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Department of Neurobiology, University of California, San Diego, La Jolla, California
Submitted 13 June 2006; accepted in final form 26 June 2007
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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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Neurons in the early stages of vertebrate visual systems have been reported to change their gain or sensitivity as the stimulus contrast changes (Baccus and Meister 2002
; Benardete and Kaplan 1999; Chander and Chichilnisky 2001; Enroth-Cugell and Robson 1966; Kim and Rieke 2001, 2003; Ohzawa et al. 1985; Rieke 2001; Sanchez-Vives et al. 2000a,b; Shapley 1997; Shapley and Enroth-Cugell 1984; Shapley and Victor 1978, 1979, 1981; Smirnakis et al. 1997; Solomon et al. 2004; Zaghloul et al. 2005). In the early visual circuit, normalization of the neural response to the temporal contrast is just one feature of a complex phenomenon known as contrast gain control, which also entails adaptation to spatial contrast and changes to neural latency and response dynamics on changes in contrast. In this study we focus exclusively on the normalization of response sensitivity to the temporal contrast (the variance over time of neural input) because this feature is common to many sensory neurons at different stages of neural processing, in different sensory modalities, and across phylogeny. For example, motion contrast normalization has been reported in motion-sensitive neurons of primates (MT) and flies (H1) (Brenner et al. 2002; Fairhall et al. 2001; Kohn and Movshon 2003). To this end we have made a simplifying reduction by using spatially uniform visual stimuli that are modulated only in time. It remains for future studies to extend these results to incorporate other aspects of contrast gain control in the retinothalamic circuit.
Some experiments indicate that changes in ionic conductances play an instrumental role in contrast normalization (Kim and Rieke 2001; Sanchez-Vives et al. 2000a). Other studies have shown that resting membrane potentials change in response to contrast (Baccus and Meister 2002; Carandini and Ferster 1997). On the other hand, recent theoretical findings have suggested that neurons could exhibit contrast normalization without actively changing either of these properties. Instead, intrinsic nonlinear properties (such as the threshold and saturation) are sufficient to reproduce contrast normalization in some models (Borst et al. 2005; Yu and Lee 2005; Yu et al. 2005).
We have recently shown that a realistic nonadapting model can exhibit contrast normalization comparable to that found in lateral geniculate nucleus (LGN) cells (Gaudry and Reinagel 2007). Unlike the models in which contrast adaptation was explored previously (Borst et al. 2005; Yu and Lee 2005; Yu et al. 2005), this model (Keat et al. 2001) produces spiking responses with the high temporal precision and spike-count reliability of LGN cells. This enabled us to test whether the effects of contrast on temporal precision, spike-count reliability, and response latency that have been reported in LGN neurons can also be reproduced in a nonadapting model.
In neurons, response properties appear to change gradually after abrupt changes in contrast. Firing rates change with both slow and rapid components (Baccus and Meister 2002; Chander and Chichilnisky 2001; Smirnakis et al. 1997). When neural gain and information transmission were estimated at a fine timescale, both were found to evolve with a time constant in the tens of milliseconds (Fairhall et al. 2001). These transient effects are suggestive of an active adaptation mechanism. Therefore we explored whether our nonadapting model cells can also produce transient changes in response properties after sudden changes in contrast.
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METHODS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Models were implemented as described in Keat et al. (2001) and as explained in detail in RESULTS and Fig. 1. A population of model cells was created by systematically varying the model parameters. These parameters are in arbitrary units of the generator signal (see Fig. 1). We created model cells by using all possible combinations of the following parameters: 
= 0.1 or 0.2; B = 3, 5, or 7; 
P = 20, 35, or 50 ms; A = 20 ms; 
a = 0.01, 0.16, 0.31, or 0.61; b = 0.02, 0.15, or 0.28; F = FX or FY. The filter functions FX and FY were taken from the X ON and Y OFF cat LGN data in Fig. 8C of Keat et al. (2001). Simulations were computed in 1-ms time steps.
