Frequency-dependent entrainment of striatal fast-spiking interneurons

Abstract

Striatal fast-spiking interneurons (FSIs) fire in variable-length runs of action potentials at 20–200 spikes/s separated by pauses. In vivo, or with 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 parvalbumin-expressing striatal FSIs in mouse brain 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 intrarun firing rate. 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. In contrast, after subtraction of the effect inherited from the previous spike, the remaining stimulus contribution to spike generation was less sharply tuned, showing a larger contribution of lower frequencies. The frequency specificity of entrainment within a run was reproduced with a phase resetting model based on experimentally measured phase resetting curves of the same FSIs. In the model, broadly tuned phase entrainment for the first spike in a run evolved into sharply tuned gamma entrainment over the next few spikes. The data and modeling results indicate that for FSIs firing in brief runs and pauses firing within runs is entrained by gamma-frequency components of the input, whereas the onset timing of runs may be sensitive to a wider range of stimulus frequency components.

NEW & NOTEWORTHY Specific types of neurons entrain their spikes to particular oscillation frequencies in their synaptic input. This entrainment is commonly understood in terms of the subthreshold voltage response, but how this translates to spiking is not clear. We show that in striatal fast-spiking interneurons, entrainment to gamma-frequency input depends on rhythmic spike runs and is explained by the phase resetting curve, whereas run initiation can be triggered by a broad range of input frequencies.

INTRODUCTION

Neurons respond unequally to different frequency components present in their synaptic input, providing selectivity that contributes to the distinct functions of each cell type. A major cell type known to possess strong frequency selectivity is the parvalbumin-expressing fast-spiking interneuron (FSI) found in the cerebral cortex, hippocampus, amygdala, and striatum (Beatty et al. 2015; Bracci et al. 2003; Goldberg et al. 2008; Golomb et al. 2007; Sciamanna and Wilson 2011; Tateno et al. 2004; Woodruff and Sah 2007). FSIs are tuned to gamma-frequency (30–100 Hz) components of their synaptic input. Gamma resonance is evident from subthreshold oscillations observed at membrane potentials near spike threshold (Bracci et al. 2003; Golomb et al. 2007), an abrupt onset of firing at a nonzero minimum frequency (Erisir et al. 1999; Mancilla et al. 2007; Martínez et al. 2017; Sciamanna and Wilson 2011; Tateno et al. 2004; Tikidji-Hamburyan et al. 2015), and modulation of spike probability that is strongest for input of nonzero frequency, a phenomenon termed spiking resonance (Beatty et al. 2015; Pike et al. 2000). In vivo, the spiking resonance of striatal FSIs likely contributes to spike coherence with gamma oscillations detected in the cortex (Sharott et al. 2009, 2012) and in the striatal local field potential (Berke 2009; van der Meer and Redish 2009). This spike-field coherence is thought to reflect a modulation of spike probability by oscillating components of the synaptic input stream shared among neurons in the local area (Wilson et al. 2018).

The resonant properties of FSIs have been explained by the balance between a relatively small persistent sodium current and a large potassium current primarily mediated by K_v3 and K_v1 channels (Erisir et al. 1999; Goldberg et al. 2008; Golomb et al. 2007; Gu et al. 2018). K_v3 channels are critical to generation of brief spikes, for which FSIs are named, whereas K_v1 channels are required to produce the abrupt onset of repetitive firing and high minimum rate. Because of its slow, voltage-dependent inactivation and deinactivation, the K_v1 current can also explain the characteristic stuttering pattern of spike runs and pauses that is particularly robust in striatal FSIs (Sciamanna and Wilson 2011).

Despite this biophysical understanding, it has remained unclear how the intrinsic resonance affects the responses of FSIs under realistic conditions, when “fast-spiking” neurons do not usually fire rhythmically (Berke 2011). Under natural conditions in vivo, the characteristic firing patterns observed in brain slice preparations are presumably disrupted by fluctuating synaptic input (Beatty et al. 2015). To understand the role of spiking resonance under natural conditions, we need to know whether it arises directly from subthreshold membrane potential oscillations or depends on runs of fast spiking. If spiking resonance is primarily of subthreshold origin, it may be prominent even during sparse firing when spike runs and pauses are brief and highly variable. Conversely, if spiking resonance depends on rhythmic firing, it could be restricted to periods of sustained synaptic input sufficient to produce longer spike runs.

The studies reported here investigated the relationship between firing patterns and spiking resonance in striatal parvalbumin-expressing FSIs, using noise stimuli applied at different direct current (DC) levels to elicit firing with varying mean rates and degrees of rhythmicity. An analysis was performed to dissect the contribution of spike sequences to spiking resonance. To aid in understanding the results, a phase oscillator model was constructed based on experimental phase resetting curve (PRC) data obtained from FSIs producing spike runs.

MATERIALS AND METHODS

Animals.

All animal procedures were prospectively approved by The University of Texas at San Antonio Institutional Animal Care and Use Committee. For identification of striatal FSIs, the studies were performed with brain slices prepared from 4- to 12-wk-old male and female parvalbumin-tdTomato mice, which were F₁ progeny of parvalbumin-Cre [B6;129P2-Pvalb^tm1(cre)Arbr/J] and Cre-dependent tdTomato reporter [B6.Cg-Gt(ROSA)26Sortm14(CAG-tdTomato)Hze/J] mice.

Brain slice preparation.

