A Two-Stage Method for Decorrelating the Errors in Log-Linear Models for Spectral Density Comparisons in Neural Spike Sequences

Abstract

In this paper, we present three log-linear models for comparing spectral density functions (SDFs) of neural spike sequences (NSSs). The logarithmic (ln) ratios of the estimated SDFs are modeled as polynomial expressions with respect to angular frequencies plus residual series with autocorrelated errors. The advantage of the proposed models is that they can be applied within certain frequency ranges. Analysis of point processes in the frequency domain can be performed to obtain estimates of the SDFs of NSSs by smoothing the mean-corrected periodograms using moving average weighting schemes. The weighting schemes may differ in the estimated SDFs. To decorrelate the error terms in the log models, we apply a two-stage method: in the first stage, the error terms are identified by choosing a suitable model, while in the second stage, the reliable estimates of the unknown parameters involved in the polynomial expressions are derived by decorrelating the data. An illustrative example from the field of neurophysiology is described, in which the neuromuscular system of the muscle spindle is affected by three different stimuli: (a) a gamma motoneuron, (b) an alpha motoneuron, and (c) a combination of gamma and alpha motoneurons. It is shown that the effect of the gamma motoneuron on the muscle spindle is shifted by the presence of the alpha motoneuron to lower frequencies in the range of [1.03, 7.6] Hz, whereas the presence of the gamma motoneuron shifts the effect of the alpha motoneuron in two bands of frequencies: one in the range of [13.5, 19.9) Hz and the other in the range of [19.9, 30.8] Hz.

1. Introduction

In practice, neural spike sequences (NSSs) are often considered as realizations of point processes (see [1,2,3,4,5,6,7,8,9,10] for relevant studies that apply such a consideration). Therefore, methods for analyzing point processes can be used to study the information contained in NSSs. Such methods in the frequency domain involve the estimation of the SDF (see [1,11,12,13,14] for relevant applications). In this context, recent studies have proposed a broad range of relevant methods. Regularization techniques for the estimation of the spectral density matrix in high-dimensional multivariate point processes have been developed [12]. Center-outward ranks and the Isometric Log-Ratio (ILR) transformation have been utilized to define a rank-based likelihood for spike trains, resulting in an efficient method for the identification of typical patterns and outliers in spike train data, thus enhancing classification and ranking tasks [13]. A multivariate generalization of the Skellam process with resetting (MSPR) has been introduced to model simultaneously recorded neural spike trains [14]. However, the aforementioned methods are oriented to multivariate point processes. Furthermore, relevant studies have focused either on estimating neuronal connectivity [12,14] or classification and ranking [13]. Methods for SDF estimation have been proposed based on the smoothing of the mean-corrected periodogram [11,15].

In this paper, we discuss log-linear models that enable the comparison of SDFs in NSSs. These models consist of polynomial expressions plus residual series with autocorrelated errors. While direct applications of log-linear models for comparing SDFs in NSSs are limited, relevant methodologies have been studied. Generalized Linear Models (GLMs) have been used to characterize the coding properties of neural populations [16,17]. A simulation-based study proposed a method for augmentation of the log-likelihood using terms measuring the dissimilarity between simulated and recorded activity, with the aim of improving the realism of simulated neural activity [18]. Two recent studies [13,19] have proposed filtered point process models that are capable of capturing both rhythmic and broadband components of neural activity. The aforementioned studies represent an ongoing effort to enhance the analysis of NSSs through spectral density estimation and statistical modeling. However, the direct application of log-linear models for comparison of spectral densities in spike sequences remains an under-explored area. In addition, the issue of autocorrelated errors (a case that can occur in recorded datasets, causing significant issues) has not been addressed. The presence of autocorrelated errors can lead to unreliable estimates for the unknown parameters of the log-linear models. An important point of this study is recognizing the information carried by the $Ia$ sensory axon of the muscle spindle to the spinal cord when it is affected simultaneously by gamma and alpha motoneurons and how this is related to the effect of a gamma motoneuron alone, as well as the effect of an alpha motoneuron alone in certain frequency bands. This can be achieved by comparing the logarithms of the ratios of the estimated SDFs using the appropriate log-linear model. By subtracting the correlated part from the data, we can derive reliable estimates of the unknown parameters involved in the log-linear models. In this study, such decorrelation is achieved using a two-stage method. In the first stage, the error terms are identified with a suitable model. In the second stage, we obtain reliable estimates for the parameters of the polynomial expressions by decorrelating the data.

An illustrative example from the field of neurophysiology is used to demonstrate the application of log-linear models. The data sets are sequences of NSSs obtained from the complex system of the muscle spindle when it is affected by (a) a gamma motoneuron, (b) an alpha motoneuron, and (c) a combination of gamma and alpha motoneurons, considered as three different experimental cases. The data and source code files are publicly available at Github repository (https://github.com/bvasil/muscle-spindle/tree/main/data/response (accessed on 20 February 2025) (datasets), https://github.com/ymichai/mspindle/tree/main (accessed on 5 May 2025) (source code files)). MATLAB R2016a and R 4.4.3 were used. The only information that we need is the response of the muscle spindle, and through comparison of the estimated SDFs of the responses in certain frequency bands, we reach useful conclusions with respect to the information that is transferred to the spinal cord through the $Ia$ sensory axon. In particular, the effect of the gamma motoneuron is shifted by the presence of the alpha motoneuron to lower frequencies in the range of $[1.03,7.6]\mathrm{H}\mathrm{z}$, whereas the presence of the gamma motoneuron shifts the effect of the alpha motoneuron to two ranges of higher frequencies: one in the range of $[13.5,19.9)\mathrm{H}\mathrm{z}$ and the other in the range of $[19.9,30.8]\mathrm{H}\mathrm{z}$.

In the Section 2, we present the notation, definitions, and SDF estimation method, which involves smoothing the mean-corrected periodogram with the help of a moving average weighting scheme. A description of the general problem is also provided. The log-linear models that involve the ratios of the estimated SDFs are discussed in the Section 3, together with the process of decorrelating the data. In the Section 4, an example from the field of neurophysiology is described along with the derived results, while in the Section 5 of this paper, the conclusions are summarized.

2. Materials and Methods

2.1. Notation, Definitions, and the Estimation of SDF

Consider the realizations $\left\{N\left(t\right),t\in \left[0,T\right]\right\}$ and $\left\{M\left(t\right),t\in \left[0,T\right]\right\}$ of two stationary point processes (SPPs). The point processes are assumed to be orderly, obey a (strong) mixing condition, and their increments are $\left\{dN\left(t\right)\right\}$ and $\left\{dM\left(t\right)\right\}$, respectively. These assumptions have been described in [20,21,22] and are approximately valid in practice. Let $p_N$ and $q_{NN}\left(u\right)$ denote the mean intensity and the second-order cumulant density of $\left\{N\left(t\right)\right\}$, respectively [20,23]. Then, the second-order SDF is defined by

$$f_{NN}\left(\lambda \right)={\left(2\pi \right)}^{-1}{p}_N+{\left(2\pi \right)}^{-1}{\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}exp\left(-i\lambda u\right)q_{NN}\left(u\right)du,-\mathrm{\infty }<\lambda <+\mathrm{\infty }$$

(1)

For large $\lambda , f_{NN}\left(\lambda \right)$ tends to a constant value; that is

$$\underset{\lambda \to \infty }{\lim }{f}_{NN}\left(\lambda \right)={\left(2\pi \right)}^{-1}{p}_N.$$

(2)

This constant value corresponds to the SDF of a Poisson point process with mean intensity $p_N.$ The SDF exists when ${\int }_{-\infty }^{+\infty }\left|q_{NN}\left(u\right)\right|du<+\infty$.

The second-order SDF of $\left\{M\left(t\right)\right\}$ is similarly defined in terms of $p_M$ and $q_{MM}\left(u\right)$.

