Measurement and nature of firing rate adaptation in turtle spinal neurons

Abstract

There is sparse literature on the profile of action potential firing rate (spike-frequency) adaptation of vertebrate spinal motoneurons, with most of the work undertaken on cells of the adult cat and young rat. Here, we provide such information on adult turtle motoneurons and spinal ventral-horn interneurons. We compared adaptation in response to intracellular injection of 30-s, constant-current stimuli into high-threshold versus low-threshold motoneurons and spontaneously firing versus non-spontaneously-firing interneurons. The latter were shown to possess some adaptive properties that differed from those of motoneurons, including a delayed initial adaptation and more predominant reversal of adaptation attributable to plateau potentials. Issues were raised concerning the interpretation of changes in the action potentials’ afterhyperpolarization shape parameters throughout spike-frequency adaptation. No important differences were demonstrated in the adaptation of the two motoneuron and two interneuron groups. Each of these groups, however, was modeled by its own unique combination of action potential shape parameters for the simulation of its 30-s duration of spike-frequency adaptation. Also, for a small sample of the very highest-threshold versus lowest-threshold motoneurons, the former group had significantly more adaptation than the latter. This finding was like that shown previously for cat motoneurons supplying fast- versus slow twitch motor units.

Appendix

R.B. Gorman

Exponentials used to describe late adaptation

Biological processes that decay over time to a constant value may often be modeled by a decaying exponential. A sum of decaying exponentials may be required if there are a number of processes with different time courses (Sawczuk et al. 1995). In the present report, the late adaptation in spike-frequency (f) after the transition spike-frequency point was modeled by a sum of exponentials:

$$ f = k + A_1 {\text{e}}^{ -t/\tau _1 } + A_2 {\text{e}}^{ - t/\tau _2 } + \ldots + A_n {\text{e}}^{ -t/\tau _n} $$

where t (in s) is the time after the first spike, K (Hz) is the offset constant (i.e., the spike-frequency that the exponentials decay to), A₁ ... A _n (Hz) are the amplitudes of each exponential, and τ1 ... τ _n (s) are the time constants.

The modeling process involved three parts: (1) an initial estimate of K; (2) an initial estimate of the other coefficients of the fit with a “peeling” procedure (Spielmann et al. 1993, Sawczuk et al. 1995); and, (3) an iterative non-linear curve fit to determine the final coefficients. The estimation of the offset, K, was difficult because most cells had not fully adapted to a constant level within the 30-s stimulation period.

Initial estimation of exponential coefficients

The process of estimating the coefficients is summarized in Fig. 14 and involved the following steps: (1) K was estimated (see below) and subtracted from the spike-frequency profile; (2) from the result of step 1, the exponential fit with the longest time constant was estimated and subtracted; (3) next, each exponential fit was estimated and subtracted in turn, from the longest to the shortest time constant, until the resultant spike-frequency profile, with all parts of the model subtracted, could not be significantly fit with another exponential (i.e., the resultant spike-frequency profile did not significantly deviate from 0 Hz). The models resulting from the initial estimation of the exponential coefficients were all inspected visually to ensure that they accurately followed the spike-frequency adaptation profile through all the later phases of adaptation. In some cases, the estimation process was repeated with a fewer or greater number of exponentials to ensure that the appropriate number of exponentials had been used.

Fig. 14

Step-by-step diagram of the “peeling” procedure used for the exponential fit. The top row of graphs for the Early, PP, Mid, and Late exponential fits represent the instantaneous spike-frequency and the bottom row the natural logarithm (ln) of the spike frequency. Note the different time scales for the four sets of panels. The procedure required the peeling of the exponentials with the largest time constant first, such that the process started with the late exponential (i.e., left side panels in figure). The steps in the estimation of the Late exponential, as indicated, were: 0 the spike-frequency profile (dots in upper Late panel), 1 subtract K (constant spike-frequency) from 0, 2 find its natural logarithm (dots in lower Late panel), 3 fit a straight line to the linear portion of the graph (thick line in lower Late panel), 4 from the straight line fit, calculate the late exponential fit (thick line in upper Late panel), 5 subtract 4 from 0 to give the component of the spike-frequency profile (dots in upper Mid panel) to be analyzed for the subsequent exponential. Steps 2 to 5 were then repeated for the mid and subsequent exponential fits. The final summed exponential fit gave a very good fit of the spike-frequency profile and is plotted (thin line) over the spike-frequency profile in the upper Late panel. In the Mid, PP, and Early upper panels note that the dots indicate the spike-frequency residual from the preceding step 5. The thick lines are as in step 4. Similarly in the lower Mid, PP, and Early panels the dots are as in step 2, and the thick lines are as in step 3

While the summed exponentials provided a good fit for most tests, only exponential fits with an amplitude greater than the spike-frequency variability could be detected accurately. In some cells with high spike-frequency variability, the average of the spike-frequency profile was used to determine the initial estimates of the coefficients for the final non-linear curve fit (see below).

