Charles J. Wilson J Neurophysiol. 118(2):855-873.

Spike timing effects of small amplitude sinusoidal currents were measured in mouse striatal spiny neurons firing repetitively.  Spike timing reliability varied with the stimulus frequency.  For frequencies near the cell’s firing rate, the cells altered firing rate to match the stimulus and became phase-locked to it. The stimulus phase of firing during lock depended on the stimulus frequency relative to the cell’s unperturbed firing rate. Interspike intervals during sinusoidal stimulation were predicted using an iterative map constructed from the cells’ phase resetting curve.  Variability of interspike intervals was reduced by stimulation at all frequencies higher than about half the cells unperturbed rate, and interspike intervals were accurately predicted by the map.  Long sequences of spike times were predicted by iterating on the map. The accuracy of that prediction varied with frequency.  Spike time predictability was highest near and during phase lock. The map predicted the phase of firing on the input and its dependence on stimulus frequency. Prediction errors, when they occurred, were of two kinds: unpredicted variation in interspike interval from intrinsic cell noise, and accumulation of prediction errors from previous interspike intervals. Each type of prediction error arose from a different mechanism, and their impact was also predicted from the phase model.  When two oscillatory input currents were presented simultaneously, striatal neurons responded selectively to only one of them, the one closest in frequency to the cell’s unperturbed firing rate.  Their spike times encoded the frequency and phase of that single oscillatory input.

Figure 1. Measurement of the phases of action potentials. A. The basics of the experiment; an eleven second trace of a repetitively firing spiny neuron has a 10 second, 20 pA, 0.5 Hz sinusoidal modulation superimposed. B. Each action potential has a phase θ defined relative to the sinusoidal current. Phase varies from 0 to 1. C. A graph of the phases of every spike in the trace shown in A. D. The histogram of phases for the same trace. E. Modulation of instantaneous firing rate for the same trace.

Figure 2. Frequency dependence of firing patterns evoked by sinusoidal current. A-D. Stimulus currents (upper trace), neuronal responses (lower trace) and spike phase distributions and their entropies  for selected stimulus frequencies in an example cell. E. Combined spike phase distribution for all stimulus frequencies tested in the same cell. Each histogram is a column, and probability is in color. Note firing twice per cycle between 3 and 6 Hz, and once per cycle from 7 to 11 Hz. In both cases the phase of entrainment increases with increasing stimulus frequency. F. Entropy of the phase distribution for all frequencies tested in the same example cell, which had an unperturbed firing rate of 9.4 spikes/s (arrow). The phase-locked frequency range was centered on the cell’s unperturbed rate. The entropy values expected if the stimulus intensity were zero are shown by the dotted line. G. Relationship between frequency of best phase locking (lowest entropy) versus unperturbed firing rate for the sample of 24 neurons. H. Entropy versus stimulus frequency (like that shown in F) but for the sample (standard errors of estimates in gray). Frequency is normalized by each cell’s unperturbed firing rate to allow comparison across cells firing at different rates. Sine wave stimulation modulates firing pattern at all frequencies, but is most effective at the cell’s unperturbed firing rate.

Figure 3. Phase locking frequencies and phases. A. Example showing average firing rate over 10 second episodes of 10, 20, and 30 pA sinusoidal stimulation. Blue line labeled FCell is the unperturbed firing rate. During phase lock, firing rate lies on the 1:1 line (firing rate equals stimulus frequency). Locking also occurs on the 2:1 line (two spikes per stimulus cycle). The range of lock, defined as the frequency range over which there is < 10% difference between firing rate and stimulus frequency. B. Average and standard errors of the locking range versus stimulus amplitude for the sample of 24 neurons (F=28.65; df=2,24; p<0.01). C. In the locking range, the phase of firing depends on stimulus frequency relative to unperturbed rate, even though at that moment the cell is firing at the stimulus frequency. D. Mean and standard errors of the firing phase at lock for 10, 20 and 30 pA sinusoidal stimulation. Locking range is defined relative to the cell’s unperturbed rate (FCell). The phase range (0.2-0.35) is the same for all amplitudes.

Figure 4. Measuring the phase resetting curve for SP neurons. A. The effect of  ±180 pA, 0.5 ms current pulses on the membrane potential trajectory and firing time of a spiny striatal neuron during repetitive firing at about 10 spikes/s. B. The change in phase (ΔISI × FCell) for a variety of pulse amplitudes applied at the same time point. The slope of the line indicates the sensitivity of the cell to applied charge at that point in the interspike interval. C. A phase resetting curve (PRC) constructed for a striatal spiny neuron by calculating the slopes at times covering the entire range of the interspike interval. Stimulus time (on the abscissa) is normalized to intrinsic phase. D. Phase resetting curves for three groups of cells, identified direct pathway neurons (dSP), indirect pathway neurons (iSP), and a sample of spiny cells of unidentified pathway (SP). E. Comparison of the average amplitude of the PRC for dSP and iSP neurons. F. Average PRC for all neurons in the sample, and 4th order polynomial fit used for predicting spiny cell responses (red line and equation).

Figure 5. Iterative maps of spike phase on sine wave stimuli. A. Measurement of phase pairs. The phases of the ith and (i+1)th action potentials on the stimulus (blue trace) and the trajectory of an unperturbed interspike interval (red trace). B. The stimulus frequency and amplitude dependence of the phase map. Increasing stimulus intensity increases the deviation of the map from the diagonal, whereas changes in stimulus frequency translates the map. C. Comparing the maps, based on the phase model, with actual stimulus phases of spike pairs in an example spiny neuron. D. Phase map error for the example shown in C. Dotted line indicates expected error for a zero-amplitude stimulus. Note especially accurate predictions for the map near the cell’s unperturbed firing rate (arrow) and at twice that frequency. E. Average and standard error of map prediction errors for the entire sample. Prediction errors were calculated twice, once (blue curve) using each cell’s individual PRC and once (red) using the average PRC for striatal spiny neurons. Stimulus frequency is normalized to each cell’s unperturbed firing rate to allow comparison across neurons.

