOVs of simulated PSPs

The error between the SICM average and the average

of a simulated PSP was calculated dividing the differ-ence between the areas of the two averages by the area

of the PSP average. The error between a set of SICM

ACOVs and the PSP ACOVs was calculated as the mean

ratio of the area differences relative to the area of the

corresponding PSP ACOV. However, as some types of

fluctuations may produce ACOVs close or near to zero

(e.g. Fig. 4F), only non-zero-area PSP ACOVs were

used in the calculation.

3.4. Implementation of the SICM parameter estimation

procedure

Parameters r and c must be assigned with arbitrary

values in the SICM-IP. Except when indicated, all the

tests in this article were performed with r and c made

equal to the values programmed in the simulations (Eq.(11)). These values of r and c were calculated from

electrical and anatomical data derived from cat spinal

cord motoneurons. The cable’s diameter was calculated

assuming that all stem dendrites (mean number per

cell�/11.7, mean stem dendrite diameter�/6.6 mm;

Cullheim et al., 1987) merged into two bundles, at

opposite extremes of the soma, and using the d3/2 law

(Rall, 1989). The membrane specific capacitance wasassumed to be Cm �/1.0 mF/cm2 and the cytoplasmic

resistivity Ri �/70 V cm (Fleshman et al., 1988). These

calculations produced the values already specified in Eq.

(11) and a cable leak conductivity of about 0.2 mS/mm.

The units employed in this work were chosen to make

the numerical values of all the parameters of the same

order of magnitude and thus reduce the likelihood of

numerical errors. All the calculations required tosimulate and analyze the PSPs were programmed in

the numerical analysis package MATALB (The Math-

works, Inc.). The tolerance values imposed to the

Simplex method were 10�6. The optimization of an

initial solution by the SICM parameter fitting algorithm

required up to about 400, 5700, 7800 and 17 000

iterations of the Simplex Method for configurations

with one, two, three and four components, respectively.To assess the quality of the parameter estimations we

calculated the difference between each estimated SICM

parameter and the corresponding simulated parameter,

and expressed the absolute difference as a fraction of the

simulated parameter value. The errors in the estimation

of the variances and covariances of the stochastic

parameters cannot be calculated in that way becausesome variances or covariances could have been simu-

lated to be zero. To overcome this difficulty, the

estimated variances and covariances for which their

simulated counterparts were zero were divided by the

mean of all the simulated variances.

4. Results

4.1. Average and ACOVs of single-component PSPs

exhibiting a single type of fluctuations, and accuracy of

the SICM

Truncated Taylor series approximations are less

accurate for larger deviations of the involved variables.

Thus, the following sections examine the ability of theSICM to handle physiological fluctuations. The tests

serve also to illustrate the information provided by

ACOVs on the type of variability exhibited by a PSP.

4.1.1. Latency-fluctuating single-component PSP

Fig. 4A�/C shows three latency-fluctuating single-

component PSPs (Lat-Fluct Single-Cpt PSPs). The

average of the PSP in panel B is shown in Fig. 4D

(circles) while three representative ACOVs of the PSPare plotted in panels E�/G (circles).

4.1.1.1. ACOV relative to the PSP rise phase. When a

Lat-Fluct PSP appears sooner, the voltage at any fixed

time during the PSP rise phase is larger (Fig. 4B, e.g. h7).

Thus a positive correlation will exist between the PSP

voltage at h7 and the voltage at any other time within the

PSP rising phase, and this is precisely what one observesin the h7-relative ACOV for 0.7B/t B/1.5 ms, in panel E.

The PSP latency fluctuations have a small impact on the

peak of the averaged PSP (t�/h21; panel B). Hence, any

rise-phase-relative ACOV is zero for that time (panel E,

t�/1.6 ms). Finally, the decay-phase of the PSP gets

smaller when the PSP leads and larger when it lags

(panel B, e.g. h37). This introduces a negative correlation

between the voltage changes of the rise and decay phasesof the PSP, as reflected in COVV(h7,t) for t �/

tPSP average’s peak�/1.6 ms, in panel E.

4.1.1.2. ACOV relative to the PSP peak time. The

minimal voltage changes occurring near the peak of a

Lat-Fluct PSP (Fig. 4B; t�/h21) makes the entire

COVV (h21,t), t � /t, practically equal to zero, as observed

in Fig. 4F.

4.1.1.3. Late or decay-phase-relative ACOVs. In a Lat-

Fluct PSP, the voltage changes at any pair of times

O. Ruiz, P. Rudomın / Journal of Neuroscience Methods 124 (2003) 1�/26 9

during the decay phase have the same direction, and

they are opposite to the changes in the PSP rising phase

(Fig. 4B). Hence, voltage variations of two points in the

PSP decay phase will be positively correlated and, at thesame time, negatively correlated with the rising-phase

points. The negative correlation between the PSP rise

and decay phases is visible as the negativity of

COVV(h37,t), for 0.6B/t B/1.6 ms, in panel G. The

positive correlation between all the decay-phase points is

too small to be seen in this plot.

4.1.1.4. Ability of the SICM to reproduce the average

and ACOVs of a Lat-Fluct Single-Cpt PSP. In order to

compare the SICM outcome with the average and

ACOVs of the Lat-Fluct Single-Cpt PSP, the parameters

utilized in the PSP simulation were introduced into theSICM (Eqs. (4) and (5)). The average and ACOVs

predicted by the SICM are plotted by the solid traces in

panels D and E�/G, respectively. As one may see, the

outcome of the SICM is very similar to the average and

ACOVs of the simulated PSP. To obtain a general view

of the SICM accuracy when reproducing different

latency-fluctuating PSPs, we tested PSPs of three shapes

(‘fast’, x�/0.5l ; ‘medium’, x�/1.0l ; and ‘slow’, x�/

2.0l ; Fig. 4A�/C), simulated in the infinite cable, and

exhibiting different magnitudes of latency fluctuations.

The errors between the SICM outcome and the actual

average, and between the SICM ACOVs and the

ACOVs of the PSP are shown in panel H. One may

see that the errors in the SICM outcome increase with

larger fluctuations. If a maximal error of 10% in the

outcome of the SICM is tolerable, the standard devia-tion of the PSP latency fluctuations should not exceed

about stdv(b )�/0.02 ms�/0.0064tm for the fast PSP,

stdv(b)�/0.05 ms�/0.016tm for the medium PSP, and

stdv(b)�/0.10 ms�/0.032tm for the slow PSP. These

values might seem too small from a numeric point of

view, but panels A�/C show that the corresponding

fluctuations in the times of onset of the PSPs are

considerable. The tests indicate also that, if errors inthe outcome of the SICM close to 10% are admissible,

the SICM should be useful to study PSPs with latency-

fluctuations in the order of stdev(b )�/50 ms, which is

comparable to the latency fluctuations in Ia EPSPs

recorded in cat spinal motoneurons, as reported by

Munson and Sypert (1979), Collatos et al. (1979), Cope

and Mendell (1982a,b).

4.1.2. PMC-fluctuating single-component PSP

Changes in the membrane resistance of the postsy-

naptic cell, between one evoked PSP and the next,

produce PSP shape fluctuations like those shown in Fig.5A�/C, mainly affecting the PSP decay phase. The

average and ACOVs of the PSP in panel B are shown

in panels D and E�/G, respectively (circles).

4.1.2.1. ACOVs of the PMC-Fluct Single-Cpt PSP. A

decrease in the membrane resistance of the postsynaptic

cell (increase in PMC) makes the PSP decay faster while

introducing little change in the rising phase (Fig. 5B).This behavior produces PSP ACOVs almost equal to

zero for early reference times (panel E), and of increas-

ing magnitude for later reference times (panels F�/G).

