Charles J. Wilson, David Barraza, Todd Troyer, and Michael A. Farries  PLoS Comput. BIol. 10:e1003612 

We used phase resetting methods to predict firing patterns of rat subthalamic nucleus (STN) neurons when their rhythmic firing was densely perturbed by noise. We applied sequences of contiguous brief (0.5-2 ms) current pulses with amplitudes drawn from a Gaussian distribution (10-100 pA standard deviation) to autonomously firing STN neurons in slices. Current noise sequences increased the variability of spike times with little or no effect on the average firing rate. We measured the infinitesimal phase resetting curve (PRC) for each neuron using a noise-based method. A phase model consisting of only a firing rate and PRC was very accurate at predicting spike timing, accounting for more than 80% of spike time variance and reliably reproducing the spike-to-spike pattern of irregular firing. An approximation for the evolution of phase was used to predict the effect of firing rate and noise parameters on spike timing variability.  It quantitatively predicted changes in variability of interspike intervals with variation in noise amplitude, pulse duration and firing rate over the normal range of STN spontaneous rates. When constant current was used to drive the cells to higher rates, the PRC was altered in size and shape and accurate predictions of the effects of noise relied on incorporating these changes into the prediction. Application of rate-neutral changes in conductance showed that changes in PRC shape arise from conductance changes known to accompany rate increases in STN neurons, rather than the rate increases themselves. 

Our results show that firing patterns of densely perturbed oscillators cannot readily be distinguished from those of neurons randomly excited to fire from the rest state. The spike timing of repetitively firing neurons may be quantitatively predicted from the input and their PRCs, even when they are so densely perturbed that they no longer fire rhythmically.

Figure 1. The effect of noise on the variability of spike times in the phase neuron model

A. The relationship between phase trajectories, phase distribution at the mean spike time, and the distribution of interspike intervals. Ten example phase trajectories are shown in blue. The red line is the mean of 5000 trajectories. The distribution of phases at the firing time (average phase =1) is shown in the histogram to the right of the trajectories and the histogram of interspike intervals is shown above. B. The evolution of the phase distribution for 5000 trajectories like the ones in A. The mean moves at the drift rate ω but the variance increases depending on the phase resetting curve Z (shown in blue). The variance of the interspike interval distribution is the same as that of the phase distribution at the firing time, scaled by 1/ω². C. The evolution of the standard deviation of the phase distribution (blue line) for the Monte Carlo simulation of the phase neuron, and the Ermentrout et al. approximation to that evolution (red line). The large black dot is the standard deviation of the interspike intervals (for mean firing rate of 1 spike/s). The approximation for standard deviation of phases is shown in the inset. D-F Effect of noise amplitude, pulse duration, and baseline firing rate on the coefficient of variation of interspike intervals in the Monte Carlo simulation (blue points) compared to the prediction of the approximation (red line).

Figure 2. Responses of STN neurons to pulsed current noise.

A. Autonomous firing of a STN neuron recorded in the absence of injected noise (upper panel) and in the presence of contiguous 1 ms current pulses (middle panel). The current injected is shown in the bottom panel. The standard deviation of the noise in this example was 90 pA. Average firing rate was unchanged. B. Histogram showing the distribution of noise pulse amplitudes. C. Higher resolution of the injected current (lower panel) and the membrane potential response to currents (upper panel) in the same cell. The capacitative transient at pulse onset and offset is restricted to a single sample (0.05 ms), the membrane changes are almost entirely capacitative and series resistance is well-compensated. D. Changes in regularity of firing during application of the same noise shown in A -C, as indicated by the autocorrelation histogram in the absence (upper panel) and the presence (lower panel) of noise. E. An example showing dependence of interspike interval CV qualitatively similar to those predicted from the phase model, for noise amplitude, pulse duration and baseline firing rate.

Figure 3. Predicting spike times from the PRC.

