INTRODUCTION

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The first two joints of the crab leg provide an excellent system for comparison of spiking and nonspiking (analog) information transfer and processing in a simple motor system, because the movement of the first two joints of the crab leg are monitored by both nonspiking and spiking proprioceptors. The nonspiking thoracic-coxal muscle receptor organ (TCMRO) spans the TC joint, while the coxo-basal (CB) joint is monitored by the spiking CB chordotonal organ (CBCTO) and by three nonspiking afferents arising from levator and depressor elastic strands. These receptors, and feedback from other leg proprioceptors, play significant roles in modifying motor output in postural control and during locomotion. They may directly modify the strength of motor neuron activity via reflex pathways or act indirectly by providing input to central pattern generating networks (Marder and Bucher 2001; Pearson 1995, 2000).

The CBCTO is a typical arthropod chordotonal organ (Mill 1976) and consists of an elastic connective tissue sheet that spans the CB leg joint (Alexandrowicz 1967; Alexandrowicz and Whitear 1957) that produces levation and depression of the more distal leg segments. The strand arises proximally from an endoskeletal peg on the rim of the coxopodite and runs distally to insert on the rim of the basiopodite between the two heads of the levator muscle. Embedded in this strand are 70-80 bipolar neurons running parallel to the long axis of the strand. These sensory neurons respond unidirectionally to stretch or relaxation of the receptor (Bush 1965), corresponding to extension of the CB joint (depression of the leg) and flexion of the CB joint (levation of the leg). Individual afferents can be classified into functional groups that respond to position, velocity, and acceleration of the joint. True tonic (position sensitive) fibers have been described that respond relatively linearly to joint position but individual fibers only respond to joint angles from the middle of the joint angle range to one extreme of joint movement and are therefore unidirectional (Bush 1965). Movement sensitive fibers (velocity and acceleration) are also unidirectional, responding to either stretch or release of the receptor and exhibit little or no tonic firing when the receptor length is held constant (Bush 1965).

A general goal of many physiological studies is to determine the input-output relationship of a system to predict the response of the system to arbitrary inputs and to assess the effects of experimental manipulations on system function. One method for specifying this relationship is to assume a structural configuration of the system that can be described by a series of differential (or other) equations and to determine the value of the coefficients of these equations that best predict system performance. A second approach is to determine the system functional F, y(t) = F[x(t)], where x(t) and y(t) are the input and output of the system, respectively. Using appropriate excitation, x(t), of the system, and observing the response y(t), the system functional F may be computed. This nonparametric, or "black box" approach to the system identification problem can be used to determine the system transfer characteristic without specifying the internal structure or mechanisms present. The actual mechanisms are replaced with a filter with exactly the same transfer characteristics as the system under study. White noise (or Wiener kernel) analysis is a nonparametric approach to systems identification (French and Marmarelis 1999; Marmarelis and Marmarelis 1978; Westwick and Kearney 1998) that has been used in the study of neurophysiological systems, such as visual (Marmarelis and Naka 1972; Sakai et al. 1988), auditory (Eggermont 1993), and mechanoreceptor systems (Dickinson 1990; French et al. 2001b; French and Wong 1977; Kondoh et al. 1995; ).

We first wished to determine the transfer functions of the spiking and nonspiking receptors to compare their linear and nonlinear transfer characteristics and to provide a framework for functional comparisons of receptor properties, synaptic transmission, and information rates in these neurons. The best fitting linear estimates of CBCTO afferent characteristics were consistent with afferent responses that had been described previously with deterministic stimuli, and these linear response properties were found to be independent of movement amplitude. Nonlinear models of CBCTO afferent responses were constructed based on the first- and second-order Wiener kernels, and second-order models were able to accurately predict the firing pattern of individual afferents.

	METHODS

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Isolated ganglion-receptor preparations of male and female green shore crabs, Carcinus maenas, were used in all experiments. The walking legs and chelea were autotomized, and the dorsal carapace, viscera, and brain were removed. The sternal artery supplying the thoracic ganglion was immediately cannulated, and the ganglion was perfused with chilled (16-17°C) oxygenated saline at a rate of 2-3 ml/min for 15-20 min before proceeding with further dissection. The saline composition was (in mM) 500 Na⁺, 12 K⁺, 20 Mg²⁺, 12 Ca²⁺, and 576 Cl, buffered to pH 7.2 with 10 mM Tris maleate (Ripley et al. 1968). The nerve to the CB chordotonal organ of the fifth leg segment was isolated, and the chordotonal organ, along with the surrounding connective tissue, was freed from its proximal attachment to the small endoskeletal peg on the rim of the coxopodite. The chordotonal organ was traced between the two heads of the levator muscle to its distal attachment on the proximal rim of the basiopodite and cut at this point. All remaining nerves from the thoracic ganglion were cut and the receptor-ganglion removed and placed in a small (5 ml) Sylgard lined plastic chamber. The chamber was continuously superfused with chilled oxygenated saline for the duration of the experiment. The connective tissue surrounding the proximal end of the CBCTO was pinned securely to the Sylgard with three to four stainless steel minutien pins. The distal end of the receptor was attached to an electromechanical puller via a small hook inserted through the connective tissue at the distal end of the CBCTO (Fig. 1A). The in situ length of the chordotonal organ was measured with a caliper when the joint was held in the middle of the physiological range (approximately 100°, Mill 1976). The resting length of the CBCTO was set to this length by mechanically adjusting the position of the puller after the preparation was pinned to the substrate. The response of the CBCTO was first tested with trapezoidal movements while observing the combined afferent response in the extracellular CBn recording. Any receptors that did not have a maintained tonic discharge or where the magnitude of the whole CB nerve response was not uniform with stretch and release of the chordotonal organ (positive and negative trapezoids), were assumed to be damaged and were not used for further analysis. The electromechanical puller was constructed from a 5" diam low-midrange speaker, and the position of the speaker cone was monitored by an optical position sensor consisting of a light source, photodiode, and an optical wedge scale (Hofmann and Koch 1985). The puller was controlled by a proportional-integro-differential (PID) controller (Hofmann and Koch 1985), operating in a length feedback mode. The frequency response of the puller is flat to a cutoff frequency (f_c) of approximately 220 Hz over a displacement range of ±1 mm, with a sight resonance occurring at 280 Hz (Fig. 1B).

