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Department of Biochemistry and Cell Biology, Rice University, Houston, Texas 77005
Submitted 30 April 2004; accepted in final form 29 June 2004
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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; Korenberg et al. 1989; Kretzberg et al. 2001). At postsynaptic sites, postsynaptic potentials (PSPs) with different temporal features and polarity are integrated into a continuous time-varying current. During impulse generation, the synaptic current interacts with yet another filter at the spike generator. At the minimum every spike generator has a time constant and a threshold. It may also exhibit a particular sensitivity to the time derivative of the local membrane potential (Galaretta and Hestrin 2001; Mainen and Sejnowski 1995), spike-frequency adaptation (Fohlmeister and Miller 1997; Fuhrmann et al. 2002; Glantz and Schroeter 2004; Liu and Wang 2001), resonance (Hunter et al. 1998; Reich et al. 1998; Schreiber et al. 2003), and/or intrinsic oscillations or bursting properties (Hunter and Milton 2003; Smith et al. 2000; Szücs et al. 2001). Although complex, the above scenario is highly oversimplified because it neglects dendritic conductances, cable properties, and the like, although it is possible to distinguish the characteristics of the spike-generating mechanism from the effects of the synaptic current through analysis of the responses to defined extrinsic currents (French et al. 2001; Knight et al. 1970; Smith et al. 2000). In a recent study (Glantz and Schroeter 2004) we found that spike-frequency adaptation substantially reduces the firing rate of steady-state and slowly varying discharges but has little or no effect on brief transient responses. This result has important implications for the temporal frequency response (Fohlmeister et al. 1977) and provided one impetus for the present study.
In this study we examine the temporal frequency response of 2 classes of crayfish visual interneurons: sustaining fibers and dimming fibers (Wiersma and Yamaguchi 1966), which arise in the second optic neuropile (medulla externa) (Kirk et al. 1982). The sustaining fibers are tonic ON neurons. They respond to increments of illumination with a high-frequency (200 to 300 imp/s) transient discharge and a lower-frequency sustained response. The dimming fibers are tonic OFF cells. They respond to illumination decrements with a transient OFF response and maintain a continuous discharge in the dark. The dimming fiber dark discharge is inhibited by increments of illumination. In the brain, sustaining fibers synapse on oculomotor neurons (Glantz et al. 1984) and the sustaining fibers form the afferent limb of the dorsal light reflex (Miller et al. 2003). The reflex consists of light-activated eye rotations that minimize the angular deviation between the dorsoventral axis of the eyestalk and the direction of skylight.
In this study, the spike-generating mechanism is first characterized by extrinsic sine-wave currents and an adaptive leaky integrate-and-fire model. Then, the visual responses elicited by drifting sine-wave gratings are analyzed with the same model, holding all parameters constant except the background current and a synaptic resistance scaling factor. The visual responses typically exhibit a high-pass temporal frequency response. Of the 2 fiber classes, dimming fibers exhibit stronger spike-frequency adaptation, and it is shown that adaptation in this cell type is the major determinant of frequency-dependent variations in gain. In the sustaining fibers, spike-frequency adaptation is weaker and the gain is more strongly influenced by frequency-dependent nonlinearities in the PSP. Spike timing precision for both visual and extrinsic current stimuli is related to the stimulus frequency and is further enhanced by the occurrence of rectification in the discharge. For the sustaining fiber visual response, excitatory postsynaptic potential (EPSP) nonlinearities are found to substantially (>10-fold) enhance the precision of spike timing compared with the response to a sine-wave extrinsic current. The encoder model and synaptic currents derived from the PSPs as presented below provide a semiquantitative description of all of the above phenomena.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Adult crayfish, Pacifastacus leniusculus, of both sexes were prepared as in Glantz and Schroeter (2004). The eyestalks were cemented into their sockets with cyanoacrylate adhesive and the hemolymph was replaced with oxygenated crayfish saline buffered with 5.0 mM Hepes at pH 7.5. During the experiments, the animal was submerged in a chamber of oxygenated saline at 15°C. The optic lobe was exposed by removal of the dorsal eyestalk cuticle and an overlying sheath.
Recording and data acquisition
Sustaining and dimming fibers were impaled in the second-optic neuropile in situ with sharp micropipettes filled with 3.0 M K+-acetate. Electrode resistances were 80 to 100 M
and the time constants (after capacity compensation) were 0.2 to 0.4 ms. Signals were led to an Axoclamp IB amplifier (Axon Instruments, Foster City, CA). The neurons were identified by location and characteristic responses to a flash of light (Waldrop and Glantz 1985a). The voltage, current, and stimulus signals were digitized at 1,000 Hz/channel with a National Instruments (Austin, TX) A/D card and LabVIEW software.
