Simmons DV, Higgs MH, Wilson CJ.  J Neurophysiol. 120:2679-2693.

The changes in firing probability produced by a synaptic input are usually visualized using the post-stimulus time histogram (PSTH).  It would be useful if postsynaptic firing patterns could be predicted from patterns of afferent synaptic activation, but attempts to predict the PSTH from synaptic potential waveforms using reasoning based on voltage trajectory and spike threshold have not been successful, especially for inhibitory inputs. We measured PSTHs for substantia nigra pars reticulata (SNr) neurons inhibited by optogenetic stimulation of striato-nigral inputs, or by matching artificial inhibitory conductances applied by dynamic clamp. The PSTH was predicted by a model based on each SNr cell’s phase resetting curve (PRC). Optogenetic activation of striato-nigral input or artificial synaptic inhibition produced a PSTH consisting of an initial depression of firing followed by oscillatory increases and decreases repeating at the SNr cell’s baseline firing rate. The phase resetting model produced PSTHs closely resembling the cell data, including the primary pause in firing and the oscillation. Key features of the PSTH, including the onset rate and duration of the initial inhibitory phase, and the subsequent increase in firing probability could be explained from the characteristic shape of the SNr cell’s PRC. The rate of damping of the late oscillation was explained by the influence of asynchronous phase perturbations producing firing rate jitter and wander. Our results demonstrate the utility of phase resetting models as a general method for predicting firing in spontaneously active neurons, and their value in interpretation of the striato-nigral PSTH.

Figure 1.  Direct pathway channelrhodopsin transfection and SNr cell identification.  A.  Fluorescent labeling of the direct pathway by viral injection in the striatum in the TAC1 Cre mouse.  Direct pathway axons project in the external globus pallidus (GPe) and entopeduncular nucleus (EP) as well as the substantia nigra, where they arborize mostly in the pars reticulata (SNr).  Axonal labeling in the SNr (inset) forms a network of small boutons throughout the neuropil.  B.  Spontaneous firing of a representative SNr cell. C. SNr cell action potentials were very brief.  D. The SNR response to hyperpolarizing current pulses was accompanied by only a small sag, even at very negative voltages.  E.  SNr neurons could fire at high rates in response to depolarizing current pulses.  F.  A frequency-intensity curve for the cell in B-E.

Figure 2. Synaptic inhibition of SNr neurons by direct pathway axons. A. Photo-evoked direct pathway IPSCs recorded from SNr neurons in whole-cell configuration using CsCl intracellular solution (gray traces). Mean photo-evoked IPSC is shown in black, illustrating the measurement of rise time (20% to 80%, orange dots) and decay time constant (from the illustrated single-exponential fit, orange line). B. Sample of photo-evoked IPSPs recorded in perforated-patch configuration (gray). IPSP amplitude is sensitive to membrane voltage at the time of onset; IPSPs earlier in the ISI (e.g. blue) occur at more negative membrane potentials and are smaller than IPSPs later in the ISI (red). C. ISI voltage trajectories with no IPSP (black), an early IPSP (blue), and a late IPSP (red). Both early and late IPSPs lengthen the ISI, but late IPSPs have a larger effect. D. Chloride reversal potential measured in the perforated-patch configuration. The IPSP onset slope was plotted versus the membrane potential. A linear fit (orange) was obtained by least-squares linear regression, and the x-intercept of the fit line (where IPSP slope = 0) was taken as E_rev.

Figure 3. PRC variation among SNr neurons. A. Three example PRCs from neurons with similar firing rates: blue, 38.6 Hz, black, 37.5 Hz, and red, 34.6 Hz. For all neurons, sensitivity to stimulus current was low at early phases, and generally peaked at a late phase before falling rapidly at phases approaching 1. Both the mean amplitude and the shape of the PRC varied among cells. B. Overall sensitivity versus mean firing rate for all cells. Colored dots correspond to the PRCs in A. Sensitivity was quantified as the mean squared PRC value across all phase bins. There was no significant correlation between mean firing rate and sensitivity (r² = 0.16, n = 16). C. Centroid versus mean firing rate for all cells. The mean centroid phase was 0.63 (SD = 0.04). There was no significant correlation between mean firing rate and centroid (r² = 0.23, n=16).

Figure 4. PSTH shapes produced by direct pathway synaptic input and conductance clamped artificial synaptic input.  A. Photo-stimulation of direct-pathway synaptic input. (Top) A photo-stimulated IPSP in a voltage trace during repetitive firing. (Middle) Spike raster of over 800 trials aligned at photo-stimulation. (Bottom) PSTH of spiking aligned to the photo-stimulation. B. Same as A, but using dynamic clamp to apply the inhibitory postsynaptic conductance (IPSG) to a single isolated neuron in the presence of glutamate and GABA receptor antagonists.

