Parameter estimation for a leaky integrate-and-fire neuronal model from ISI data

Abstract

The Ornstein-Uhlenbeck process has been proposed as a model for the spontaneous activity of a neuron. In this model, the firing of the neuron corresponds to the first passage of the process to a constant boundary, or threshold. While the Laplace transform of the first-passage time distribution is available, the probability distribution function has not been obtained in any tractable form. We address the problem of estimating the parameters of the process when the only available data from a neuron are the interspike intervals, or the times between firings. In particular, we give an algorithm for computing maximum likelihood estimates and their corresponding confidence regions for the three identifiable (of the five model) parameters by numerically inverting the Laplace transform. A comparison of the two-parameter algorithm (where the time constant tau is known a priori) to the three-parameter algorithm shows that significantly more data is required in the latter case to achieve comparable parameter resolution as measured by 95% confidence intervals widths. The computational methods described here are a efficient alternative to other well known estimation techniques for leaky integrate-and-fire models. Moreover, it could serve as a template for performing parameter inference on more complex integrate-and-fire neuronal models.

Statistical Models of Spike Trains

“…However, before we dive into these different approaches for computing and maximizing the likelihood, it is worth noting a surprising and important fact about the likelihood in this model: the logarithm of the likelihood turns out to be concave as a function of many of the model parameter values θ (Paninski et al, 2004c;Paninski, 2005;Mullowney and Iyengar, 2007). This makes model fitting via maximum likelihood or Bayesian methods surprisingly tractable, since concave functions lack any suboptimal local maxima that could trap a numerical optimizer.…”

“…However, the stochastic calculus leads directly to partial differential equation (Risken, 1996;Paninski et al, 2004c) or integral equation (Siegert, 1951;Buoncore et al, 1987;Plesser and Tanaka, 1997;Burkitt and Clark, 1999;DiNardo et al, 2001;Paninski et al, 2007a;Mullowney and Iyengar, 2007) methods for computing the likelihood. For example, it is well-known (Tuckwell, 1989) that the probability density of the next spike satisfies…”

“…Alternately, p(τ |θ) solves a number of integral equations of the form Siegert, 1951;Ricciardi, 1977;Plesser and Tanaka, 1997;Burkitt and Clark, 1999;DiNardo et al, 2001;Paninski et al, 2007a;Mullowney and Iyengar, 2007), where the kernel function K(τ, t) and the left-hand-side f (t) may be computed as simple functions of the model parameters θ. In the case of constant current and conductance (I(t) = I, g(t) = g), this integral equation may be solved efficiently by Laplace transform techniques (Mullowney and Iyengar, 2007); more generally, the equation may be solved by numerical integration methods (Plesser and Tanaka, 1997;DiNardo et al, 2001) or by direct matrix inversion methods (Paninski et al, 2007a) (the latter methods also lead to simple formulas for the derivative of the likelihood with respect to the model parameters; this gradient information is useful for optimization of the model parameters, and for assessing the accuracy of the parameter estimates).…”

“…In the case of constant current and conductance (I(t) = I, g(t) = g), this integral equation may be solved efficiently by Laplace transform techniques (Mullowney and Iyengar, 2007); more generally, the equation may be solved by numerical integration methods (Plesser and Tanaka, 1997;DiNardo et al, 2001) or by direct matrix inversion methods (Paninski et al, 2007a) (the latter methods also lead to simple formulas for the derivative of the likelihood with respect to the model parameters; this gradient information is useful for optimization of the model parameters, and for assessing the accuracy of the parameter estimates). Thus, to summarize, stochastic process methods may be used to derive efficient schemes for computing the likelihood in this model.…”

Optimal Design for Estimation in Diffusion Processes from First Hitting Times

“…Partly due to the applications of this problem in theoretical neuroscience, a lot has been written on the subject, for example in [23,7,13,18], as well as in our own work [11], where we deal with the the more exotic case of a sinusoidal driving force. The literature on estimation from hitting times of the class of diffusion models we consider asserts that one parameter in particular, the characteristic time (i.e., the time constant of the LIF model), is the most difficult to estimate and has by far the widest confidence bounds.…”