Stability of point process spiking neuron models

Abstract

Point process regression models, based on generalized linear model (GLM) technology, have been widely used for spike train analysis, but a recent paper by Gerhard et al. described a kind of instability, in which fitted models can generate simulated spike trains with explosive firing rates. We analyze the problem by extending the methods of Gerhard et al. First, we improve their instability diagnostic and extend it to a wider class of models. Next, we point out some common situations in which instability can be traced to model lack of fit. Finally, we investigate distinctions between models that use a single filter to represent the effects of all spikes prior to any particular time t, as in a 2008 paper by Pillow et al., and those that allow different filters for each spike prior to time t, as in a 2001 paper by Kass and Ventura. We re-analyze the data sets used by Gerhard et al., introduce an additional data set that exhibits bursting, and use a well-known model described by Izhikevich to simulate spike trains from various ground truth scenarios. We conclude that models with multiple filters tend to avoid instability, but there are unlikely to be universal rules. Instead, care in data fitting is required and models need to be assessed for each unique set of data.

Appendices

Appendix A: Datasets

Table 1 Details of the datasets used in this paper

Appendix B: Diagnostic Maps

Fig. 9

Stability maps for two FLF models (Eq. (4)) F_θ using the diagnostic of Gerhard et al. (2017) and our updated diagnostic. For each value of θ, we (i) simulated a 10 sec. long spike train from F_θ, and deemed the model unstable if it generated over 900 spikes in the last second, (ii) produced the diagnostic curve and determined from it if the model was stable/fragile/divergent, and (iii) plotted θ against the outcomes in (i) and (ii). a Reproduction of the stability map in Gerhard et al. (2017) Fig. 4, where F_θ is an FLF model with β₀ = − 5.3 and h(t) = β₁ ⋅B₁(t) + β₂ ⋅B₂(t) + Dip(t), where B₁(t) = e^−t/0.02, B₂(t) = e^−t/0.1 and Dip(t) is a negative window function modeling a 2 msec. refractory period, θ = (β₁, β₂), and filter length T_h = 0.2 sec. The maps suggest that the diagnostic is mostly reliable, except in small regions of the parameter spaces. b Our updated diagnostic for the same model matches the simulation better. c Stability map using the diagnostic of Gerhard et al. (2017) for F_θ an FLF model with with β₀ = − 4, h(t) = β₁ ⋅B₁(t) + β₂ ⋅B₂(t). Basis B₁(t) and B₂(t) are the same as Fig. 2a. θ = (β₁, β₂), and filter length T_h = 0.35 sec. d Our updated diagnostic for the same model matches the simulation better

Appendix C: Simulation algorithms

Algorithms 1, 2, and 3 generate Izhikevich-xx, FLF, and FNF datasets, respectively.

Appendix D: Misc. results

Derivation of Eq. (7)

$$\begin{array}{@{}rcl@{}} &&\sum\limits_{t_{j*} \in (t-T_{h}, t_{1*})} h(t - t_{j*})\\ &&\approx \mathop{\mathbb{E}}_{N} \left[ {\int}_{t- T_{h}}^{t_{1*}} h(t - \tau) \mathrm{d} N(\tau) \right] \end{array} $$

(10)

$$ = \mathop{\mathbb{E}}_{N_{\Delta}} \left[ \mathop{\mathbb{E}}_{N|N_{\Delta}} \left[ {\int}_{t- T_{h}}^{t_{1*}} h(t - \tau) \mathrm{d} N(\tau) \left| N_{\Delta}\right. \right] \right] $$

(11)

$$ \stackrel{t - \tau = u}{=} \mathop{\mathbb{E}}_{N_{\Delta}} \left[ \frac{N_{\Delta}}{t_{1*} - t + T_{h}} {\int}_{t - t_{1*}}^{T_{h}} h(u) \mathrm{d}u \right] $$

(12)

$$ = A_{0} {\int}_{t - t_{1*}}^{T_{h}} h(u) \mathrm{d}u $$

(13)

where $N_{\Delta } = N_{(t-T_{h}, t_{1*})}$ is the number of spikes in (t − T_h, t_1∗), and A₀ is the mean firing rate in that time window. In Eq. (11), the inner expectation is taken over spike count conditioned on a fixed number of spikes in the interval (t − T_h, t_1∗). If the filter h(u) is estimated by e^h(u) − 1 in Gerhard et al. (2017), the error will be larger if h(u) is not close to 0. Because the point process itself is unknown, the firing rate function is approximated under the assumption that it is a homogeneous Poisson process. For homogeneous Poisson process, if the number of events is fixed, they distribute evenly in the interval, which leads to Eq. (12).

The time rescaling theorem

Let $Z_{i} = {\int }_{t_{i-1}}^{t_{i}} \lambda _{0}(t) dt$, where t_i are spike times, Z_i are time integral transformed intervals. Time rescaling theorem states that if λ₀(t) is the firing rate of the true model, then Z_i are iid and Z_i ∼Exp(1). The goodness-of-fit test checks how close the distribution of transformed intervals from estimated model is to the unit exponential distribution.

Fig. 10

aIzhikevich-burst synthetic dataset spike trains and simulated spike trains from FLF, FNF_S(k = 4), FNF_M(k = 4). Both type of models can generate busts similar to those in the dataset. b Fitted filters of the FNF_S models with number of spikes k = 1,...,9. When k > 3, the filters are very close to each other since further spikes will not make too much contribution to the future firing rate, thus will only affect the filter shape slightly. The fitted FLF filter overlaps with the FNF_S filters with k ≥ 5, suggesting that these models are functionally similar. c Fitted filters of an FNF_M model with k = 4: the filters are substantially different, which suggests that past spikes of different order have different effects on the firing rate. A likelihood ratio test comparing the FNF_S(k = 5) and FNF_M(k = 5) models favors the FNF_M model (p ≪ 0.001)