Michael A. Farries and Charles J. Wilson J. Neurophysiol. 108:1838-1855.

Experimental evidence indicates that the response of subthalamic neurons to EPSPs is well described by their infinitesimal phase response curves (iPRC).  However, the factors controlling the shape of that iPRC--and hence controlling the way subthalamic neurons respond to synaptic input--are unclear.  We developed a biophysical model of subthalamic neurons to aid in the understanding of their iPRCs; this model exhibited an iPRC type common to many subthalamic cells.  We devised a method for deriving its iPRC from its biophysical properties that clarifies how these different properties interact to shape the iPRC.  This method revealed why the response of subthalamic neurons is well approximated by their iPRCs and how that approximation becomes less accurate under strong fluctuating input currents.  It also connected iPRC structure to aspects of cellular physiology that could be estimated in simple current clamp experiments.  This allowed us to directly compare the iPRC predicted by our theory to the iPRC estimated from the response to EPSPs or current pulses in individual cells.  We found that theoretically predicted iPRCs agreed well with estimates derived from synaptic stimuli, but not with those estimated from the response to somatic current injection.  The difference between synaptic currents and those applied experimentally at the soma may arise from differences in the dynamics of charge redistribution on the dendrites and axon.  Ultimately, our approach allowed us to identify novel ways in which voltage-dependent conductances interact with AHP conductances to influence synaptic integration that will apply to a wide range of cell types.

Figure 1.  Behavior of the SK model.  

A, Time course of Ca2+ sensor activation (blue trace) and active SK conductance (red trace) in relation to the action potential (black trace).  B, Peak SK conductance achieved as a function of firing rate in the model.  The firing rate was controlled by application of DC current (-7 pA, 16 pA, 80 pA, 160 pA, and 350 pA).

Figure 2.  iPRC of the model.

A, Autonomous activity generated by the model.  B, Response of the model to a synaptic current waveform (black trace).  [peak current, decay t.c., EPSP ampl]  The gray trace shows the membrane potential trajectory of the model in the absence of input.  C, iPRC of the model.  D, Detail of the iPRC at very early input phases (black curve) plotted with the membrane potential trajectory (gray trace) on the same horizontal scale.  Note that the oscillation cycle begins (and ends) during the action potential's rising phase, where the rate of rise is maximal, not at the action potential peak.  E, Normalized phase shift induced by brief synaptic current waveforms at very early phases.  The synaptic currents decayed with time constants of 0.3 ms (blue circles) and 0.15 ms (red circles). [EPSP ampl, peak current]  The black curve shows the iPRC.  F, Threshold of the first poststimulus spike relative to the threshold of spontaneous spikes, plotted as a function of input phase.  Gray circles denote trials on which the spike was triggered less than 10 ms after EPSP onset, before the synaptic current waveform had decayed to zero.

Figure 3.  Reduction of the full STN model.

A, Selection of a cycle segment that excludes the spike.  The black trace shows the voltage trajectory of one complete oscillation cycle; the gray trace shows the continuation of that trajectory into the preceding and succeeding cycles.  The green circle marks the beginning of the spike-free segment, 2.29 ms after the start of the cycle; the membrane potential at that point becomes the resetting potential.  The red circle marks the point where the membrane potential crosses spike threshold, defining the end of the segment.  B, Model AHP.  The K+ delayed rectifier (blue) and SK (red) conductances are plotted as a function of time since the beginning of the cycle, starting at the resetting time (2.29 ms).  The sum of the KDR conductance in the first 30 ms of the cycle and the SK conductance of the entire cycle (starting at the resetting time and ending at threshold crossing) is defined as the AHP conductance; note that this excludes the small KDR activation that occurs just before spike threshold is crossed.  Inset, the AHP current resulting from this conductance.  C, Treatment of the Na+ inactivation gating variable, h.  The actual trajectory h(t) (red) differs considerably from the steady-state value of h (blue) determined by the membrane potential trajectory V(t).  To deal with this issue, we use the trajectories h(t) and V(t) for t > 52 ms to define h(V) for V > -66 mv (inset, green curve).  D, Non-AHP membrane currents plotted as a function of membrane potential, assuming that all gating variables save h remain at their steady-state values; h is given by the modified function h*(V).  The Na+ current (red) combines the persistent and inactivating Na+ conductances, but the visible part of the curve is dominated by the persistent Na+ current.  The thick black curve plots the sum of all non-AHP membrane currents.  E, Spontaneous membrane potential trajectory of the full model (black) and reduced model (red).  Inset, detail of the trajectories as they approach spike threshold, showing a very small deviation.  F, iPRCs of the full model (black) and reduced model (red).  Obviously, the reduced model iPRC is only defined over 0.015 ≤ ϕ ≤ 0.998.

Figure 4.  iPRC of the reduced model in the absence of AHP currents. 

