Michael A. Farries and Charles J. Wilson  J. Neurophysiol. 108:1822-1837.

Infinitesimal phase response curves (iPRCs) provide a simple description of the response of repetitively firing neurons, and may be used to predict responses to any pattern of synaptic input.  Their simplicity makes them useful for understanding the dynamics of neurons, when certain conditions are met.  For example, the sizes of evoked phase shifts should scale linearly with stimulus strength, and the form of the iPRC should remain relatively constant as firing rate varies. We measured the PRCs of rat subthalamic neurons in brain slices using corticosubthalamic EPSPs (mediated by both AMPA- and NMDA-type receptors) and injected current pulses, and used them to calculate the iPRC.  These were relatively insensitive to both the size of the stimulus and to the cell’s firing rate, suggesting that the iPRC can predict the response of STN cells to extrinsic inputs.  However, the iPRC calculated using EPSPs differed from that obtained using current pulses.  EPSPs (normalized for charge) were much more effective at altering the phase of subthalamic neurons than current pulses.  The difference was not attributable the extended time course of NMDAR-mediated currents, being unaffected by blockade of NMDARs.  The iPRC provides a good description subthalamic neurons' response to input, but iPRCs are best estimated using synaptic inputs rather than somatic current injection.

Figure 1.  Measuring the synaptic iPRC in an example cell.

A, Example trace showing the regular autonomous activity of this cell.  This cell had an autonomous firing rate of 6.6 Hz and ISI CV of 0.06.  B, Example trace showing an ISI containing an evoked EPSP (black) plotted with example spontaneous ISIs (gray) aligned to the time of the first spike in each ISI.  C, Total phase shift induced by EPSPs plotted as a function of the phase at which the EPSP arrived.  Black circles indicate trials in which the entire EPSP fell within the stimulated ISI.  Gray circles show trials in which a spike was fired during the EPSP; in these cases the stimulated ISI contained only a fraction of the total EPSP.  The solid blue line marks the maximum possible phase shift given the input phase.  D, Measuring EPSP amplitude.  Black trace of the main panel shows average EPSP evoked at input phases of 0.2 - 0.5.  The red trace plots the average of spontaneous ISIs for each trial, where the average spontaneous ISI for a given trial has been aligned to the first spike of the stimulated ISI for that trial.  Inset, time derivative of the average EPSP, used to select the time at which the EPSP is judged to be complete.  That time is the point at which the rate of rise decayed back to the prestimulus rate of rise (marked here with a horizontal line); the red circle marks our judgment of when the EPSP is over in both the inset and main panel.  E, iPRC for this cell, where total phase shift has been normalized by the average EPSP amplitude to give the phase shift per mV of stimulus-induced potential change.  The gray circles, denoting trials in which the stimulated ISI contained only part of the EPSP, are normalized only by the fraction of the EPSP amplitude that fell within that ISI.  The red trace is the result of smoothing these points with a Gaussian filter.  F, Threshold of the first poststimulus spike relative to the spontaneous spike threshold, plotted as a function of input phase.  Stimuli delivered at late input phases tend to trigger spikes at a lower threshold, and most trials exhibiting lower thresholds are those in which the spike was fired during the EPSP (gray circles).

Figure 2.  Overview of synaptic iPRCs recorded in subthalamic cells.   

A, iPRCs estimated from the response to EPSPs in 89 cells (thin gray traces) plotted with the median synaptic iPRC (thick black trace) and the 25th and 75th percentiles of the iPRC range (thin black traces).  B, Individual iPRCs and iPRC quartiles, as in A, including only trials containing the entire EPSP.

Figure 3.  Effect of EPSP amplitude on synaptic iPRCs.

A, Histogram of EPSP amplitudes in our data set.  The colored bars shown below the histogram mark the range of EPSPs sizes included in the 4 groups we divide our data into to investigate the effect of EPSP size.  B, Average iPRCs estimated from EPSPs of 4 different amplitude ranges.  C, Probability of type I error when testing the null hypothesis that the means in these 4 groups are all equal by one-way analysis of variance.  This analysis is repeated over narrow ranges of input phases (each 0.02 phase units wide), and the probability of mistakenly rejecting the null hypothesis that the means are equal in each narrow phase range is plotted (the abscissa for each point is the midpoint of its phase range).  The red line shows the p = 0.05 criterion for rejecting the null hypothesis.  D, Probability of type I error when testing the null hypothesis that the regression slope of the relationship between normalized phase shift and EPSP size is equal to zero.  As in C, this analysis is repeated on data taken from narrow phase windows (0.02 units wide) that collectively cover the ISI; the red line again shows the p = 0.05 criterion.  E, Relationship between normalized phase shift and EPSP size at relatively early phases.  Each point represents data from one cell; the ordinate is the normalized phase shift of that cell's iPRC averaged over input phases <0.6, while the abscissa is the size of the EPSP used to estimate that iPRC.  The regression line for the relationship between these factors is shown in red.  F, Same as E, but using normalized phase shifts averaged over late input phases (ϕ > 0.85).

