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Neuroscience Program, Department of Biological Sciences, Ohio University, Athens, Ohio
Submitted 16 February 2007; accepted in final form 10 July 2007
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ABSTRACT |
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INTRODUCTION |
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; Clarac 1991; Pearson 1995) and the accurate encoding and transmission of sensory information may be a crucial determinant of sensorimotor system performance. In most proprioceptors, limb movement parameters such as position, velocity and acceleration are transduced by the receptor and then encoded and transmitted to the CNS as a train of action potentials. In some arthropod systems, however, information transmission occurs solely by decremental conduction of graded potentials in nonspiking afferent neurons (Bush and Roberts 1968, 1971; Cannone 1987; Cannone and Nijland 1989; DiCaprio 2003; Heitler 1982; Paul and Bruner 1999; Ripley et al. 1969). In addition, many central motor networks in arthropods also process information with nonspiking neurons or by graded interactions between spiking neurons (Burrows 1979; DiCaprio 1989; Graubard 1978; Graubard et al. 1980; Laurent and Burrows 1989; Paul and Mulloney 1985; Seigler and Burrows 1980). The position and movement of the first two leg joints in the crab is monitored by proprioceptors that use nonspiking (graded) and spike-mediated transmission for information encoding and transmission.
We have previously assessed the information rate of the graded signals from the thoracic-coxal muscle receptor organ (TCMRO) that spans the basal (TC) leg joint. The two nonspiking afferent neurons arising from the TCMRO are capable of information rates of about 2,500 bits/s at a bandwidth of 200 Hz (DiCaprio 2004). Preliminary estimates of the information rate of the nonspiking afferents from the elastic levator and depressor strand receptors that monitor the coxobasal (CB) joint are also in excess of 2,000 bits/s (DiCaprio 2001). The movement of the CB joint is also monitored by the CB chordotonal organ (CBCTO). The CBCTO is a typical crustacean chordotonal organ (Mill 1976), which consists of an elastic strand that spans the CB joint with 70–80 bipolar afferent neurons embedded within or close to the strand (Alexandrowicz and Whitear 1957; Mill 1976). CBCTO afferents respond unidirectionally to stretch or relaxation of the receptor (Bush 1965), corresponding to extension or flexion of the CB joint. The responses of individual afferents can be classified into functional groups that signal the position, velocity, and acceleration of the joint (Bush 1965; Gamble and DiCaprio 2003).
The nonspiking TCMRO afferents and the spiking CBCTO afferents mediate similar negative feedback (resistance) reflexes in their respective motor neuron pools (Bush 1965; Bush and Roberts 1968; Cannone and Bush 1980; El Manira et al. 1991; Le Ray et al. 1997a,b). Although the response bandwidth of the two receptors is relatively similar (DiCaprio 2004; Gamble and DiCaprio 2003), the information rates of the nonspiking TCMRO afferents are approximately an order of magnitude greater than information rates reported for spiking mechanoreceptor afferents (DiCaprio 2004; French et al. 2001; Juusola and French 1997).
We used the stimulus reconstruction method to calculate a lower bound on the information rate of CBCTO afferents because this method is relatively simple to implement and computationally efficient, providing an efficient survey of the information rate of a large sample of CBCTO afferents. An additional set of afferent recordings were used to directly estimate the information transfer rate of CBCTO afferents from calculation of the spike train entropy (Strong et al. 1998). The timing precision of CBCTO afferent firing was also assessed and the variation in timing of individual spikes in response to mechanical stimulation of the receptor was found to be extremely small. Spikes evoked by current injection into the soma had markedly smaller jitter, indicating that a major cause of the timing jitter is due to the mechanotransduction process.
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METHODS |
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). Briefly, the dorsal carapace was removed and the thoracic ganglion was perfused by the sternal artery with chilled (16–17°C) oxygenated saline (Ripley et al. 1968) at a rate of 3–5 ml/min for 15–20 min before proceeding with further dissection. After removal of the remoter, depressor and levator muscles of the fifth leg, the nerve to the CB chordotonal organ was exposed and the CBCTO and associated connective tissue were freed from proximal attachment on the dorsal rim of the coxopodite. The distal attachment of the CBCTO on the proximal rim of the basiopodite was cut after the in situ length of the receptor was measured at midjoint angle. The receptor ganglion was then removed and placed in a small (5-ml) plastic chamber that was continuously superfused with chilled oxygenated saline (3–5 ml/min) for the duration of the experiment. The proximal end of the CBCTO, with surrounding connective tissue, was pinned securely to a Sylgard substrate with four to five stainless steel pins to mimic the normal attachment at the coxa. The distal end of the receptor was attached to an electromechanical puller and the resting length of the CBCTO set to the measured in situ length. The maximum length change (peak-to-peak amplitude) applied to the CBCTO was 0.4 mm. This corresponds to a joint angle range of about 30°, which is close to the maximum amplitude of CB joint movement (30–40°) observed during lateral walking in Carcinus (Clarac and Coulmance 1971).
The electromechanical puller was constructed from a 5-in.-diameter speaker controlled by a proportional integrodifferential (PID) controller (Hofmann and Koch 1985) operating in length feedback mode. The frequency response of the puller is flat to a cutoff frequency (fc) of about 220 Hz over a displacement range of ±0.8 mm.