The mouse was deeply anesthetized with isoflurane and euthanized by decapitation. Parasagittal brain slices (300 μm) were prepared by standard methods with a Vibratome (Leica VT1000S). The cutting solution contained (in mM) 2.5 KCl, 1.25 NaH₂PO₄, 0.5 CaCl₂, 10 MgSO₄, 10 d-glucose, 26 NaHCO₃, and 202 sucrose (pH 7.4), which was bubbled with 95% O₂-5% CO₂ and chilled on ice before use. After cutting, slices were stored in artificial cerebrospinal fluid (ACSF) containing (in mM) 126 NaCl, 2.5 KCl, 1.25 NaH₂PO₄, 2 CaCl₂, 2 MgSO₄, 10 d-glucose, and 26 NaHCO₃ bubbled with 95% O₂-5% CO₂. The slice storage ACSF (but not the recording ACSF) also contained (in mM) 0.005 l-glutathione, 1 Na pyruvate, and 1 Na ascorbate. The slices were maintained at room temperature until use.

Recording.

A slice was transferred to the recording chamber and superfused with ACSF heated to 34°C. tdTomato-expressing striatal interneurons were identified by epifluorescence and Dodt contrast imaging with an Olympus BX51WI microscope. All recordings were obtained with the gramicidin-perforated patch method. Patch pipettes (~5 MΩ) were pulled from borosilicate glass (Warner Instruments G150F-4) and filled with patch solution containing (in mM) 140 K methylsulfate, 7.5 NaCl, 10 HEPES, and 0.1 phosphocreatine, with ~0.6 μg/ml gramicidin D (MP Biomedicals). After the on-cell recording configuration was established, the access resistance was allowed to decrease to ~50 MΩ before an experiment began. The apparent amplitude of action potentials increased during perforation, and overshoot of 0 mV was usually attained. Current injection and recording were performed with a MultiClamp 700B amplifier (Molecular Devices). The main experimental stimuli were created off-line with Mathematica (Wolfram) and were applied with a dedicated computer running RTXI (the Real-Time eXperiment Interface, http://rtxi.org) with a National Instruments PCIe-6251 A/D board. The voltage data were low-pass filtered at 10 kHz and sampled at 20 kHz.

Initial cell characterization.

To verify that each recorded neuron was physiologically fast spiking, and to measure the rheobase, the cell was injected with a 10-s current ramp from 0 to 1,000 pA. A few cells did not fire at 1,000 pA and were injected with a longer ramp of the same slope. All cells included in the data set showed narrow action potentials (~0.3-ms width at half-height) with prominent fast afterhyperpolarizations, and most began firing abruptly at high frequency, often after a period of fast, noisy oscillations. The rheobase was taken as the ramp current at the time of the first spike.

Noise stimulation.

To characterize spiking resonance across a range of firing patterns, each neuron was stimulated with a set of 5-s current steps without and with noise. The stimuli were scaled based on each cell’s rheobase. In the main experiment, steps were presented at DC levels of 1, 1.1, and 1.2 × rheobase, each without and with noise. The noise was made up of contiguous 0.5-ms square current pulses with amplitudes drawn from a Gaussian distribution (SD = 0.1 × rheobase). The current steps were applied at a 15-s interval, and the sequence of six steps was repeated up to 20 times. For data analysis, the range of stimulus episodes analyzed was limited to the period when the mean firing rates for each stimulus condition were stable.

Analysis of spike entrainment.

All analyses were performed with Mathematica. Spikes were detected on their rising phase, and the time of each spike was recorded at the detection point. Because most cells showed spike frequency adaptation during the initial portion of each current step, subsequent analysis was restricted to the steady-state responses during the second half of the steps (2.5–5 s).

Spike entrainment was measured essentially as described by Beatty et al. (2015). Frequency bands centered at 10–300 Hz (10-Hz bandwidth at half-amplitude, 5-Hz spacing) were extracted from the noise stimuli by Fourier transformation, multiplication by a Tukey window function, and inverse Fourier transformation. For each frequency component, the stimulus phase, θ(t), was taken as a linear ramp from 0 to 1 interpolated between successive upward zero-crossings. Each spike time, t_j, was converted to the corresponding stimulus phase, θ_j = θ(t_j), and a normalized vector sum or resultant vector (R) was calculated:

$$\mathbf{R}=\frac{{\sum }_{j=1}^n\text{cos}\left[2{\text{πθ}}_j\right]+i{\sum }_{j=1}^n\text{sin}\left[2{\text{πθ}}_j\right]}{n}$$ (1)

For each frequency, the vector strength (VS) and the vector phase were obtained as the magnitude and angle of R. For measurement of spiking resonance, the peak of the VS spectrum was fitted with a piecewise bilinear function. The resonance frequency (f_res) was taken as the intersection of the rising and falling segments, and the resonance strength (s_res) was taken as the relative increase in VS from 0.5f_res to f_res:

$$s_{\text{res}}=\frac{\text{VS}\left(f_{\text{res}}\right)}{\text{VS}\left(0.5f_{\text{res}}\right)}-1$$ (2)

Contribution of regular interspike intervals to entrainment.

This analysis identified the contribution of the previous spike times to the resultant vector R, given the observed interspike intervals (ISIs). The analysis was performed as described above, except that the actual spike times were replaced by a surrogate data series comprised of a set of potential spike times. To obtain each value in the surrogate data, an actual spike time and an actual ISI were selected randomly and independently from the data and were added to generate the potential spike time. For each frequency, the analysis yielded a resultant vector R_P quantifying the entrainment expected to be inherited from the previous spike time, based on the harmonic relationship between the average intrarun firing rate and the stimulus frequency. It was assumed that the total modulation of spike probability by each frequency component of the stimulus is a linear combination of 1) the effect mediated by the timing of the previous spike and the cell’s tendency to fire regular ISIs and 2) the remaining, direct effect of the stimulus. Thus the direct effect of the stimulus was estimated by the difference vector R_D = R − R_P, and the corresponding vector strength was obtained as the magnitude of R_D.

