(725c) A Quantitative Assessment of the Izhikevich Neuron Model Against Experimental Data

Neurons communicate with each other in a network by generating and transferring action potentials. The exact phenomenon through which these action potentials encode the information contained in presynaptic activities is still unknown. To understand this phenomenon of neural coding using rigorous mathematical analysis is an important problem in computational and theoretical neuroscience. One of the challenges in attempting such analysis is to develop computationally efficient mathematical models of neurons which can predict action potentials observed in experimental conditions, both qualitatively and quantitatively. Recent efforts towards developing single neuron models show the capability of mathematically simplified low-order models in capturing broad range of qualitative behavior of biological neurons. The obvious benefits of using these low-order models in the analysis of network of neurons for neural coding are their mathematical simplicity and existence of small number of model parameters which can easily be extracted from experimental data. Other benefits are in implementing neural prostheses where these models can be simulated at low costs. In this work, we analyze the quantitative predictive capability of one such model, the Izhikevich single neuron model [1], using the benchmark set of experimental data available from “The quantitative single-neuron modeling competition” [2,3,4]. Our analysis reveals the prospective applicability of this model in aforementioned applications. In addition, it may serve as a benchmark comparison of the performance of this model against other existing models.

The dynamical behavior of the Izhikevich single neuron model is expressed by two coupled non-linear ordinary differential equations with state-based reset conditions. Existence of these reset conditions shows discontinuities in the membrane potential trajectory predicted by the model. To make a quantitative assessment of the model, a part of the data can be used to estimate unknown model parameters and the rest of the data can be used to validate the model predictability against the experimental data. To estimate unknown parameters of the model, we introduce stochasticity in the model and take the well-known approach of maximum likelihood estimation (MLE). The objective here is to maximize the joint probability distribution of the occurrence of an experimentally observed sequence of action potentials over these unknown parameters. For computing the joint probability distribution, we derive the two dimensional Fokker-Planck equation [5], a convective-diffusive (hyperbolic-parabolic) partial differential equation, using the stochastic Izhikevich model. For computational efficiency, we approximate the two dimensional Fokker-Planck equation with a one dimensional Fokker-Planck equation coupled with two ordinary differential equations. We solve the Fokker-Planck equation at the time of occurrence of action potentials, in a given sequence of action potentials, by implementing the method of lines with a 5-point upwind scheme [6].  Finally, estimation of parameters is performed by computing and maximizing the joint probability distribution of the first passage times (the time of occurrence of action potentials) using the MATLAB optimization toolbox.

In the presentation, we will first show the validation of the MLE approach used in estimating model parameters against simulated data sets. For this, we simulate the stochastic Izhikevich model with given parameters and generate 50 realizations of spike trains, each consisting of 300 action potentials. We estimate 9 model parameters of the Izhikevich model using the first 200 action potential timings for 100 realizations. Average values of parameters over these 50 realizations are then used to predict the next 100 action potential timings in all realizations. This establishes the validation and efficacy of the MLE approach used in estimating model parameters. Next, we show the efficacy of the model predictability against experimental data available from the competition. For this, we use two-third of the data (200 action potentials) in each data set of available 13 data sets from the competition to estimate 9 model parameters of the Izhikevich model. Average values of parameters over 13 data sets are then used to predict timings of the rest of the one-third of the data in each data set. Finally, we demonstrate the efficacy of the model in predicting experimentally observed action potentials by making the benchmark comparison stated in the competition.

1. Izhikevich, E. M. Dynamical systems in neuroscience. The MIT Press, 2007.
2. Jolivet et.al. The quantitative single-neuron modeling competition. Biol Cybern, vol. 99, 417-426, 2008.
3. http://www.incf.org/community/competitions/spike-time-prediction/2009 : Quantitative single-neuron modeling 2009.
4. Gerstner et.al. How good are neuron models? Science, vol. 326, no. 5951, 379-380, 2009.
5. Gihman and Skorohod. Stochastic differential equations. Springer, 1972.
6. Schiesser and Griffiths. A compendium of partial differential equation models. Cambridge University Press, 2009.