2.2. A Way of Estimating the Mean-Corrected Periodogram

Let $\left\{N\left(t\right),t\ge 0\right\}$ be a stationary point process, which is observed over a finite interval of length $T$. The second-order mean-corrected periodogram of $N\left(t\right)$ is defined by

$${\widehat{I}}_{NN}^{\left(T\right)}\left(\lambda \right)={\displaystyle \frac{1}{2\pi H_2^{\left(T\right)}\left(0\right)}}{\left|{\widehat{d}}_N^{\left(T\right)}\left(\lambda \right)\right|}^2, \lambda \ne 0$$

(3)

where ${\widehat{d}}_N^{\left(T\right)}\left(\lambda \right)$ is the finite mean-corrected Fourier–Stieltjes transform of the increments of the point process, given by

$${\widehat{d}}_N^{\left(T\right)}\left(\lambda \right)={\int }_0^T{h}_T\left(t\right)exp\left(-i\lambda t\right)\left\{d_N\left(t\right)-{\widehat{p}}_{N}dt\right\}.$$

(4)

The quantity $H_2^{\left(T\right)}\left(0\right)$ is defined by

$$H_2^{\left(T\right)}\left(0\right)={\int }_0^T{h}_T^2\left(t\right)dt,$$

where $h_T\left(t\right)$ is a data window or a taper. More details about the mean-corrected periodogram and data windows are given in [15,24]. In practice, (4) can be approximated by the following expression:

$${\widehat{d}}_N^{\left(T\right)}\left(\lambda \right)\approx {\sum }_{t=0}^{T-1}{h}_T\left(t\right)exp\left(-i\lambda t\right)\left\{N\left(t+1\right)-N\left(t\right)-{\widehat{p}}_N\right\},$$

(5)

where ${\widehat{p}}_N=N\left(T\right)/T$ is an estimate of the mean intensity. By $N\left(T\right),$ we denote the number of events of $\left\{N\left(t\right)\right\}$ in the interval $[0,T]$. For more details regarding this approach, we refer the reader to [11]. The discretization approach is helpful in using the FFT algorithm for the computation of the mean-corrected periodogram.

As the mean-corrected periodogram is not a good estimate of the SDF (see [15]), we use a moving average weighting scheme over $2m_1+1$ frequencies to improve its properties, as follows:

$$\begin{gathered} f_{NN}^{\left(T\right)}\left({\lambda }_j\right)={\left(2m_1+1\right)}^{-1}{\sum }_{r=-m_1}^{m_1}{\widehat{I}}_{NN}^{\left(T\right)}\left({\lambda }_{j+r}\right),{\lambda }_j={\displaystyle \frac{2\pi j}{T}},j=\mathrm{1,2},\dots ,n, \\ wheren=\left[\left(T-1\right)/2\right]and\left[·\right]denotestheintegralpart. \end{gathered}$$

(6)

It can be proven, from (6), that the distribution of $f_{NN}^{\left(T\right)}\left(\lambda \right)$ is asymptotically of the following form ([11]):

$$f_{NN}^{\left(T\right)}\left(\lambda \right)~f_{NN}\left(\lambda \right){{\rm X}}_{2k_1}^2/2k_1, k_1=2m_1+1,$$

(7)

where ${{\rm X}}_{2k_1}^2$ denotes a chi-square distribution with $2k_1$ degrees of freedom. Moreover, from (7), for large $k_1$, the estimate ${log}_{10}{f}_{NN}^{\left(T\right)}\left(\lambda \right)$ asymptotically follows a normal distribution, with mean ${log}_{10}{f}_{NN}\left(\lambda \right)$ and variance given by

$${\displaystyle \frac{S}{T}}{c}_P{\displaystyle \frac{{log}_{10}e}{2m_1+1}},$$

(8)

where $C_p={\displaystyle \frac{{\int }_0^1{\left\{h\left(t\right)\right\}}^{4}dt}{{\left[{\int }_0^1{\left\{h\left(t\right)\right\}}^{2}dt\right]}^2}}$ and S is the extended sample interval obtained by adding zeroes to the end of the observed interval $T$ until a power of 2 is obtained, enabling the use of the FFT algorithm [15,25]. As $C_p\ge 1$, it is obvious from (8) that there is a small increase in the value of the variance. This, however, significantly reduces the phenomenon of leakage by using a data window in (5) [24]. Through choosing a suitable value of $m_1$ in (6), we can reduce the variance of ${log}_{10}{f}_{NN}^{\left(T\right)}\left(\lambda \right)$ and obtain a good estimate of ${log}_{10}{f}_{NN}\left(\lambda \right)$. In addition, an approximate 95% confidence interval of ${log}_{10}{f}_{NN}^{\left(T\right)}\left(\lambda \right)$ can be found, as follows:

$$\begin{gathered} {log}_{10}{f}_{NN}^{\left(T\right)}\left(\lambda \right)-1.96\left({log}_{10}e\right){\left({\displaystyle \frac{S}{T}}{C}_p{\displaystyle \frac{1}{2m_1+1}}\right)}^{1/2}<{log}_{10}{f}_{NN}\left(\lambda \right)< \\ {log}_{10}{f}_{NN}^{\left(T\right)}\left(\lambda \right)+1.96\left({log}_{10}e\right){\left({\displaystyle \frac{S}{T}}{C}_p{\displaystyle \frac{1}{2m_1+1}}\right)}^{1/2}. \end{gathered}$$

(9)

An approximate 95% confidence interval for a Poisson point process can be constructed by setting $f_{NN}^{\left(T\right)}\left(\lambda \right)={\widehat{p}}_N/2\pi$ in (9).

Figure 1 represents the base 10 logarithms of the estimated SDFs for the response of the $Ia$ sensory axon of the muscle spindle when it is affected by (a) a gamma motoneuron, (b) an alpha motoneuron, and (c) a combination of gamma and alpha motoneurons. In case (a), the number of neural spikes in the interval of length $T=\mathrm{15,870}\mathrm{ms}$ is $N\left(T\right)=538$. Notably, 10% of the data $\left\{N\left(t+1\right)-N\left(t\right)-{\widehat{p}}_N\right\},t=\mathrm{0,1},2,\dots ,\left(T-1\right)$ at the beginning and 10% at the end are multiplied by a split bell cosine data window [25]. Thus, the value of $C_p$ is equal to 1.116 for $p=0.2$[25]. Then, the sample $T$ is extended to $S=\mathrm{16,384}=2^{14}$ by adding 514 zeroes in the end. This permits the use of the Sande–Tukey Radix 2 form of the FFT for the computation of ${\widehat{d}}_N^{\left(T\right)}\left(\lambda \right)$ in (5) [26]. The estimate of the SDF is then obtained from (6) by choosing $m_1=40$. The black dotted line in the middle of Figure 1a corresponds to ${log}_{10}\left({\widehat{p}}_{{\rm N}}/2\pi \right)$, while the red solid lines above and below this value correspond to the approximate confidence limits calculated using (9). As ${log}_{10}\left({\widehat{p}}_{{\rm N}}/2\pi \right)$ corresponds to ${log}_{10}$ of the estimate of the SDF of a Poisson point process, it is clear that ${log}_{10}{f}_{NN}^{\left(T\right)}\left(\lambda \right)$ differs from ${log}_{10}\left({\widehat{p}}_{{\rm N}}/2\pi \right)$ in the frequency range of $[10,35]\mathrm{H}\mathrm{z}$. As the angular frequency $\lambda$ becomes larger, the ${log}_{10}{f}_{NN}^{\left(T\right)}\left(\lambda \right)$ value tends to ${log}_{10}\left({\widehat{p}}_{{\rm N}}/2\pi \right).$

In case (b), similar results hold as in case (a), except that $N\left(T\right)=358$. Figure 1b is interpreted the same as Figure 1a, except that the estimate of the mean intensity ${\widehat{p}}_{{\rm N}}$ is different. The estimate ${log}_{10}{f}_{NN}^{\left(T\right)}\left(\lambda \right)$ is quite different than ${log}_{10}\left({\widehat{p}}_{{\rm N}}/2\pi \right)$ in the frequency range of $[0,70]\mathrm{H}\mathrm{z}$ and, as $\lambda$ becomes larger, the former tends to ${log}_{10}\left({\widehat{p}}_{{\rm N}}/2\pi \right)$.

