Fig 1.
Regular and irregular oscillations.
A) LOST infers (red) the oscillatory modulation (black), if any, underlying the spikes (vertical ticks) in a spike train, using only the spiking data. The oscillation amplitude stabilizes after the first few spikes are observed. B) We refer to spectrally narrow (wide) oscillations as regular (irregular), shown by the blue (pink) oscillation with OCV = 0.02 (0.22). The correlation between the current and past phase decays slowly (quickly) as a function of lag in regular (irregular) oscillations.
Fig 2.
LIF neuron with stimulus-triggered change in firing rate.
Phase inference for simulated LIF neuron driven by irregular oscillation with an average firing rate of 40Hz and a 15Hz modulating oscillation with OCV of 0.13. 60 trials of 1 second duration were used for LOST. A) Samples of the representative AR parameters. B) Top, samples of the TAE (cyan), the mean (red), and the empirical PSTH used to initialize TAE knot locations (green), and bottom, samples (cyan) of the spike history and mean (red). C) Example of inferred latent oscillation. The RL of the inferred oscillation is 0.65 ± 0.02, while the spike train has an R of 0.22 ± 0.01 with the ground truth oscillation.
Fig 3.
Interaction of spike history and latent state.
Two inhomogeneous, history dependent spike trains, with first spike train A) generated with stochastic oscillational modulation at 10Hz and OCV at 0.18, and spiking at 40Hz, and the second spike train F) without modulation, spiking at 60Hz. Gibbs samples from 4 (3) separate LOST for first (second) spike train in B-E (G-I). In B-E and G-I, we show the 3 representative sampled quantities, frequency and modulus of the slowest root and the amplitude of the inferred latent state on the left, and the ISI distribution and spike history term on the right. The ground truth, sampled mean and 5-95% intervals of the sampled spike history term plotted in black, red and cyan, respectively. The differences between the different LOST runs is in how spike history knots were chosen, or if a fixed spike history function was used. For the first spike train A, B uses the standard knot choices, C uses a constant history set to 0 and D, a fixed spike history set to a time-compressed ground-truth history. For the second spike train, G uses the standard knot choice, H the ground truth and I the standard knots moved farther out in time. For the first spike train, misspecification of the history may lead to a false inference of a flat latent state, a latent state with high uncertainty in its parameters or the correct latent state. For the second spike train, a misspecified history can lead to a latent state with significant amplitude but with uncertain parameters, making the inference inconclusive.
Fig 4.
Spike trains without oscillatory modulation.
Simulated spike trains at 34Hz (A-D) and 60Hz (E-H), with 4 types of post-spike refractory profiles. Each panel shows Gibbs samples on left and on the right, an example spike train at the top, and the ISI histogram and spike history function as in Fig 1. The expected output of LOST is a nearly flat latent state, which is the case in A, B, D, E, G, H, accompanied by large variability in the other sampled parameters. In C and F, the amplitude of the latent state does not approach 0, but rather continues to fluctuate while taking larger values.
Fig 5.
A) limiting case where models are not identifiable. The spike train in A is nearly periodic, but generated without an oscillatory modulation. When using the standard knots, the latent state amplitude approaches 0, B, but when we use a fixed history that is very different than the ground truth, C, the sampled parameters appear to be very certain, and an oscillation at 62Hz, nearly the spiking frequency, 64hz, is inferred. For cases where the spike train itself is nearly periodic, the spike train may be described equally well with various combinations of spike histories and an oscillation at the spiking frequency.
Fig 6.
Effect of data size on posterior samples.
2 simulations of spike train with spiking around 60Hz, modulated with a 20Hz oscillation with OCV around 0.14 and 1 second long trials, but with different refractory period profiles, A and E. Layout of the figures is the same as in Fig 3, but with B-D and F-G showing different number of trials used, B) 8, C) 15 trials, and D) 40 for spike train in A and F) 5, G) 60 and H) 80 trials for spike train in E. For the smallest number of trials, the posterior is noticeably more variable, but the amplitude of the latent state approaches 0, meaning the latent state is almost decoupled from the spiking. We can see that the posterior narrows as more trials are used.
