Biol. Cybernetics 17, 99--108 (1975) @ by Springer-Verlag 1975
Perturbation Effects on Stability of Gravity Receptors D. P. O'Leary, J. P. Segundo, and J. J. Vidal Departments of Anatomy, Surgery, and Computer Sciences, and Brain Research Institute, University of California, Los Angeles, California, USA Received: May 14, 1974
Abstract Stability properties of gravity receptor transduction mechanisms were investigated by recording action potentials from the eighth nerve of cats maintained at two spatial positions both with and without an added click perturbation. The cells were demonstrated to present directional sensitivity, multivaluedness and wandering of mean rate trajectories. To the wandering were fitted appropriate stability boundaries corresponding to recent theories of finite time stability. Results supported the hypothesis that physiological stimuli result in local deformations of a flexible trampoline-like macula.
Introduction
Gravitational sensing by vertebrate inner ear receptors is ascribed to the utricular and saccular maculae, composed of an approximately planar array of ciliated hair cells covered by gelatine and loaded with calcite crystals (i.e. otoconia) whose specific gravity is greater than that of the endolymphatic fluid bathing the maculae. Both changes in the direction of the gravitational vector and linear accelerations displace the otoconia, producing a shearing force in the gelatinous layer that, by acting upon the cilia, constitutes the physiological stimulus. Information about the overall stimulus is coded and transmitted centrally in the form of spike discharge timings and correlations in the ensemble of firstorder neurons innervating the receptors. Certain afferents are defined as directionally sensitive if they respond to gravitation during a maintained head position with a stationary discharge rate, and, furthermore, if that rate varies systematically as a function of head position (e.g., Fernfindez et al., Loe et al., Vidal et al.) In spite of their stationarity, these afferents exhibit some variation or wandering of discharge rates even while the head is maintained at a static position. Furthermore, the average discharge rate associated with a particular static position often differs between successive returns to that orientation, i.e., the position-discharge relation is multivalued. The wandering and multivaluedness pose significant
questions and suggest that gravity receptors could show varying degrees of stability in their equilibrium states under certain conditions of stimulation. Confidence in some degree of time-invariance is implicit in any experimental study of sensory systems, whether it be essentially qualitative in the classical physiological fashion or based on the use of inputoutput analytical techniques derived from systems and control theory. All these procedures are useful for the identification of a model when the system's output behavior is reproducible upon repeated applications of a uniform stimulus. Certain sensory systems such as gravity receptors exhibit afferent ~responses that are both multivalued in magnitude and nonstationary in the sense of exhibiting "wandering" even when analyzed with repeated uniform inputs. This report will describe a new approach for characterizing such systems in terms of the relative stabilities of the underlying receptor states which determine the multivalued and nonstationary behavior. The stability properties of a system can be tested experimentally by observing its response to a perturbation. In descriptive terms, a system is in a "stable" state when it responds to the perturbation with a small displacement of its output trajectory followed by a return to, or almost to, the original state. In contrast, a system is in an "unstable" state when it responds with transitions to new states that are more stable than the original one. The system states from which output trajectories originate are characterized according to specific stability criteria, both in the presence and absence of an experimental perturbation. An analysis of the stability properties of gravity receptors may have particular physiological significance in the sense that they exhibit great sensitivities and operate normally in the presence of ongoing perturbations, e.g., external vibration; head movements or vascular pulsations. Furthermore, stability concepts have been applied extensively in the design of sensors and controllers used in inertial
100 navigation, and, as noted by Barlow, vestibular receptors show striking structural and functional similarities to inertial navigation sensors. The multivalued and wandering properties of gravity receptors might be characterized in terms of the relative stabilities of the afferent responses during maintained positions both with and without an added perturbation applied experimentally. This approach required several choices. Because stability is often used only in an intuitive context, it was necessary to choose a set of definitions of stability types applicable to neurophysiological data (see Theory). Furthermore, an experimental perturbation had to be chosen that would produce a significant effect without resulting in irreversible changes. We therefore chose to apply loud clicks in the ipsilateral auditory canal, with the rationale that auditory stimuli have been shown previously to affect vestibular reflex pathways (Parker et al.) and that our preliminary experiments indicated that such clicks did indeed affect reversibly the responses of gravity-sensitive afferents. A description of the mechanism of this auditory-vestibular interaction, although interesting, is not the purpose of this report; the clicks should be viewed merely as nonspecific perturbation for the purpose of classifying afferent responses within the stability framework to be described in Theory. In short, the purposes of this report are to describe and discuss experiments which explore transduction in the gravity receptor system, and to consider a theory of finite time stability that could be applicable to investigations of vestibular functions.