In case (c), the number of neural spikes in the interval of length $T=\mathrm{11,360}\mathrm{ms}$ is $N\left(T\right)=356$. The interval $T$ is extended to $S=\mathrm{16,384}$ by adding 5024 zeroes at the end of the initial data. The estimate of the SDF is obtained from (6) by choosing $m_1=30$. Figure 1c is interpreted similarly to Figure 1b, except that the estimate of ${\widehat{p}}_{{\rm N}}$ is different in this case. The estimate ${log}_{10}{f}_{NN}^{\left(T\right)}\left(\lambda \right)$ is quite different from those in the previous two cases, and tends to ${log}_{10}\left({\widehat{p}}_{{\rm N}}/2\pi \right)$ as $\lambda$ becomes larger.

The SDF estimates presented in Figure 1 were obtained using the MATLAB R2016a software [27].

2.3. A Description of the General Problem

In this subsection, we discuss the general problem of comparing two SDFs using log-linear models, in which the residual series has autocorrelated errors. The presence of autocorrelation in the residual series leads to the production of unreliable estimates of the parameters. By decorrelating the initial data, we can obtain reliable estimates of the unknown parameters involved in the log-linear model. The whole procedure is performed using a two-stage method [28,29].

In the first stage, the natural logarithm (ln) of the ratio of the estimates of the SDFs can be described by an $\mathcal{l}\mathrm{t}\mathrm{h}$ order log-linear model of the form:

$$ln{Y}_t=a_0+a_{1}t+a_2{t}^2+\dots +a_{\mathcal{l}}{t}^{\mathcal{l}}+v_t, \mathcal{l}=1,\dots ,4$$

(10)

where $Y_t=f_{NN}^{\left(T\right)}\left({\lambda }_t\right)/f_{MM}^{\left(T\right)}\left({\lambda }_t\right);{\lambda }_t=2\pi t/S;t=\mathrm{1,2},\dots ,n$. By $f_{MM}^{\left(T\right)}\left({\lambda }_t\right)\mathrm{a}\mathrm{n}\mathrm{d}{f}_{MM}^{\left(T\right)}\left({\lambda }_t\right)$ we denote the estimates of the SDFs and, by $v_t$, we denote a residual series with autocorrelated errors. The mean value of $v_t$ is zero, and its variance is ${\sigma }_v^2$.

Through the identification of a suitable model for the residual series $v_t$, we can estimate its unknown parameters with the help of a valid regression model. Then, we can derive the estimated residual series, ${\widehat{v}}_t,$ and, thus, an estimate of the uncorrelated residual series $e_t$ can be obtained, where $e_t=$ $v_t{-\widehat{v}}_t$.

In the second stage, we subtract the ${\widehat{v}}_t$ from both sides of (10) to obtain

$$\begin{gathered} ln{Y}_t-{\widehat{v}}_t=a_0+a_{1}t+a_2{t}^2+\dots +a_{\mathcal{l}}{t}^{\mathcal{l}}+v_t{-\widehat{v}}_t \\ =a_0+a_{1}t+a_2{t}^2+\dots +a_{\mathcal{l}}{t}^{\mathcal{l}}+e_t,\mathcal{l}=1,\dots ,4,t=\mathrm{1,2},\dots ,n. \end{gathered}$$

(11)

Hence, the expression (11) is a valid regression model with respect to $t,t^2$, $\dots ,t^{\mathcal{l}}$, as $e_t$ represents a sequence of uncorrelated errors [30]. The quantity $ln{Y}_t-{\widehat{v}}_t$ represents the decorrelated data.

More details of this general problem are given in the following sections, where special cases of (10) are discussed when analyzing the three experimental neurophysiological situations.

The proposed methodology is carried out in five steps for the general case and six steps for the changepoint case (see Section 4.2.2). The aforementioned steps are summarized in the flow charts presented in Figure 2 and Figure 3, respectively.

• Step 1—Estimation of SDF: The estimate of the SDF for each signal is obtained through the mean corrected periodogram (Equation (6));
• Step 2—Model fitting to the ratio of the spectra: The ratio of the estimated SDFs of the two examined signals is calculated, and a log-linear model is fitted to the real data. Fitted values are subtracted from the real values to obtain the residual series $v_t$. Estimates of the autocorrelation function (ACF) and the partial autocorrelation function (PACF), with their 95% confidence intervals, are also obtained. The ACF indicates whether the residual series are autocorrelated or not, while the PACF is used to determine a suitable model to describe the autocorrelation structure of the residual series $v_t$. The number of residuals that fall outside the confidence limits of the PACF indicates the order of the model that is required for the description of the residual structure;
• Step 3—Estimation of the unknown parameters of the AR model: Calculations based on the residual values $v_t$ are performed. Estimates of the unknown parameters of the AR model are obtained, which will be used for decorrelation of the initial correlated data;
• Step 4—Reliable estimates for the unknown parameters of the log-linear model: Reliable estimates for the unknown parameters of the log-linear model of the decorrelated data are obtained. The ACF, PACF, and Q–Q plots are also calculated, and a normality test is performed in the case of uncorrelated residual series $e_t$;
• Step 5—Estimation of the 95% prediction intervals: Estimates of the 95% prediction intervals for the log-linear model of the decorrelated data are obtained.

3. The Proposed Log-Linear Models

As mentioned above, the comparisons between the muscle spindle responses are carried out based on the estimated SDFs. Three log-linear models are presented for comparison of the estimated SDFs using the natural logarithm of their ratio in the three experimental cases. The models were selected in accordance with [31,32,33]. The correlation coefficients, partial autocorrelation coefficients, and corresponding $AR\left(1\right)$–$AR\left(2\right)$ models for the residual series were estimated in accordance with [25,34].

3.1. The $\gamma –\alpha \gamma$ Case

Figure 1c shows that the presence of the alpha motoneuron shifts the effect of the gamma motoneuron to lower frequencies in the range of $[1.03,7.6]\mathrm{H}\mathrm{z}$. Therefore, the estimated SDF for the response of the muscle spindle in the presence of the gamma motoneuron in the frequency range of $[5.4,12.0]\mathrm{H}\mathrm{z}$ was compared with that in the presence of a combination of gamma and alpha motoneurons in the frequency range of $[1.03,7.6]\mathrm{H}\mathrm{z}$. The aforementioned frequency ranges are presented in Figure 4. In this case, the proposed log-linear model is given by

$$ln{Y}_t=a_0+at+b{t}^2+c{t}^3+d{t}^4+v_t,t=\mathrm{1,2},\dots ,108$$

(12)

where $Y_t$ denotes the ratio of the estimated SDFs, and $v_t$ is the residual series. The angular frequency ${\lambda }_t$ is defined by ${\lambda }_t=2\pi t/S$ and corresponds to the frequency $t/S$, where $S=\mathrm{16,384}\mathrm{ms}$. In both frequency ranges (i.e., $[1.03,7.6]\mathrm{H}\mathrm{z}$ and $[5.4,12.0]\mathrm{H}\mathrm{z}$), the frequencies $t/S$ correspond to the index values $t=\mathrm{1,2},\dots ,108$.

Estimates of the unknown parameters $a_0,a,b,c,$ and $d$ can be derived using the least squares method in the model (12). These estimates and their standard deviations are given in

Table 1. Results obtained from the fourth-order model given by (12).

Coefficient	Estimate	Std. Error	t-Value	Pr (>|t|)
$a_0$	8.50408778 × 10−1	2.9384156640 × 10−2	28.94106	<2.22 × 10−16
$a$	−2.11292204 × 10−4	3.6987624613 × 10−3	−0.05713	0.95456
$b$	−7.07267755 × 10−4	1.3709407880 × 10−4	−5.15900	1.2100 × 10−6
$c$	1.43736266 × 10−5	1.8856341793 × 10−6	7.62270	1.2699 × 10−11
$d$	−7.34781793 × 10−8	8.5831713996 × 10−9	−8.56073	1.1490 × 10−13
Residual Std. Error	0.057713392	Degrees of freedom	103
Multiple R-Squared	0.8676911	Adjusted R-squared	0.8625517
F-statistic	168.86869	Degrees of freedom	4 and 103
p-value	<2.2204460 × 10−16

. The proposed model has an Akaike Information Criterion (AIC) value of $-302.72$, while the linear model demonstrates an AIC value of $-164.93$. Therefore, the log-linear model fits the real values much better.

