Construction of the dynamic model of SCI rehabilitation using bidirectional stimulation and its application in rehabilitating with BCI

Abstract

Patients with complete spinal cord injury have a complete loss of motor and sensory functions below the injury plane, leading to a complete loss of function of the nerve pathway in the injured area. Improving the microenvironment in the injured area of patients with spinal cord injury, promoting axon regeneration of the nerve cells is challenging research fields. The brain-computer interface rehabilitation system is different from the other rehabilitation techniques. It can exert bidirectional stimulation on the spinal cord injury area, and can make positively rehabilitation effects of the patient with complete spinal cord injury. A dynamic model was constructed for the patient with spinal cord injury under-stimulation therapy, and the mechanism of the brain-computer interface in rehabilitation training was explored. The effects of the three current rehabilitation treatment methods on the microenvironment in a microscopic nonlinear model were innovatively unified and a complex system mapping relationship from the microscopic axon growth to macroscopic motor functions was constructed. The basic structure of the model was determined by simulating and fitting the data of the open rat experiments. A clinical rehabilitation experiment of spinal cord injury based on brain-computer interface was built, recruiting a patient with complete spinal cord injury, and the rehabilitation training and follow-up were conducted. The changes in the motor function of the patient was simulated and predicted through the constructed model, and the trend in the motor function improvement was successfully predicted over time. This proposed model explores the mechanism of brain-computer interface in rehabilitating patients with complete spinal cord injury, and it is also an application of complex system theory in rehabilitation medicine.

Supplementary Information

The online version contains supplementary material available at 10.1007/s11571-022-09804-3.

Introduction

Spinal cord injury (SCI) involves damage to the structure and function of the spinal cord caused by trauma, disease, etc., which can cause motor, sensory dysfunction in patients (Ditunno et al. 1994; Kirshblum et al. 2011). After surgery or acute treatment, patients require long-term rehabilitation treatment for functional recovery. According to the American Spinal Cord Injury Association Injury Scale A (AIS A), complete SCI patients showed complete loss of motor and sensory function below the injury plane, and although the patients underwent 6 months to 1 year of rehabilitation training, their motor function still showed low recovery rate (Wirz et al. 2006; Marino et al. 2011). Generally, there is a complete loss of function of the spinal nerve pathway in these patients, and the possibility of nerve remodeling is very low. Hence, promoting axonal regeneration may be a new research direction. However, if the nerve cells in the central nervous system of the adult mammal, including those in the spinal cord, are damaged, it is extremely difficult to regenerate (Ry 1928). Current studies have shown that the central nervous system can regenerate only under certain conditions (Barnabe-Heider and Frisén 2008; Okano and Yamanaka 2014). The special microenvironment in the injured area critically restricts the axonal regeneration of the nerve cells (Fehlings and Weidner 2016). After SCI, the affected area produces glial scarring, which hinders the development of the nerve cells (Gaudet and Fonken 2018). At the same time, factors that inhibit nerve cell growth will be produced in the injured area, and the secretion of factors that promote nerve cell growth will be reduced.

There are presently three main technical routes for the rehabilitation of SCI: (1) Stimulating the peripheral neurons, through exercise training of the effectors, such as muscle of limbs, by generating a continuous upward signal to the injured area of the spinal cord (Aravind et al. 2019; Wirz et al. 2005; Baunsgaard et al. 2018); (2) Stimulating the injured area of the spinal cord by applying top-down signals from the central nervous system through transcranial direct current /magnetic stimulation (Yamaguchi et al. 2016; Yoon et al. 2014; Awad et al. 2015; Calvert et al. 2019); (3) Stimulating the injured area of the spinal cord bidirectionally by applying top-down and bottom-up signals achieved by fusing exercise therapy with the magnetic stimulation/spinal cord electrical stimulation (Litvak et al. 2007), or implementing new technologies such as brain computer interface (BCI) (Donati et al. 2016; Shokur et al. 2018). In previous studies, the rats with SCI were subjected to unidirectional simulation training and after the completion of the training, the microenvironment in the SCI area was analyzed by dissecting and extracting it. There was a significant improvement in the microenvironment involving the reduction in the concentration of the inhibiting factor and an increase in the concentration of the enhancing factor. Hence, the unidirectional stimulation was proven to improve the microenvironment of the SCI region to a certain extent and promote the recovery of function. However, the therapeutic effect of unidirectional stimulation on patients with complete SCI is very limited. In other studies, the rats subjected to a bidirectional combined brain electric/magnetic stimulation and exercise bidirectional of the SCI area showed a better improvement in the motor function and microenvironment, compared to those subjected to a unidirectional stimulation (Petrosyan et al. 2015; Hou et al. 2014). Another bidirectional stimulation rehabilitation method, namely the rehabilitation based on BCI, reported some positive results in recent years. In the application of motor function rehabilitation for stroke patients (Biasiucci et al. 2018) and motor function rehabilitation for cerebral palsy patients (Xie et al. 2021), BCI rehabilitation system has achieved positive clinical effects. In the BCI-based rehabilitation training in SCI, the injury area is simultaneously stimulated by the descending control signal and the ascending sensory signal. A clinical study carried out by Donati et al. on six patients with complete motor injury using BCI-based rehabilitation technology, found significant improvement in the lower limb motor function after long-term training (Donati et al. 2016).