A, Voltage trajectory generated by the reduced model without AHP currents.  It is expressed as a function of phase, as defined by the shortened period of the model without the AHP: phase θ = t/T1.  This illustrates the relationship between the PRC and the voltage trajectory in this model.  Blue circles show the state just prior to stimulation at two distinct input phases (θ = 0.3 or 0.9) and red circles show the state immediately after stimulation, assuming that the stimulus current is a delta function that causes an instantaneous change in the membrane potential.  B, iPRC of the reduced model without AHP currents.  Inset, phase θ as a function of membrane potential.

Figure 5. iPRC of the reduced model with AHP currents.

A, Voltage trajectories generated by the reduced model without AHP currents (red trace) and with AHP currents (blue trace).  The lower dashed line marks the postspike resetting potential, corresponding to a phase (θ) of 0.027 in the AHP-free model, the minimum phase for which the relationship between V and θ is initially defined.  The AHP currents pull the membrane potential below this level, into a region where θ is not defined.  The upper dashed line marks the spike threshold.  Note that the phase θ at V_reset is not the same as the resetting phase in model's natural phase coordinate (ϕ = 0.015), because the period has changed (from T2 with AHP currents intact to T1 in the absence of the AHP) while treset remains fixed at 2.29 ms.  B, Extension of the phase model to include θ < 0.027.  Phase θ for V < V_reset can be defined by integrating Equation 1 (with gAHP(t) = 0) backwards in time from (t_reset, Vr_eset) and dividing time by T1.  The resulting function V_θ(θ) is shown in the main panel and covers the full voltage range encountered during the AHP and beyond; the black circle marks the starting point for integration at θ = 0.027 / t_reset = 2.29 ms.  This extension alters the structure of the phase model's state space, shown schematically in the inset.  Accessible states are no longer limited to the circle, and if a hyperpolarizing input pushes θ below zero (strictly speaking, below the resetting θ of 0.027), the model enters a range of θ corresponding to V < V_reset rather than looping back into late phase states.  C, The AHP-free iPRC, extended to deal with the AHP current.  D, Trajectory of the AHP-free phase variable θ as a function of the natural phase variable ϕ when the AHP is present.  E, Value of z1 as a function of ϕ as the AHP current sweeps θ through the range of values shown in panel D.  Inset shows the indirect effect of the interaction between AHP currents and the IV-defined iPRC z1, given by equation 8.  F, Predicted iPRCs of the reduced model without the AHP (red curve) and with the AHP (green curve), compared to the iPRC of the full model (thick black curve).

Figure 6.  Predictive power of phase models. 

A, Normalized PRCs generated by the full STN model using EPSPs of varying amplitude.  EPSPs were driven by synaptic current waveforms with a decay time constant of 1 ms.  Inset, normalized synaptic PRCs generated by a phase model using the iPRC shown in Fig. 2C, D.  EPSPs are the same as in the main panel.  B, Comparison of normalized synaptic PRCs generated using large (8 mV) and small (0.5 mV) EPSPs in the full model and the phase model (subset of data plotted in panel A).  Normalized PRCs of the full model are shown in red (large EPSP) and black (small EPSP); those of the phase model are shown in light red (large EPSP) and gray (small EPSP).  C, Difference between the normalized synaptic PRC generated by the full model and those generated by three different phase models, using 8 mV EPSPs.  This difference, the "iPRC error," is shown for the iPRC of Fig. 2C, D (green curve, "full model iPRC," also used in panels A and B of this figure), the iPRC of the reduced model, given by equation 9 (red curve), and the IV-derived iPRC with an applied AHP conductance (blue curve).  D, Absolute value of the iPRC error averaged over input phase, plotted as a function of the size of the EPSP used to generate normalized synaptic PRCs.  E, Models driven by a noisy current waveform.  Top, voltage trace generated by the full model.  Middle, rasters showing spike times predicted by each of the three phase models.  Bottom, current applied to the models.  F, Histograms of the spike time prediction error for the full model iPRC (top row), reduced model iPRC (middle row), and applied AHP model (bottom row), when driven by current waveforms with standard deviations of 10 pA (left column), 50 pA (center column), and 125 pA (right column); the mean current was +40 pA in all cases.  Changing the current standard deviation changed the CV of the ISI distribution while leaving the firing rate largely (but not entirely) unchanged.  Note the change in time scale across columns.  G, Median magnitude of spike time prediction error plotted as a function of CV, for the full model iPRC (green), the reduced model iPRC (red), and the applied AHP model (blue).  CV was varied by changing the applied current standard deviation from a minimum of 10 pA to a maximum of 250 pA while the mean current level was held constant at +40 pA.  The mean firing rate varied from 15.1 Hz to 18.6 Hz.  H, Median magnitude of spike time prediction error plotted as a function of mean firing rate, for the full model iPRC (green), the reduced model iPRC (red), and the applied AHP model (blue).  The firing rate was varied by changing the mean current from a minimum of -10 pA to a maximum of +200 pA while the standard deviation was held constant at +50 pA.  The CV also changed considerably as the mean current level was altered, ranging from 0.16 (+200 pA) to 0.46 (-10 pA).  However, the CV at a mean current of -10 pA was something of an outlier; the CV at the next mean level tested (+20 pA) was 0.29 and continued to decline as the mean current increased.