Phase resetting curve estimation.

The PRC quantifies the sensitivity of a neuron to input arriving during rhythmic firing, measuring the change in spike time per unit of injected charge as a function of the position in the ISI (phase) at which the input arrives. Striatal FSIs alternate between rhythmic firing (runs), during which the PRC should apply, and pauses, when subsequent firing is not predicted in any obvious way by the PRC. Thus we estimated the PRCs of FSIs based on the ISIs within spike runs, excluding the pauses. As shown in results, the frequency of resonant firing is approximated well by the mode of the ISI distribution (T_mode) (see Fig. 2E). Thus a cutoff ISI of 1.5 × T_mode was used to distinguish the intrarun ISIs from the pauses. PRCs were then measured essentially as described by Wilson et al. (2014), using the intrarun ISIs during noise stimulation at a DC level of 1.2 × rheobase. The noise stimulus spanning each ISI was subdivided into a set of equal-length bins, where the number of bins was set to T_mode/0.5 ms, rounded to the nearest integer. The charge injected (Q) was determined for each bin. The PRC values (Z) were then determined by a multiple-regression analysis in which the independent variables represented Q for each bin and the dependent variable was the ISI length. The regression coefficients were normalized by the mean ISI to provide the Z values (cycles/pA ms). The Z values were expressed as a function of phase (ϕ) on the interpolated-phase assumption (i.e., that the expected value of ϕ for a spiking oscillator perturbed with zero-mean noise is equal to the time from the first spike divided by the total length of the perturbed ISI). In most cells, the estimated PRC terminated with one or two Z values < 0. This is a known error associated with PRC estimation (Polhamus et al. 2011), which results from phase uncertainty near the end of the ISI in the presence of intrinsic noise. Because the neuron is expected to lose sensitivity to external input as the action potential is initiated, these bins were removed from the analysis and the corresponding Z values were taken as 0.

Fig. 2.

Analysis of spiking resonance. A: example of spike response to a noise stimulus, showing an individual frequency band (center frequency = 70 Hz, bandwidth = 10 Hz) extracted from the stimulus with a Tukey window function (left inset; see materials and methods). Red vertical lines indicate the alignment of each spike on the stimulus component. Top right inset shows the convention for assigning spike phases (ϕ) with respect to a stimulus component. Bottom right inset illustrates the vector representation of spike phases. The 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. Black dashed lines indicate the bilinear fit used to measure the resonance frequency and strength at each direct current (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. intrarun firing rate (reciprocal of the mode of the interspike interval distribution).

Simulation of spike runs.

A two-mode model including a linear-nonlinear (L-N) component (paused mode) and a phase oscillator component (run mode) was constructed to simulate the initiation and continuation of spike runs. The noise stimulus applied to the model was similar to the experimental stimuli, with a pulse width of 0.5 ms and a standard deviation of 50 pA. Simulations were performed with a time increment of 0.05 ms.

For the L-N mode, the noise current was convolved with a linear filter, F(Δt), to produce an activation variable, S(t). The linear filter was constructed from the spike-triggered average current (I_STA), subtracting the corresponding average current triggered on the potential spike times described above (I_PSTA) and reversing the waveform in time to represent the forward transformation from current to S(t):

$$F\left(\text{Δ}t\right)=I_{\text{STA}}\left(-\text{Δ}t\right)-I_{\text{PSTA}}\left(-\text{Δ}t\right)$$ (3)

$$S\left(t\right)=I_{\text{noise}}\left(t\right)\ast F\left(\Delta t\right)$$ (4)

The mean rate of run initiation, r, was represented by an exponential function of S, where r₀ was set to reproduce the mean experimental interrun interval of ~60 ms obtained in the experimental data at a DC level of 1.2 × rheobase in the absence of noise, and the exponent, k, was set to reproduce the mean interrun interval of ~33 ms in the presence of noise.

$$r\left(S\right)=r_0\text{exp}\left(kS\right)$$ (5)

Simulation trials (each representing a run of 10 spikes) were started at random times within the noise stimulus. The first spike time was produced by a rate-modulated Poisson process, drawing a random value (0 < z < 1) from a uniform distribution at each time step and registering the spike time if z < r(S)∆t. After producing the first spike, the model switched to the phase-oscillator mode.

The phase-oscillator mode was simulated using the average PRC from 14 FSIs, each resampled by linear interpolation at 100 phases. The rising and falling portions of the average PRC were fitted separately with fourth-order polynomial functions, constrained to intercept the same peak point. The model PRC was then constructed as a composite of the rising function, the falling function, and the final zero-valued segment.

$$Z\left(\varphi \right)=\left\{\begin{array}{c}-0.000121884\varphi -0.00735335{\varphi }^2+0.0247837{\varphi }^3-0.0147992{\varphi }^4,\varphi <0.86\\ 0.248874-2.29133\varphi +5.63961{\varphi }^2-5.38062{\varphi }^3+1.78264{\varphi }^4,0.86\le \varphi <0.97\\ 0,\varphi \ge 0.97\end{array}\right\}$$ (6)

At the first spike time (generated by the L-N mode), the phase, ϕ, was initialized at 0, representing the first spike reset. The phase then evolved according to the phase equation:

$$\frac{\text{d}\varphi }{\text{d}t}=\text{ω}+I\left(t\right)Z\left(\varphi \right)$$ (7)

When ϕ = 1, a spike time was collected and ϕ was reset to 0. Each simulation was continued until 10 spike times were collected, and the next trial was started at a different random time in the stimulus waveform.