Our team designed a BCI-based rehabilitation robot, and performed 4-week rehabilitation training for a complete SCI patient. After the training, the strength level of the left quadriceps femoris muscle at the patient’s lower extremity increased from 0 to 2. In the follow-up visits, the muscle strength of the patient improved more significantly. Although some progress has been made in the study of diffusion tensor imaging (Petersen et al. 2012; Martin et al. 2016) and blood markers, a method for efficiently detecting the growth of the human spinal nerve cells is still elusive. This makes the study of the motor function recovery and the internal mechanism of patients with complete SCI difficult. Many previous studies have explored models such as the brain-spinal cord association and motor neuron excitability after SCI (Venugopal et al. 2012; Lu and Tian 2014; Lu 2015), but such studies is limited by the dearth of direct experimental evidence. Currently, the BCI and other technologies applied to patients with complete SCI have shown a rehabilitation effect. However, it is still unclear as to whether this effect is related to nerve growth or nerve function remodeling, thus limiting the further improvement of this technology and its application in more patients with complete SCI. This paper devises a new research method to explore the rehabilitation mechanism of SCI based on the BCI.

SCI and its rehabilitation are relevant to the theory of complex systems. As a neural pathway of signal transmission, the spinal cord has a high order, which is destroyed after injury due to trauma or disease. Rehabilitation training attempts by the continuous application of various stimuli on the boundary of the SCI area tending to change the boundary conditions and to promote the restoration of the SCI system to that of the original order. Assuming that the SCI area is a complex system, rehabilitation training would change the microenvironmental growth/inhibiting factor concentration of the SCI area by stimulating its boundary, which would in turn change the complex system. Therefore, the mathematical model for SCI rehabilitation can be constructed using complex system theory. If the recovery curve of the SCI patient is consistent with the results predicted by the mathematical model, the model can explain the mechanism of BCI rehabilitation to some extent.

The main work of this paper was to establish a mathematical model for the rehabilitation of SCI by using the open data of SCI rats and the complex system theory. The model included two parts. (1) Based on the previous knowledge, stimulating the SCI region would change the concentration ratio of the growth and inhibiting factors in the microenvironment. Thus, a micro-growth dynamics model was designed for the growth of the spinal cord axon cells into the target cells. (2) The mapping model of evaluating the micro-growth dynamics and macro-motor function was established combining the public data from the rat experiments, to describe the trend of motor function changes with time under different rehabilitation methods. Finally, we expected to apply the rat model to evaluate human clinical rehabilitation. This paper adopts the above steps to compare the results predicted by the model with the results of a patient with complete SCI after rehabilitation based on a BCI. Through comparative analysis, this paper focuses on the predictability of the model, especially the prediction of a patient’s 285-day rehabilitation process. We discussed the basic idea of model construction and further explored the rehabilitation mechanism of BCI in SCI.

Materials and methods

Public rat data

Petrosyan HA et al. found that long-term combinatorial treatment of spinal electromagnetic stimulation (sEMS) and exercise training in the rats with SCI led to the continuous strengthening of the transmission of lumbar motor neurons and hindlimb muscles with further improvement in the motor function (Petrosyan et al. 2015). In this study, the rats were divided into four groups: injury group (SCI), electrical stimulation group (SCI, sEMS), exercise training group (SCI, Exercise), electrical stimulation + exercise group (SCI, sEMS + Exercise).

