Structural Reconstruction And Fast Algorithms Of Hodgkin-Huxley Neural Network

The structural connectivity of neural networks in the brain is important to the understanding of network function but it is generally challenging to be directly measured.Meanwhile,causal inference measures are widely used as statistical techniques to detect the interactions among the network nodes(neurons).Since the neural network system is high-dimensional,nonlinear,and random,some important issues remain to be clarified for a successful reconstruction.First,what is the relationship of causal interactions among different inference methods and how these inferred causal interactions are related to the structural connectivity of networks? Besides,there are several practical constraints in actual applications,such as limited data length and hidden nodes in the network.Then,how to design an effective causal inference framework?It requires a large amount of data in the above causal inference methods.However,it is challenging to measure signals of a large-scaled neural network for a long time in experiment,and it is also hard to directly measure the ground truth of the structural connectivity.Based on these real problems,we consider the classical Hodgkin-Huxley(HH)neural network.The HH neuron model is proposed to describe the detailed generation of action potentials in the squid's giant axon,and is regarded as the foundation of all spiking neural models.However,the HH equations are highly nonlinear that one has to use numerical simulations in practice,e.g.,the Runge-Kutta(RK) methods.When an HH neuron generates an action potential(a spike),the nonlinear HH equations during the spike period become stiff and prohibit the use of large time steps for numerical integration.Standard numerical schemes such as RK methods can only allow a small time step,and the evolving will be highly inefficient.Then how to design fast algorithms to efficiently evolve HH neural network?In this work,not limited to the neural networks,we focus on nonlinear networks with pulse signals as measured output.We address the above issues based on several intensively utilized causal inference measures,i.e.,timedelayed correlation,time-delayed mutual information,Granger causality,and transfer entropy.We theoretically show the relationship among causal interactions inferred by these four statistical measures.Using spike train of HH neurons,the relationship among the causal inference methods is numerically verified,and the structure connectivity of HH neural network is successfully reconstructed.We observe that the stiffness in the HH equations are quite different,for example,between the spike and non-spike regions.Based on this phenomenon,we design fast algorithms that can maintain accurate membrane potential trajectories and accurate statistical properties of HH neurons,respectively.Compared with the RK method,these fast algorithms can use time steps one order of magnitude larger,while achieving high-precision solutions.This high accuracy and efficiency can be robustly obtained and do not depend on the dynamical regimes,connectivity structure or the network size.