Identifying the pulsed neuron networks' structures by a nonlinear Granger causality method

Abstract

Background: It is a crucial task of brain science researches to explore functional connective maps of Biological Neural Networks (BNN). The maps help to deeply study the dominant relationship between the structures of the BNNs and their network functions.

Results: In this study, the ideas of linear Granger causality modeling and causality identification are extended to those of nonlinear Granger causality modeling and network structure identification. We employed Radial Basis Functions to fit the nonlinear multivariate dynamical responses of BNNs with neuronal pulse firing. By introducing the contributions from presynaptic neurons and detecting whether the predictions for postsynaptic neurons' pulse firing signals are improved or not, we can reveal the information flows distribution of BNNs. Thus, the functional connections from presynaptic neurons can be identified from the obtained network information flows. To verify the effectiveness of the proposed method, the Nonlinear Granger Causality Identification Method (NGCIM) is applied to the network structure discovery processes of Spiking Neural Networks (SNN). SNN is a simulation model based on an Integrate-and-Fire mechanism. By network simulations, the multi-channel neuronal pulse sequence data of the SNNs can be used to reversely identify the synaptic connections and strengths of the SNNs.

Conclusions: The identification results show: for 2-6 nodes small-scale neural networks, 20 nodes medium-scale neural networks, and 100 nodes large-scale neural networks, the identification accuracy of NGCIM with the Gaussian kernel function was 100%, 99.64%, 98.64%, 98.37%, 98.31%, 84.87% and 80.56%, respectively. The identification accuracies were significantly higher than those of a traditional Linear Granger Causality Identification Method with the same network sizes. Thus, with an accumulation of the data obtained by the existing measurement methods, such as Electroencephalography, functional Magnetic Resonance Imaging, and Multi-Electrode Array, the NGCIM can be a promising network modeling method to infer the functional connective maps of BNNs.

Keywords: Integrate-and-fire model; Network structure identification; Nonlinear granger causality; Radial basis function.

Fig. 1

A 6-mode SNN simulation (a) 6-node SNN’s structure (b) multivariate response data generated by the network simulation (after sampling the pulse sequences of the neurons)

Fig. 2

Identification results of 6 node network structure by the LGCIM and the NGCIM. a The network connection structure identified by the NGCIM with the Gaussian kernel function. b The network connection structure identified by the LGCIM

Fig. 3

A structure of an RBF network

Fig. 4

A schematic drawing of the RBF learning process

Fig. 5

Schematic drawing of conditional causality. For the Granger causalities analysis from y to x, there is a direct causality and an indirect causality via z. All direct connections are denoted by solid lines, and indirect connections are represented by dash lines