This study used motor function indicators (regularity index, the base of support, stride length, and mean paw intensity) and biochemical indicators (FR-labeled cells) to evaluate the rehabilitation of rats with SCI. Table 1 shows the motor function indices of rats with SCI under different rehabilitation modes after 5 weeks. Table 2 shows the biochemical evaluation indices of rats with SCI under different rehabilitation modes after 9 weeks. And the results showed that there was no statistical difference in the final rehabilitation results (motor function Indicators and Biochemical Indicators) between the rats receiving single stimulation training (SCI, sEMS/SCI, Exercise) and those without stimulation training (SCI). In addition, the motor function Indicators and Biochemical Indicators of group (SCI, sEMS + Exercise) were significantly higher than those of the other three groups (SCI/SCI, sEMS/SCI, Exercise).

Table 1.

The evaluation indices of the motor function

Subjects	Regularity index (%)	Base of support	Stride length	Mean paw intensity
SCI	85.6 ± 2.2	34 ± 1.3	91.4 ± 2.6	63.8 ± 2.95
SCI, sEMS	87 ± 1.46	33.6 ± 1.45	92.3 ± 2.1	63.2 ± 2.69
SCI, Exercise	86 ± 1.42	35 ± 1.8	91.4 ± 1.69	64.6 ± 2.68
SCI, sEMS + Exercise	91.6 ± 1.56	30.5 ± 1.0	94.8 ± 1.4	69.4 ± 2.6

Table 2.

Biochemical evaluation indicators

SCI	SCI, sEMS	SCI, Exercise	SCI, sEMS + Exercise
116 ± 6.8	108 ± 5.7	111 ± 5.5	152 ± 5.7

FR-labeled L1 L2 cells: Kieh’s study found that the activity of the commissural interneurons located in the L1–L2 segment and projected to the L4/L5 motor neurons mediates the CPG function of the rats (Kiehn and Ole 2006). The number of the middle-level FR-labeled cells in the L1–L2 segment of rats correlated with some research results. These results suggested that synaptic plasticity was crucial for coordinating and recovering the motor function after SCI (Blits et al. 2003; Courtine et al. 2007; Petrosyan et al. 2013).

BCI system and clinical trial

Our BCI rehabilitation system used the combination of BCI and lower-limb robots to perform sports rehabilitation training on the lower limbs of the patient. A complete SCI patient (male, 42 years old, T10 injury, AIS A, BCI-naïve) participated in the rehabilitation training of the system. Our experiment was carried out in the First Affiliated Hospital with Nanjing Medical University, Jiangsu Province, China, and approved by the hospital ethics committee. The patient provided informed consent before the experiment. The patient participated in 20 days of training, for 20 min each day. The lower limb motor function of the patient was evaluated by the international standard Lower extremities motor score (LEMS) (see Fig. 1).

Fig. 1.

Lower limb robot rehabilitation system based on BCI

After 20 days of training, the LEMS of the patient was increased from 0 to 3, where the right quadriceps femoris increased from 0 to 1, the right iliolumbar muscle strength increased from 0 to 2. After that, the patient was transferred to other hospitals for rehabilitation treatment. After the 40th week, their muscle strength increased significantly, the LEMS of the patient increased to eight. The level of the muscle strength of the left/right quadriceps femoris and left/right iliolumbar muscle increased to two.

The relationship between LEMS in different time periods during rehabilitation training for the patient is shown in Table 3:

Table 3.

LEMS evaluation time and Score

Time/week	1	2	8	9	40
LEMS	0	0	1	3	8

The establishment of the models

After SCI, the concentration evolution of the growth and inhibitory factors should meet Fick’s first law. In rehabilitation training, the electrical stimulation of the central nervous system and peripheral nerve would affect the secretion of the growth and inhibiting factors. The specific expression was shown in Eq. 1.

$$\begin{array}{c}\frac{\partial {\rho }_1}{\partial t}=D_1{\mathrm{\nabla }}^2{\rho }_1-k_{-1}{\rho }_1+\sum _{j=1}^{N_T}{\sigma }_1\delta (r-r_j^T)\\ \frac{\partial {\rho }_2}{\partial t}=D_2{\mathrm{\nabla }}^2{\rho }_2-k_{-2}{\rho }_2+\sum _{j=1}^{N_G}{\sigma }_2\delta (r-r_j^G)\\ \end{array}$$ 1

where ρ₁ and ρ₂ represented the concentrations of the growth and inhibiting factors (unit μM), respectively. D₁ and D₂ represented the diffusion coefficients of the two factors (unit μm²s⁻¹), which were constants. k₋₁ and k₋₂ represented the linear attenuation coefficients (unit s⁻¹) of the two factors, which were constants. σ₁ and σ₂ represented the release rate of the two factors (unit μMs⁻¹). δ is the Dirac function. r^T and r^G represented the location of target cells and glial scar respectively (where the target cells mainly secreted the growth factors and the glial scar secreted the inhibitory factors). N^T and N^G represented the number of target cells and glial scar cells respectively.