The aspect of the PMC-fluctuating PSP ACOVs is

clearly different from the ACOVs of a latency-fluctuat-

ing PSP (Fig. 4E�/G). Then ACOVs should be useful in

qualitative analysis of experimental PSPs.

4.1.2.2. Ability of the SICM to reproduce the average

and ACOVs of a PMC-Fluct Single-Cpt PSP. Panels D

and E�/G of Fig. 5 (solid traces) show the average and

ACOVs of the SICM set with parameters equivalent to

those used in the simulation of the records in panel B.

One may see that the differences between the SICM

ACOVs and the actual ACOVs (circles) are minimal for

earlier ACOVs (panels E�/F) but increase for subsequentreference times (panel G). The errors in the SICM

outcome, for fast, medium and slow PSPs exhibiting

PMC fluctuations of different magnitudes, are shown in

panel H. Most errors are below 10%, and attain 13%

only for the maximal possible variability that a nor-

mally-distributed PMC may exhibit before taking zero

or negative values (CV(g )�/0.25). A normal distribution

for PMC fluctuations seems a reasonable assumptionbecause of the thousands of synapses affecting the PMC

and because of the Central Limit Theorem. Hence, one

may conclude that physiological conditions producing

PMC-Fluct Single-Cpt PSPs, can be handled by the

SICM with maximal errors of about 10%.

4.1.3. Amplitude-fluctuating single-component PSP

4.1.3.1. ACOVs of the Amp-Fluct Single-Cpt PSP. TheACOVs of an amplitude-fluctuating single-component

PSP (Amp-Fluct Single-Cpt PSP) have all similar

shapes, and they are equal to that of the averaged PSP

(not shown). These features, together with the behavior

of the ACOVs for latency-fluctuating and PMC-fluctu-

ating PSPs (Sections 4.1.1 and 4.1.2), can be used as

diagnostic tools to classify experimental PSPs.

4.1.3.2. Ability of the SICM to reproduce the average

and ACOVs of an Amp-Fluct Single-Cpt PSP. The

particular role of a , as a multiplicative factor in Eq.

(1), results in an exact representation of amplitude-

fluctuating PSPs by means of the Taylor series approx-

imation. Hence, single- or multiple-component Amp-

Fluct PSPs will be represented by the SICM without

error regardless of the span or probability distributionof their amplitude fluctuations. However, the biophysics

of the represented synaptic response does restrict the

span of amplitude fluctuations. Thus, the amplitude

O. Ruiz, P. Rudomın / Journal of Neuroscience Methods 124 (2003) 1�/2610

variability of a PSP component cannot be arbitrarily

large without making the component reverse its sign in

some instances. An approximated upper limit for the CV

of these restricted fluctuations can be obtained con-

sidering a stochastic amplitude constrained to fluctuate

between 0 and a maximal value A . From all the

probability distributions that such stochastic variable

may exhibit, those with the largest variances will be

those where the amplitude jumps between values near

the extremes 0 and A . Under these circumstances the

coefficient of variation of the amplitude will be near 1

(Bernoulli distribution with P (0)�/P (A )�/0.5). Conse-

quently, amplitude fluctuations close to or larger than

CV(a ):/1 would be meaningless in the SICM. This

Fig. 5. Simulated single-component PSP generated by a synaptic connection exhibiting normally-distributed fluctuations in the postsynaptic

membrane conductivity (PMC-Fluct Single-Cpt PSP). Same format as Fig. 4.

O. Ruiz, P. Rudomın / Journal of Neuroscience Methods 124 (2003) 1�/26 11

variability limit has been included in the criteria for

rejecting spurious SICM solutions specified in Appendix

B.

4.2. SICM-IP: analysis of single-component noiseless

PSPs

The simplest problem that one can pose to the SICM-

IP is to identify the parameters and fluctuations of

simulated single component PSPs, when the PSPs

exhibit a single type of fluctuations, in absence of noise.

Three Single-Cpt PSPs (one Amp-Fluct, one Lat-Fluct

and one PMC-Fluct) were simulated with the infinitecable model (Eq. (1)) and analyzed with the SICM

parameter fitting algorithm (Section 2.3.2; results not

shown). In the three cases tested, the algorithm

succeeded in the identification of the type and magni-

tude of the PSP simulated fluctuations, with errors less

than 10%, and averaging 0.6%.

4.3. SICM analysis: multiple-component noisy PSPs

The ensuing test for the SICM-IP was the identifica-

tion of two-component PSPs, produced in a homoge-

neous infinite cable, displaying different combinations

of amplitude, latency and PMC fluctuations, in presence

of noise. In particular, we focused on the following

questions: (i) Is the SICM-IP able to estimate the

number of components in a PSP exclusively from theaverage and ACOVs of the PSP? and (ii) What is the

performance of the SICM-IP when the records of the

simulated PSPs contain different levels of noise?

4.3.1. Multiple-component noisy PSPs: simulated

situations

Three PSPs were simulated on the infinite cable

model, intending to represent the situations schematizedin Fig. 6. In each case, a sensory neuron (Ia) has

synaptic contacts with a target neuron at two different

locations (x1�/0.5l and x2�/1l ).

4.3.1.1. Amplitude-fluctuating two-component PSP

(Amp-Fluct Two-Cpt PSP; Fig. 6A). In this situation,

the sole source of PSP variability is the stochastic

fluctuations of transmitter release from the afferent

terminals. The resulting PSP is thus made of twocomponents displaying uncorrelated amplitude fluctua-

tions.

4.3.1.2. Amplitude- and latency-fluctuating two-

component PSP (Amp/Lat-Fluct Two-Cpt PSP; Fig.

6B). As above but, in addition, two different pools of

presynaptic inhibitory interneurons make synaptic con-

tact with the afferent axon (Int1 and Int2). The firing ofthese interneurons delays the propagation of the action

potential along the two axonal branches without block-

ing conduction or decreasing transmitter release, be-

cause the location of the contacts is far from the

terminals. This arrangement introduces independent

latency fluctuations in both components of the PSP.

4.3.1.3. Combined-fluctuation two-component PSP

(Comb-Fluct Two-Cpt PSP; Fig. 6C). Again, the two

components of the PSP fluctuate in amplitude because

of the transmitter release variability at each presynaptic

terminal. In addition, the postsynaptic neuron receives a

barrage of synaptic inputs from different pools of

excitatory and inhibitory neurons (Int3 and Int4) which

are the source of PMC fluctuations. Two other pools of

inhibitory interneurons have presynaptic contacts withthe afferent axon (Int1 and Int2). Int1 has contacts also

with the postsynaptic neuron. Its activity reduces the

amplitude of the first PSP component because of

presynaptic inhibition while, at the same time, increases

the membrane conductance of the postsynaptic cell. This

action results in a negative correlation between the

amplitude of the first PSP component and the PMC.

Int2 contacts the other branch of the afferent axon attwo different locations: one ‘upstream’*/far from the

terminals*/and the other at the terminals themselves. In

the first case, the firing of Int2 introduces a delay on the

Fig. 6. Diagrams showing the connections intended to produce the PSPs analyzed in Section 4.3, Table 1 and Figs. 7�/9. See text for explanation.

O. Ruiz, P. Rudomın / Journal of Neuroscience Methods 124 (2003) 1�/2612

action potential propagation; in the second, it reducesthe amount and/or probability of transmitter release.

Thus, the spontaneous firing of Int2 introduces a

negative correlation between the amplitude and the

latency fluctuations of the second PSP component.

