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1Department of Biological Sciences, Ohio University, Athens, Ohio; and 2Tierphysiologie, Zoologisches Institut, University of Cologne, Cologne, Germany
Submitted 23 September 2006; accepted in final form 5 December 2006
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Muscles show great variation in how quickly they reach their steady-state length (Prosser 1973
). One extreme is muscles that produce large twitches in response to single motor neuron spikes (Kuffler 1942). Summation of these large twitches typically results in these muscles reaching steady state after a relatively small number of spikes. These muscles also typically relax quickly, which means that if motor neuron firing decreases, the muscles quickly reach the steady-state length appropriate to the new frequency. In these muscles length therefore well mirrors changes in motor neuron spike frequency and spike frequency can be said to code for contraction amplitude. These muscles, which typically produce muscle action potentials in response to each motor neuron spike, are called spiking muscles (Prosser 1973).
The other extreme is muscles that produce small contractions in response to individual motor neuron spikes (Kuffler and Williams 1953a,b). These muscles can require scores to hundreds of motor neuron spikes to reach steady state. These spike numbers can be significant fractions or even longer than physiological motor neuron bursts (Morris and Hooper 1997) and thus much or all of the muscle's contraction can be in the nonsteady-state domain. In this domain the muscles relax little between motor neuron spikes. In many cases so little relaxation occurs between each spike that, to a first approximation, the summed contraction amplitude is equal simply to the sum of the individual contraction amplitudes that each individual spike induced (in the limit in which zero relaxation occurs between each spike, the individual contractions would summate in a staircase-like fashion, with horizontal steps between the amplitude increases induced by each individual spike, and summed contraction amplitude exactly equaling the sum of the individual contraction amplitudes). In muscles in which facilitation or similar history-dependent effects are absent and thus the contraction increase each spike induces is constant, muscle contraction amplitude at any time during the period in which interspike relaxation is small would thus roughly equal the contraction increase each spike induces x the number of spikes that have been received by that time, and thus spike-number codes for contraction amplitude in a very simple manner (Morris and Hooper 1997). In muscles in which facilitation or similar history-dependent effects on twitch amplitude occur, spike-number dependency is more complicated because not only the number of spikes, but also the changing amplitude of the contractions they induce (which may depend on spike frequency) must be followed. Although these muscles are still, strictly speaking, spike-number–dependent muscles provided little relaxation occurs between spikes (because in this case, again, only the sum of twitch amplitudes, not the effects of interspike relaxation on summed amplitude, need be considered in calculating achieved contraction amplitude), clearly in these more complex muscles much of the conceptual and computational simplifications afforded by spike-number dependency disappears (see also DISCUSSION). An important consequence of the most simple type of spike-number coding, in which twitch amplitude does not show facilitation or similar history-dependent effects, is that, provided mean spike frequency is maintained, short-term changes in spike frequency scarcely alter contraction amplitude. Spike-number–dependent muscles, which generally do not produce action potentials, are called graded or tonic muscles (Prosser 1973).
Predicting muscle response to motor neuron activity is impossible without knowing where on this response continuum the muscle lies. Furthermore, the structure of the neural networks driving motor neuron activity presumably profoundly depends on whether spike frequency, spike number, or some combination of the two controls contraction amplitude. We report here that in the stick insect Carausius morosus spike-number codes for extensor muscle contraction amplitude across the large majority of the physiological range of motor neuron spike frequencies, that extensor motor neuron bursts are highly irregular, and that motor neuron bursts with different patterns of spiking activity can produce nearly identical muscle contractions.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Neural recordings during treadmill walking
This technique was extensively described elsewhere (Gabriel et al. 2003). All legs except the right middle were amputated at midcoxa, the animal was attached dorsal side up to a foam platform, and middle leg retraction and protraction were prevented with dental cement (Protemp II, ESPE). Platform height was adjusted so that at midstance femur–tibia and tibia–treadwheel angles were 90°. The thorax was then opened and gut, fat, and connective tissue were removed. Recordings from nerve nl3 (Graham 1985; Marquardt 1940) were obtained using a monopolar hook electrode and transferred to a personal computer using Spike2 software after digitization with a Micro1401 (both from Cambridge Electronic Design). Throughout the experiment the thorax was filled with C. morosus saline (in mM: NaCl 178.54, CaCl2·2H2O 7.51, KCl 17.61, MgCl2·2H2O 25, HEPES 10, pH 7.2) (Weidler and Diecke 1969). Tactile stimulation of the antennae or abdomen was used to induce walking. Six walking sequences from five animals were used in this work.
Muscle stimulation and recording
The same preparation as described earlier was used except dental cement was used to prevent all femur movement, the tibia was fixed to the side of the platform with an insect pin, and the extensor muscle tendon was severed below the femur–tibia joint (Fig. 1A). The tendon was then attached to an Aurora Scientific Inc. (ASI) Dual-Mode Lever System, a combined movement and force transducer in which the experimenter sets the maximum force the system delivers to the muscle. On muscle contraction the ASI delivers the amount of force necessary to prevent muscle shortening (isometric recording) until the set maximum force is reached. At muscle force levels greater than this force, the ASI delivers only the set maximum force and the muscle shortens against this constant load (isotonic recording). Maximum force was always set to the minimum that allowed the muscles to fully relax between contractions induced by stimulations mimicking fast extensor tibiae (FETi) motor neuron activity observed during isolated middle-leg walking. Muscle contraction was induced by 1-ms pulses delivered through a model MI401 stimulus isolation unit (designed by the Elektroniklabor Tierphysiologie at the University of Cologne) and a bipolar stainless steel electrode on nl3 insulated with petroleum jelly. Current-pulse amplitude was set slightly (10–20%) above contraction threshold.