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Fig. 4C. For each of 200 model cells, the values of , B, P, A, a, and b were randomly selected from the above-listed values. The filter function was described by a summation of Gaussians with a positive peak between 20 and 40 ms and a negative peak of half the amplitude between 35 and 65 ms. The widths of the first and second peaks were randomly chosen to be between 7 and 15 and between 15 and 24 ms, respectively. We required the filter function at time t = 0 to be <0.05 of the maximum filter amplitude. All filter functions were ON type; calculations using OFF-type filters showed similar results (not shown).
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Model results are compared with data recorded from LGN neurons, as described elsewhere (Gaudry and Reinagel 2007). Experimental methods were essentially according to Reinagel and Reid (2000). Briefly, cats were anesthetized with sodium pentothal (2–4 mg·kg–1·h–1, intravenous). Animals were ventilated using an endotracheal tube. Electrocardiogram, electroencephalogram, temperature, expired CO2, and oxygen in blood were continually monitored. All surgical and experimental procedures were in accordance with National Institutes of Health and United States Department of Agriculture guidelines and were approved by the UCSD Institutional Animal Care and Use Committee. Parylene-coated tungsten electrodes (AM Systems, Everett, WA) were inserted through a 0.5-cm-diameter craniotomy over the LGN. Waveforms were analyzed off-line to isolate single-unit responses (Fee et al. 1996). Stimuli were spatially uniform random binary flickers presented on a custom-built photopic LED array at 125 frames/s. The same binary pattern was scaled about the mean to obtain 11, 33, and 100% contrast, where contrast is defined as the SD of the luminance over the mean. Only steady-state responses (after 5 s at fixed contrast) are shown.
Gain calculations
First we estimated the filter at each contrast as the spike-triggered average. The filters for the three contrasts were normalized to the amplitude of their first peaks. The stimulus at each contrast was convolved by its corresponding filter to estimate the instantaneous stimulus strength. The input–output function was defined as the relation between the observed probabilities of spiking and the instantaneous stimulus strength. For each cell, we fit the input–output function to the data from the 100% contrast condition to the following sigmoidal equation
 | (1) |
In this equation, the variables A, G, and S describe, respectively, the amplitude, slope, and horizontal offset of the nonlinear function. We then fit the input–output functions at the other two contrasts holding A constant. (Estimates of A were noisy at low contrasts, and fitting A did not improve the quality of fits.) We excluded a data set from our analysis if there were <100 spikes in the response or if the R2 value associated with this fit was <0.90. We define the gain and offset as G and S from the sigmoidal fits. In our data both gain and offset changed with contrast, but changes in offset were not systematic and not predictive of information transmission.
Spike-count variability measure
In Fig. 6D, we report the trial-to-trial variability in spike count by calculating the Allan Factor (AF). The Allan Factor is equal to the average squared difference in spike counts (N) between consecutive trials divided by twice the mean spike count (µ) across trials
 | (2) |
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Spike-timing jitter measures
We measured the trial-to-trial jitter in spike timing by the width of the peak of the average cross-correlation between sequential trials. We defined the jitter as the half-width of this peak at half-maximum. We considered this measure invalid if the peak was not at t = 0 or if the peak was not smooth. Specifically, we smoothed by a 5-ms moving average and defined the smoothness as 1 minus the summed squared error between the smoothed and raw correlograms. We used a criterion of smoothness 
0.75, which eliminated from this analysis 2.8% of our simulated data sets (73 of 2,592).
Response latency
Spike-triggered averages (STAs) were computed from responses of each model cell to each contrast. The latency was defined as the time to the shortest latency peak of the STA. In a separate analysis (not shown) we replicated the results of Fig. 4C using two event-based methods. First, we computed STAs instead from the times of discrete firing events (peristimulus time histogram peaks) according to Berry et al. (1997), where the time of each event was defined as the average time of the first spike in the event. Second, we compared the times of all firing events that were shared across both contrast conditions, as described earlier. Both calculations yielded similar trends (not shown). We note that the trends in latency observed in the model did not depend on the sign of the filter (not shown).
