The patterns and condition of theta phase precession

Although a neuron may have preferred phase during theta where it is more likely to fire, the exact phases at which individual spike events occur often vary. Importantly, the phase variation between spikes is not altogether random; in some cases, it exhibits consistent pattern. One intriguing example is the theta phase precession in which spikes fired by a neuron advance unidirectionally towards earlier phases from one theta cycle to the next. Theta phase precession was first reported in hippocampal place cells when the animal transversed the “place field” of a “place cell” (O'Keefe & Recce 1993). Subsequently, it was also demonstrated in non-spatial contexts and indeed related with transiently intensified firing (Harris et al 2002, Huxter et al 2008). Will the phase precession persist in the Kcnq3^-/- neurons of a differed theta phase preference?

Firstly, I examined the relations between the instantaneous firing rates of the spikes and their phases in theta cycles. The instantaneous rate for a spike was derived from counting the number of spikes in the two theta cycles that centered at the inquired spike event. The

instantaneous firing rate of a spike was classified as “low rate” (of one spike per 2 cycles) or

“high rate” (of ≥ 3 spikes per 2 cycles). In both Kcnq3^+/+ and Kcnq3^-/- pyramidal cells, the average phase of spikes at high firing rate was more advanced in theta cycle than phases at

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low firing rate: in Kcnq3^+/+ neurons, mean phase advanced from 164° ± 13° at low rate to 146° ± 15° (Fig. 3.8A, B; circular mean ± 95% confidence interval; “low rate”: n = 1676 events, “high rate”: n = 2703 events; p < 0.0001,Watson-Williams test for circular data) while in the knockout group, from 130° ± 7° at low rate to 116° ± 7° at high rate (Fig. 3.8A, B;

circular mean ± 95% confidence interval; “low rate”: n = 9178 events, “high rate”: n = 13920 events; p < 0.0001, Watson-Williams test for circular data). Moreover, Kcnq3^-/- putative cells fired consistently at earlier theta phases than wild-type (Harrison-Kanji test for two-way classification of circular data with factors “genotype” and “rate level”: the effect of

“genotypes”, χ² (2, N = 27477) = 28.80, p < 0.0001; the effect of “rate level”, χ² (2, N = 27477) = 12.82, p < 0.01). This result confirms the phase-rate correlation in the firing of pyramidal cells during theta as reported in previous studies. Furthermore, it suggests that such spike-phase dynamic is preserved in the knockout hippocampus while the exact phases, compared to wild-type, are systematically advanced.

Figure 3.8: Spike theta phases in relation to instantaneous firing rate and the rate's derivative. (A) Polar histogram plots and scatter plots of the firing of CA1 pyramidal cells in reference to the phases of theta oscillation in CA1 LFP. Each dot shows the mean phase (0° and 360° for theta peak, 180° for theta trough) and modulation depth (mean resultant length, represented by the distance from the center of the plot) of the spikes from each putative neuron at low (1 spike per 2 cycles) or high (≥ 3 spikes per 2 cycles) firing rate. Histograms show the distributions of mean phases sorted in 12 bins. (B) Firing advanced to earlier phase of theta at high rate compared to low rate in both genotypes. Plots represent circular mean ± 95% confidence interval. In Kcnq3^-/-: “low rate”: n = 9178 events, “high rate”: n = 13920 events, p < 0.0001; in Kcnq3^+/+: “low rate”: n = 1676 events, “high rate”: n = 2703 events, p <

0.0001; Watson-Williams test for circular ANOVA.

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Next, I examined the firing events spanning multiple adjoining theta cycles. Although theta phase precession is often studied in association with the transversal of place field in place cells, phase precession also occurs during non-spatial behavioral states and the phase advancement can be measured by phase-position correlation and phase-time correlation (Harris et al 2002, O'Keefe & Recce 1993). To examine the change of spike phase in all recorded units and irrespective of behavioral states, I adopted the phase-time correlation. In both Kcnq3^+/+ and Kcnq3^-/- pyramidal cells, firing events with systematic advanced theta phases could be observed (Fig. 3.9A, B). But not all spiking sequences during theta showed phase precession (Fig. 3.9A, B). A circular-linear fit of phase and spike timing was applied to estimate the strength and slope of phase-time correlation. For both groups of neurons, more than half of the events were with negative phase-time slopes which indicated an advancement in the spike phases (63% of events in Kcnq3^+/+ and 54% of events in Kcnq3^-/-; the analysis included only events with mean vector length >= 0.3 in the circular–linear fit of phases and timestamps). The average slope of these phase-advanced spike events in the wild-type was significantly steeper than in knockout (mean

± SEM: Kcnq3^-/-: -0.02 ± 0.02, n = 552 events; Kcnq3^+/+: -0.13 ± 0.04, n = 99 events; p < 0.01, student t-test; Fig. 3.9C, D). These results suggested that theta phase precession existed in Kcnq3 knockout pyramidal cells, but the extent of advancement was less pronounced than in wild-type.