Visual stimuli
Stimulus procedures were similar to those in previous studies (Glantz et al. 1995). Drifting sine-wave gratings were presented on a Hitachi display oscilloscope (as in Fig. 1A, Os) driven by a Picasso Image Synthesizer (Innesfree, Cambridge, UK) at a refresh rate 90 Hz. The gratings were projected to the surface of a fiber-optic lens (FOL) that converted the 2-dimensional image to a concave spherical projection spanning 120°. The crayfish eye was fixed at the optical center of the concave surface. The optimal stimulus was determined at the start of an experiment by varying the grating orientation, direction of motion, and spatial wavelength. A typical configuration was a vertical grating at a spatial wavelength of 30 to 40°. The contrast was adjusted (typically between 0.2 and 0.5) to elicit large but unsaturated responses.
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Extrinsic current was injected through the recording electrode by a bridge circuit. Currents consisted of pulses, 0.5 to 1.0 s duration at 0.2-Hz repetition rate or continuous sine waves. The sine-wave trajectory had the form: Isin = I0 + If [sin (2
ft)], where I0 is the mean current and If/I0 was typically between 0.2 and 1.0. I0 was chosen to elicit a continuous discharge of 5 to 20 imp/s and samples of responses to I0 alone were acquired for each set of measurements. The injected currents were monitored at a port on the Axoclamp IB amplifier.
Computational methods
SPIKE TRAIN ISOLATION AND PSP EXTRACTION.
The light- and current-elicited impulse trains were separated from the PSP or slowly varying potential by a wavelet-discrimination routine (Glantz and Schroeter 2004; Johnson et al. 2000) and the impulse train was converted to a binary sequence of ones and zeros. For the light-elicited responses, the PSP was separated from the superposed spike train by subtraction (French et al. 2001). The binary spike train was used to locate the spike positions and the spikes were removed from the voltage trace by interpolation.
CHARACTERIZATION OF THE DISCHARGE. Responses of the neuron and of the model were expressed as a time-varying impulse rate or poststimulus time histogram (PSTH). The impulse rate at a particular point in time (ti) was calculated as the inverse of the time between the spikes surrounding ti. The PSTH was calculated with approximately 30 bins per stimulus cycle and smoothed by a moving window average of 2–3 bins. The gain and phase were determined for neural and simulated responses from the fundamental of the discrete Fourier transform of the PSTH and the PSP. For current-elicited responses, gain and phase were measured relative to the sinusoidal current waveform. For light-elicited responses, gain and phase were measured relative to the PSP.
Throughout the text we use the term rectification to describe a neural response (firing rate or PSP) that is invariant over a fraction (>0.2) of the stimulus cycle while the stimulus itself varies. For measurements of the SD of spike times at the onset of a rectified discharge, we selected responses in which firing was completely suppressed for a fraction of the stimulus cycle.
ADAPTIVE LEAKY INTEGRATE-AND-FIRE MODEL.
The adaptive leaky integrate-and-fire encoder (Glantz and Schroeter 2004; Koch 1999) is given by
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m is the membrane time constant, and Rin is the input resistance. When V reaches the impulse threshold (Vth), an impulse is produced and V is set to Vrest. Gadapt is a postimpulse shunting conductance (Koch 1999), which is initially zero and incremented after each impulse by Ginc. Gadapt then decays exponentially with time constant adapt
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CURRENTS DERIVED FROM THE MEMBRANE POTENTIAL OR PSP.
The extracted potential provided the basis for computing the synaptic current. Is is given by
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m are identical to the parameters of Eq. 1. The synaptic conductance Gs(V) is a sigmoidal function of the instantaneous potential (V) (Waldrop and Glantz 1985a). As the potential varies from 0 to 40 mV, Gs normalized by the resting input conductance Grest (equal to 1/Rin) varies from 0 to 1.5. ks represents the ratio between the synaptic resistance and the input resistance (Rin) at the spike-initiating zone. For the visual stimuli, Is was calculated from the PSP using Eq. 3 for input to Eq. 1.
OPTIMIZATION OF MODEL PARAMETERS.
Equations 1 and 2 specify a 7-parameter (Ib, Ginc, adapt, Vth, tref, m, Rin) model that generates firing times given an input current. For current stimuli Is is set to zero. For visual stimuli, Ie is set to zero and Is is calculated from Eq. 3. The refractory period (tref) was typically fixed at 4 to 5 ms and was not varied during optimization. For each cell, an optimum fit of the model to the observed response was discovered in 3 stages. A preliminary optimization of the visual response was used to establish a range for Vth. This was followed by a thorough optimization of the model for the responses to sine-wave currents (constrained to a range of Vth values). The parameters so determined were used to simulate the response to sine-wave gratings, varying only the background current Ib (to reflect variations in the background discharge) and the resistance ratio ks.
The best fit of model to experimental responses was found using the Nelder–Mead methods in the MATLAB optimization toolkit and software produced in house as described in Glantz and Schroeter (2004). The fit of the model to the data was evaluated by the root mean square error (RMSE) normalized to the mean firing rate and the linear correlation coefficient. The RMSE = 
{sqrt [(Fd – Fm)2]}/n, where Fd is the observed firing rate, Fm is the model instantaneous rate, and n is the number of data points. Optimization was always performed on a family of responses (
12) elicited by variations in the frequency of extrinsic current. Our preliminary studies suggested that the model is too simple to provide more than a rough estimate of the time-varying firing rates in some conditions, although the simulations do provide a framework for evaluating the interaction between the dynamic properties of the spike generator and the nonlinear synaptic currents elicited by visual stimulation.