A-C. Examples of phase resetting curves calculated in a single neuron for σNoise from 9 to 50 pA. The estimate of the PRC improves as σ_Noise is increased, up to about 50 pA, and then is stable. D. The proportion of ISI variance accounted for by the injected noise (R² for the regression) increases and then stabilizes near 80%. E. For the group of 21 neurons tested, the proportion of the ISI variance accounted for by the noise follows a similar profile. Noise levels of 60 pA and above were equivalent, with ~ 80% of the ISI variance being predicted by the phase model interacting with the injected noise. F. The measurement of Sensitivity (integral of the square of the PRC) increases in parallel with R², and is constant at higher levels of injected noise. G. Histogram of Sensitivities for neurons in the sample. H. The phase model predicts spike times with high accuracy. A sample of intracellular recording is shown, with action potentials times predicted by the phase model shown by dashed red lines.

Figure 4. The range of PRC shapes seen in STN neurons.

Phase resetting curves calculated by noise regression are in the left column. The phase resetting curve obtained using synaptic stimulation is shown in the center column, and the phase-normalized average interspike membrane potential trajectory is shown in the right hand column. Phase resetting curves measured in the two ways are similar, and are associated with differences in the average membrane potential trajectories.

Figure 5. Prediction of CV in STN neurons with changes in noise amplitude and duration.

A. An example showing the relationship between CV and noise amplitude for 1 ms duration pulses. Data points are shown in black, and the curve for the prediction of the phase model approximation based on this cell’s sensitivity is in red. Goodness of fit was measured as the proportion of the variance in the data accounted for by the approximation. B. Distribution of goodness of fit for the CV versus noise amplitude for the sample of 21 neurons. C. The relationship between CV and pulse duration for the same cell as in A, displayed with the prediction. D. The distribution of goodness of fit for pulse duration across neurons in the sample.

Figure 6. Prediction of CV in STN neurons with changes in firing rate.

A. PRCs calculated for a single STN neuron at three different rates. In all neurons, large increases in firing rate by injection of constant current resulted in a shift of the peak of the PRC to later phases, and an increase in the overall amplitude of the PRC. Centroids of the PRCs are indicated by red arrows. B. The changes in mean Sensitivity (integral of the square of the PRC) with changes in current imposed by constant current injection for the sample of 12 neurons. C. Average phase centroid versus mean rate in the same sample. D. Changes in average interspike interval membrane potential trajectory with rate in the same cell shown in A. E. Fit of the phase neuron approximation for CV for the same neuron, based on the experimental value of the PRC sensitivity for each firing rate. Inset is the histogram of R2 for the fit obtained for each cell in the sample.

Figure 7. The effect of artificial rate-neutral conductance increases.

A. Contour diagram of firing rate varying conductance clamp-applied somatic conductance at a range of subthreshold reversal potentials. X marks two rate-neutral points at 1 and 2 nS conductance levels. B. PRCs calculated at the with rate-neutral applied conductances marked in A. Centroids of the PRCs are at the red arrows C. Membrane potential trajectories corresponding to the PRCs shown in B. D. Average (and standard errors) of the PRC centroid for 9 neurons tested with varying but rate-neutral values of g_Leak.

Figure 8. The multiple regression method for calculating the PRC, and the choice of phase interpolation. 

A.  The use of pulsed noise provides a natural set of independent stimuli at each time slice (corresponding to pulse duration) during the ISI.  Different ISIs are indicated by different colors.  B.  The PRC and its standard errors calculated by multiple regression on a Monte Carlo simulation using fixed time steps and phase interpolation based on the mean ISI.  The true PRC used by the simulation is shown as a red line. Note the consistent error near the end of the PRC.  C.  The standard error of the PRC estimates as a function of phase.  The error increases dramatically at large phases.  D. The strategy for phase interpolation.  Two phase versus time trajectories from the Monte Carlo simulation used in B and C are shown in black.  One is the trajectory a very short ISI, and one for one of the longest ISIs in the simulation.  The linear phase versus time estimate used for calculating the phase of current pulses based on mean interval is shown as a dotted red line.  This estimate is good at early times in the ISI but fails at longer times.  The phase interpolations we employed is shown as solid red lines.  Although less accurate at short times they are more accurate later, at times when the PRC values are higher and the noise is more influential.  E.  The PRC calculated using our interpolation method, overlaid by the true PRC for the simulation (red).  F.  The standard error of the estimates of the PRC in E.  Note lower overall error, and the even distribution of error across the ISI.