Biological
Cybernetics
© SpnngerVerlag 1986
A Spike Generator Mechanism Model
Simulates Utricular Afferents Response to Sinusoidal Vibrations
R. W. Budelli, E. Soto, M. T. GonzalezEstrada, and O. Macadar
Departamento de Ciencias Fisiologicas, lnstituto de Ciencias de la Universidad Autónoma de Puebla, Puebla,
Pue., México and Instituto de Investigaciones Biológicas, Montevideo, Uruguay.
Abstract. Using a model of spike generator mechanism (SGM) with a variable threshold we simulate the
responses of articular afferents to sinusoidal vibrations. It reproduces the phase locking characteristics
(bifurcations diagrams) and the stimulus frequency firing rate relationships of different types of utrieular
affereuts. We estimate the model parameters selecting the values which best fit the experimental results and
we compare them with those from basic mechanisms involved in articular codification.
INTRODUCTION
In a previous paper (Budelli and Macadar 1979) we studied the response of articular afferents to sinusoidal
vibrations. We classified these fibers in three types. Type I afferents fire spontaneously, respond to tilts in a
phasictonic way and follow stimulation frequency, increasing or decreasing their firing rate when vibrated
sinusoidally. Type II afferents differ from type I in relation to response to vibrations. They always increase
the firing rate when vibrated, whatever the stimulation frequency. Type 111 afferents do not fire
spontaneously. Our interest is to understand how the differences between type I and II afferents are
originated.
Several mechanisms produce utricular codification (Aidley 7971; Davis 1961): stimulus modification,
transducdon, afferent synapses and spike generation. All of them may be involved in the generation of
response characteristics from the different types of utricular afferents. In this paper we study the capacity of
the spike generation mechanism ISOM) to produce these characteristics and, furthermore, the differences
between type I and II fibers. Using a model of SGM with variable threshold (Perkel et al. 1981; Torras 1985)
which is in good agreement with the classic theory of spike generation (Mounteaettle 1980; Aidley 1971;
Adrian and Zottermau 1926; Erlanger and Gasser 1937), we simulate the responses of articular afferents to
sinusoidal vibrations. With different values of the model's parameters we may reproduce the behavior of the
different types of afferents. The values of these parameters (compared with those experimentally determined)
may indicate which process is or is not involved in the physiological basis of repetitive firing of utricular
afferents.
The Model
Firstly, we assume that the place where spikes are generated is a well defined region of the afferent fiber, on
which synaptic potentials exert a direct influence. We also assume that the membrane potential and the
threshold is uniform in this region. These conditions define a unieompaetmeutal model (Perkel et al. 1981). It
also includes a relative refractory period (RRP) following each spike (Fig. 1) modelled through a time
dependent increase of the threshold (T in Fig. 1):
Fig. 1. The model The membranepotcntial V Is a function of the time t, The threshold T increases after each spike (S).
The afferent will produce a spike when the membrane potential (V) reaches the threshold (T). The decay of the threshold
to its asymptotic potential (T
0
) follows an exponential function
where To is the steady state threshold and f is a decreasing function with limit 0 when t tends to infinit. In our
simulations we will use:
f (t)=b exp((tt
0
)τ),
where to is the time of the last spike and 6 is reset after each spike to the actual value of fit) plus a fixed
amount (B). These last assumptions are conventional; 1201 we shall use them for two reasons. On the ono
hand, computations are asier with exponential functions, sincef(t+dt)=f(t) exp(dt) and we may calculate ,fat
fixed intervals multiplying it by the fixed mount exp(dt) without using exponentials each time. On the other
hand, under these assumptions, the model is equivalent, regardless of a linear filter, to the leaky integrator
(Seharstein 1979; Budelli 1985). We may therefore interpret the results in relation to the biophysical basis of
this model.
The membrane potential of the region where the spikes are generated is a function of the synaptic
activity which roduces the generator potential. Since the synaptic activity depends upon the stimulus, the
membrane potential of the Region where the spikes generate is a function of the stimulus. As a first
approach, in order to compare the model with the experimental results, we will suppose that the relation
ship between stimulus and generator potential is linear. If we assume that the filter transforming the stimulus
in the generator potential is a leaky in tegrator, with the same time constant of the evolution ofthe threshold,
the model reduces itselfto the classical leaky integration and fire model (Scharstein 1979; Budelli 1985). We
will work with the relationship between the generator potential and the spikes, and thus we will not deal with
the transfer function between the stimulus and the generator potential. Methods: We simulate the behavior of
the model for different parameters with standard computational methods (Perfect et al. 1981) using a Digital
computer PDP 11/34.