The growth direction and growth rate of axons were related to the varying gradient of chemical concentration. The chemical concentration gradient of the growth factor had a significant effect on the growth of axons, while the chemical concentration gradient of the inhibiting factor exclusively affected the growth of axons. Assuming that the growth cone of the axon is a particle and considering its slow growth rate, the effect of inertial force could be ignored. Its dynamic equation was expressed as follows:

$$\begin{array}{c}\frac{d{\mathbf{r}}_j^A(t)}{\mathit{dt}}=\frac{1}{\mu }F({\mathbf{r}}_j^A,t),j=1,2,\dots ,N_A\\ F({\mathbf{r}}_j^A,t)=\sum _{i=1}^2{\lambda }_i{\mathbf{p}}_i,{\mathbf{p}}_i=\left(\frac{\partial {\rho }_i}{\partial x}\mathbf{i}+\frac{\partial {\rho }_i}{\partial y}\mathbf{j}\right)\frac{∥\mathrm{\Delta }{r}_j∥}{{\rho }_{\sum }},{\rho }_{\sum }=\sum _{i=1}^2{\rho }_i\\ \end{array}$$ 2

where, r^A represented the position of the axon growth cone, which was a two-dimensional vector; N_A was the number of axon growth cone; λ_i was the constant of proportionality (the growth factor was positive, while the inhibiting factor was negative). Where the y-direction was the growth direction of spinal cord nerve cells, the x-direction was perpendicular to the y-direction, and ∇ρ_i was the i-th gradient of concentration factor; $∥\mathrm{\Delta }r∥=\sqrt{\mathrm{\Delta }{x}^2+\mathrm{\Delta }{y}^2}$; ρ_Σ was the location of the j axon growth cone and two types of concentration factors The sum $F\left({\mathit{r}}_j^A,t\right)$ represented the chemotactic force of the j-th axon growth cone.

It was assumed that the BCI or other rehabilitation methods would affect the secretion of the target tissue-related growth and inhibiting factor of the glial scar production, that is, acting on σ₁ and σ₂. Its function equation was as follows:

$$\begin{array}{cc}{\sigma }_1\hfill & ={\sigma }_{10}Tr\left(t\right)\\ {\sigma }_2\hfill & ={\sigma }_{20}/Tr\left(t\right)\end{array}$$ 3

where:

$$Tr\left(t\right)=\left\{\begin{array}{cc}{K}_{\mathit{tr}}\hfill & t>t_{\mathit{tr}}\\ 0\hfill & t\le t_{\mathit{tr}}\end{array}\right)$$ 4

In the equation, K_tr represented the proportion of the influence of rehabilitation training on the growth and inhibiting factors. t_tr represented the starting point for training.

According to the research, the neural synapses could also transmit the electrical conduction signals without fully binding to the target cells. The existing research proved that axon synapses could also improve the neurological function in different degrees without point-to-point connections, such that after SCI, the inaccurate synaptic connections could restore some functions in the patient (Hamid and Hayek 2008).

The main parameters affecting the recovery of the nerve function were the growth distance of the nerve cell axon to the target cells, denoted as r_y in the model explained above. To construct a model from the microscopic axon growth to macroscopic motor function recovery, we need to to establish the relationship between parameter r_y and motor function score Fun(t):

$$\begin{array}{c}Fun\left(t\right)=Fun\left({\mathbf{r}}_y\left(t\right)\right)\\ {\mathbf{r}}_y\left(t\right)=\left\{r_{y1}\left(t\right),r_{y2}\left(t\right),r_{y3}\left(t\right),\dots ,r_{y{N}_A}\left(t\right)\right\}\\ r_{\mathit{yi}}\left(t\right)={\mathbf{r}}_{\mathit{yi}}^A\left(0\right)-{\mathbf{r}}_{\mathit{yi}}^A\left(t\right)i=1,2,3,\dots ,N_A\\ \end{array}$$ 5

where Fun(t) is the patient’s motor function score; r_yi(t) was the distance component of the i-th axonal growth cone growing towards the target cell; ${\mathit{r}}_{\mathit{yi}}^A\left(t\right)$ was the y-axis position of the i-th axon growth cone at time t; N_A was the number of axon growth cone. Therefore, Eq. 5 represented the relationship between the motor function of the patient after SCI and the time of external stimuli.