Information calculations
We calculated the visual information from spike trains by using the direct method (Strong et al. 1998), implemented exactly as in Reinagel and Reid (2000). This standard method allows us to directly estimate, in a model-independent way, the extent to which spike trains contain information that discriminates among the visual stimuli presented in the experiment or simulation. We used this method because it makes no assumptions about which features of the visual stimulus are encoded, nor about which aspects of the neural response carry that information. In detail, we represented the model responses as time-binned spike trains, using a fixed bin size of 
= 2 ms. Due to the afterpotential, it was rare to observe two spikes within one bin. The value of a bin was defined to be zero if no spikes occurred within the bin or one if any spikes occurred. For steady-state information calculations (Fig. 5), entropies were measured for words of length L = 1 bin (2 ms), although trends were the same as we varied the bin length. We calculated both the average noise entropy 
Hnoise
, which estimates the trial-to-trial variability of responses during fixed stimuli, and the average total entropy Htotal, which estimates the variability of responses across all stimuli in the ensemble. Entropy values were averaged across time points during the steady-state stimulus (from t values of 5 to 10 s relative to a change in stimulus contrast). The mutual information was defined as the difference in these entropies: I = Htotal – Hnoise. We correct for finite data size according to the method of Strong et al. (1998). Specifically, we fit a second-order polynomial to 1/(fraction of data) versus the entropy estimates, and evaluate the polynomial at 1/(fraction of data) = 0. We required that the total correction for finite data size was <10% and the second-order term of this correction was <1%.
Analysis of responses as a function of time after contrast change
We define a stimulus sequence as a 40-s stimulus in which contrast changes every 10 s, transitioning from high to medium to low to medium contrast in that order. Different sequences differ in the binary flicker pattern, but not the contrast transition pattern. We presented one model cell with 216 different stimulus sequences (each shown once) and 686 additional stimulus sequences (each repeated 512 times). The time-varying firing rates were calculated by averaging the firing rate in each time bin over the 216 unique stimulus sequences (Fig. 7A). Results shown are from averaging in overlapping boxcar windows of length 8 ms.
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We calculated the time-varying noise entropy separately from each of the 686 repeated stimulus sequences and averaged the entropies across stimulus sequences. We calculated the time-varying total entropy separately from each of 128 blocks of 512 unique sequences and averaged the entropies across blocks. Entropies and information were calculated for words of length L = 1 bin, for every 2-ms time step relative to the time of contrast change. Results shown are from averaging the information rates in overlapping boxcar windows of 12 bins (24 ms).
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RESULTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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). Here, we explore why contrast normalization occurs in this model. Additionally, we show that the nonadapting model also exhibits several other properties of LGN responses, some of which have previously been thought to give evidence of active adaptation. A generative model for spike trains
Responses of LGN neurons can be compactly described by a nonlinear model with few parameters (Fig. 1; Keat et al. 2001). This model has been shown to successfully reproduce responses of LGN neurons to white noise stimuli, including the reliability and precision of spikes. This is only one of several nonlinear models that can produce contrast normalization with fixed model parameters, but it was the only model we tested that also exhibited changes in spike timing (precision and latency) with contrast, as subsequently discussed further.
The model first convolves the stimulus with a linear filter. The generator potential is equal to the convolved stimulus plus noise; the amplitude and time constant of this noise are defined by the model parameters a and A, respectively. A spike is generated whenever the generator signal crosses a threshold . Each time a spike occurs, a negative afterpotential is added to the generator potential, such that the threshold is crossed repeatedly during sustained excitatory stimuli. The amplitude, time constant, and variability in the amplitude of the negative afterpotential are defined by the model parameters B, P, and b, respectively. This model is closely related to, but distinct from, the leaky integrate-and-fire model of Pillow et al. (2005).