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). This observation indicates that either we were in all cases stimulating the same set of axons or, if in different experiments different sets were being stimulated (in some, all three axons, in others, only FETi), this variation scarcely altered muscle response. The possibility that in some experiments SETi or common inhibitor (CI) may have been coactivated with FETi is thus unlikely to have any effect on the data presented here. Two nl3 nerve stimulation paradigms were used. In the first, the nerve was stimulated with bursts containing between five and 90 spikes, each delivered at 50, 117, 181, and 250 Hz. In the second, the nerve was stimulated once for each FETi motor neuron action potential in the nerve recordings recorded earlier in walking animals. Nerve stimulations and induced muscle contractions were simultaneously recorded with Spike2. Seven extensor muscles were used in this work, each driven by all six of the walking sequences recorded earlier from the five other animals.
Muscle fatigue
Compared with the other stepping sequences, one sequence (stepping sequence C in Fig. 2 of Hooper et al. 2006) consisted of very high frequency, high spike-number bursts that repeated at very short intervals. Comparison of the contractions induced by motor neuron bursts with equivalent spike frequencies and spike numbers early and late in this sequence showed that most of the muscles fatigued during this sequence (as assessed by equivalent motor neuron bursts inducing smaller contractions as the sequence progressed). The motor neuron bursts in this sequence were also unusually regular and thus the appearance of fatigue easy to identify; fatigued contractions were excluded from all analyses except that shown in Fig. 7D1, in which muscle response was irrelevant. In no muscles were more than 19 (11%) of the 178 contractions analyzed here excluded; on average 16 (9%) were excluded. In one muscle fatigue was not observed and thus for this muscle the data from all 178 contractions were analyzed; the data from this muscle were the same as those of the other six muscles [e.g., its contraction rises overlaid when plotted vs. spike number (Fig. 4, A and B) and contractions with very similar rises could be produced by very different spike trains (Fig. 6)]. As such, exclusion of the fatigued contractions from the other muscles did not appear to have any effect on the results. A 1-min rest interval was given between each stepping sequence and, after sequence C was played, full recovery (as assessed by comparing contractions induced by similar motor neuron bursts from stepping sequences played before and after sequence C) appeared to occur within this interval. The other stepping sequences were much more variable, with interspersed small and large contractions and, in particular, did not have long periods of very rapid stepping. No sign of fatigue (assessed as above) occurred in any other stepping sequence.
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Nerve stimulation and muscle contraction data were transferred to Kaleidagraph (Synergy Software) and analyzed using Kaleidagraph built-in functions or macros written by the authors. With respect to these analyses, two important issues are defining time 0 and muscle contraction velocity (rise slope). Figure 1, B–D shows the stimulation trains and contractions for two tonic (B, C) and one real (D) burst. An important concern was the considerable delay between the beginning of nerve stimulation (traces marked "Stim") and the beginning of muscle shortening (top traces). This delay occurred because the muscle could not begin to shorten until it developed sufficient force (traces marked "Force") to overcome the maximum force level set on the Aurora transducer. This delay posed the question of whether to use the beginning of the nerve stimulation, force production, or muscle shortening as time 0 in these experiments. For both tonic and real spike bursts, how long it took for muscle shortening to begin varied between muscles for identical stimulations and inside muscles for different patterns of stimulation. However, in most cases these variations, particularly within a muscle, were relatively small for tonic stimulations and, in any case, the analyses performed on these experiments were not affected by these variations. In these experiments the first spike of the stimulation was therefore defined as time 0. Real spike bursts showed much greater variation in how many spikes occurred before muscle shortening began. Examination of the force and movement traces showed that this variation occurred because of large burst-to-burst variations in how long it took for force to begin to develop and how long it took for the force level to reach the Aurora preset value at which muscle shortening could begin. To ensure that all muscles were analyzed from a common and functionally important time, for real spike bursts we defined spike zero as the first spike after muscle shortening had begun.
The second concern was defining contraction muscle contraction velocity (rise slope). For long stimulations contraction velocity (rise slope) decreases, eventually to zero, as the contraction approaches steady state (particularly evident in Fig. 1B). Our goal was to measure contraction velocities (rise slopes) in only the initial linear range. For tonic stimulations we therefore adjusted by hand the amount of the contractions included in the fits until they were highly linear as evidenced by values of R2 
0.995 (rectangles, Fig. 1, B and C). This procedure could not be used for contractions induced by stimulations mimicking walking motor neuron bursts because contraction velocity (rise slope) nonlinearity could arise either from increased contraction amplitude or variable motor neuron firing. To resolve this difficulty we noted by eye that no natural contractions appeared to show the characteristic leveling off of velocity (rise slope) that occurs at large contraction amplitudes (see Fig. 1B) until 
0.15 s after the time at which the contractions had achieved an amplitude of 0.05 mm (dashed horizontal line with arrow, Fig. 1D). For the natural contractions we therefore performed linear fits to the portion of the contractions between contraction beginning and 0.1 s after the time at which the contraction had achieved an amplitude of 0.05 mm (rectangle, Fig. 1D).
Muscle model
Each spike induced a constant-amplitude input that lasted 0.02 s. Model time steps were 0.00011 s; 181 time steps thus occurred in 0.02 s and each time step 0.0055249 mm was added to the previous time step's amplitude to give the next time step amplitude. In the absence of relaxation, this would result in each spike inducing a linear contraction lasting 0.02 s and resulting in an amplitude increase of 1 mm (181 time steps x 0.0055249 mm/time step = 1 mm). However, during each time step a relaxation process whose amplitude was proportional to the previous time step's amplitude also occurred (in the absence of spike input, such a "proportional" decrease results in exponential relaxation). The relaxation time constant was 1 s and thus at each time step 0.00011 x the previous time step's amplitude was also subtracted from the previous time step's amplitude. During the 0.02 s after each spike, these two processes result in an initial steep rise in amplitude whose slope then decreases as contraction amplitude increases. This change in slope during the rise induced by each spike is too small to be seen in the expanded insets in Fig. 2A, but is the reason that the rise induced by each spike is greater earlier than it is late in the contractions (given that because contraction amplitude is greater late in the contractions, the amount of relaxation occurring in the time steps during the spike-induced input is therefore also greater). After the spike induced rise ceases at 0.02 s, only the relaxation process is active, which results in contraction amplitude exponentially decreasing with a 1-s time constant.
