Predicting stimulus-locked single unit spiking from cortical local field potentials

Abstract

The rapidly increasing use of the local field potential (LFP) has motivated research to better understand its relation to the gold standard of neural activity, single unit (SU) spiking. We addressed this in an in vivo, awake, restrained mouse auditory cortical electrophysiology preparation by asking whether the LFP could actually be used to predict stimulus-evoked SU spiking. Implementing a Bayesian algorithm to predict the likelihood of spiking on a trial by trial basis from different representations of the despiked LFP signal, we were able to predict, with high quality and fine temporal resolution (2 ms), the time course of a SU's excitatory or inhibitory firing rate response to natural species-specific vocalizations. Our best predictions were achieved by representing the LFP by its wide-band Hilbert phase signal, and approximating the statistical structure of this signal at different time points as independent. Our results show that each SU's action potential has a unique relationship with the LFP that can be reliably used to predict the occurrence of spikes. This "signature" interaction can reflect both pre- and post-spike neural activity that is intrinsic to the local circuit rather than just dictated by the stimulus. Finally, the time course of this "signature" may be most faithful when the full bandwidth of the LFP, rather than specific narrow-band components, is used for representation.

Fig. 1

Relating the LFP to SU spikes. SUs were well isolated, as demonstrated by (a) SU1381's well-defined waveform (all spikes shown), and (b) lack of refractory spikes in the interspike interval (ISI) distribution. (c) The co-recorded LFP's power spectrum showed a peak around 5–10 Hz, decaying ∼1/f². (d) Top panel: the temporal relation between SU spikes (balck dots) and the co-recorded LFP in the wide-band original representation is shown for a set of trials. Acoustic stimulus is presented during the interval marked by the horizontal gray bar. Botton panel: single trial describing the interaction SU-LFP; inset shows the LFP around each of the spikes in the trial. (e) The STPDT was obtained by realigning the LFP to the time of each spike, for all spikes in half the trials, and computing the probability that the LFP fell into a specific bin at a specific time relative to the spike. Here and in later figures, darker colors represent higher probabilities, while lighter colors denote lower probabilities. Under the independent algorithm, the distribution at each time bin relative to the spike was assumed independent from any other time bin. Spike-triggered average (STA) is shown as well (dash-gray). (f) The negative deflection in the STPDT contrasted with the temporally uniform a priori distribution based on random times in the trial (PDT). Under the Markov algorithm, the (g) STPDT and (h) PDT became 3-dimensional matrices, with the shading now representing the transition probability between the LFP at time t and t+Δt

Fig. 2

Predicting SU spiking from the LFP's Hilbert components. (a) LFP trace of a single trial and the co-recorded spikes (vertical hashes). (b) Original LFP STPDT (same trials as in Fig. 1(e)). (c) Hilbert amplitude representation of the same trial as in Fig. 1(d). (d) Hilbert amplitude STPDT. (e) correspondent Hilbert phase representation, and the corresponding (f, upper panel) STPDT and (f, lower panel) PDT. Dashed boxes outline the time window [t_ini t_final] around t′=0 in the probability distributions that were convolved with the X trajectory (black line) for the Bayesian prediction (Eq. (1)). (g) spiking likelihood at each time point, t′, which ran from 50 ms to 550 ms, for each trial not used in estimating the STPDT. (h) The sum across trials of the likelihoods, normalized to agree with the pre-stimulus spontaneous firing rate, generated the predicted activity (black line). (i) The predicted rate was compared on a time-by-time basis with the experimental firing rate (gray histogram in (H). Dashed lines represent slopes of 2 and 0.5, the limits for valid predictions

Fig. 3

Quantifying the prediction quality and the selection of the optimal window. CC for original LFP representation (a), Hilbert phase representation (b) and Hilbert amplitude (c). Darker colors represent better prediction quality (high CC). From the triangle of quality values for different window combinations, we selected the one that outperformed the other intervals (black circles). Possible intervals that did not pass the slope condition are represented by “x”, indicating no assigned value

Fig. 4

Comparison of predicted PSTHs across σ, representation (original, phase and amplitude) and algorithm (independent and Markovian). (a) Each panel shows the predicted PSTHs for SU1381 derived using the three representations with their respective optimal windows. The three top panels show the best prediction provided by the independent algorithm when the experimental PSTH (gray histogram) was smoothed with σ=1, 2 and 4 ms, respectively. The bottom panel shows the case with Markov algorithm and σ=2; note that the amplitude representation did not provide an acceptable slope in this case. (b) Each panel shows the comparison of the CC derived for different representations across the SU population. The plus symbol represents the median across SUs, and the star indicates the example, SU1381