The connective tissue sheath surrounding the CB nerve was removed with fine forceps along most of the nerve to permit intracellular recording from the axons of individual CBCTO afferents. Intracellular recordings were made with microelectrodes filled with 2 M KAc and amplified by a bridge electrometer (NPI SEC-05L). In several experiments, a second intracellular electrode was used to record from the same axon at a site 7–9 mm distant from the first electrode. Placement of the second electrode was varied until a recording was obtained with a strict one-to-one spike correlation with a fixed latency. Single CBCTO afferents are not individually identifiable 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 (Bush 1965; Gamble and DiCaprio 2003).
In several experiments we were also able to obtain recordings from single afferent cell bodies. This was possible because 10–12 of the CBCTO afferents have their cell bodies located in a loose matrix of connective tissue that is located off the main axis of the receptor strand close to the proximal insertion of the receptor, although the dendrites of these cells originate in the CBCTO elastic strand (see inset in Fig. 8). The basic preparation was modified to include a small Sylgard block under this connective tissue to stabilize the region around the cell bodies with four to six cactus spines (diameter <0.05 mm), taking care not to damage the afferent dendrites or to interfere with the normal movement of the CBCTO. Intracellular recording was relatively easy in this fixed region, and spikes could readily be evoked by intracellular current injection in DCC mode. In two preparations, we were also able to maintain the intracellular recording while applying length changes of sufficient amplitude to elicit CBCTO afferent spikes.
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Computation of information rate (lower bound)
The stimulus reconstruction technique (Bialek 1991; Rieke et al. 1997) provides a relatively simple and efficient method for estimating the lower bound on the information transmission rate of spiking neurons. The detailed theoretical background and methodology for this technique has been provided in several recent papers and reviews (Borst and Theunissen 1999; Gabbiani and Metzner 1999; Rieke et al. 1997; Roddey and Jacobs 1996; Theunissen et al. 1996). Briefly, the output of the system, a train of action potentials, is convolved with a filter function to provide a linear estimate of the input signal that elicited the spike train. The difference between this estimate of the input signal and the real signal is the noise in the reconstruction, which arises from noise in the system, system nonlinearity, and other factors. The initial computational task required to implement this method is to calculate the best filter, or reconstruction kernel, that minimizes the mean-squared error between the stimulus and the estimate of the stimulus, thereby minimizing the noise in the reconstruction. The reconstruction kernel is
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H(
)R(t–)d.
Any difference between the real stimulus and the stimulus estimate will therefore result in noise that will reduce the information rate. The noise, N(f), in the reconstruction is N(f) = S(f) – Sest(f) (Borst and Theunissen 1999; Gabbiani and Metzner 1999; Theunissen et al. 1996). The lower bound on the information transfer rate is then calculated from the Shannon formula (Shannon 1948; Shannon and Weaver 1949) for the information capacity, R, of a continuous channel
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Direct calculation of information rate
A direct method for estimating of the information rate of spiking neurons was previously described (Reinagel and Reed 2000; Strong et al. 1998). This method makes no assumptions regarding the nature of the neural code because the estimates of mutual information are based on the probability distributions of the neural response. The information rate is determined by stimulating the system with a random input (white noise) and then calculating the total entropy of the resultant spike train. The method is implemented by dividing time into bins, 
t, where t is sufficiently small to allow only one spike in each time bin. The probability distribution of all binary words contained in the spike train is the total entropy
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t) is the set of all possible words of length L bins with a bin width t, and P(w) is the probability that a specific word occurs in the response. The total entropy represents the complete response to the input and is thus the upper bound on the information rate of the system. This maximum information rate will be achieved only if the responses to repeated presentation of the same random stimulus are identical, that is, if there is no noise in the system. The noise entropy is estimated from these repeated stimulus trials in the same manner as for the total entropy, and averaged over all trials. The total entropy and noise entropy (at a given t) are plotted with respect to the reciprocal of word length, and the estimate of each entropy is obtained by extrapolating these plots to infinitely long word length (Reinagel and Reed 2000; Strong et al. 1998). The difference between the total and noise entropy, RD(t) = Htotal – Hnoise, is the information rate in bits/s. A bin width t = 0.5 ms, equal to the average temporal precision of CBCTO spikes (see RESULTS), was used in all calculations and only zero or one spike was present in each bin.
The CBCTO was driven with band-limited white noise for a period of 500–550 s and neurons that maintained a relatively constant mean firing rate for 
450–500 s were selected for further analysis. The CBCTO was stimulated with a 4- to 5-s period of random movement that was then repeated sequentially without pauses for 110 cycles and the last 90–100 cycles (after the afferent had adapted and the firing rate was stable) were used for the calculation of the noise entropy. The correction for the error due to finite data set size (Strong et al. 1998) in all cases was <5% of the entropy estimates obtained by the extrapolation to infinite word length. In addition, estimates of the total and noise entropies were also made with shorter or longer data segments (±20 s), and in all cases the difference between the three estimates was <3%.