To test the effects of contrast in this model, we used a temporal binary white noise stimulus. The stimulus was shown repeatedly at each of three contrasts. Responses of a model cell to 100, 33, and 11% contrast stimuli are shown in Fig. 2, A, B, and C, respectively. Responses to all contrasts were generated using a single set of fixed model parameters. In the following sections we will describe the responses of fixed-parameter model cells in terms of contrast normalization and information transmission, as well as the contrast dependence of response reliability, precision, and latency. We will compare the qualitative trends in the model to previous results from real LGN neurons.
Model cells can exhibit contrast normalization
Although this model has fixed parameters at all contrasts, it exhibits contrast normalization due to its nonlinearity (Gaudry and Reinagel 2007). To measure the gain of the model we fit a linear–nonlinear cascade to the responses at each contrast condition, a method routinely used to characterize gain of visual neurons (Baccus and Meister 2002; Chander and Chichilnisky 2001; Kim and Rieke 2001; Sakai et al. 1995; Zaghloul et al. 2005). Briefly, neural data are fit by a filter and an input–output function. The filter is defined as the spike-triggered average, which is taken as an approximation of the stimulus feature to which the neuron is sensitive. The convolution of the stimulus with the filter is evaluated at each time step of the stimulus; this can be thought of as the stimulus strength. An input–output function is defined by the observed probability of spiking as a function of the stimulus strength. Differences in gain at different contrasts can be measured by comparing the amplitudes of the recovered filters (if the input–output function is fixed) or by comparing the recovered input–output functions (if the filter amplitude is normalized). We have analyzed our model responses in both ways.
Here, we show the latter approach. For one example model cell, we scaled all of the recovered filters to the same amplitude (Fig. 3 A), such that gain changes were contained within differences in the input–output functions. The observed input–output function is steeper for lower-contrast stimuli than for higher contrasts (Fig. 3B). We fit a sigmoid equation to the input–output function at each contrast to calculate the gain (see METHODS). The gain for this model cell at low, medium, and high contrast is equal to 0.100, 0.079, and 0.054, respectively. Therefore the model cell exhibits contrast normalization, despite the fact that the same parameters were used to generate responses to all contrasts. We note that the model also shows changes in filter shape, to be discussed in the following text.
Next, we simulated model responses using many different sets of parameters (see METHODS) to generate a large representative population of model cells. For any given model cell the parameters were fixed for all contrasts. We compared the gain of each model cell at 100% contrast to its gain at either 33 or 11% contrast (Fig. 4 A).
As reported previously (Gaudry and Reinagel 2007), for some model cells, the responses showed little or no contrast normalization: the gain was relatively independent of contrast (symbols near thick line). For other model cells, the gain increased by nearly the same factor that the contrast decreased (triangles near dashed line, and circles near thin line).
We express the extent of contrast normalization by 
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= 1. If the gain does not change with contrast, the contrast normalization = 0. We note that could in principle be <0 (gain paradoxically increased at high contrast) or >1 (gain increased by more than stimulus contrast decreased), but for real LGN neurons these values are rarely observed.
The population of model cells exhibited a range of contrast normalizations , broadly distributed between zero and one (Fig. 4B). The extent of the contrast normalization decreased as we increased the amplitude of the noise added to the convolved stimulus a (compare symbols of different colors in Fig. 4, A and B). The dependence of on model parameters will be subsequently described further. The range of contrast normalization in LGN neurons (Gaudry and Reinagel 2007; data reproduced in Fig. 4, A and B, black symbols) is comparable to the range we observed in our population of model cells.
In neurons, contrast normalization can be accompanied by a decrease in latency with increasing contrast. In our model cells, the linear filters as estimated by the spike-triggered average had both shorter latency and shorter duration at higher contrasts (Fig. 4C; see also Fig. 2D), consistent with results from neurons (Benardete et al. 1992; Chander and Chichilnisky 2001; Shapley and Victor 1978; Smirnakis et al. 1997). Recall that for any given model cell, the linear filter used to generate responses was fixed at all contrasts.