RESULTS

Current-clamp recordings were obtained from striatal FSIs in brain slices from parvalbumin-tdTomato mice, with the gramicidin-perforated patch method used to maintain intact firing properties (see materials and methods). The rheobase of each cell was estimated with a slow current ramp (100 pA/s), and the cell was then stimulated with noiseless and noisy current steps (5 s each) scaled to DC levels of 1, 1.1, and 1.2 × rheobase. Analysis was restricted to the steady-state response in the second half of each step (2.5–5 s). An example of the responses is shown in Fig. 1.

Fig. 1.

Effects of direct current (DC) and noise on stuttering. The steady-state firing of a typical striatal fast-spiking interneuron is shown at 3 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. Insets show the noisy subthreshold oscillations observed during pauses (expanded from the red boxed regions).

Most cells produced a stuttering firing pattern at DC levels near rheobase. The lengths of the spike runs were variable but increased with greater DC. During pauses, as reported previously (e.g., Bracci et al. 2003), many cells showed subthreshold membrane potential oscillations with frequencies similar to the intrarun firing rates (Fig. 1, insets), although their amplitudes and periodicity were variable. Adding noise reduced the regularity and length of the runs and the length of the pauses, making the stuttering patterns less distinct but still apparent in most recordings.

Characterization of spiking resonance.

To measure spiking resonance, defined as the frequency-dependent modulation of spike probability by components of the broadband noise stimulus, the noise was filtered into a set of band-pass components. The phase of each spike was then identified with respect to each frequency component (Fig. 2A; see materials and methods). From the spike phases, a vector average (resultant vector) was computed for each frequency. The length of the resultant vector, or vector strength (VS), quantifies the strength of phase entrainment, and the angle (vector phase) is the average phase of spiking. A peak in the VS spectrum at nonzero frequency indicates that the neuron had spiking resonance. The noise stimulus was a series of contiguous 0.5-ms pulses and thus had a flat power spectrum up to 500 Hz, so the frequency dependence of VS over the illustrated range did not result from variation of stimulus power. As the DC level was increased, the peak of the VS spectrum shifted slightly to the right and became sharper (Fig. 2B). The largest effects of DC were seen below the resonance frequencies, where higher DC levels reduced the VS.

The spectra of vector phase also showed a pattern consistent with spiking resonance (Fig. 2C). The phases were defined with respect to a sine wave, so a phase of 0.25 (dashed line in Fig. 2C) indicates that the cell fired most often at the peak of the stimulus current (i.e., with zero phase lag). The phase spectra were typical of resonant neurons, showing a transition from phase lead (phase < 0.25) to phase lag near the resonant frequency. The frequency of zero lag has occasionally been used as a measure of resonance (e.g., Fuhrmann et al. 2002) and has been referred to as the phasonance frequency (Rotstein 2017). Like the resonance frequency, the phasonance frequency increased with the DC level.

To measure each VS spectrum, we fit the peak with a bilinear function, taking the resonance frequency (f_res) as the intersection of the rising and falling segments. Across the 14 cells tested, with data obtained at one to three DC levels each, the values of f_res were either similar to or higher than the mean firing rates under the corresponding stimuli (Fig. 2D) but were consistently similar to the intrarun firing rates, measured as the reciprocal of the mode of the ISI distribution (Fig. 2E). The f_res values showed a large heterogeneity among FSIs, ranging from ~20 to 120 Hz. They also showed a smaller change in individual cells as the DC level was varied from 1 to 1.2 × rheobase. The phasonance frequencies showed similar patterns and were highly correlated with f_res (r = 0.95), but the values were slightly lower (87 ± 10% of f_res).

Spike runs and spiking resonance.

In a neuron firing sparsely, with spikes driven by input fluctuations, spiking resonance can arise from subthreshold mechanisms (Richardson et al. 2003). In qualitative support of this possibility, the presence of gamma-frequency membrane potential oscillations leading up to spikes suggests a potential subthreshold origin of spiking resonance in FSIs (Bracci et al. 2003; Schulz et al. 2011). However, rhythmic spike runs provide another, potentially stronger mechanism of resonance that may dominate when these firing patterns emerge (Agüera y Arcas et al. 2003; Beatty et al. 2015; Wilson 2017). In this mechanism, ionic currents triggered by action potentials, rather than subthreshold oscillations, provide a temporal window of opportunity for perturbation of the next spike time, and the sequence of these windows underlies spiking resonance.

To investigate how spiking resonance depends on spike runs, we segregated each spike train into runs separated by pauses, which were defined by an ISI greater than 1.5 times the mode (Fig. 3A). For this analysis, single spikes preceded and followed by a pause were considered as runs of length 1. The run lengths increased with the DC level of the stimulus, particularly in the absence of noise but also when noise was present (Fig. 3B). The resonance strength (s_res), measured as the relative increase in VS from 0.5f_res to f_res based on the fits described above, was compared to the mean run length (Fig. 3C). Individual FSIs consistently showed low s_res at DC levels producing short runs but had much larger s_res at higher DC levels generating longer runs. These data suggest that spike runs made a large contribution to spiking resonance.

Fig. 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(f_res)/VS(0.5f_res) − 1 (where VS is vector strength and f_res 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).

Dissecting the influence of sequential spikes.