Spike-timing measurements
The precision of spike timing evoked by random stimulation was assessed by calculating the jitter in the spike timing during repeated random stimulation of the receptor. The jitter in spike timing was defined as the SD of the individual spike times that were evoked by stimulus features (events) at discrete times during the stimulus presentation. To locate these event times, a peristimulus time histogram (PSTH) with a 1-ms bin width was constructed for the final 80 trials to eliminate from this analysis changes in spike timing associated with adaptation. A threshold was applied to the PSTH to select the times in each stimulus cycle that elicited spikes in 20% of the trials to provide sufficient data for calculation of the SD of the spike times. These times were stored and the time of the spikes at each stimulus time, relative to the cycle start time, was extracted from each stimulus cycle, using a window of ±3 ms around each time. This time window was chosen because it is less than the minimum interspike interval of about 5.5 ms observed for CBCTO afferents. The analysis software also checked for the occurrence of more than one spike within each time window and multiple spikes were never present within this interval in all trials in all experiments. Spike occurrence times were determined by application of a simple (voltage) threshold crossing to the intracellular waveform using linear interpolation between sample points. The underlying temporal resolution of Spike2 data files is independent of the channel sample rate and was set to 2 µs.
All programs for the reconstruction method and the direct method were written in the CED Spike2 script language (v. 5.13). Results were plotted using Grapher (Golden Software, V4) and Axum (MathSoft, v. 5.03C) and the figures were prepared in Canvas (v. 8.04, Deneba Systems).
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RESULTS |
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). A purely velocity sensitive (phasic) afferent is silent at any length and responds to constant velocity stretch or release of the receptor with a firing rate that is proportional to the velocity of the movement. In contrast, mixed position–velocity-sensitive afferents respond similarly during stretch or release of the CBCTO, but also maintain a tonic firing rate in (approximate) proportion to the length of the CBCTO. The maximum firing rate for such afferents can be up to 300 spikes/s for a 100-ms duration, 0.3-mm-length change. The firing rate of most CBCTO afferents adapt to maintained random stimulation, with a 30–60% decline in firing rate occurring in the first 15–30 s after the onset of stimulation. After this period, the firing rate of most afferents remained relatively constant (<10% change) for the remainder of the stimulation. All measurements were made during this stable period, and afferents that did not maintain a stable firing rate were not used in the work presented here.
Lower-bound estimates
A typical stimulus reconstruction derived from the response of a velocity-sensitive CBCTO afferent is shown in Fig. 1A. In the 165-ms record shown here, the average firing rate of the afferent was 44 spikes/s. Due to the rather long intervals between evoked spikes relative to the bandwidth of the random stimulus (gray trace), the estimate of the stimulus (black trace) does not appear to be very accurate during this time. The quality of the reconstruction improved in a short interval where the firing rate transiently increased to about 72 spikes/s (Fig. 1B), due to the higher spike density in this segment of the data. The reconstruction filter and the signal-to-noise ratio (SNR) are shown in Fig. 1, C and D. The maximum SNR occurs in a broad peak centered around 120 Hz, which is close to the frequency where the (linear) gain for CBCTO afferents is maximum (Gamble and DiCaprio 2003). The SNR for this neuron was typical for the CBCTO afferents and the maximum ranged from 0.1 to 2.2. The mean firing rate for this afferent was 44 spikes/s and the lower bound on the information rate was 77 bits/s. The lower bound on the information rate was strongly correlated (n = 49, R = 0.935, P < 0.001) with the mean firing rate of the afferent (Fig. 2). In this sample of 49 velocity- or position–velocity-sensitive afferents, the information rate ranged from 43 to 169 bits/s as the mean firing rate varied from 20 to 100 spikes/s. This corresponds to an average of 1.93 bits/spike (SD 0.23).
We have previously shown that the input–output transfer characteristics of the CBCTO afferents are not affected by changes in the amplitude, and thus power, of the stimulus (Gamble and DiCaprio 2003). As long as the mean firing rate of a given afferent remained relatively constant, the lower-bound information rate also did not depend on the power of the input stimulus (data not shown). However, stimulus bandwidth did affect the information transfer rate for most afferents because stimulation of the receptor with lower bandwidth noise (140 vs. 220 Hz) resulted in a decreased level of afferent adaptation (8 of 10 neurons) and thus a higher overall mean firing rate, with the increase in firing rate ranging from 5 to 40%.
Figure 3 shows examples of the stimulus reconstruction for a velocity-sensitive afferent when stimulated with 220- and 140-Hz bandwidth noise. With 140-Hz bandwidth noise, the average firing rate of this afferent increased from 34 to 53 spikes/s. This higher spike density resulted in fewer gaps in the spike train reconstruction and thus an improved quality of the reconstruction. The peak (linear) gain for this afferent occurred at a frequency of about 150 Hz, which was higher than the average cutoff frequency of about 100 Hz for velocity-sensitive CBCTO afferents (Gamble and DiCaprio 2003). The maximum SNR for this afferent was 0.65 with 220-Hz bandwidth stimulation and increased to about 2.5 with 140-Hz bandwidth stimulation and the maximum SNR occurred at a lower frequency (
100 Hz) with 140-Hz bandwidth stimulation. In other experiments (n = 8) where the afferent firing rate increased when 140-Hz bandwidth stimulation was used, similar increases in SNR, and thus the lower-bound estimate of information rate, were also observed.
