Synthetic whole-nerve action potentials for the cat E. de Boer PhysicsLaboratory,Ear, Noseand Throat Clinic (KNO), Wilhelmina Hospital,Amsterdam,1',[etherlands (Received 30 June 1975) In responseto a tone burst, a pattern of activity is set up in the set of auditory nerve fibers.A simplified model is describedin this paper that can representthis pattern both with respectto the number of nerve fibers involved and the detailed time courseof the firings. The parametersof the model concerning frequencyselectivityand latencyare adaptedfrom measurements with the so-calledreversecorrelation technique.Further propertiesincorporatedin the model are saturationfor excitationover 40 dB above thresholdand high sensitivity.When specificassumptionsare made regardingthe contributionof the firing of each nerve fiber to the whole-nervepotential,the waveformof the action potential (AP) can be synthesizedfrom the model.The theory givesa quantitativeaccountof the dependence of the AP latency upon stimulusintensity.Two main contributingfactorsare frequencyselectivityand latencydistributionof nerve fibers. ExperimentalAP amplitudesshow a more complexcourseas a function of stimulusintensity than the theory predicts.Severalpossibleimprovementsof the method are discussed.SinceAP recordings are often usedfor diagnosticpurposes,a separatesectionof the paper is dedicatedto the connection betweenthe propertiesof auditory nerve fibersand of the whole-nerveAP in abnormalears. SubjectClassification:65.42, 65.35.
INTRODUCTION
Many studies have been dedicated to the fundamental
properties of responses of auditory nerve fibers (e.g., Kiang et al.,
1965; Pfeiffer and Kim, 1972; Rose et al.,
1971; Evans, 1972). The results have been used to obtain general overviews and these have served to construct theories about psychophysical phenomena in man
(Sicbert, 1968; Goldstein, 1973; Colburn, 1973; Duif.huis, 1973). The advent of electrocochleography in man (e.g., Aran et al., 1971; Eggermontet al., 1974a) has revived interest in the whole-nerve action potential. The question may now be asked to what extent the properties of the whole-nerve action potential can be described
in terms
of the behavior
of the ensemble
of
primary auditory neurons (cf. Teas et al., 1962; Odzamar, 1973; Biondi et al., 1974). The present paper describes an attempt at the computation of synthetic action potentials based on a simplifi•ed representation of the dynamics of auditory nerve fibers.
their firings. The theory contains specific assumptions about the way the instantaneous probability of firing of a nerve fiber depends upon the waveform of the excita-
tion. The rate function, characteristic (b), is the central link here. Furthermore, assumptions are made about the sensitivities of the auditory nerve fibers. The final step in the computation leads from the nervous activity to the whole-nerve action potential in the form as it can be picked up by a gross electrode in the vicinity of the auditory nerve. In this step the actual waveshape
of the unit function, characteristic (c), is needed. The steps in the computations are outlined in Fig. 1.
The theory presented here differs in several aspects from earlier attempts to explain the properties of wholenerve action potentials. Those theories have mostly been of a phenomenological nature, and sought to explain
the whole-nerve potential in terms of the (physiologically measurable) contributions from various sets of primary auditory neurons.
The three
basic
characteristics
concern
(a) the frequency selectivity evident from the responses of auditory nerve fibers,
(b) the rate function, i.e.,
the way the firing probabil-
ity of a nerve fiber depends on the intensity of the stimulus,
(c) the unit function, i.e.,
the signal waveform that
every firing of one particular nerve fiber contributes to the whole-nerve action potential.
Experiments for this type
of research have been reported, e.g., by Teas et al.
(1962), Eggermontand Odenthal(1974), Eggermontet al. (1974b), and Elberling (1973, 1974). Our theory starts directly from properties of auditory nerve fibers and is more comparable to theories as reported by
'•klzamar(1973)andBiondiet al. (1974). A mostexplicit use is made of what we know about the transformation and encoding processes in the cochlea for the
most general type of acoustical stimulus (de Boer, 1973). From this it is evident that our theory is synthetic in character.
Characteristic (a) is the most important one. The theory makes use of a fairly complete description of the pattern of excitation in the cochlea, computed for a tone burst as stimulus. The extent of the region of excitation is determined by frequency selectivity, the timing is a function of the latencies shown by fibers of different characteristic
frequencies.