Figure preparation
Plots were generated using Kaleidagraph (Synergy Software) and figures were prepared in Canvas (ACD Systems).
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Review of slow, spike-number–dependent muscles
The muscles of many invertebrate systems in which the neuronal basis of motor pattern generation has been well investigated are graded [e.g., movement is produced by graded muscles in leech swimming, Aplysia feeding, the crustacean pyloric, gastric mill, cardiac sac, esophageal, and swimmeret systems, and crab ventilation (Davis 1968; Mason and Kristan 1982; Morris and Hooper 1997; Selverston et al. 1976; Weiss et al. 1992; Young 1975)]. Extensor muscles in locust (Hoyle 1978; Bässler 1996a) the related stick insect Cuniculina impigra (Bässler 1996a,b) and Carausius morosus (Bässler 1996a) similarly do not produce action potentials but instead show only summating extrajunctional potentials in response to motor neuron input and therefore contract in a graded fashion. However, most vertebrate skeletal muscles are not graded (Prosser 1973) and the properties of graded muscles may thus be unfamiliar to many readers and thus we briefly compare the two muscle types.
In both types large contractions result from summation of the twitches each individual motor neuron spike induces. Early in a motor neuron burst the amplitude of the twitches is larger than the relaxation that occurs between spikes and thus the twitches temporally summate and overall contraction increases. As the burst continues both types eventually achieve (for tonic motor neuron activity) a steady-state contraction. Under both isometric and isotonic recording conditions a large part of this equalization occurs because interspike relaxation amplitude increases as (isometric conditions) force increases or (isotonic conditions) muscle length decreases. Decreases in twitch amplitude can also play a role in this equalization process in two ways. First, under both isometric and isotonic conditions twitch amplitude can decrease if motor neuron spike frequencies are sufficiently high that excitation–contraction processes (e.g., intracellular calcium concentrations) begin to saturate. Second, under isotonic conditions twitch amplitude decreases because, as the muscle shortens, the muscle approaches the length on the length–tension curve at which muscle force equals muscle load. In both cases and for both muscle types, steady-state force/length is a function of motor neuron spike frequency.
The difference between the two muscle types is the rapidity with which steady state is attained. In nongraded muscles each motor neuron spike induces twitches that are large enough that in response to tonic stimulation the muscles reach steady state within a few (generally 
10) spikes. That these muscles produce large twitches does not, a priori, imply anything about their relaxation rate. However, the fact that they produce large unitary contractions gives them the option of relaxing quickly, in that even with considerable interspike relaxation, large twitches can still summate to produce relatively large contractions, and many such muscles do relax relatively quickly. In response to changing spike input, their contraction therefore both increases and decreases rapidly concomitantly with spike frequency. Although some nongraded muscles have history-dependent effects that complicate this simple description [e.g., latch, in which adding a single extra spike to a train induces large, long-lasting changes in contraction amplitude (Abbate et al. 2000; Grottel and Celichowski 1999; Lee et al. 1999a,b; Thomas et al. 1999; Van Luteren and Sankey 2000)], to a first approximation motor neuron spike frequency consequently largely codes for their contraction amplitude.
Graded muscles, alternatively, produce small, often very small, twitches and can often take scores to hundreds of spikes to achieve steady state. Unlike nongraded muscles, graded muscles must relax slowly (relative to physiological motor neuron interspike intervals) to achieve large contractions. Provided history-dependent processes that change twitch amplitude, such as facilitation (see INTRODUCTION and DISCUSSION), are not present (which is the case for the muscle investigated here), a consequence of this combination of small twitch amplitude and slow relaxation is that contraction amplitude during their initial rises of these muscles often depends on spike number, not spike frequency (see following text). A final general point is that graded muscles typically do not produce muscle action potentials and nongraded muscles do. However, this is not the salient difference between the two muscle types because spiking muscles that produce small twitches could be graded and nonspiking muscles that produce large twitches nongraded. The salient difference is whether twitch amplitude is large enough that it takes only a few spikes to reach steady-state contraction amplitudes.
To make these points clear, Fig. 2 shows data from a simple model in which each spike induces a contraction that linearly rises for 0.02 s to reach a nominal amplitude of 1 mm and amplitude relaxes exponentially with a 1-s time constant (for further model details see METHODS). Although this model is very simple it well captures many of the salient features of slow muscle contraction and has been successfully used in investigations of slow crustacean pyloric muscles (Morris and Hooper 1997, 1998). Figure 2A shows the model's response to 7-s-duration tonic stimulations at 10, 25, 50, and 100 Hz. In all these stimulations the model contracts much more slowly than would a nongraded muscle (the dotted red line schematically indicates the contraction expected for 100-Hz stimulation of a nongraded muscle, in which a fused or nearly fused tetanus would be achieved within the first five to ten spikes).
The contractions consist of an early linear rise in amplitude that then levels off. The initial linear rise occurs because summated amplitude is small early in the contractions. Because relaxation is an exponential function of amplitude, little relaxation therefore occurs between each spike. The twitches consequently summate in a staircase-like fashion (Fig. 2A, left inset; the black traces show the contraction induced by the first motor neuron spike, the model's relaxation during the subsequent interspike interval, and the contraction induced by the second spike). Note that nongraded muscles, because of their rapidly reaching steady state in response to tonic drive, cannot produce such slow rises in response to tonic motor neuron firing—for these muscles such slow rises require either a matching slow increase in motor neuron spike frequency or sequential activation of additional motor units (Desmedt and Godaux 1977; Monster and Chan 1977).