A continuous decrease in firing rate occurred in 32% of the afferents in experiments using 220-Hz bandwidth stimulation and some of these afferents stopped firing completely during the course of stimulation. However, when the CBCTO was driven with 140-Hz bandwidth noise, the mean firing rate of the afferents was usually stable for long periods. In addition, the afferents that stopped firing completely during stimulation of the receptor with 220-Hz noise often maintained a constant firing rate with 140-Hz stimulation. It should be noted that although stimulation of the CBCTO with 140-Hz noise usually increased the mean firing rate of individual afferents when compared with 220-Hz bandwidth stimulation, it did not change the range (20–80 spikes/s) of steady-state afferent firing rates. Because a constant firing rate is critical for the direct estimate of information rate and timing jitter, we used 140-Hz bandwidth noise for CBCTO stimulation for the remainder of the experiments.
Direct measurement of information rate
Although the stimulus reconstruction method is a computationally simple and fast method for estimating the information rate of spiking neurons, it provides only a lower-bound estimate. We therefore directly measured CBCTO afferent information rate to make a more appropriate comparison with the prior direct estimates of the information rates of the nonspiking proprioceptive afferents in this system. An additional 24 preparations were used that yielded usable recordings from 34 afferents. The CBCTO was driven with 140-Hz bandwidth noise for 550–600 s to produce a period of steady-state firing of 500 s. The receptor was also driven with repeated identical 5-s-duration noise sequences for 110 trials to estimate the noise entropy. The lower-bound estimate was also calculated (as above) for these 34 afferents using the same stimulus data set.
The direct estimates were, as expected, greater than the lower-bound estimates. For example, the lower-bound estimate of the information rate for a velocity-sensitive afferent calculated using 140-Hz bandwidth random stimulation was 74 bits/s with a mean firing rate of 35 spikes/s. The total entropy of the afferent spike train estimated using the same data set was 197 bits/s and the noise entropy was 83 bits/s. The information rate for this afferent was therefore RD = 197 – 83 = 114 bits/s. The range of directly estimated information transfer rates for the set of 34 afferents was 24 to 278 bits/s over a range of mean firing rates of 8 to 65 spikes/s, or an average of 3.98 bits/spike (SD 0.81). The directly estimated information rates were again correlated with the mean firing rate of the afferent (Fig. 4). The dotted line in Fig. 4 shows the regression line for the lower-bound estimates for the same afferents.
The stimulus reconstruction method assumes linear coding of information, even though the encoding scheme may be nonlinear (Reike et al. 1997), whereas the direct method makes no assumptions about the coding scheme used by the system. Calculating the information rate using both methods on the same data set therefore allows one to calculate the percentage of information in a spike train that can be accounted for by linear means alone (Chacron 2006). The percentage of the information (
I) that can be recovered from the spike train by a linear decoder is given by I = RLB/RD. For the 32 CBCTO afferents where both rates were calculated, I = 0.59, SD = 0.10.
CBCTO afferent spike-timing precision
The repeated stimulation of the CBCTO required to calculate the noise entropy also allowed the spike-timing precision of CBCTO afferents to be determined. Depending on the mean firing rate for the afferent, there were 100–250 individual stimulus "events" that evoked spikes during the stimulus. The reliability of the spikes (number of spikes/number of trials) at each stimulus event was calculated and stimulus times that produced a spike reliability of 20% were selected to ensure sufficient data for calculation of the SD of the spike times.
A small portion of the complete PSTH for a velocity-sensitive afferent is shown in Fig. 5A and the histogram of the spike times at a single stimulus time (marked with an asterisk in Fig. 5A) is shown in Fig. 5B. The jitter for the spikes at this stimulus time was 0.36 ms. The 5.2-s-duration stimulus evoked 111 spikes with a reliability of 20% and the average timing jitter for these events was 0.461 ms (SD 0.097), with a range of 0.23 to 0.76 ms (Fig. 5C). The average spike-timing jitter for all 34 experiments analyzed was 0.539 ms (SD 0.062) with a range of 0.40 to 0.65 ms (Fig. 5D).
We examined the relationship between jitter and the mean firing rate to determine whether these parameters were correlated. We have previously shown that the average jitter for all spikes in a given experiment is negatively correlated with the mean firing rate of the afferent (Billimoria et al. 2006). In the data set used in the experiments presented here, the same negative correlation was also observed (cross-correlation (CC) = –0.399, P = 0.02; data not shown). Another relationship could exist between the reliability of spike firing and timing jitter because spikes evoked less frequently by a specific stimulus feature might occur with relatively high jitter. We define reliability in this context as the percentage of spikes evoked at a specific time during the stimulus in repeated stimulus trials. Figure 6 B shows the relationship between reliability and jitter using pooled data from 16 afferents constituting 2,003 discrete stimulus events that evoked spikes with a reliability of >15%. There is a negative correlation between reliability and jitter, but the correlation was not significant (CC = –0.276, P = 0.226). Distribution of the reliability of the spikes evoked at specific stimulus times is relatively uniform and this was typical of the distribution in single experiments as well. We have also made preliminary measurements of stimulus features preceding each evoked spike such as the peak velocity of movement, initial and final length of the receptor, and the total range of movement, but have yet to find any significant correlation between these stimulus parameters and spike-timing jitter or reliability.
Our calculation of the spike-timing precision was based on intracellular recordings from axons of single afferents in the CB nerve and therefore includes several mechanisms that may contribute to the overall jitter, such as mechanotransduction mechanisms, the spike initiation process, and axonal conduction. To isolate these potential noise sources, we used a modified preparation that allowed for simultaneous paired intracellular recordings from an afferent axon in the CB nerve and the soma or axon of the same afferent. In addition, the soma recordings allowed for the direct stimulation of a single afferent with intracellular current injection, independent of CBCTO mechanical stimulation. In a few experiments, stable intracellular recordings were also made from the soma while the CBCTO was driven with random movement. Preliminary data from these experiments were presented previously (Billimoria et al. 2006; DiCaprio and Billimoria 2005).