The number of active nerve
fibers andthe timing of their firings are importan•determinants of the whole-nerve action potential. Auditory nerve fibers show a stochastic character in
1030
J. Acoust.Soc.Am., Vol. 58, No. 5, November1975
The theoretical developments should be confronted with results of physiological experiments. To this aim we made recordings of the action potentials produced by the auditory nerve in cats. The electrodes were implanted on or near the round window membrane and the experiments
were carried
out under free-field
conditi-
tions. Despite the variability implicit in this method, the results are useful for judging the quality of prediction of the theory.
Copyright¸ 1975 by the Acoustical Societyof America
1030
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 132.206.205.6 On: Wed, 17 Dec 2014 00:55:53
1031
de Boer: Synthetic whole-nerve action potentials
NEURAL UNIT r
:rUNC•O,I
"•
r
•
FIRING
1031
L
• PROBABILITY •
,r-
'i•. ........
!:• EXCITATORY ENVELOPE [,. ........ ,• ., :ACTIVITY: SIONAL •: •:........
..... .............. F,
e',l,I
USED
"--} .....
G,
¾,l,I
CONCEPTS
P,l,• G2 OBSERVED ACTION POTE
I P2(') '••
NTIA L
,
I
ACOUSTICAL
I
STIMULUS
x(t•
i
I
I
G#
•
. (t)
NECESSARY PARAMETERS
FIG. 1. Model and flow diagram of the computations. The "neural unit" represents transduction in a primary auditory neuron. The input is the acoustical stimulus, the output the probability of firing pt(t). The secondstage of the computationinvolves the summationof neural activity, the third the convolutionwith the unit function. In the top row of this figure are listed some of the conceptsused in this work; these are explained in Sec. I of the text. The lower part of the figure indicates the parameters that must be inserted;
these are described in Sec. III.
I. DYNAMICS OF NEURAL UNITS; BASIC
We will assume this principle to hold true for toneburst input signals as well.
ASSUMPTIONS
The frequency selectivity
exhibited by auditory nerve
tion of continuous stimulation with white noise by the re-
In the study reported in this paper we replace the set of auditory nerve fibers by a limited set of •eural u•its. The properties of these neural units are deduced from experimental observations and represented in a stylized form. The left-hand part of Fig. 1 shows in diagrammatical form how we conceive of the properties of these
verse correlation method(de Boer, 1973). The im-
neural units.
pulse response recovered with this method is called the reverse correlation function, abbreviated revcor function. The freq.uency response associated with a revcor function agrees very well with the pure-tone tuning
input to all neural units. For each neural unit the stimulus is first filtered by a linear filter, shown by the leftmost column in Fig. 1. The impulse response of
fibers
has several
features
characteristic
of a linear
system (cf. Evans, 1974a; Evans and Wilson, 1973). The impulse response of the filter associated with a particular nerve fiber can be measured under the condi-
The acoustical stimulus x(t) is common
the filter F• associated with neural unit number i is
curve for the same fiber (de Boer, 1969, 1973), and the
designatedby hi(t). The set of standardfunctionshe(t)--
waveshape of the function shows approximately the same number of oscillations as the PST histogram for clicks. Furthermore, a revcor function indicates that a certain minimum time interval should elapse between stimulation and response, and this interval corresponds well with the latency evident from the click PST histogram. These agreements suggest that for each fiber the acoustical stimulus filtered by the appropriate filter is a good predictor for the firing probability. The revcor function of the fiber should be the impulse response of the
with i from i to N--is
filter. This concepthas been proven to be correct (de Jongh, 1972; de Boer, 1973) for noise input signals.
modeled after experimentally
ob-
tained revcor functions of auditory nerve fibers (de Boer, 1973; de Boer and de Jongh, to be published). The parameters of these functions, notably those concerning frequency selectivity and latency, are made to depend upon the resonance frequency f• in the same manner as the parameters of measured revcor functions depend on the characteristic frequencies of the fibers
tested. In their turn, the resonance frequencies fi are distributed uniformly over the (logarithmic) frequency scale. The output of the filtering stage, the filtered stimulus, can be found by the convolution integral.