The leveling off occurs because, again because relaxation is exponential, relaxation slope increases with amplitude. As amplitude increases, the model therefore relaxes more rapidly and, thus further, during each interspike interval. Because relaxation continues to occur during the rises each spike induces (see METHODS), twitch amplitude also decreases as summed contraction amplitude increases. These two processes result in the contraction, eventually reaching an amplitude at which interspike relaxation and twitch amplitudes are equal (Fig. 2A, right inset; with the parameters used here the equalization primarily arises from increased interspike relaxation amplitude), at which time shortening ceases. The amplitude at which this occurs depends on spike frequency because interspike interval decreases with spike frequency. For higher spike frequencies greater amplitudes must therefore be achieved before interspike relaxation and twitch amplitudes are equal.
During the initial phases of the contractions the relaxation slope is sufficiently small that changing interspike interval only slightly changes summated amplitude. This point is illustrated in the orange lines in the left inset of Fig. 2A, which show the effect of having the second spike occur at half the delay as it did in the black trace. Comparing the amplitude achieved after this higher-frequency spike to that achieved at the slower spike frequency shows that doubling spike frequency only slightly increased the amplitude achieved after two spikes. It is this property that leads to the contraction amplitude of graded muscles being spike-number dependent during the initial phases of their contraction, in that during this period contraction amplitude approximately equals twitch amplitude x the number of spikes the muscle has received. This point is made graphically in Fig. 2B, which plots the contractions in Fig. 2A versus spike number instead of time. The low spike-number portions (oval) of all the contractions now overlie, demonstrating how during this phase of the contractions, contraction amplitude depends only on the number of spikes the model has received, not the frequency with which they were delivered.
Figure 2, C and D shows fits to the initial linear portions (defined as with other tonic stimulations; see METHODS) of the contractions in Fig. 2, A and B. Figure 2, E and F plots the slopes of these fits versus spike frequency. This analysis shows that, when plotted in the time domain (Fig. 2, C and E), the initial velocity (rise slope) of graded muscle contractions should double with each doubling of motor neuron spike frequency. The analysis in the spike-number domain (Fig. 2, D and F) shows that, even in the linear range, spike-number dependency is not perfect (because the spikes do fall earlier in the relaxation phase for higher spike frequencies, contractions driven by higher-frequency spike trains do rise slightly faster). However, the increase in rise per spike with increasing spike frequency is much less (a 30% increase between 10 and 100 Hz, Fig. 2F) than the increase in rise/s (a 956% increase, Fig. 2E). The increase in rise per spike is also nonlinear, becoming less at high spike frequencies.
SUMMARY. In slow muscles, when muscle shortening is plotted versus spike number, contractions induced by tonic nerve stimulation at different spike frequencies should overlie at small spike numbers (the spike-number–dependent domain). At high spike numbers the curves should separate and contraction steady-state amplitude should double as spike frequency doubles. When plotted versus time, the velocities (rise slopes) of the initial linear portion of the contractions should double as spike frequency doubles. When plotted versus spike number, the velocities (rise slopes) of the initial linear portion of the contractions should change relatively little as spike frequency increases.
Carausius extensor muscles do not fulfill all theoretical predictions, but these differences only increase the muscles' spike-number dependency
To determine whether extensor muscles were spike-number dependent, we performed the analyses in Fig. 2 with seven real muscles by stimulating the extensor muscle's motor nerve, nl3, with constant spike frequency trains. Figure 3A repeats the model data in the spike-number domain. The responses of all seven muscles were similar; Fig. 3, B1–B4a shows data for four of them. Comparing the model and real contraction data shows that they differ considerably. In the model the traces overlaid only for small spike numbers and the steady-state amplitudes at different spike frequencies doubled with each spike frequency doubling. The 50-Hz curves in muscles 6 and 7 do have smaller steady-state amplitudes than those of curves for the faster spike frequencies. However, in these two plots at the faster spike frequencies, and in the other plots for all spike frequencies, steady-state amplitude in general does not increase with increasing spike frequency, with 250 Hz having lower amplitudes (at matching spike numbers) than 117 and 181 Hz in muscles 1 and 4; 117 and 181 Hz having similar amplitudes in muscle 1; and all three high-frequency curves having similar amplitudes in muscles 6 and 7. To demonstrate unambiguously how different these data are from those in Fig. 2, in Fig. 3B4b the data of Fig. 3B4a were scaled so that steady-state amplitudes are proportional to spike frequency (i.e., the 117-Hz steady-state amplitude is 117/50 = 2.34 times greater than 50 Hz). When this is performed, there is once again an early period of spike-number dependency (oval) and then a divergence of the four curves as spike number increases.
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Regardless, it is important to stress that the differences between the responses of the model and the real muscles actually make the extensor muscle more, not less, spike-number dependent in that spike number fairly well predicts contraction amplitude for all spike frequencies and at all spike numbers, not just low spike numbers as was the case in Fig. 3A. To see this, note that in perfect spike-number–dependent contractions, the equation Amp = A0(1 – e–spike number/
) well fits the data for all spike numbers. Using this equation to fit all the data in Fig. 3B4a results in a fit (open circles) with an R2 value of 0.87 and a maximum error in the high spike-number region of only 45% (double-headed arrow). In the model (Fig. 3A), alternatively, contraction amplitude in the high spike-number (steady-state) region doubles with each spike frequency doubling, and thus the fivefold change in spike frequency used in Fig. 3B4a would result in a 500% change in contraction amplitude at high spike numbers. This increased ability in the real data of spike number to predict contraction amplitude at all spike numbers, not just small ones, is also demonstrated by performing the same fit on the "data" in Fig. 3B4b (open circles), which results in an R2 value of only 0.51 and a maximum error in the high spike-number region of 210% (double-headed arrow).
SUMMARY. Unlike the model shown in Fig. 2, Carausius extensor muscles tend to achieve similar contraction amplitudes when driven by tonic spike trains at different spike frequencies. However, the primary result of this difference is that spike number is a good predictor of contraction amplitude at all spike numbers, not just low ones.