Figure 7.  Computation of the theoretically predicted iPRC in an example cell.

A, Example of the current clamp experiment used to obtain an AHP-free voltage trajectory.  A current pulse (-40 pA , bottom trace) initiated near the end of the ISI returns the membrane potential to a hyperpolarized level after the AHP has decayed (top trace).  A segment of the post-pulse voltage trajectory, starting at a potential below the minimum achieved during the autonomous ISI (green circle) and ending at spike threshold (red circle), is used to estimate the instantaneous IV curve.  Inset, voltage trajectory immediately after a spike, showing our choice of resetting point (gray circle) 2 ms after the beginning of the ISI.  B, Initial estimate of the IV curve (thick black curve), defined as the time derivative of the voltage trajectory segment shown in A divided by the effective capacitance (128 pF) and plotted as a function of voltage.  Inset, membrane currents flowing during the ISI.  The estimated IV curve and the voltage trajectory of the autonomous ISI are used to generate an estimate of the current flowing through voltage-dependent conductances during the ISI (red trace).  The total current flowing during the ISI is estimated from the time derivative of the ISI trajectory divided by the capacitance (black trace); the difference (blue trace) between the total current and the IV prediction serves as an initial estimate of the AHP current.  C, Estimate of the AHP conductance as a function of time since the last spike.  An initial estimate (gray trace) is generated from the difference current of panel B by assuming that the current is generated by a K⁺ conductance.  The decay phase of this estimate, g_diff, is fitted with the exponential function aExp[-(t - t₀)/τ] + b; the decay phase starts at 25 ms and ends at 130 ms.  The final AHP conductance estimate, g_AHP (black trace), is computed by subtracting the asymptotic value (b) of the fitted function from g_diff and setting g_AHP to aExp[-(t - t₀)/τ] for all t > 130 ms.  D, Final IV curve estimate (black curve) corrected by computing the AHP current from gAHP(t) and ISI voltage trajectory, subtracting the AHP current from the total current to get the current supplied by non-AHP conductances, and plotting that current as a function of voltage.  This is equivalent to taking the part of the AHP difference current (B, blue trace) that is not accounted for by g_AHP, treating it as a function of voltage, and adding it to the original estimate of the IV curve.  The original estimated IV curve (B) is replotted here (gray curve) for comparison.  Inset, detail of the hyperpolarized portion of these IV curves.  E, iPRC of the cell without AHP currents, computed from the IV curve f(V) and the AHP-free voltage trajectory (inset). The AHP-free oscillation period was computed by integrating forward and backward in time from t=2 ms and V=-61.5 mV. The AHP-free oscillation period (T1) is defined as the time when this trajectory crosses spike threshold (-39.4 mV) plus the time from threshold crossing to cycle end measured in this cell (0.2 ms).  F, Theoretically predicted iPRC (red curve) plotted with the iPRC estimated from EPSPs (green curve) and current pulses (black curve).  The normalized synaptic and current pulse PRCs were converted from units of phase shift per mV of ΔV to phase shift per pC of charge delivered by dividing the original curves by the capacitance (128 pF).

Figure 8.  Comparison of measured and predicted iPRCs.

 A, Synaptic (green), current pulse (black), and theoretically predicted (red) normalized PRCs.  Thick lines are average PRCs, thin lines show mean ± SEM.  The theoretically predicted curves were generated by simulating the response to the measured synaptic current waveform for each cell using its theoretically predicted iPRC, z₂(ϕ), and normalizing by EPSP size.  These predicted normalized synaptic PRCs differ only slightly from the iPRCs (z₂) themselves, when expressed in the same units.  The sample for synaptic and theoretically predicted PRCs included 11 cells; current pulse PRCs were obtained in a subset of these cells (n = 9).  B, Probability of type I error when testing the null hypothesis that the average synaptic (green circles) or current pulse (black circles) PRC is equal to the average theoretically predicted PRC, using paired t tests.  The red line marks the p = 0.05 criterion level.  The current pulse PRCs were tested against theoretically predicted current pulse PRCs, not the theoretically predicted synaptic PRCs shown in panel A (although the difference between the two predictions was very small).  C, Synaptic (green), current pulse (black), and theoretically predicted (red) normalized PRCs of an example cell.  D, As in C, but from a different cell.  E, Hyperpolarized region of the IV curves of the cells shown in panels C (orange curve) and D (blue curve).  Inset, AHP-free iPRCs (z₁) derived from these IV curves.  The AHP-free iPRC of the cell shown in C (orange curve) is multiplied by 0.5 so that the two iPRCs can be more easily compared in the same plot.  F, Simulated voltage trajectory measured in the axon (50 μm from the soma) following somatic current injection (black trace; 2 ms, +50 pA) or a synaptic conductance change (green trace; alpha function with 1 ms time constant, 2 nS peak conductance, 0 mV reversal potential) occurring halfway down the length of the dendrite.