Figure 5a presents the real data (black curve; i.e., $ln{Y}_t,t=\mathrm{1,2},\dots ,108$) and the fitted fourth-order model (red curve). Figure 5b shows the residual series when the fitted fourth-order model is subtracted from the real data.

Figure 6a shows the estimated autocorrelation coefficient function (ACF) for the residual series shown in Figure 5b. A strong correlation is clearly observed among residual values. Figure 6b presents the estimated partial autocorrelation coefficient function (PACF) of the residual series. The first value of the estimated PACF is very significant, and, thus, the correlation amongst the residual values can be modeled by an autoregressive model of order 1; namely, $AR\left(1\right)$. The presence of a strong correlation between the values of the residuals indicates that the estimates of the parameters $a_0,a,b,c,d,$ and $\phi$ are not reliable, where $v_t=\phi v_{t-1}+{\epsilon }_t,$ and ${\epsilon }_t$ is a sequence of uncorrelated errors. In the next section, we present a method for obtaining reliable estimates of the parameters $a_0,a,b,c,d$, and $\phi$.

3.2. The $\alpha –\alpha \gamma$ First Case

3.2.1. First Approach

Figure 1b shows that there is an increase in power of the estimated SDF in the frequency range of $[2.6,9.0]\mathrm{H}\mathrm{z}$. This refers to the situation where an alpha motoneuron affects the muscle spindle. Almost the same pattern (see Figure 1c) is observed in the situation where the muscle spindle is affected by a combination of gamma and alpha motoneurons. However, this pattern occurs in the frequency range of $[13.5,19.9)\mathrm{H}\mathrm{z}$. Therefore, the estimated SDF of the response of the muscle spindle in the presence of an alpha motoneuron in the frequency range of $[2.6,9.0]\mathrm{H}\mathrm{z}$ will be compared with that in the presence of a combination of gamma and alpha motoneurons in the frequency range of $[13.5,19.9)\mathrm{H}\mathrm{z}$. This indicates that the presence of a gamma motoneuron shifts the effect of the alpha motoneuron at higher frequencies in the range of $[13.5,19.9)\mathrm{H}\mathrm{z}$. The aforementioned frequency ranges are presented in Figure 7. In this case, the proposed log-linear model is given by

$$ln{Y}_t=a+b{t}^2+v_t,t=\mathrm{1,2},\dots ,105$$

(13)

where $Y_t$ denotes the ratio of the estimated SDF, and $v_t$ is a residual series. The frequencies $t/S,t=\mathrm{1,2},\dots ,105$ correspond to both frequency ranges (i.e., $[2.6,9.0]\mathrm{H}\mathrm{z}$ and $[13.5,19.9]\mathrm{H}\mathrm{z}$), where $S=\mathrm{16,384}\mathrm{ms}$.

Estimates of the unknown parameters $a$ and $b$ can be derived using the least squares method in the model (13). These estimates and their standard deviations are given in

Table 2. Results obtained from the quadratic model given by 13.

Coefficient	Estimate	Std. Error	t-Value	Pr (>|t|)
$a$	−6.24145146 × 10−1	6.76522941 × 10−3	−92.25779	<2.22 × 10−16
$b$	2.92669467 × 10−5	1.35596128 × 10−6	21.58391	<2.22 × 10−16
Residual Std. Error	0.04607669	Degrees of freedom	103
Multiple R-squared	0.81893777	Adjusted R-squared	0.81717988
F-statistic	465.865184	Degrees of freedom	1 and 103
p-value	<2.220446 × 10−16

. The proposed model has an Akaike Information Criterion (AIC) value of $-344.31$, while the linear model demonstrates an AIC value of $-447.46$. Therefore, the linear model fits the real values much better. However, the log-linear model provides a better approximation of the normal distribution of errors $e_t$ (see Section 4.2), while its fitting is still very satisfactory.

Figure 8a presents the real data (black curve; i.e., $ln{Y}_t,t=\mathrm{1,2},\dots ,105$) and the fitted quadratic model (red curve). Figure 8b shows the residual series when the fitted quadratic model is subtracted from the real data.

Figure 9a shows the estimated ACF for the residual series shown in Figure 8b. A strong correlation is clearly observed amongst the residual values. Figure 9b shows the estimated PACF for the residual series. In this case, there is again a very significant first value in the estimated PACF, indicating that the correlation amongst the residual values can be modeled using an $AR\left(1\right)$ model. This implies that the estimates of the unknown parameters $a$, $b,$ and $\phi$ are not reliable, where $v_t=\phi v_{t-1}+{\epsilon }_t,$ and ${\epsilon }_t$ is a sequence of uncorrelated errors. Reliable estimates of the parameters $a$, $b,$ and $\phi$ will be discussed in the next section.

3.2.2. Second Approach

If we look carefully at Figure 8a, we can see that there is a changepoint in the $\alpha -\alpha \gamma$ first case data. To check that there is a changepoint in the $\alpha -\alpha \gamma$ first case data in the frequency range of $[13.5,19.9)\mathrm{H}\mathrm{z}$, we use the changepoint analysis R 4.4.3 package, as described in [35]. The obtained changepoint corresponds to the index value $t=62\left(17.27\mathrm{H}\mathrm{z}\right)$, as shown in Figure 10. For more details about the analysis of the changepoint models, we refer the reader to [36]. Thus, the frequency band $[13.5,19.9)\mathrm{H}\mathrm{z}$ can be separated into two parts: one in the range of $[13.5,17.3)\mathrm{H}\mathrm{z}$, referred to as $\alpha -\alpha \gamma \left(1\right)$, and the other in the range of $[17.3,19.9)\mathrm{H}\mathrm{z}$, referred to as $\alpha -\alpha \gamma \left(2\right)$. In the $\alpha -\alpha \gamma \left(1\right)$ case, the proposed log-linear model is given by

$$ln{Y}_t=a_1+b_{1}t+v_{1_t},t=\mathrm{1,2},\dots ,62$$

(14)

where $Y_t$ denotes the ratio of the estimates of the SDFs, and $v_{1_t}$ is a residual series. The index values $t=\mathrm{1,2},\dots ,62$ correspond to the frequencies in the range of $[13.5,17.3)\mathrm{Hz}$.

Estimates of the unknown parameters $a_1$ and $b_1$ were obtained by applying the least squares method in the model (14). The estimates and their standard deviations are given in

Table 3. Results obtained from the quadratic model given by (14).

Coefficient	Estimate	Std. Error	t-Value	Pr (>|t|)
$a_1$	−0.6330233	0.0079985	−79.143	<2 × 10−16
$b_1$	0.0011431	0.0002208	5.178	2.75 × 10−6
Residual Std. Error	0.03111	Degrees of freedom	60
Multiple R-squared	0.3088	Adjusted R-squared	0.2973
F-statistic	26.81	Degrees of freedom	1 and 60
p-value	2.751 × 10−6

. The proposed model has an Akaike Information Criterion (AIC) value of $-250.39$, while the linear model demonstrates an AIC value of $-325.78$. Therefore, the linear model fits the real values much better. However, the log-linear model provides a better approximation of the normal distribution of errors $e_t$ (see Section 4.2), while its fitting is still very satisfactory.

Figure 11a shows the real data (black curve; i.e., $ln{Y}_t,t=\mathrm{1,2},\dots ,62$) and the fitted first-order model (red curve). Figure 11b presents the residual series obtained when the fitted first-order model is subtracted from the real data.