For
J. Acoust Soc. Am., Vol. 58, No. 5, November 1975
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 132.206.205.6 On: Wed, 17 Dec 2014 00:55:53
1032
de Boer: Synthetic whole-nerve action potentials
1032
creases from zero to a constant asymptotic value
the ith neural unit the outputsignal yi (t) is
(saturation).
x(t-
(h) To simplify the representation of the case in which the stimulus is repeatedly presented with random phase •O,a special assumption is invoked: within
This signal yi(t) is called excitatory signal. The second stage of a neural unit consists of a proba-
each cycle of oscillation with a (nearly) constant
bilistic pulsegeneratordrivenby the signalyi(t). We assumethat the probability of firing p•(t) is a unique
amplitude, the firing probability is a rectified
function R(z) of the quantity z, which is proportional to
result, the average firing probability, averaged over all phase values •, becomes solely a func-
yi(t):
p•(t)=R(z),
(2)
linear transform
of the excitatory signal.
As a
tionof the envelope e•(t)of y•(t). (i) The rate functiondescribedunder assumption(e)
with
also describes the dependence of the average firing probability on the envelope value of the excitatory
z = k• y• (t) .
signal.
The function R(z) should show two main characteristics: rectification and saturation.
For negative values of the
excitatory signal the firing probability is assumedto be zero. For high positive values of the excitatory signal the probability tends to saturate so that the dynamic range of excitation over which the probability goes from barely discernible values (e.g., 0.05) to the onset of saturation is not larger than 40 dB, just as is true for
(j) Each neural unit has its sensitivity adjusted in such
a way that the thresholdof response(definedas the point at which the firing probability is 0.1) lies at the absolute threshold of audibility at the pertinent resonance frequency f•.
(k) Each firing of a neural unit gives a contribution to
in Eq. 2 are chosen in such a way that for each neural
the whole-nerve action potential A(t) with the same waveform, the unitfunction u(t). The waveform of u(t) is biphasic, with an average of zero. In the
unit the threshold for neural response, e.g., p•(t)= O.1, correspondsto an input level of x(t) equalto the thresh-
most general case the contributions from the neural units are not equal; the contribution from the
old of audibility at the resonance frequency f• for that
ith neural unit is expressedby c• ßu(t).
auditory nerve fibers (Kiang, 1968). The constantsk•
particular neural unit. This is a restriction that has been built in for reasons of simplicity.
(1) In the larger part of the computations all coefficients c• are taken as equal.
The assumptions on which our computationsare based are listed below; explanatory remarks and justification
(m) Adaptation is represented by a linear filtering process applied to the neural activity function
are presented later.
(i.e., the weightedsum of all firing probabilities).
(a) The set of primary auditorynerve fibers is replaced by a limited set of N "neural units," each of which has properties adapted to the average of the subset of nerve fibers which it represents.
(b) The frequency selectivity exhibited by the responses of auditory nerve fibers is represented by a process of linear filtering in the neural units. Each
neural
unit contains
a linear
filter
as its first
A short discussion on these assumptions follows. The first three assumptions imply a linear action of the frequency-selective mechanism. Many data on the detailed relation
between
the filtered
stimulus
waveform
are
in
agreement with this notion. However, many manifesta-
tions of cochlear nonlinearity are knownas well (e.g., Goldstein and Kiang, 1968; Sachs and Kiang, 1968;
Goblick and Pfeiffer, 1969). In the present context we
element. The properties of these filters are described by the set of impulse response functions
are most interested in the initial buildup of the neural
h•(t) (i = 1, 2, ..., N). Thesefunctionsare called standardfunctions. The outputsignals y•(t) of the
equal to several cycles.
linearities of the "essential" type (Goldstein, 1967) ex-
filters form the set of excitatory signals.
ercise a substantial influence on this buildup.
(c) The impulse response is invariant to the type of stimulation, be it tonal, stochastic, or transient.
excitation
for a tone-burst
stimulus
with
a rise
time
It is not known whether nonIn any
event, similarly rapid changes in the filtered signal occur during stimulation with noise, and the linear ap-
proximationis sufficientlyusefulin this case (deJongh,
(d) The neural units act independently of one another. (e) The second element of a neural unit is a proba-
bilistic pulse generator. The probabilityp•(t)that a pulse is generated by a neural unit is solely a
functionof the instantaneousvalue of z = k•. y•(t). The functional dependence is symbolized in the
form of the rate function R(z).
Compare Eq. 2.
(f) The rate function is the same for all neural units.
(g) The rate function R(z) is zero for negative values of z.
For positive values of z the rate function in-
Another type of nonlinearity results from mechanical overloading (cf. Rhode, 1971; Rhode and Robles, 1974). For the genesis of the whole-nerve action potential this type of nonlinearity will probably not be significant, since it will affect only neural units that are in satura-
tion. These arguments give us confidence in trying out the theory in linear terms only--details about nonlinearity can later be filled in, if need be.