Previously (Glantz and Schroeter 2004) we found that the model is robust in simulations of responses to current pulses. Modest variations in one parameter (e.g., m) can be offset by changes in another parameter (Rin) that has the opposite effect on impulse rate, with little or no effect on the RMSE. Thus one should not attach too much significance to particular parameter estimates outside the context of the whole set.
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Spike-frequency adaptation is a characteristic feature of impulse generation in most neurons (Benda and Herz 2003; Liu and Wang 2001). The effect of adaptation is seen in the dimming and sustaining fiber responses to an extrinsic pulse of constant current as shown in Fig. 1, B and C. The discharge exhibits a transient peak rate (Fp) and the firing rate rapidly declines to a plateau phase for both neuron classes. The magnitude of spike-frequency adaptation (
F) is the difference between Fp and the mean impulse rate of the plateau phase. The percentage change in rate [(F/Fp) x 100] varied from 25 to 70% among the cells we examined. In Fig. 1, adaptation was 51% in the dimming fiber and 36% in the sustaining. This difference was characteristic of the 2 cell populations (56 ± 11% for dimming fibers, 40 ± 15% for sustaining fibers).
Responses to sine-wave current
The spike trains elicited by sine-wave current provide a measure of the frequency response of the impulse generator. At low stimulus frequencies (<1.0 Hz) the firing rate elicited by extrinsic sine-wave current varies smoothly with the current for both cell types, as in Fig. 2, A and E. Above 1.0 Hz the impulses cluster around the time of maximum depolarization as shown in Fig. 2, B, C, and G and the discharge may contain brief bursts or doublets. At the highest stimulus frequencies (5–11 Hz) the discharge is rectified. No spikes are seen in the response during the times surrounding the stimulus minima (Fig. 2, D and H). The tendency toward rectification at higher stimulus frequencies is a function of If/I0, the ratio between the current stimulus amplitude, and the mean current. Rectification is diminished or absent if If/I0 is <0.2.
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Gain and phase analysis indicates the frequency dependence of the spike generator's sensitivity to current and the influence of frequency on the timing of the response. The gain and phase of the sustaining and dimming fiber impulse trains are shown in Fig. 4 as open circles. The simulated gain and phase are shown by the solid lines in the figure. The gain of the 2 cells (Fig. 4, A and C, open circles) are similar (40 to 50 imp/s/nA) at low temporal frequencies and both cells exhibit a frequency-dependent increase in gain. The dimming fiber's gain increase is about 3.7-fold, whereas that in the sustaining fiber is 1.9-fold. In the simulated response, this difference in frequency-dependent gain is largely attributed to the difference in the strength of spike-frequency adaptation. Setting Gadapt to zero in Eq. 1 substantially increases the simulated low-frequency gain as shown in Fig. 4, A and C (broken lines) and diminishes the frequency-dependent gain variation.
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The simulated gain and phase are well correlated to those of the measured responses. For the dimming fiber, the correlation coefficient r = 0.99 for gain and 0.95 for phase. For the sustaining fiber, r = 0.92 for gain and 0.99 for phase. Similar results for the 2 cell populations are summarized in Table 2.
Light-elicited synaptic potential
The synaptic potentials shown in Fig. 5 are averaged responses elicited with drifting sine-wave gratings and isolated from the associated action potentials by subtraction (as described in METHODS). As the positive phase of the stimulus enters the receptive the sustaining fiber depolarizes (as in Fig. 5, A–C) and the dimming fiber hyperpolarizes (as in Fig. 5, E–G). Three aspects of the PSP are particularly relevant to the temporal frequency response of the discharge. These are the polarity (EPSPs in sustaining fibers and inhibitory postsynaptic potentials [IPSPs] in dimming fibers), magnitude, and nonlinearity. Sustaining fiber PSPs elicited by drifting gratings are rarely linear. In general (10/12 cells) the segment of the stimulus below the mean intensity is associated with little or no voltage variation, whereas the positive phase of the stimulus elicits a depolarization related to the local stimulus intensity (as in Fig. 5A). At the lowest stimulus frequencies, the membrane potential changes slowly (relative to m), the capacitative currents are small and the simulated currents (from Eq. 3, dotted traces in Fig. 5, A and E) parallel the voltage trajectory for both cell types.
