Fig 1.
Schematic overview of the nonlinear Hawkes process and the quasi-renewal (QR) approximation.
The quasi-renewal approximation can be used to semi-analytically obtain steady-state firing rates of general, nonlinear Hawkes processes (PP-GLMs). Left: In the nonlinear Hawkes process, the conditional intensity of the point process, , is a function of the whole spiking history (see Eq (3)). It is modeled as a nonlinear function (here, an exponential function) of a linear convolution of the previous spike history with a spike-history filter η(s) plus a constant offset I0. The dependence of the instantaneous firing rate on all previous spikes results in a non-renewal process. There are no closed-form solutions for even the first-order statistics of general, nonlinear Hawkes processes. Right: In the quasi-renewal approximation, the conditional intensity is modeled as a combination of the effect of the most recent spike ti and a term involving the average over the whole spike history before the most recent spike (see Eq (5)). This term includes the average firing activity in the past A0 ≡ A(t − s), which is filtered with the quasi-renewal filter γ(s) and added to the spike-history filter η(τ) of the most recent spike at t − τ. This predicts the instantaneous inter-spike interval density P0(τ) from which the steady-state firing rate can be obtained as the inverse of the expected inter-spike interval E[τ]. The self-consistent solutions for which an assumed average history of A0 leads to an equivalent predicted steady-state rate are fixed points of the transfer function defined in the quasi-renewal approximation.
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.
Fig 3.
Stochastic dynamics of nonlinear Hawkes PP-GLMs endowed with absolute refractory periods: Classification based on transfer functions in the quasi-renewal approximation.
The quasi-renewal approximation provides a predicted firing rate of a neuron model f(A0) based on an assumed average firing rate in the past A0 (see “Materials and Methods”). This defines an iterative equation whose fixed points represent the steady-state firing rates. A qualitative classification of the dynamical behaviors is based on the location and stability of the fixed points. Note that throughout the study, we are assuming that there exists an absolute refractory period. Thereby, the maximum firing rate of any model is limited by a maximal firing rate λmax. We define a steady-state firing rate to be unphysiological if it exceeds λthr = 0.9 × λmax (gray area). Given an absolute refractory period, there always exists at least one stable fixed point. If it is the only one and below λthr, the model is classified as stable (top, left). If the only stable fixed point is above λthr, the model is divergent (top, center). If there are two stable fixed points, one above and one below λthr, the model is classified as “fragile” (metastable), indicating that the (physiological) low-rate fixed point is only transiently stable. Expected divergence times E[Tdiv] will depend on the distance between the fixed points. To provide a complete classification framework, we also need to consider the case of two or more stable fixed points, although the latter case seems to be rarely encountered in our experience. In case of two or more stable fixed points below the threshold, the model is classified as stable (bottom, left). Its dynamics is predicted to be multi-stable with steady-state rates fluctuating around two fixed points. If all stable fixed points lie above the threshold the model is considered “divergent” (bottom, center). Any case for which there are multiple stable fixed points both below and above λthr are considered “fragile” (bottom, right).
Fig 4.
A framework to assess stability and dynamics of stochastic spiking neuron models.
Stability of models estimated from physiological data is analyzed using the quasi-renewal approximation. When stable models are desired for sampling or simulations, stability constraints can be included in model estimation. Three types of dynamics can be distinguished: First, “stable” models have steady-state firing rates that are in the physiological range. Spike trains can be safely generated from the model. Second, “divergent” models have a steady-state firing rate that is very close to the maximally allowed firing rate. In this case, stabilization constraints can be added to the maximum-likelihood optimization problem to constrain the feasible parameter space to non-divergent models. Finally, model dynamics can be classified as “fragile”, indicating metastable dynamics. While there is a steady-state firing rate at physiological firing rates, there are additional steady-state rates at unphysiological high rates. A simulation that is started with physiological initial conditions may remain in the low-rate regime for a while, but will ultimately visit the unphysiological rate. The framework may provide an estimate of the expected escape or “divergence” time, E[Tdiv]. Depending on E[Tdiv], the model can be effectively treated as “stable” or “divergent” based on the typical time scales that would be relevant for simulation. In any of the three cases, the model (or its stabilized variant) is evaluated based on standard model selection and goodness-of-fit tests before any inference is made.
Fig 5.
Predicted stability reflects simulation outcomes: Exponential spike-history filter.
(A) Spike trains are simulated from a nonlinear Hawkes model with baseline firing rate c and an auto-history kernel η(s) = J exp(−s/τ) with amplitude J, time constant τ = 0.02 s, and absolute refractory period τref = 2 ms. The QR approximation qualitatively predicts three classes of dynamics (separated by thick lines). Color indicates average divergence times estimated from simulations. In the dashed region, no finite divergence times were observed. (B) An example of a spike-history filter for each of the dynamics is shown together with a spike raster of simulated activity. Purely inhibitory filters produce stable dynamics (top, J = −1, c = 5 s−1). For fragile models, after an episode of irregular firing, dynamics switches into a tonic firing mode close to the limit frequency 1/τref (middle, J = 1, c = 5 s−1). For divergent models, the only stable fixed point is close to the limit frequency (bottom, J = 3, c = 5 s−1). After a brief transient, the model switches to the tonic firing mode at limit frequency. All dynamics observed in simulations are in accordance with the prediction from the QR approximation.
Fig 6.