Precision, reliability, and information
Like LGN neurons, most model cells showed at least some decrease in information transmission at lower contrast (Fig. 5 A; Gaudry and Reinagel 2007). In the LGN, we found that the spike-count variability and temporal jitter of LGN responses both increased as stimulus contrast decreased. The model we have implemented here is able to replicate the spike-count and spike-timing variability of neural data at fixed contrast (Keat et al. 2001), but it had not been shown whether this model, with fixed parameters, would show changes in spike-count variability and temporal jitter with changes in contrast.
We quantified the spike-count variability of model responses to repeated stimuli by calculating the Allan Factor (see METHODS). The Allan Factor is high if the spike count is highly variable; a Poisson spike train has an Allan Factor of one. In model responses, we find that the spike-count variability increases as the stimulus contrast decreases (Fig. 5B), as is the case for LGN neurons (Gaudry and Reinagel 2007). Like LGN cells, the model responses often exhibit sub-Poisson spike-count variability.
Our model cells had spike-timing jitter of a few milliseconds, well within the range of values measured under identical conditions from LGN cells. The temporal precision of model responses degraded as contrast decreased (see Fig. 2). We measured the jitter in spike timing by the width of the peak in the cross-correlation between responses of consecutive trials (see METHODS). As the stimulus contrast decreased, the temporal jitter increased (Fig. 5C), as was also found in LGN data (Gaudry and Reinagel 2007).
We previously reported that the extent of contrast normalization was correlated with the preservation of information in LGN cells (R2 = 0.43; P < 0.01, replotted in Fig. 5D) and also in a population of model cells (Gaudry and Reinagel 2007). We extend this to a larger population of model cells here (Fig. 5D; R2 = 0.76; P < 0.01) and note that this correlation was considerably stronger in the model population than in the real cells.
We used our population of model cells to explore whether the preservation of information was related to a preservation of either temporal precision or spike-count reliability. To measure the change in information rate with contrast, we define the information ratio as the information rate in bits/spike during the lower contrast divided by that at the higher contrast. The information ratio = 1 if the information rate is constant across contrasts and <1 if information rate decreases at lower contrasts. To measure the change in spike-count variability we define the Allan Factor ratio as the Allan Factor at the higher contrast divided by that at the lower contrast. Defined this way, the Allan Factor ratio will be <1 if spike-count becomes more variable as the contrast decreases. We found a significant but very weak correlation between the Allan Factor ratio and the information ratio for model cells (Fig. 5E, black symbols; R2 = 0.0086; P < 0.01). This suggested that effects on spike-count variability are not crucial for the beneficial effects of contrast normalization. We went back to previous LGN data to test this prediction. We found no significant correlation between information ratio and Allan Factor ratio in LGN cells (Fig. 5E, red symbols; R2 = 0.021; P = 0.22), consistent with the model results.
To measure the change in temporal precision with contrast, we defined the jitter ratio as the temporal jitter at the higher contrast divided by that at the lower contrast. The jitter ratio will be <1 if spike timing becomes more variable as the contrast decreases. Among our model cells, the jitter ratio was strongly and significantly correlated with the information ratio (Fig. 5F, black symbols; R2 = 0.73; P < 0.01); the jitter ratio was also correlated with the contrast normalization index (R2 = 0.52; P < 0.0001; not shown). In LGN neurons, however, the jitter ratio was not correlated with either the information ratio (Fig. 5F, red symbols; R2 = 0.0025; P = 0.70) or the contrast normalization index (R2 = 0.002; P = 0.75; not shown).
