Practical Nonlinear Analysis Using Pseudo-Random Impulse Train Inputs : A Case Study of the Crayfish Brain Responses(Physiology)

7 Pages·1991·4.1 MB·English

by  Yoshio Watanabe

ZOOLOGICAL SCIENCE 8: 847-853 (1991) © IWl Zoological Society ofJapan
Practical Nonlinear Analysis Using Pseudo-Random
Impulse Train Inputs: A Case Study of
the Crayfish Brain Responses
Yoshio Watanabe
Division ofFunctional Neurobiology, Centerfor Neurobiology and
Molecular Immunology, School ofMedicine,
Chiba University, Chiba 280, Japan
—
ABSTRACT This paper outlines an extension ofthe Krausz method for nonlinear system identifica-
tion inwhichthe Poissonrandomimpulsetrainsareusedasthetestinputsandthesystemisdescribedby
the Volterra functional expansion. The extension concerns improvement ofaccuracy in the nonlinear
system identification that the test inputs are not perfectly random, in other words, their spectra are
non-white. A formula forthe improvement ofthesystem kernels in the functional expansion aregiven
undersuch input condition. Thisapproach isillustratedandevaluatedwith analysesofthe responsesin
the crayfish brain activated by the pseudo-random impulse train stimuli. When the input spectral
property was apparently deviated from white, the accuracy ofthe system identification was efficiently
improved through the present procedure. Even if the input impulse trains are not perfectly random,
theycan be usedasthe test inputsinordertoidentifynonlinearsystemsthrough thesimpleextensionof
the Krusz method.
be the white noise in the Krausz method, spon-
INTRODUCTION
taneous spike trains which are observed in some
White noise analysis has recently been a power- central nervous system and not so perfectly ran-
ful tool for investigation ofinformation processing dom could be employed as the test inputs for
in living neuron networks. In the Lee-Schetzen nonlinear analyses of neuron networks in it.
method for identification of non-linear systems, An extension of the Wiener filter for linear
the Wiener kernels for the functional expansion system identification to nonlinear system analysis
can be evaluated by the multiple cross correlations had been investigated first by Tick [6]. This
between "the white and Gaussian noise input" and approach brings about the same result as the
the resultant system output [1]. In another extension of the Lee-Schetzen method using the
approach of nonlinear system identification, Gaussian white noise inputs to the use of the
Krausz has derived a similar method in which the Gaussian non-white inputs in identification ofnon-
Volterra kernels are obtained from the multiple linear systems. Several authors have described
^r^s-correlations between "the Poisson random almost same approaches as the Tick's one [7-10].
impulse train input" and the resultant system out- In thisstudy, an extension ofthe Krauszmethod to
put, because the ideal Poisson process is the white the use of non-white inputs was tried with the
noise [2]. The Poisson random impulse train pseudo-random impulse train inputs in the analysis
inputs have been practically used in the linear and of the nerve spike train transmissions in the
nonlinear analyses of the nerve signal transmis- crayfish brain.
sions in the crayfish brain [3-5]. Ifwe are released
from the strict condition that the test inputs must MATERIAL AND METHODS
Accepted June 20. 1991 The experimental date used in the present com-
Received Februarv 19. 1991 putations were maink gathered m the brain (cere-
848 Y. Watanabe
bral ganglion) and partly in the abdominal gang- in the brackest [2, 5].
lion of the crayfish (Procambarus clarkii). The The extension ofthe Wiener filter theory to the
stimuli to the brain were applied to the nerve nonlinearsystem identification, ortheextensionof
bundle in the optic peduncle. At the same time, the Lee-Schetzen method to the use of the non-
the nerve spike trains of the brain outputs were white Gaussian input, has been given in the fre-
recorded extracellularly from the large descending quency domain by several authors [6-10].