Figure 12a presents the estimated ACF for the residual series shown in Figure 8b, from which a strong correlation is clearly observed among the residual values. Figure 12b shows the estimated PACF for the residual series. Again, this shows that the first value of the estimated PACF is very significant, indicating that the correlation amongst the residual values can be modeled using an $AR\left(1\right)$ model. This implies that the estimates of the unknown parameters $a_1$, $b_{1,}$ and $\phi$ are not reliable, where $v_{1_t}=\phi v_{1_{t-1}}+{\epsilon }_{1_t,}$ and ${\epsilon }_{1_t}$ is a sequence of uncorrelated errors. Reliable estimates of the parameters $a_1$, $b_1$, and $\phi$ will be given in the next section.

In a similar way, we can analyze the $\alpha -\alpha \gamma \left(2\right)$ data. In this case, the proposed log-linear model is given by

$$ln{Y}_t=a_2+b_{2}t+v_{2_t},t=\mathrm{63,64},\dots ,105$$

(15)

where $Y_t$ denotes the ratio of the estimated SDFs, and $v_{2_t}$ is a residual series. The index values $t=\mathrm{63,64},\dots ,105$ correspond to the frequencies in the range of $[17.3,19.9)\mathrm{H}\mathrm{z}$.

Estimates of the unknown parameters $a_2$ and $b_2$ were obtained by applying the least squares method in the model (15). The estimates and their standard deviations are given in

Table 4. Results obtained from the quadratic model given by (15).

Coefficient	Estimate	Std. Error	t Value	Pr (>|t|)
${\mathit{a}}_2$	−0.4455818	0.0131921	−33.78	<2 × 10−16
${\mathit{b}}_2$	0.0022146	0.0005223	4.24	0.000124
Residual Std. Error	0.0425	Degrees of freedom	41
Multiple R-squared	0.3048	Adjusted R-squared	0.2879
F-statistic	17.98	Degrees of freedom	1 and 41
p-value	0.0001239

. The proposed model has an Akaike Information Criterion (AIC) value of $-145.63$, while the linear model demonstrates an AIC value of $-179.33$. Therefore, the linear model fits the real values much better. However, the log-linear model provides a better approximation of the normal distribution of errors $e_t$ (see Section 4.2), while its fitting is still very satisfactory.

Figure 13a presents the real data (black curve; i.e., $ln{Y}_t,t=\mathrm{63,64},\dots ,105$) and the fitted first-order model (red curve), whereas Figure 13b shows the residual series when the fitted first-order model is subtracted from the real data.

Figure 14 presents the estimated ACF and PACF for the residual series shown in Figure 13b, from which a strong correlation among the values of the residuals is clearly observed (see Figure 14a); furthermore, the first value of the estimated PACF (see Figure 14b) is significant, indicating that the correlation among the residual values can be modeled using an $AR\left(1\right)$ model. The presence of correlation in the residual series implies that the estimates of the unknown parameters $a_2$, $b_2,$ and $\theta$ are not reliable, where $v_{2_t}=\theta v_{2_{t-1}}+{\epsilon }_{2_t}$, and ${\epsilon }_{2_t}$ is a sequence of uncorrelated errors. Reliable estimates of the parameters $a_2,b_2$, and $\theta$ will be discussed in the next section.

3.3. The $\alpha –\alpha \gamma$ Second Case

In this case, we compare the estimated SDF for the response of the muscle spindle when it is affected by an alpha motoneuron in the frequency range of $[9.3,20.5]\mathrm{H}\mathrm{z}$ with the estimated SDF for the response of the muscle spindle when it is affected simultaneously by gamma and alpha motoneurons in the frequency range $[19.9,30.8]\mathrm{H}\mathrm{z}$. The aforementioned frequency ranges for each signal are presented in Figure 15. The proposed log-linear model for this case is given by

$$ln{Y}_t=a_0+at+b{t}^2+c{t}^3+d{t}^4+v_t,t=\mathrm{1,2},\dots ,178$$

(16)

where $Y_t$ denotes the ratio of the estimated PS, and $v_t$ is a residual series. The angular frequency ${\lambda }_t$ is defined by ${\lambda }_t=2\pi t/S$ and corresponds to a frequency of $t/S$, where $S=\mathrm{16,384}\mathrm{ms}$. Here, the index values $t=\mathrm{1,2},\dots ,178$ correspond to the frequencies in the ranges of $[9.3,20.5]\mathrm{H}\mathrm{z}$ and $\left[19.9,30.8\right)\mathrm{H}\mathrm{z},$ respectively.

Estimates of the unknown parameters $a_0,a,b,c,$ and d were obtained by applying the least squares method to the model (16). The estimates and their standard deviations are given in

Table 5. Results obtained from the model given by (16).

Coefficient	Estimate	Std. Error	t-Value	Pr (>|t|)
${\mathit{a}}_0$	−3.06599633 × 10−1	2.35705707 × 10−2	−13.00773	<2.22 × 10−16
$\mathit{a}$	1.97784140 × 10−3	1.81372982 × 10−3	1.09048	0.27702
$\mathit{b}$	−2.36808412 × 10−4	4.10412459 × 10−5	−5.77001	3.584 × 10−8
$\mathit{c}$	3.03725688 × 10−6	3.44035850 × 10−7	8.82832	1.146 × 10−15
$\mathit{d}$	−1.00040256 × 10−8	9.53543397 × 10−10	−10.49142	<2.22 × 10−16
Residual Std. Error	0.06078627	Degrees of freedom	173
Multiple R-Squared	0.80823249	Adjusted R-squared	0.80379856
F-statistic	182.28352	Degrees of freedom	4 and 173
p-value	<2.220446 × 10−16

. The proposed model has an Akaike Information Criterion (AIC) value of $-361.16$, while the linear model demonstrates an AIC value of $-484.87$. Therefore, the log-linear model fits the real values much better.

Figure 16a presents the real data (black curve; i.e., $ln{Y}_t,t=\mathrm{1,2},\dots ,178$) and the fitted fourth-order model (red curve). Figure 16b shows the residual series when the fitted fourth-order model is subtracted from the real data.

Figure 17a shows the estimated ACF for the residual series shown in Figure 16b, from which a strong correlation among the residual values is clearly observed. Figure 17b presents the estimated PACF of the residual series. The first two values of the estimate of PACF are significant, which indicates that the residual series can be modeled using an autoregressive model of order 2, that is, $AR\left(2\right)$. The presence of a strong correlation amongst the residual values indicates that the estimates of the parameters $a_0,a,b,c,d,$ ${\phi }_1$, and ${\phi }_2$ are not reliable, where $v_t={\phi }_1{v}_{t-1}+{\phi }_2{v}_{t-2}+{\epsilon }_t$, and ${\epsilon }_t$ is a sequence of uncorrelated errors. Reliable estimates of these parameters are discussed in the next section.

All the computations and graphical presentations in this section were executed using the Time Series Analysis (TSA) and Changepoint libraries in the R 4.4.3 package [37].

4. Reliable Estimates for the Parameters of the Log-Linear Models

In this section, we discuss reliable estimates for the unknown parameters of the log-linear models with autocorrelated errors described in the previous section.

4.1. The $\gamma –\alpha \gamma$ Case

To find reliable estimates for the unknown parameters of Equation (12), we use a two-stage method [28,29]. In the first stage, we obtain a reliable estimate of the parameter $\phi$ while, in the second stage, we obtain reliable estimates of the parameters $a_0,a,b,c$, and $d$ using the estimated $\widehat{\phi }$. This approach can be summarized as follows:

Setting $t-1$ in the place of $t$ in (12), we have

$$\begin{gathered} ln{Y}_{t-1}=a_0+a(t-1)+b{(t-1)}^2+c{(t-1)}^3+d{(t-1)}^4+v_{t-1}, \\ t=\mathrm{2,3},\dots ,108 \end{gathered}$$

(17)

If we multiply (17) by $\phi$ and subtract the new equation from (12), we obtain

$$\begin{gathered} ln{Y}_t=\phi {ln{Y}_{t-1}+a}_0\left(1-\phi \right)+a\left(t\right)+\alpha \phi \left(t-1\right)+b{t}^2-b{\phi \left(t-1\right)}^2 \\ +c{t}^3-c{\phi \left(t-1\right)}^3+d{t}^4-d{\phi \left(t-1\right)}^4+{\epsilon }_t, \\ \mathrm{where}{\epsilon }_t=v_t-\phi v_{t-1},t=\mathrm{2,3},\dots ,108 \end{gathered}$$

(18)

Then, ${\epsilon }_t$ is a sequence of uncorrelated errors, and, hence, the expression (18) is a valid regression equation model with respect to $ln{Y}_{t-1},t,t^2$, $t^3,$ and $t^4$ [30]. This enables us to estimate the parameter $\phi$ reliably. In particular, this estimate is given by $\widehat{\phi }=0.901914$.