Assumption(e) has been shownto be valid for a stim-
J. Acoust. Soc. Am., Vol. 58, No. 5, November1975
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1033
de Boer:Syntheticwhole-nerve actionpotentials
1033
ulus like white noise; the probability of firing follows
ing frequencies. We see later how the effect of allowing
the fluctuationsof the filtered stimulus (de Jongh, 1972).
a more gradual distribution can be predicted.
The same holds true for a signal consisting of several
sinusoids(Bruggeet al., 1969). It doesnot matter very much whether the nerve fiber is driven into saturation, the details of the excitatory waveform still appear in the firing probability. There seems to be a fast adaptation process present that tends to normalize the sensi-
tivity to the value of maximal excitation. Since this (hypothetical) mechanism is very fast, the responseto clicks displays surprisingly little changes when the
stimulusintensity is raised to valuesthat causesaturation (cf. Kiang et al., 1965). The notion of an infinitely fast normalization process is implicit in our assumptions (e)-(i). This statement needs some extra explanation. One method to avoid contamination of physiological action potentials by co-
chlear microphonic(CM) potentials consists of present-
In relation with assumptions (k) and (1), it is known that nerve fibers with a high characteristic frequency
(CF) contributemore to the whole-nerveactionpotential than fibers with low CF (Teas et al., 1962; Elberling, 1974; Eggermont et al., 1974b). But there is no agreement between the reported studies as to manner and extent of this property. This is readily understood when we realize how many factors contribute: spectral composition of the stimulus, electrical geometry with respect to the electrode, species of animal tested, the in-
fluence of secondaryresponses(like N•.), etc. On the basis of the available data it is not possible to ascertain whether it is justified to use the same function u(t) throughout. In order to simplify our model to the ex-
treme, we use a uniform unit functionu(t), and, in addition, we initially chooseall coefficientsc• to be equal.
ing the stimulus repeatedly with slowly varying phase.
This is certainly an oversimplification, but it allows us
That is, the envelope e•(t) of the stimulus remains the
to retain a good overview of the relations between model
same during the repetitions, but the phase of the fine
parameters and the properties of synthetic action po-
structure is changingcontinuously. Whenthe onset of the stimulus is sufficiently gradual--and we deliberately
tentials.
set out to meet this criterion--the
Assumption (m) is covered in the next section.
response functions
,
y•(t) will all showthe sametendency. All thesefunctions can then be described in terms of an envelope
e•(t) anda fine structurejust as the stimulus:
II. OUTLINE SYNTHETIC
OF STEPS IN THE COMPUTATION ACTION POTENTIALS
OF
The steps in the computations can be followed from
x(t)= e=(t). cos(w0 t+$),
(3)
yt(t)= e•(t). cos[wt(t) ßt + •5].
Fig. 1. The starting point is the stimulus x(t).
We
must choosea sinusoldwith a slow rise (and decay), so that the representation of the filtered waveforms y•(t)
What is important here is that during the repetitions of
in terms of envelope and fine structure as indicated by
the stimulus eachYt(t) keeps the same envelopeand the same fine structure describedby (,)t(t). t, but varies
Eq. 3'is valid. This canbe realizedby makingthe
only in the phase •5. Note that t as it appears here is always referred to the onset of the appropriate stimulus
signal. Our assumption(e) says that each excitation functionyt(t) must be treated accordingto Eq. 2 in order to find the appropriate.firing probability pt(t). Consider nowthe values of yi(t) andPt(t) at oneparticular instant of time t during all these repetitions.
The values
of yi(t) vary accordingto a cos•)function. As a result of assumptions(e)-(i), the firing probabilityPt(t) also
variesas a function cos•),andthemaximalvaluee•(t) of the probability is then related to the envelopee•t(t) of the y (t) signalthroughthe same relation, Eq. 2. The averagefiring probabilitypt(t), averagedover all values of ½, nowbecomesa constantfractionof e}(t). By this expedient we have removed details about the fine struc-
ture of the signalsx(t), y,(t), andpt(t) andwe are left
onset sufficiently gradual (see Sec. IIi). The waveforms of the excitatory signals are computed
from the convolution integral, Eq. 1. The firing probabilities are then found by applying Eq. 2. In view of what has been said about using stimuli with random varying phase •, the effective probability is found from
e•(t)by applyingEq. 2 to the envelope e••(t) of the signal y•(t). In fact, a reduction factor--the average of cos• over a half-cycle--should be included, but this is left out.
