In this section, the study’s background is presented first, followed by its contributions and the paper’s organization.

Fractional calculus (FC) is the generalization of traditional calculus to real order by allowing derivatives and integrals of non-integer (fractional) order [7,8]. Over recent years, FC has emerged as a powerful tool for modeling various processes in diversified fields [9–11] including physics [12], engineering [13,14], control systems [15], neuroscience [16,17], and biology [18] due to its promising features that inherently capture memory and hereditary effects. FC has been exploited to effectively model the biological processes and disease dynamics, such as monkeypox virus [19], hepatitis B transmission dynamics [20], measles infection [21], diabetes mellitus [22], and PD with therapeutic involvement [23].

Different mathematical models of the PD have been proposed. For instance, Qi et al. modeled dopamine metabolism [24], Braatz and Coleman developed an insulin resistance model [25], Baston et al. modeled the levodopa medication effect [26], Al-Tuwairqi and Badrah modeled the innate and adaptive immune responses to PD [27], Parakkal Unni et al. developed a gait freezing model in PD [28], Elfouly presented an improved PD model with Hopf bifurcation analysis [29], Yang et al. modeled the degradation dynamics αS in PD [30], and Kuznetsov and Kuznetsov modeled the αS transportation dynamics in PD [31]. PD is linked with αS aggregation, which is related to the dysfunctional ubiquitin–proteasome system. Sneppen et al. modeled the proteasome dynamics [6] and discovered that when αS accumulates in the form of oligomers and becomes a burden for the protein degradation machinery, which is responsible for the clearance of misfolded proteins. As per the authors’ exhaustive literature survey, most existing models describing the interaction between the proteasome degradation pathway and αS fibril formation have been developed through integer order systems. These classical models provide useful insights into protein aggregation and clearance dynamics, but they often fail to capture the memory effects and hereditary dynamics. The hereditary and memory-driven features of FC may help better to capture the dynamics of Proteasome-Fibril interactions in PD.

Generally, the machine learning-based neural architectures have demonstrated promising performance in disease dynamics [32,33], including PD diagnosis [34]. However, the nonlinear autoregressive exogenous networks (NAREN) have shown superior results over traditional machine learning algorithms and famous neural architectures. The superior performance of the NAREN over its counterparts motivated the authors to investigate the solution dynamics of F-PFIM using NAREN. The summary of the related work is provided in Table 1.

In this section, the design of the F-PFIM for the onset and progression of PD is presented.

The integer-order proteasome model for the onset and progression dynamics of PD is described by three differential classes, which show concentrations of fibrils (F), proteasomes (P), and the proteasome fibril complex (C). F is characterized by the accumulation of misfolded and aggregated proteins in the cell, like the αS protein in the PD. When αS accumulates in the form of oligomers and F, it becomes a burden for the protein degradation machinery (PDM) in the cell. PDM is responsible for the clearance of misfolded proteins, and P is part of the PDM. When αS fibril binds with P of the PDM, immediately C is formed, which facilitates the exclusion of F from the cell. The governing mathematical relations of the model are presented in Eqs. (1)–(3), where t represents the changes in the concentration levels of F, P, and C [6]. The graphical description of the proteasome fibril workflow in terms of the normal pathway and PD pathway is given in Fig. 1. (1)dFdt=α1+P−βFP (2)dPdt=ε−P−βFP+γC (3)dCdt=βFP−γCwhere the variables involved in the proteasome model are defined as: α denotes the protofilaments influx, β is the constant linked with the proteasome-fibril complex formation, γ represents fibril degradation time in the complex, and ε is the proteasome production rate.

This study aims to develop the fractional order Proteasome model by incorporating the concept of fractional derivatives into the integer model presented in (1)–(3). Fractional calculus allows us to compute the derivatives of real order (fractional order). Fractional derivatives are the generalization of the conventional integer derivatives and can be defined in different ways. Although in recent times, new definitions of fractional derivatives have been introduced [35–37], the commonly used definitions, including Grunwald-Letnikov (GL), Riemann-Liouville (RL), and Caputo (Ca), for the fractional order σ are provided here in Eqs. (4)–(6), respectively [38]. (4)D0,tσGLy(t)=limx→0,mx=tx−σ∑j=0m(−1)j(kj)y(t−jx) (5)D0,tσRLy(t)=1Γ(s−σ)dsdts∫0t(t−τ)s−σ−1y(τ)dτ,s−1<σ<s (6)D0,tσCay(t)=1Γ(s−σ)∫0t(t−τ)s−σ−1y(s)(τ)dτ,s−1<σ<s

Now, describe the fractional-order proteasome model, F-PFIM, by three fractional differential classes, as shown in Eqs. (7)–(9). (7)DσCaF=α1+P−βFP (8)DσCaP=ε−P−βFP+γC (9)DσCaC=βFP−γC