It has been pointed out, however, that paradoxes may arise from application of these concepts in specific real cases (La Salle and Lefschetz). For example, a system may be capable of moving gradually toward a desired state which therefore is asymptotically stable, but if oscillations are large, it may not operate acceptably. Conversely, a system may never remain very close to a desired state, i.e., the latter may be unstable but, if oscillations are not too large, its performance may still be acceptable. These observations led to the development of the concept of "practical stability" or stability within bounds acceptable for operation (La Salle and Lefschetz). What constitutes acceptable bounds must of course be decided operationally; for example, walking across a canyon on a tightrope requires more stringent boundaries than using a wider bridge.
fix (-~)ll
to
T
to
tl
T
Theory Stability is an important factor in many systems, but the term is often used only in its intuitive context and w i t h o u t regard for the various theories and nomenclatures (e.g. Hahn). In the classical work of Lyapunov, a system is considered stable if for every small arbitrary deviation e > 0 from a given state there exists a sufficiently small range 6 > 0 of initial condition such that all trajectories that deviate by less than e from a desired state have initial conditions that are bounded by 6. This definition is general and applies to perturbed systems with or without forcing inputs, in contrast to other definitions such as "bounded input/output stability" or those restricted to linear systems. Lyapunov's analysis was extended later to systems whose solutions were not known by showing that they are stable if mathematical functions satisfying appropriate criteria could be found.
to
I to
t~
""
T
1 T
Fig. 1. Two dimensional graphical illustration of finite time stabilities in a system 2=f(u,x,t). On the abscissa, a time interval extending from to to the finite value 7~ on the ordinate, the norm of either the system trajectory (in a-c) or the perturbation (in d). a. The trajectory starts at ~0 within e and either remains within (simply stable system) or moves beyond (unstable system) the bound 7. b. A quasi-contractively stable system's trajectory starts within c~, remains within 7, and after a settling time t~ remains within a ~ greater than c~. c. A contractively stable system's trajectory starts within c~, always remains within 7, and after t t remains within a ~ smaller than c~. d. The ongoing perturbation u(x,t) is bounded, i.e., its norm always remains within
101
Subsequently, Weiss and Infante recognized that the theory could be made even more realistic by considering systems which operate over finite, as opposed to infinite, time intervals since, in any real system, practical stability during a finite time is sufficient for the completion of any desired task. They also derived simple criteria for this type of behavior. The finite time analogue of Lyapunov stability is defined for a case with no perturbations (Fig. la) as the situation where, with bounded initial conditions, all trajectories remain bounded during the required period. Conversely, the system is unstable if, under the same conditions, they do not remain bounded. These definitions were then extended to systems subjected to perturbing forces. Three definitions were proposed which express separate and progressively narrowing conditions of finite time stability under perturbations. Simple stability under perturbing forces implies essentially that, if both the initial conditions and the perturbations are small, all trajectories will remain bounded during the required period. Quasi-contractive (Fig. lb) and contractive (Fig. lc) stability imply, in addition to simple stability, that the bounds on the trajectories become narrower after a certain settling time. If the late boundary is smaller than the initial one, the stability is contractive and not quasi-contractive. The different forms of stability differ operationally in the sense that a contractively stable control system requires less input energy to result in an acceptable output than a simply stable one (Weiss, personal communication): the intuitively acceptable reason is that, under conditions where the target state is the origin, a contractively stable system is heading toward the origin as a matter of course and without the benefit of the control, while this is not necessarily the case with simply stable systems. These definitions were expressed more rigorously by Weiss and Infante. To illustrate the concept of finite-time practical stability in the context of continuous dynamical systems, let us consider a system of N first-order real differential equations:
2 = f(x, t)
(1)
defined over the finite interval of time J = [to, t o + T). The initial state vector X(to) is Xo. The function f is assumed to be smooth enough in x and in the time interval J (i.e., in t e J ) to assure existence and uniqueness of solutions over J , as well as continuous dependence on the initial conditions at t o. (It is not required, however, that f(0, t)=0.) Prespecifying a norm II" l] (a "norm"
being a real number that represents what, in vector spaces, is referred to as "length") and two bounds c~ and 7, the system [Eq. (1)] is stable with respect to the reference set (e, 7, II' I], to, T) with e-- 10. These recursively determined normalized rank scores formed, for each bin-rate trajectory, a time series whose points were either within or outside of a trend-free region bounded by the values plus and minus 1 on the ordinate. (As described in
103 gravity-sensitive ones, and that it is quite possible that other classes of afferents would display quite differei]t properties (e.g., Macadar et al., submitted for publication).