Consider now the case where a stimulus is repeated many times with identical waveform and the resulting action potentials are averaged. Then the average ac-
tion potential A(t) is composedof contributionsdue to the firings of all neural units at all instants previous to t:
with the concept that the envelopes of these signals play the most important part.
N
In short, the average firing
•o
(4a)
probability •t (t) is directly proportional to the envelope e•(t), whichin its turn can be foundfrom Eq. 2 by sub-
stitutingkt e•(t)for z. Notethatthe samemethodcan-
(note: pt(t)= 0 for t c•,
h•(t)=0, with co•= 2•rf•.
for t0, R(z)=O,
forz-
INTRODUCTION
Many studies have been dedicated to the fundamental
properties of responses of auditory nerve fibers (e.g., Kiang et al.,
1965; Pfeiffer and Kim, 1972; Rose et al.,
1971; Evans, 1972). The results have been used to obtain general overviews and these have served to construct theories about psychophysical phenomena in man
(Sicbert, 1968; Goldstein, 1973; Colburn, 1973; Duif.huis, 1973). The advent of electrocochleography in man (e.g., Aran et al., 1971; Eggermontet al., 1974a) has revived interest in the whole-nerve action potential. The question may now be asked to what extent the properties of the whole-nerve action potential can be described
in terms
of the behavior
of the ensemble
of
primary auditory neurons (cf. Teas et al., 1962; Odzamar, 1973; Biondi et al., 1974). The present paper describes an attempt at the computation of synthetic action potentials based on a simplifi•ed representation of the dynamics of auditory nerve fibers.
their firings. The theory contains specific assumptions about the way the instantaneous probability of firing of a nerve fiber depends upon the waveform of the excita-
tion. The rate function, characteristic (b), is the central link here. Furthermore, assumptions are made about the sensitivities of the auditory nerve fibers. The final step in the computation leads from the nervous activity to the whole-nerve action potential in the form as it can be picked up by a gross electrode in the vicinity of the auditory nerve. In this step the actual waveshape
of the unit function, characteristic (c), is needed. The steps in the computations are outlined in Fig. 1.
The theory presented here differs in several aspects from earlier attempts to explain the properties of wholenerve action potentials. Those theories have mostly been of a phenomenological nature, and sought to explain
the whole-nerve potential in terms of the (physiologically measurable) contributions from various sets of primary auditory neurons.
The three
basic
characteristics
concern
(a) the frequency selectivity evident from the responses of auditory nerve fibers,
(b) the rate function, i.e.,
the way the firing probabil-
ity of a nerve fiber depends on the intensity of the stimulus,
(c) the unit function, i.e.,
the signal waveform that
every firing of one particular nerve fiber contributes to the whole-nerve action potential.
Experiments for this type
of research have been reported, e.g., by Teas et al.
(1962), Eggermontand Odenthal(1974), Eggermontet al. (1974b), and Elberling (1973, 1974). Our theory starts directly from properties of auditory nerve fibers and is more comparable to theories as reported by
'•klzamar(1973)andBiondiet al. (1974). A mostexplicit use is made of what we know about the transformation and encoding processes in the cochlea for the
most general type of acoustical stimulus (de Boer, 1973). From this it is evident that our theory is synthetic in character.
Characteristic (a) is the most important one. The theory makes use of a fairly complete description of the pattern of excitation in the cochlea, computed for a tone burst as stimulus. The extent of the region of excitation is determined by frequency selectivity, the timing is a function of the latencies shown by fibers of different characteristic
frequencies.
The number of active nerve
fibers andthe timing of their firings are importan•determinants of the whole-nerve action potential. Auditory nerve fibers show a stochastic character in
1030
J. Acoust.Soc.Am., Vol. 58, No. 5, November1975
The theoretical developments should be confronted with results of physiological experiments. To this aim we made recordings of the action potentials produced by the auditory nerve in cats. The electrodes were implanted on or near the round window membrane and the experiments
were carried
out under free-field
conditi-
tions. Despite the variability implicit in this method, the results are useful for judging the quality of prediction of the theory.
Copyright¸ 1975 by the Acoustical Societyof America
1030
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 132.206.205.6 On: Wed, 17 Dec 2014 00:55:53
1031
de Boer: Synthetic whole-nerve action potentials
NEURAL UNIT r
:rUNC•O,I
"•
r
•
FIRING
1031
L
• PROBABILITY •
,r-
'i•. ........