The parameters of the three simulated PSPs may be

found in Table 1.

4.3.2. Multiple-component noisy PSPs: SICM-IP

analysis

The average and ACOVs of the three two-component

simulated PSPs (using samples of 2000 records) were

calculated either from records without noise (VARN �/

0) or from noisy records (VARN �/0.0001, 0.001 and0.01 mV2). The estimated PSP averages and ACOVs

were processed with the SICM-IP (Analysis, Section

2.3), setting the arbitrary parameters (r , c ) with their

simulated values (Eq. (11)). Thirteen initial solutions

with one, two, three and four components were fitted by

the SICM parameter optimization algorithm. The fitted

solutions were analyzed with the x2-test (Appendix D)

and inspected with the criteria listed in Appendix B. Theresults could be classified into three groups. First, in

some cases, the x2-test determined that a single solution,

from the 13 tested, was appropriate to fit the data (e.g.

low-noise amplitude-fluctuating two-component PSP

(Amp-Fluct Two-Cpt PSP)). The same solution was

selected also by the criteria described in Appendix B.

Second, in other cases the x2-test determined that two or

more solutions with different number of components

were appropriate, e.g. low-noise amplitude- and latency-

fluctuating two-component PSP (Amp/Lat-Fluct Two-

Cpt PSP) and low-noise combined-fluctuation two-

component PSP (Comb-Fluct Two-Cpt PSP). In these

cases, a solution could still be singled out by means of

two criteria: either (i) parsimoniously, by choosing the

valid solution with the smaller number of components;

or (ii) by keeping the valid solution with the minimal

ERRAverage. For the PSPs analyzed so far, both criteria

pointed towards the same- and right-solution. The third

case was observed for high-noise PSPs (noise standard

deviations of 0.001 mV2 or larger). In this case, the x2

indicated that most or all solutions were appropriate to

fit the data, regardless of their number of components

and of larger differences in ERRAverage. For high-noise

PSPs also Appendix B criteria failed to select a single

solution, and even the best fittings exhibited mean

estimated-parameter errors exceeding 10%. These results

showed that SICM-IP is unable to provide a solution for

Table 1

Parameters of the simulated two-component noisy PSPs

Parameter Amp-Fluct

Two-Cpt PSP

Amp/Lat-Fluct Two-

Cpt PSP

Combined (Amp, Lat and PMC)-Fluct

Two-Cpt PSP

Fixed parameters and mean values of the stochastic parameters

X1 0.50 0.50 0.50

X2 1.00 1.00 1.00

/a1 0.547 0.547 0.90

/a2 1.425 1.425 1.50

/b1 0.40 0.40 0.40

/b2 0.40 0.40 0.40

/g 0.20 0.20 0.20

Variances of the stochastic parameters

VAR(a1), amplitude variance of the first component 0.0120 0.0120 0.0324

VAR(a2), amplitude variance of the second component 0.0810 0.0810 0.0450

VAR(b1), latency variance of the first component 0 0.0004 0

VAR(b2), latency variance of the second component 0 0.0004 0.0025

VAR(g ), PMC variance 0 0 0.0008

Covariances of the stochastic parameters

COV(a1, b1) 0 0 0

COV(a1, a2) 0 0 0

COV(a1, b2) 0 0 0

COV(a1, g ) 0 0 �/0.0036

COV(b1, a2) 0 0 0

COV(b1, b2) 0 0 0

COV(b1, g ) 0 0 0

COV(a2, b2) 0 0 �/0.0075

COV(a2, g ) 0 0 0

COV(b2, g ) 0 0 0

O. Ruiz, P. Rudomın / Journal of Neuroscience Methods 124 (2003) 1�/26 13

high-noise PSPs. Hence, PSPs with noise levels larger

than 0.0001 mV2 were not considered further.

The SICM-IP estimated parameters, calculated from

the three low-noise PSPs (VARN �/0.0001 mV2) are

listed in Table 2 (second to fourth columns). The fixed

and mean values of the stochastic parameters were

estimated with high accuracy (compare with Table 1).

The estimation of the variances and covariances of the

stochastic parameters were less accurate, but never-

theless reflected well the PSP parameters fluctuating

during the simulations. Altogether, the estimated para-

meters exhibited average/maximal errors of 0.67/8.3%

for the Amp-Fluct Two-Cpt PSP, 3.20/35% for the

Amp/Lat-Fluct Two-Cpt PSP, and 4.1/28% for the

Comb-Fluct Two-Cpt PSP (with (r ,c )SICM

�/(r ,c )simul). The errors in the three PSP estimated

parameters averaged 3%.

The fifth column of Table 2 presents the results of the

SICM analysis of the Comb-Fluct Two-Cpt PSP, when

the arbitrary parameters (r ,c) were set with different

values to those programmed in the simulation. These

results are considered in Section 4.3.4.

4.3.3. Multiple-component noisy PSPs: Interpretation of

the SICM-IP results

This section interprets the SICM estimated para-

meters of the simulated PSPs assuming that the compo-

sition and fluctuations of the PSPs were not known in

advance.

Figs. 7�/9 show the averages (panel B, circles) and six

representative ACOVs (panels A1�/A6, circles) of thelow-noise PSPs (VARN �/0.0001 mV2). Fig. 7 pertains

to the Amp-Fluct Two-Cpt PSP, Fig. 8 to the Amp/Lat-

Fluct Two-Cpt PSP, and Fig. 9 to the Comb-Fluct Two-

Table 2

SICM-IP estimated parameters for the three two-component noisy PSPs

Parameter Amp-Fluct Two-Cpt PSP

(r , c )SICM�/(r , c )simul

Amp/Lat-Fluct Two-Cpt PSP

(r , c )SICM�/(r , c )simul

Combined (Amp, Lat and PMC)-Fluct Two-Cpt PSP

(r , c )SICM�/(r , c )simul (r , c )SICM�/(1,1)

Fixed parameters and mean values of the stochastic parameters

X1 0.499 0.511 0.502 0.49

X2 0.997 1.004 1.003 0.99

/a1 0.54 0.57 0.93 1.6

/a2 1.4 1.4 1.5 2.9

/b1 0.40 0.40 0.40 0.40

/b2 0.40 0.40 0.41 0.40

/g 0.20 0.20 0.20 0.32

Variances of the stochastic parameters

VAR(a1) 0.011 0.011 0.034 0.10

VAR(a2) 0.08 0.08 0.036 0.15

VAR(b1) 0.0000027 0.00035 0.0000002 0.0000003

VAR(b2) 0.000081 0.00054 0.002 0.0022

VAR(g ) 0.0000025 0.0000014 0.00087 0.0021

Covariances of the stochastic parameters

COV(a1, b1) �/0.0000035 �/0.00033 0.00003 0.000012

COV(a1, a2) 0.00044 0.0017 0.00034 0.011

COV(a1, b2) �/0.00024 �/0.00037 �/0.00033 �/0.00077

COV(a1, g ) 0.000034 0.000033 �/0.0038 �/0.01

COV(b1, a2) 0.000028 0.00045 �/0.0000017 �/0.000063

COV(b1, b2) 0.000014 �/0.00013 0.0000007 0.000012

COV(b1, g ) �/0.0000018 0.0000085 �/0.0000032 �/0.0000013

COV(a2, b2) 0.000068 0.0000017 �/0.0054 �/0.013

COV(a2, g ) �/0.000056 �/0.000059 0.00019 �/0.00068

COV(b2, g ) �/0.0000064 �/0.000019 �/0.00002 0.000029

Second and third columns list the estimated parameters of the PSPs described in Figs. 7 and 8, respectively, calculated when the SICM arbitrary

parameters (r , c ) were set with the same values utilized in the simulations. Fourth and fifth column presents the results for the Comb-Fluct Two-Cpt

PSP (Fig. 9), when the SICM-IP was set with arbitrary parameters either equal to or different from the simulated values.

