Integrate-and-fire (IF) model neurons ignore the rapid spike generation process, instead emphasizing sub-threshold integration dynamics (Burkitt, 2006
). IF neurons are thus more realistic than the simplest neural models, like McCulloch-Pitts neurons or coincidence detectors (McCulloch & Pitts, 1943
; Mikula & Niebur, 2003
), which makes them a computationally efficient model that can predict spiking times recorded in some physiological experiments reasonably well (Kim, Sripati, Vogelstein, Armiger, Russell, & Bensmaia, 2009
). Extensions of the classical integrate-and-fire model, introduced more than a century ago (Lapicque, 1907
), expand the applicability of the model to more biophysically realistic firing patterns (Izhikevich, 2003
; Brette & Gerstner, 2005
; Tonnelier, Belmabrouk, & Martinez, 2007
; Mihalas & Niebur, 2009
In the traditional IF neuron, the membrane voltage is reset to a predetermined value after each spike, and each interspike interval is independent of the previous. This lack of history dependence limits the complexity of the spiking behavior the model can generate. For instance, adaptation and bursting behavior are found in many biological neurons but cannot be explained by the classical IF neuron. Mihalas and Niebur (2009)
extended the IF neuron model by adding a variable threshold and an arbitrary number of spike-induced currents (Hille, 1992
), all with linear dynamics. This generalized integrate-and-fire neuron is capable of producing spiking/bursting, tonic, phasic or adapting responses, depolarizing and/or hyperpolarizing afterpotentials etc
.. The equations governing the time evolution of the membrane potential are solved analytically between spiking, allowing (but not requiring) event-based implementations of the model. Furthermore, the analytical solution is obtained very efficiently since the model dynamics can be written as a diagonalizable set of linear differential equations. In this study, we focus on this generalized leaky IF (LIF) model, because it is the only one of the cited extensions of the LIF models that maintains linearity of the state variable dynamics.
An important problem in computational neuroscience is the determination of the parameters of a neuronal model given the available experimental data (Prinz, 2007
). We assume that only spike trains (i.e.
, the sequence of time points when a neuron spiked) are available but not voltage traces; this is the case in extracellular recordings. Parameter optimization is then accomplished by minimizing a suitably chosen cost function. Choosing a cost function is not easy; for instance, using the benchmark (the “Г− factor”) for spike-train prediction proposed by the INCF Quantitative Single-Neuron Modeling 2009 competition
as the cost function (Jolivet, Kobayashi, Rauch, Naud, Shinomoto, & Gerstner, 2008
) results in large numbers of local minima. Finding the global minimum then requires substantial computational resources (Rossant, Goodman, Platkiewicz, & Brette, 2010
A natural cost function for parameter fitting is the maximum likelihood estimator of the observed sequence of spike times. Paninski et al. (2004)
showed that the negative log-likelihood function of the stochastic LIF neuron model is convex and that its unique global minimum can thus be reached using gradient descent techniques.
As mentioned, the complexity of biological spike trains exceeds what can be generated by the stochastic LIF neuron but the generalized LIF model can generate a much richer set of behaviors (Mihalas & Niebur, 2009
). Assuming that noise appears only in the threshold, we have shown that the likelihood function of the generalized LIF model can be computed by solving the one-dimensional Fokker-Planck equation (Russell et al, in press
). In the present study, we first generalize that result by deriving the Fokker-Planck equation of the likelihood function with noise in both membrane voltage and threshold. For computational efficiency, we then derive a reduced, one-dimensional Fokker-Planck equation for the likelihood function, in this case assuming noise only in the membrane voltage. We then show that numerical accuracy is substantially improved by computing the likelihood function of the generalized LIF model by the Volterra integral equation method (Paninski, Haith, & Szirtes, 2008
). Convexity of the negative log-likelihood function is not guaranteed after inclusion of the variable threshold but we show that in practice, common optimization methods converge for this generalized LIF neuron.