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2V/s) the simulated current computed for the transient phase is much larger (by about 3-fold) than that derived from the plateau phase. The comparable ratio for the voltage is 1.5. As the stimulus frequency increases further, the sustaining fiber PSP amplitude declines but dV/dt and the associated peak current remain high, as in Fig. 5C. In contrast, the dimming fiber response (as in Fig. 5, E–G) is dominated by the IPSP that drives the membrane potential negative to rest. At the lowest temporal frequencies (0.1 Hz, Fig. 5E) the computed synaptic current (Fig. 5E, dotted trace, right-hand ordinate) is much smaller than that in the sustaining fiber because the PSP is smaller and the dimming fiber Rin is much larger. At stimulus frequencies of 1.0 and 11.0 Hz, an increase in dV/dt in the PSP is associated with a corresponding increase in computed current (as in Fig. 5F). However, at 11.0 Hz the maximum dV/dt in the dimming fiber PSP is about 10% that of the sustaining fiber and the peak current is about 5% that in the sustaining fiber (Fig. 5G).
The above results indicate that the dimming fiber synaptic signals are smaller than those in sustaining fibers and the nonlinearity has a different frequency dependence. To quantify these differences the PSPs were analyzed with a discrete Fourier transform and the results for the first 3 harmonics are presented in Fig. 5, D and H. In both cells the fundamental (open circles and solid lines) is low-pass, declining by 50 to 60% between 0.1 and 11.0 Hz. The important difference between the 2 cell types is the behavior of second harmonic (2nd h) as the stimulus frequency increases. In the sustaining fiber 2nd h increases relative to the fundamental (Fig. 5D, broken line) whereas in the dimming fiber (Fig, 5H, broken line), 2nd h declines relative to the fundamental as the stimulus frequency increases. Although the absolute magnitudes varied between cells, the pattern of frequency-dependent harmonic amplitudes shown in Fig. 5, D and H is consistent across the 2 cell populations. Thus between 0.1 and 5.0 Hz, the ratio of the 2nd/1st harmonic increases from 0.22 ± 0.16 to 0.48 ± 0.20 in sustaining fibers, whereas in dimming fibers the comparable ratios are 0.25 ± 0.15 and 0.23 ± 0.04. Because the higher harmonics contribute substantially to dV/dt (compare Fig. 5C and Fig. 5D), they provide evidence of large transient synaptic currents at high stimulus temporal frequencies. We will represent the PSP harmonics by an index of nonlinearity (Smith et al. 2000) Inon, which is the ratio of (2nd h + 3rd h)/1st h, where h indicates the harmonic. At stimulus frequencies near 10 Hz Inon is about 80% larger in sustaining fibers than in dimming fibers (Table 3). As will be shown below, the large harmonics in the PSP and the associated synaptic currents have a strong influence on the firing rate.
Visually elicited discharge of dimming fibers
The responses of a dimming fiber to drifting sine-wave gratings of various frequencies are shown in Fig. 6. The discharge commences as the positive phase of the stimulus exits the receptive field as in Fig. 6, A–D. The rectified response of this neuron reflects the alternation of inhibition and excitation as the positive phase of the stimulus moves in and out of the receptive field. A notable feature of the response to high temporal frequencies (as in Fig. 6D) is the scalloplike waveform of the PSP, with impulses confined to about 20% of the stimulus period. At 0.1 Hz the discharge occupies 60% of the stimulus period.
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The simulations of the visual responses (Fig. 6, E–H, dark broken lines) approximate the duration of inhibition and the mean rate of the "dark discharge." The timing and amplitude of the OFF response are approximately correct in Fig. 6, F and H but not in Fig. 6, E and G. For 10 stimulus frequencies (0.1 to 11 Hz) the correlation between the measured and model PSTHs was 0.86 and the RMSE was 22% of the mean rate (0.44 SDs). Similar results were obtained for the 8 dimming fibers we examined, as shown in Table 2.
Sustaining fiber visual response
The responses of a sustaining fiber to drifting sine-wave gratings of various frequencies are shown in Fig. 7. Sustaining fiber excitation reflects the passage of the positive contrast phase of the stimulus through the receptive field. The sustaining fiber discharge (as in Fig. 7, A–D) exhibits rectification similar to that seen in the PSP (as in Fig. 5, A–C). At low stimulus frequencies there is nearly a linear relation between the time-varying membrane potential and the impulse rate. The magnitude of the cross-correlation is r = 0.90 to 0.97 (normalized to the PSP autocorrelation function). As the temporal frequency increases, so does the nonlinearity of the PSP and the discharge, and the intensity of the cross-correlation declines (r = 0.42 for a 10-Hz stimulus). At higher stimulus frequencies, the sustaining fiber discharge (like that of dimming fibers) is confined to a small fraction (
15%) of the stimulus period. The PSTHs (Fig. 7, E–H, solid lines) reveal the frequency-dependent variations in peak firing rate and the phase locking at the highest frequencies. The simulations generally capture the time course of the discharge. For 11 different stimulus frequencies the RMSE was equal to 17% of the mean firing rate. The results are representative of sustaining fiber data as a whole (as in Table 2).
