Matthew H. Higgs & Charles J. Wilson (2019) Frequency-dependent entrainment of striatal fast-spiking interneurons. J Neurophysiol 122: 1060–1072, 2019.

Striatal fast-spiking interneurons (FSIs) fire in variable-length runs of action potentials at 20-200/s separated by pauses.  In vivo, or with artificial fluctuating applied current, both runs and pauses become briefer and more variable.  During runs, spikes are entrained specifically to gamma-frequency components of the input fluctuations. We stimulated striatal FSIs in slices with broadband noise currents added to direct current steps, and measured spike entrainment across all frequencies. As the constant current level was increased, FSIs produced longer runs and showed sharper frequency tuning, with best entrainment at the stimulus frequency matching their intra-run firing rate.  The precision of gamma-entrainment was poorest for the first spike in a run, and increased over the first few spikes. We separated the contributions of previous spikes from that of the fluctuating stimulus, revealing a strong contribution of previous action potentials to gamma-frequency entrainment. Entrainment within a run and its frequency specificity was reproduced using a phase-resetting model based on experimentally measured phase resetting curves of the same FSI neurons.  The phase resetting model could not account for the onset of runs or single spiking, which was also less dependent on gamma frequency components of the stimulus fluctuations.  The spectral sensitivity of history-independent spike generation had prominent components at lower frequencies.  For FSI neurons firing in brief runs and pauses, firing within runs is entrained by gamma frequency components of the input, whereas the timing and duration of runs may be sensitive to a wider range of stimulus frequency components.

Figure 1. Effects of DC and noise on stuttering. The steady-state firing of a typical striatal FSI is shown at three DC levels, each without noise (left) and with noise (right). Increasing the DC level produced longer spike runs, whereas noise disrupted the runs and pauses. The insets show the noisy subthreshold oscillations observed during pauses (expanded from the red boxed regions).

Figure 2. Analysis of spiking resonance. A. Example of spike response to a noise stimulus, showing an individual frequency band (fcenter = 70 Hz, bandwidth = 10 Hz) extracted from the stimulus using a Tukey window function (inset; see Materials and Methods). Red vertical lines indicate the alignment of each spike on the stimulus component. Upper right inset shows the convention for assigning spike phases with respect to a stimulus component. Lower right inset illustrates the vector representation of spike phases. Red arrow is the vector average (resultant vector), and its length is the vector strength (VS). B. Spectra of VS with respect to each frequency component of the stimulus, at each mean current. The black dashed lines indicate the bilinear fit used to measure the resonance frequency and strength at each DC level. C. Spectra of vector phase. D. Resonance frequency vs. mean firing rate. The lines connect data points obtained from the same cell at different DC levels. E. Resonance frequency vs. intra-run firing rate (reciprocal of the mode of the ISI distribution).

Figure. 3. Analysis of run length. A: identification of spike runs, defined as sequences of spikes separated by interspike intervals (ISIs) 􏰄1.5 times the mode of the ISI distribution. Green marks indicate the first spike in each run. B: effect of direct current (DC) on mean run length in each cell, without noise (left) and with noise (right). Data are from the steady-state portion of each stimulus. Lines connect data points from the same cell at each DC level. Note the consistent increase in run length with more DC and the shortening of runs by noise, particularly at the higher DC levels. C: resonance strength, defined as VS(fres)/VS(0.5fres) 􏰃 1 (where VS is vector strength and fres is resonance frequency), vs. mean run length. Note the increases in resonance strength with run length in the individual cells, as well as the overall correlation across all cells and DC levels (r=0.60).

4. Contribution of sequential spikes to spiking resonance. A. Method for analysis of potential spike times. Each trace shows a randomly chosen ISI (black bar) added to a randomly chosen spike time (green bar) to give a potential spike time (green dashed line). B. Top: example of spike-triggered average current (STA, black line) compared to potential-spike-triggered average current (PSTA, green line). The PSTA has a lower noise level than the STA because 10 potential spike times were chosen for each actual spike time. The PSTA matches the early portion of the damped oscillation seen in the STA. Bottom: difference between the STA and the PSTA. Only the fast transient and one preceding negative excursion remain in the difference current. C. Polar plots illustrating the analysis of spiking resonance for the actual spike times (left) and the potential spike times (middle), and the vector differences (right). Each point represents a resultant vector measuring the phase distribution with respect to one frequency component of the stimulus. The magnitude of the vector is the vector strength (VS, distance from the origin), and the angle of the line connecting the point to the origin is the vector phase. The illustrated data were obtained at a DC level of 1.2 ´ rheobase. Red and blue symbols indicate the 70 Hz and 35 Hz frequency components, respectively. D. Original VS spectrum and difference VS spectrum from the data in C. The sharper, higher peak seen in the original spectrum compared to the difference spectrum indicates that sequences of spikes contributed to spiking resonance. E. Summary data comparing the peak frequencies (top plots) and resonance strengths (bottom plots) for the original VS spectra to the corresponding results for the difference spectra. The data show that spike sequences contributed to the change in resonance frequency associated with increasing DC, and were largely responsible for the increase in resonance strength.

