In this article, we propose a 6N-dimensional stochastic differential equation (SDE), modelling the activity of N coupled populations of neurons in the brain. This equation extends the Jansen and Rit neural mass model, which has been introduced to describe human electroencephalography (EEG) rhythms, in particular signals with epileptic activity. Our contributions are threefold: First, we introduce this stochastic N-population model and construct a reliable and efficient numerical method for its simulation, extending a splitting procedure for one neural population. Second, we present a modified Sequential Monte Carlo Approximate Bayesian Computation (SMC-ABC) algorithm to infer both the continuous and the discrete model parameters, the latter describing the coupling directions within the network. The proposed algorithm further develops a previous reference-table acceptance rejection ABC method, initially proposed for the inference of one neural population. On the one hand, the considered SMC-ABC approach reduces the computational cost due to the basic acceptance-rejection scheme. On the other hand, it is designed to account for both marginal and coupled interacting dynamics, allowing to identify the directed connectivity structure. Third, we illustrate the derived algorithm on both simulated data and real multi-channel EEG data, aiming to infer the brain's connectivity structure during epileptic seizure. The proposed algorithm may be used for parameter and network estimation in other multi-dimensional coupled SDEs for which a suitable numerical simulation method can be derived.

翻译：本文提出了一个6N维随机微分方程（SDE），用于模拟大脑中N个耦合神经元群体的活动。该方程扩展了Jansen和Rit神经质量模型——该模型最初用于描述人类脑电图（EEG）节律，特别是具有癫痫活动的信号。我们的贡献有三方面：首先，引入该随机N群体模型并构建可靠高效的数值模拟方法，扩展了针对单一神经群体的分裂过程。其次，提出改进的序贯蒙特卡洛近似贝叶斯计算（SMC-ABC）算法，用于推断连续和离散模型参数，其中离散参数描述网络内的耦合方向。该算法进一步改进了先前基于参考表的接受-拒绝ABC方法，该方法最初用于单一神经群体的推断。一方面，所提出的SMC-ABC方法减少了基本接受-拒绝方案的计算成本；另一方面，其设计同时考虑了边缘动力学和耦合交互动力学，从而能够识别有向连接结构。第三，我们通过模拟数据和真实多通道脑电图数据验证所提算法，旨在推断癫痫发作期间大脑的连接结构。该算法可推广至其他可构建合适数值模拟方法的多维耦合随机微分方程中的参数与网络估计。