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The gain and phase of the visual responses for both fiber types (Fig. 8, A, B, D, and E, open circles) qualitatively resemble those of current-elicited responses, including a frequency-dependent increase in gain and a small phase lead relative to the PSP (Fig. 8, B and E). Because of the PSP nonlinearity, the increase in dV/dt with stimulus frequency exceeds that of a pure sine wave. Thus the frequency-dependent increase in gain is typically higher in visual than in current-elicited responses. The gain of the sustaining fiber visual response increases 3.8-fold between 0.1 and 11.0 Hz. The comparable figure for the current-elicited response of the same cell was 1.9-fold (as in Fig. 4C). In general we found a frequency-dependent gain increase of about 6- to 7-fold in the visual response of both cell populations (as in Table 3). The simulations of the gain functions (Fig, 8, A and D, solid lines) gave excellent correlations (r = 0.99 for the dimming fiber and r = 0.98 for the sustaining fiber). The phase of the model dimming fiber response, however, was less accurate (as in Fig. 8B, open circles vs. solid line) and the correlation (r = 0.74) was low.
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In the sustaining fiber, spike-frequency adaptation is smaller and setting Gadapt to zero in the simulated response (Fig. 8D, dots and broken line) has a smaller effect on gain. Because of the nonlinearities in the sustaining fiber PSP, the simulated synaptic current increases about 2-fold between 0.1 and 11.0 Hz (as in Fig. 8F) and the peak firing rate exhibits a similar 2-fold increase, as shown in Fig. 7, E and H. Because the PSP declines by about 50% over the same stimulus frequency interval (as in Fig. 5D), the ratio of current variation to voltage variation at 11.0 Hz is about 4-fold that at 0.1 Hz. This is consistent with the observed 3.8-fold increase in gain between 0.1 and 11.0 (as in Fig. 8D, open circles). Removing Gadapt from the simulation produces a small reduction in the phase lead (as in Fig. 8, B and E, broken lines) in all simulated responses.
In general we found that the frequency-dependent gain variations in the dimming fibers were strongly influenced by spike-frequency adaptation. Because the same adaptation is expressed in the current-elicited discharge, one might expect that the gain variations in the 2 regimes would be similar and this was the case, as shown in Table 3. In sustaining fibers, the gain was more sensitive to frequency-dependent variations in the synaptic current resulting from the increase in PSP nonlinearity with frequency. Because these variations are absent in the current-elicited response, one might expect a greater frequency-dependent gain increase in the sustaining fiber visual response compared with the current-elicited response and this was also the case, as shown in Table 3. Examination of the PSPs at the frequency of maximum gain indicates that sustaining fibers typically exhibit a higher degree of nonlinearity than that of dimming fibers, as shown in Table 3.
Spike timing precision
For transient visual stimuli, crayfish oculomotor neurons determine the timing and direction of eye movements in accordance with the timing of the first one or 2 motoneuron impulses (Miller et al. 2003). These impulses are primarily elicited by sustaining fiber to motoneuron synapses. Thus the timing precision of the sustaining fiber discharge can have important functional consequences. The timing precision for sustaining and dimming fibers has been examined by 2 methods. The first is an investigation of the autocorrelation function of the impulse train at a time lag equal to one stimulus period. Because the autocorrelation function reflects all of the spikes in the impulse train, it provides a measure of the repeatability of the entire response from one cycle to the next. The second method examines the variance in the time of the first spike of a rectified discharge and of the time of the PSP threshold crossing.
Spike timing from autocorrelation functions
The impulse train autocorrelation function provides a measure of the precision of the impulse times as indicated in the magnitude of the peak at a time lag equal to the stimulus period and the width of this peak at half maximum amplitude (i.e., the half width). If the impulse train contained one impulse per stimulus cycle at intervals of exactly one stimulus period, the correlogram would have an amplitude of 1.0 and a half width of zero. Jitter in the timing of that one impulse would reduce the amplitude and increase the half width. Variability in the number of impulses/cycle or the timing of those impulses will also diminish the correlation amplitude and increase the half width. Figure 9 shows samples of 2 autocorrelograms from a sustaining fiber's visual response at different temporal frequencies (note differences in scales between the 2 panels). As the stimulus temporal frequency increases from 1.0 to 7.2 Hz, the amplitude increases about 6-fold and the half width declines from 50 to 1.5 ms. Figure 10 shows the peak height (open circles) and half width (open squares) for a dimming fiber and 2 different sustaining fibers as a function of stimulus frequency. Because the half width is coupled to the stimulus frequency we normalized the half width to the stimulus period.
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). Because IPSPs and large EPSPs elicit a rectified discharge pattern, we also considered the influence of rectification on spike timing precision.