When driven by natural single-leg walking motor nerve stimulations, spike number excellently predicts the amplitudes of the rise phase of the resulting muscle contractions
These data with artificial spike trains suggest that extensor muscle amplitude should be spike-number dependent across a very wide range of spike frequencies and at large spike numbers. It was nonetheless possible that in natural bursts, which we will show later to be highly irregular in their internal structure, some form of history dependency and spike patterning might reveal that spike number is not a good predictor of contraction amplitude. To resolve this issue, we stimulated the nl3 nerves of seven extensor muscles with 178 natural bursts recorded during single-leg walking and performed a series of analyses to test whether spike number well predicted contraction amplitude during contraction rises. Figure 4 shows this analysis for one muscle (the one shown in Fig. 3B3).
We first plotted the initial (the portion of the contraction between contraction beginning and 0.1 s after the time the contraction had achieved an amplitude of 0.05; see METHODS) rises of all 159 of this muscle's usable (see Muscle fatigue in METHODS) contractions (Fig. 4A). The contraction rises were visually very different, ranging from contractions showing slow and irregular initial rises (most of the black and red contractions and many of the blue; see following text for explanation of color assignments) to ones showing very strong and rapid initial rises (the green and purple contractions).
The part of the neuron bursts that drove these contractions, with their wide range of initial velocities (slopes) and contraction shapes, had very different durations and patterns of spike firing. For instance, the initial rise portion of all the purple contractions ended within 0.1 s of contraction beginning, and thus only spikes before this time could have contributed to their initial rise. For the red and black contractions, alternatively, all spikes occurring before 0.15 to as much as 0.3 s after contraction beginning contributed to the initial rises of the contractions. Spike patterning during these early parts of the bursts also showed great variation, with spiking during the initial rise period for the purple and green contractions being relatively regular and high frequency, but spiking during the initial rise period for the red and black (and some blue) contractions generally showing an initial bout of rapid spiking followed by varying lengths of very low frequency spiking before a final period of rapid firing.
If this muscle is spike-number dependent, the prediction is that these wide velocity (slope) and shape variations would be appreciably reduced when the contractions were plotted against spike number. That is, if the muscles were perfectly spike-number dependent, regardless of the frequency and the pattern with which spikes are delivered, all contractions would have equal amplitudes after an equal number of spikes had occurred. Figure 4B, in which the data in Fig. 4A are plotted versus spike number, shows that this prediction is almost perfectly upheld, with almost all the traces now completely overlying (in the replotting, each contraction retained the color it had in Fig. 4A and thus, for instance, the black traces in Fig. 4, A and B are the same set of contractions). Exactly similar results, in which the initial portions of the contractions showed large variation when plotted against time, but almost perfectly overlay when plotted against spike number, were observed in the other six muscles as well (data not shown).
Although the visual impression of increased overlap of the contractions when plotted versus spike number supports the contention that the extensor muscle is spike-number dependent, this is nonetheless merely a visual comparison and thus subject to the usual problems of observer bias and interpretation. To remove any ambiguity and to prove that the contractions become more similar when plotted versus spike number, we performed two analyses on the data in Fig. 4, A and B to quantify the magnitude of the changes between the two plots and to determine whether the changes were statistically significant. In the first we performed linear fits to each contraction rise in Fig. 4A to compare quantitatively the extent to which the wide variety of contraction velocities (rise slopes) in Fig. 4A were reduced when the contractions were instead plotted versus spike number (Fig. 4B). The fits to the contraction rises in Fig. 4A had slopes varying between 0.4 mm/s (the fit to the black trace marked with an asterisk) and 3.4 mm/s (the most rapidly rising purple contraction). We then sorted the contractions into five classes on the basis of fit slope (slope 0.4 to <1, black traces; 1 to <1.6, red traces; 1.6 to <2.2, blue traces; 2.2 to <2.6, green traces; 2.8 to <3.4, purple traces; all mm/s). We then plotted for each contraction velocity (rise/s) class (the different colors) the slope of each contraction in Fig. 4A that was in that class (open circles) (Fig. 4C). Thus the bottom open circle in the first column of circles (the 0.4 to <1 column) in Fig. 4C is the slope of the lowest velocity (lowest rise slope) contraction in Fig. 4A (the black contraction marked with the asterisk). The other circles in this column are the velocities (rise slopes) of the fits to the other black contractions in Fig. 4A. Similarly, the open purple circles in the last column of circles (the 2.8 to <3.4 column) are the velocities (rise slopes) of the fits to all the purple contractions in Fig. 4A. The closed circle and error bars to the right of each column of open circles are the mean contraction velocity (rise slope) and SD of that column's open circles. A 3.7-fold difference in contraction velocity (rise slope, rise/s) is present in the averaged data (mean of black data, 0.8 ± 0.2 mm/s; mean of purple data, 2.96 ± 0.2 mm/s). We also tested whether the data in the different classes differed from each other (e.g., whether the purple open circles differed from the green, blue, red, and black open circles). Not surprisingly, because the data were purposely sorted into nonoverlapping classes, each class's data differed from those of every other class at P < 0.001 except for one comparison (0.4 to <1 vs. 1 to <1.6), which differed at P < 0.002.
We then performed linear fits on the data in Fig. 4B to obtain rise per spike data for each contraction and plotted (Fig. 4D) these rise per spike data sorted according to which rise/s class they corresponded to. That is, the data in the first (black) column in Fig. 4D are the rise per spike slopes of the most slowly rising class of contractions (the first, black column in Fig. 4C) when the contractions are plotted versus time. For instance, the bottom data point in the open black circle column in Fig. 4C, with a slope of 0.4 mm/s when plotted versus time, had a slope of 0.007 mm/spike (the bottom data point in the open black circle column in Fig. 4D) when plotted versus spike number. When this was done, what had been a 3.7-fold range was reduced to a 1.1-fold range and the data in none of the five classes statistically differed from one another. This analysis shows that all these contractions, with their very different contraction velocities (rise slopes) when plotted versus time, nonetheless have identical rises per spike, independent of the frequency or pattern with which the spikes were delivered, when plotted versus spike number. For all these contractions the amplitude at any time in the contraction's initial rise can therefore be calculated simply by multiplying the average rise each spike induces (for the muscle shown in Fig. 4, 0.015 ± 0.002 mm/spike) by the number of spikes the muscle has received by that time, and thus spike number codes for initial rise amplitude in this muscle.