Results, the ensemble of time series for all trajectories at each stimulus state was used to characterize the finite time stability of that state by fitting appropriate boundaries.) For each particular trajectory, the average of the normalized values was used as a measure of its wandering. Because certain variations in bin-rates, e.g., those symmetrical about their average (or trajectory-rate), may not be detected by the Kendall trend test (Otnes and Enochsen) and still constitute wandering under our definition, the coefficients of variation of both the bin-rates and of the interspike intervals were also computed for all trajectories. Tests were always at the 95 To confidence level unless otherwise specified. tt is important to point out that we have analyzed only a small sample of a single functional class of afferents, i.e., the tonic
Results
The perturbing dicks had a significant augmenting effect upon the directional sensitivity (DS), defined as a systematic variation of rate discharge as a function of head position. This is illustrated in Fig. 2 where each point represents a trajectory-rate, i.e.,
N-10-12
32
(a)
104-
30
iO'Zk
,m as
(~ IO0--I
D
LU
N-10-3
W 98-
IE
Por?urbotIon OFF
Perlur botlon ON
6~
~"
6"
6"
~~
Per~rbaffon ON
;~
;.
6"
SUCCESSIVE SPATIAL POSITIONS
;-
Per lurbotlo~ O FF
;.
;.
;~
SUCCESSIVE SPATIAL POSITIONS
N-10-9
(c)
" ~ 27.
,,o. 26 84
bJ Z4
Z
2 3 84
22
6"
$"
6"
6"
~"
6"
SUCCESSIVE SPATIAL POSITIONS
Fig. 2a-c. Relative influence of perturbations on the directional sensitivity, wandering and multivaluedness of three different gravitysensitive afferents. Abscissa: Spatial position of the head in degrees about the roll axis with perturbations ON and OFF. (Points corresponding to different stations at the same position, i.e. 5 degrees, are shifted laterally slightly to avoid superpositions.) Ordinate: Mean impulse rate during maintained head position; marks represent trajectory-r~/tes and error bars indicate standard deviations of 1 sec bin-rates. Temporal order of recordings: 9 = 1, 9 = 2, 9 = 3, A = 4 and [] = 5. The mean slope of the differences between rates at the positions 0 degrees and 5 degrees represents the directional sensitivity for each cell; the scatter of marks at each position represents the multivaluedness; and the error bar represents the wandering during each trajectory. There were three types of perturbation effects, a Type I: Trajectory rates at 5 degrees with perturbations ON significantly differed from those at 5 degrees without perturbations; no significant effects at 0 degrees, b Type II. Trajectory rates at 0 degrees with perturbations O N significantly differed from those at 0 degrees without perturbations; no significant effects at 5 degrees, c Type III: No significant effect of perturbations on trajectory rates.