In both the light- and current-elicited responses shown in Fig. 10, the amplitude of the autocorrelation typically increases with stimulus frequency, but the current response yielded uniformly small amplitudes (the maximum was <0.2), as shown by the open circles in Fig. 10, A, C, and E. At low stimulus frequencies the half widths (Fig. 10, squares) are roughly comparable (50–100%) to the stimulus period in most cells. The results show that large reductions in the half width (corresponding to better timing precision) can accompany rectification of the discharge (solid bar at top of panel). Similar decreases in autocorrelation function half width are also observed near the resonant frequency fo, both in the absence (Fig. 10E) or presence of rectification (Fig. 10B). The smallest half widths occur in conjunction with the discharges elicited by the largest and steepest PSPs (as in Fig. 10D, same cell as in Fig. 7). Conversely, if the same neuron is subjected to a small sine-wave extrinsic current (0.4 x threshold current) at stimulus frequencies much less than resonance (as in Fig. 10C), the half width does not fall below 30% of the stimulus period. Thus a small half width, and corresponding increased precision can be achieved by high dV/dt, a rectified discharge, or as the stimulus frequency approaches resonance. For the data set as a whole the joint effect of stimulus frequency and rectification was as follows: for stimuli at 0.1 to 0.5 Hz the occurrence of rectification reduces the half width from 0.70 ± 0.30 stimulus periods to 0.46 ± 0.17 stimulus periods. For stimuli at 6 to 11 Hz rectification reduces the half width from 0.23 ± 0.12 to 0.05 ± 0.03 stimulus periods. These results are qualitatively similar to related findings of Hunter and Milton (2003) and Schreiber et al. (2003). The autocorrelation functions of the simulated responses were quite similar to those of the experimental data (r = 0.95 for both the peak intensity and half width). This result implies that the frequency-dependent variations in the structure of the simulated impulse trains approximate those of the neural response.
Spike time variance
For rectified responses, an additional measure of spike precision is the variance in the time of the first impulse after the onset of the discharge (SDspk), which is related to the variance in the time at which the PSP reaches threshold (SDPSP) (Mar
álek et al. 1997). In rectified responses, we computed the variance of the times at which the membrane potential reaches a fixed potential just below the impulse threshold (the exact value may not be critical because the depolarizing voltage trajectories are parallel to one another). The temporal jitter in the PSP (SDPSP) varies between 0.7 and 5.0% of the stimulus period, as shown in Fig. 11A (filled circles). The figure indicates that the variance in PSP threshold crossing is an approximately constant fraction of the stimulus frequency. The variance of spike times was very similar to that of the voltage times, as shown by a plot of the ratio SDspk/SDPSP (Fig. 11A, open circles). The average ratio SDspk/SDPSP = 1.06 ± 0.55. This ratio exhibits a weak but significant (P < 0.05) correlation to the log of stimulus frequency (r = 0.33). Thus at low stimulus frequencies, spike time precision is slightly better than the precision of the PSP, as suggested by numerical simulations of integrate-and-fire neurons (Marálek et al. 1997). At higher frequencies, however, spike timing precision appears to be limited by factors in addition to the timing precision of the PSP (e.g., channel noise; White et al. 2000). For model spike times, SDspk/SDPSP (Fig. 10A, x symbols) is 1.69 ± 2.35. Much of the discrepancy between simulated and experimental data was produced by a few highly divergent model SDspk values (x symbols at top of Fig. 11A).
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; Poliakov et al. 1997; Warzecha et al. 2003), as shown in Fig. 11B. In the figure, SDspk is seen to vary inversely with the rate of depolarization. This relationship between SDspk and dV/dt was similar for sustaining and dimming fibers, for experimental (open circles) and simulated (x symbols) responses and for all temporal frequencies of the visual stimulus. Four of the SDspk observations in Fig. 11B are between 0.5 and 1.0 ms. Given that the sampling rate was 1,000 Hz, these observations suggest that a higher sampling rate is necessary to obtain an unbiased estimate of spike time variance at the highest stimulus temporal frequencies.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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), as noted below. Gain control
The gain is primarily influenced by adaptation and PSP nonlinearity. Stimulation by extrinsic square-wave and sine-wave currents reveals spike-frequency adaptation of the encoder which is 30 to 40% greater in dimming fibers than in sustaining fibers. This difference is statistically significant (P < 0.01). In both cell types, the frequency response of the discharge is high-pass or band-pass and the discharge phase leads the current. The results in Figs. 2–4 indicate that in response to sine-wave currents, gains of both sustaining and dimming fiber impulse trains increase with increasing stimulus frequency. Thus frequency-dependent gain is a property of the spike generator. Our simulations (as in Fig. 4) imply that spike-frequency adaptation is the principal determinant of the frequency-dependent gain. Adaptation disproportionately reduces the firing rate at low stimulus frequencies for both dimming fibers and sustaining fibers. The gain reduction is smaller in the sustaining fibers because of the smaller degree of spike-frequency adaptation. The frequency dependence of encoder gain has been described in a variety of neurons in crayfish and Limulus (Fohlmeister et al. 1977; Terzuolo and Knox 1971), in spiders (French et al. 2001), and in mammalian cortical neurons (Smith et al. 2000). Furthermore, gain control by encoder adaptation or feedback inhibition is well documented (Benda and Herz 2003; Fohlmeister et al. 1977; Knight et al. 1970; Rauch et al. 2003).