Although the preceding analysis showed that the velocities (rise slopes) of the contractions in Fig. 4A became the same when plotted against spike number, that alone does not show that the contraction rises overlap. For instance, in theory the slopes could all have been the same when plotted against spike number, but the different contraction classes could still have been shifted horizontally relative to each other when plotted in this manner (e.g., all the purple contractions in Fig. 4B could have laid to the left of the black contractions–that is, the intercepts of the curves could have occurred at different spike numbers). Visual inspection of Fig. 4B shows that this is not the case, although visual inspection alone does not quantify the extent to which the overlap of the various contraction classes increases when plotted versus spike number. To quantify the increased degree of overlap, we drew by eye minimum area hexagons that enclosed the traces of each rise/s class in Fig. 4A [the black dashed lines in Fig. 4A show the hexagon that enclosed the slowest rising (black) contraction class]. The insets in Fig. 4, A and B show the enclosing hexagons of all the contraction classes in both the time and spike-number domains; much greater overlap occurs when the contractions are plotted against spike number. This analysis allowed us to quantify the amount of overlap between the hexagons corresponding to the different rise/s classes in Fig. 4A. The hexagons in the time domain had overlaps ranging from 1% (purple and black hexagons) to 31% (blue and red hexagons) with a mean overlap of 14 ± 12%, whereas those in the spike-number domain had overlaps ranging from 39% (red and black hexagons) to 75% (green and blue hexagons) with a mean overlap of 55 ± 11%, and the two data sets differed at the P < 0.0001 level. Thus plotting contraction initial velocities (rise slopes) against spike number results in an almost fourfold increase in overlap, again showing that spike number codes for contraction amplitude in this muscle.
SUMMARY. Although the natural contractions had a wide variety of initial contraction velocities (rise slopes) and shapes, all contraction rises overlaid extremely well when plotted versus spike number. Quantitative analyses of the data supported these visual observations and, in particular, showed that contractions that rise with different slopes when plotted versus time nonetheless had identical rises per spike when plotted versus spike number.
In the model, different spike patterns can give rise to very similar contraction rises
Figure 4A shows that contractions driven by stimulations mimicking natural spike bursts have a wide variety of initial velocities (rise slopes), ranging from slopes of 1 to 3 mm/s. The bursts that gave rise to these different burst were, of course, very different. In particular, the more rapidly rising (e.g., purple) contractions were driven by bursts with much higher spike frequencies during the part of the burst giving rise to these initial portions of the contractions than were the more slowly rising (e.g., black) contractions. This is not an argument that the muscles are not spike-number dependent in all cases; Fig. 4B clearly shows that they are. Spike-number dependency does not state that a muscle will not achieve a higher contraction amplitude more quickly when spikes are delivered more quickly, but rather just that (absent facilitation or other history-dependent phenomena), no matter how quickly or slowly the spikes are delivered, that the muscle will always achieve the same amplitude after the same number of spikes are received (contraction amplitude equals twitch amplitude x spike number).
However, further consideration of this issue raises the related issue of whether the bursts that give rise to contractions with similar rises (e.g., the purple contractions) have similar spike patterning within them. This issue arises because a key requirement for a muscle to show spike-number dependency is that its time constants are long compared with the spike intervals present in its input (it is this property that results in relatively little relaxation occurring between spikes). These muscles are thus, compared with their inputs, very low pass filters, and thus would filter out high-frequency variations in spike patterning. Thus, provided overall spike frequency is maintained, the velocity (rise slope) of a spike-number–dependent muscle should be relatively independent of fine-scale variation in spike timing within its motor neuron's burst. Figure 5 demonstrates this property by showing the response of the model used in Fig. 2 to six patterns of 11 spikes arranged in different ways. In Fig. 5, A and B, different spike frequencies continue for relatively long periods (in A, 25 Hz for 0.2 s followed by 100 Hz for 0.05 s; in B, 100 Hz for 0.05 s followed by 25 Hz for 0.2 s). As expected, each of these two spike patterns induces contractions with two different slopes and the two spike patterns induce contractions with different shapes (in Fig. 4A an initial slow rise followed by a late rapid rise; in Fig. 4B an initial rapid rise followed by a late slow rise). Thus even in a spike-number–dependent muscle (see Fig. 5E), changing spike frequency in a sustained manner will result in different contraction velocities (rise slopes). Referring again to Fig. 4A, the fast spiking that gives rise to the initial fast rise in Fig. 5B would correspond to the fast average spiking that gave rise to the purple contractions, and the slow spiking that gave rise to the initial slow rise in Fig. 5A would correspond to the slow average spiking that gave rise to the black contractions. Similarly, contractions that showed multiple slopes in their rises (several of the black and red contractions) would have been produced by bursts that had sustained periods of low- and high-frequency spiking (analogous to the sustained periods of slow and fast spiking present in the bursts in Fig. 5, A and B).
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SUMMARY. Driving the model used to create Fig. 2 suggests that, provided the bursts have approximately the same spike numbers and overall spike frequency, contractions with very similar rises can be produced by bursts with differing spike patterns.