Fig 7.
Comparison with other models of spiking and the AR order.
Spiking noise and oscillation irregularity affect the quality of phase inference. A, B) Clockwise from top, spike train (ticks) produced by the regular oscillation plotted above, the resultant lengths when oscillation inferred using 3 different history lengths using GLM along with the RL from LOST inference of same data, and the history weights b (dashed red) and their confidence intervals (purple) for GLM using history length 720ms. Modulating oscillations have OCV = 0.02 and 0.2 in A) and B), respectively. C) Sample CIF inference of data shown in A) using LOST (top trace), and GLM (bottom trace), with black line the duplicated ground truth CIFs. For GLM, the dashed (solid) line is the inference obtained from using 720ms (180ms) of spiking history. D) Comparison of how different AR orders for the LOST model affect inference performance for irregular oscillations.
Fig 8.
Spike train modulated with fast oscillation and a slower square-wave fluctuation.
For a sample trial, top shows inferred, square-wave modulation of spiking (black) and the inferred latent state (red). Bottom shows inference of faster rhythm by taking the slow signal shown above to be a known signal, and re-running LOST.
Fig 9.
Mixture of modulated and non-modulated trials.
A), B), C) 0%, 25% and 40% of the trials are weakly modulated. Left shows posterior distribution of π1 and the black line is the ground truth value, red line is the mode of the marginal posterior. Right shows inferred modulation state, weak or strong, for each of the 80 trials. Black dots are ground truth, red dots are inferred state. Below them are indicators (+−) of whether inferred matches ground truth.
Fig 10.
Two simultaneously-recorded motor cortical neurons: Slow fluctuations in firing rate for neuron 1 and 2.
A) rasters from sample trials for the two neurons. There are noticeable holes in the rasters of the neuron 1, and neuron 2 also appears to have a similar slower fluctuation that is less noticeable. B, C) the PSTHs for the two neurons (top) show they both increase their activity around the time of PULL onset (0ms), and power spectral density of latent state show slow fluctuations on the order of 1-2 Hz. Correlation coefficient at 0 lag for the slow fluctuations inferred from neuron 1 and 2 are weakly correlated, with a correlation coefficient 0.15, p < 1 × 10−3, boot strap test.
Fig 11.
Two simultaneously-recorded motor cortical neurons: Simultaneous inference of theta oscillations using inferred slow fluctuation from Fig 10 as a known signal.
A) SP histograms for neurons 1 and 2 with circular statistics R = 0.13 ± 0.01 and 0.08 ± 0.01. B) power spectral densities of inferred oscillation for the 2 neurons showing power in the theta range. C) The cross-correlation function of the inferred latent oscillations of the two neurons, grey traces from when trials shuffled. D) The spike-spike cross correlation function shows very little oscillatory structure.
Fig 12.
Motor cortical neuron (neuron 1, Figs 10 and 11) shows changing relationship to LFP theta oscillation in different trials.
A) The posterior distributions of the mixture weight of the weakly modulated L trials, and relative modulation strength of the strongly modulated H trials. B) PSTHL (blue) and PSTHH (pink) overlap one another, showing no difference in average firing rate. C) LFP theta amplitudes calculated for all L and H trials on a per trial basis. Difference between cumulative distributions of per trial LFP theta amplitude not significant, p < 0.78, 2 sample KS-test. D) Top, the 33 (187) trials classified as weakly (strongly) modulated trials, blue (pink) dots, top. Bottom, the SP histogram calculated from L (H) trials, left (right). Inference of modulation strength using the spike train alone corresponds to the modulation strength to LFP theta. Circular statistics RL = 0.06 ± 0.01 and RH = 0.16 ± 0.01 are significantly different, bootstrap test p < 7 × 10−4.