Maximum Likelihood Estimation of a Stochastic Integrate-and-Fire Neural Model

Part of Advances in Neural Information Processing Systems 16 (NIPS 2003)

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Liam Paninski, Eero Simoncelli, Jonathan Pillow


Recent work has examined the estimation of models of stimulus-driven neural activity in which some linear filtering process is followed by a nonlinear, probabilistic spiking stage. We analyze the estimation of one such model for which this nonlinear step is implemented by a noisy, leaky, integrate-and-fire mechanism with a spike-dependent after- current. This model is a biophysically plausible alternative to models with Poisson (memory-less) spiking, and has been shown to effectively reproduce various spiking statistics of neurons in vivo. However, the problem of estimating the model from extracellular spike train data has not been examined in depth. We formulate the problem in terms of max- imum likelihood estimation, and show that the computational problem of maximizing the likelihood is tractable. Our main contribution is an algorithm and a proof that this algorithm is guaranteed to find the global optimum with reasonable speed. We demonstrate the effectiveness of our estimator with numerical simulations.

A central issue in computational neuroscience is the characterization of the functional re- lationship between sensory stimuli and neural spike trains. A common model for this re- lationship consists of linear filtering of the stimulus, followed by a nonlinear, probabilistic spike generation process. The linear filter is typically interpreted as the neuron’s “receptive field,” while the spiking mechanism accounts for simple nonlinearities like rectification and response saturation. Given a set of stimuli and (extracellularly) recorded spike times, the characterization problem consists of estimating both the linear filter and the parameters governing the spiking mechanism.

One widely used model of this type is the Linear-Nonlinear-Poisson (LNP) cascade model, in which spikes are generated according to an inhomogeneous Poisson process, with rate determined by an instantaneous (“memoryless”) nonlinear function of the filtered input. This model has a number of desirable features, including conceptual simplicity and com- putational tractability. Additionally, reverse correlation analysis provides a simple unbi- ased estimator for the linear filter [5], and the properties of estimators (for both the linear filter and static nonlinearity) have been thoroughly analyzed, even for the case of highly non-symmetric or “naturalistic” stimuli [12]. One important drawback of the LNP model,