The timing variability during axonal transmission was assessed by making paired recordings for the same afferent axon in the CB nerve (7- to 9-mm electrode spacing) during mechanical stimulation of the receptor. The jitter that occurred during axonal transmission was then assessed by calculating the SD of the interspike interval from the two recording sites (Fig. 7B). For a total of 3,020 evoked spikes, there were no failures of transmission along the axon and the average timing jitter, 6.7 µs, was extremely small. In a total of six such experiments, the average jitter was 7.8 µs (SD 2.6). In four additional experiments where one electrode was placed in the soma of the afferent and spikes were evoked by current injection, the average jitter at the axonal recording site was also <10 µs. It would therefore appear that there is no significant contribution of spike propagation to the overall timing jitter of CB afferents.
The most likely sources of noise in this system must therefore reside in the biomechanics of the receptor, the transduction channel dynamics, and coupling to the receptor structure, or timing variability in spike initiation due to noise in voltage-dependent sodium and potassium channels along with any other intrinsic currents present in these afferents. In experiments where spikes were evoked by current injection into the soma, the mean jitter for the soma spikes was small (0.098 ms, SD 0.009, n = 6). This jitter is an order-of-magnitude higher than the jitter introduced during spike propagation, but it accounts for only about 20% of the jitter observed during mechanical stimulation. An example of the results of one such experiment is shown in Fig. 7C. Note that in contrast to the jitter in axonal spikes (Fig. 7B), the distribution of the jitter for the 111 soma spikes evoked at discrete times by current injection is not normally distributed. In this instance, the mean jitter was 0.102 ms, the median jitter was 0.08 ms, and this skewed distribution was observed in all six experiments.
In two experiments, spikes were evoked by current injection in the soma and also by mechanical stimulation of the receptor (Fig. 7C). The difference in the median jitter of 0.08 ms with electrical stimulation and the overall mean jitter of 0.56 ms for mechanically evoked spikes would indicate that the receptor biomechanics and mechanotransduction channel kinetics are the likely dominant factors producing jitter in a CBCTO afferent timing. It is also possible that there is some mechanical instability in the puller system that we cannot measure because the position of the electromechanical puller is monitored at the output shaft of the device, rather than directly at the attachment to the CBCTO. However, the ratio of the signal-to-noise power of the position transducer signal is >107 over the stimulus frequency range. In other terms, the root-mean-squared amplitude of the noise in the position monitor with a white noise driving signal to the system is <0.2 µm, which is less than the minimum movement amplitude required to elicit spikes from CBCTO afferents. The precision of the puller was also evaluated by applying a threshold to the position monitor signal and calculating the jitter of these threshold crossing times. In all experiments, the jitter in the movement monitor measured in this manner ranged from 4 to 6 µs. This indicates that variability in the puller output was not an important source of temporal jitter. However, because we cannot directly monitor the position of the receptor at the attachment point to the puller, any instability in the mechanical coupling between the puller shaft and the receptor strand (a 4-mm-long, 0.35-mm-diameter stainless steel pin) could contribute to the timing jitter, although changing the pin length or diameter did not alter the timing jitter in response to CBCTO movement.
Relationship between jitter and information rate
A recent study of the information rate of grasshopper auditory receptors (Rokem et al. 2006) demonstrated that the direct estimates of information rate were negatively correlated with spike-timing jitter. Figure 8 shows the relationship between jitter and information rate (Fig. 8A) and noise entropy (Fig. 8B) for CBCTO afferents. The information rate is negatively correlated with mean jitter (CC = –0.397, P = 0.024) but the noise entropy is not correlated with the mean jitter (CC = 0.0485, P = 0.776). A confounding factor in our data set is that CBCTO afferents exhibited a large (eightfold) range of mean firing rates and mean firing rate is positively correlated with the information rate (Fig. 4) as well as with total and noise entropies (P < 0.001). In the study by Rokem et al. (2006), their data set was restricted to a much smaller ( twofold) range of afferent firing rates to minimize the effect of mean firing rate on information rate (Borst and Haag 2001). To account for the large range of firing rates, and guided by our preliminary modeling study (Billimoria et al. 2004), we plotted the noise entropy as a function of mean firing rate and mean jitter (Fig. 8C). Multiple linear regression on these two variables (the shaded plane in Fig. 8C) reveals that these two independent variables can predict the noise entropy of the afferent (R = 0.708, Prate < 0.001; Pjitter = 0.04).
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DISCUSSION |
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; Chapman and Smith 1963; Hofmann and Koch 1985; Hofmann et al. 1985) and with random stimulation using nonlinear systems identification techniques (Gamble and DiCaprio 2003; Kondoh et al. 1995). In the crab, chordotonal afferents respond over a wide range of frequencies (0–200 Hz), with the maximum (linear) gain occurring at frequencies of 100–120 Hz (Gamble and DiCaprio 2003). We used the stimulus reconstruction technique (Bialek 1991; Rieke et al. 1997) to estimate a lower bound on the information transfer rate of CBCTO afferents. Using an additional data set, we calculated estimates of the information transfer rate for the spike-train entropy (Strong et al. 1998) as well as the lower-bound estimate for comparison. The spike-timing precision of CBCTO afferents in response to dynamic stimulation was also determined using mechanical stimulation of the receptor and electrical stimulation of single afferents with current injection.