104
the overall mean rate of the spike train at a particular stimulus state, and each sub-figure shows data from one cell only. Each state is characterized by one of two possible positions and the presence or absence of clicks. The vertical bars represent the standard deviation of the population of bin-rates in that trajectory. The DS was measured as the average of the absolute values of the slopes connecting points at two positions. For all of 10 cells used for the DS analysis, it was greater with the perturbation ON than O F F : this result would occur only once in 2 9 cases on the basis of chance. In the guitarfish, mechanical jitter can increase, decrease or even reverse the DS (Macadar et al., submitted for publication). Three basic types of "gravity-sensitive" afferents were found in the DS analysis. In the first type (Fig. 2a), at the 5-degrees position there was an increase in trajectory-rates with perturbations ON relative to those without perturbations, as indicated by the Wilcoxen test. At 0 degrees differences were not significant. In the second type (Fig. 2b), at 0 degrees there was a significant decrease in trajectoryrates with perturbations ON relative to those without perturbations. At 5-degrees differences were not significant. I n t h e third type (Fig. 2c), trajectory-rates with perturbations ON were not significantly different from those without them at either of the two positions. A multivaluedness was encountered that exceeded what could be expected (at the stated confidence level) simply because of the wandering. Figure 3 illustrates for a typical cell that the multivaluednesses, i.e., the variability between trajectories, was greater than the wandering, i.e., the variability within each trajectory. The quantitative and statistical evidence for this assertion rests, as mentioned, upon the H-statistic from the Kruskal-Wallace nonparametric analysis-of-variance test. This test was applied to 6 cells at each of the 6 stimulus states and the
5o 28-
.."
"
.m
27-
0]
2
26-
! 25-
,b
i~
2'o
2's
3o
TIME (see.)
~176 t 25-e...q
m-4
:
,:.
, -
v" -
.~.' ' " . . 4 "
/
]~
b
" ~. ".." ""X ..
"
.
o
It
o.,II ~"
k"
/ """~
9
....-
,
",
R:../
23 Z I.d
22
~
ib
i~
2b
~5
-~o
TIME (sec.)
Fig. 3a and b. Trajectories of mean 1 sec bin-rates for cell N - 1 0 9 with perturbations ON. The means from the three trajectories at 0 degrees (b); or from those at 5 degrees, (a) are derived from three significantly different statistical distributions, indicating multivaluedness in the means as indicators of spatial position. The three trajectories at 5 degrees (a) are clustered along the ordinate with an overall m e a n that is significantly greater than that for 0 degrees (b), indicating a significant sensitivity to spatial position
Table I. H-statistics determined from the Kruskal-Wallis test and chi-square significance levels (p < 0.05) used as a test of significant multivaluedness for 6 cells at each stimulus state. States that were not significantly multivalued are indicated by asterisks Cell
H-Statistics 0~
N - 10-3 N - 10-7 N-10-9 N-10-10 N-10-12 N- 7-12
2.22* 51.76 77.76 7.73 13.78 43.16
Perturbation O N 5~ 18.04 37.86 36.58 t 1.91 12.29 54.00
0~
0~
24.20 28.77 77.15 7.80 25.52 52.80
24.49 31,65 30,28 16,12 11.58 31.95
Perturbation O F F 5~ 2.71 * 45.50 13.18 44.17 0.75* 46.93
Sample
Degrees
0~
size N
of freedom k
2 Xk,O.05
0.08* 62.24 35.47 11.36 0,55" 32.41
19 31 28 32 24 24
1 2 4 1 1 2
3.84 5.99 9.49 3.84 3.84 5.99
105
Perturbation ON
I00
Per turboflon OFF
80 (_3 C-- 6Oif)
m 4020-
0
0~
5"
0~
0~
5"
0~
SUCCESSIVE SPATIAL POSITIONS
Fig. 4. H-statistics computed from the Kruskal-Wallis test for two Type III cells: 9 = N - 1 0 - 9 ; (D = N-10-10. The magnitude of the H-statistic at each position is an estimator of the spread of the multivaluedness in the ensemble of trajectories at that position. The error bars indicate the 95% confidence interval of the H-statistic as determined from its approximately chi-square distribution. H-statistics at positions with perturbations O N are significantly different (at p < 0 . 0 5 level) from those at corresponding positions without them by the chi-square test
resulting H-statistics and chi-square statistics at the 95% level (p
Perturbation Effects on Stability of Gravity Receptors D. P. O'Leary, J. P. Segundo, and J. J. Vidal Departments of Anatomy, Surgery, and Computer Sciences, and Brain Research Institute, University of California, Los Angeles, California, USA Received: May 14, 1974
Abstract Stability properties of gravity receptor transduction mechanisms were investigated by recording action potentials from the eighth nerve of cats maintained at two spatial positions both with and without an added click perturbation. The cells were demonstrated to present directional sensitivity, multivaluedness and wandering of mean rate trajectories. To the wandering were fitted appropriate stability boundaries corresponding to recent theories of finite time stability. Results supported the hypothesis that physiological stimuli result in local deformations of a flexible trampoline-like macula.