To properly compare the model to LGN data, it is therefore necessary to isolate the effect of contrast normalization on information preservation. In the model, the partial correlation of contrast normalization with information ratio (holding jitter ratio constant) was highly significant (R2 = 0.50 P < 0.0001). Therefore like LGN neurons, the model can exhibit information preservation due to normalization, independent of any effects of contrast on temporal precision. We note that after factoring out the role of jitter, the strength of the correlation between information ratio and contrast normalization was similar in model and data (R2 = 0.50 vs. 0.43).
Given that contrast affected gain, temporal precision, and spike-count reliability in both LGN cells and model cells, it was not clear a priori which of these most accounted for changes in information rates across contrast. From this analysis we conclude that the effect of contrast normalization on information preservation in the LGN is not due to an underlying correlated effect on precision or reliability of responses. The primary determinant of information conservation in the LGN is contrast normalization.
How model parameters influence contrast normalization
To develop better intuitions about why contrast normalization arises in the absence of active adaptation, we separately varied each parameter of the model holding all other parameters at fixed values (see METHODS). We find that the contrast normalization depends most strongly on the generator potential's noise term a (Fig. 6E; also compare symbols of different colors in Fig. 4B). Model cells with less noise exhibited more contrast normalization. This is consistent with the finding in other models that a balanced change in both the excitatory and inhibitory background firing rates can give rise to gain control (Chance et al. 2002).
The contrast normalization is also weakly dependent on parameters of the negative afterpotential, including its amplitude (B), time constant (P), and noise (b) (Fig. 6, B, C, and F, respectively). Our interpretation of how contrast normalization arises in the model will be further considered in the DISCUSSION.
Model cells with fixed parameters exhibit transitory changes
Sensory neurons exhibit transitory changes in their response properties immediately after an abrupt change in contrast. These gradual changes suggest an active adaptation process, although transitory effects on the timescale of integration time of the filter are expected even in a fixed, linear system. Like LGN neurons, the model cells contained a biphasic linear filter that integrated stimuli for 
100 ms (e.g., see Fig. 3A). Thus it is not surprising that model cells exhibit smoothing of transitions as well as overshooting and ringing after abrupt changes in contrast (Fig. 7). In particular, the firing rate overshoots the steady state when contrast changes (Fig. 7A), and the information rate dips below the eventual steady state particularly for transition from high to medium contrast (Fig. 7B). Additional slow changes, on the order of seconds, occur in neurons but not in the model.
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DISCUSSION |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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In general, many systems that combine linear filtering with a threshold exhibit contrast normalization due to saturation of responses at high contrast (Yu et al. 2005). Contrast normalization arises in the present model by a complex interaction of spiking threshold, noise, and refractoriness.
We illustrate a short sample of response generation in a model cell in Fig. 8. For high-contrast stimuli (red curves), the generator potential h(t) tends to cross threshold well before the peak of the filtered stimulus g(t). The negative afterpotential causes repetitive firing [i.e., h(t) crosses threshold again] for the strongest stimulus events. Nevertheless, the cumulative effect of the negative afterpotential is to truncate responses, such that the number of spikes during the peak in g(t) increases much less than linearly with the size of the peak. Therefore even though g(t) scales linearly with contrast, spiking responses do not.
The afterpotential reduces the probability of repeated firing within peaks of g(t). As we decrease the amplitude or duration of the afterpotential, the gain increases at both contrasts, but the effect is larger for high contrast because there are more large peaks (Fig. 8). Therefore the difference in gain becomes smaller, such that contrast normalization decreases (Fig. 6, B and C). At the extreme, when the afterpotential is small and fast compared with fluctuations of g(t), the model fires repetitively for the duration of an excitatory input, such that the responses at both high and low contrast are approximately linear with the size of the peak in g(t) and contrast normalization is lost. Although this is consistent with the proposed role for sodium channel inactivation in contrast normalization (Kim and Rieke 2003), it is important to keep in mind that the afterpotential in this model is an abstract formal concept that encompasses all forms of membrane and synaptic refractoriness in every stage of processing from the photoreceptor to LGN spike generation.