Lower-bound estimates of the information transfer rate of CBCTO afferents determined for a sample of 49 afferents using 220-Hz white noise mechanical stimulation of the receptor ranged from 45 to 170 bits/s (mean firing rates range 20–85 spikes/s). Similar estimates of the lower-bound information rate have been obtained in other mechanoreceptors such as the spider VS-3 slit sense organ (10–80 bits/s, mean firing rate range 5–50 spikes/s; French et al. 2001) and cricket cercal afferents (75–220 bits/s, mean firing range 75–135 spikes/s; Roddey and Jacobs 1996). A summary of information transfer rate estimates for a variety of sensory systems is given in Borst and Theunissen (1999). The lower-bound information rate for CBCTO afferents was correlated with the mean firing rate of the afferent because a higher spike density results in a more accurate estimate of the input stimulus. This relationship is expected based on theoretical considerations of the maximum information capacity of a spiking neuron (McKay and McCulloch 1952; Reike et al. 1997; Stein 1967) and has also been observed in the H1 neuron of the fly visual system (Borst and Haag 2001).
Direct estimates of information rate of CBCTO afferents
In a separate set of experiments, the information transfer rate of CBCTO afferents was estimated using both stimulus reconstruction methods to provide a lower bound and by a direct method that estimates spike-train entropy. The chordotonal organ was stimulated with 140-Hz bandwidth white noise because this lower bandwidth more often elicited spikes trains with a constant mean firing rate, compared with 220-Hz noise bandwidth, although the range of firing rates produced was very similar with both stimulus protocols. The lower-bound estimates in these experiments ranged from 24 to 165 bits/s, whereas the information rates estimated using the direct method ranged from 35 to 278 bits/s and both estimates were correlated with the mean firing rate of the afferents (Figs. 2 and 4). When these information rates were normalized by the mean firing rates, the lower-bound estimates averaged 1.93 bits/spike (SD 0.23), whereas the direct estimates averaged 3.98 bits/spike (SD 0.81). These normalized information rates are similar to the normalized rates observed in other primary afferent neurons (Roddey and Jacobs 1996; for summaries see Borst and Theunissen 1999; Bura
as and Albright 1999).
The stimulus reconstruction method assumes a linear coding of information (Reike et al. 1997), whereas the direct method makes no assumptions about the coding scheme used by the system. Calculation of the information rate using both methods on the same data set therefore allows one to determine the percentage of information in the afferent spike trains that can be accounted for by linear means alone (Chacron 2006). The percentage of information that can be extracted by a linear decoder was 59%, indicating that a significant fraction of information encoded in CBCTO spike trains would require a nonlinear decoder for complete recovery.
CBCTO afferent spike-timing precision
The timing of CBCTO afferent spikes elicited by dynamic stimulation is extremely precise. In a single experiment (Fig. 5C) the range of jitter for all evoked spikes was 0.23–0.76 ms with a mean of 0.461 ms (SD 0.097, n = 111). In pooled data, spikes evoked by a single stimulus event could occur with a jitter of <0.25 ms and the mean jitter for the spikes in 34 afferents evoked by mechanical stimulation of the receptor was 0.539 ms (SD 0.062) with a range of 0.40 to 0.65 ms. Precise spike timing, especially in response to dynamic inputs, has been found to be a feature in many afferent neurons and at higher levels of signal processing in the nervous system (Berry and Meister 1998; Fellous et al. 2001; Lestienne 2001; Mainen and Sejnowski 1995; Rokem et al. 2006; Uzzell and Chichilnisky 2004). The jitter in CBCTO afferents is similar to the timing jitter observed in other sensory systems (Aldworth et al. 2005; Berry et al. 1997; Billimoria et al. 2006; Kreiman et al. 2000; Mainen and Sejnowski 1995; Reinagel and Reed 2000, 2002), although at the low end of the range of values observed in previous studies.
CBCTO afferent spike-timing jitter was not significantly correlated with the mean afferent firing rate or with the reliability of the evoked spikes. However, we expected that the information rate and the noise entropy of the afferents would be correlated with jitter, with the information rate decreasing and the noise entropy increasing as jitter increases. This relationship was demonstrated in the analysis of information rates of afferents in the grasshopper auditory system (Rokem et al. 2006). However, although the direct information rate for CBCTO afferents was negatively correlated with jitter, the noise entropy was not correlated with jitter (P = 0.776, Fig. 8). Our data set contained a large range of afferent firing rates and there was a significant correlation between mean firing rate and the estimates of direct information rate, total entropy and noise entropy. When the noise entropy was plotted as a function of mean firing rate and mean jitter, a multiple linear regression on these two variables could predict the noise entropy (Fig. 8C) and shows that noise entropy increases as spike-timing jitter increases.