Introduction
Gravitational sensing by vertebrate inner ear receptors is ascribed to the utricular and saccular maculae, composed of an approximately planar array of ciliated hair cells covered by gelatine and loaded with calcite crystals (i.e. otoconia) whose specific gravity is greater than that of the endolymphatic fluid bathing the maculae. Both changes in the direction of the gravitational vector and linear accelerations displace the otoconia, producing a shearing force in the gelatinous layer that, by acting upon the cilia, constitutes the physiological stimulus. Information about the overall stimulus is coded and transmitted centrally in the form of spike discharge timings and correlations in the ensemble of firstorder neurons innervating the receptors. Certain afferents are defined as directionally sensitive if they respond to gravitation during a maintained head position with a stationary discharge rate, and, furthermore, if that rate varies systematically as a function of head position (e.g., Fernfindez et al., Loe et al., Vidal et al.) In spite of their stationarity, these afferents exhibit some variation or wandering of discharge rates even while the head is maintained at a static position. Furthermore, the average discharge rate associated with a particular static position often differs between successive returns to that orientation, i.e., the position-discharge relation is multivalued. The wandering and multivaluedness pose significant
questions and suggest that gravity receptors could show varying degrees of stability in their equilibrium states under certain conditions of stimulation. Confidence in some degree of time-invariance is implicit in any experimental study of sensory systems, whether it be essentially qualitative in the classical physiological fashion or based on the use of inputoutput analytical techniques derived from systems and control theory. All these procedures are useful for the identification of a model when the system's output behavior is reproducible upon repeated applications of a uniform stimulus. Certain sensory systems such as gravity receptors exhibit afferent ~responses that are both multivalued in magnitude and nonstationary in the sense of exhibiting "wandering" even when analyzed with repeated uniform inputs. This report will describe a new approach for characterizing such systems in terms of the relative stabilities of the underlying receptor states which determine the multivalued and nonstationary behavior. The stability properties of a system can be tested experimentally by observing its response to a perturbation. In descriptive terms, a system is in a "stable" state when it responds to the perturbation with a small displacement of its output trajectory followed by a return to, or almost to, the original state. In contrast, a system is in an "unstable" state when it responds with transitions to new states that are more stable than the original one. The system states from which output trajectories originate are characterized according to specific stability criteria, both in the presence and absence of an experimental perturbation. An analysis of the stability properties of gravity receptors may have particular physiological significance in the sense that they exhibit great sensitivities and operate normally in the presence of ongoing perturbations, e.g., external vibration; head movements or vascular pulsations. Furthermore, stability concepts have been applied extensively in the design of sensors and controllers used in inertial
100 navigation, and, as noted by Barlow, vestibular receptors show striking structural and functional similarities to inertial navigation sensors. The multivalued and wandering properties of gravity receptors might be characterized in terms of the relative stabilities of the afferent responses during maintained positions both with and without an added perturbation applied experimentally. This approach required several choices. Because stability is often used only in an intuitive context, it was necessary to choose a set of definitions of stability types applicable to neurophysiological data (see Theory). Furthermore, an experimental perturbation had to be chosen that would produce a significant effect without resulting in irreversible changes. We therefore chose to apply loud clicks in the ipsilateral auditory canal, with the rationale that auditory stimuli have been shown previously to affect vestibular reflex pathways (Parker et al.) and that our preliminary experiments indicated that such clicks did indeed affect reversibly the responses of gravity-sensitive afferents. A description of the mechanism of this auditory-vestibular interaction, although interesting, is not the purpose of this report; the clicks should be viewed merely as nonspecific perturbation for the purpose of classifying afferent responses within the stability framework to be described in Theory. In short, the purposes of this report are to describe and discuss experiments which explore transduction in the gravity receptor system, and to consider a theory of finite time stability that could be applicable to investigations of vestibular functions.