axons in the circumesophageal connective and Although the test input must be Gaussian and
intracellular^ in the brain [4, 5, 11]. The input- white in the Lee-Schetzen method, only Gaussian-
output data of the abdominal ganglion cells were ity ofit is necessary in the extended method. The
obtained by stimulation of axon bundles in the bases of this extension are that the odd-order
interganglionic connective and extracellular re- momentsofthe Gaussianprocessalwaysvanish [9]
cording from the axons in the ganglionic roots. and that the even-order moments ofit are decom-
The nerve cells in which the experimental data posedinto the product ofitssecond ordermoment
were obtained were named by means of alpha- [12].
numeric specification (Table 1). The alphabet in The higher order moments of the Poisson pro-
each cell name indicates whether the cell is in the cess can be also decomposed into the product of
brain (B-) orin the abdominal ganglion (A-). The the second ordermoment ofit [2]. Therefore, it is
stimuliwere generatedwith arandompulse gener- considered that, when the spectrum ofthe input is
ator [3], then the pulse trains which had various non-white and also notsofarfromwhite, thesame
degree of "non-white" spectral properties were procedure as the Gaussian input case can be used
picked up for the present purpose (Fig. 1 and in order to improve the accuracy in nonlinear
Table 1). system identification by the Krausz method, as
approximation. In this study, the improvement of
Theory the system kernels was performed in the time
In the Krausz method that the Poisson random domain by the reasons mentioned in the discus-
impulsetraininputisusedasthetestinputx(t), the sion. The time domain version ofthe extension is
system output y(t) is given by given as follows
y(t)=G +Gi+Gr (1) °n) hn(zu ..., Zn) (4)
using the functionals ri). Tn)dzi...dzn,
G =h where 4>xy is the rc-th order cross-correlation func-
tion between the input and the output, hn is the
hl{z)x{t-z)dz (2) improved n-th order system kernel and &, the
auto-correlation function of the input.
h2(Tu z2)x(t-zl)
Computation
•X[t— Z2)dzXdz2, Signal intensitywas set to be 1 in a bin (10msec
where hQ is the mean value ofy(t), and hi and h2 width) where the nerve impulse (or the stimulus
are the first and second order system kernels, pulse) was present and, otherwise, to be 0. First,
respectively. The n-th order system kernel can be the unimproved Volterra kernels up to the third
obtained by the multiple cross-correlation order were calculated with the equation-(3) and
the improved kernels of the same orders were
hn(zu ..., zn) -^rE\[y(t)-J]Gk] (3) derived with the equation-(4). The computation
with the original Krausz method was described
x(t-zi)...x(t-zn)\ , n#r2=¥...#r„, detailedlyin the previouspaper[5]. Theequation-
where Ais the mean rate ofthe input impulses and (4) can be transformed by discretization into a
E indicates the expectation of the content system of linear algebraic equations, whose coef-
| |
Neural Analysis bv Pseudo-Random Stimuli 849
ficients construct the //-fold block Toeplitz matrix. were calculated. E denotes time average.
j j
Several methods were tried in order to solve the Then improvement ofthe model output wasevalu-
systems of the linear equations corresponding to ated by percentage of (e-e) in £ for each input-
the present input-output data. These systems of output pair. The power spectra of the test inputs
the linear equations can be solved by use of the were calculated from the normalized auto-
regular structure of the nested block Toeplitz correlation functions of them through the Fourier
matrix and also any of the usual iterative methods transform. And the deviation of each input spec-
[13, 14]. trum was evaluated also by means of the MSE
Next, the unimproved model output z(f) and the between it and the spectrum of the white noise of
improved one z{t) were computed respectively unit power (Fig. 1).
with the euqations-(l) and -(2) from each input- Computations in this work had been begun with
output pair used in the above kernel computation. a mini-computor HITAC 10-11 and were carried
Then the both model outputs were compared with out on several personal computors (Hitachi B-16/
the output oi the same pair. i.e. the brain output B-32 series and others).