View larger version (26K): [in this window] [in a new window]   Fig. 1. A: schematic diagram of the experimental preparation. The thoracic ganglion, chordotonal nerve, and the CB chordotonal organ (CBCTO) are isolated and continuously superfused with oxygenated saline in a small volume chamber. The proximal end of the CBCTO is securely pinned to the Sylgard substrate of the chamber with 3-4 stainless steel pins while the distal end of the receptor is attached to the output shaft of an electromechanical puller. The electromechanical puller is constructed from a small loudspeaker and the position of the output shaft is monitored with a optical position sensor. The puller is in a length feedback control loop controlled by a proportional-integro-differential (PID) controller. Extracellular recordings of CBCTO afferent activity were made with a suction electrode placed on the mid-point of the chordotonal nerve (CBn) and intracellularly from individual CB afferents (CBa) close to the point where the CB nerve enters the thoracic ganglion. B: power spectrum (left) and output amplitude probability density function (pdf) of the puller system. The response of the puller is flat out to the cutoff frequency of the applied noise (220 Hz) and the amplitude distribution of the output is Gaussian (solid line) when driven with 220 Hz white noise.

The total range of movement of the CB joint is approximately 100° (Mill 1976), and the normal range of CB joint movement during lateral walking in Carcinus is approximately 40° for the leading and trailing legs (Clarac and Coulmance 1971). For the size of animals used in these experiments (approximately 4- to 5-cm carapace width), a length change of 0.4 mm corresponds to a joint angle movement of approximately 30°, which was the typical length change (peak-to-peak amplitude) applied to the CBCTO.

The ganglionic sheath at the point where the CB nerve (CBn) enters the ganglion was removed with fine forceps to permit intracellular recording from individual CBCTO afferents. Intracellular recordings were made with microelectrodes filled with 2 M KAc and amplified by a bridge electrometer (NPI SEC-05L). An extracellular suction electrode was placed on the CBn mid-way between the ganglion and the receptor to monitor whole nerve activity. The intracellular recordings were confirmed to be from CB afferents by spike-triggered averaging of the CBn recording. Inspection of the shape and amplitude of the extracellularly recorded CB afferent spike also allowed confirmation of the recording being from a unique afferent in each preparation to eliminate duplicate recordings. It was not possible to individually identify single afferents based on physiological criteria, and all afferents were simply classified by their response to trapezoidal stimulation of the CBCTO. Classifications were made with respect to directional sensitivity (stretch or release of the CBCTO, corresponding to depression or levation of the leg, respectively) and as position-, velocity-, mixed position-velocity-, or acceleration-sensitive afferents.

All signals were digitized on-line using a CED Power1401 laboratory interface (16-bit A/D converter, ±5 V range, 0.4 µs conversion time) with intracellular and extracellular recordings sampled at 12.5 kHz, while the position output of the feedback controller was sampled at 2.5 kHz. White noise was generated by a 31-bit pseudo-random number generator clocked at 10 kHz, resulting in a pseudorandom sequence length of >200,000 s. The digital output of this generator was filtered to the desired bandwidth with a variable 8-pole low-pass filter (Wavetek 852), DC-offset, and amplified as required. Trapezoidal stimuli were generated by a custom-built waveform generator with variable rise/fall time, amplitude, and duration.

White noise analysis

The mathematical basis for the Wiener approach to systems identification has been discussed in several reviews (French and Marmarelis 1999; Marmarelis and Marmarelis 1978; Westwick and Kearney 1998), and an abbreviated description of the theory follows.

In classical linear systems theory, the input-output relationship of a time-invariant linear system is described by the convolution integral
where h() is the impulse response of the system. This integral states that the output of the system, y(t) can be written as a weighted sum of the past inputs, x(t), where the weighting function at each time lag is h(). If h() is known for a linear system then we may predict the system response to any input by application of the convolution integral. In practice however, most, if not all, biological systems contain significant nonlinearities.

The Wiener analysis of nonlinear systems is based on Volterra's (1959) approach to functional identification of a finite memory nonlinear system, where the relationship between x(t) and y(t) can be described by the Volterra series
where k₀, k₁(), k₂(₁, ₂), k₃(₁, ₂, ₃) ... are the zero-, first-, second-, third-, ... , order kernels. Note that the first-order term is exactly the same as the convolution integral for a linear system. The difficulty in using the Volterra series is that the nth order interaction depends on all kernels of order higher than n, and the estimation of the system kernels is therefore not practical. Wiener showed that if the input to the system was Gaussian white noise, a series expansion could be constructed with mutually orthogonal terms, and a convenient and computationally practical scheme for measuring the system kernels could be implemented. The input-output relationship for a system can then be written as
where G_m are orthogonal functionals if the input to the system, x(t), is a zero-mean Gaussian white noise signal and h_m is the mth order Wiener kernel.