In the second stage, we set $Z_t=ln{Y}_t-\widehat{\phi }$ $ln{Y}_{t-1},V_t=t-\widehat{\phi }$ $\left(t-1\right),X_t=t^2-\widehat{\phi }$ ${(t-1)}^2,Y_t=t^3-\widehat{\phi }$ ${\left(t-1\right)}^3,W_t=t^4-\widehat{\phi }$ ${\left(t-1\right)}^4,$ and $e_t=v_t-\widehat{\phi }{v}_{t-1}$, and, thus, obtain the following equation:

$$Z_t=a_0\left(1-\widehat{\phi }\right)+a{V}_t+b{X}_t+c{Y}_t+d{W}_t+e_t,t=\mathrm{2,3},\dots ,108$$

(19)

Expression (19) is a valid regression equation, from which estimates of the parameters $a_0^{\prime },a,b,c,$ and $d$ can be obtained, where $a_0^{\prime }=a_0\left(1-\widehat{\phi }\right)$.

An alternative method for estimation of the unknown parameters $a_0,a,b,c,$ and d can be derived from (12) by decorrelating the real data, as follows:

$$\begin{gathered} ln{Y}_t-\widehat{\phi }{v}_{t-1}=a_0+at+b{t}^2+c{t}^3+d{t}^4+v_t-\widehat{\phi }{v}_{t-1}= \\ {a}_0+at+b{t}^2+c{t}^3+d{t}^4+e_t,t=\mathrm{2,3},\dots ,108 \end{gathered}$$

(20)

Both methods provide similar results for the estimates of the unknown parameters $a_0,a,b,c,$ and d. The values of these estimates, with their standard deviations derived from (20), are given in

Table 6. Results obtained from Equation (20).

Constant	Estimate	Std. Error	t-Value	Pr (>|t|)
$a_0$	1.01231699	1.38967653 × 10−2	72.84551	<2.22 × 10−16
$a$	−1.85923157 × 10−2	1.765304245 × 10−3	−10.53207	<2.22 × 10−16
$b$	−6.51206952 × 10−5	6.60325763 × 10−5	−0.98619	0.32637
$c$	5.84503339 × 10−6	9.16623405 × 10−7	6.37670	5.3463 × 10−9
$d$	−3.55283094 × 10−8	4.21099460 × 10−9	−8.43704	2.2836 × 10−13
Residual Std. Error	0.0271533	Degrees of freedom	102
Multiple R-Squared	0.9716365	Adjusted R-squared	0.9705242
F-statistic	873.541783	Degrees of freedom	4 and 102
p-value	<2.220446 × 10−16

.

Figure 18a presents the estimated ACF, whereas Figure 18b shows the estimated PACF of the error series $e_t$. Both figures verify that $e_t$ is a sequence of uncorrelated errors.

Figure 19a presents the Q–Q plot for the error series $e_t$. Applying the Shapiro–Wilk test [30] to the series $e_t$, we obtain $W=0.984$ with a p-value of $0.215$. This implies that the null hypothesis—stating that the error series $e_t$ follows a normal distribution—cannot be rejected.

Substituting the estimates of the parameters $a_0,a,b,c,$ and $d$ into (12), we find

$$\begin{gathered} ln{Y}_t-\widehat{\phi }{v}_{t-1}={\widehat{a}}_0+\widehat{a}t+\widehat{b}{t}^2+\widehat{c}{t}^3+\widehat{d}{t}^4+v_t-\widehat{\phi }{v}_{t-1}= \\ {\widehat{a}}_0+\widehat{a}t+\widehat{b}{t}^2+\widehat{c}{t}^3+\widehat{d}{t}^4+e_t,t=\mathrm{2,3},\dots ,108. \end{gathered}$$

(21)

The decorrelated data $ln{Y}_t-\widehat{\phi }{v}_{t-1}$ are produced by subtracting the estimated autocorrelated errors ${\widehat{v}}_t=\widehat{\phi }{v}_{t-1}$ from the real data ${lnY}_t$. Figure 19b shows the smoothed curve of the fourth-order model with the 95% predictive intervals. Only five values of the modified data are outside the 95% predictive intervals, indicating that the fit of the model (21) is quite satisfactory. The statistical test results provided in Table 6 are also valid, as the error series $e_t$ follows a normal distribution.

We know that the effect of a gamma motoneuron results in lengthening of the central sensory region of the muscle spindle, leading to increased firing rate of the $Ia$ response (see [38]).

It is obvious, from our results, that the gamma effect on the muscle spindle is shifted by the presence of the alpha motoneuron from the frequency range of $[5.4,12.0]\mathrm{H}\mathrm{z}$ to the range of $[1.03,7.6]\mathrm{H}\mathrm{z}$. This indicates that the mechanistic function of the inner fibers of the muscle spindle continue to affect the terminals of the sensory axon and produce action potentials in the new range of frequencies at almost the same rate, in a manner depending on frequency as a fourth-order polynomial function.

4.2. The $\alpha –\alpha \gamma First$ Case

4.2.1. First Approach

If we apply the two-stage method mentioned above, we find the following:

In the first stage, working as in the $\gamma –\alpha \gamma$ case, from expression (13) we obtain the following regression equation:

$$\begin{gathered} ln{Y}_t=a\left(1-\phi \right)+\phi ln{Y}_{t-1}+b{t}^2-b{\phi \left(t-1\right)}^2+{\epsilon }_t, \\ t=\mathrm{2,3},\dots ,105 \end{gathered}$$

(22)

where ${\epsilon }_t$ is a residual series with uncorrelated errors. This is a valid regression equation from which an estimate of the parameter $\phi$ is found. The value of the estimate is equal to $\widehat{\phi }=0.779354$. The predicted error values of $v_t$ are calculated using the equation ${\widehat{v}}_t=\widehat{\phi }{v}_{t-1}$.

In the second stage, by subtracting ${\widehat{v}}_t$ from both sides of (13), we obtain

$$\begin{gathered} ln{Y}_t{-\widehat{v}}_t=a+b{t}^2+v_t-{\widehat{v}}_t=a+b{t}^2+e_t, \\ t=\mathrm{2,3},\dots ,105 \end{gathered}$$

(23)

The values $e_t=v_t-{\widehat{v}}_t$ are the estimated uncorrelated errors ${\epsilon }_t$. Expression (23) is a valid regression equation, from which estimates of the unknown parameters $a$ and $b$ can be derived. These estimates, along with their standard deviations, are given in

Table 7. Results obtained from Equation (23).

Coefficient	Estimate	Std. Error	t-Value	Pr (>|t|)
$a$	−6.26750309 × 10−1	4.20319573 × 10−3	−149.11281	<2.22 × 10−16
$b$	2.91575577 × 10−5	8.38429201 × 10−7	34.77641	<2.22 × 10−16
Residual Std. Error	0.02831701	Degrees of freedom	102
Multiple R-Squared	0.9222205	Adjusted R-squared	0.9214579
F-statistic	1209.398702	Degrees of freedom	1 and 102
p-value	<2.220446 × 10−16

.

Figure 20a shows the estimated ACF, whereas Figure 20b presents the estimated PACF of the error series $e_t$. Both figures verify that $e_t$ is a sequence of uncorrelated errors.

Figure 21a presents the Q–Q plot for the error series $e_t$. The Shapiro–Wilk test gives $W=0.983$ with corresponding $p\text{-}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}=0.21$. This implies that the null hypothesis—stating that the error series $e_t$ follows a normal distribution—cannot be rejected.