Large scale analysis suggests that extensor motor neuron bursts do not have a common internal structure
The work in the model suggested that contractions with very similar rises could be induced by bursts with different fine-scale spike patterns. We therefore investigated whether the portions of the bursts that gave rise to the contraction beginnings in each rise/s class in Fig. 4A had similar spike patterns and how these patterns changed as contraction velocity (rise slope) changed. In the first analysis we divided the spikes occurring before and during the initial rise into 0.025-s bins and then calculated the mean spike frequency in each bin. When averaged in this manner, the spike frequency profiles were largely flat for all rise/s classes (data not shown). Two consistent large-scale changes did occur as rise/s class increased. First, burst duration decreased. This change was expected because more slowly rising contractions took longer both to reach the Aurora force/movement set point and to reach the amplitude used to define rise ending (METHODS). Second, average burst spike frequency increased as rise/s increased, from about 100 Hz for contractions that rose 0.4 to 1 mm/s to about 200 Hz for contractions that rose 2.8 to 3.4 mm/s. This change was again expected because more rapidly rising contractions should be driven by higher spike frequency bursts. However, a very large spike frequency range was present in each bin in all panels, almost always 100 Hz and often close to 300 Hz. We also tested individually whether spike frequency and time into the burst were correlated for the spikes occurring before and during the initial contraction rise for each of the 159 motor neuron bursts that gave rise to usable contractions in this muscle. This analysis showed that no correlations were present in any rise/s class except the most rapidly rising (2.8 to <3.4 mm/s) one, in which 58% of the bursts showed a decreasing spike frequency late (after time 0; see Fig. 1D) in these initial portions of the bursts (data not shown). This lack of correlation in most motor neuron bursts, and the large ranges and SDs in the binned data, showed there was great variability in the initial portions of the different bursts, even when sorted into classes on the basis of producing contractions with similar rises/s. These analyses thus suggest that there is no canonical spiking pattern in the initial portions of extensor motor neuron burst (spike frequency does not, for instance, smoothly and similarly rise in all bursts as the bursts progress) and that therefore [given that many contractions have similar initial velocities (rises slopes)] bursts with different spike patterns produce contractions with similar rises.
Contractions with very similar rises can be produced by very different spike bursts
The above analysis examining how spike frequency varies across many bursts does not examine spike-by-spike patterning within individual bursts, an important issue given that the modeling work in Fig. 5 suggests that different spike patterns can produce contractions with very similar rises. To test whether this was true of real muscles, we examined all 159 usable contractions of the muscle shown in Fig. 3B3 to identify six pairs of contractions with similar rises (Fig. 6, A–F). Each panel shows the contractions (the smoothly rising red vs. blue lines), the spike trains that gave rise to the contractions, and the spike frequency profiles of the trains. In each case the spiking pattern and spike frequency profiles appear very different, even though the contraction rises are very similar.
However, although the spike patterns appear different, as with Fig. 4, it is difficult to state without quantification how great a difference this visual impression actually represents. One method to quantify the differences between them is to calculate the normalized distances between the spike-frequency profiles. This could have been performed with the spike-frequency plotting convention used in Fig. 6, A–F, but to make these distances comparable to work in other systems (Zhurov and Brezina 2006), for these calculations we replotted the data in a slightly different manner (Fig. 6, G–I). In these new plots of the data, instead of each spike pair's spike frequency being plotted as a point at the time of the last spike of the pair and the points then connected with lines (the slanting lines connecting the points in Fig. 6D), instead each spike pair's spike frequency is plotted as lasting throughout the duration of the pair's interspike interval, and changing instantaneously from spike pair to spike pair (the "skyscraper" plotting convention shown in Fig. 6G). For example, the spike frequency of the red spikes marked with asterisks in Fig. 6G is 77 Hz and the corresponding portion of the red curve has a value of 77 Hz for all times between these two spikes. The red curve then jumps to the spike frequency of the next spike pair (333 Hz) and maintains this value for all times between these next two spikes (comparison of these same data points in Fig. 8D shows that in the plotting convention of Fig. 8D each spike frequency is represented by a single data point, plotted at the time of the second spike in each spike pair, and these points are connected by slanting lines). The data from each spike burst were plotted in the fashion shown in Fig. 8G (red, blue curves) and the normalized distance between the two curves was calculated (see figure legend for details). The normalized distances between the contractions induced by the two spike trains were also calculated. When these calculations were performed for each of the spike train pairs shown in Fig. 6, A–F and the data averaged, the mean normalized spike frequency distance was 0.51 ± 0.06 and the mean normalized contraction difference was 0.03 ± 0.01. These normalized distances are directly comparable and show that the changes in spike patterning in Fig. 6, A–F are about 17-fold greater than the corresponding changes in contraction rise.
Although these analyses show that the spike pattern changes are much larger than the contraction rise changes, they do not show how different the spike pattern pairs are relative to all possible rearrangements of the spikes in the bursts. This comparison can be performed in two ways. In the first, the spikes are rearranged to maximize the distance between the two spike patterns while keeping interspike intervals constant. Figure 6H shows this procedure for the data in Fig. 6G; the rearrangement results in one burst's spike frequency monotonically declining, whereas the other's monotonically increases. When the spike intervals are thus rearranged, the normalized distance between the two is 0.78. The normalized distance between the real spike patterns (Fig. 6G) is 0.49. The distance between the real spike bursts is thus 63% of the maximum distance that can be achieved by rearranging the interspike intervals present in the two bursts.
The second comparison is to rearrange the spikes and also alter interspike interval to maximize the distance between the two spike patterns. This procedure has the drawback that the maximum distance that can be obtained is not well defined because it can always be increased further by allowing the two bursts to have greater maximum spike frequencies. We chose here to limit maximum spike frequency in the rearranged bursts to 333 Hz, the maximum spike frequency observed in real extensor motor neuron bursts. The result of this rearrangement is that one burst fires tonically at 333 Hz for all spike intervals except the last and the second burst fires tonically at 333 Hz for all spike intervals except the first (Fig. 6I). The normalized distance between these two spike patterns is 1.6 and thus the distance between the real spike patterns (0.49) is 31% of the maximum distance that can be achieved by rearranging both spike intervals and spike frequencies. When similar rearrangements were performed on the spike patterns in Fig. 6, A–F and the results averaged, the distances between the real bursts had a mean of 58 ± 6% of the maximum distance that could be achieved by rearranging spike intervals (as in Fig. 6H) and 28 ± 3% of the maximum distance that could be achieved by rearranging spike intervals and spike frequencies (as in Fig. 6I). Taken together, these analyses thus show that the differences between the spike patterns in Fig. 6, A–F are truly large compared with the changes in the rises of the contractions they induce and are truly large compared with spike pattern rearrangements that produce maximally different bursts.