The first three Wiener functionals are

where P is the power density of the input white noise x(t). The set of Wiener kernels h_m characterizes the system and allows the prediction of the system response to any arbitrary input. The zero-order kernel describes the DC response of the system, the first-order kernel is the best fitting (in the mean squared error sense) linear response of the system, and the second-order kernel describes the nonlinear interaction of two inputs in the past on the output of the system in the present and so on for higher order kernels.

There are numerous methods for the calculation (or estimation) of the Wiener kernels operating directly in the time domain or after transformation to the frequency domain (French and Marmarelis 1999; Marmarelis and Marmarelis 1978). We used the method of cross-correlation (Lee and Schetzen 1965) to calculate the Wiener kernels, which is simple to implement and reasonably efficient given the computational power of standard laboratory computers. All computations were performed by programs written in the Spike2 (CED, Version 4.03) script language. The cross-correlation method is still commonly used and was relatively easy to implement with the Spike2 data analysis software. Use of the Spike2 software also allowed the convenient addition of model predictions and other derived parameters to the original Spike2 raw data files.

The maximum noise bandwidth applied to the CBCTO was 220 Hz, which was greater than the frequency of the maximum gain for most CBCTO afferents. This limit also served to minimize the error in kernel estimation due to excessive input bandwidth, while reducing the error associated with the finite width of the autocorrelation function of bandlimited white noise (Marmarelis and Marmarelis 1978). All system kernels were computed for a 30-ms time range (75 time lags at a sample interval of 0.4 ms), which was sufficiently long for all kernels to decay to zero. While there is no explicit rule for determination of the record length required to estimate the Wiener kernels, longer records will reduce the error in the estimate (Marmarelis and Marmarelis 1978). In addition, the number of data points (input-output pairs) must be greater than the number of free parameters in the kernels. The use of 75 time lags to calculate the zero-, first-, and second-order kernels results in 2,926 free parameters. All calculations of first- and second-order kernels were computed over a 15- to 20-s time interval or 37,500-50,000 data points. First- and second-order kernels were not smoothed and were plotted with Axum graphing software (Ver 5.0c, Mathsoft).

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Recordings were made from 74 CBCTO afferents obtained from 20 preparations. CBCTO afferents are directionally sensitive (Bush 1965), and approximately equal numbers of levation and depression sensitive units were recorded. Within each of these two broad classes, individual afferents were classified based on their response to trapezoidal (ramp-and-hold) stimulation into position sensitive (n = 4), mixed position/velocity (n = 25), velocity (n = 41), and acceleration sensitive (n = 4) afferents. Position-sensitive units responded with tonic discharge proportional to the length of the CBCTO, velocity units fired only during the ramp (constant velocity) phase of the movement, position-velocity afferents had both tonic and phasic response components, and acceleration units responded only when there was a change in movement velocity. Single CBCTO afferents could not be individually identified, but in each experiment, comparison of the spike triggered average of the extracellular recording of the afferent from the CB nerve was used to eliminate duplicate penetrations of the same afferent.

Typical recordings from two CBCTO afferents are shown in Fig. 2. The first (Fig. 2A) is a velocity sensitive afferent that responds to lengthening of the CBCTO corresponding to depression of the CB joint. This afferent only fired during the positive ramp (constant velocity) phase of the movement, and there was no tonic discharge during periods of constant length. The second record (Fig. 2B) is from a position-velocity-sensitive afferent that has both phasic and tonic response components to a trapezoidal stretch. This afferent fired phasically during the negative velocity phase of the ramp and fired tonically when the length of the receptor was decreased. The small amplitude potentials at the end of the ramp were presumably due to primary afferent depolarization (PAD) input from other CBCTO afferents (Cattaert et al. 1992; El Manira et al. 1991).

View larger version (18K): [in this window] [in a new window]   Fig. 2. Response of CBCTO afferents to trapezoidal stretch of the receptor. A: purely velocity sensitive afferent that responds to stretch of the chordotonal organ, corresponding to depression of the CB joint. B: position-velocity sensitive afferent that responds phasically to shortening of the chordotonal organ, corresponding to levation of the joint. This afferent did not fire tonically at the initial length of the receptor, which was equivalent to the length of the receptor at the middle of the joint angle range. Tonic firing of this afferent was proportional to the (shorter) length of the receptor.

Most of the afferents responded to white noise stimulation with a higher mean rate of firing when the stimulation was first applied and then maintained a decreased, but relatively constant, mean firing rate for the remainder of the stimulation (Fig. 3). In this example, the initial mean firing rate was approximately 85 Hz, and the rate decreased over the next 10-12 s and then remained relatively stable at 52 ± 3.5 (SD) Hz for the remaining 35 s of stimulation. All calculations of first- and second-order kernels were computed during the constant mean rate period over a 15- to 20-s time interval that included 600-1,200 spikes. Calculation of the system kernels over different nonoverlapping time intervals or for larger intervals produced similar kernels in all cases.