Figure 5. Operation of the phase model. The left, middle, and right columns illustrate simulations of a noiseless phase model (see eq. 3) where the synaptic conductance waveform G(t) (top traces) arrives early, midway, and late in the ISI. The synaptic conductance produces a synaptic current I(t) (upper middle traces), which alters the phase trajectory (lower middle traces). The lower traces show the trajectory of the voltage function of phase, ν(φ), which determines the driving force for I(t).

Figure 6. PSTH produced by the phase model of an SNr neuron. A. Applied IPSG waveform and experimental PSTH, shown for a duration equal to the mean unperturbed ISI. B. Phase trajectories of the noiseless phase model, and the corresponding PSTH. The model produced a pause in firing similar to that seen in the cell data, but the spike rate increased more steeply after the pause, and the overshoot was higher and sharper compared to the experimental data. C. Phase trajectories of the same model, but incorporating fast noise producing ISI jitter, and the corresponding PSTH. Addition of fast noise smoothed the recovery and overshoot of spike rate at the end of the pause.

Figure 7. Predictions of pause duration and area by the noisy phase models. A. Model pause duration versus experimental pause duration for each cell. Pause duration was measured as the time from the stimulus until the PSTH recovered to the baseline firing rate. Model pause durations (mean = 15.8 ± 6.2 ms) were highly correlated with the corresponding experimental data (mean = 13.8 ± 5.6 ms) (r² = 0.876). B. Model pause area versus experimental pause area for each cell. Pause area was measured by subtracting the baseline rate, and then integrating the rate change across the duration of the pause. Model pause areas (mean = -0.312 ± 0.155 spikes) were highly correlated with the corresponding experimental data (mean = -0.252 ± 0.130 spikes) (r² = 0.916).

Figure 8. Desynchronization of phase and damping of PSTH oscillations by two kinds of noise. A1 – A4: Phase trajectories for models initialized at a phase of zero (i.e. synchronized). B1 – B4: Model PSTHs for corresponding simulations initialized at uniformly distributed phases (0-1). In a noiseless model (A1, B1), phase trajectories remain synchronized forever, and there is no damping of the PSTH. In a model with jitter (A2, B2), the trajectories desynchronize gradually, but the effect is insufficient to explain the experimental PSTH damping seen in the late portion of the illustrated data (B5). A model with slow noise, or wander, (A3, B3) predicts desynchronization and PSTH damping more adequately, but does not explain the shape of the first overshoot response (see Figure 7). A model with both jitter and wander (A4, B4) predicts both the shape of the primary response and the damping of the secondary oscillation.

Figure 9. A quantitative explanation of PSTH oscillation damping by jitter and wander. A-D: Experimental PSTH data fitted with equation 9 (the amplitude envelope) and equation 10 (the damped sine wave). The jitter (j = ISI CV) and wander (w = firing rate CV) were measured from periods of unperturbed firing interspersed in each experiment. A. Cell 1: j = 0.065, w = 0.051. B. Cell 2: j = 0.071, w = 0.083. C. Cell 3: j = 0.104, w = 0.057. D. Cell 4: j = 0.130, w = 0.136.

Figure 10. Perturbation of the phase probability distribution explains the PSTH. A. PSTH for 2 nS inhibitory synaptic conductance in a phase model with the average SNr neuron PRC. Four specific times in the histogram are marked with dotted lines. B. The time course of the inhibitory conductance. C. The net rate of change of phase for the neuron at all phases consists of two components, a constant one (in this case 25 cycles/s) that is the unperturbed firing rate, and a time- and phase-varying one caused by the synaptic current acting through the PRC. Before and after the stimulus current phase advance consists of the constant component alone. At various times during the stimulus current, the phase advance is slowed or even reversed on trials according to their phases at the time of stimulus arrival. D. The probability distribution of phase at the four marked times. Before the stimulus, phase is uniformly distributed. The value of the PSTH is proportional to the phase probability density at phases approaching 1 (asterisks). At the peak of the stimulus, trials at late phases are delayed, producing a compaction at phases around 0.85, and a rarefaction in the distribution at later phases. After the stimulus is complete, the entire distribution, distorted by the effects of the stimulus, moves to the right at the unperturbed rate, producing alternating increases and decreases in firing rate. The phase bin width of 0.025 corresponds to 1 ms at the unperturbed oscillation rate of the model neuron.

Figure 11. The effect of firing rate on the PSTH and phase probability distribution. A-C. PSTHs (left) and phase probability distribution at the peak of the synaptic conductance (right) for model cells firing at 5 spikes/s (A), 25 spikes/s (B) or 50 spikes/s (C). The dotted line in each PSTH indicates the peak of the synaptic conductance. Horizontal lines with arrowheads show the width of the initial trough. D. The trough-width (recovery time) for model neurons at rates varying from 5 to 50 spikes/s. The dotted line is a linear fit to the points, with slope 0.30 and intercept (minimum trough width) of 4.2. E. The rate of change of phase versus phase for three firing rates, plotted for the time corresponding to the peak of the synaptic conductance.