O. Ruiz, P. Rudomın / Journal of Neuroscience Methods 124 (2003) 1�/2614

Cpt PSP. One may observe that: (i) for each PSP, the

shape of its ACOVs is different from the PSP average,

excluding the possibility that the PSPs could be an Amp-

Fluct Single-Cpt response (Section 4.1.3); (ii) the

ACOVs of the three PSPs do not resemble those in

panels E�/G of Fig. 4 or Fig. 5, then the PSPs are not

Lat-Fluct or PMC-Fluct Single-Cpt responses (Sections

4.1.1 and 4.1.2); (iii) the ACOVs of the three PSPs in

Figs. 7�/9 are different, suggesting that the composition

and/or fluctuations of the three PSPs differ from each

other. The details of the PSP composition and fluctua-

tions were provided by the SICM-IP (Table 2). First of

all, the SICM analysis found the three low-noise PSPs to

be made of two components (as simulated), one

originated at 0.5l and the other at 1.0l from the

postsynaptic soma (Table 2: rows X1 and X2). The

estimated parameters can be used to reconstruct the

averages of the identified PSP components, as has been

done in panel B of Figs. 7�/9 (continuous and dotted

traces). Note the matching with the simulated compo-

nents (squares and triangles, respectively).

An inspection of the SICM-IP estimated variances

indicates that amplitude fluctuations were the major

source of variability in the three PSPs (Table 2; second

to fourth columns). This conclusion is consistent with

the simulations (Table 1). The row VAR(b1), in Table 2,

shows the latency fluctuations estimated for the prox-

imal component in the three PSPs. The SICM-IP results

indicate that the proximal component of the Amp/Lat-

Fluct Two-Cpt PSP exhibited much larger latency

fluctuations than the proximal components of the other

two PSPs (more than 100 times). The latency fluctua-

tions of that component might result from presynaptic

influences delaying the action potential in the first

branch of the Amp/Lat-Fluct Two-Cpt PSP. Those

presynaptic influences would be absent from the prox-

Fig. 7. SICM-IP analysis of a simulated Amp-Fluct Two-Cpt PSP. (A1 to A6) Six representative ACOVs out of 50 considered in the analysis. Circles:

ACOVs of the simulated PSP. Continuous trace: ACOVs produced by the fitted SICM. (B) Average of the PSP (circles) and average of the fitted

SICM (bold continuous trace). The squares and triangles represent the actual averages of the simulated proximal and distal components, respectively.

Continuous and dotted traces represent the estimated PSP component averages, reconstructed from the fitted SICM parameters. The PSP average

and ACOVs are made of 50 points; each point corresponds to one of the reference times selected for the calculation of the ACOVs. The sampled

points are less dense throughout the PSP decay phase to reduce the memory requirements and processing time.

O. Ruiz, P. Rudomın / Journal of Neuroscience Methods 124 (2003) 1�/26 15

imal axonal branch in the other two PSPs. Note also

that the latency fluctuations estimated for the distal

component of the Amp/Lat-Fluct Two-Cpt PSP,

VAR(b2), are comparable to those of the proximal

component, VAR(b1). Then, the same factors should

be acting on the two branches of the presynaptic axon

producing the Amp/Lat-Fluct Two-Cpt PSP. This con-

clusion matches the simulated situation (Fig. 6B).

Altogether, an inspection of the latency variabilities

estimated for the components of the three PSPs (rows

VAR(b1) and VAR(b2)) indicates that the distal compo-

nent of the Comb-Fluct Two-Cpt exhibited the largest

latency fluctuations (fourth column of Table 2). Hence

one would conclude that the distal component of the

Comb-Fluct Two-Cpt PSP is the most affected by a

putative presynaptic mechanism.Let us now consider the PMC variability estimated by

the SICM-IP for the three PSPs (Table 2, row VAR(g),

second to fourth columns). The PMC fluctuations of the

Comb-Fluct Two-Cpt PSP are much larger (more than

300 times) than those of the two other PSPs. This PMC

fluctuation could have been produced either by changes

in the intensity of the synaptic bombardment from one

time to the next, or because the same amount of

background synaptic activity became intermittently

synchronized and desynchronized. These factors would

be almost absent in the other two connections. Again,

this interpretation matches the simulated situations.

Table 2 lists also the covariances between the stochas-

tic parameters of the three PSPs, as estimated by the

SICM-IP (second to fourth columns). The majority of

these parameters are smaller, suggesting that indepen-

dent mechanisms introduced the different fluctuations in

each PSP. Yet the magnitudes of COV(a1,g ) and

COV(a2,b2) in the Comb-Fluct Two-Cpt PSP are

distinctly larger than the others. The negative correla-

tion between the amplitude of the first component and

the PMC (COV(a1,g)�/�/0.0038 pC mS/mm) suggests a

mechanism acting simultaneously on the presynaptic

terminals and the postsynaptic cell membrane for the

Comb-Fluct Two-Cpt PSP. A common modulating

mechanism is also suggested by the negative correlation

between the amplitude and latency fluctuations of the

second component (COV(a2,b2)�/�/0.0054 pC ms). All

Fig. 8. SICM-IP analysis of a simulated Amp/Lat-Fluct Two-Cpt PSP. Same format as Fig. 7.

O. Ruiz, P. Rudomın / Journal of Neuroscience Methods 124 (2003) 1�/2616

these inferences match the simulated conditions (Section

4.3.1 and Table 1). Furthermore, the numerical values of

the identified covariances are not far from the simulated

values (�/0.0036 pC mS/mm and �/0.0075 pC ms,

respectively).

In summary the SICM-IP, in its present status, was

able to correctly determine the composition and fluctua-tions of the three simulated two-component PSPs

‘recorded’ in presence of low-level background noise

(VARN �/0.0001 mV2). The parameters of the PSPs

were estimated with maximal errors attaining 35%, and

with a mean error smaller than 3%.

4.3.4. Multiple-component noisy PSP: SICM-IP analysis

with r and c unknown

The preceding analysis was conducted setting the

SICM arbitrary parameters, r and c , with the same

values employed in the simulations. This way ofchoosing r and c is of course impossible when the

geometry and electrical characteristics of the studied

neuron are unknown. To examine how the lack of

knowledge of the true values of r and c affects the

SICM-IP results, we repeated the SICM analysis of the

Comb-Fluct Two-Cpt PSP (Fig. 9) using c�/1.0 nF/mm

and r�/1.0 MV/mm, instead of their simulated values

(Eq. (11)). We found that the SICM-IP also converged

to a solution, and that this solution was made of two

components, similarly as when (r ,c )estimation�/(r ,c)simu-

lation. The (r ,c )estimation"/(r ,c)simulation solution is pre-

sented in the fifth column of Table 2. There one may see

that the numerical values of the majority of the

estimated parameters are different from the

(r ,c )estimation�/(r ,c)simulation fitting (and from the simu-

lated values in Table 1). Parameters with similar values,

regardless of the r and c used in the SICM-IP, were: the

estimated electrotonic locations of the two identified

components (X1 and X2; compare fourth and fifth

columns), the mean latencies of the components (rows

b1 and b2); and the variances of the component latencies

(VAR(b1) and VAR(b2)). The other estimated fluctua-

tions (amplitude and PMC) only match the simulated

fluctuations when expressed in relative terms. Thus,

Fig. 9. SICM-IP analysis of a simulated two-component PSP displaying combined amplitude, latency and PMC fluctuations (Comb-Fluct Two-Cpt

PSP). Same format as Fig. 7.