Substituting the estimates for $\widehat{a}$ and $\widehat{b}$ into (23), we have

$$\begin{gathered} ln{Y}_t-{\widehat{v}}_t=\widehat{a}+\widehat{b}{t}^2+v_t-{\widehat{v}}_t=\widehat{a}+\widehat{b}{t}^2+e_t, \\ t=\mathrm{2,3},\dots ,105 \end{gathered}$$

(24)

Figure 21b shows the smoothed curve of the quadratic model passing through the decorrelated data $ln{Y}_t-{\widehat{v}}_t$, together with the 95% predictive intervals. Only six values of the decorrelated data are outside the 95% predictive intervals, indicating that the fit of model (24) is quite good. The statistical test results provided in Table 7 are considered to be valid, as the error series $e_t$ follows a normal distribution.

It is known that active muscle contraction is produced by the activation of alpha motoneurons [38]. Each time an alpha spike stimulates the parent muscle, it would be expected that the muscle will contract as well as the muscle spindle because it lies in parallel with the extrafusal fibers, leading to the interruption of the spindle’s discharge.

The presence of the gamma motoneuron shifts the effect of the alpha motoneuron from the frequency range of $[2.6,9.0]\mathrm{H}\mathrm{z}$ to the range of $[13.5,19.0]\mathrm{H}\mathrm{z}$. It is clear that the muscle spindle does not respond to the effect of the alpha motoneuron at low frequencies and starts producing action potentials gradually at higher frequencies with a rate that depends on frequency, in particular, in a manner that can be modeled using a quadratic function.

4.2.2. Second Approach

The $\alpha –\alpha \gamma \left(1\right)$ Case

A similar approach can be described for the $\alpha –\alpha \gamma \left(1\right)$ data as in the case of the first approach of the $\alpha –\alpha \gamma$ case. In the first stage, we obtain the following equation from expression (14):

$$\begin{gathered} ln{Y}_t=a_1\left(1-\phi \right)+\phi ln{Y}_{t-1}+b_{1}t-b_1\phi \left(t-1\right)+{\epsilon }_{1_t}, \\ t=\mathrm{2,3},\dots ,62 \end{gathered}$$

(25)

where ${\epsilon }_{1_t}$ is a residual series with uncorrelated errors. This is a valid regression equation, from which an estimate of the unknown parameter $\phi$ can be derived. In particular, this estimate is equal to $\widehat{\phi }=0.540223$.

In the second stage, by subtracting ${\widehat{v}}_{1_t}$ from both sides of (14), we obtain

$$\begin{gathered} ln{Y}_t-{\widehat{v}}_{1_t}=a_1+b_{1}t+v_{1_t}-{\widehat{v}}_{1_t}=a_1+b_{1}t+e_{1_t}, \\ t=\mathrm{2,3},\dots ,62 \end{gathered}$$

(26)

Expression (26) is a valid regression equation, from which estimates of the unknown parameters $a_1$ and $b_1$ can be obtained. These estimates, along with their standard deviations, are given in

Table 8. Results obtained from Equation (26).

Coefficient	Estimate	Std. Error	t-Value	Pr (>|t|)
$a_1$	−0.64200843	0.00678814	−94.57797	<2.22 × 10−16
$b_1$	0.00137370	0.00018585	7.39129	5.8785 × 10−10
Residual Std. Error	0.02555751	Degrees of freedom	59
Multiple R-squared	0.48077658	Adjusted R-squared	0.47197619
F-statistic	54.631238	Degrees of freedom	1 and 59
p-value	5.87851 × 10−10

.

The estimated ACF and PACF for the error series $e_{1t}$ are shown in Figure 22, which verifies that $e_{1_t}$ is a sequence of uncorrelated errors.

Figure 23a shows the Q–Q plot for the error series $e_{1_t}$. The Shapiro–Wilk test gave $W=0.977$ with the corresponding $p\text{-}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}=0.29$. This implies that the null hypothesis—stating that the error series $e_{1t}$ follows a normal distribution—cannot be rejected.

Substituting the estimates ${\widehat{a}}_1$ and ${\widehat{b}}_1$ into (26), we have

$$\begin{gathered} ln{Y}_t-{\widehat{v}}_{1_t}={\widehat{a}}_1+{\widehat{b}}_{1}t+v_{1_t}-{\widehat{v}}_{1_t}={\widehat{a}}_1+{\widehat{b}}_{1}t+e_{1_t}, \\ t=\mathrm{2,3},\dots ,62 \end{gathered}$$

(27)

Figure 23b presents the smoothed curve of the first-order model passing through the decorrelated data $ln{Y}_t-\widehat{\phi }{v}_{t-1}$, together with the 95% predictive intervals. Only one value of the decorrelated data falls outside the 95% predictive intervals, indicating that the fit of model (27) is quite satisfactory. The statistical test results provided in Table 8 are considered valid, as the error series $e_{1_t}$ follows a normal distribution.

The $\alpha –\alpha \gamma \left(2\right)$ Case

The analysis of the $\alpha –\alpha \gamma \left(2\right)$ data was carried out in the same manner as for the α–αγ(1) data.

In the first stage, from Equation (15), we obtain

$$\begin{gathered} ln{Y}_t={\alpha }_2\left(1-\theta \right)+\theta ln{Y}_{t-1}+b_{2}t-b_2\theta \left(t-1\right)+{\epsilon }_{2_t}, \\ t=\mathrm{64,65},\dots ,105 \end{gathered}$$

(28)

where ${\epsilon }_{2_t}$ is a residual series with uncorrelated errors. As (28) is a valid regression equation, an estimate of the unknown parameter $\theta$ can be obtained. In particular, this estimate was found to be $\widehat{\theta }=0.769608$.

In the second stage, by subtracting ${\widehat{v}}_{2_t}$ from both sides of (15), we obtain

$$\begin{gathered} ln{Y}_t-{\widehat{v}}_{2_t}=a_2+b_{2}t+v_{2_t}-{\widehat{v}}_{2_t}=a_2+b_{2}t+e_{2_t}, \\ t=\mathrm{64,65},\dots ,105 \end{gathered}$$

(29)

Expression (29) is a valid regression equation, from which estimates of the unknown parameters $a_2$ and $b_2$ can be derived. These estimates, along with their standard deviations, are provided in

Table 9. Results obtained from Equation (29).

Coefficient	Estimate	Std. Error	t Value	Pr (>|t|)
$a_2$	−0.44415798	0.00894584	−49.64963	<2.22 × 10−16
$b_2$	0.00224432	0.00035003	6.41174	1.247 × 10−7
Residual Std. Error	0.02749600	Degrees of freedom	40
Multiple R-squared	0.50684496	Adjusted R-squared	0.49451608
F-statistic	41.110394	Degrees of freedom	1 and 40
p-value	1.247004 × 10−7

.

The estimated ACF and PACF for the error series $e_{2_t}$ are shown in Figure 24, which verifies that $e_{2_t}$ is a sequence of uncorrelated errors.

Figure 25a provides the Q–Q plot for the error series $e_{2_t}$. The Shapiro–Wilk test gives $W=0.965$ with corresponding p-value of $0.229$. This implies that the null hypothesis—stating that the error series $e_{2_t}$ follows a normal distribution—cannot be rejected.

Substituting the estimates ${\widehat{a}}_2$ and ${\widehat{b}}_2$ into (29), we have

$$\begin{gathered} ln{Y}_t-{\widehat{v}}_{2_t}={\widehat{a}}_2+{\widehat{b}}_{2}t+v_{2_t}-{\widehat{v}}_{2_t}={\widehat{a}}_2+{\widehat{b}}_{2}t+e_{2_t}, \\ t=64,65,\dots ,105 \end{gathered}$$

(30)

Figure 25b presents the smoothed curve of the first-order model passing through the decorrelated data $ln{Y}_t-{\widehat{v}}_{2t}$, together with the 95% prediction intervals. Only two values of the decorrelated data are clearly outside the 95% predictive intervals, while a third one is also outside but very close to the lower interval, indicating that the fit of the model (30) is satisfactory. The statistical test results provided in Table 9 are considered valid, as the error series $e_{2_t}$ follows a normal distribution.