It has been pointed out, however, that paradoxes may arise from application of these concepts in specific real cases (La Salle and Lefschetz). For example, a system may be capable of moving gradually toward a desired state which therefore is asymptotically stable, but if oscillations are large, it may not operate acceptably. Conversely, a system may never remain very close to a desired state, i.e., the latter may be unstable but, if oscillations are not too large, its performance may still be acceptable. These observations led to the development of the concept of "practical stability" or stability within bounds acceptable for operation (La Salle and Lefschetz). What constitutes acceptable bounds must of course be decided operationally; for example, walking across a canyon on a tightrope requires more stringent boundaries than using a wider bridge.
fix (-~)ll
to
T
to
tl
T
Theory Stability is an important factor in many systems, but the term is often used only in its intuitive context and w i t h o u t regard for the various theories and nomenclatures (e.g. Hahn). In the classical work of Lyapunov, a system is considered stable if for every small arbitrary deviation e > 0 from a given state there exists a sufficiently small range 6 > 0 of initial condition such that all trajectories that deviate by less than e from a desired state have initial conditions that are bounded by 6. This definition is general and applies to perturbed systems with or without forcing inputs, in contrast to other definitions such as "bounded input/output stability" or those restricted to linear systems. Lyapunov's analysis was extended later to systems whose solutions were not known by showing that they are stable if mathematical functions satisfying appropriate criteria could be found.
to
I to
t~
""
T
1 T
Fig. 1. Two dimensional graphical illustration of finite time stabilities in a system 2=f(u,x,t). On the abscissa, a time interval extending from to to the finite value 7~ on the ordinate, the norm of either the system trajectory (in a-c) or the perturbation (in d). a. The trajectory starts at ~0 within e and either remains within (simply stable system) or moves beyond (unstable system) the bound 7. b. A quasi-contractively stable system's trajectory starts within c~, remains within 7, and after a settling time t~ remains within a ~ greater than c~. c. A contractively stable system's trajectory starts within c~, always remains within 7, and after t t remains within a ~ smaller than c~. d. The ongoing perturbation u(x,t) is bounded, i.e., its norm always remains within
101
Subsequently, Weiss and Infante recognized that the theory could be made even more realistic by considering systems which operate over finite, as opposed to infinite, time intervals since, in any real system, practical stability during a finite time is sufficient for the completion of any desired task. They also derived simple criteria for this type of behavior. The finite time analogue of Lyapunov stability is defined for a case with no perturbations (Fig. la) as the situation where, with bounded initial conditions, all trajectories remain bounded during the required period. Conversely, the system is unstable if, under the same conditions, they do not remain bounded. These definitions were then extended to systems subjected to perturbing forces. Three definitions were proposed which express separate and progressively narrowing conditions of finite time stability under perturbations. Simple stability under perturbing forces implies essentially that, if both the initial conditions and the perturbations are small, all trajectories will remain bounded during the required period. Quasi-contractive (Fig. lb) and contractive (Fig. lc) stability imply, in addition to simple stability, that the bounds on the trajectories become narrower after a certain settling time. If the late boundary is smaller than the initial one, the stability is contractive and not quasi-contractive. The different forms of stability differ operationally in the sense that a contractively stable control system requires less input energy to result in an acceptable output than a simply stable one (Weiss, personal communication): the intuitively acceptable reason is that, under conditions where the target state is the origin, a contractively stable system is heading toward the origin as a matter of course and without the benefit of the control, while this is not necessarily the case with simply stable systems. These definitions were expressed more rigorously by Weiss and Infante. To illustrate the concept of finite-time practical stability in the context of continuous dynamical systems, let us consider a system of N first-order real differential equations:
2 = f(x, t)
(1)
defined over the finite interval of time J = [to, t o + T). The initial state vector X(to) is Xo. The function f is assumed to be smooth enough in x and in the time interval J (i.e., in t e J ) to assure existence and uniqueness of solutions over J , as well as continuous dependence on the initial conditions at t o. (It is not required, however, that f(0, t)=0.) Prespecifying a norm II" l] (a "norm"
being a real number that represents what, in vector spaces, is referred to as "length") and two bounds c~ and 7, the system [Eq. (1)] is stable with respect to the reference set (e, 7, II' I], to, T) with e-- 10. These recursively determined normalized rank scores formed, for each bin-rate trajectory, a time series whose points were either within or outside of a trend-free region bounded by the values plus and minus 1 on the ordinate. (As described in
103 gravity-sensitive ones, and that it is quite possible that other classes of afferents would display quite differei]t properties (e.g., Macadar et al., submitted for publication).