This section first presents the details of the fractional Adams method (FAM) used to generate synthetic data for the machine learning knowledge-driven NAREN-LM scheme to analyze the solution dynamics of the F-PFIM presented in Section 3. A fractional order differential equation (FDE) is generally defined as in Eq. (10). (10)DσCay(t)=g(t,y(t)),y(t0)=y0

The implementation of FAM to solve a general FDE given in Eq. (10) required three steps: summation, prediction, and correction. These three stages are mathematically expressed in Eqs. (11)–(13), respectively [39,40]. (11)Ωm=bb,0g(t0,y0)+∑l=1m−1am−lg(tl,yl) (12)ympr=y0+hσ[Ωm−a0g(tm−2,ym−2)+2a0g(tm−1,ym−1)] (13)ymcr=y0+hσ[Ωm+a0f(tm,ympr)]

The NAREN-LM is developed in Matlab using ‘ntstool’ by considering the following specifications: layer size 10, log sigmoid activation function, input and feedback delay of 1:4, and Levenberg-Marquardt optimization algorithm for backpropagation of weights to optimize the NAREN. The dataset consists of the time-series solutions generated from the proposed fractional-order model, where the input features correspond to the delayed time-series states, and the target outputs represent the system variables at subsequent time steps. The dataset generated through FAM is given to NAREN-LM by arbitrarily splitting into train, test, and validation sets with a proportion of 80%, 10%, and 10%, respectively. The selection of the NAREN hyperparameters is based on preliminary experimentation and validation performance to achieve a balance between model accuracy and overfitting prevention. The block diagram of the NAREN is presented in Fig. 2.

This section presents the results of the NAREN-LM for the solution dynamics of F-PFIM in terms of various graphical illustrations for different fractional orders, i.e., σ = {0.89, 0.90, 0.99, 1}. For numerical experimentation, the values of the parameters involved in the F-PFIM presented in (7)–(9) are selected as: α = 25, β = γ = ε = 1 [28].

To investigate the convergence of the NAREN-LM scheme for the solution dynamics of the F-PFIM, learning curves are plotted in terms of training, testing, and validation, and the results are presented in Figs. 3–5a for σ = 1, 0.95, and 0.90, respectively. It is witnessed from the convergence plots that NAREN-LM achieves the best validation performance in the range of 10⁻⁶ to 10⁻⁸ for all considered σ variations, endorsing the convergent performance of the NAREN-LM neural architecture for F-PFIM.

The performance evaluation of the NAREN-LM is further conducted through histogram analyses, and the results are presented in Figs. 3–5b for σ = 1, 0.95, and 0.90, respectively, where the error is distributed over 20 bins, and the vertical line in the plots represents zero error, indicating the optimal error value. It is seen that most of the errors are distributed in a few bins, with the majority in a single bin. From the results, it is evident that most of the error instances lie in the range of 10⁻³ to 10⁻⁵ for all considered σ variations, endorsing the accurate performance of the NAREN-LM scheme from F-PFIM.

To get deeper insight into the performance of the NAREN-LM for the solution dynamics of F-PFIM, error correlation and input-error correlation analyses are conducted, defining the proportional trends, and the results are presented in Figs. 3–5c for σ = 1, 0.95, and 0.90, respectively, in case of autocorrelation. While input-error cross-correlation results are provided in Figs. 3–5d for σ = 1, 0.95, and 0.90, respectively. A strong positive error autocorrelation is observed for all the F-PFIM variations demonstrated in Figs. 3–5c. The NAREN-LM exhibits positive cross-correlation between the error and the input of the model for σ = 1, and a negative cross-correlation for σ = 0.95 and 0.90, as demonstrated in Figs. 3–5d.

The hyperparameter conditions play a fundamental role in the performance of any machine learning algorithm. In this regard, the variation in the hyperparameters of the NAREN-LM over the epochs during the training process is demonstrated in Figs. 3–5e for σ = 1, 0.95, and 0.90, respectively. It is observed from Fig. 3e that NAREN-LM is trained at epoch 1000 with a gradient of 0.00025641 corresponding to the step size (mu) of 1E−07 for F-PFIM with σ = 1. Similarly, in the case of σ = 0.90, the NAREN-LM is trained at epoch 1000 with a gradient of 0.00025641 corresponding to the mu 1E−06. Further, the results show that no validation checks are encountered during the training process.