RESULTS
There are two types of spontaneously firing utricular afferente with different responses to sinusoidal vibra
tions (Budelli and Macadar 1979). When type I afferents are stimulated with frequencies close to their
spontaneous firing rate, the s s spikes always discharge close to a fixed phase of the stimulus (phase lacking)
(Fig. 2 left); the cycle histogram presents a sharp peak and the fiber may increase or decrease its firing rate
as a function of the stimulus frequency. Nevertheless, hen the stimulus value is increased
above a critical value, the phase locking paradoxically disappears and the cycle histogram becomes flatter
Fig. 2. Response of typosl and it utncular aRCrcnts. The left column shows results from a type I afferent and the right
one, from a type 11. The first row shows raw records of responses to emasondal annotation: type l ancient phaselocks
and type 11 does not. The second row shows the corresponding cycle histograms Thethnd row
showsoyclehnstogramscorrespondmg to a larger stimulus: the type 1 afferent loses the phase locking
(Fig. 2 left, bottom). Type IT afferents do not show phase locking (Fig. 2 right), the cycle histogram does not
present sharp peaks and the firing rate can not be reduced below its spontaneous rate.
Some results from type 1 afferents can be reproduced by the leaky integrator model (Rescigno et al.
1970; Stein et al. 1972; Barbi et al. 1975; Poggio and Tome 1977; Ascoliet al. 1977; Rescigno 1978;
Angelini and Petracchi 1978; Angelina et al. 7980; Zeevi and Bruekstein 1981). Some aspects of the behavior
of type 11 afferents can be reproduced by the perfect integrator model (Bayly 1968; Knight 7972; Stein et al.
1972). Nevertheless, this model produces meaningless results. For example: a negative stimulus will produce
a fixed delay of the next spike quite independently of the moment in which it was released. In order to present
a more convincing model we have to be able to simulate bath types of afferents with a single model, selecting
different numerical values for the parameters. Furthermore, we are attempting to estimate the model
parameters, looking for those that fit the best to the experimental results. This estimation will provide
information related to the physical baste of the spike generation mechanism.
Phase Locking
We determined the parameters of the stimulus (amplitude, A, and frequency, f) under which phase locking
occurs, for different time constants (r) of the function
f(t)=K'exp((tt
o
)/τ),
where to is the time of firing of the last spike and
K=B Σ
i
exp((t
o
ti)/ τ) i=∞,…,0,
where t, are the firing times of the previous spikes and B is the increase of the threshold following each spike.
We usually understand by phaselocking the stable firing of a spike in a fixed phase of each cycle of the
stimulus. This concept was extended to the case where n spikes fire in fixed phases of m cycles, and we say
there is a (n, m) phase locking.
The bifurcation diagrams (Fig. 3) show the regions of the stimulus parameters (A, f) where some simple m,
in) phase lockings are produced for different rates (τ/P) between the model time constant (r) and the
spontaneous firing period (P). The stimulus amplitude is expressed as a percentage of the asymptotic value of
the threshold (To). The stimulus frequency is expressed as a percentage of the firing rate. In general, the
regions of phase locking do not overlap, corresponding to "Arnold's tongues", which in turn correspond to
the results predicted when using small stimulus for a wide range of firing models (Arnold 1965). This figure
also shows that for small time constants (r'P=0.1) the phase lockingregions are narrow and that they widen
ass, P increases. For intermediate values of the time constant (τ/P=1) the regions widen at small amplitudes
of the stimulus, but they narrow at larger amplitudes. With further increases of the time constant (τ /P>3) the
regions vanish.
These results show that the model reproduces the behavior of both types of afferents. Obviously, it
simulates the behavior of type I fibers, since our model is equivalent to the leaky integrator. For i,P=1. it also
simulates the lost of (1,1) phase locking for a stimulus frequency close to the firing rate when the stimulus
amplitude increases above a critical value (Fig. 2, Fi&3 from Budelli and Macadar 1979; Ange
Fig. 3. Bifurcation diagrams, The figure shows for xT equal l0 10, 3.1.03, and 01 the bdurcenon diagrams Them arc
several types of phaselocking on, am where n os the number of,p,k,, produced in fixed phases of or cycles
Fig.4. Minimum frequency obtained with u, 1)phaselocking as a function of TIP
lini and Petracchi 1978). For TIP > 3 it also reproduces the results of type II afferents. Since the regions of
phase locking in these cases are very narrow, the probability of selecting the stimulus parameters in the (1,1)
phase lacking region is very low. Furthermore, when in these units phase locking is obtained (Fig. 3 from
Macadar and Budelli 1984) it is so for frequencies above the spontaneous firing rate.
Figure 4 shows the minimum frequency with (1,1) phase locking as a function of τ/P. This figure allows
an estimation of τ/P. If, for example, the minimum frequency allowing (1,1) phase locking is 70 %, we may
estimate 0.3 < τ/P < 1. Since, as we have reported, most of type I fibers can not phase lock with frequencies
below 70% of their spontaneous firing (Budelli and Macadar 1979), we estimate TIP > 0.3.