During a run, each spike activates intrinsic currents that combine with the extrinsic current input to determine the time of the next spike. This phenomenon, sometimes called spike history, can have large effects on stimulus-response relationships in models and in real neurons (Agüera y Arcas et al. 2003; Mease et al. 2014). To dissect the influence of a preceding spike from the remaining effect of the stimulus, we created a set of potential spike times by adding randomly chosen ISIs to randomly chosen spike times from the same stimulus episodes (Fig. 4A; see materials and methods). These are times at which an action potential is statistically likely, based on the time of a preceding action potential. They should be subject to the same spike history effects as the times of actual spikes. The average stimulus currents associated with the real and potential spike times (STA and PSTA, respectively) are illustrated in Fig. 4B. Under conditions producing spike runs, the STA typically showed two noticeable cycles of oscillation. However, the early oscillatory component also appeared in the PSTA, indicating that this was the average current associated with the previous spikes, irrespective of the presence of the reference spike. Thus the PSTA was subtracted from the STA to obtain the difference current associated with the reference spike. The difference current preserved the final dip and peak of the STA but did not show the preceding oscillation. This analysis indicates that sequences of two or more spikes contributed to the oscillatory component of the stimulus-spike relationship.

Fig. 4.

Contribution of sequential spikes to spiking resonance. A: method for analysis of potential spike times. Each trace shows a randomly chosen interspike interval (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) compared with potential-spike-triggered average current (PSTA). 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 1 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 (center) and the vector differences (right). Each point represents a resultant vector (R) measuring the phase distribution with respect to 1 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 direct current (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 with the difference spectrum indicates that sequences of spikes contributed to spiking resonance. E: summary data comparing the peak frequencies (top) and resonance strengths (bottom) 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.

To determine how sequential spikes affected our band-pass analysis of spiking resonance, the phases of the potential spike times were determined with respect to each component of the stimulus, yielding a resultant vector R_P for each frequency (Fig. 4C, center). Because the amplitude of each stimulus component was small, we assumed that the perturbation of spike probability mediated by the previous spike was additive with the remaining effect of the stimulus. Thus we subtracted each R_P from the original resultant vector R to obtain a difference vector R_D representing the effect of the stimulus that was not accounted for by the previous spike (Fig. 4C, right). From the polar plots in Fig. 4C, we see that the magnitudes of R_D (i.e., the distance of each point from the origin) were less sharply frequency dependent compared with those of the original R. The reason for this can be understood by examining the 70 Hz and 35 Hz points on the polar plots. At 70 Hz, R_P and R_D show approximate phase alignment, indicating that the previous spike and the subsequent stimulus worked together to modulate spike probability. In contrast, at 35 Hz, R_P and R_D are out of phase, showing that the previous spike and the subsequent stimulus had opposing effects. Thus the net VS was reduced at 35 Hz.

The overall effect of previous spikes on the VS spectrum is shown in Fig. 4D, comparing the original spectrum (from R) to the difference spectrum (from R_D). This analysis indicates that sequences of two or more spikes sharpened the peak in the original spectrum. The difference spectrum shows the spiking resonance that could be expected for spikes following pauses, which do not benefit from the spike history effect. Spiking resonance is present but is weaker and broader, especially in the direction of low frequencies.

To quantify these effects in each cell, at each DC level, the difference spectra were measured as described above and characterized by their peak frequency and resonance strength as described above. The values obtained were compared with the corresponding data for the original spectra (Fig. 4E). For the original spectra, the peak frequencies and resonance strengths consistently increased with the DC level [P = 0.0006 and 0.00004, respectively (n = 13), for 1.1 vs. 1.2 × rheobase]. In contrast, the difference spectra did not show any significant relationship between DC and peak frequency or resonance strength. Compared with the original spectra, the difference spectra showed lower resonance strength at 1.1 and 1.2 × rheobase (P = 0.0002 and 0.00008, respectively). These data indicate that sequences of spikes were largely responsible for the strength and stimulus dependence of spiking resonance.

Input sensitivity during spike runs—the phase resetting curve.

The results described above suggest that a spike-driven entrainment process occurring within runs strengthens spiking resonance. To understand this process, we first characterized how the stimulus affects spike timing during runs. Considering the rhythmically firing neuron as an oscillator, the appropriate measure of spike time perturbation is the PRC, which quantifies the change in ISI length produced by external current delivered at each phase of the spiking oscillation (Perkel et al. 1964; Reyes and Fetz 1993; Stiefel and Ermentrout 2016).

We estimated the PRC of each neuron based on the intrarun ISIs (defined by lengths <1.5 times the mode of the ISI distribution) during noise stimulation at a DC level of 1.2 × rheobase, using a multiple regression method to compute the PRC (Wilson et al. 2014; see materials and methods). An example PRC is shown in Fig. 5A. The illustrated PRC has a very small negative lobe at early phases and a much larger positive lobe starting just beyond a phase of 0.5, indicating that the neuron was most sensitive to current arriving at late phases. The estimated PRCs for the sample of FSIs are superimposed in Fig. 5B. It is apparent that the PRC shapes and amplitudes were similar among these cells, so for use in a model (see below) the data were resampled and averaged. The average PRC shows a small negativity at phases up to ~0.4, followed by a larger positive lobe. The presence of a negative lobe identifies the PRC as type 2 (Hansel et al. 1995) and means that depolarizing stimuli arriving early in the ISI cause a small delay of the next spike time. This PRC shape is typical of neurons with class 2 excitability, which show a minimum frequency of rhythmic firing in response to constant inputs (Brown et al. 2004; Ermentrout 1996; but see Ermentrout et al. 2012).

Fig. 5.

The phase resetting curve (PRC). A: example PRC computed from the perturbation of intrarun interspike intervals (ISIs) in a fast-spiking interneuron (FSI) (see text for details). The PRC shows a very slight negativity for phases up to ~0.5, followed by a much larger positive lobe 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 direct current 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.