Figure 4.  Effect of intrinsic firing rate on synaptic iPRCs.

A, Histogram of intrinsic firing rates in our data set.  The colored bars shown below the histogram mark the range of firing rates included in the 4 groups we divide our data into to investigate the effect of this factor.  B, Average iPRCs estimated from firing rates of 4 different ranges.  C, Probability of type I error when testing the null hypothesis that the means in these 4 groups are all equal by one-way analysis of variance, like the analysis performed for EPSP amplitude in Figure 3C.  The red line shows the p = 0.05 criterion for rejecting the null hypothesis.  D, Probability of type I error when testing the null hypothesis that the regression slope of the relationship between normalized phase shift and firing rate is equal to zero; the red line again shows the p = 0.05 criterion.  E, Relationship between normalized phase shift and firing rate at relatively early phases.  Each point represents data from one cell; the ordinate is the normalized phase shift of that cell's iPRC averaged over input phases <0.5, while the abscissa is the intrinsic firing rate of that cell.  The regression line for the relationship between these factors is shown in red.  F, Same as E, but using normalized phase shifts averaged over late input phases (ϕ > 0.65).

Figure 5.  Higher-order iPRCs estimated using EPSPs.

A, Median second-order synaptic iPRC (thick black trace) plotted with the 25th and 75th percentiles of the second-order iPRC range (thin black traces).  B, Effect of EPSP amplitude on second-order iPRCs.  Thick traces show the average second-order iPRCs estimated from EPSPs of 4 different amplitude ranges.  The EPSP amplitude groups are the same as in Fig. 3A, B: <1.5 mV (black), 1.5 - 2.5 mV (blue), 2.5 - 3.5 mV (green), and >3.5 mV (red).  The thin black traces show the mean iPRC for <1.5 mV EPSP ± standard error of the mean, illustrating that is it is not necessarily different from zero (in fact, it is not; see text).  C, Effect of firing rate on second-order iPRCs.  The average second-order iPRCs for 4 different ranges of firing rate are shown. The firing rate groups are the same as in Fig. 4A, B: <6 Hz (black), 6 - 9 Hz (blue), 9 - 14 Hz (green), and >14 Hz (red).  D, Average normalized phase shift as a function of time since the EPSP.  Each point represents the average normalized phase shift of all ISIs (taken from all cells combined) that began within a narrow (1 ms wide) range of times since the EPSPs.  ISIs that might have contained some of the EPSP (i.e., because the spike defining the start of the ISI was fired before the EPSP was over) are excluded, just as they are excluded from the second-order iPRCs shown in A - C.  The red line shows an exponential fit to these data, limited to ISIs initiated >30 ms after the EPSP (this avoids what could be construed as a brief rising phase of this response).

Figure 6.  Measuring the current pulse iPRC in an example cell.

This is the same cell whose synaptic data are shown in Figure 1. A, Total phase shift induced by current pulses (each 2 ms in duration) plotted as a function of the phase at which the pulse was delivered.  Red circles show the response to +75 pA pulses, blue circles show the response to -75 pA pulses.  Gray circles show trials in which a spike was fired during the pulse; in these cases the stimulated ISI contained only a fraction of the total pulse.  The diagonal black line marks the maximum possible phase shift given the input phase.  B, Measuring membrane potential changes caused by current pulses.  Black traces show average membrane potential trajectories following +75 pA and -75 pA pulses delivered at input phases of 0.2 - 0.5.  The red trace plots the average of spontaneous ISIs for each trial, where the average spontaneous ISI for a given trial has been aligned to the first spike of the stimulated ISI for that trial.  The red and blue circles mark the points at which the membrane potential changes are measured.  C, iPRC for this cell derived from both positive (red) and negative (blue) current pulses, where total phase shift has been normalized by the pulse-induced membrane potential change.  The gray circles, denoting trials in which the stimulated ISI contained only part of the current pulse, are normalized only by the fraction of the pulse that fell within that ISI.  The red and blue traces show the smoothed iPRCs derived from +75 pA and -75 pA current pulses, respectively.  The black trace shoes the smoothed iPRC derived from all pulses combined.  D, Synaptic (green trace) and current pulse (black trace) iPRCs measured in this cell, plotted together.