Figure 6. Predicting phase sequences and histograms by map iteration. The order of phases in the sequence is indicated by arrows. A. Fixed points in the map (left) arise when the map (red line) crosses the identity line (black dotted line). One of the crossings will be stable, and phase sequences (middle) approach that point. This is phase-lock. The steady state phase distribution (right) consists of only that point, the phase at lock. B. At sufficiently high stimulus frequency there is no longer a fixed point, but a sequence of stimulus phases. Phases associated with the closest approach of the map to the identity line occur more than others, because the stimulus causes the cell to speed up at these phases, reducing the difference between the stimulus frequency and the firing rate. C. At stimulus frequencies too slow to have a fixed point in the map, the closest approach of the map to the identity line occurs at earlier phases, at which the stimulus is most effective at slowing the cells firing. The cell fires faster at other parts of the sequence. D. When the stimulus frequency is half the cell’s firing rate, the cell fires twice on each stimulus cycle. Each of the two stimulus phases at which the cell fires maps on to the other.

Figure 7. Testing the phase model predictions of full firing sequences. Predictions were made by iterating the map as in Figure 6, from a starting phase measured from the first spike in the sequence. A. Predicted spike times and actual spike times for an example neuron at selected frequencies. Predicted spike times are shown as red lines above each trace. The lowest frequency (0.5 Hz) shows an entire trace; the rest are two seconds of data taken half-way through the trace, to allow spike-by-spike comparison. Predictions are very good at some frequencies (e.g. 11 and 21 Hz) and poor at others (e.g. 4.5 and 15 Hz). B. Mean prediction error at all stimulus frequencies for the example cell shown in A, calculated as the mean absolute value of the difference in stimulus phase of every spike and its corresponding prediction. Two error estimates were calculated, one using the cell’s own PRC (blue) and once using the average PRC for the sample of striatal spiny neurons (red). Black line is the expected value and standard error for a zero amplitude stimulus (uniform stimulus phase distribution). C. Average and standard error of prediction error for the entire sample of striatal spiny neurons, calculated using each cell’s own PRC (blue) or the average for the sample (red). Chance performance is indicated by black line.

Figure 8. The slope of the iterative map determines accumulation of error. A. At phase lock in the presence of noise, the phase of the ith spike is described as a probability distribution with variance σi2. During the interspike interval, exposure to noise introduces additional variability with variance σn2. If the slope of the map is larger than zero, some of the variance of the ith interspike interval will propagate to the next, and errors will accumulate. The rate of accumulation is equal to the slope of the map. B. The Lyapunov Exponent is the mean value of log of the slopes of all visited points on the map. It predicts the frequencies at which errors will accumulate, with a value of zero meaning summation of errors over time, and with negative values associated with small values of slope and frequencies at which error accumulation will be minimal. Note the similarity between the graph in B and the mean errors shown in Figure 7.

Figure 9. Two-frequency stimulation of an example SP neuron. A&B. Constant current pulses were used to evoke episodes of repetitive firing at 5-75 spikes/s, and two different 20 pA sine wave currents were superimposed, one at 26 Hz and one at 45 Hz, either alone or together. A. With the cell firing at about 26 spikes/s the interspike voltage trajectory is dominated by spike currents, and the influence of the sine wave stimulus is detectible but not clearly related to firing. With the constant current reduced just enough to not evoke firing (gray), the influence of the two sine wave stimuli on the membrane potential is apparent. The timing of action potentials (dotted lines) does not correspond with peaks in the subthreshold membrane potential waveform. However, the spikes are phase-locked to the subthreshold membrane response to the 26 Hz sine wave component as apparent when it is presented alone (red trace). B. Similarly, when the cell is firing at about 45 spikes/s, action potentials are phase locked to the membrane response to the 45 Hz component (shown alone in blue), even though the spiking trace is recorded in the presence of both sine waves. C-E. Quantification of phase locking to the individual sine wave components. C. Phase distribution of firing at 45 spikes/s on the 45 Hz component of the stimulus current waveform. D. Phase distribution of firing at 26 spikes/s on the 26 Hz sine wave component. E. Entropy of the phase distributions of firing on the 26 Hz (red) and 45 Hz (blue) components at all firing rates.

Figure 10. Average two-frequency stimulation results for the sample of 17 neurons. A. Average and standard error of entropy for the histograms of phase at 26 Hz (red) and 45 Hz (blue) at firing rates from 5 to 75 spikes/s. Dotted lines indicate the frequencies of the stimulus sine waves. The small sinusoidal currents were effective at structuring the firing of the neurons, when cells were firing at rates near the stimulus frequency.  B. The entropy changes in A are not caused by spurious coherence due to rate similarity. The entropy of the phase distribution obtained with a zero-amplitude stimulus showed a much smaller effect. C&D. Comparison of the combined phase distribution for all cells shows a strong entrainment near 0.2, as expected for phase locking when the sine wave stimulus is present (C) but not when it is absent (D). E&F. There is little interference between the effects of the two sine wave stimuli. E. The entropy profile for the simultaneous 26 and 45 Hz stimulus is similar to the profile for a 26 Hz stimulus alone at most firing rates. There is only a small increase in entropy (decrease in the effectiveness of the 26 Hz component) when the 45 Hz component is present.  F. A similar comparison for the 45 Hz component. The presence of the 26 Hz stimulus slightly reduces phase locking to the 45 Hz component, especially at firing rates less than 45 spikes/s.