Simulating frequency-dependent entrainment.

To investigate the basis of entrainment during spike runs, we constructed a model with two modes: a linear-nonlinear (L-N) mode simulating the onset of each spike run and governed by the difference STA shown in Fig. 4 and a phase-oscillator mode based on the average PRC, which produces the following spike times in the run. The operation of the model is described in materials and methods and illustrated in Fig. 6. For the L-N mode, the linear filter was the difference STA, reversed in time to describe the forward transformation from current to a variable predicting spike probability. The biphasic shape of the linear filter accounts for the frequency selectivity of the FSI that remains in the absence of recent spikes. The noise current was convolved with the filter to generate a filtered stimulus variable, S(t), and the probability of starting a spike run was represented by an exponential function of S. Upon producing the first spike, the model entered the phase-oscillator mode governed by the phase equation: dϕ/dt = ω + I(t)Z(ϕ). On each trial the phase equation was integrated numerically until spikes 2–10 were collected, and the process was repeated to produce 10,000 runs of 10 spikes each. The model spike times were analyzed by identifying the phase of each spike with respect to the stimulus frequency components, as described above.

Fig. 6.

Simulating spike runs. The figure illustrates the generation of 1 spike run. A: noise current [I(t)] providing input to the model. The direct 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 postspike 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.

The results of the simulations are illustrated in Fig. 7, focusing on stimulus components centered at 25 Hz (0.5ω), 50 Hz (ω), and 75 Hz (1.5ω). Histograms of spike phases show that for the first spike in a run the depth of modulation of spike probability was similar for the three frequencies (Fig. 7A, top). This is consistent with the broad spectrum of spiking resonance shown in Fig. 4D, which is expected for spikes that are not influenced by previous spikes. However, for spikes occurring later in each run, the 50-Hz stimulus component (corresponding to the intrarun firing rate) had an increased influence on spike probability, whereas for the 25- and 75-Hz components the depth of modulation decreased (Fig. 7A, middle and bottom). To quantify the evolution of entrainment, the VS was computed for each spike in the run, for each stimulus frequency (Fig. 7B). For the first spike in the run, VS was similar for the three highlighted frequencies. The VS for the 50-Hz component then increased from spikes 2 to 4, whereas the VS for the 25- and 75-Hz components decreased.

Fig. 7.

Evolution of entrainment across simulated spike runs. A: model spike phases with respect to the 25-Hz, 50-Hz, and 75-Hz band-pass components of the stimulus. The 50-Hz component matches the oscillation frequency (ω) of the phase-oscillator model. Top: the phase distributions for the first spikes, produced by the linear-nonlinear component of the model. Middle and bottom: the distributions for spikes 2 and 10, respectively. The modulation of spike probability was initially similar for the 3 frequencies but progressively strengthened at 50 Hz and weakened at 25 and 75 Hz. B: vector strength (VS) vs. spike number for each frequency. The VS values approached steady state by spike 5. For the resonant frequency (50 Hz) the approach to steady state was approximately exponential, whereas for some other frequencies (i.e., 25 Hz) 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. The yellow line is the deterministic map for a pure sine wave input matching the average frequency and amplitude of each band-pass 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.

This process of frequency-dependent spike entrainment or disentrainment can be understood based on iterative maps comparing the phases of sequential spikes. Figure 7C shows the maps for the 25-Hz, 50-Hz, and 75-Hz components of the stimulus. Figure 7C, top, compares the phases of the second and first spikes in a run, relative to each input component. For the 50-Hz input, the deterministic map crosses the identity line (Figure 7C, top) at two points. These are fixed points where sequential spike phases on the pure sine wave remain the same. The lower fixed point at a phase of ~0.22 is stable, whereas the upper fixed point is unstable. During rhythmic firing, the phases are attracted toward the stable fixed point, as illustrated in Fig. 7D for a single 50-Hz sine wave input of larger amplitude, which produces a larger curvature in the map. With the broadband noise stimulus, the attraction is balanced by diffusion caused by the other input components, resulting in a broad distribution of phases around the fixed point. In contrast, for input frequencies sufficiently far from ω (e.g., 25 and 75 Hz), the deterministic map has no stable fixed point. In these cases, each spike is followed by a next spike at a different phase, and the initial concentration of phase probability dissipates as the run proceeds.

The net effect of the entrainment process is illustrated by comparing the VS spectra for spikes 1, 2, 3, 4, and 10 (Fig. 7E). For spike 1, the frequency tuning was broad (similar to the difference spectrum in Fig. 4D). Over the next several spikes the resonance progressively sharpened, approaching a steady state resembling the overall VS spectra from FSIs (e.g., Figure 4D, original spectrum) by around spike 4. This run length is well within the range of our experimental data obtained with noise (Fig. 3). Thus the simulations demonstrate that the frequency-selective entrainment produced by a phase oscillator with a typical FSI PRC can account for the spiking resonance associated with longer spike runs.

The simulations did not address the increase in resonance frequency observed with higher DC. Although it was possible to run the model with a DC offset, the changes in intrarun firing rate did not correspond well to those observed in real neurons (data not shown), indicating a limitation of the model in this regard. However, the absence of any manipulations of subthreshold properties (i.e., the L-N mode) or oscillation frequency in the model demonstrated that increased run length was a sufficient explanation for the stronger resonance observed with higher DC levels in the real neurons.

Effects of PRC shape.