Each exponential fit was estimated by calculating the linear regression of the natural logarithm (ln) of the spike-frequency profile, after K and any longer time constant exponential fits were subtracted. The slope of the regression equals −1/τ, and the intercept equals ln A, so the coefficients τ and A could be estimated from the regression. The linear regression was inspected visually and restricted to the linear portion of the ln spike-frequency profile: i.e., after the exponentials with shorter time constants affected the linearity, and before the profile crossed the spike-frequency axis. For these decaying exponentials, the time constant was always positive, as was K, while the amplitude was positive except for the exponential fit that described the PP phase of the spike-frequency profile.

Estimation of K

Most cells had not fully adapted to a constant level within the 30 s of stimulation. So K was estimated from the differential of the second half of the spike-frequency profile (f, 1-s bins). The process of estimating K was described by the following equations:

$$ f = K + A{\rm e}^{-t/\tau}$$

(1)

$$ {\rm d}f/{\rm d}t = -A/\tau{\rm e}^{-t/\tau}$$

(2)

$$ K = f + \tau {\rm d}f/{\rm d}t $$

(3)

The second half of the average spike-frequency profile (Eq. 1) consisted of only one exponential (late). Therefore, differentiation of f (Eq. 2) removed the offset (K) and scaled the amplitude (A multiplied by −1/τ), but the time constant of the exponential fit (τ) was not changed, enabling the late exponential time constant to be estimated. The exponential fit of the differentiated profile was then rescaled (i.e., multiplied by −τ), subtracted from f, and the resultant, when averaged, gave an estimate of K (Eq. 3). The average of the spike-frequency profile was used because differentiation of the spike-frequency was too noisy. Only the second half of the profile was used, to reduce the effects of other time constants on the fit of the longest time constant. In the few cases in which this method did not give a good estimate of K, a value was chosen that was slightly less than the final average spike-frequency of the spike-frequency profile.

Non-linear curve fit

The final values of the coefficients for the summed exponential fits (K, A, and τ for each exponential) were determined by non-linear curve fitting using the Levenberg-Marquardt algorithm, which searched for coefficient values to minimize chi-square (Σ (f_fit − f_raw)²). This iterative process was finished when the decrease in chi-square between iterations was <0.001. The curve-fitting algorithm required the initial estimates of the coefficients (see above) and the equation of the fit, with K forced to be >0.

Time course of exponential fits

For each adaptation test, the time course of each exponential fit was determined. The end time of each exponential fit was defined as the time when that exponential was contributing less than 1% of the total summed exponential fit. For each test of the PP cells, the minimum spike-frequency before the PP and the maximum spike-frequency of the PP were determined from the summed exponential fit.

Multiple regression analysis

The program to calculate the multiple regressions of AP shape parameters versus spike-frequency used a step-down algorithm, which started with three parameters (see below) for initial adaptation and all five for late adaptation. It then progressively removed the least significant parameters one by one (i.e., even when the removed parameter may have been significant in the single-parameter regression), and recalculated the regression between each step until all parameters left made a significant contribution to the regression (Table 5). A statistical constraint was that the multiple regression of the first five APs (for traditional and delayed initial adaptation) could only start with a maximum of three shape parameters. In this instance, multiple regressions of all combinations of three of the five parameters were calculated using the above algorithm to find the best combination of regression parameters (Table 4). Note also that for the first 5 APs, the shape parameters for each AP were regressed against the spike-frequency subsequent to that AP.

The regressions were performed on the average parameters and spike-frequency for each cell type. Note that the algorithms for performing the multiple regressions removed (i.e., excluded) parameters which did not contribute significantly to the regressions, and used only those that contributed significantly. These significant parameters were used to simulate the actual (i.e., directly measured) average spike-frequencies throughout the initial (see Fig. 13a) and later (Fig. 13b, c) phases of adaptation. To simplify the analysis, we did not compare the degree of significance of the parameters included in the regressions, and as a result, their subsequent description gives equal weight to them. Instead, the analysis compared the relative representation of the spike versus AHP shape parameters in the multiple regressions, between the cell groups.