The visual response to sine-wave illumination reflects the interaction of the encoder mechanisms and the synaptic current. The most potent feature of the PSP with respect to the gain of the discharge is the rate of depolarization (Hunter and Milton 2003; Poliakov et al. 1997). Figure 5 shows that as the stimulus frequency increases, there is an increase in dV/dt of the sustaining fiber synaptic potential. The computed peak synaptic current increases even though the PSP fundamental amplitude declines. These relationships form the basis of the frequency-dependent increase in gain of the visual response. The nonlinearity of the PSP contributes to the frequency-dependent increase of dV/dt, the associated synaptic current, and the gain. The PSP rectification contributes to the gain because it minimizes the response at the trough of the stimulus and provides an interval for the dissipation of Gadapt and Na+-inactivation. As shown in Fig. 8D, the model suggests that frequency-dependent variations in the PSP can account for substantial variations in gain that are nearly independent of spike-frequency adaptation. At stimulus frequencies between 0.3 and 2.0 Hz, the sustaining fiber EPSP undergoes a transition from a graded response to what appears to be a "triggered" event. This event is associated with a large increase in the rate of depolarization, the magnitude of the simulated synaptic current, and the nonlinearity of the response. These triggered events in turn control the time course of the discharge and elicit a frequency-dependent increase in gain beyond that seen in the response to extrinsic current (as in Table 3). In dimming fibers, the PSP transients are smaller and slower than those in sustaining fibers. The PSP nonlinearity makes only a minimal contribution to firing rate gain, and the maximum gain for the visual response is similar to that seen with the extrinsic current stimulus (Table 3). Thus the control of the gain of the discharge may reflect spike-frequency adaptation (Benda and Herz 2003; Fohlmeister et al. 1977) as in most dimming fibers or frequency-dependent variations in synaptic current, as in most sustaining fibers. In fly motion detectors (Haag and Borst 1997) the visual response exhibits a frequency-dependent increase in gain associated with a corresponding increase in PSP nonlinearity. This result is analogous to our findings in sustaining fibers. In the spike trains of spider mechanoreceptors (French et al. 2001), a frequency-dependent increase in gain reflects encoder dynamics as in dimming fibers.
The nonlinear relationship between the sine-wave visual stimulus and the sustaining fiber EPSP has a basis in the sustaining fiber afferent pathway (Glantz and Nudelman 1976; Pfeiffer and Glantz 1989; Waldrop and Glantz 1985b). The afferents are controlled by delayed feedback inhibition and they can oscillate synchronously during strong excitation. Thus sinusoidal illumination may act like a forcing function applied to an oscillatory system (Szücs et al. 2001) and the nonlinear PSP may be the result.
Timing precision
The timing precision of spikes is important in several contexts including information transmission (Johnson et al. 2000) and the synchronization of neural populations (Fuhrmann et al. 2002). The autocorrelation studies suggest that timing precision of the entire impulse train is influenced by rectification of the discharge but this influence is most strongly expressed at higher stimulus frequencies. Between 6 and 11 Hz rectification reduces the autocorrelation half width by about 80% compared with an unrectified discharge. At low frequencies the comparable reduction is only 35%. The relationship between stimulus frequency, rectification, and timing precision can be clarified by reference to Figs. 6 and 7. At low stimulus frequencies, the rectified discharge occupies 50–60% of the stimulus period and the membrane potential remains close to threshold for much of this interval. In this circumstance spike generation is strongly influenced by uncorrelated synaptic activity and ion channel noise (White et al. 2000). The onset of the discharge may exhibit high temporal precision but in the autocorrelation function, these events are averaged with later, less precisely timed spikes and it is the average that determines the correlation half width. At higher stimulus frequencies there are fewer spikes/cycle and the discharge occupies about 15–20% of the stimulus period. By confining a smaller number of spikes to a smaller fraction of the stimulus cycle, the discharge at the higher stimulus frequency is certain to exhibit an increase in timing precision. In an unrectified response, many of the spikes are not so confined and the discharge is less precisely timed. All of these considerations are particularly relevant to the autocorrelation function because it represents an average of the timing fidelity of all the spikes in the impulse train. With strong rectification, timing precision can approach one part in 300 relative to the stimulus period, as shown in Fig. 10D.
Spike-frequency adaptation can influence timing precision through its effects on the resonant frequency (Schreiber et al. 2003) and the timing precision of the resonant response. Furthermore, adaptation can increase timing precision by contributing to rectification of the discharge. Adaptation effects rectification through forward masking (Liu and Wang 2001). After a high-frequency burst of impulses near the peak of the stimulus, the enhanced shunt conductance suppresses the discharge during the subsequent trough of the stimulus. As noted above, the silent period associated with rectified responses may increase spike timing precision through mechanisms other than spike-frequency adaptation.
Resonance can also be a powerful determinant of spike timing precision (Hunter and Milton 2003; Knight 1972; Schreiber et al. 2003). In this study we found, however, that if the discharge rectifies at a low stimulus frequency as in Fig. 10, A and F, stimulation at the higher resonant frequency may add little to the spike timing precision. This result is in accord with evidence (Hunter and Milton 2003) that the influence of resonance on timing precision is limited to an intermediate range of stimulus amplitudes.