O. Ruiz, P. Rudomın / Journal of Neuroscience Methods 124 (2003) 1�/26 17

regardless of the values given to r and c , one has

CV(a1)�/0.2, CV(a2)�/0.13, and CV(g)�/0.15. We also

observed that the averages of the PSP components,

reconstructed from the (r ,c )estimation"/(r ,c)simulation so-lution, were equal to those obtained with

(r ,c)estimation�/(r ,c )simulation (not shown).

These results show that the lack of information on the

simulated values of r and c : (i) did not prevent the

SICM-IP converging to a solution consistent with the

simulated PSP; (ii) made the SICM-IP missing the right

values of a; g; and most of the variances and covariances

of the stochastic parameters; (iii) the estimated physicaldistances to the synaptic locations (x1 and x2) were

different to their simulated counterparts; however, the

electrotonic distances to the identified contacts (X1 and

X2) matched the simulated values; (iv) the estimated

parameters allowed a calculation of CV(a ) and CV(g )

close to the simulated values; and (v) the SICM-IP

found values for b and VAR(b) matching the simulated

values.

4.4. SICM analysis of a ‘non-SICM’ PSP

We explored also the behavior of the SICM-IP when

analyzing a PSP simulated, not in an infinite cable, but

in a structure a step closer to an actual neuron: a finite,

tapering, non-homogeneous cable excited by alpha

current pulses (henceforth, a ‘non-SICM’ PSP). Specificaims were: (i) to test if the SICM-IP was able to

converge when the analyzed PSP has been produced in

a structure with geometry and electrical characteristics

different from those embedded in the SICM, (ii) to

explore how one could interpret the SICM parameters

estimated for this non-SICM PSP, and (iii) to illustrate

the minor issue that nothing in the model or the

identification procedure restricts the SICM-IP to theanalysis of excitatory PSPs.

We set up a simple compartmental model of a

hypothetical neuron consisting of a sphere (representing

the soma; first compartment) and a seven-compartment

tapering cable representing the dendritic arbor of the

neuron. The axon of a presynaptic inhibitory neuron

was assumed to have three contacts with the postsynap-

tic cell: at the somatic, sixth and eighth compartments.

The model parameters were deliberately chosen not to

resemble the arbitrary parameters used in the current

implementation of the SICM (Section 3.1). The synaptic

inputs to the compartmental model were also different

from those considered by the SICM: instead of infini-

tesimal-width Dirac delta currents, we simulated ‘alpha’

current pulses with a peak time of 0.15 ms and a half

width of about 0.38 ms. Uncorrelated amplitude fluc-

tuations of each component and PMC fluctuations were

simulated to yield CV(a1)�/CV(a2)�/CV(a3)�/0.20 and

CV(g )�/0.091. Background noise (variance�/0.0001

mV2) was added to the 2000 simulated records of this

compartmental-model IPSP (C-PSP). The average and

some ACOVs of the C-PSP, estimated from the noisy

records, are shown in panels B and A1�/A6 of Fig. 10.

The SICM-IP was carried out as before, fitting ten

initial solutions with configurations of one, two, three

and four components. The x2-test of the fitted solutions

indicated than none could explain the data thoroughly.

This reflects the fact that the homogeneous infinite cable

cannot reproduce exactly the PSPs produced in a non-

homogeneous, finite structure. Nevertheless a SICM

solution for the C-PSP could be still singled out by

means of the criteria described in Appendix B, and

provided useful information on the C-PSP. This solution

is described next.Fig. 10B shows the SICM-IP reconstructed averages

of the components identified in the C-PSP (continuous

traces). Note first that the SICM-IP found the right

number of components in the C-PSP (three). Note also

that the three estimated-component averages resemble

the actual components (squares, triangles, diamonds)

but exhibit some differences, particularly in the decay

phase of the first component and the rising phase of the

third. Considering that the SICM has no means to

reproduce exactly the voltage transients in a non-SICM

structure, the reconstructed averages seem nonetheless

acceptable.

The bars in Fig. 10C1 show the SICM-IP estimation

of the mean latencies of the C-PSP components. The

mean latencies match the simulated values (b1�/b2�/

b3�/0.4 ms). The SICM-IP also provides an estimation

of the latency fluctuations per component (lines over the

bars represent one standard deviation above and one

Fig. 10. SICM-IP analysis of an amplitude- and PMC-fluctuating three component IPSP simulated in a compartmental model of a tapering finite

structure, receiving alpha-function synaptic currents (C-PSP). (A1�/A6) Six representative ACOVs out of 120 considered in the analysis. Circles:

ACOVs of the simulated IPSP. Continuous trace: ACOVs produced by the fitted SICM. (B) Average of the IPSP (circles) and average of the fitted

SICM (bold continuous trace). The squares, triangles and rhombi represent the average of the simulated proximal, medial and distal components,

respectively. Continuous traces represent the estimated PSP component averages, reconstructed from the fitted SICM parameters. The PSP average

and ACOVs are made of 303 points separated by a constant sampling period. (C1�/C4) SICM-IP estimated parameters of the C-PSP. (C1) Latency of

the ‘synaptic’ current transients producing the three identified components (bars). Same horizontal scale as B. The estimated latency fluctuations per

component are indicated by the lines over the bars. These lines represent one standard deviation above and one below the mean of the corresponding

parameter. (C2) Electrotonic distances to the source of the identified components. (C3) Membrane conductivity of the postsynaptic cell: mean value

and estimated fluctuations as indicated in C1. (C4) Electrical charge associated with the synaptic currents identified by the SICM-IP (mean values and

fluctuations as indicated in C1).

O. Ruiz, P. Rudomın / Journal of Neuroscience Methods 124 (2003) 1�/2618

Fig. 10

O. Ruiz, P. Rudomın / Journal of Neuroscience Methods 124 (2003) 1�/26 19

below the mean of the corresponding parameter). The

estimated latency fluctuations are zero for the first

component and quite small for the second, consistent

with the simulated fluctuations (VAR(b1)�/VAR(b2)�/

0). However, the SICM-IP assigned latency fluctuations

to the third component which were not present in the

simulation (VAR(b3)�/0). The variability estimated for

the third component exceeds the SICM limit to faith-

fully represent a latency fluctuating PSP (Section 4.1.1

and criterion 2 in Appendix B). Then, if lacking a priori

information on this PSP, one might say that the third

component has been, or is close to having been wronglyidentified, and one should be careful with its interpreta-

tion.

Given the different structure and ‘synaptic’ currents

of the C-PSP and the SICM, one should expect few

SICM estimated parameters to match the numerical

values of the simulated C-PSP parameters. Thus, for

example, the estimated location of the first PSP compo-

nent (X1�/0.56; Fig. 10C2) is clearly different from thesimulated location (somatic, X�/0). The reason for this

discrepancy is that, in order to account for the slow

rising phase of the first C-PSP component, the SICM-IP

positioned the corresponding current source farther

than it was simulated. Another difference between the

identified and simulated parameters is the relation

exhibited by the electrical charge carried by the three

‘synaptic’ currents. The mean charges injected by theSICM to reproduce the C-PSP components exhibit the

relation 1:2:5 (panel C4), while the amplitudes of the

simulated currents in the compartmental model were

1:1.75:1.75. The largest difference between the estimated

and simulated ratios occurred for the distal PSP

component. This is because the current injected in the

last compartment of the finite structure spreads only

towards the soma and requires less magnitude, for thesame somatic effect, than the infinite cable. In any case,

the charge parameters estimated by the SICM-IP allow

one to estimate the relative variability of each compo-

nent’s amplitude: CV(a1)�/0.20, CV(a2)�/0.30,

CV(a3)�/0.26. These estimates are not far from the

simulated values (0.20, for the three components). Thus,

the SICM-IP would have been determined that the three

components in the C-PSP were amplitude fluctuating, assimulated.

