Maximum Likelihood Parameter Estimation in a Stochastic Resonate-and-Fire Neuronal Model Jun Chen * , Jose Suarez † , Peter Molnar ‡ and Aman Behal § * Department of Electrical Engineering and Computer Science University of Central Florida, Orlando, Florida 32817 Email: lay2008@knights.ucf.edu † Department of Electrical Engineering and Computer Science University of Central Florida, Orlando, Florida 32817 Email: jnsuarez@knights.ucf.edu ‡ Department of Zoology, Division of Natural Sciences University of West Hungary, Savaria Campus, Karolyi Gaspar ter 4. Szombathely, H-9700, Hungary Email: pmolnar@pminfonet.com § Department of EECS and NanoScience Technology Center University of Central Florida, Orlando, Florida 32817 Email: abehal@mail.ucf.edu Abstract—Recent work has shown that resonate-and-fire model is both computationally efficient and suitable for large network simulations. In this paper, we examine the estimation problem of a resonate-and-fire model with random threshold. The model parameters are divided into two sets. The first set is associated with subthreshold behavior and can be optimized by a nonlinear least squares algorithm. The other set contains threshold and reset parameters and its estimation is formulated in terms of maximum likelihood formulation. We evaluate such a formulation with detailed Hodgkin-Huxley model data. Keywords-resonate-and-fire; parameter estimation; maxi- mum likelihood; simulated annealing. I. I NTRODUCTION One essential issue in computational neuroscience is to characterize the relationship between neural output recording and the input current to the cell, [1]. To reproduce the behav- ior of neurons, the spiking neuron model has been widely used. Although a detailed Hodgkin-Huxley [2] neuron model could mimic the neuronal dynamics more accurately, it is computationally inefficient in parameter estimation and model simulation. Therefore, spiking model is the first choice for large network simulation. In [3], a second order model is introduced to be able to reproduce almost all types of firing patterns in-vivo and keep the computation efficient at the same time. In authors’ previous work, a linear- in-the-parameters presentation of aforementioned quadratic model is developed and allows us to identify experimentally obtained data. However, a small flaw in the assumptions prevents us from further research. Although the quadratic model is proven to be biologically meaningful, it can only qualitatively reproduce the firing pattern, i.e., the model can not quantitatively represent the upstroke of the spike unless it is assumed that the parameters are voltage-dependent [4]. To address this issue, a modification is made to this model. Instead of assuming voltage-dependent parameters, we remove the quadratic term in the model and consider it to be a representation only for subthreshold dynamics. Since now we narrow our interest zone down to the subthreshold region, the mismatch in the upstroke does not affect our identification any more. Another reason to consider sim- plifying the model instead of additional assumption is, by the removal, the model dynamics become linear and can be treated analytically, without hurting the ability to replicate multiple firing patterns. In order to match the experimentally observed data to a particular parametric model , many parameter estimation methods have been introduced for estimating parameter values from the recorded voltage trace. In [5], an adaptive exponential integrate-and-fire neuron model is manually hand-tuned to fit a detailed Hodgkin-Huxley based model. Although the result in [5] shows good match between the proposed model and the detailed Hodgkin-Huxley based model, such trial-and-error approach depends mainly on researcher’s experience and thus is labor-intensive. In [6], a database of single-compartment model neurons is con- structed by exploring the entire parameter space – this approach is only practical when the parameter space has a low dimension. In [1], a maximum likelihood problem is formulated with a stochastic integrate-and-fire model. The maximum likelihood formulation can be also found in other literature, e.g., [7]. In this paper, we consider a resonate-and-fire model with stochastic threshold which is assumed to be a Gaussian random variable. By assuming that the threshold is the only component that has a stochastic property, the membrane potential is therefore deterministic and solvable. Here, we