(698e) Optimal Parameter Estimation of Stochastic Izhikevich Single Neuron Model Using Experimental Inter-Spike Interval Data

A control-theoretic approach of a neuroprosthetic system requires an appropriate mathematical model for representing cortical neurons. It is a well known fact in neuroscience that the central nervous system carries information in the form of action potential trains [1] generated by neurons. More specifically, the time between two action potentials, or the Inter-Spike-Interval (ISI), carries most of the neural information [2]. This suggests a need for an appropriate single neuron model that can predict these action potential intervals or ISIs reasonably [3], and is a computationally efficient building block for subsequent control-theoretic analysis of the closed-loop neuroprosthetic system. To choose an appropriate neuron model for control-theoretic study of a neuroprosthetic system, it is necessary to first validate the efficacy of these models using experimentally recorded single neuron data [4]. Also, estimation of several unknown parameters within these models using experimental data is necessary for making use of these models in a closed loop context.

In most experimental studies, ISIs are the only available experimental data to validate a model as well to estimate model parameters. One of the major challenges in the estimation of model parameters and model validation using ISIs is the large variability obtained in the spike intervals, as well as the lack of information about synaptic input currents. Very few results have been reported for estimating model parameters using ISI data only [5, 6] for the leaky integrate-and-fire model. These results use model generated ISIs for validating the model parameters estimation, and are limited to ISIs that varies in a very small range. In experimentally recorded data, however ISI variations are typically large and impose a practical limitation on these methods. Recently, experimentally recorded ISIs from a primate study have been used to validate and estimate model parameters of the Izhikevich single neuron model [7]. However in this work, it has been assumed that the synaptic currents are deterministic in nature.

Experimental evidences shows that the synaptic input currents to motor cortex neurons possess stochastic characteristics [8]. To capture this behavior, we propose to build upon our previous deterministic model and incorporate the stochastic nature of synaptic currents in the existing Izhikevich single neuron model for theoretical validation of single neuron dynamics using experimental ISIs data. For this purpose, we use ISIs data recorded from a single cortical neuron from a primate study [9]. To the best of our knowledge, this is the first time this stochastic model is validated and optimal model parameters are estimated using experimental ISIs data from a primate study.

In order to estimate model parameters, we represent synaptic inputs in the form of standard Weiner process with time invariant mean and variance. Based on this, we define a first passage time to represent model estimated ISIs and thus formulate a well known first passage time problem. In order to estimate model parameters using experimental ISIs, we solve the first passage time problem by maximizing a log maximal likelihood function over unknown model parameters as well as synaptic inputs. We define the maximal likelihood function as the product of the first passage time probability density function over the number of available ISIs. To estimate the first passage time probability density function, we apply the Itô formula [10] to the Izhikevich model and compute the transition probabilities numerically. With this, we solve the nonlinear constrained optimization problem of log maximal likelihood function by implementing primal-dual interior point method in MATLAB and estimate model parameters as well as synaptic input currents. Further, we estimate confidence intervals for our estimated model parameters. Estimation of reasonable model parameters using this method may serve as a template for studying and developing a model of ensemble cortical neurons for neuroprosthesis applications.

[1] Kandel et.al. Principles of Neural Science. Mc Graw Hill, 2000.

[2] M. Abeles. Time is precious. Science, Vol. 304, 523-524, 2004.

[3] Jolivet et.al. Special issue on quantitative neuron modeling. Biological Cybernatics, Vol. 99, 237-239, 2008.

[4] Jolivet et.al. Special issue on quantitative neuron modeling. Biological Cybernatics, Vol. 99, 417-426, 2008.

[5] S. Ditlevsen and P. Lansky. Parameters of stochastic diffusion processes estimated from observations of first-hitting times: Application to the leaky integrate-and-fire neuronal model. Physical Review E., Vol. 76, 1-5, 2007.

[6] P. Mullowney and S. Iyengar. Parameter estimation for a leaky integrate-and-fire neuronal model from ISI data. J. Comput. Neurosci., Vol. 24, 179-194, 2008.

[7] Kumar et.al. Optimal parameter estimation of the Izhikevich single neuron model using experimental ISI data. Proceedings of the American Control Conference, Baltimore, MD, 2010.

[8] Faisal et.al. Noise in nervous system. Nature Reviews, Vol. 9, 292-303, 2008.

[9] Schieber, M.H. and Hibbard, L.S. How somatotopic is the motor cortex hand area? Science 261:489-492, 1993.

[10] J. Michael Steele. Stochastic Calculus and Financial Applications. Springer, 2001.