The noise parameter a had by far the strongest correlation with contrast normalization among our model cells. Adding noise to g(t) increases the probability that a just-subthreshold stimulus will evoke a spike in a given trial, but decreases the probability that a just-superthreshold stimulus will evoke a spike. Therefore the probability of spiking is less dependent on input strength; the slope of the input–output function becomes shallower with noise a (gain decreases). This effect is stronger at lower contrast because the stimulus-induced fluctuations in g(t) are smaller. Therefore the gains at high and low contrast become more similar as noise increases, i.e., there is less contrast normalization. This feature in the model is in qualitative agreement with the data showing that the effects of subthreshold noise on spiking depend on contrast (Demb et al. 2004). However we note that the noise parameter a in the model encompasses noise of this form distributed throughout the neural circuit, not only in the cell from which we recorded.
How does this translate to information transmission? In the extreme of very low noise, where contrast normalization is greatest, the low-contrast stimuli elicit spikes only at very strong peaks in g(t), such that information in bits/spike is comparable to that obtained with high-contrast stimuli. As noise increases, the times of threshold crossings during low-contrast stimuli can be dominated by noise, thereby decreasing the mutual information about the stimulus in bits/spike. For high-contrast stimuli the same noise has less effect on the threshold crossings and thus less effect on information transmission.
We note that these mechanisms are distinct from other nonlinear models that also produce contrast normalization. Yu et al. (2005) described a nonlinear model, which exhibits contrast normalization but does not optimize information transmission across contrasts (Yu et al. 2005). We found contrast normalization in linear–nonlinear–Poisson–refractory model cells (results not shown) and in the leaky integrate-and-fire model of Pillow et al. (2005). Those models did not exhibit the contrast-dependent changes in spike-timing precision and spike-count reliability found in real cells, at least not with the parameters we tested (K Gaudry, P Reinagel, and J Pillow, personal communication).
Contrast gain control
In the early visual system, contrast normalization is just one aspect of the well-defined phenomenon known as contrast gain control. In addition to normalization, in contrast gain control the dynamics of the cell change with stimulus contrast (Baccus and Meister 2002; Benardete et al. 1992; Chander and Chichilnisky 2001; Kim and Rieke 2001; Shapley and Victor 1978, 1981; Victor 1987; Zaghloul et al. 2005). We generated model responses using a fixed filter at all contrasts and found that the latency of the recovered filters depends on the stimulus contrast (Fig. 4C). We attribute this to the positive threshold-crossing structure of the model (Fig. 1), which effectively introduces a derivative of the filter into the generation mechanism. We do not think this arises from bias in the shape of the filter due to estimation by the spike-triggered average (Pillow and Simoncelli 2003; Schwartz et al. 2002). Using spatially patterned stimuli, it is possible to dissociate the luminance contrast of stimuli from their efficacy in driving a cell. In retinal ganglion cells, the changes in latency associated with contrast gain control are predicted by stimulus contrast, not driving strength (Shapley and Victor 1978). We have not tested this in our model cells, but we expect that such an effect can be accounted for only by explicitly modeling the presynaptic neural populations (for example, using a nonspiking, rectifying generative model for each contributing bipolar cell). In general, effects of contrast could easily be uncoupled from driving strength if normalization occurs in an earlier neuron in the circuit or if normalization arises more from effects of noise than from effects of refractoriness.
Transient changes after contrast shifts
After the stimulus contrast is changed, neural firing rates quickly change and then gradually converge to their steady-state values. These transient changes in firing rate include a fast component (Baccus and Meister 2002; Chander and Chichilnisky 2001; Smirnakis et al. 1997), which occurs within 100 ms of a change in stimulus contrast (Shapley 1997; Victor 1987). The slow component of the firing rate adaptation occurs on the order of 1–10 s (Baccus and Meister 2002; Brown and Masland 2001; Chander and Chichilnisky 2001; Smirnakis et al. 1997; Solomon et al. 2004). This slow adaptation has been attributed to a gradual change in cells' baseline membrane potentials (Baccus and Meister 2002; Solomon et al. 2004). The time constants of the slow component have been related theoretically to Bayesian optimal strategies for detecting that the contrast has changed (DeWeese and Zador 1998). Consistent with this theory, the slow time constant depends on the duration of the contrast periods (Manookin and Demb 2006).