The membrane conductance of the C-PSP ‘cell’ was

found to fluctuate with a CV(g)�/0.098 (Fig. 10C3).

This value is almost equal to the simulated variability:

CV(g1)�/. . .�/CV(g8)�/0.091. Therefore, if no infor-

mation on the simulation parameters and variability

were available to the user, he could nevertheless infer

that part of the C-PSP fluctuations would have beenproduced by changes in the membrane resistance of the

postsynaptic cell.

In summary, the SICM-IP analysis of this non-SICM

PSP (i) produced estimated parameters different from

the parameters of the simulated PSP; (ii) was wrong or

inconclusive with respect to the latency fluctuations

estimated for the distal component; and (iii) indicated,

according to the x2-test, that the analyzed PSP was notperfectly reproducible by the homogenous infinite cable.

Nevertheless, the SICM-IP; (iv) converged to a solution;

(v) found the right number of components in the C-PSP;

(vi) determined that the synaptic transients generating

the components occurred simultaneously but at three

different locations; (vii) found that the major fluctua-

tions in the C-PSP occurred in the amplitude of the three

components and in the PMC; (viii) allowed an approxi-mated estimation of the coefficient of variation of the

amplitudes and PMC; and (ix) provided an approximate

reconstruction of the averages of the C-PSP compo-

nents.

5. Discussion

This article has argued for the physiological relevanceof analyzing amplitude, latency fluctuations and PMC-

dependent shape fluctuations in the components of

evoked PSPs. To perform this analysis, we propose the

use of the ACOVs of the PSP and a stochastic model of

the postsynaptic response. The model presented in this

work, the SICM-IP, represents the components of a PSP

by the response of a linear homogeneous infinite cable

to brief current pulses injected at different locations. Yetneurons are non-homogeneous, finite in length, rami-

fied, possess non-linear voltage-dependent membrane

conductances, and synaptic transients are not Dirac

deltas. The following sections discuss the applicability,

limitations, caveats and future work to be done in the

model and the parameter identification procedure.

5.1. SICM analysis of PSPs produced in realistic

structures

Even if the SICM-IP characterize a PSP in quantita-

tive terms, a numerical match between the estimated

parameters and the biophysical parameters of the

analyzed system cannot be expected for most situations.

The first cause of disagreement is that*/even when a

physiological structure could be faithfully representedby a homogeneous infinite cable (perhaps in the case of

a muscle fiber, or the main dendrite of a pyramidal

cell)*/the cable parameters r and c will be, in general,

unknown. The second is simply that neurons are not

infinite cables excited by infinitesimal-width current

pulses. The tests performed in Sections 4.3.3 and 4.4

suggest that, when the underlying geometry and elec-

trical parameters of the studied neuron are not known,one should take into account the SICM parameters in

the following decreasing order of confidence: (i) the

estimated averages of the PSP components; (ii) the

O. Ruiz, P. Rudomın / Journal of Neuroscience Methods 124 (2003) 1�/2620

means of the component latencies; (iii) the relative

variability of the PMC and component amplitudes;

(iv) the correlations between the stochastic parameters,

and the variances of the component latencies; (v) theelectrotonic locations of the identified components; and

(vi) the numerical values of the SICM parameters.

Future work should explore more realistic stochastic

models of synaptic potentials and currents, e.g. voltage

transients in finite structures (Rall, 1967, 1969; Jack and

Redman, 1971b). However, refined models will necessa-

rily imply additional parameters to estimate, and one

might find that the information conveyed by the averageand ACOVs of the PSP is insufficient for the identifica-

tion of all the parameters. At that time, to obtain more

information on the electrotonic structure of the studied

cell, one might consider the model response to current-

(or voltage-) pulses, and modify the identification

procedure to include these data in the parameter

estimation.

Finally, even if designed for single-fiber PSPs, theSICM-IP could be also useful in the study of multi-fiber

PSPs. In this case, the PSP components would corre-

spond to the action of nerve bundles with different

conduction-velocity fibers, or to the activation of

interposed interneurons. In such circumstance, of

course, the electrotonic locations of the synaptic sites

would be meaningless and they would be used by the

model only to specify the shape of the components.

5.2. Estimating the number of components in a PSP

Assessing the number of underlying components in a

data set is usually the major difficulty in a finite-mixture

analysis problem (Hasselblad, 1966; Stricker et al., 1994;

McLachlan and Peel, 2000). The choice of this number

may be easy if the peaks or clusters corresponding to

each component are clearly visible from the data, butsuch situation is rather atypical (and avoided in the

examples simulated in this paper). The usual strategy to

assess the number of components in a finite mixture is to

fit solutions with increasing number of components and

to stop when the fittings cease to improve. In a number

of cases, the SICM-IP behaved in this way. In other

cases, the fitting error reached a minima for solutions

with the right number of components and then increasedfor solutions with more components. A second observa-

tion to be mentioned at this point is that different initial

solutions*/with similar number of components*/lead

almost invariably to different final solutions. Third:

even if in some cases a spurious component*/not

present in the simulation*/was much reduced by the

SICM-IP, its amplitude was almost never made equal to

0. These three observations suggest that the SICMparameter space possess many local optima. Multiple-

optima problems are not easy to solve and call for more-

or-less heuristic methods (Gershenfeld, 1999). Up to this

time, we have faced the local-optima problem in the

SICM-IP by (i) making the SICM parameter optimiza-

tion algorithm to fit as many different initial solutions,

with as many different numbers of components, aspossible; and (ii) establishing and applying the criteria in

Appendix B to eliminate spurious solutions (Section

5.3). Future work should evaluate the advantages of

using other optimization methods, e.g. Genetic Algo-

rithms, for the estimation of the SICM parameters.

5.3. Consistency tests for the SICM solutions

Tests of preliminary SICM-IP versions showed that asole consideration of ERRAverage was unable to assess

the right number of components in a PSP. This led us to

look for other features in the SICM to reject wrong

solutions. Our search produced the criteria described in

Appendix B. These criteria were conceived as a-poster-

iori tests to be applied on the fitted solutions. Yet, in the

future, some or all of these criteria could be incorpo-

rated into the SICM parameter optimization algorithm.To illustrate how this could be done, the criterion

limiting the magnitudes of the estimated fluctuations

(number 2 of Appendix B) has been already included in

the error functional (Eq. (10)) by means of a penalty

function (g ). The usual way to introduce penalty

functions in optimization algorithms is by adding them

to the error functional (Jacoby et al., 1972). However,

the tests of the SICM-IP performed in this work showthat the incorporation of g as a factor in the error

functional is also feasible. Further work should be done

to incorporate other criteria in the manner offering the

best results.

At any rate, the penalty function in Eq. (10) not only

enlarged the error of invalid solutions but also improved

the convergence of some initial solutions. In fact, the

inclusion of the excessive-variability criterion in theerror functional made it possible to identify the three-

component non-SICM PSP of Section 4.4. This result

suggests that the incorporation of other criteria in the

SICM parameter optimization algorithm, as well as

additional information on the system under study, may

improve the general performance of the SICM-IP.