By simulating the nerve cell growth of the untreated, electrical stimulation treatment, exercise training treatment, electrical stimulation + exercise training treatment of four groups of rats, this paper uses nlinfit function in Matlab to fit the relationship between exercise function score and r_y.

Model simulation

Rat model simulation

1. The initial setting of the rat model parameters satisfied the following principles:

2. The rat model was a proportional model whose scale and time could be measured according to the actual situation;

3. The diffusion coefficient, attenuation coefficient, initial release rate, dissociation constant, and other parameters in the rat model had not been accurately determined by biochemical experiments. In this paper, the above parameters were adjusted based on the reference (Petrosyan et al. 2015), to maintain consistency in the simulation of the four groups of the rat model.

For rats in the experiment: When they were not trained, K_tr was 1; when they received electrical stimulation and exercise rehabilitation training, K_tr was 1.31; when they received electrical stimulation rehabilitation training, K_tr was 0.93; when they received exercise rehabilitation training, K_tr was 0.96. These values met the FR-labeled cell concentration ratio between the different training as well as untrained groups.

The Alternating direction implicit (ADI) algorithm was used concentration field calculation, and the boundary conditions were set using the Lattice Boltzmann Method (LBM) (see Table 4).

Table 4.

The basic parameters and initial values of the rat model

Rat model initial parameters
Parameter	Value	parameter	Value
The minimum value of the x-axis	0	The minimum value of the y-axis	0
The maximum value of the x-axis	79	The maximum value of the y-axis	79
Meshing step	1	Move step	1
Diffusion coefficient of the growth factor	0.21	Attenuation coefficient of the the growth factor	0
Diffusion coefficient of the inhibiting factor	0.23	Attenuation coefficient of the inhibiting factor	0.01
The initial release rate of the growth factor (nMs-1)	0.93	Dissociation constant	1
The initial release rate of the inhibiting factor (nMs-1)	1	lambP1	1
Ucoeff = 1	1	lambP2	− 1
The x-axis coordinate of the colloidal scar center	40	Colloidal scar width	36
The y-axis coordinate of the colloidal scar center	40	Colloidal scar height	40
Model training time (seconds)	14,000

Simulation of the human model.

The human model parameters and initial conditions of the dynamic model of human SCI rehabilitation based on the BCI are shown in Table 5. The construction of the human model was consistent with that of the rat model, and only the diffusion coefficient, attenuation coefficient, and initial release rate were slightly adjusted. In the action equation, the value of K_tr was 1.31 for the AIS A patients in the training. This was consistent with the parameters of the action equation of rats under electrical stimulation + exercise rehabilitation training, that is, in the process of rehabilitation training based on the BCI, the patient’s SCI area received bidirectional stimulation. The start time of patient training K_tr was determined according to the actual time of the patient.

Table 5.

The basic parameters and initial values of the human model

Human model initial parameters
Parameter	Value	Parameter	Value
The minimum value of the x-axis	0	The minimum value of the y-axis	0
The maximum value of the x-axis	79	The maximum value of the y-axis	79
Meshing step	1	Move step	1
Diffusion coefficient of the growth factor	0.22	Attenuation coefficient of the growth factor	0
Diffusion coefficient of the inhibiting factor	0.23	Attenuation coefficient of the inhibiting factor	0.01
The initial release rate of the growth factor (nMs-1)	1.3	Dissociation constant	1
The initial release rate of the inhibiting factor (nMs-1)	1.15	lambP1	1
Ucoeff = 1	1	lambP2	− 1
The x-axis coordinate of the colloidal scar center	40	Colloidal scar width	42
The y-axis coordinate of the colloidal scar center	40	Colloidal scar height	50
Model training time (seconds)	15,000

Result

Rat model simulation results

According to the rat model proposed above, the concentration distribution of the growth and inhibiting factors were calculated in the four groups (SCI/SCI, sEMS/SCI, Exercise/SCI, sEMS + Exercise) after 9 weeks and the growth of the axon cells was simulated. The concentrations of growth and inhibiting factors in the four groups showed gradient distribution, and reached the highest near the target cells and glial scar respectively. The concentration gradient of the growth and inhibitor factors jointly acted on the growth path and speed of the axon cells.