Figure 5. The phase resetting curve (PRC). A. Example PRC computed from the perturbation of intra-run ISIs in a FSI (see text for details). The PRC shows a very slight negativity for phases up to approximately 0.5, followed by a much larger positive phase that falls off abruptly at the right side. This PRC shape indicates that stimulus current arriving in the late portion of each ISI was primarily responsible for changes in ISI length. B. PRC data from 14 FSIs, each at a DC level of 1.2 ´ rheobase (gray lines), the average PRC (black line), and a composite function fitted to the average PRC (red line; see Materials and Methods). The fitted curve was used for the simulations described below.

Figure 6. Simulating spike runs. The figure illustrates the generation of one spike run. A. Noise current providing input to the model. The DC current was incorporated in the baseline firing rate of the model and does not appear explicitly. B. For simulation of each spike run, the model starts in the linear-nonlinear (L-N) mode, which represents the cell in the paused state. The noise current is convolved with a linear filter (top inset) to produce a filtered stimulus variable S(t). The linear filter was constructed as the time-reversed, average difference STA (see Fig. 4B) from 14 cells. The value of S is transformed to spike rate by an exponential nonlinearity (bottom inset) with parameters chosen to account for the mean pause lengths observed without and with the noise current. The model remains in the L-N mode until a spike is produced by a rate-modulated Poisson process. C. Phase oscillator mode. Upon producing the first spike, the oscillator is initialized at 𝜑 = 0, representing the post-spike reset. The phase 𝜑(t), then advances at a rate ω + I(t) Z(𝜑), where ω = 50 Hz and Z(𝜑) is the fitted average PRC. The subsequent spike times are indicated by the resets from 𝜑 = 1 to 𝜑 = 0. For analysis, the spike times are aligned with the frequency components of the noise stimulus.

Figure 7. Evolution of entrainment across simulated spike runs. A. Model spike phases with respect to the 25 Hz, 50 Hz, and 75 Hz bandpass components of the stimulus. The 50 Hz component matches the oscillation frequency (ω) of the phase-oscillator model. Top row shows the phase distributions for the first spikes, produced by the L-N component of the model, and the second and third rows show the distributions for spikes 2 and 10. The modulation of spike probability was initially similar for the three frequencies, but progressively strengthened at 50 Hz and weakened at 25 and 75 Hz. B. VS versus spike number for each frequency. The VS values approached steady-state by spike 5. For the resonant frequency (50 Hz, green line) the approach to steady state was approximately exponential, whereas for some other frequencies (i.e. 25 Hz, blue line) the VS alternated between higher and lower values before steady state was attained. C. Maps of sequential spike phases (top, spike 2 vs. spike 1; bottom, spike 10 vs. spike 9). Black dots indicate spike phases during the noise stimulus. Yellow line is the deterministic map for a pure sine wave input matching the average frequency and amplitude of each bandpass frequency component. The identity line is shown in red, and the point where the deterministic map intersects the identity line from top to bottom is a stable fixed point. During the runs, the spike phases are attracted toward the stable fixed point for the 50 Hz stimulus component. In contrast, there is no fixed point for the 25 and 75 Hz components, and the spike phases disperse as the runs proceed. D. Illustration of the entrainment process for a single 50 Hz sine wave input of larger amplitude, showing how sequential spike phases evolve toward the fixed point. E. VS spectra for spikes 1, 2, 3, 4, and 10. The spectral peak was relatively broad for spike 1 and progressively sharpened, producing clear first and second harmonic peaks.

Figure 8. Entrainment simulated with different PRC shapes. Spike runs were simulated as described above, except that the initial spike times were chosen randomly without using the L-N model component. A. PRC shapes used for simulations: the mean FSI PRC (used for Fig. 7), a flat PRC representing a perfect integrator, a type-1 (always positive) cosine function, and a symmetrical type-2 (negative and positive) sine function. B. Vector strength versus spike number. In each case the VS for spike 1 is low because the spike times were random, giving random initial phases. For the three shaped PRCs, frequency-selective entrainment developed across the runs. In contrast, with the flat PRC, the VS approached a small value that was similar for all frequencies. With each PRC, the VS levels approached steady state around spike 5. C. VS spectra for spike 10. Each of the three shaped PRCs produced a peak at the frequency of the phase oscillator (ω = 50 Hz), but only the FS PRC generated a second harmonic peak. In addition, the FS PRC produced higher VS at frequencies above the second harmonic.