!:• EXCITATORY ENVELOPE [,. ........ ,• ., :ACTIVITY: SIONAL •: •:........
..... .............. F,
e',l,I
USED
"--} .....
G,
¾,l,I
CONCEPTS
P,l,• G2 OBSERVED ACTION POTE
I P2(') '••
NTIA L
,
I
ACOUSTICAL
I
STIMULUS
x(t•
i
I
I
G#
•
. (t)
NECESSARY PARAMETERS
FIG. 1. Model and flow diagram of the computations. The "neural unit" represents transduction in a primary auditory neuron. The input is the acoustical stimulus, the output the probability of firing pt(t). The secondstage of the computationinvolves the summationof neural activity, the third the convolutionwith the unit function. In the top row of this figure are listed some of the conceptsused in this work; these are explained in Sec. I of the text. The lower part of the figure indicates the parameters that must be inserted;
these are described in Sec. III.
I. DYNAMICS OF NEURAL UNITS; BASIC
We will assume this principle to hold true for toneburst input signals as well.
ASSUMPTIONS
The frequency selectivity
exhibited by auditory nerve
tion of continuous stimulation with white noise by the re-
In the study reported in this paper we replace the set of auditory nerve fibers by a limited set of •eural u•its. The properties of these neural units are deduced from experimental observations and represented in a stylized form. The left-hand part of Fig. 1 shows in diagrammatical form how we conceive of the properties of these
verse correlation method(de Boer, 1973). The im-
neural units.
pulse response recovered with this method is called the reverse correlation function, abbreviated revcor function. The freq.uency response associated with a revcor function agrees very well with the pure-tone tuning
input to all neural units. For each neural unit the stimulus is first filtered by a linear filter, shown by the leftmost column in Fig. 1. The impulse response of
fibers
has several
features
characteristic
of a linear
system (cf. Evans, 1974a; Evans and Wilson, 1973). The impulse response of the filter associated with a particular nerve fiber can be measured under the condi-
The acoustical stimulus x(t) is common
the filter F• associated with neural unit number i is
curve for the same fiber (de Boer, 1969, 1973), and the
designatedby hi(t). The set of standardfunctionshe(t)--
waveshape of the function shows approximately the same number of oscillations as the PST histogram for clicks. Furthermore, a revcor function indicates that a certain minimum time interval should elapse between stimulation and response, and this interval corresponds well with the latency evident from the click PST histogram. These agreements suggest that for each fiber the acoustical stimulus filtered by the appropriate filter is a good predictor for the firing probability. The revcor function of the fiber should be the impulse response of the
with i from i to N--is
filter. This concepthas been proven to be correct (de Jongh, 1972; de Boer, 1973) for noise input signals.
modeled after experimentally
ob-
tained revcor functions of auditory nerve fibers (de Boer, 1973; de Boer and de Jongh, to be published). The parameters of these functions, notably those concerning frequency selectivity and latency, are made to depend upon the resonance frequency f• in the same manner as the parameters of measured revcor functions depend on the characteristic frequencies of the fibers
tested. In their turn, the resonance frequencies fi are distributed uniformly over the (logarithmic) frequency scale. The output of the filtering stage, the filtered stimulus, can be found by the convolution integral.
For
J. Acoust Soc. Am., Vol. 58, No. 5, November 1975
Redistribution subject to ASA license or copyright; see http://acousticalsociety.org/content/terms. Download to IP: 132.206.205.6 On: Wed, 17 Dec 2014 00:55:53
1032
de Boer: Synthetic whole-nerve action potentials
1032
creases from zero to a constant asymptotic value
the ith neural unit the outputsignal yi (t) is
(saturation).
x(t-
(h) To simplify the representation of the case in which the stimulus is repeatedly presented with random phase •O,a special assumption is invoked: within
This signal yi(t) is called excitatory signal. The second stage of a neural unit consists of a proba-
each cycle of oscillation with a (nearly) constant
bilistic pulsegeneratordrivenby the signalyi(t). We assumethat the probability of firing p•(t) is a unique
amplitude, the firing probability is a rectified
function R(z) of the quantity z, which is proportional to
result, the average firing probability, averaged over all phase values •, becomes solely a func-
yi(t):
p•(t)=R(z),
(2)
linear transform
of the excitatory signal.
As a
tionof the envelope e•(t)of y•(t). (i) The rate functiondescribedunder assumption(e)
with
also describes the dependence of the average firing probability on the envelope value of the excitatory
z = k• y• (t) .
signal.
