On the stability and dynamics of stochastic spiking neuron models: Nonlinear Hawkes process and point process GLMs
Fig 2
Point-process models estimated from physiological data can pass common goodness-of-fit tests, but simulated activity may diverge.
(A) Neurons in the stomatogastric ganglion (STG) of the crab show rhythmic bursts of spike patterns. Each line shows a random 2-second segment of the data from one neuron aligned to the first spike of a burst. The spike-history filter is estimated following the procedure in [6]. The neuron model passes commonly used goodness-of-fit tests, such as those based on the time-rescaling theorem [24, 25]. Here, the Kolmogorov-Smirnov test is shown for rescaled inter-spike intervals to come from an exponential distribution with unit mean. The null hypothesis that observed spikes are coming from the estimated model is not rejected (P > 0.05). When sampling spike trains from the model, the model regenerates the rhythmic, bursty activity that is qualitatively matched to the training data. (B) Similar analysis for single-unit activity from neocortical recordings in a person with pharmacologically intractable focal epilepsy [30]. Each line corresponds to a random ten-second segment of spontaneous activity during interictal periods, i.e., outside seizures. The estimated spike-history filter shows a refractory period and an excitatory rebound. The model passes commonly used goodness-of-fit tests (P > 0.05). When stochastic samples are generated from the model, spiking activity diverges to a periodic firing pattern at the maximally allowed frequency given the absolute refractory period (here, 2 ms). For some sampled realizations, this divergence can happen very early in the simulated trial (e.g., trial 5). Therefore, simulated activity from the model is unphysiological. It does not match statistics of the spike train in the training data (mean firing rate, inter-spike interval statistics) despite passing the goodness-of-fit test. (C, D) Additional examples of single-unit activity from monkey cortex, areas PMv and M1 [31, 32]. Each line represents a steady-state movement preparation period preceding visual cues leading to execution of reach and grasp actions. Although spike-history filters appear typical in both examples, and goodness-of-fit tests are passed, simulated activity diverges into unphysiological firing rates in one case (first example) and remains physiological in the other.