Fig. 5

Population summary of CC prediction quality, relative to the wide-band Hilbert phase representation using the independent algorithm, $W_{\mathit{\text{phase}}}^{\mathit{\text{ind}}}$. Upward triangles mark the population median, and error bars depict the interquartile range. For this and all subsequent figures, σ was set to 2 ms. $W_{\mathit{\text{phase}}}^{\mathit{\text{ind}}}$ significantly outperformed other algorithms, except for $W_{\mathit{\text{orig}}}^{\mathit{\text{Mark}}}$ and $W_{\mathit{\text{phase}}}^{\mathit{\text{Mark}}}$ and ${\mathit{\text{WSp}}}_{\mathit{\text{phase}}}^{\mathit{\text{ind}}}$ (Wilcoxon signrank test for significant difference from 1; $W_{\mathit{\text{org}}}^{\mathit{\text{ind}}}/W_{\mathit{\text{phase}}}^{\mathit{\text{ind}}}$: signed rank statistic S=22, number of differences nd=24, p=0.0003; $W_{\mathit{\text{ampli}}}^{\mathit{\text{ind}}}/W_{\mathit{\text{phase}}}^{\mathit{\text{ind}}}$: S=4, nd=24, p=0.00003; $W_{\mathit{\text{orig}}}^{\mathit{\text{Mark}}}/W_{\mathit{\text{phase}}}^{\mathit{\text{ind}}}$: S=73, nd=22, p=0.05; $W_{\mathit{\text{phase}}}^{\mathit{\text{Mark}}}/W_{\mathit{\text{phase}}}^{\mathit{\text{ind}}}$: S=77, nd=23, p=0.06; $W_{\mathit{\text{ampli}}}^{\mathit{\text{Mark}}}/W_{\mathit{\text{phase}}}^{\mathit{\text{ind}}}$: S=4, nd=16, p=0.0009; and ${\mathit{\text{WSp}}}_{\mathit{\text{phase}}}^{\mathit{\text{ind}}}/W_{\mathit{\text{phase}}}^{\mathit{\text{ind}}}$: S=103, nd=26, p=0.07). The double asterisks indicate significance at the p<0.01 level

Fig. 6

“Signature” SU-LFP interactions for two different co-recorded SUs, SU1516 (dark gray) and SU1517 (light gray). (a) and (c) The respective ISI distributions both lacked refractory spikes. (b) Spikes from SU1516 inhibited SU1517, as evidenced by the asymmetric cross correlation with no intervals at ∼2 ms. The gray line shows the trial shuffled correlation. (d) and (f) The respective spike waveforms where clearly stereotyped. (e) and (g) The respective STPDTs were noisy, but distinct from one another. (h) and (i), The respective predicted PSTHs approximately matched their corresponding experimental PSTHs, which showed an excitatory response for SU1516 and inhibitory response for SU1517

Fig. 7

Summary of wide-band phase predictions by the independent algorithm. (a) Six different examples of predictions spanning a wide range of CCs shows the algorithm can be successful for a wide range of SU response time courses. (b) Cumulative distribution of CC across 37 stimulus-driven cells with valid wide-band phase predictions. SUs from A are identified by their respective symbols. (c) The optimal window for prediction for each SU is depicted as a horizontal line at that SU's corresponding CC (lines for SUs in panel A are darker). There was no apparent relation between the timing of the optimal window and the CC. (d) Across the population, the initial time of the optimal window was mostly pre-spike, whereas the final times were mostly post-spike

Fig. 8

Dependence of predictions on LFP frequency band. (a) PSTH predictions for SU1381 using the independent algorithm for the Hilbert phase within the θ-, β-, and γ- bands of the LFP. The corresponding STPDTs and PDTs were constructed for each frequency band in a similar way as in the wide-band case. To test whether success in the wide-band algorithm could be explained simply by the increased information carried by independent frequencies in the wider frequency range, a combined band (C-band) prediction was constructed. It was derived by the direct product of the θ-, β, and γ- likelihoods. Note that in this example the predicted PSTH dramtically overestimates the magnitude of the excitatory transient, but does conserve the timing of the peak. (b) Population summary of the narrow-band and combined-band predictions, relative to the independent wide-band phase prediction. The figure shows median and interquartile ranges for the Hilbert amplitude (open circles) and Hilbert phase representations (filled squares) in each narrow-band case. For the C-band, only SUs with valid predictions in all three narrow-band ranges were included, and the slope criterion was not applied. In all cases, the ratios were significantly less than 1, indicating the independent wide-band phase again outperformed the other representations (Wilcoxon signrank test for significant difference from 1; θ-phase: S=23, nd=28, p=0.00004; θ-ampli: S=3, nd=14, p=0.0006; β-phase: S=14, nd=24, p=0.0001; β-ampli: S=0, nd=16, p=0.0004; γ-phase: S=23, nd=23, p=0.0005; γ-ampli: S=15, nd=14, p=0.02; C-phase: S=2, nd=19, p=0.0002). (c) In the γ-band only, the amplitude component performed significantly better than the phase on a SU-by-SU basis (θ-band: S=13, nd=14, p=0.01; β-band: S=41, nd=19, p=0.03; γ-band: S=31, nd=17, p=0.03). Asterisks indicate values significantly different from 1 (**: p<0.01; *: p<0.05)