View larger version (15K): [in this window] [in a new window]   Fig. 3. Stimulation of the CBCTO with 220-Hz bandwidth white noise. A: 50 s recording of a single CBCTO afferent during white noise stimulation. The top record is the mean frequency of this afferent computed from the times of occurrence of action potentials using a 400 ms moving window. The mean firing rate of this afferent was higher just after the onset of movement and then slowly declined to a relatively constant average rate of 52 ± 3.5 (SD) Hz firing for 35 s. B: response of the afferent in A to random movement shown at an expanded time scale.

First-order linear response of CBCTO afferents

The first-order kernel of an afferent is analogous to the impulse response of a linear system and represents the best-fitting linear estimate of the systems response (Marmarelis and Marmarelis 1978). The first-order kernels were calculated by cross-correlation of the input (receptor length) with the output spike train, where the output spikes were considered to be delta functions produced by applying a simple level-crossing threshold to the intracellular recording. The units of this kernel are spike density, that is, spikes · mm¹ · s¹. A positive kernel value therefore denotes an increase in firing frequency while a negative value indicates a decrease in firing rate. The alternative interpretation of a first-order kernel for a spiking cell is that it is the average input that precedes each spike, as the cross-correlation integral reduces to this form when the output spike train is considered to be a sequence of delta functions. The time axis for the first-order kernel (or the impulse response function of a linear system) is the time preceding the generation of a spike (negative time) but is customarily plotted as presented here with increasing time lags () on the x axis. An example of a first-order kernel for a velocity sensitive afferent is shown in Fig. 4. The afferent responds with a strong phasic discharge on stretch of the receptor (depression), and the first-order kernel has an initial positive component corresponding to an increase in firing rate followed by a smaller negative peak. The first-order kernel for an afferent that fired phasically on release of the CBCTO (levation) would have a first-order kernel of similar shape but reversed in sign on the y axis, i.e., the negative peak would occur first (see Fig. 5C). The first-order kernels and their associated gain functions for the four classes of afferent response properties are shown in Fig. 5. In linear systems theory, the Fourier transform of the impulse response is the transfer function (gain and phase) of the system in the frequency domain. Gain functions for all afferents were calculated by taking the Fast Fourier Transform (1024-point FFT) of the first-order kernel.

View larger version (7K): [in this window] [in a new window]   Fig. 4. Response of a velocity sensitive afferent and corresponding 1st-order Wiener kernel. This is a velocity sensitive afferent responding to stretch of the receptor (A). The 1st-order Wiener kernel for this afferent (B) was calculated by cross-correlation of the input (receptor length) with the output spike train. The units for the kernel are spike density, with a positive value denoting an increase in firing rate while a negative value indicates a decrease in firing. The 1st-order kernel for a spiking cell may also be intrepreted as the average input (length) that precedes the occurrence of a spike. The 1st-order kernel for this and other CBCTO afferents decayed to 0 in approximately 20 ms.

View larger version (14K): [in this window] [in a new window]   Fig. 5. First-order Wiener kernels and gain for each CBCTO response class. The 1st-order kernels for afferents sensitive to position (A), mixed position-velocity (B), pure velocity (C), and pure acceleration (D) responses are shown in the right panels. The corresponding gain functions for each afferent type are give to the right of the 1st-order kernels. The bandwidth of the white noise stimulus was 220 Hz in all cases, with a peak-to-peak amplitude of 0.4 mm.

The first afferent (Fig. 5A) is a position-sensitive unit that responded to ramp stretches with a maintained firing during the tonic position of the ramp, although there was a small phasic increase in firing rate when the CBCTO was stretched at high velocity. The gain function for this afferent was similar to a low-pass filter with a constant gain up to approximately 40 Hz. The weak additional phasic response of this afferent to high velocity stretch of the CBCTO is reflected in the increasing gain at frequencies above 40 Hz until the gain then decreased above the cutoff frequency (f_c) of approximately 110 Hz. The first-order kernel and gain plot for a position-velocity afferent is shown in Fig. 5B. This afferent had a relatively constant gain at frequencies below 10-15 Hz after which the gain increased with a slope of approximately 20 db/decade, which is characteristic of a differentiator, with a f_c of 110 Hz. The phasic response component for this afferent was elicited by a decrease in length of the CBCTO (see Fig. 2B), which accounts for the initial negative peak of the first-order kernel.

An afferent with a pure velocity response to ramp stimulation, that is, with no tonic firing at a constant receptor length, had the first-order kernel shown in Fig. 5C. This kernel is symmetric around the x axis, which is characteristic of a purely velocity sensitive afferent. The response of this afferent to trapezoidal stimulation was similar to the one shown is Fig. 2A, but with the phasic response occurring on the falling phase of the ramp (decreasing receptor length). The gain increased at 20 dB/decade up to the f_c of 80 Hz for this afferent, characteristic of a first-order high-pass filter or differentiator. The final response type observed were afferents that were sensitive to the acceleration of the receptor. These units were also directionally sensitive, responding to positive or negative acceleration with one or two spikes at the beginning or end of the ramp. Similar to the velocity sensitive afferents, the slope of the gain curve increased with increasing frequency to a f_c of 140 Hz, but in this case the slope was close to 40 dB/decade, which is characteristic of a second-order high-pass filter.