5.4. Linearity

The SICM was constructed assuming linearity both in

the combination of the PSP components, and in the

combination of the components with the background

noise. Non-linear summation of the components in an

PSP may occur if one or more components activate

postsynaptic voltage-dependent conductances, or if

depolarization during an EPSP diminishes the electro-motive force of the ions producing the EPSP compo-

nents. Yet, the small depolarizations associated with

most single-fiber PSPs in the Central Nervous System

O. Ruiz, P. Rudomın / Journal of Neuroscience Methods 124 (2003) 1�/26 21

are probably insufficient to introduce these non-linear-

ities in the combination of PSP components.

Non-linear summation of the components and synap-

tic noise may occur if the membrane resistance of thepostsynaptic cell changes with the background noise.

But membrane resistance changes caused by variations

of the background synaptic activity, is precisely the

factor considered in this work to account for PSP shape

fluctuations (PMC fluctuations). Thus, providing that

PMC-fluctuations are slow (changing from a PSP to the

next, but remaining more or less constant throughout

each PSP), the non-linear effects of the backgroundsynaptic activity on the PSP under study are already

accounted for in the SICM.

5.5. Comparison with methods in use

First, it should be pointed out that the SICM-IP is not

intended for the analysis of spontaneous PSPs. This is

because the SICM contemplates the fluctuating post-synaptic response to a well-defined presynaptic action

potential, either produced by the experimenter, or

monitored and used to trigger the PSP acquisition.

The SICM-IP has both similarities and differences with

respect to current methods for the analysis of PSP and

PSP fluctuations. These are considered in the following

paragraphs.

5.5.1. Cable models

It is evident that the SICM-IP includes the electro-

tonic characterization of a PSP in terms of an infinite

cable model excited by brief current sources. This

characterization is similar to that pioneered by Rall

(Jack et al., 1975; Rall, 1989; Holmes and Rall, 1992).

5.5.2. Standard deviation of the PSP and comparison

with its average (Rudomın et al., 1975; Jack et al., 1981)

This method considers the fluctuations of the entire

PSP waveform to classify a PSP as a pure amplitude-

fluctuating single-component response or not. A similar

analysis, based on the autocovariances of a PSP, can

match or exceed this classification (Section 4.1). How-

ever, finer classifications or the analysis of combined-

fluctuation and/or multiple-component PSPs require a

model like the SICM.

5.5.3. Quantal or non-quantal amplitude-distribution

analysis of the PSP amplitude (Del Castillo and Katz,

1954; Auger and Marty, 2000; Clements and Silver, 2000;

Edwards et al., 1976; Stricker et al., 1994; Stricker and

Redman, 1994)

These approaches take into account only a limited

region of the postsynaptic response*/usually the PSPpeak*/and are aimed to disclose the value and separa-

tion of the amplitude steps exhibited by the synaptic

response. The SICM makes no assumptions on the

probability distribution underlying the PSP fluctuations.

Thus, the SICM-IP can be regarded as a different but

complementary approach to the amplitude distribution

analysis.

5.5.4. Amplitude-distribution analysis and partial

averaging of PSP records (Jack et al., 1981)

This approach separates the noisy records of the PSP

after their peak amplitude and average independently

each subset of records. The comparisons of the partial

averages can indicate the presence or absence of

different-shape contributions to the PSP and the pre-sence of components with different electrotonic origins.

The SICM-IP separates the PSP components (i) based

not only on their shapes but also in their latencies and

amplitude fluctuations; and (ii) operates without any

assumption on the distribution of the PSP fluctuations.

5.5.5. Principal components analysis of PSPs (Astrelin et

al., 1998)

This method is aimed to estimate the shape of theamplitude-fluctuating components in a PSP and their

amplitude distributions. However, contrary to the

SICM-IP, the possibility of latency and/or shape fluc-

tuations is not contemplated.

5.6. Effects of background noise

Noise degrades the accuracy of any method and theSICM-IP is not an exception. Our results show that the

SICM parameter estimations become less accurate with

increasing levels of noise. Further work should compare

the noise sensitivity of the SICM-IP with that of other

methods. The comparison could be done by assessing

the minimal signal variability, relative to noise magni-

tude, which allows a proper analysis by each method. A

signal-variability to noise-magnitude ratio may bedefined, for example, as the quotient between the

maximal value of the standard deviation of the PSP

and the noise standard deviation. Thus, for the two-

component PSPs analyzed in Section 4.3.3, the proper

operation of the SICM-IP required the PSP standard

deviation to be about ten times the noise standard

deviation. Signal-to-noise StdDev ratios equal to or

larger than 10 are not common for primary afferentEPSPs recorded from motoneurons in vivo (Clammann

et al., 1991); however, they might be attained with whole

cell clamp or in in-vitro recordings.

5.7. Closing remarks

It is clear that the SICM-IP possess limitations and

requires further improvements. Our main interest so farhas been to show its feasibility. Future improvements of

the model and the identification procedure are expected,

as well as applications to real PSPs. We hope also that

O. Ruiz, P. Rudomın / Journal of Neuroscience Methods 124 (2003) 1�/2622

this work may prompt new developments in the analysis

of synaptic potentials in more comprehensive ways.

Acknowledgements

The authors are indebted to Jose Bargas (IFC,UNAM, Mexico) and Emilo Salinas for valuable com-

ments. We wish also thank the anonymous reviewers

whose comments allowed us to significantly improve this

text. Partly supported by grants NIH NS09196 (USA)

and CONACyT (Mexico) to P. Rudomın.

Appendix A: Analytical expressions of the derivatives of

the homogeneous infinite cable response to a Dirac delta

current pulse

Define:

Ksq(t; uk)‹

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir

c(t � bk)

s

Ke(t; uk)‹1

2ffiffiffip

p e�(rcx2k)=4(t�bk)�(g=c)(t�bk)

Kgc(t; uk)‹

�g

c�

rcx2k

4(t � bk)2

�

Then, the analytical expressions for V (t ,uk ) and its

derivatives are:

V (t; uk)�akKsq(t; uk)Ke(t; uk)

Va(t; uk)�Ksq(t; uk)Ke(t; uk)

Vb(t; uk)�ak

�r

2cKsq(t; uk)(t � bk)2

�Ksq(t; uk)Kgc(t; uk)

�Ke(t; uk)

Vg(t; uk)��ak

t � bk

cKsq(t; uk)Ke(t; uk)

Vaa(t; uk)�0

Vab(t; uk)��

r

2cKsq(t; uk)(t � bk)2

�Ksq(t; uk)Kgc(t; uk)

�Ke(t; uk)

Vag(t; uk)��t � bk

cKsq(t; uk)Ke(t; uk)

Vbb(t; uk)�a

��

r2

4c2Ksq(t; uk)3(t � bk)4

�r

cKsq(t; uk)(t � bk)3

�rcx2

k

2(t � bk)3Ksq(t; uk)

�r

cKsq(t; uk)(t � bk)2Kgc(t; uk)

�Ksq(t; uk)Kgc(t; uk)2

�Ke(t; uk)

Vbg(t; uk)�a

�1

cKsq(t; uk)�

r

2c2Ksq(t; uk)(t � b)

�t � bk

cKsq(t; uk)Kgc(t; uk)

�Ke(t; uk)

Vgg(t; uk)�a(t � bk)2

c2Ksq(t; uk)Ke(t; uk)

Appendix B: Criteria to reject invalid SICM solutions

First criterion. Reject those multiple-component solu-

tions for which ERRAverage (Eq. (10)) and/or ERRACOVs

exceed those of a single-component solution. ERRACOVs

can be defined as:

ERRACOVs(u;J;COVV (h; t))

�1000Xl

j�1

½½COVV (hj; t; u;J)�COVV (hj; t)½½ (B1)

Second criterion. Reject those solutions where ampli-tude, latency or PMC fluctuations exceed the represen-

tational capabilities of the SICM: CV(ak )�/0.8,

StdDev(bk )/(Xktm)�/0.015, or CV(g )�/0.24 (Sections

4.1.1, 4.1.2 and 4.1.3).