In the fly motion-sensitive neuron H1, Fairhall et al. (2001) showed that information transmission efficiency (in bits/spike) dropped transiently just after a decrease in contrast, recovering gradually. This result was highly suggestive that an active adaptation process is required to restore coding efficiency after a change in stimulus statistics. In light of the two time constants discussed earlier, it is of interest that both the gain change and the information optimization in H1 occurred within tens of milliseconds, much faster than the firing rate adaptation that was on the order of seconds.
Numerous investigations have searched for changing neural properties underlying contrast adaptation, including changes in gain of membrane voltage (Baccus and Meister 2002; Kim and Rieke 2001; Rieke 2001; Sanchez-Vives et al. 2000a; Zaghloul et al. 2005), shifts in mean membrane potential relative to threshold (Carandini and Ferster 1997; Zaghloul et al. 2005), and recruitment of inhibitory circuitry (Chance et al. 2002; Murphy and Miller 2003; Prescott and De Knoninck 2003). Intracellular studies in the retina concluded that ganglion cell's spiking generation mechanism adapts to stimulus contrast on timescales of seconds (Kim and Rieke 2001; Zaghloul et al. 2005). An increased number of sodium channels become inactive after a high-contrast stimulus is presented, resulting in a decreased sensitivity to stimulus fluctuations (Kim and Rieke 2001).
Nevertheless, the necessity of adaptive mechanisms to explain transient changes has been questioned (Borst et al. 2005; Shapley 1997; Yu et al. 2005). Here we show that a model with no adapting parameters is capable of producing the fast component of firing rate change after a change in stimulus contrast (Fig. 7A). A well-established fact about linear signal processing is that a biphasic filter will produce overshooting such as observed in the firing rates of visual neurons. In our simulations, biphasic filters were necessary and sufficient to obtain this effect (not shown). The duration of the firing-rate transient is limited by the length of the filter, 50–100 ms. The model does not replicate the slow component of firing rate adaptation, suggesting that this slow component is not required for contrast normalization or information optimization. In our model cells, gain changes (not shown) and optimization of information transmission (Fig. 7B) both occur on this fast timescale and both occur without any change in model parameters. Because of the very large data sets required, we have yet to obtain comparable measurements from the biological neurons.
In summary, many sensory neurons exhibit contrast normalization (normalization to input variance). Often this is cast in the vocabulary of adaptation, as if neurons are actively changing their properties in response to the contrast of recent stimuli to optimize information coding. We show here that realistic spiking model cells with fixed parameters can qualitatively account for normalization to temporal contrast across the range observed in LGN neurons. Our results do not rule out a role of active adaptation for contrast normalization in the LGN, but we do show that many important effects of contrast arise automatically from linear filtering and the nonlinear nature of spike generation, consistent with the fact that contrast normalization is rapid or instantaneous in the early visual system.
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ACKNOWLEDGMENTS |
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TOPABSTRACTINTRODUCTIONMETHODSRESULTSDISCUSSIONGRANTSACKNOWLEDGMENTSREFERENCES |
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Address for reprint requests and other correspondence: P. Reinagel, Department of Neurobiology, University of California, San Diego, 9500 Gilman Drive #0357, La Jolla, CA 92093-0357 (E-mail: preinagel{at}ucsd.edu)
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) and 11% contrast (
). In the absence of gain control, gain would be constant (x = y, thick, solid line). Three- or 9-fold gain changes correspond to the thin, solid line or the dashed line, respectively. Black symbols are data from the lateral geniculate nucleus (LGN) (Gaudry and Reinagel 2007