The time response for element 1 of the F-PFIM, i.e., F (Fibrils), is presented in Figs. 3–5f for σ = 1, 0.95, and 0.90, respectively. The small values of target output errors demonstrate the accuracy of the NAREN-LM for the temporal dynamics of F-PFIM for all considered σ = variations

To further validate the performance of the NAREN-LM, time response graphs along with the corresponding absolute errors are plotted for all three elements of F-PFIM, i.e., F, C, and P, and are demonstrated in Fig. 6 for σ = 1, 0.99, 0.98, and 0.97. Similarly, Figs. 7 and 8 provide the corresponding plots for σ = 0.96, 0.95, 0.94, and 093, and σ = 0.92, 0.91, 0.90, and 0.89, respectively. The three components of the F-PFIM (F, C, and P) are separately plotted to effectively decipher the temporal variations due to a change in fractional order. Figs. 6–8 clearly demonstrate the effect of fractional order on the temporal dynamics of the Proteasome Fibril interaction model. It is observed from Figs. 6–8a that the F of the F-PFIM exhibits high amplitude and sharply peaked oscillations for σ = 1, with a monotonic decrease in oscillation amplitude as σ = decreases from 1 to 0.89. Further, the largest F peak for σ = 1 indicates strong aggregation bursts, whereas σ = 0.89 shows strong attenuation with peaks reduced by more than half. Importantly, it is seen that the oscillation persists for all fractional order variations. Similarly, it is observed from Figs. 6–8c that the peak concentration of C consistently decreases as σ decreases from 1 to 0.89, with the highest peak for σ = 1 showing strong C formation (binding of F with P). Figs. 6–8e show the temporal dynamics of P with respect to fractional order variations. It is seen that the maximum P concentration drops significantly as the σ = decreases from 1 to 0.89, with peaks becoming progressively smaller. Moreover, the consistent overlapping of the FAM outcomes with the NAREN-LM results, as well as the corresponding low values of absolute errors, endorse the accuracy of the machine learning-driven NAREN-LM for modeling the dynamics of F-PFIM.

In order to have a clear understanding of the temporal evolutions of the F, C, and P, and how they relate to each other, results are demonstrated in Fig. 9 for σ = 1, 0.95, and 0.90. Since the detailed analyses of the fractional dynamics of the F, C, and P are already discussed in Figs. 6–8. Fig. 9 only focuses on describing the F, C, and P interaction. All of these three components collectively make a controlling system that works for the clearance of excess or unnecessary protein from the cell to maintain cell hemostasis and demonstrate oscillatory actions determined by a negative feedback loop. A phenomenon of mutual exclusion is observed in Fig. 9, which shows high levels of free P are linked with low levels of F due to the normal functioning of the degradation machinery. On the other hand, the high quantity of F is linked with low levels of free P because most of the P is in the form of C (Fibril-Proteasome Complex). This demonstrates that the concentration of P looks like sharp and narrow spikes, as shown in Fig. 9, while the concentration of F displays a slow accumulation. During the oscillatory period, the P levels are very low, which means the cell degradation machinery is unable to clear the toxic aggregates of protein. The spikes in P for a short period of time have a significant role in clearance and partial recovery, but it is not enough for all the aggregates. For extended time periods, this instability of accumulated proteins and partial clearance leads to slow and progressive degeneration that may cause PD.

In this work, a novel fractional-order Proteasome Fibril Interaction model, F-PFIM, is presented for the onset and progression dynamics of PD, represented by three fractional differential classes, showing concentrations of Fibrils, Proteasomes, and Proteasome Fibril Complex (F, P, and C). Machine learning knowledge-driven nonlinear autoregressive exogenous networks backpropagated with Levenberg-Marquardt optimization, NAREN-LM, are presented to analyze the dynamics of the proposed F-PFIM for different fractional orders.

The NAREN-LM accurately modeled the temporal evolution dynamics of the F-PFIM. The three components (F, P, and C) of the F-PFIM collectively make a controlling system that works for the clearance of excess or unnecessary protein from the cell to maintain cell hemostasis. When the P levels are very low, and F accumulation is high, the cell degradation machinery becomes overburdened and is not able to fully clear the toxic aggregates of protein. For extended time periods, this instability of accumulated proteins and partial clearance leads to slow and progressive degeneration that may be associated with the onset of PD. As the process of aggregation is very slow, the PD progression is slow as well. Regarding the fractional dynamics of the F-PFIM, the peak concentration of F, C, and P consistently decreases as σ decreases from 1 to 0.89, with the highest peak for σ = 1 showing strong aggregation bursts in the case of F and strong C formation (binding of F with P).

The convergence and reliability of the NAREN-LM are verified by the learning curves for training and testing, histogram, and correlation analyses. Moreover, the comparative analyses of the NAREN-LM with the FAM outcomes validate the accuracy of the machine learning knowledge framework through consistent overlapping of the NAREN-LM and FAM outcomes, along with low values of corresponding absolute errors. The presented machine learning knowledge driven approach is a supervised learning scheme that requires labeled training data. Therefore, the accuracy of the NAREN-LM is dependent on the reliability of the numerical baseline algorithm that is used to generate training data.

Future studies would extend the application of a machine learning knowledge-driven framework to model the dynamics of other, more complex and complicated biological systems. Moreover, fractional dynamics would be explored in the modeling of other neurodegenerative diseases to get a better understanding of the complex neurodegenerative disorders.