The spike-timing precision will therefore influence the overall information rate due to the increase in noise entropy as jitter increases. Recall that the noise entropy is evaluated by repeatedly driving the system with a specific input signal, dividing the response into (small) time bins so that there is only one spike per bin, and calculating the probability of occurrence of the unique "words" in the response (see METHODS). If the jitter is zero, that is, if all spikes occur exactly at the same time during each stimulus trial, all of the n-bit "words" formed from the spike train will be identical in each trial (assuming that the reliability for all evoked spikes is 100%), and the noise entropy will be minimum. If, however, there is jitter in the spike timing, additional "words" may then be present in the response probability distribution, resulting in an increase in the noise entropy. This effect is most pronounced when the magnitude of the jitter is close to the bin width used to create the response words. With respect to a single spike, the worst case will occur when the mean time of a spike is exactly at a bin start or end time. Adjacent bins will therefore contain a spike in 50% of the trials, producing two words instead of one in the output probability distribution, albeit at a lower probability of each word. For example, if a given spike pattern (word) occurred with a probability of 0.1 and the jitter was zero, the contribution of this word pattern to the noise entropy would be 0.332 bits. If the mean time of a single spike in the word occurred with finite jitter at a bin boundary, two different words would now be present in the distribution, each with a probability of 0.05, resulting in an increase in noise entropy to 0.432 bits due to the presence of the two different spike patterns.
Clearly, this increase in noise entropy will occur only when the bin width is similar to the average jitter for the evoked spikes. This was illustrated in an analysis of insect auditory afferents where the jitter of experimentally evoked spikes was artificially increased (Rokem et al. 2006). When additional jitter with a flat probability distribution (±0.5 or ±1.0 ms) was added to an experimental response set that had a mean jitter of 0.45 ms, the information rate decreased as the jitter was artificially increased when the bin width used was <1 ms because the magnitude of the jitter was now similar to the bin time interval. When the information rate was calculated using bin widths >1 ms (equal to the maximum added jitter), the information rate estimates were independent of the average jitter and essentially identical to the original (0.45 ms average jitter) data set.
Spike-timing jitter has also been shown to decrease the accuracy with which stimulus features that trigger spikes can be determined (Aldworth et al. 2005; Rokem et al. 2006) and these measures are improved if the spike train is dejittered before the estimation of the spike triggered average. This was demonstrated in studies of the timing precision of sensory interneurons in the cricket cercal system (Aldworth et al. 2005) where the timing jitter was about 5 ms. However, when the jitter is small compared with the autocorrelation time of the stimulus or, similarly, to the temporal width of the reconstruction filter, these effects have been demonstrated to be negligible (Chacron 2006). In the experiments presented here, the average jitter is extremely small (0.54 ms) and is much smaller than the autocorrelation time of the white noise stimulus (15.7 ms) and the typical width of the reconstruction filters (20 ms) used for the lower-bound estimate. The small jitter in CBCTO afferent timing will therefore result in a negligible error in spike-triggered stimulus averages and in the lower-bound estimates of information transfer (Krieman et al. 2000) in this system.
Origin of timing jitter
Paired intracellular recordings from CBCTO afferents indicate that the jitter likely arises from the mechanotransduction and spike-initiation processes (Fig. 7). CBCTO afferents are capable of very accurate spike timing (average jitter of 80–100 µs) when spikes are generated directly with intracellular current injection into the soma. When the same cell was excited by mechanical stimulation of the CBCTO, the mean jitter increased to 0.56 ms, indicating that, although some jitter is associated with the ion channels mediating the production of action potentials, the major source of timing error is likely associated with mechanotransduction channels or perhaps the biomechanics of the receptor. Fluctuations in ion channels have been demonstrated to be a principal source of intrinsic noise in neurons (Diba et al. 2004; Manwani et al. 1999; Steinmetz et al. 2001; White et al. 1998) and a detailed analysis of the effect of noise on timing precision in retinal ganglion cells indicates that voltage-gated channel noise, along with synaptic noise, is a significant factor in producing variability in spike timing (van Rossum et al. 2003). This conclusion is supported by the finding that neuromodulators that increase or decrease the membrane conductance of CBCTO afferents and thus decrease or increase the amplitude of the membrane potential noise, respectively, also alter the magnitude of the jitter in CBCTO afferents (Billimoria et al. 2006). The electromechanical puller could also introduce some response variability, but the noise amplitude in the puller output is very low (0.2 µm), less than the minimum amplitude of CBCTO movement required to elicit afferent spikes. In addition, the jitter in the timing of events produced by applying a threshold to the movement record is also extremely precise (4–6 µm). However, these small noise sources could be amplified by the transduction process or receptor biomechanics, which could then contribute to some fraction of the overall spike-timing jitter.
Contrast with information rates of nonspiking afferents
The information transfer rates of the two nonspiking proprioceptive afferents from the TCMRO that monitors the TC joint are >4,500 bits/s at a 500-Hz bandwidth (DiCaprio 2004) and about 2,500 bits/s over the 200-Hz bandwidth that encompasses the response range of the spiking CBCTO afferents (Gamble and DiCaprio 2003). The TCMRO is the only receptor monitoring the basal TC joint and, given the continuous nature of the graded signaling, these high rates are not unexpected. A preliminary estimate of the information transfer rates for the nonspiking afferents of the two elastic strand receptors that monitor the movement of the CB joint in parallel with the CBCTO has also been determined (DiCaprio 2001, 2004; Ludwar and DiCaprio, unpublished observations) and are 2,000 bits/s. The information transfer rate of the spiking afferents is therefore approximately only 5–15% of the information rate of the nonspiking afferents over the same bandwidth.