Another approach to estimate the phase shift along multiple theta cycles is by comparing the phases between spikes from the onset of a theta firing sequence and those from the offset (Harris et al 2002). When phase precession was examined in the context of the place field transversal in previous studies, the acceleration of a place cell firing during theta (i.e. the “onset”) correlated with the entry of its place field while the “offset” where the firing rate decreased corresponded to the exit of the field. The phase shift can be thus revealed from the phase difference between “onset” and “offset”, irrespective of the strict point-wise correlation between phases and timestamps. I first detected firing events that exhibited transient acceleration and deceleration across multiple (>= 5) theta cycles. For each such firing sequence, the segment of spike train that monotonically accelerated along consecutive theta cycles was assigned as the “onset” while the “offset” was defined the cessation part of the firing where the number of spikes in each theta cycles decreased to zero. The mean phase of the spikes was 199°

± 38° during the onset and at the offset, 99° ± 22° in Kcnq3^+/+ pyramidal cells (circular mean ± 95% confidence interval). For spikes from Kcnq3^-/- cells, mean phase of the onset spikes was 148° ± 31° and of the offset, 90° ± 10° (Fig. 3.10A, B). In both genotypes, the spike phases of

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the pyramidal cells during theta cycles advanced considerably from onset to offset (Fig. 3.10B;

Kcnq3^-/-: p < 0.0001, n = 452 events; Kcnq3^+/+: p < 0.0001, n = 97 events; Watson-Williams test for circular data). Moreover, during the acceleration of the firing, the extent of phase advancement in both Kcnq3^+/+ and Kcnq3^-/- pyramidal neurons were correlated with the

Figure 3.9: Single-trial phase precession. Examples of phase variation over successive theta cycles from wild-type (A) and from Kcnq3^-/- cells (B). Each dot represents one spike and the inset values show the slope and mean resultant length (R) of circular-linear regression on phases against spike timestamps.

Examples on left column show significant negative phase-time correlation, suggesting phase precession.

(C) The distribution of phase slopes for all spiking events that spanned multiple theta cycles. Lighter (light blue for Kcnq3^+/+, orange for Kcnq3^-/-) versus darker (dark blue for Kcnq3^+/+, red for Kcnq3^-/-) colors differentiate events of weaker correlation (R < 0.3) from those of stronger correlation (R ≥ 0.3). (D) Comparing the slopes of spike phases during theta cycles between two genotypes; **, p < 0.01, student t-test.

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derivative of firing rate (Fig. 3.10C, D): the faster the firing of a neuron accelerated, the further the phases of spikes advanced (Pearson’s linear correlation between the slope of mean phases over the theta cycles and the derivative of firing rate: Kcnq3^-/-: r = -0.24, p < 0.0001, n = 257 events; Kcnq3^+/+: r = -0.27, p = 0.04, n = 57 events).

Figure 3.10: Advancement of theta phases along successive theta cycles. (A) Example spike trains from Kcnq3^+/+ and Kcnq3^-/- units during theta. Subsets of spikes were defined as occurring at the onset (blue) or at the offset (yellow) of successive theta cycles (selection criteria seen in method). (B) Comparison of the theta phases of spikes at the onset versus offset in each genotype. The circular mean phase and 95% confidence interval were calculated for all spikes defined as occurring at onset (blue) or at the offset (yellow), in both Kcnq3^+/+ and Kcnq3^-/- pyramidal cells. Kcnq3^-/-: ****p < 0.0001, n = 452 events; Kcnq3^+/+: p < 0.0001, n = 97 events; Watson-Williams test for circular data. (C) Calculation of the firing rate derivative. Upper: theta oscillations (grey line) and the timestamps of spikes (black ticks); bottom: the derivative of firing rate calculated by linear regression of spike counts against the theta cycle sequence. (D) Slope of mean phase over successive theta cycles as a function of the firing rate's derivative in both genotypes. Pearson’s linear correlation between the slope of mean phases over theta cycles and the derivative of firing rate: Kcnq3^-/-: r = -0.24, p < 0.0001, n =257 events; Kcnq3^+/+: r = -0.27, p = 0.04, n = 57 events).

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