# Parameter Estimation of a Class of Neural Systems with Limit Cycles
## 1 Introduction
### Parameter Estimation in Neural Model  
System parameter estimation has received substantial attention in several decades [3; 4; 5; 6; 7]. One reason is its broad application to mechanical systems [8; 9], neuron systems [10], communication systems [11], structural systems [12], power systems [13], etc. Existing black-box parameter identification methods for nonlinear systems are simple and universal, but the identification accuracy and convergence of the identification algorithm are not of universal significance. Therefore, accurately identifying the system parameters of the spiking neuron models under finite measurement presents a far more challenging modeling problem. Examples of parameter identification methods for neural modelsinclude the time/frequency-domain method [14], maximum likelihood method [15], self-organizing state-space-model approach [16], heuristic optimization method [10], and so on. In [14], a time-domain method and a frequency-domain method are applied to estimate parameters of a neuronal model consisting of a soma coupled to a uniform dendritic cylinder, where the time-domain method is more prone to estimation errors in the cable parameters. [PERSON] et al. [16] presented an adaptive sampling algorithm using the self-organizing state-space model to estimate parameters in a Hodgkin-Huxley-type model of single neurons and to achieve reduced variance of parameter estimates. [PERSON] et al. [15] proposed a maximum likelihood methodology for parameter estimation in a leaky integrate-and-fire neuronal model with the only available data being the interspike intervals or the times between firings. In [10], a global heuristic search method using in-vitro and in-vivo electrophysiological data was explored for identification of an Izhikevich-type neuron model. In spite of these advances, little attention on parameter estimation of the FitzHugh-Nagumo (FHN) model has been received, including Bayesian statistical approaches [17; 18], and least-squares algorithms [19; 20]. In [17], a Bayesian framework was proposed for drift parameter estimation of the stochastic FHN model. [PERSON] and [PERSON] [18] employed nonlinear filtering for periodic, time-varying parameter estimation, which was illustrated in estimation of the external voltage parameter in the FHN model. In comparison, from the recursive estimation point of view, we perform parameter inference for the FHN model with data generated from the model in this paper. [PERSON] and [PERSON] [19] presented two methodologies based on the least-squares algorithm to estimate the parameters of the FHN model. The two methods were only suitable for the case with noncontinuous input current stimulus, at the cost of a linear integral filter. [PERSON] et al. [20] employed the recursive least-squares algorithm for parameter estimation of the FHN model, which requires the first and second time derivatives of the membrane potential and the input current stimulus being continuously differentiable. It is well known that the stochastic gradient algorithm is a class of important stochastic approximation methods, which have received much attention and have been widely used in different systems, such as Hammerstein systems [21], Wiener systems [22] and sampled systems [23]. Though the stochastic gradient algorithm has a slower convergence rate than the least-squares algorithms, but it requires less computational effort. In this paper, to improve the convergence rate of the stochastic gradient algorithm, we extend the innovation concept in [24] and explore the multiinnovation stochastic gradient algorithm for parameter estimation of the FHN neuron system. The parameterized FHN model in [20] can also be performed by using the proposed multiinnovation recursive least-squares algorithm in this paper and a better accuracy of the parameter estimation will be obtained due to the use of past innovations and repeated available data.