Dynamics and coupling of fractional-order models of the motor cortex and central pattern generators

Abstract

Objective. Fractional calculus plays a key role in the analysis of neural dynamics. In particular, fractional calculus has been recently exploited for analyzing complex biological systems and capturing intrinsic phenomena. Also, artificial neural networks have been shown to have complex neuronal dynamics and characteristics that can be modeled by fractional calculus. Moreover, for a neural microcircuit placed on the spinal cord, fractional calculus can be employed to model the central pattern generator (CPG). However, the relation between the CPG and the motor cortex is still unclear. Approach. In this paper, fractional-order models of the CPG and the motor cortex are built on the Van der Pol oscillator and the neural mass model (NMM), respectively. A self-consistent mean field approximation is used to construct the potential landscape of the Van der Pol oscillator. This landscape provides a useful tool to observe the 3D dynamics of the oscillator. To infer the relation of the motor cortex and CPG, the coupling model between the fractional-order Van der Pol oscillator and the NMM is built. As well, the influence of the coupling parameters on the CPG and the motor cortex is assessed. Main results. Fractional-order NMM and coupling model of the motor cortex and the CPG are first established. The potential landscape is used to show 3D probabilistic evolution of the Van der Pol oscillator states. Detailed observations of the evolution of the system states can be made with fractional calculus. In particular, fractional calculus enables the observation of the creation of stable modes and switching between them. Significance. The results confirm that the motor cortex and CPG have associated modes or states that can be switched based on changes in the fractional order and the time delay. Fractional calculus and the potential landscape are helpful methods for better understanding of the working principles of locomotion systems.

A new biological central pattern generator model and its relationship with the motor units

“…Next, we introduce a fractional-order neuron model. In this work, we use the Grunwald-Letnikov (G-L) fractional derivative (Lu 2019(Lu , 2020, and obtain the following fractional-order CPG model with synaptic transmission,…”

“…Nevertheless, although the mammalian locomotion networks have been widely studied, understanding the coupling association in these networks is still an open problem (Kiehn 2016;Roberts et al 2012). In addition, the relationship between the CPG and the motor cortex has been explored in earlier studies based on the integral-order models (Lu et al 2015;Lu and Tian 2014) and the fractional-order ones (Lu 2020). The previous investigations showed that there exists parameters space which can make the synchronization of the motor cortex optimum when the CPG is added into the motor cortex.…”

Numerical simulation of a fractional stochastic delay differential equations using spectral scheme: a comprehensive stability analysis

“…There is a well-established exercise of assimilating delays into dynamic systems to narrative for phenomena such as biological reaction times, finite propagation speeds, and communication delays in control systems 11 , 12 . A rich array of mathematical challenges and opportunities arises from the interaction between fractional calculus, stochasticity, and time delays 13 , 14 . As well as for theoretical purposes, understanding the behavior of FSDDS has practical applications in fields such as biology, ecology, economics, finance, and engineering 15 .…”

A Novel Double-Layered Central Pattern Generator-Based Motion Controller for the Hexapod Robot

“…Meanwhile, the CPG-based method can control various forms of locomotion, perform smooth transition, and easily integrate abundant sensing signals [17,18]. Fractional calculus and the investigation of fractional-order systems have been extensively studied in the last decades [19][20][21]. Fractional central pattern generator (CPG) models for locomotion systems have been researched in the reference [22,23].…”