Figure 7.  Properties of current pulse iPRCs.

A, iPRCs estimated from the response to current pulses in 66 cells (thin gray traces) plotted with the median current pulse iPRC (thick black trace) and the 25th and 75th percentiles of the iPRC range (thin black traces).  B, Comparison of first-order iPRCs estimated using depolarizing (red) and hyperpolarizing (blue) current pulses in cells where both pulse polarities were used (63 cells).  Thick traces show the mean, while the thin traces show the mean ± SEM.  C, Comparison of second-order iPRCs estimated using depolarizing (red) and hyperpolarizing (blue) current pulses in cells where both pulse polarities were used, and limited to a subset of data that allowed us to see average second-order iPRCs at relatively late input phases (51 cells).  Thick traces show the mean, thin traces show the mean ± SEM.  D, Threshold of the first poststimulus spike relative to the spontaneous spike threshold plotted as a function of input phase, with data from positive (red) and negative (blue) current pulses. Thick traces show the mean, thin traces show the mean ± SEM.  E, Relationship between normalized phase shift and the relative spike threshold of the first poststimulus spike following positive current pulses, averaged over late input phases (ϕ > 0.8).  Each point represents data from one cell. The regression line for this relationship is shown in red.  F, Comparison of iPRCs estimated from EPSPs (green) and current pulses (black). Thick traces show the mean, thin traces show the mean ± SEM.

Figure 8.  Effect of APV on iPRCs.

A, Synaptic iPRCs measured before (blue) and during (red) bath application of 50 µM APV, n = 14. In panels A - E, thick traces show the mean, thin traces show the mean ± SEM.  B, Raw synaptic iPRCs measured before (blue) and during (red) application of APV.  These are averages of PRCs that have not been normalized by EPSP size.  C, Current pulse iPRCs measured before (blue) and during (red) application of APV, n = 9.  D, Pairwise difference between synaptic iPRCs measured in control (blue) or APV (red) and current pulses measured in control conditions, n = 14.  Control current pulse iPRCs were subtracted from both control and APV synaptic iPRCs to maximize the sample size (current pulse iPRCs were collected in APV for only 9 cells) and because the purpose was to determine whether NMDAR blockade made synaptic iPRCs more like control current pulse iPRCs.  E, Second-order synaptic iPRCs measured before (blue) and during (red) application of APV.  F, Average normalized phase shift as a function of time since the EPSP, in control (blue) and APV (red).  Each point represents the average normalized phase shift of ISIs starting within a 1 ms time bin.  The solid lines show exponential fits.

Figure 9.  Examples of individual iPRCs derived from synaptic (thick green trace) and current pulse (thin black trace) data.

Figure 10.  Synaptic currents evoked by stimulation in the internal capsule. 

A, Example of the time derivative of an averaged EPSP (black), proportional to the perisomatic current generated by the synaptic input, plotted with the EPSC recorded in the same cell in voltage clamp (red).  The EPSP derivative is plotted on an inverted scale to facilitate comparison; this EPSP was 3.1 mV.  The EPSC is an average of 10 traces, recorded at a holding potential of -64 mV.  B, Histograms of decay time constants for EPSP derivatives (black, n = 89 cells, bin size 0.2 ms) and EPSCs (red, n = 47 cells, bin size 0.5 ms).  One EPSC measurement (τ = 1.1 ms) is concealed behind the histogram of EPSP derivative time constants, while another falls outside the range of the plot (τ = 30.8 ms); all the others (n = 45) are visible.  C, Relationship between the size EPSP recorded in current clamp and integral of the EPSC recorded in voltage clamp.  Black circles are from recordings with low series resistance (<30 MΩ), gray circles are from recordings with high series resistance (≥30 MΩ), and the red line is the linear fit to all the data.  D, Normalized EPSP amplitude as a function of membrane potential, including only cells with measurable EPSPs that covered a voltage range that included the reference potential of -57 mV (n = 66 cells).  EPSP amplitudes were normalized to give an average amplitude of 1 mV at -57 mV (see Methods).  Gray circles are individual EPSPs, red line is the linear fit.