The model simulates the spontaneous firing of utricular afierents (types I and II). With an adequate
selection of parameters we were also able to simulate the phase lacking of type I fibers and its absence in
type II afferents. Quantitative changes in the model parameters may produce qualitative changes in its
behavior. Furthermore, we showed that for many pairs of integers the model produces (m, n) phase locking.
Although this result was not reported in our experimental papers, it may probably accounts for some
multimodal cycle histograms (Budelli and Macadar 1979).
Stimulus Frequency  Firing Rate Relationship
When type I and II afierents were defined, one of the criteria we considered was the capacity of type I fibers
to increase or decrease their rate as a function of the stimulus frequency. On the other hand, type II afferents
always increase their firing rate whatever the stimulation frequency. Since for τ/P<1 the model produces a
(1,7) phase locking region reaching fre quency values below the firing rate, we can be certain that for these
values of τ/P it reproduces this behavior of type I aBerents. Nevertheless, we do not know what happens for
larger values of τ/P.
Figure 5 shows this relationship for TIP 10 and far several amplitudes of the stimulus. The broken
straight line passing through (1, 1) corresponds to points where the (1,1) phase locking may be produced.
Since there are no simulation points on this line, we confirmed that the (7,1) phase locking region is very
narrow. Nevertheless, the curves change their slope near this line, tending to decrease the angle of
intersection. In this case all the stimuli increased the firing rate of the model proportionally to their
amplitude.
Figure 6 shows the results for τ/P=3. In this case, there is no proportional relationship between the
increase in firing rate and the stimulus amplitude. The curves follow partially the phase locking lines. This
tendency is notorious for the (1,1) phase locking, but it is also clear far the (1, 2) and (2,1) ones. There are
also several points in the lines corresponding to phase lockings (3,1), (3, 2), and (2, 3).
Figure 7 shows the same kind of curves for τ/P=1. Here, the tendency of the simulation points to appear
in the phase locking lines is even greater. Besides the (1,1), (1, z), and (z,1) phase soakings, (z, 3), (3, z),
(3,1), and even (3,4) and (3, 5) became notorious. Moreover, this is the largest value of τ/P producing a clear
decrease of the firing rate for some frequencies (30, 90, and 190) and stimulus amplitudes.
The curves for τ/P=0.3 (Fig. 8) also have many points in the phase locking lines. For this time constant it is
possible to produce reductions in the firing rate of up to 30% for the (1,1) phase locking up to 20% for (2,1)
and up to 10 % for (1,2)
Fig. 5. Firing rate against stimulus frequency for
τ
/P=10. It shows this relationship for different amplitudes of the
stimulus, shown on each curve in relation to the difference between the asymptotic values of the membrane potential and
the threshold
Fig. 6. Firing rate against stimulus frequency for τ/P=3.
Fig. 7. Firing rate against stimulus frequency for τ/P=1.
Fig. 8. Firing rate against stimulus frequency for τ/P=0.1.
Fig. 9. Firing rate against stimulus frequency for τ/P=0.3.
or τ/P=0.1 (Fig. 9) the (1,1), (1, 2), and (2, 3) phase locking curves are notorious. 1, a smaller degree
ma
timulus frequency relationships are similar to those described for
typ
iscussion
imulation experiments show that the SGM may reproduce several characteristics of the oracular afferenls
he stimulus frequency firing rate curves show that a sinusoidal stimulus may increase (or decrease) the rate
decreases in the rate may be produced by stimuli which have negative and positive phases.
F
ny points appear in the (2, 1) and (8, 5) lines. This result shows that it is easier to obtain phase locking by
decreasing the Firing rate than by increasing it This result does not coincide with our experimental results,
thus indicating τ/P must be larger than 0.1.
It is worth noting that the firing rate  s
e I afierents if i~P=0.3 and to type H ones if τ/P=3. Results for τ/P=1 probably correspond to intermediate
results mentioned in our experimental papers (Budelli and Macadar 1979, 1981; Macadar and Budelli 1984.).
D
S
response to sinusoidal stimulation. Furthermore, an estimation of the model parameters is possible, selecting
the values τ/P which best fit the experimental results With τ/P=1 the model simulates type I aBerents and
with τ/P=3, type II. The fact that a model with different values of its parameters can reproduce the behavior
of both types of afierents is a strong argument on its behalf, since there is no need to introduce different
processes for different types of fibers. Besides, it is worth noting that quantitative changes in the parameters
produce qualitative changes m the response.