The upper frequency limits (cutoff frequency) for the afferents presented in Fig. 5 are typical for others of similar type, although some velocity- and acceleration-sensitive afferents (n = 6) had a cutoff frequency very close to, or perhaps slightly greater than, the 220 Hz bandwidth of the applied noise. It was difficult or impossible in these cases to determine the actual cutoff frequency of these afferents. It is therefore likely that some of the CBCTO afferents have a cutoff frequency of 200 Hz or greater, but we cannot resolve this issue given the maximum frequency of our puller system.

Stimulus amplitude and afferent characteristics

To determine if the first-order response properties of CBCTO afferents were dependent on the amplitude of CBCTO movement, we applied white noise stimulation of different amplitudes during recordings of afferents of several response types. A velocity sensitive afferent that responded to shortening of the receptor (levation of the joint) was stimulated with three amplitudes of movement (±0.1, ±0.3, and ±0.6 mm peak-to-peak amplitude, 7°-40° equivalent angle range) at a bandwidth of 220 Hz. The first-order kernels and associated gain functions calculated for each movement amplitude are shown in Fig. 6. The first-order kernels calculated for the two larger movement amplitudes were very similar. The amplitude and width of the initial negative peak of the kernel calculated with low-amplitude movement was also similar, but the following positive peak was smaller (gray line). The gain functions calculated from these kernels are all characteristic of a velocity sensitive afferent with a slope close to 20 dB/decade (Fig. 6B). The only difference is seen in the gain curve for the low-amplitude movement (gray line), where the corner frequency was approximately 85 Hz compared with a corner frequency of approximately 100 Hz for the two larger movement amplitudes. The length (L) of the receptor at a given frequency () and amplitude (A) of movement is L(t) = A × sin (t) and the velocity of movement (dL/dt) is therefore A ×  cos (t). The range of movement velocity at a constant bandwidth of applied white noise will therefore decrease in proportion to the amplitude of the signal. As this afferent was velocity sensitive and the applied noise for all amplitudes had a bandwidth of 220 Hz, the range of velocities in this signal decreased with decreasing amplitude. The observed decrease in corner frequency of this purely velocity sensitive afferent can therefore most likely be attributed to the decreased range of velocity. This is illustrated by the probability density function for the velocity of the applied movement (Fig. 6, inset), where only 25% of the velocities present in the largest amplitude input were present in the low-amplitude noise. To determine the true cutoff frequency of this neuron at low movement amplitudes, the bandwidth of the signal would have to be increased to maintain the range of velocities present in the higher amplitude movements. As the puller system was operating at its maximum bandwidth, we could not compensate for this effect.

View larger version (14K): [in this window] [in a new window]   Fig. 6. First-order response properties with different amplitudes of stimulation. A velocity sensitive afferent was stimulated with 220 Hz bandwidth noise at 3 amplitudes of movement (±0.1, ±0.3, and ±0.6 mm; peak-to-peak amplitude). The 1st-order kernels for the 2 largest amplitudes are plotted as the black lines, while the 1st-order kernel calculated with low-amplitude stimulation is shown in gray. The corresponding gain functions for each amplitude of stimulation are very similar for the 2 largest amplitudes (black lines), while the gain for the low-amplitude movement (gray line) follows the 2 others, but has a lower cutoff frequency. Inset: the probability density function for the velocity of the applied signals at each amplitude.

Second-order kernels

The second-order Wiener kernel, h₂(₁,₂), provides a measure of the second-order nonlinear interactions between the input signal at past times (₁ and ₂) and their effect on the output of the system in the present. For example, an off-diagonal positive peak in the second-order kernel at (₁, ₂) denotes an increase in output (firing rate) due to the interaction between the two parts of the input signal at these time lags. When the time lags are equal (₁ = ₂), the kernel describes the amplitude-dependent nonlinearities (Marmarelis and Marmarelis 1978). Second-order kernels were calculated for CBCTO afferents of all response types by cross-correlation of the input with the spike train output. In all cases, the second-order kernel was nonzero, indicating that there was a nonlinear component to the overall response of the afferents.

The second-order kernels for afferents representing each of the response types shown in Fig. 4 are presented in Fig. 7. The kernels are presented as three dimensional (3-D) plots, with the z axis representing changes in firing rate (units are spikes · mm² · s²) while the x and y axes are the two time lags 30 ms. All of the kernels had their largest amplitude components on the diagonal (₁ = ₂) with smaller amplitude off-diagonal peaks or troughs. The terms on the diagonal are the amplitude-dependent nonlinearities, while the off-diagonal terms are nonlinear interactions between the input at different time lags.

View larger version (54K): [in this window] [in a new window]   Fig. 7. Second-order Wiener kernels for each CBCTO response class. The 2nd-order kernels for position, position-velocity, velocity, and acceleration sensitive afferents are shown as three-dimensional (3-D) surface plots. The 2 time axes are the 2 time lags, 1 and 2, and the z axis is the amplitude of the 2nd-order kernel with units of spikes·mm2·s2. The maximum values for the time axes are 30 ms.

The second-order kernel for a position-sensitive afferent consisted of a single positive peak lying on the diagonal with smaller negative off-diagonal valleys. The kernels for the remaining afferents of each response type had large positive and negative components on and off the diagonal. Unlike the reversal in polarity of the first-order kernels associated with the directional sensitivity of an afferent, the relative order of peaks and troughs in the second-order kernel is independent of the directional sensitivity of the afferent. For example, velocity-sensitive afferents responding to increasing or decreasing length of the receptor (positive or negative velocity response) all had a second-order kernel similar to the one shown in Fig. 7.