Frequency-dependent spike entrainment depends on the neuron’s differential sensitivity to input arriving at different phases of the spiking oscillation, as characterized by the PRC. To illustrate this, we compared simulations using the FSI PRC to otherwise identical simulation runs using three artificial PRC waveforms (Fig. 8A). The flat PRC defines a perfect integrator neuron, which responds equally to input arriving at any phase of the spiking oscillation. The type 1 (always positive) cosine PRC is a canonical form representing a class 1 neuron operating near its saddle-node bifurcation (Ermentrout 1996). The type 2 sine PRC is in the same category as the FSI, but its two lobes are of equal size and its peak is at an earlier phase compared with the FSI PRC. For ease of comparison, the three artificial PRCs were scaled to the same mean squared amplitude as the FSI PRC. Thus the different PRCs are predicted to produce ISIs with the same variance in response to the same noise stimuli (Ermentrout et al. 2011; Wilson et al. 2014). Because the artificial PRCs apply only to neurons firing repetitively, each simulation trial was initialized by starting a run at a random time point in the noise stimulus.

Fig. 8.

Entrainment simulated with different phase resetting curve (PRC) shapes. Spike runs were simulated as described above, except that the initial spike times were chosen randomly without using the linear-nonlinear model component. A: PRC shapes used for simulations: the mean fast-spiking interneuron (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 (VS) vs. spike number. In each case the VS for spike 1 is low because the spike times were random, giving random initial phases. In simulations using the 3 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 3 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.

The evolution of entrainment across the simulated spike runs is illustrated in Fig. 8B, and the VS spectra for spike 10 are shown in Fig. 8C. The flat PRC allows a slight entrainment, because external input does alter the model’s rate of phase advance, but the final VS is independent of input frequency, showing that frequency-selective entrainment will not occur unless the cell’s input sensitivity varies across the ISI. In contrast, the type 1 cosine and type 2 sine models both show maximal entrainment at input frequencies around ω, with lower VS at other frequencies. The type 2 sine PRC produces the strongest entrainment at f = ω. This is expected because the sine PRC has a single frequency component of one cycle per ISI (approximately speaking, f = ω) and no DC (0 Hz) component, and thus the input component at f = ω has the largest influence on spike timing. Compared with these purely sinusoidal PRCs, the FSI PRC produces an intermediate VS at f = ω but also an entrainment peak at f = 2ω and overall higher values of VS at frequencies above that. These differences presumably reflect the presence of higher-frequency components in the PRC shape, which confer responsiveness to corresponding stimulus frequencies (Goldberg et al. 2013). In particular, the second harmonic component in the PRC will contribute to the VS at f = 2ω, and the higher-frequency components required to produce the rapid fall-off at the right side of the PRC are expected to increase the VS at higher frequencies. Thus the distinctive shape of the FSI PRC confers sensitivity to a wide range of fast signals.

DISCUSSION

Firing patterns of striatal FSIs.

When made to fire by constant-current pulses, striatal FSIs stutter, firing variable-length runs of spikes separated by pauses. Stuttering is associated with the existence of a minimum firing rate below which the cell cannot fire rhythmically. This rate differs across neurons, ranging from ~20 to 120 spikes/s in this sample or even as high as 250 spikes/s (Beatty et al. 2015). The intrarun firing rate increases with the amplitude of constant current, but the minimum rate remains relatively constant over repeated episodes of stimulation. The average firing rates of FSIs vary much more continuously and over a larger range, because runs become longer and pauses shorter with increases in injected current. In the absence of fluctuating input, the transitions from pause to run and vice versa are controlled mainly by gradual changes in the inactivation state of a potassium current (Golomb et al. 2007; Rush and Rinzel 1995) that has been identified as K_v1 (Sciamanna and Wilson 2011).

In vivo, FSIs continue to exhibit bursts that resemble runs and pauses seen in slices, but both run and pause durations are much more variable and runs are intermixed with isolated spikes (Berke 2011). Presumably, the mechanisms underlying stuttering continue to operate in vivo, but both pauses and runs are truncated by synaptic input. A similar disruption of the stuttering firing pattern was seen previously with artificial synaptic conductance or current barrages (Beatty et al. 2015; Klaus et al. 2011) and here with noisy barrages of current pulses. In these cases, the transitions from pause to burst and from burst to pause happen too often to be determined by inactivation of K_v1 current but rather are triggered by fluctuations in synaptic current that transition the cell between simultaneously stable resting and rhythmic firing states (Golomb et al. 2007; Rush and Rinzel 1995).

Previous studies of spiking resonance suggested that near rheobase the FSI responds selectively to input frequency components in the gamma range, and specifically those that match each cell’s own minimum (intrinsically preferred) firing rate (Beatty et al. 2015; Sciamanna and Wilson 2011). This would be consistent with the dynamical systems description of the FSI presented by Rush and Rinzel (1995) and Golomb et al. (2007), in which stuttering is viewed as noise-induced transitions between the firing and resting states at a Hopf bifurcation. In these models, the membrane resonant frequency of the stable state and the rate of repetitive firing at the same input current are nearly the same. The transition to the firing state would be most likely when the input frequency matches the peak frequency of the subthreshold resonance. In this view, the FSI is responding to a narrow range of frequencies at all times, even when stuttering. Runs would occur when the gamma component of the cell’s input is large, and firing would be entrained by the same gamma signal.