The PSP analysis also suggests that the most important determinant of timing precision of the first spike in a train is the stimulus frequency. The stimulus frequency controls the rate of depolarization and the temporal jitter of the PSP and these act directly on the spike generator. Our results indicate that the SD of the PSP time to threshold is about 1 to 2% of the stimulus period and that the spike time SD is closely related to that of the PSP. Although the simulated spike time SDs only approximate those of the neurons, they have a similar relationship to the stimulus frequency. This similarity suggests that spike timing precision may be the result of the interaction of a noisy PSP and a deterministic encoder.
The timing precision of the visually elicited discharge is also influenced by PSP nonlinearity. In the model, this nonlinearity exerts its effects through an increase in the dV/dt term in Eq. 3. As can be seen in Fig. 11B, both the observed and simulated responses show a strong inverse relation between spike time variance and the rate of depolarization of the PSP for a 1,000-fold range of dV/dt. This relationship reflects the interaction of spontaneous membrane noise and the stimulus-elicited synaptic current. At low rates of depolarization (1–10 mV/s), the membrane potential will spend hundreds of milliseconds in a depolarized but subthreshold state. In this condition spike timing is strongly influenced by spontaneous membrane potential fluctuations and channel noise (White et al. 2000). As the rate of depolarization increases, an increasing proportion of the synaptic current is associated with dV/dt and the instant of threshold crossing is increasingly dominated by the PSP trajectory. Similar finding are described in Poliakov et al. (1997) and related results are implicit in the findings of Mainen and Sejnowsky (1995) and Hunter and Milton (2003). In fly motion detectors (Warzecha et al. 2003) there is an inverse relationship between spike time variance and the rate of presynaptic depolarization. The effect of PSP nonlinearity also manifests itself in the autocorrelation function of visually elicited sustaining fiber spike trains. As seen in Fig. 10, D and F, the half width of the peak declines dramatically (and timing precision increases) as the EPSP becomes nonlinear near stimulus frequencies of 1 or 2 Hz.
Evaluation of the adaptive integrate-and-fire model
The simulations provided only approximations of the time-varying firing rates, with an RMSE of 21–27% for both fiber types and for both extrinsic current and visual stimuli. This is a much poorer fit than the 5–15% RMSE for responses to square-wave stimuli (Glantz and Schroeter 2004). However, the model was invaluable in several areas. In most neurons the model provided excellent estimates of the frequency-dependent variations in gain that were essential to the interpretation of the results. The high correlations between the autocorrelation features of experimental and simulated responses implies that the model contains the essential elements responsible for timing precision, e.g., the conversion of variations in dV/dt into correlated variations in impulse times and the ability to form rectified responses at the correct phase. Additionally, the model enabled us to explore the influence of spike-frequency adaptation on the temporal frequency response, as well as estimate the synaptic currents associated with the PSPs and the influence of those currents on the firing rate. A review of the experimental and model responses suggests several possible sources of error. Foremost are the possibilities that a neglected neural mechanism such as cable conduction (Fohlmeister and Miller 1997) or the deinactivation of Na+ channels by hyperpolarization (Borisyuk et al. 2002) might influence the frequency response. A second source of error is the variability of the neuronal firing rate. The CV of the mean rate was typically about 0.3, whereas the model is completely deterministic, containing no "noise term" to simulate this variability. Finally the model has no variable to accommodate the tendency of some cells to fire in bursts (Smith et al. 2000). All of these omissions follow from the motivation to keep the model as simple (and as transparent) as possible.
The link between stimulus temporal frequency, gain, and precision has important functional implications for the crayfish optomotor system. Crayfish oculomotor reflexes are elicited by changes in the distribution of light in the dorsal visual field (Miller et al. 2003). In the motoneurons, which mediate these reflexes, the direction of eye movement is determined by the first one or 2 impulses in the cells of the motoneuron ensemble. Because this activity is largely controlled by sustaining fiber to motoneuron synapses (Glantz et al. 1984), the gain and precision of the sustaining fiber discharge have a direct bearing on behavior.
In conclusion, crayfish visual interneurons exhibit variations in gain and spike timing precision related to the frequency of sine-wave stimulation. These phenomena are influenced by spike-frequency adaptation and nonlinearities in the PSP. An adaptive integrate-and-fire model approximates the frequency response and timing precision of both the current- and light-elicited discharges and simulates the influences of adaptation and PSP nonlinearity. Thus these phenomena reflect the operation of adaptive integrate-and-fire mechanisms. Because the PSP nonlinearity entails a high rate of depolarization, it can increase the precision of spike timing by 10- to 100-fold compared with the response to pure extrinsic current sinusoids at the same temporal frequency. Spike-frequency adaptation (Benda and Herz 2003; Schneider 2003) and frequency-dependent PSP nonlinearities (Fortune and Rose 2001; Fuhrmann et al. 2002; Sakai 1992; Szücs et al. 2001) occur in a wide variety of neural systems and their effects may be particularly significant where high gain or/and precision are required.
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