This difference in information transfer between nonspiking and spiking neurons is not unexpected, given the limitations on information rate imposed by the finite firing rates and timing precision of spiking neurons (MacKay and McCulloch 1952; Reike et al. 1997). A similar study of information transfer rates of spider slit sensilla, a mechanoreceptor detecting cuticular strain, estimated an information transfer rate of over 2,000 bits/s for the graded receptor potential, but this was reduced to 200 bits/s for the resultant spiking afferent output (Juusola and French 1997). We have made some preliminary measurements of the graded receptor potential in CBCTO afferents in response to mechanical stimulation. The amplitude of the receptor potential is low when recorded in the soma (<2 mV), presumably due to attenuation along the 400- to 500-µm length of the dendrite. It appears that the spike initiation region is located in the dendrite, as indicated by the larger depolarization required to elicit spikes from the soma with current injection compared with the potential recorded with mechanical stimulation of the receptor. This appears to be similar to the distal spike initiation zone found in spider V3 slit sensillae mechanoreceptors (Juusola and French 1997). In spite of the limited access to the site of mechanotransduction, the signal-to-noise ratio of the soma membrane potential was >100 over most of the stimulation bandwidth, which corresponds to an information transfer rate on the order of 1,500 bits/s, although this is probably an underestimate of the actual information rate after the transduction stage.
Although the data transfer rates are much lower for spiking versus nonspiking afferents, they may not adequately represent the complete information stream input to the motor neuron pools that receive this afferent information. This is explained by the fact that each of the motor neurons controlling the CB joint receives from two to six (monosynaptic) inputs from different CBCTO afferents (El Manira et al. 1991; Le Ray et al. 1997a,b). Thus although the degree of redundancy in this parallel convergent input is not known, it is likely that the total information input to CB joint motor neuron pools is higher than the rate of a single afferent.
Are the information rates determined here and previously for the nonspiking TCMRO afferents relevant in the functional context of the sensory motor system? We might assume that a higher information rate is "better" in a given system because information not present in the input, or later corrupted by noise, cannot be recovered by the receiver. At a minimum, the information rate characterizes the magnitude of the stimulus information that can be encoded by these proprioceptors and the extent to which this information is faithfully transmitted (or degraded) before transynaptic transformation, resultant motor neuron firing, and ultimately the generation of muscle force resulting in movement of the limb.
A more functional approach considers the parameters that determine the information rates for both spiking and nonspiking receptors found in the leg control system. Both types of receptors have a similar response bandwidth of around 200 Hz and their linear transfer functions are similar (DiCaprio 2003; Gamble and DiCaprio 2003). The major difference in the receptor transfer functions is that the spiking CBCTO afferents can be divided into three classes based on their linear transfer characteristics, that is, position, velocity, and acceleration (Gamble and DiCaprio 2003), whereas the two nonspiking TCMRO afferents are both broadly tuned transfer functions characteristic of a low-pass filter (DiCaprio 2003). In this regard, the ensemble of CBCTO afferents can encode stimulus features similar to those of the TCMRO, with the major difference being a lack of an acceleration component in the response of the nonspiking afferents.
As shown here, the information rate for the CBCTO afferents is 10-fold less than the information transfer rate of the nonspiking receptors due to the inherent limitations on information transmission imposed by the finite spiking rate sensory neurons. This limitation is highlighted by our preliminary measurements of the high information rate of the receptor potential of these afferents before spike initiation. In contrast, the extremely high information rate of the nonspiking TCMRO afferents is due to the extremely high signal-to-noise ratio (104 over a 0- to 200-Hz bandwidth) of the graded and continuous membrane potentials propagated by these large-diameter axons. Although the information transfer rates for graded TCMRO afferents and the spiking CBCTO afferents differ by an order of magnitude, these very different values still reflect an underlying precision, accuracy, and repeatability in neural encoding and transmission in both systems. This is evident in the very small spike-timing jitter present in CBCTO afferent spike trains and the high SNR of the nonspiking afferents. This precision in both systems is perhaps the more important functional parameter for these proprioceptors because it underlies the information rate estimates arising from the minimal variation in both CBCTO spike trains and TCMRO graded-potential–evoked receptor stimulation. This precision and fidelity may therefore be reflected in the subsequent activation of postsynaptic motor neurons in the motor control system. Further investigation of the spike-mediated and graded synaptic transfer functions, the transformation to motor neuron activation, and muscle contraction will be required to properly place these and earlier results in an appropriate functional context.
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GRANTS |
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ACKNOWLEDGMENTS |
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Present addresses: C. P. Billimoria, Hearing Research Center, Dept. of Biomedical Engineering, Boston University, Boston, MA 02215; B. Ch. Ludward, Dept. of Neuroscience, Mount Sinai School of Medicine, One Gustave L. Levy Place, New York, NY 10029.
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FOOTNOTES |
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Address for reprint requests and other correspondence: R. A. DiCaprio, Department of Biological Sciences, Ohio University, Athens, OH 45701 (E-mail: rdicaprio1{at}ohiou.edu)
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ABSTRACT
INTRODUCTION
e, was subtracted from the individual spike times tei. Jitter (SD) in the spike times for the 80 spikes shown here is 0.36 ms. C: probability histogram for the jitter for the 111 spikes with a reliability 