T
of firing depending upon its frequency and the model parameters. Consequently, a change in the frequency
of discharge produced by a periodic stimulus does not indicate the presence of a maintained polarization at
the level where the spikes are produced. Perkel et al. (1964) have shown that an increase in frequency of a
hyperpolarizing stimulus may paradoxically increase the firing rate of the neuron. Nevertheless, all these
changes of rate correspond to values below the basal rate (Segundo 1979; DiezMartinez et al. 1983). When
excitatory stimulation is used, similar results are obtained, but the mean effect is an increase of the rate
(Segundo and Kohn 1987; DiezMartinez and Segundo 1983). In our model, we prove that both increases or
Moat of the neuronal models show phaselocking. Au exception is the integrate and fire model (Knight
1972; Rescigno et al. 1970). Probably, this general behavior is a consequence of ge
neral topological
the
egulation of the repetitive firing of the vestibular
afie s, simulated in our model by an increase of the threshold. They may be related to the dynamics of the
Na`
hysical basis (the charge of the membrane capacitor), we prefer not to
e model parameters.
These estimated values can be compared with those experimentally determined for the different mechanisms
inv
ould like to thank Drs. O. DiceMarnncz, J A Brig, B Holmgrcm and R. Lara for
reading the manuscript. We also appreciate the artistic help of R. Bcraal and R. Cardozo and the
971) The Physiology of ...table calls. Cambridge University Press
ngelini F, Pelrucchl D (1978) Significance of phaselocking mennmemenls as a test for neural cneodlag
timulation. Biol Cybern 36:137142
orems (Arnold 1965; Rescigno 1978). In this case, any model will share some general characteristics of
the phaselocking regions in the frequencyamplitude diagram (Glass and Mackey 1979; Glass and Perez
1982; Guevara et al. 7981; Grasman 1984). Then, in order to select between models, we must compare
mostly geometrical characteristics. Since our model shares this behavior, we have to compare the
bifurcations diagrams it produces with those experimentally determined. They are different in our model and
in those of Glass and Perez (1982) or Grasman (1984).
Several mechanisms could be involved in the r
rent
channel (Gunman et al. 1980), the Car' dependent K' channel (Meech and Standen 1975; Partridge
1982), the Acurrent channel Connor 1985), or the charge of the membrane capacitor (Hodgkin 1948; Stein
ct al. 1972)_ The determination of the time constants by the simulation of lhc experimental results can be
compared with the time constants of these processes in order to determine which one might be responsible
for the characteristics just examined.
Our model, in the conditions used m the snmulafions. behaves like the leaky integrator (Budelh 7985). Since
the leaky integrator model implies a p
use it in order to avoid the subsequent interpretation of the meaning of tire parameters: the time constant is
that of the membrane (resistance times capacity). Nevertheless, since both models behave in the same way,
we may check if our estimation of the time constant correspond to those experimentally determined in the
afferent fiber. Forukawa et al. (197R) recorded poet synaptic potentials from the nerve terminals from the
gold fish sacculus. The decay of these potentials establishes a superior boundary for the time constant of the
membrane of around 1 ms. On the other hand, we estimate τ/P between 0.3 and 3. Sine. the values of P are
usually larger than 20 ms, it must be r>6ms, reaching values even higher than 60 ms. These considerations
show that the values of r necessary to simulate the experimental results are always larger than the time
constant of the membrane. Consequently, we have to reject the biophysical basis of the leaky integrator (the
charge of the membrane capacitance) as the regulator mechanism of the repetitive discharge of the utricular
afferents. Then, we will have to look elsewhere to find the mechanism accountable for the regulation of
repetitive firing of utricular afferents. They can consist of changes in excitability related to the dynamical
behavior of the sodium current channel, or slow changes in the potassium permeability.
In this paper we show how the simulation results can be used for the estimation of th
olved in the process we are dealing with. This is a powerful method for testing alternative hypothesis
about the physical basis of any characteristic. Usually, the result can be the rejection of a hypothesis, since a
numerical coincidence between parameters estimated by a model and a basic process does not prove that the
latter is responsible for the general behavior of the system. In our case we rejected the hypothesis that the
repetitive firing of vestibular afferents is regulated by the charge of the membrane capacitor. Usually the
rejection of a hypothesis is disturbing. But, as Ironware (1902) argued "this feeling is noujustified. The
physicist who has just given up one of his hypothesis should, on the contrary, rejoyce; since by the same
token he has found an unexpected ocassion of discovery. Has the discarded hipothesis been fruitless? Far
from that, we might say, it has provided more benefit than a well stablished hypothesis. It has pointed to the
decisive experiment; and furthermore, if the experiment was the product of hazard without any supporting
hypothesis, nothing extraordinary would be detected; and we would simply catalogue another fact without
reaching any conclusion.”
Acknowledgements. We w
critically
editorial one of G Payas.
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