SUMMARY. Contractions that have visually very similar rises can be produced by spike patterns that are visually very different. Quantitative analyses support these observations.
Return maps show that the initial portions of single middle leg walking extensor motor neuron bursts have no fine-scale order
Although these data demonstrate that different spiking patterns can give rise to similarly rising contractions in stick insect extensor muscles, because they consist of only 12 contractions they do not prove there is no fine-scale pattern present in most extensor motor neuron bursts. To examine this question we constructed return maps on the beginnings (the portions giving rise to the contraction rises) of the motor neuron bursts. Return maps are a standard plotting technique in dynamic systems analysis that reveal relationships between past and present events in time sequences by repeatedly plotting, time step by time step, a system's value at that time step versus the system's value at a given amount of time before that time step (i.e., for all values of t plotting the value at time t vs. the value at time t minus a certain value). If there is some relationship between values at different time points, this procedure transforms time series data into characteristic geometric shapes that depend on system properties and the time offset value chosen. The utility of this plotting technique is that often regularities present in the data that are not obvious when inspecting the time series will become obvious in the return map.
Although in neurobiology this technique has been primarily used in theoretical and modeling work, it has also been used to analyze experimental data in the decapod crustacean pyloric system to show that the motor neuron bursts have repeatable and characteristic interspike interval profiles (Szucs et al. 2003). When used for this purpose, the technique is slightly altered in that, instead of repeatedly plotting the values of the system (in the case of neurons, the membrane potential) time step by time step, what is repeatedly plotted instead are the spike frequencies of each adjacent spike pair, that is, the spike frequency defined by the second and third spikes is plotted versus that defined by the first and second, the spike frequency defined by the third and fourth spikes is plotted versus that defined the second and third, and so forth. Inasmuch as this technique may not be familiar to all readers, and to demonstrate its utility in revealing temporal patterns in time series data, we will first demonstrate this technique using artificial data. Figure 7A1 shows the spiking activity and spike frequency profile of a neuron that fires spikes in its bursts in the following pattern: 250 Hz, 50 Hz; 250 Hz, 50 Hz, and so forth for five repetitions followed by a final 250-Hz spike pair. In all cases a 10% randomness was added to the data so that multiple spike frequencies would be present. Figure 7A2 shows the return map of these data. This plot is constructed by using the spike frequency of the first spike pair (about 250 Hz) as the abscissa (x) value and the spike frequency of the second spike pair (about 50 Hz) as the ordinate (y) value, thus giving rise to one point in the lower right data cluster. The next point is then plotted by using the spike frequency of the second spike pair (about 50 Hz) as the ordinate value and the spike frequency of the third spike pair (about 250 Hz) as the abscissa value, thus giving rise to one of the points in the upper left data cluster. This process is repeated for all the spike pairs in every burst (in this and all analyses in this figure, interspike intervals corresponding to interburst intervals were discarded). This analysis results in two clusters of points, indicating that in this neuron rapidly firing spike pairs always follow slowly firing spike pairs and slowly firing spike pairs always follow rapidly firing spike pairs.
Figure 7, B1, B2, C1, and C2 shows the same analysis for a perfect half-parabolic burster (the neuron's spike frequency profile follows the ascending branch of a parabola, Fig. 7B1) and a half-parabolic burster with added noise (Fig. 7C1). Because in Fig. 7B1 the spike frequencies perfectly follow the parabola, in the return map for these data (Fig. 7B2) each apparently single data point is actually three data points (one for each burst in Fig. 7B1). In Fig. 7C2, alternatively, except for the first two low-frequency spike pairs, the spike pairs do not segregate into clusters because of the introduced noise. However, it is still clear that there is a progressive increase in spike frequency in these bursts as the burst progresses.
Figure 7D1 shows that no pattern is apparent in a return map (using spikes before or during the initial contraction rises) when data from all 178 bursts in our data set are included. This is not necessarily surprising because these bursts give rise to contractions with a wide range of velocities (rises/s) (Fig. 4A). Figure 7D2 shows the return map of only the bursts in the highest rise/s (purple) class. No clustering (similar to that seen in Fig. 7A2) or obvious ordering (similar to that seen in Fig. 7, B2 and C2) is present. To confirm this observation, return maps (using spikes before or during the initial contraction rises) of each burst of the individual experiments in each rise per spike classes were made. In none of these bursts was consistent obvious clustering or data ordering present, nor were n + 1 and n correlated in a majority of the bursts. This analysis indicates that no consistent spiking pattern appears to be present within the portions of extensor motor neuron bursts giving rise to the rising portions of extensor muscle contractions.
idák compensations for 10 simultaneous comparisons were made.
= 249.11 Hz. When this is done across the entire time interval of the 2 curves (–0.04 to 0.12 s), the 2 spike frequency profiles in G are separated by a distance of 93 Hz and the corresponding contractions in D by a distance of 0.0013 mm. These numbers cannot be directly compared because the average amplitudes of the spike frequency profiles and the contractions profiles are so different (the average spike frequency of the blue and red traces in G is 189 Hz, whereas the average of the contraction profiles in D is 0.11 mm). Raw distances were therefore normalized by dividing them by the average of the parameters that gave rise to them. When this normalization is performed, the spike patterns in D have a normalized distance of 0.49 (93 Hz/189 Hz) and the contractions of 0.012 (0.0013 mm/0.11 mm). Because normalization also removes any difficulties associated with the spike patterns and contraction pairs in A–F having different average values, it also allows averaging normalized distances across these panels.