In contrast, our results suggest that the synaptic signal required to initiate a run of firing in the FSI and that required to phase-entrain the cell during a run may not be identical. Once firing a run, the neurons did show a strong resonance effect, responding to coherence between the gamma component of the current input and the previous history of spiking at the same rate. The occurrence of previous action potentials in a run created an oscillatory change in input sensitivity that could interact with a frequency component of the input signal to entrain the neuron’s firing. The entrainment accumulated over several action potentials (up to ∼4), producing stronger entrainment under conditions generating long runs. This mechanism could produce reliable entrainment, even when the input component near the resonant frequency was small and buried in a noisy barrage of signals over a wide range of other frequencies. On the other hand, run initiation did not show a strong preference for input signals in the gamma frequency range, although that transition was driven most effectively by nonzero frequency components. The different frequency selectivity for run initiation versus spike timing within a run provides an opportunity for interaction between high- and low-frequency components of striatal afferent signals that control the activity of FSIs.

The importance of PRC shape.

Although the minimum firing rate constrains the frequency of spiking resonance, the PRC shape provides the mechanism of frequency-dependent entrainment. In the hypothetical case of a perfect integrator neuron with a flat PRC, the spiking rhythm would have no impact on the cell’s input sensitivity, and thus the cell would not selectively lock to input at a frequency matching its own firing rate. The curvature of the PRC creates the curvature of the phase map, and when the stimulus frequency is similar to the cell’s natural firing rate that curvature creates a stable fixed point, or locking phase. Compared with the broad-shaped PRCs of many class 1 neurons, the PRCs of FSIs show highly phase-dependent responses, with weak negative sensitivity over the first half of the ISI followed by strong positive sensitivity peaking late in the cycle. These PRC shapes produce substantial curvature in the phase map, even for small sinusoidal inputs such as the individual frequency bands from our broadband noise stimuli, and this allows locking to occur.

In addition to the general type 2 shape with negative and positive lobes, the sharp fall of the PRC is also important for the frequency response properties of FSIs. For transient input of small amplitude, it has been shown that the poststimulus time histogram is proportional to the derivative of the PRC, but reversed in sign and in time (Gutkin et al. 2005). Thus a PRC falling sharply at the right side provides a rapid, short-latency response to transient input. Similarly, it was shown that for small-amplitude white noise input the I_STA is proportional to the derivative of the PRC but reversed in sign (Ermentrout et al. 2007). Thus the falling phase of the PRC gives rise to the fast positive peak of the STA, whereas the rising phase of the PRC produces the slower negative portion of the STA. Although this relationship is altered with input currents of finite amplitude, it illustrates the close connection between the PRC and the input selectivity of the neuron. In general, fast features in the PRC provide sensitivity to fast components in the cell’s input, and sinusoidal components of the PRC waveform are the primary determinants of spike responses to corresponding input frequencies (Goldberg et al. 2013; Tiroshi and Goldberg 2019).

Functional implications.

In the striatal network, spiny projection neurons (SPNs) and FSIs share input from the corticostriatal and thalamostriatal pathways (Kemp and Powell 1971; Kocsis et al. 1977; Lapper et al. 1992; Smith et al. 2004). Based on striatal local field potentials (Berke 2009; van der Meer and Redish 2009) and recordings from individual corticospinal neurons (Cowan and Wilson 1994; Stern et al. 1997), this input carries a broad range of frequency components including gamma oscillations. However, SPNs and FSIs may respond very differently to the shared input because of their differential frequency sensitivity. SPNs are most sensitive to oscillatory inputs at a frequency near their own firing rates (Beatty et al. 2015; Wilson 2017). Because SPN firing rates are usually low (although they can fire very rapidly for brief periods during responses), this probably explains the observation that SPN firing in vivo is usually not phase-locked to gamma components of the field potential. FSIs are specifically sensitive to gamma oscillations in their input and can entrain their firing to them even when they are a small proportion of the total input signal. Their firing is often phase-locked to gamma oscillations in the striatal local field potential, under the same circumstances in which SPN firing is mostly related to lower-frequency oscillations. The corticostriatal and thalamostriatal pathways are both heterogeneous, and it is not known for certain whether FSIs and SPNs receive synaptic input from exactly the same subtypes of afferent neurons, but even if they do their differential frequency sensitivity is enough to ensure that they respond to distinctly different signals.

In addition to their direct cortical and thalamic input, SPNs receive feedforward signals from striatal interneurons, among which FSIs appear to play a large role (Assous and Tepper 2019; Bennett and Bolam 1994). FSIs are particularly effective on an individual basis. It has been estimated that two to four FSIs make synapses onto each SPN (Tepper and Koós 2017), each providing strong, mostly somatic feedforward inhibition (Koos et al. 2004; Taverna et al. 2007). Because of the resonant properties of FSIs, it might be expected that they would transmit gamma oscillations from their input stream to SPNs, in the form of fast somatic inhibition. However, the firing patterns of FSIs do not simply reflect the gamma content of the input. As shown here, the onset of runs is regulated by a much broader range of input frequency components, and especially lower frequencies. Low-frequency inputs (including DC) also regulate the length of runs, as evidenced by the longer runs obtained at higher DC levels. Thus the firing patterns of FSIs depend on low-frequency signals affecting run onset and offset as well as gamma-frequency inputs that entrain the FSI during the run. The intrinsic frequency sensitivity of FSIs also shows considerable cellular heterogeneity as well as a dependence on the overall level of excitation. Thus the firing pattern of FSIs and their impact on the striatal output will reflect an interaction between oscillatory signals spanning a broad range of frequencies.

GRANTS

This work was supported by National Institute of Neurological Disorders and Stroke Grant NS-097185.

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

M.H.H. and C.J.W. conceived and designed research; M.H.H. performed experiments; M.H.H. analyzed data; M.H.H. and C.J.W. interpreted results of experiments; M.H.H. and C.J.W. prepared figures; M.H.H. drafted manuscript; C.J.W. edited and revised manuscript; M.H.H. and C.J.W. approved final version of manuscript.