Assuming that there is a changepoint in the $\alpha –\alpha \gamma$ data at a frequency of $17.27\mathrm{H}\mathrm{z}$, Figure 18b and Figure 20b show the logarithms of the ratios of the SDF estimates fitted by two different straight lines with positive trends. This indicates that the muscle spindle produces action potentials gradually at two different rates, which depend linearly on the frequency.

4.3. The $\alpha –\alpha \gamma$ Second Case

In this case, we follow the steps described for analysis of the $\gamma –\alpha \gamma$ data.

In the first stage, we obtain the following equation from expression (16)

$$\begin{gathered} ln{Y}_t=a_0\left(1-{\phi }_1-{\phi }_2\right)+{\phi }_{1}ln{Y}_{t-1}+{\phi }_{2}ln{Y}_{t-2}+\alpha t-\alpha {\phi }_1\left(t-1\right)- \\ \alpha {\phi }_2\left(t-2\right)+b{t}^2-b{\phi }_1{\left(t-1\right)}^2-b{\phi }_2{\left(t-2\right)}^2+c{t}^3-c{\phi }_1{\left(t-1\right)}^3 \\ -c{\phi }_2{\left(t-2\right)}^3+d{t}^4-d{\phi }_1{\left(t-1\right)}^4-d{\phi }_2{\left(t-2\right)}^4+{\epsilon }_t, \\ t=\mathrm{3,4},\dots 178 \end{gathered}$$

(31)

where ${\epsilon }_t=v_t-{\phi }_1{v}_{t-1}-{\phi }_2{v}_{t-2}$ is a residual series of uncorrelated errors. This is a valid regression equation, from which estimates of the unknown parameters ${\phi }_1$ and ${\phi }_2$ can be derived. In particular, ${\widehat{\phi }}_1=1.113922$ and ${\widehat{\phi }}_2=-0.228612$.

In the second stage, by subtracting ${\widehat{v}}_t$ from both sides of (16), we obtain

$$\begin{gathered} ln{Y}_t-{\widehat{v}}_t=a_0+at+b{t}^2+c{t}^3+d{t}^4+v_t-{\widehat{v}}_t= \\ {a}_0+at+b{t}^2+c{t}^3+d{t}^4{+e}_t,\hspace{1em}t=\mathrm{3,4},\dots 178 \end{gathered}$$

(32)

where $e_t=v_t-{\widehat{\phi }}_1{\widehat{v}}_{t-1}-{\widehat{\phi }}_2{\widehat{v}}_{t-2}$. This is a valid regression equation, from which estimates of the unknown parameters $a_0,a,b,c,$ and $d$ can be derived. These estimates, along with their standard deviations, are given in

Table 10. Results obtained from Equation (31).

Coefficient	Estimate	Std. Error	t-Value	Pr (>|t|)
$a_0$	−3.7749757 × 10−1	1.1331616 × 10−2	−33.31366	<2.22 × 10−16
$a$	4.6681041 × 10−3	8.4080734 × 10−3	5.55193	1.0614 × 10−7
$b$	−2.5989716 × 10−4	1.8528108 × 10−5	−14.02718	<2.22 × 10−16
$c$	3.0240552 × 10−6	1.5280872 × 10−7	19.78981	<2.22 × 10−16
$d$	−9.5555194 × 10−9	4.1899271 × 10−10	−22.80593	<2.22 × 10−16
Residual Std. Error	0.0253854	Degrees of freedom	171
Multiple R-squared	0.9587728	Adjusted R-squared	0.9578084
F-statistic	994.186305	Degrees of freedom	4 and 171
p-value	<2.220446 × 10−16

.

The estimated ACF and PACF for the error series $e_t$ are shown in Figure 26, which verifies that $e_t$ is a residual series of uncorrelated errors.

Figure 27a presents the Q–Q plot for the error series $e_t$. The Shapiro–Wilk test gives $W=0.978,$ with a corresponding p-value of $0.007$. This suggests that the null hypothesis—stating that the error series $e_t$ follows a normal distribution—must be rejected.

Substituting the estimates ${\widehat{a}}_0$, $\widehat{a}$, $\widehat{b}$, $\widehat{c},$ and $\widehat{d}$ into (32), we have

$$\begin{gathered} ln{Y}_t-{\widehat{\phi }}_1{v}_{t-1}-{\widehat{\phi }}_2{v}_{t-2}={\widehat{a}}_0+\widehat{a}t+\widehat{b}{t}^2+\widehat{c}{t}^3+\widehat{d}{t}^4+e_t, \\ t=\mathrm{3,4},\dots ,178 \end{gathered}$$

(33)

Figure 27b shows the smoothed curve of the fourth-order model passing through the decorrelated data $ln{Y}_t-{\widehat{\phi }}_1{v}_{t-1}-{\widehat{\phi }}_2{v}_{t-2}$, together with the 95% predictive intervals. Only eight values of the decorrelated data are outside the 95% predictive intervals, indicating that the fit of model (33) is satisfactory. The statistical test results provided in Table 10 must be interpreted cautiously, as the error series $e_t$ does not follow a normal distribution.

In this case, the presence of the gamma motoneuron shifts the effect of the alpha motoneuron from the frequency range of $[9.3,20.5]\mathrm{H}\mathrm{z}$ to the range of $\left[19.9,30.8\right]\mathrm{H}\mathrm{z}$. According to Figure 27b, which presents the logarithm of the ratio of the SDF estimates, there is a small shift in the power values in the beginning and a larger shift as the frequency increases. This indicates that initially, the muscle spindle produces action potentials at a lower rate, which increases at higher frequencies and varies with the frequency in a manner described by a fourth-order polynomial function.

Thus, the flexibility of the system to cope with a more complicated case, which involves the simultaneous effect of two stimuli, allows for its explanation via a more realistic mechanistic model, which contains three different ranges of frequencies (as described above).

All computations and graphics in this section were completed using the TSA library in R 4.4.3 [37].

5. Conclusions

We described three log-linear models for comparison of the estimated SDFs for neural spike trains with autocorrelated errors. The estimated SDFs were obtained from mean-corrected periodograms using simple moving average weighting schemes. However, the existence of autocorrelation leads to estimates for the unknown parameters of the log-linear models that are not reliable. Therefore, a two-stage procedure was applied for decorrelation of the data, allowing for the determination of reliable estimates for the unknown parameters. An illustrative example from the field of neurophysiology was presented, consisting of three experimental cases in which a muscle spindle is affected by the following: (a) a gamma motoneuron, (b) an alpha motoneuron, and (c) gamma and alpha motoneurons simultaneously. It was shown that the effect of the gamma motoneuron in the presence of the alpha motoneuron was shifted to lower frequencies in the range of $\left[1.03,7.6\right]\mathrm{H}\mathrm{z}$, whereas the presence of the gamma motoneuron shifted the effect of the alpha motoneuron to higher frequencies in the ranges of $\left[13.5,19.9\right)\mathrm{H}\mathrm{z}$ and $\left[19.9,30.8\right]\mathrm{H}\mathrm{z}$. The mathematical expression of the ratio of the estimated SDFs was found to follow a fourth-order model for the $\gamma –\alpha \gamma$ case in the range of $\left[1.03,7.06\right]\mathrm{H}\mathrm{z}$, a quadratic model for the $\alpha –\alpha \gamma$ case in the range of $\left[13.5,19.9\right)\mathrm{H}\mathrm{z}$ (or two straight lines with positive trends if we assume that there is a changepoint at $17.27\mathrm{H}\mathrm{z}$), and a fourth-order model for the $\alpha –\alpha \gamma$ case in the range of $\left[19.9,30.8\right]\mathrm{H}\mathrm{z}$.

The muscle spindle is a complex receptor that receives many inputs and responds with two different sensory axons. In future work, we would like to consider the case that the muscle spindle is affected by more inputs—for example, two gamma motoneurons and an alpha motoneuron simultaneously—and responds with both sensory axons (i.e., the $Ia$ primary and the $II$ secondary axon). The aim of such research would be to compare the information that is transferred to the spinal cord and, from there, to the central nervous system by the $Ia$ primary sensory axon and the $II$ secondary sensory axon.