The function R(z) should show two main characteristics: rectification and saturation.
For negative values of the
excitatory signal the firing probability is assumedto be zero. For high positive values of the excitatory signal the probability tends to saturate so that the dynamic range of excitation over which the probability goes from barely discernible values (e.g., 0.05) to the onset of saturation is not larger than 40 dB, just as is true for
(j) Each neural unit has its sensitivity adjusted in such
a way that the thresholdof response(definedas the point at which the firing probability is 0.1) lies at the absolute threshold of audibility at the pertinent resonance frequency f•.
(k) Each firing of a neural unit gives a contribution to
in Eq. 2 are chosen in such a way that for each neural
the whole-nerve action potential A(t) with the same waveform, the unitfunction u(t). The waveform of u(t) is biphasic, with an average of zero. In the
unit the threshold for neural response, e.g., p•(t)= O.1, correspondsto an input level of x(t) equalto the thresh-
most general case the contributions from the neural units are not equal; the contribution from the
old of audibility at the resonance frequency f• for that
ith neural unit is expressedby c• ßu(t).
auditory nerve fibers (Kiang, 1968). The constantsk•
particular neural unit. This is a restriction that has been built in for reasons of simplicity.
(1) In the larger part of the computations all coefficients c• are taken as equal.
The assumptions on which our computationsare based are listed below; explanatory remarks and justification
(m) Adaptation is represented by a linear filtering process applied to the neural activity function
are presented later.
(i.e., the weightedsum of all firing probabilities).
(a) The set of primary auditorynerve fibers is replaced by a limited set of N "neural units," each of which has properties adapted to the average of the subset of nerve fibers which it represents.
(b) The frequency selectivity exhibited by the responses of auditory nerve fibers is represented by a process of linear filtering in the neural units. Each
neural
unit contains
a linear
filter
as its first
A short discussion on these assumptions follows. The first three assumptions imply a linear action of the frequency-selective mechanism. Many data on the detailed relation
between
the filtered
stimulus
waveform
are
in
agreement with this notion. However, many manifesta-
tions of cochlear nonlinearity are knownas well (e.g., Goldstein and Kiang, 1968; Sachs and Kiang, 1968;
Goblick and Pfeiffer, 1969). In the present context we
element. The properties of these filters are described by the set of impulse response functions
are most interested in the initial buildup of the neural
h•(t) (i = 1, 2, ..., N). Thesefunctionsare called standardfunctions. The outputsignals y•(t) of the
equal to several cycles.
linearities of the "essential" type (Goldstein, 1967) ex-
filters form the set of excitatory signals.
ercise a substantial influence on this buildup.
(c) The impulse response is invariant to the type of stimulation, be it tonal, stochastic, or transient.
excitation
for a tone-burst
stimulus
with
a rise
time
It is not known whether nonIn any
event, similarly rapid changes in the filtered signal occur during stimulation with noise, and the linear ap-
proximationis sufficientlyusefulin this case (deJongh,
(d) The neural units act independently of one another. (e) The second element of a neural unit is a proba-
bilistic pulse generator. The probabilityp•(t)that a pulse is generated by a neural unit is solely a
functionof the instantaneousvalue of z = k•. y•(t). The functional dependence is symbolized in the
form of the rate function R(z).
Compare Eq. 2.
(f) The rate function is the same for all neural units.
(g) The rate function R(z) is zero for negative values of z.
For positive values of z the rate function in-
Another type of nonlinearity results from mechanical overloading (cf. Rhode, 1971; Rhode and Robles, 1974). For the genesis of the whole-nerve action potential this type of nonlinearity will probably not be significant, since it will affect only neural units that are in satura-
tion. These arguments give us confidence in trying out the theory in linear terms only--details about nonlinearity can later be filled in, if need be.
Assumption(e) has been shownto be valid for a stim-
J. Acoust. Soc. Am., Vol. 58, No. 5, November1975
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1033
de Boer:Syntheticwhole-nerve actionpotentials
1033
ulus like white noise; the probability of firing follows
ing frequencies. We see later how the effect of allowing
the fluctuationsof the filtered stimulus (de Jongh, 1972).
a more gradual distribution can be predicted.