impulse train (Fig.3). Last of all. the MSE (mean
square error) between the unimproved model out-
RESULTS
put z(t) and the brain output y(t)
The input power spectra of the input-output
e=E\(y(t)-z(t)Y\ (6)
data used in this study are not flat, in other words,
and the MSE between the improved model output non-white at a glance; but their deviations arc not
and the same brain output as used above so large as over ±3 db (Fig. 1). Accordingly, it
seems that these input-output data are suitable to
e=E\{y(t)-z(t))' (7) improve the system kernels with the present exten-
-4 B03-Cell
20 30 4050
A50 Cell
20 30 4050 20 30 40 50
1 (J
frequency (Hz)
Fig. 1. Power spectra of the random pulse trams used as the test inputs tor identification <>( the impulse signal
transfer systems in the crayfish central nervous systems The broken line in each graph of the spectrum indicates
the unit power level of the white noise
850 Y. Watanabe
sion of the Krausz method, i.e. the equation-(4). 45
The MSEs of the deviations of the input spectra,
the MSEs of the abrrations of the model outputs
which are calculated with the system kernels up to 40
the second orderd and up to the third order and
the perentages of the model output improvement
are given in Table 1. The MSE of the third order ~
model was usually smaller than the MSE of the 15
second order one for each input-output pair in 3
a.
both unimproved and improved cases. And the 3
O
improvement percentage ofthe third order model 10
was mostly larger than the second order one and Oo5 o
had a tendency to increase depending upon aug- £
mentation of the deviation of the input power £
spectrum (Fig. 2).
The upper half of Fig. 3 illustrates the third
order model outputs before (a) and after (b)
kernelimprovementforthedataoftheB04-cell, in £
which relatively large improvement, 9.76%, is aberra0t.i5on of i1n0put spec1t.r5a (MSE2).0
obtained. Some large aberrations could be recog-
nized at several places in the unimproved model -15
output (Fig. 3a) and they decreased noticeableyin
the improvedone (Fig. 3b). The lowerhalfofFig.
3 shows the effect of improvement of the kernels
-20
up to the third order for the data of the B03-cell.
In this case, the MSEofthe third ordermodelwas Fig. 2. Relationship between the deviations (MSEs) of
extremely small, the percentage of improvement the test input spectra from the unit power level of
was also small, 2.56%, and the large aberrations the white noise and the improvements (%) of the
modeloutputsofthesecondorder(openmarks)and
could notbe discriminatedoverthewholerangeof
thethirdorder(filledmarks)fortheresponsesofthe
the model outputs before (Fig. 3c) and after (Fig. braincells(circles) andtheabdominalganglioncells
3d) improvement of the system kernels. (squares) (see Table 1).
Table 1. The MSEs of the deviations of the input power spectra, the MSEs of the aberrations in the
unimproved and improved model outputs of the second order and the third order and the
improvement percentages ofthe model outputs. Asterisks indicate the exceptional cases (see Fig.
4). Units of the MSEs are not comparable between the inputs and the model outputs.
MSEs of MSEs of 2nd order outputs improvement MSEs of 3rd order outputs improvement
Cell
names input of 2nd order of 3rd order
spectra unimproved improved outputs (%) unimproved improved outputs (%)
B03 0.49 19.87 19.79 0.40 9.00 8.77 2.65
B39 0.87 29.65 28.73 3.10 21.42 20.63 3.69
B38 1.01 30.42 29.81 2.01 24.70 23.40 5.26
A13 1.03 26.93 25.38 5.76 14.83 13.37 9.84
A50 1.04 18.14 17.15 5.46* 11.45 13.49 -17.81*
B40 1.19 25.19 24.13 4.20 15.71 13.63 13.24
B02 1.45 28.80 26.36 8.47* 38.73 21.67 44.05*
B04 1.50 23.87 22.62 5.24 11.67 10.53 9.76
Neural Analysis bv Pseudo-Random Stimuli 851
U
I J
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Fig. 3. Comparisons between the unimproved model output (a) and the improved one (b) for the response of the
B04-cell and between the unimproved (c) and improved model outputsforthe response ofthe B()3-cell. In each
record, lower solid trace represents the stimulus pulse train, the upper dotted trace represents the brain output
impulse train and the upper solid trace shows the model outputs calculated with the unimproved (a and c) and
improved (bandd) kernelsuptothethirdorder. Horizontalscale: 200msec. Verticalscale: Unitimpulse height
(inapplicable to stimuli).