Rate and time to divergence: Exponential spike-history filter.
(A) Observed steady-state firing rate versus predicted steady-state firing rate for every model classified as “stable” in Fig 5. Each dot corresponds to a model. Dashed line indicates equality. Linear correlation coefficient is ρ = 0.9996. (B) Observed versus predicted divergence times for all models classified as “fragile” in Fig 5. Note the logarithmic axes. The QR approximation provides an approximation of the divergence times. The divergence time of a simulation was defined as the end of the first two-second interval in which the average rate exceeded λthr. Therefore, estimated divergence times cannot be below 2 s (gray area).
Fig 7.
Predicted stability reflects simulation outcomes: Sum-of-exponentials spike-history filter.
(A) Spike trains are simulated from a nonlinear Hawkes model with fixed baseline c = 5 s−1 and an auto-history kernel consisting of two exponentials with amplitudes Jr, Ja, and corresponding time constants τr = 0.02 s and τa = 0.1 s. Observed divergence times for simulated spike trains are color-coded (same scale as in Fig 5). In the dashed region, no finite divergence times were observed. (B) Auto-history kernels for three different parameter values (models a, b and c), displaying irregular, bursty and divergent dynamics, respectively, consistent with the prediction of the QR-approximation except for model b (see main text).
Fig 8.
Rate and time to divergence: Sum-of-exponentials spike-history filter.
(A) Observed steady-state firing rate versus predicted steady-state firing rate for every model classified as “stable” in Fig 7. Each dot corresponds to a model. Dashed line indicates equality. Linear correlation coefficient is ρ = 0.50. While steady-state rates are correctly predicted for most models, discrepancies arise due to limiting assumptions of the QR approximations when Jr ≫ 0 and Ja ≪ 0. However, the qualitative prediction of the dynamics as stable remains correct. (B) Observed versus predicted divergence times for all models classified as “fragile” in Fig 7. Note the logarithmic axes. Dashed line indicates diagonal. The QR approximation provides an approximation of the divergence times. The divergence time of a simulation was defined as the end of the first two-second interval in which the average rate exceeded λthr. Therefore, estimated divergence times cannot be below 2 s (gray area). The data suggest a power-law dependence between predicted and observed divergence times (Pearson’s correlation coefficient ρ = 0.94).
Fig 9.
The quasi-renewal approximation predicts stability for complex (physiologically plausible) model parameters.
(A) Single-unit activity (SUA) from multi-electrode recordings from monkey cortical area PMv [31] (same unit as in Fig 2C). Spike waveforms are shown (mean waveform in green) and indicate well-sorted SUA. (B) Estimated spike-history kernel using maximum-likelihood estimation. The kernel exhibits a relative refractory period followed by an excitatory rebound. (C) The transfer function predicted by the QR approximation. There is a single stable fixed point at ≈ 500 s−1. The model is therefore classified as “divergent”. (D) The QR approximation predicts the maximum-likelihood estimate (MLE) (center dot) to be divergent. Color indicates average divergence times in simulations for variations of the baseline rate c (y-axis) and scaled versions of the filter relative to the integral of eη(s) − 1 (x-axis). Thick lines indicate the separation between areas for which the QR approximation predicts stability, fragility, or divergence. Overall, estimated divergence times from simulations agree with the qualitative predictions.
Fig 10.
Neuron models can be stabilized with constrained maximum-likelihood estimation.
(A) Estimated spike-history kernels using maximum-likelihood estimation (blue) and the constrained (stabilized) maximum-likelihood estimation (red). Data are from single-unit activity (SUA) recordings from monkey cortical area PMv [31]. (B) Power of predicting spiking activity on test data for both the maximum-likelihood estimate (MLE) and the stabilized MLE. The receiver operating characteristic (ROC) curve is shown for predicting spikes in 1 ms time bins with false positive rate (FP, x-axis) and true positive rates (TP, y-axis). Diagonal line indicates chance-level prediction. Predictive power is defined as PP = 2 ⋅ AUC − 1 with AUC being the area under the curve. Perfect spike prediction corresponds to PP = 1. Both models predict spikes equally well. (C) Log-likelihood evaluated on test data. Both MLE and stabilized MLE models preserve information about spike times. Model log-likelihoods (in bits per second) are relative to a homogeneous Poisson process with the correct spiking rate. (D) Kolmogorov-Smirnov test of rescaled inter-spike intervals following the time-rescaling theorem. Both MLE and the stabilized MLE pass the goodness-of-fit test (P > 0.05). (E) Recorded spike trains and simulated spike trains from estimated models. Top: Randomly selected 10 s intervals of neural activity part of the training data (black). Simulating from the unconstrained MLE model (blue) leads to quickly diverging firing rates (Tdiv ≈ 10 s). Spiking activity from the stabilized MLE (red) remains finite and physiological. Simulating from the MLE with a reset condition after each spike (green, [34]) leads to non-divergent firing rates, but the firing rate of the training data is not matched. (F) Inter-spike interval statistics of real and simulated spiking activity for which rates were non-divergent. Same colors as in (D). The stabilized MLE qualitatively reproduces the training data ISI distribution. (G) Autocorrelation of recorded and simulated activity. (H) Serial ISI correlations of real and simulated spiking activity. The stabilized MLE accurately reproduces correlations in the training data. Simulating the MLE with a reset condition (green) leads to a renewal process, hence vanishing correlations at non-zero lags.