An improved multi-swarm particle swarm optimizer for optimizing the electric field distribution of multichannel transcranial magnetic stimulation

Highlights

• In order to optimize the electric field distribution in the brain to meet the treatment expectations of multichannel transcranial magnetic stimulation (mTMS), this paper presents a novel multi-swam particle swarm optimizer (NMSPSO) to optimize the current configuration of coil array in mTMS.

• Three improvement strategies are used in the NMSPSO algorithm to balance the exploration and exploitation abilities. These three strategies and their role are as follows: (a) Information exchange strategy: Guarantee the rational flow of information in the population. (b) Learning strategy: Tradeoff between population diversity and convergence rate. (c) Mutation strategy: Enable the population to quickly jump out of the local optimal solution.

• NMSPSO algorithm is examined on a set of well-known benchmark functions and the results show that the NMSPSO has better performance than many particle swarm optimization variants.

• By optimizing the current configuration of the coil array in mTMS with NMSPSO algorithm, the electric field can be focused at the target and the electric field focusing degree can be improved compared to several other PSO variant algorithms.

Abstract

Multichannel transcranial magnetic stimulation (mTMS) is a therapeutic method to improve psychiatric diseases, which has a flexible working pattern used to different applications. In order to make the electric field distribution in the brain meet the treatment expectations, we have developed a novel multi-swam particle swarm optimizer (NMSPSO) to optimize the current configuration of double layer coil array. To balance the exploration and exploitation abilities, three novel improved strategies are used in NMSPSO based on multi-swarm particle swarm optimizer. Firstly, a novel information exchange strategy is achieved by individual exchanges between sub-swarms. Secondly, a novel leaning strategy is used to control knowledge dissemination in the population, which not only increases the diversity of the particles but also guarantees the convergence. Finally, a novel mutation strategy is introduced, which can help the population jump out of the local optimum for better exploration ability. The method is examined on a set of well-known benchmark functions and the results show that NMSPSO has better performance than many particle swarm optimization variants. And the superior electric field distribution in mTMS can be obtained by NMSPSO to optimize the current configuration of the double layer coil array.

Introduction

Transcranial magnetic stimulation (TMS) is a non-invasive, painless technique [1,2], which can be used to treat depression, epilepsy and other neurological diseases. The principle of it is that the pulse magnetic field generator produces a variable field, and the magnetic field lines go through the scalp, skull and brain tissue. Then the pyramidal neurons are activated due to current induced by pulse magnetic field in the brain. In 1980s, Baker et al. [3] successfully stimulated the cerebral cortex by magnetic field, demonstrating the feasibility of TMS.

Conventional TMS coils such as circular and figure-8 coil have been widely used commercially, but their common disadvantage is that they need to be manually moved until the desired response is achieved. Since Ruohonen and Ilmoniemi [4,5] presented the theory of stimulus targeting with multi-coils in 1998, the multichannel transcranial magnetic stimulation (mTMS) has attracted widespread attention. The electric field can be shaped and targeted in mTMS without the coil being moved because the electric field value is jointly generated by a plurality of coils. However, due to too many coils, how to determine the coil array current configuration so that the coil array could induce appropriate electric field to stimulate the preset areas in the cortex and improve the focus of the electric field is still the key point for mTMS [6,7]. To solve this problem, this paper introduces an improved particle swarm optimization (PSO) algorithm to optimize the current configuration of the double layer coil array to obtain a better electric field distribution.

PSO is one of nature-inspired evolutionary algorithms proposed by Kennedy and Eberhart in 1995, which imitates the foraging behavior of bird flocks [8,9]. The individual in a population is called a particle without mass and volume, and is used to perform a search of the solution space. During the optimization process, particle adjusts their flight speed and position according to their historical best position and the global best position. Although the search pattern of each particle is quite simple, the interaction between individuals makes the search behavior of the population intelligent and efficient. It has been widely applied to solve many scientific and real-world problems for the benefit of ease of implementation and effectiveness [[10], [11], [12]].

Similar to the other nature-inspired evolutionary algorithms, PSO also has a lot of limitations in solving complex multimodal problems [13,14]. Exploration (global search) and exploitation (local search) are the two important factors that influence the performance of the PSO [15,16]. Exploration means that the particle searches around the current optimal solution to find the better, while it is easy to cause premature convergence and trap the population to a local optimum. Exploitation means that the population has a high explore in the search space, while excessive exploration wastes computational resources and reduce the algorithm effectiveness. Therefore, it is difficult for PSO to tradeoff between exploration and exploitation.

To address these problems mentioned above, several variants of PSO were developed. Shi and Eberhaet [17] introduced a new parameter inertia weight $\omega$ and linearly changes during the optimization process to dynamic adjustment the relationship between exploration and exploitation. In 1999, Kennedy [18] analyzed the impact of different neighborhood topologies on PSO performance and found that the random connected topology was better than other topologies in solving complex multimodal problems. Liang and Suganthan [19] proposed a dynamic multi-swarm particle swarm optimizer (DMSPSO), the population is randomly divided into multiple sub-swarms and every regrouping period the population regroups randomly for multiple sub-swarms to exchange information. Higashi et al. [20] utilized Gaussian mutation to improve the spatial search ability of particles in PSO, which is used in other nature-inspired evolutionary algorithms. Wang et al. [21] proposed an adaptive mutation strategy for PSO to dynamically choose Gaussian mutation, Cauchy mutation and Lévy mutation during the evolution. Although these PSO variants have better global search capabilities and higher convergence accuracy than traditional PSO algorithms, the performance of tradeoff between the exploitation and exploration is still unsatisfactory for solving complex optimization problems.

In this paper, a novel multi-swarm particle swarm optimizer (NMSPSO) is proposed. Three novel improvement strategies are introduced in NMSPSO, the first strategy is that the population is divided into many sub-swarms and the information transfer between sub-swarms is achieved by individual exchanges between each other along with the evolutionary process. The second strategy is to assign different neighborhood topologies to particles with different fitness values in sub-swarm, which not only improves the population diversity but also guarantees the convergence of the population. The last is a novel mutation strategy, which can help the population jump out of the local optimum for better exploration ability.

The paper structure is arranged as follows. In Section 2, the optimization problem of electric field distribution and PSO algorithm are described. In Section 3, we introduce the three improvement strategies and framework of NMSPSO algorithm. In Section 4, some parameter settings and performance of three improved strategies of NMSPSO was discussed, and the optimization performance of NMSPSO algorithm was verified by comparison with many PSO variants based on a set of well-known benchmark functions. In Section 5, the performance of NMSPSO algorithm in optimizing the current configuration of the double layer coil array was verified. Finally, some conclusions and future work are presented in Section 6.