The three-dimensional representation and contour distribution of the concentration distribution of growth and inhibiting factor in the three groups (SCI/SCI, sEMS/SCI, Exercise) are shown in Fig. 2, respectively. The maximum concentration of the growth factor in the three groups was below 15. The growth path of these three groups of axon cells is shown in Fig. 2A—c, B—c, C—c and the mean and standard deviation of axon cell growth components r_y in these three groups were different (SCI: 37.94 ± 28.09; SCI, sEMS: 34.36 ± 31.46; SCI, Exercise: 46.16 ± 29.03). Three groups of results shows that some axon cells grew towards the target cells around the glial scar, and only a small part of them grew towards the target cells.

Fig. 2.

Schematic diagram of growth, inhibiting factor concentration distribution and axon cell growth. A/B/C corresponds to the SCI/SCI, sEMS/SCI, Exercise group. Take figure A as example, A—a is the concentration distribution diagram of the promotion factor; A—b is the contour map of factor concentration; A—c is the growth chart of axon cells, where the ring represents the glial scar

The three-dimensional representation and contour distribution of the concentration distribution of the growth and inhibiting factors in SCI, sEMS + Exercise group is shown in Fig. 3a and Fig. 3b, respectively. The maximum concentration of the growth factor was below 20. The growth path of the axon cells is shown in Fig. 3c, and the mean and standard deviation of growth component r_y of axon cells in SCI, sEMS + Exercise group was 55.28 ± 19.28. The results showed that most axon cells grew around the glial scar and grew towards the target cells. The mean value of r_y in SCI, sEMS + Exercise group was significantly higher than that in the other three groups (SCI/SCI, sEMS/SCI, Exercise), and most axon cells in this group grew to the lower end of the glial scar.

Fig. 3.

Schematic diagram of the concentration distribution of the growth, and inhibiting factors as well as the axon cell growth in the SCI, sEMS + Exercise group. Where, a is the concentration distribution diagram of the promotion factor; b is the contour map of factor concentration; c is the growth chart of axon cells, where the ring represents the glial scar

We used Matlab to fit the r_y and motor function score Fun. The effect of each neuron on the motor function was uncertain, so the power exponential model was used for fitting, as shown in Eq. 6.

$$Fun\left(t\right)=K_{\mathit{ur}}\left(r_{y1}^p+r_{y2}^p+r_{y3}^p+\dots +r_{y{N}_A}^p\right)$$ 6

where, N_A was the number of axon growth cone; K_tr was the proportional coefficient; p denoted the power ratio that needs to be fitted; r_y denoted the component of the nerve cell axons growing along the spinal cord.

Through fitting, the power rate value was found to be unexpectedly close. The power rate of the SCI group was 0.74, 0.70 for the SCI, sEMS group, 0.76 for the SCI, Exercise group, 0.73 for the SCI, sEMS + Exercise group. The power rates of the four groups of simulations were close to three-quarters, within the allowable error range.

According to complex system theory, the scale-dependent networks of organisms should fulfill the “three-quarter power law” (West and Brown 2005; Mitchell 2011). For example, the tree trunk perimeter, plant growth rate, body temperature in animals, metabolic law of reptiles in the macroscopic field, and the metabolic transport process at the molecular level within the cell in the microscopic field, the change rate of biological DNA, and the proportional relationship between the tumor growth rate and weight are in line with the “three-quarter power law.” Hence, this theory is opined to have the potential of “unifying the whole biology “ (Grant 2007).

The scale of the complex network composed of spinal nerve cells was assumed to be set of R and the functional electrical conduction was generated. After SCI, the set of new network r generated by the growth of the nerve cell axons to the target area could also produce certain functional electrical conduction. According to the “three-quarter power law” in the biological world, the effect of functional electrical conduction could be assumed to be related to the 3/4 power rate of the growth distance to the target area. Therefore, this paper established the following model assumptions on the mapping relationship between the micro axon growth and macro motor function.

$$Fun\left(t\right)=K_{\mathit{ur}}\left(\sum _{i=1}^{N_A}{r}_{\mathit{yi}}{\left(t\right)}^{\frac{3}{4}}\right)$$ 7

where Fun(t) and r_yi(t) were variables related to time t; N_A was the number of axons; K_ur was the proportional coefficient; r_yi was the component of nerve cell axons growing along the spinal cord.