Results, the ensemble of time series for all trajectories at each stimulus state was used to characterize the finite time stability of that state by fitting appropriate boundaries.) For each particular trajectory, the average of the normalized values was used as a measure of its wandering. Because certain variations in bin-rates, e.g., those symmetrical about their average (or trajectory-rate), may not be detected by the Kendall trend test (Otnes and Enochsen) and still constitute wandering under our definition, the coefficients of variation of both the bin-rates and of the interspike intervals were also computed for all trajectories. Tests were always at the 95 To confidence level unless otherwise specified. tt is important to point out that we have analyzed only a small sample of a single functional class of afferents, i.e., the tonic
Results
The perturbing dicks had a significant augmenting effect upon the directional sensitivity (DS), defined as a systematic variation of rate discharge as a function of head position. This is illustrated in Fig. 2 where each point represents a trajectory-rate, i.e.,
N-10-12
32
(a)
104-
30
iO'Zk
,m as
(~ IO0--I
D
LU
N-10-3
W 98-
IE
Por?urbotIon OFF
Perlur botlon ON
6~
~"
6"
6"
~~
Per~rbaffon ON
;~
;.
6"
SUCCESSIVE SPATIAL POSITIONS
;-
Per lurbotlo~ O FF
;.
;.
;~
SUCCESSIVE SPATIAL POSITIONS
N-10-9
(c)
" ~ 27.
,,o. 26 84
bJ Z4
Z
2 3 84
22
6"
$"
6"
6"
~"
6"
SUCCESSIVE SPATIAL POSITIONS
Fig. 2a-c. Relative influence of perturbations on the directional sensitivity, wandering and multivaluedness of three different gravitysensitive afferents. Abscissa: Spatial position of the head in degrees about the roll axis with perturbations ON and OFF. (Points corresponding to different stations at the same position, i.e. 5 degrees, are shifted laterally slightly to avoid superpositions.) Ordinate: Mean impulse rate during maintained head position; marks represent trajectory-r~/tes and error bars indicate standard deviations of 1 sec bin-rates. Temporal order of recordings: 9 = 1, 9 = 2, 9 = 3, A = 4 and [] = 5. The mean slope of the differences between rates at the positions 0 degrees and 5 degrees represents the directional sensitivity for each cell; the scatter of marks at each position represents the multivaluedness; and the error bar represents the wandering during each trajectory. There were three types of perturbation effects, a Type I: Trajectory rates at 5 degrees with perturbations ON significantly differed from those at 5 degrees without perturbations; no significant effects at 0 degrees, b Type II. Trajectory rates at 0 degrees with perturbations O N significantly differed from those at 0 degrees without perturbations; no significant effects at 5 degrees, c Type III: No significant effect of perturbations on trajectory rates.