The two exceptional cases were observed in the was difficult to predict the behaviours of the third
present study (asterisks in Table 1). For the B02- order model outputs from the features of the
cell. the MSE ofthe unimprovedthirdordermodel second order models in these exceptional cases.
output was largerthan the MSEofthe unimproved Generally speaking, it can be concluded the
second order one, but the MSE of the third order larger the deviation of the input power spectrum,
model output drastically decreased after improve- the more the model output of any order is im-
ment of the third order system kernel (Table 1, proved (Fig. 2).
and Fig. 4a. 4b). It maybe noticed that the data of
the B02-cell has been obtained in the brain im-
DISCUSSION
mediatelyafterapplication and removal ofpicroto-
xin (10 \M). Although the equation for the extension of the
The exception of another type occurred in an Wiener filter to the nonlinear case can be derived
abdominal ganglion cell (A50): the improvement in the time domain, most authors have given the
of the third order kernel caused remarkable de- equation in the frequency domain [7-9). They
terioration of the model output (Table 1, and Fig. should naturally consider that it can be solved
4c. 4d). In this case, the MSE ofthe second order practically by use of the last Fourier transform
model output was very small beforehand with However, the latest author has pointed out that
construction of the third order one (Table 1). It the computationsofthe high-orderspectral density
—
852 Y. Watanabe
y^JkjnjU
— "t^'^™
i_L j i i mi i y
ti :
.L.LlL'piJ^jL^^
J 1 1 1111 1 L
UJJ
m
*$&*£
^ir
u
U
J I III I i_L J I I I I I
SIM
El IJF^iW
i I I III II U
Fig. 4. The exceptional casesthat the drasticchangesoccurredin the modeloutputs afterimprovementofthethird
ordersystem kernels, shownbyasterisksinTable 1. The modeloutputsbefore (a) andafter(b)improvementof
thesystem kernelsuptothethirdorderfortheresponseofthe B02-cell. Themodeloutputsbefore (c) andafter
(d) improvement ofthe system kernels up to the third orderforthe response ofthe A50-cell. Horizontal scale:
200msec. Vertical scale: Unit impulse height.
functions are usually very expensive and time- In spite of the non-Gaussian property of the
consuming [10]. In addition, moving average pseudo-random impulse train inputs used in this
techniques as the data window and the spectral study, the present results indicate that the im-
window are indispensable for stable computation provement of the system kernels with the equa-
of the frequency domain version of the equation- tion-^) can reduce efficiently the identification
(4) by use of the discrete data time series and the errorsresultedfromthenon-whitespectralproper-
discrete Fouriertransform [8]. Consequently, very ty of the present test inputs. And the kernel
high sampling rate must be employed in order to estimation admits of further improvement when
reproduce sharply each impulse in the model out- effects ofthe odd-order moments ofthe test input
put after application ofsuch moving average tech- can be taken into account in computation of the
niques. Therefore, the processing of very large system kernels, because the odd-ordermomentsof
number of the data points is not avoidable in the pseudo-random impulse train do not vanish
identification of the impulse output nonlinear sys- usually. But, the effects of the corrections of the
tems in the frequency domain. However the second order system kernels by the third order
Krausz method can be used without any moving moments of the inputs were very small in the
average techniques in orderto identifythe impulse present computation; for instance, about 0.2% for
input-output nonlinear systems [5]. Thus the ex- the input-output pair ofthe B02-cell. In any case,
tension ofthe Krausz methodwas tried in the time it is considered that spontaneously evoking nerve
domain. spike trains in some living neuron network can be
Neural Analysis bv Pseudo-Random Stimuli 853
used as the test inputs for nonlinear analyses of 563-567.
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extension of the Krausz method, even it" they are identification. Int. J. Control. 20: 557-592.
8 French. A. S. (1976) Practical nonlinear system
not so highly random. analysisbyWienerkernel estimation in the frequen-
cy domain. Biol. Cybern.. 24: 111-119.
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