Model responses

After calculation of the first- and second-order kernels, the response of the afferent was modeled by convolving the first- and second-order kernels with a 3- to 5-s portion of the data set following the time interval over which the kernels were computed. As the model prediction has units of spike density, the model spike train was constructed by applying a threshold to the model output and registering a spike for every positive threshold crossing (Marmarelis and Marmarelis 1978). The threshold value was chosen to maximize the value of the index, [a/(n + m  a)], where a is the number of correctly predicted spikes, n is the total number of spikes in the experimental response, and m is the total number of spikes in the model response (Kondoh et al. 1995). The error bound for correct spike prediction at a given threshold was ±3 ms, representing the width of the spikes plus a minimal refractory period.

The prediction of the response of a position-velocity sensitive CBCTO afferent was based on the nonlinear model of the system consisting of the first- and second-order Wiener kernels (Fig. 8). In the example shown here, the second-order model generated 129 spikes compared with 129 spikes generated by the afferent, with 120 "correct" spikes for an accuracy (correctly predicted spikes/real spikes) of 93%. Use of the first-order model only for the prediction resulted in a slightly lower accuracy of 83%. This is not necessarily unexpected, as inspection of the K₁ and K_1,2 models indicates that inclusion of the second-order response component (K₂) mainly accounts for the directional (rectifying) nature of the afferent response along with an amplitude correction, while the (linear) high-pass first-order kernel primarily determines the frequency response of the afferent. For the set of 20 afferents where first- and second-order models were computed (including examples all response types), the average model accuracy was 86 ± 7.4%. The accuracy of the model prediction for most of these afferents was constant over the entire time of stimulation. For example, for a velocity-sensitive afferent that maintained a constant mean firing rate for over 50 s of stimulation (55 ± 2 Hz), the average model accuracy was 90 ± 3% when calculated for seven nonoverlapping 1.5-s intervals in a 50-s stimulation period.

View larger version (28K): [in this window] [in a new window]   Fig. 8. The Wiener kernel model for a velocity sensitive afferent. The length of the CBCTO (len) and the actual output of the afferent (CBa) are shown in the bottom and top records, respectively. The system output calculated by convolving the 1st- and 2nd-order Wiener kernels with the input are K1 and K2, and the complete 2nd-order model resulting from the sum of K1 and K2 is shown in the K1,2 record. The 2nd-order Wiener model was converted to a spike train (gray spike train) by setting a threshold function that was adjusted to give the best fit to the real output. The accuracy of the model output calculated for a 2-s interval was 91%.

When models were computed for afferents that did not have a constant rate of firing (in contrast to the example shown in Fig. 2), the accuracy of the model prediction and the ratio of model to experimental spikes varied during the period of stimulation. For example, the second-order Wiener model was computed for a velocity-sensitive afferent that had an initial (transient) mean firing rate of 70 Hz, which then declined to, and remained stable at, approximately 55 Hz for 15 s, after which the mean rate decreased steadily to 40 Hz. The kernels used to compute the model response were determined from data taken during the sustained 55-Hz mean firing rate interval. When the spike train was predicted for the afferent 2 s after this interval, when the mean rate was still 55 Hz, the model accuracy was 91% and the ratio of model to real spikes was 1.01. During this time period, the model therefore predicts approximately the same number of spikes that were actually generated with an accuracy of 91%. If the model prediction was made earlier in the data record when the mean rate was higher (70 Hz), the model accuracy was 93%, but the ratio of model to real spikes for this early time interval was 0.95. The model during this period of higher mean firing rate therefore predicts a similar percentage of experimental spikes correctly, but slightly underestimates the number of real spikes that were produced by the afferent. When the model was run later in the experimental record (+20 s), when the average firing rate had decreased to 45 Hz, the accuracy of the model predication was still very high (98%), but the ratio of model to real spikes increased to 1.26. The model now predicts 26% more spikes than were actually generated by the afferent, but 98% of the real spikes were predicted correctly.

This was a consistent finding in all experiments where the mean firing rate of the afferent declined during the stimulation compared with a relatively long period of constant firing for other afferents (see Fig. 3). The first-order kernels measured for early, middle, and late time periods for the afferent described above were very similar and the second-order kernels had a similar pattern of peaks and troughs, indicating that the transfer characteristics of the afferent had not changed markedly during the stimulation period. The decline in firing rate may be due to adaptation of this afferent during maintained stimulation, and this nonstationary property of the system cannot be captured by the system kernels (Marmarelis and Marmarelis 1978). The similar accuracy of the model predictions, but the changing ratio of experimental to model spikes, presumably reflects the decreasing probability of spike generation (adaptation) to the same preferred input during the period of stimulation.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

The transfer characteristics of spiking afferents from the CB chordotonal organ in the crab were determined using white noise analysis based on the Wiener methodology. Previous studies of several arthropod chordotonal organs that used deterministic stimuli, primarily sine waves and trapezoids, showed that the afferents can be classified in to several functional classes based on their tonic and phasic responses to deterministic stimulation. Linear estimates of receptor response properties have also been made based on deterministic stimuli (Chapman and Smith 1963; French et al. 1972). The difficulty with using sine or trapezoidal stimulation is that, while it may provide an intuitive picture of the response to quasi-physiological inputs, it is (experimentally) time intensive and does not characterize the nonlinear properties of the system. In addition, the long time course of an experiment required to fully characterize an afferent may lead to errors based on the degradation of the preparation over time. White noise provides a powerful test signal for this type of analysis in that it is a global stimulus, encompassing all frequencies and inputs within the bandwidth of the random driving signal. This data can be used to construct a mathematical description of the system incorporating both linear estimates of system function and the nonlinear terms needed to adequately model the properties of the system. This method is also extremely efficient, as the required data can be obtained by stimulation of the receptor for a relatively short period of time.