Equation 7 constituted the core equation of this study. To understand the effects of different rehabilitation training modes on the motor function of rats more intuitively, the motor evaluation indices of the rats with SCI under different rehabilitation modes from 2 to 9 weeks were calculated according to the mapping model of three-quarter power rate, as shown in Fig. 4A. During the 2nd–4th week of treatment, the growth rate of the simulated motor function indices in the SCI, sEMS + Exercise group. During 5 weeks of treatment, the growth rate of the simulated motor function indices in the SCI groups slowed or stopped. During the 6–9 weeks of treatment, except for SCI, the simulated motor function indices of the exercise group increased at the 6th week; there was no significant change in the motor function indices of the other groups compared to the 5th week. Figure 4B is the result of exercise evaluation index of the open rat SCI research. As shown in the figure, the four motor function evaluation indices of SCI, sEMS + Exercise group in the 4 and 5 weeks were better than those of the other three groups. From the point of view of the four motor function evaluation indices in the 9 weeks, after the treatment group stopped the treatment, the motor function indices of SCI, sEMS + Exercise group were better than those of the other groups, and there were no significant changes in the motor function evaluation indexes compared to the 5 weeks.

Fig. 4.

A The simulated rehabilitation index score of the rat model and the real rehabilitation index score of the rat. B refers to research by Petrosyan et al. (2015)

Human model simulation results

Before rehabilitation training based on BCI, the human model predicts that the patient cannot improve their muscle strength. After rehabilitation training based on BCI, the human model predicted that the patient had significant muscle strength improvement after 1–2 weeks of training. After the rehabilitation training based on BCI was completed, the human model predicted the emergence of the platform period, and after that the motor function of the patient would be further improved, there were two platform periods as our model predicted.

According to the human model, we calculated the concentration distribution of growth and inhibiting factor and the growth of axon cells in the patient with SCI in 285 days. The concentrations of growth and inhibiting factor in the three platforms showed a gradient distribution. The concentration gradient of the growth and inhibitor factors jointly acted on the growth path and speed of the axon cells.

The three-dimensional representation and contour distribution of the concentration distribution of growth and inhibiting factor in the three platform periods are shown in Fig. 5, respectively. Figure 5A—a/b shows that the concentration curve of the growth in the first platform period only reached about Y axis 40. Figure 5B—a/b shows that the concentration curve of the growth factor in the second platform period reached about Y axis 53. Figure 5C—a/b shows that the concentration curve of the growth factor in the third platform period reached about Y axis 56. From the first platform period to the third platform period, the diffusion degree of the growth factor gradually increases. The growth path of axon cells in the three platform periods is shown in Fig. 5A—c/B—c/C—c, Fig. 5A—c shows that only a few axon cells bypass the glial scar in the first platform period. Figure 5B—c shows that some axon cells bypass the glial scar in the second platform period, but only a few axon cells grow to the target cells. Figure 5C—c shows that more axon cells in the third platform period bypass the glial scar and grow to the target cells. The mean and standard deviation of the growth component r_y of axon cells in the three platform periods were different (the first platform period: 7.62 ± 18.14; the second platform period: 15.01 ± 24.02; the third platform period: 21.84 ± 27.59).

Fig. 5.

Schematic diagram of growth, inhibiting factor concentration distribution and axon cell growth. A/B/C corresponds to the first/second/third platform period. Take figure A as example, A—a is the concentration distribution diagram of the promotion factor; A—b is the contour map of factor concentration; A—c is the growth chart of axon cells, where the ring represents the glial scar

The comparison between the predicted results of the human model and LEMS of the actual rehabilitation of the patient is shown in Fig. 6. As shown in the figure, the human model predicts that there are three platform periods during the rehabilitation of the patient, and the overall prediction results are highly consistent with the actual situation of the patient. Although the data of the patient after 10 weeks of training were missing due to the transfer of hospital treatment, the LEMS of actual rehabilitation of the patient after 40 weeks was consistent with the prediction results of the human model.

Fig. 6.

The LEMS of the patient’s lower limbs predicted by the human model and the real of the patient’s LEMS

Discussion

We try to apply complex system theory to SCI rehabilitation, based on the following considerations: Firstly, SCI is a process from order to disorder, and SCI rehabilitation is a process from disorder to order. The research on the evolution of the system from disorder to order is the core research field of complex systems. Secondly, after SCI, the system is a dynamic state, and constantly receiving external electrical stimulation and biochemical stimulation, which can be considered as an open system far from equilibrium. For the evolution of such systems, the framework of complex system theory should be appropriate. Finally, there is a gap between the functional evaluation of rehabilitation medicine and the microphysiological state after SCI. To establish the mapping relationship between macro-functional indicators and micro-physiological, chemical indicators and imaging examination is very worthy of further study.