Third criterion. Several wrong solutions produced

PSP components with an inflection or bump in the rising

phase of the reconstructed component average. This

cannot correspond to a single component as understoodby the SICM (a single transient in a passive structure).

Hence, solutions with bimodal components or with

components displaying an inflection in their rising phase

must be rejected.

Fourth criterion. Several wrong solutions produced

biphasic components (i.e. an initial positive or negative

phase, followed by an opposite-sign phase). These

responses cannot be produced in a linear, passivestructure, by either positive or negative current pulses.

Therefore, solutions with biphasic components must be

rejected.

O. Ruiz, P. Rudomın / Journal of Neuroscience Methods 124 (2003) 1�/26 23

Fifth criterion. The SICM-IP may introduce small,

spurious, components to improve the general fitting of

the data. For the current implementation, we defined a

‘smallness index’ as the area of each estimated compo-

nent divided by area of the largest estimated component.

For the cases analyzed in the present work, ‘smallness

index’ of 0.06 or less were associated with invalid

solutions.Sixth criterion. In the SICM no two current sources,

operating with the same delay, can be located at the

same distance from the recording site (otherwise, the

two current sources would be accounting for the same

component). A means to quantify the similarity (in time

and space, but not amplitude) of two components may

be the calculation of:

where V (t; uj�;Jj�)normis the reconstructed average of the

j-th PSP component, normalized to make its peak

amplitude equal to 1; and

V (t; uj�;Jj�)normV (t; uk�;Jk�)norm is the internal product

of the vectors V (t; uj�;Jj�)normand V (t; uk�;Jk�)norm:/

For the tests performed in the present article,

similarity indexes calculated with Eq. (B2) equal to

0.96 or larger were associated with invalid solutions.

Appendix C: Calculation of the SICM parameter

covariance matrix

Let COVV(h, t)* be the autocovariances of the PSP

(obtained from experimental data) arranged as the

successive rows of the matrix:

COVV (h; t)�

�

COVV (h1; t1)� COVV (h1; t2)� � � � COVV (h1; tl)�COVV (h2; t1)� COVV (h2; t2)� � � � COVV (h2; tl)�n n :::COVV (hm; t1)� COVV (hm; t2)� COVV (hm; tl)�

2664

3775 (C1)

Then, Eq. (5) must satisfy:

VT

h;uJVt;u�COVV (h; t)� (C2)

For a u specified, Eq. (C2) may be regarded as an

overdetermined set of linear equations with unknown

VT

h;uJ and with a matrix of (known) coefficients given

by:

Vt;u�

Va1(t1; u1) Va1

(t2; u1) � � � Va1(tl ; u1)

Vb1(t1; u1) Vb1

(t2; u1) � � � Vb1(tl ; u1)

n n nVas

(t1; us) Vas(t2; us) � � � Vas

(tl ; us)

Vbs(t1; us) Vbs

(t2; us) � � � Vbs(tl ; us)Xs

k�1

Vg(t1; uk)Xs

k�1

Vg(t2; uk) � � �Xs

k�1

Vg(tl ; uk)

266666666664

377777777775

One may thus estimate the unknown VT

h;uJ as the

Least Squares solution to the system in Eq. (C2), or by

means of:

VT

h;uJ�COVV (h; t)�(Vt;u)�

where (Vt;u)� is the Moore-Penrose pseudoinverse of

matrix Vt;u (Campbell and Meyer, 1979). Repeating theprocedure, one may solve the last equation for the

unknown J or, in terms of the pseudoinverses:

J�(VT

h;u)�(COVV (h; t)�(VT

h;u)�) (C3)

The above estimation of J does not incorporate any

constraint on the symmetry or the positive definiteness

of J (which every covariance matrix must satisfy).

Actually, when the number of points per ACOV was

larger than the number of curves (i.e. when COVV(h,t)*,in Eq. (C1), is a rectangular matrix because hƒ/t) the

first estimation of J (Eq. (C3)) never produced a

positive definite matrix. The requisite for positive

definiteness of J may be introduced into the algorithm

by means of the polar decomposition of matrices. Polar

decomposition expands a matrix into the product of two

matrices, J�J�J2; the first of which is positive definite

and given by:

J��

ffiffiffiffiffiffiffiffiffiJ

TJ

q(C4)

(Strang, 1986). The second matrix, J2, associated with a

‘rotation’ of the first, is not considered in the algorithm.

Note that discarding J2 does not represent a loss ofinformation. On the contrary, Eq. (C4) is a means to

introduce the requisite of positive definiteness in the

estimation of J. Finally, when the number of points

jV (t; uj�;Jj�)normV (t; uk�;Jk�)normj

(V (t; uj�;Jj�)normV (t; uj�;Jj�)norm

� V (t; uk�;Jk�)normV (t; uk�;Jk�)norm)=2(B2)

O. Ruiz, P. Rudomın / Journal of Neuroscience Methods 124 (2003) 1�/2624

making the ACOVs is equal to the number of curves

(t�/h), the matrix of PSP ACOVs is itself positive

definite (Eq. (C1)) and the first estimation of J (Eq.

(C3)) usually produced also a positive definite matrix. Inthis case, the J2 becomes equal to or almost equal to the

identity matrix, and discarding it has little or no effect

on the estimation of J.

Appendix D: Assessing the goodness-of-fit of a SICM

solution

The appropriateness of a model fitting can be assessed

by evaluating the probability that the error between the

model outcome and the data occurred by chance,assuming that the data were obtained from the model

(x2-test; Press et al., 1992). The error between the SICM

outcome and the average and ACOVs of a PSP is

defined as:

x2�Xl

i�1

(V (ti; u;J) � V (ti)�)2

s2(V (ti)�)�

Xm

j�1

Xl

i�1

(COVV (hj; ti; u;J) � COVV (hj; ti)�)2

s2(COVV (hj; ti)�)(D1)

where the numerators represent the squared differences

between the SICM average and the PSP average, and

between the SICM ACOVs and the ACOVs of the PSP.

The denominator in the first sum is the variance of the

PSP-average estimation, taken here to be equal to the

variance of the noise: s2/(V (ti)�)/:/VARN . The denomi-

nator of the second sum is the variance of the ACOV

estimation, mainly resulting from the effects of noise:s2(COVV (hj ,ti)*)):/s2(COVN(ti�/hj)). This last term

can be calculated as described by Bendat and Piersol

(1971) (Eq. 6.76), as:

s2(COVN(ti�hj)):2

BT(COVN(0)2�COVN(ti�hj)

2)

where B is the filter bandwidth (for a�/0.81 in Eq. (12),

B�/665 Hz), T is the duration of each record, and

COVN(0) and COVN(ti�/hj) represent the autocovar-

iance of the noise evaluated for times 0 and ti�/hj ,respectively.

For a fitting error defined as in Eq. (D1), the

probability that a certain x2 is observed follows a chi-

squared distribution with f degrees of freedom (Press et

al., 1992), where:

f �number of points in the PSP average

�(number of PSP ACOVs

�number of points per ACOV)

�number of SICM fitted parameters

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