104
the overall mean rate of the spike train at a particular stimulus state, and each sub-figure shows data from one cell only. Each state is characterized by one of two possible positions and the presence or absence of clicks. The vertical bars represent the standard deviation of the population of bin-rates in that trajectory. The DS was measured as the average of the absolute values of the slopes connecting points at two positions. For all of 10 cells used for the DS analysis, it was greater with the perturbation ON than O F F : this result would occur only once in 2 9 cases on the basis of chance. In the guitarfish, mechanical jitter can increase, decrease or even reverse the DS (Macadar et al., submitted for publication). Three basic types of "gravity-sensitive" afferents were found in the DS analysis. In the first type (Fig. 2a), at the 5-degrees position there was an increase in trajectory-rates with perturbations ON relative to those without perturbations, as indicated by the Wilcoxen test. At 0 degrees differences were not significant. In the second type (Fig. 2b), at 0 degrees there was a significant decrease in trajectoryrates with perturbations ON relative to those without perturbations. At 5-degrees differences were not significant. I n t h e third type (Fig. 2c), trajectory-rates with perturbations ON were not significantly different from those without them at either of the two positions. A multivaluedness was encountered that exceeded what could be expected (at the stated confidence level) simply because of the wandering. Figure 3 illustrates for a typical cell that the multivaluednesses, i.e., the variability between trajectories, was greater than the wandering, i.e., the variability within each trajectory. The quantitative and statistical evidence for this assertion rests, as mentioned, upon the H-statistic from the Kruskal-Wallace nonparametric analysis-of-variance test. This test was applied to 6 cells at each of the 6 stimulus states and the
5o 28-
.."
"
.m
27-
0]
2
26-
! 25-
,b
i~
2'o
2's
3o
TIME (see.)
~176 t 25-e...q
m-4
:
,:.
, -
v" -
.~.' ' " . . 4 "
/
]~
b
" ~. ".." ""X ..
"
.
o
It
o.,II ~"
k"
/ """~
9
....-
,
",
R:../
23 Z I.d
22
~
ib
i~
2b
~5
-~o
TIME (sec.)
Fig. 3a and b. Trajectories of mean 1 sec bin-rates for cell N - 1 0 9 with perturbations ON. The means from the three trajectories at 0 degrees (b); or from those at 5 degrees, (a) are derived from three significantly different statistical distributions, indicating multivaluedness in the means as indicators of spatial position. The three trajectories at 5 degrees (a) are clustered along the ordinate with an overall m e a n that is significantly greater than that for 0 degrees (b), indicating a significant sensitivity to spatial position
Table I. H-statistics determined from the Kruskal-Wallis test and chi-square significance levels (p < 0.05) used as a test of significant multivaluedness for 6 cells at each stimulus state. States that were not significantly multivalued are indicated by asterisks Cell
H-Statistics 0~
N - 10-3 N - 10-7 N-10-9 N-10-10 N-10-12 N- 7-12
2.22* 51.76 77.76 7.73 13.78 43.16
Perturbation O N 5~ 18.04 37.86 36.58 t 1.91 12.29 54.00
0~
0~
24.20 28.77 77.15 7.80 25.52 52.80
24.49 31,65 30,28 16,12 11.58 31.95
Perturbation O F F 5~ 2.71 * 45.50 13.18 44.17 0.75* 46.93
Sample
Degrees
0~
size N
of freedom k
2 Xk,O.05
0.08* 62.24 35.47 11.36 0,55" 32.41
19 31 28 32 24 24
1 2 4 1 1 2
3.84 5.99 9.49 3.84 3.84 5.99
105
Perturbation ON
I00
Per turboflon OFF
80 (_3 C-- 6Oif)
m 4020-
0
0~
5"
0~
0~
5"
0~
SUCCESSIVE SPATIAL POSITIONS
Fig. 4. H-statistics computed from the Kruskal-Wallis test for two Type III cells: 9 = N - 1 0 - 9 ; (D = N-10-10. The magnitude of the H-statistic at each position is an estimator of the spread of the multivaluedness in the ensemble of trajectories at that position. The error bars indicate the 95% confidence interval of the H-statistic as determined from its approximately chi-square distribution. H-statistics at positions with perturbations O N are significantly different (at p < 0 . 0 5 level) from those at corresponding positions without them by the chi-square test
resulting H-statistics and chi-square statistics at the 95% level (p