Afferent responses to deterministic stimuli

The response properties of crustacean chordotonal organ afferents have been described in previous studies (Bush 1965; Mill 1976) and are similar to other arthropod chordotonal organs (Büschges 1994; Field and Matheson 1998; Hofmann et al. 1985; Matheson 1990; Mill 1976). The responses of the crab CBCTO afferents in this study were essentially identical to previous studies using deterministic stimulation, and examples of all response types previously described were obtained. The first-order kernels and associated gain functions calculated from white noise stimulation of each of these response types was consistent with the responses of the afferent elicited by trapezoids (Figs. 4 and 5).

Afferents that were primarily position-sensitive with little or no phasic response at low velocities of stretch had essentially monophasic kernels characteristic of a low-pass filter. The gain for these afferents was constant for low frequencies with a slight increase observed at frequencies above 40-50 Hz, indicating a velocity sensitivity at these higher frequencies. In contrast, an afferent that only responded to position would be expected to have a constant gain up to the cutoff frequency, but we never encountered any position sensitive afferents that did not exhibit some small phasic response when stretched at high velocity.

Afferents that responded with both phasic and tonic components to trapezoidal stretch had first-order kernels with asymmetric positive and negative peaks. The gain functions for these afferents were flat to around 10 Hz and the gain increased with a slope of 20 dB/decade above these frequencies, which is characteristic of a differentiator, and hence velocity sensitivity. Afferents with a pure velocity response to stretch (no tonic firing) had symmetrical first-order kernels with a corresponding gain function characteristic of a first-order high-pass filter with the gain increasing with increasing frequency with a slope of 20 dB/decade. Afferents that were sensitive to the acceleration of the chordotonal organ had triphasic first-order kernels and a high-pass gain function, although with a slope of approximately 40 dB/decade, characteristic of a second-order high-pass linear filter. We never encountered other mixed response types, specifically the mixed velocity-acceleration sensitive units that have been described in the locust femoral chordotonal organ (Kondoh et al. 1995). The maximum gain for all afferents occurred at similar frequencies (80-100 Hz) for all of these response types, except for a small number of afferents that had maximum gains close to, or possibly greater than, the bandwidth of the applied noise. When the CBCTO was driven with different amplitudes of white noise, the linear response properties of the afferents were found to be independent of stimulus amplitude.

Nonlinear terms and models

All of the afferents tested had nonzero second-order Wiener kernels (Fig. 7), indicating that there were nonlinear components to their response to CBCTO movement. The primary contribution of the second-order term in the Wiener expansion to the overall system response was to account for the directional sensitivity (rectification) that is seen with deterministic stimulation of all response types.

The first- and second-order Wiener kernels were sufficient to model the response of CBCTO afferents with reasonable accuracy. For afferents that responded to white noise stimulation with a relatively constant mean firing rate, the mean accuracy of the spiking output derived from the model was (86 ± 7.4%, n = 20), where accuracy was measured as the percentage of the experimental spikes that were correctly predicted by the model. In most instances, the total number of spikes generated by the model was close to the number of spikes generated by the afferent, and there were relatively few false positives in the prediction. The only exceptions were for models of afferents that did not maintain a relatively constant mean firing rate during the period of white noise stimulation. The response of these afferents was characterized by a steadily decreasing mean firing rate during stimulation. Second-order models predicted the experimental output with similar accuracy compared with models for constantly firing cells. However, although the accuracy of the model could be quite high with respect to predicting real spikes, the model predicted fewer spikes than were actually produced during earlier portions of the data set when the mean firing rate was high. A greater number of model versus real spikes were predicted later in the stimulation period when the mean firing rate had decreased. The Wiener method is only valid for systems that are stationary, that is, the characteristics of the system should not change appreciably over time. We assume that the changes observed in over and underestimation of real spike number reflects adaptation of the afferent, which is a time varying property of the system.

Comparison with other systems

Several previous studies have also used white noise analysis to characterize the response properties of mechanoreceptor afferents. Common features of these investigations include high-pass linear properties for afferents with a phasic response to deterministic stimulation, with the maximum gain occurring at a relatively high-frequency (80-100 Hz or greater) and, in the case of directionally sensitive receptors, the second-order kernel for the afferent accounted for the rectification of the response.

The strain-sensitive campaniform sensilla responsible for the detection of deformation in the wings of flies were described using similar techniques (Dickinson 1990). The receptor was driven with Gaussian white noise and the first-order kernels calculated. The system was modeled with an LN cascade, that is, a linear filter (first-order kernel) followed by a static (zero-memory) nonlinearity constructed by comparing the real output of the system to the prediction based on first-order model and fitting a polynomial to this relationship (Hunter and Kornenberg 1986; Marmarelis and Marmarelis 1978). The accuracy of the model predictions was poor (mean square error = 108%) for the linear term only but improved markedly (mean square error = 26%) for the LN cascade model. These error estimates were made by comparing the spike density predictions of the LN cascade model with the average spike density function obtained from repeated experimental responses. When the second-order model was used to predict the experimental spike train produced by applying a threshold to this function, the accuracy of the spike prediction was >90%. The gain functions for the afferents were all high-pass in nature, with increasing slope of approximately 20 dB/decade up to a cutoff frequency of approximately 150 Hz. In this system, as in the crab CBCTO, the second-order kernel accounted for the directional selectivity (rectification) of the sensory response.