The same holds true for a signal consisting of several
sinusoids(Bruggeet al., 1969). It doesnot matter very much whether the nerve fiber is driven into saturation, the details of the excitatory waveform still appear in the firing probability. There seems to be a fast adaptation process present that tends to normalize the sensi-
tivity to the value of maximal excitation. Since this (hypothetical) mechanism is very fast, the responseto clicks displays surprisingly little changes when the
stimulusintensity is raised to valuesthat causesaturation (cf. Kiang et al., 1965). The notion of an infinitely fast normalization process is implicit in our assumptions (e)-(i). This statement needs some extra explanation. One method to avoid contamination of physiological action potentials by co-
chlear microphonic(CM) potentials consists of present-
In relation with assumptions (k) and (1), it is known that nerve fibers with a high characteristic frequency
(CF) contributemore to the whole-nerveactionpotential than fibers with low CF (Teas et al., 1962; Elberling, 1974; Eggermont et al., 1974b). But there is no agreement between the reported studies as to manner and extent of this property. This is readily understood when we realize how many factors contribute: spectral composition of the stimulus, electrical geometry with respect to the electrode, species of animal tested, the in-
fluence of secondaryresponses(like N•.), etc. On the basis of the available data it is not possible to ascertain whether it is justified to use the same function u(t) throughout. In order to simplify our model to the ex-
treme, we use a uniform unit functionu(t), and, in addition, we initially chooseall coefficientsc• to be equal.
ing the stimulus repeatedly with slowly varying phase.
This is certainly an oversimplification, but it allows us
That is, the envelope e•(t) of the stimulus remains the
to retain a good overview of the relations between model
same during the repetitions, but the phase of the fine
parameters and the properties of synthetic action po-
structure is changingcontinuously. Whenthe onset of the stimulus is sufficiently gradual--and we deliberately
tentials.
set out to meet this criterion--the
Assumption (m) is covered in the next section.
response functions
,
y•(t) will all showthe sametendency. All thesefunctions can then be described in terms of an envelope
e•(t) anda fine structurejust as the stimulus:
II. OUTLINE SYNTHETIC
OF STEPS IN THE COMPUTATION ACTION POTENTIALS
OF
The steps in the computations can be followed from
x(t)= e=(t). cos(w0 t+$),
(3)
yt(t)= e•(t). cos[wt(t) ßt + •5].
Fig. 1. The starting point is the stimulus x(t).
We
must choosea sinusoldwith a slow rise (and decay), so that the representation of the filtered waveforms y•(t)
What is important here is that during the repetitions of
in terms of envelope and fine structure as indicated by
the stimulus eachYt(t) keeps the same envelopeand the same fine structure describedby (,)t(t). t, but varies
Eq. 3'is valid. This canbe realizedby makingthe
only in the phase •5. Note that t as it appears here is always referred to the onset of the appropriate stimulus
signal. Our assumption(e) says that each excitation functionyt(t) must be treated accordingto Eq. 2 in order to find the appropriate.firing probability pt(t). Consider nowthe values of yi(t) andPt(t) at oneparticular instant of time t during all these repetitions.
The values
of yi(t) vary accordingto a cos•)function. As a result of assumptions(e)-(i), the firing probabilityPt(t) also
variesas a function cos•),andthemaximalvaluee•(t) of the probability is then related to the envelopee•t(t) of the y (t) signalthroughthe same relation, Eq. 2. The averagefiring probabilitypt(t), averagedover all values of ½, nowbecomesa constantfractionof e}(t). By this expedient we have removed details about the fine struc-
ture of the signalsx(t), y,(t), andpt(t) andwe are left
onset sufficiently gradual (see Sec. IIi). The waveforms of the excitatory signals are computed
from the convolution integral, Eq. 1. The firing probabilities are then found by applying Eq. 2. In view of what has been said about using stimuli with random varying phase •, the effective probability is found from
e•(t)by applyingEq. 2 to the envelope e••(t) of the signal y•(t). In fact, a reduction factor--the average of cos• over a half-cycle--should be included, but this is left out.
Consider now the case where a stimulus is repeated many times with identical waveform and the resulting action potentials are averaged. Then the average ac-
tion potential A(t) is composedof contributionsdue to the firings of all neural units at all instants previous to t:
with the concept that the envelopes of these signals play the most important part.
N
In short, the average firing
•o
(4a)
probability •t (t) is directly proportional to the envelope e•(t), whichin its turn can be foundfrom Eq. 2 by sub-
stitutingkt e•(t)for z. Notethatthe samemethodcan-
(note: pt(t)= 0 for t c•,
h•(t)=0, with co•= 2•rf•.
for t0, R(z)=O,
forz-