Based on the above considerations, we constructed a mapping model from the microscopic process of axon regeneration of spinal cord neurons to the improvement of macro motor function through nonlinear equations, and simulated the recovery process of SCI by using the open data of rats and the clinical data of the patient.

The results from the simulation in the rat model showed that there was an improvement in the motor function in all the groups. Groups 2 and 3 have a platform period in the 4–5 weeks. During the 2–5 weeks of treatment, there was no significant difference in the motor function between the 2nd and 3rd groups at any point in time, and the 4th groups showed significant improvement in the motor function. Except for the enhancements in the motor function in the 6th week in the case of the third group, the other groups maintained their motor function for at least 4 weeks after the treatment was terminated. These results show that use of sEMS + Exercise therapy can effectively improve the cumulative motor function, while the use of sEMS/Exercise training alone has negligible effect, which is consistent with the findings of Petrosyan et al. The accurate prediction of the time point and speed of the functional improvement indicated that the rehabilitation model of SCI in rats, achieved in this study, is effective. The concentration gradient of the growth/inhibitor affects the growth of axons in the nerve cells. After the bidirectional stimulation applied to improve the microenvironment, the motor function of rats was improved. The microscopic and macroscopic model of the system conforms to the “three-quarter power law” indicating that the SCI area conforms to the basic characteristics of complex system networks, especially the complex network of biological systems (West and Brown 2005).

Using the rat model and combined with the human characteristic parameters, we constructed the human SCI model. From the prediction results of the model for the patient’s 285-day rehabilitation trend: Firstly, the prediction of the motor function changes of the patient before and after BCI rehabilitation training is highly consistent with the actual evaluation results. Secondly, the model predicts that the patient has three platform periods. According to the patient’s oral statement, he actually has at least two platform periods, and the final rehabilitation muscle strength of the patient was consistent with our model prediction. Based on the above three points, our model has good predictability for the long-term rehabilitation process of the patient.

The human model and the rat model meet the three-quarter power rate, which is consistent with the explanation of the ‘three-quarter power law’ in complex network theory. In the human model, the macroscopic variables (LEMS) and the microscopic variables (axon growth length of nerve cells) are connected by the ‘three-quarter power law’ relationship of the complex network of organisms, and the quantitative relationship between the microscopic nonlinear equation and the evolution of the macroscopic system through the complex system network theory is established. It is a valuable application of complex system theory in medicine and engineering.

The basic connotation of the model is that the complex network formed by the microscopic physiological structure determines the evolution process of the macroscopic state of the system. It is worth noting that our model is not a simple fitting of clinical data, but the model with microstructure, which is explanatory and extensible. At present, the model is still relatively rough in the specific construction of micro-model and macro-model. However, if the model is used as a framework model, through the construction of micro-scale nonlinear equations, combined with complex network theory, and the macro-functional assessment-based model mapping this overall idea will have a guiding significance for the model establishment of chronic disease evolution, body injury rehabilitation, functional growth and development.

The human model we constructed was based on the rat model, and the clinical data of the patient were used for modeling and simulation. We put forward the hypothesis that BCI can improve the microenvironment of SCI area, promote the growth of nerve cell axons, and further improve the recovery rate of motor function in patients. The successful prediction of the model on the rehabilitation process of the patient provides a certain basis for exploring the hypothesis. Although compared with the clinical results of the patient, it is preliminarily proved that the model has certain value, there are still many problems to be studied. For example, in the combined treatment of electrical stimulation and exercise training, bidirectional stimulation does not occur at the same time, but the BCI can make bidirectional stimulation affect the SCI area at the same time during the treatment.

The model we constructed can fully combine medical imaging and biochemical tests in the future. The three-dimensional space of the model and the approximate initial concentration of the injured area of the patient were determined by imaging and cerebrospinal fluid, so that the model could integrate individual differences and correct the initial conditions of the model.

Supplementary Information

Below is the link to the electronic supplementary material.

Author contributions

ZC, YL, XF, PL: designed dynamic model. ZC, WC, SZ, XF: designed Brain-computer interface system. YL, JL, SZ, XW, XW: conducted the clinical trial.

Funding

This trial was funded by the Nanjing Municipal Science and Technology Bureau (Grant No. of 2019060002). The funding bodies had no role in the study design, data collection, analysis, and interpretation of data. This project was funded by Wuxi “Taihu Talent Plan” medical and health high-level talents project (Grant No.WXTTP2020008).

Data availability

All data generated or analysed during this study are included in this published article (and its supplementary information files).

Declarations

Conflict of interest

No financial interests.