A similar study of the response characteristics of an arthropod chordotonal organ afferents (Kondoh et al. 1995) used Wiener kernel analysis to characterize the response properties of locust femoral chordotonal organ (fCO) afferents. The responses of the fCO afferents were similar to the crab CBCTO in that they responded to position, position-velocity, velocity, or acceleration. The only major difference in the locust fCO was the presence of velocity-acceleration afferents, which were not found in the CBCTO. When the fCO was stimulated with broadband white noise (bandwidth, 117 Hz), the first- and second-order kernels were also similar to the kernels calculated for the crab CBCTO, as were the corresponding gain functions. The second-order kernels for the fCO afferents were similar to the CBCTO kernels and responsible for the directional selectivity of the fCO afferents in that the output of the model was essentially rectified with the inclusion of the second-order kernel. Second-order model predictions of spike train output also had a similar degree of accuracy (approximately 90%) as the model results reported here. Kondoh et al. (1995) made the observation that response characteristics of fCO afferents change as the stimulation bandwidth is increased, that is, an afferent that exhibited a low-pass (position-sensitive) response with low bandwidth (28 Hz) noise was revealed to have position-velocity characteristics when higher bandwidth noise was used to drive the fCO. This would of course be true as well for the crab CBCTO. Consider the position-velocity afferent shown in Fig. 5B. If this afferent were driven with white noise with a bandwidth of 20 Hz, one would only observe the low-pass (positional) portion of the response, and the velocity sensitivity would only be seen when higher frequencies were employed. The parallel with deterministic stimulation is the application of a slow ramp to such an afferent, where one observes a slowly increasing rate of firing until the new receptor length is reached. Application of faster ramps would reveal an additional phasic component to the response during the change in length (the velocity component).

The response properties of cricket cercal filiform sensilla were also evaluated using white noise stimulation (Kondoh et al. 1991). These receptors were found to have differentiating (biphasic) first-order kernels resulting in high-pass linear gain functions with a peak frequency of 106 Hz. The filiform hairs are directionally sensitive and the second-order kernel accounted for the directional sensitivity of the response by providing half-wave rectification. The second-order kernels for these afferents were also similar in shape to the kernels for the velocity-sensitive CBCTO afferents presented here.

Other invertebrate mechanoreceptors that have been analyzed with white noise techniques include the femoral tactile spine of the cockroach (French and Kuster 1981; French et al. 1972) and the spider slit-sense organs (French et al. 2001a; Juusola and French 1995). In these receptors, the linear response estimates were also high-pass in nature with gain increasing with increasing frequency of stimulation. However, the gain functions of these systems, although high-pass, were characteristic of a fractional differentiator, as the gain increased at a constant rate of <20 dB/decade. Fractional differentiation, where the gain increases as the kth power of frequency (0 < k < 1) has been observed in a wide variety of sensory receptors (French 1984; Thorson and Biederman-Thorson 1974). The physical basis for this relationship is unknown, but has been suggested to result from a distributed parameter system where the signal passes though several elements (filters) with different time constants (Thorson and Biederman-Thorson 1974). This relationship was not evident in the CBCTO afferent gain functions or for the locust, cricket, and fly mechanoreceptor afferents cited earlier.

Comparison with nonspiking responses of TCMRO afferents

White noise was also used to determine the Wiener kernels for the two nonspiking afferents (S and T fibers) of the TCMRO (DiCaprio 2003, companion paper). The major difference with respect to the linear estimates of receptor characteristics was that S and T fibers had very similar impulse responses and the gain functions were broadly tuned. Thus there is no subdivision of afferent response with respect to the length of the receptor and derivatives of length (velocity and acceleration) and the rather broadly tuned S and T afferents cover a wide frequency range. However, the maximum frequency response of the S and T afferents, although high, with a cutoff frequency in the range of 40-80 Hz, is still less than the maximum frequency response of the majority of CBCTO afferents (90-110 Hz).

A subset of the CBCTO afferents also respond to acceleration of the receptor, and this type of response is not seen in the linear portion of the S and T afferent response. However, in response to trapezoidal stimulation, the rise time of the membrane potential of the nonspiking afferents is faster than the rise of the imposed length change, but the possible correlation between the transient acceleration of the receptor and this parameter of the response to deterministic stimuli has not been investigated. An additional locus for acceleration sensitivity may be found in the P fiber response, as this afferent fires a single spike on initial movement of the TCMRO during a constant velocity stretch (Wildman and Cannone 1990).

Although the CBCTO afferents respond to different movement parameters (position, velocity, and acceleration) and therefore have different first-order characteristics in comparison with the broadly tuned nonspiking TCMRO afferents, both receptors are important components of the leg motor control system and make monosynaptic and polysynaptic connections with leg motor neurons and interneurons (Clarac et al. 2000; El Manira et al. 1991; Skorupski 1992). However, this difference in the individual receptor tuning may not lead to functional differences at the motor system level. Given that crustacean leg motor neurons can receive 2-5 inputs from spiking chordotonal afferents, this postsynaptic convergence of different receptor types would provide an effective broadband input to the motor neurons.