Modeling convergent ON and OFF pathways in the early visual system
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For understanding the computation and function of single neurons in sensory systems, one needs to investigate how sensory stimuli are related to a neuron’s response and which biological mechanisms underlie this relationship. Mathematical models of the stimulus–response relationship have proved very useful in approaching these issues in a systematic, quantitative way. A starting point for many such analyses has been provided by phenomenological “linear–nonlinear” (LN) models, which comprise a linear filter followed by a static nonlinear transformation. The linear filter is often associated with the neuron’s receptive field. However, the structure of the receptive field is generally a result of inputs from many presynaptic neurons, which may form parallel signal processing pathways. In the retina, for example, certain ganglion cells receive excitatory inputs from ON-type as well as OFF-type bipolar cells. Recent experiments have shown that the convergence of these pathways leads to intriguing response characteristics that cannot be captured by a single linear filter. One approach to adjust the LN model to the biological circuit structure is to use multiple parallel filters that capture ON and OFF bipolar inputs. Here, we review these new developments in modeling neuronal responses in the early visual system and provide details about one particular technique for obtaining the required sets of parallel filters from experimental data.
KeywordsLN model extension Multiple pathways Spike-triggered analysis Retina ON–OFF ganglion cells
1 1 Introduction
Assessing the relationship between the sensory stimulus and the neuronal responses and identifying the underlying biological processes are central goals in the study of sensory systems. One way of addressing these questions is by construction of suited model descriptions that aim at quantitatively mapping the stimulus-response relation while simultaneously capturing the relevant neuronal dynamics (Gerstner and Kistler 2002; Dayan and Abbott 2005; Herz et al. 2006).
Here, we will review recent developments for modeling the spiking activity of retinal ganglion cells in response to visual stimulation. These models extend the widely used LN model approach that aims at describing neural responses in terms of a linear filter and a subsequent nonlinear transformation. Recent experiments in the retina have shown that specific response features of certain types of neurons intimately rely on the convergence of parallel processing pathways, which are the result of synaptic inputs from both ON-type and OFF-type bipolar cells. This convergence of parallel pathways with markedly different stimulus-processing characteristics can be captured by models with several linear filters in parallel. Extending the LN model in such a way brings about new data-analytical challenges for obtaining the parameters from experiments. We will begin by revisiting single- and multi-filter LN models and different techniques for extracting their parameters from data. After reviewing applications of the LN model to the retina and summarizing recent related experimental findings, we will provide details about the fitting procedure for one particular multi-pathway model that captures the effects of convergent ON and OFF pathways on first-spike latencies.
2 2 The LN modeling approach
2.1 2.1 Single-filter models
A primary advantage of the LN model is the fact that obtaining the model parameters—the shape of the filter and the nonlinear transformation—can easily be achieved by a reverse correlation analysis with a stimulus that has a Gaussian (or otherwise spherically symmetric) distribution of intensity values (Chichilnisky 2001). In fact, the filter is then simply obtained as the spike-triggered average, i.e., the average of all stimulus segments that generated spikes. The nonlinearity can be subsequently determined, for example, by creating a histogram of the measured neuronal response over the computed filter output y(t). The ease of obtaining the model parameters from experimental data and their straightforward interpretation have made the LN model uniquely popular for modeling stimulus-response relationships of neurons in many sensory systems.
It is important to keep in mind, though, that the modeling of neuronal responses in terms of filters and transformations has an intrinsic phenomenological nature, aimed primarily at providing an accurate description of the signal-processing characteristics and less at capturing the individual biophysical processes that underlie the input-output relation. Nonetheless, this approach can be combined with biophysically inspired components, such as spike generation dynamics (Keat et al. 2001; Paninski et al. 2004; Pillow et al. 2005; Gollisch 2006) or gain control (Shapley and Victor 1978; Victor 1987; Berry et al. 1999). Explicitly incorporating parallel processing pathways for ON and OFF signals, as will be discussed below, represents a similar biologically inspired extension. Before diving into this topic, we review generic phenomenological approaches to multi-filter models.
2.2 2.2 Multi-filter models
Reducing a neuron’s receptive field to a single linear filter has proven too restrictive in many examples. A straightforward remedy is to replace the linear filter in the first model stage by a set of parallel linear filters. Correspondingly, the subsequent nonlinearity becomes a nonlinear function that takes as input all the filter outputs from the first stage and produces a single variable as the output (Fig. 1b). Similar to the single-filter LN model, this multi-filter model draws part of its appeal from the existence of simple and elegant techniques for parameter estimation from experiments. Statistical analysis techniques, such as the “neuronal modes” approach (Marmarelis 1989; Marmarelis and Orme 1993; French and Marmarelis 1995) and, in particular, spike-triggered covariance as a straightforward extension of the spike-triggered average (de Ruyter van Steveninck and Bialek 1988; Touryan et al. 2002; Schwartz et al. 2006) have proved expedient for promoting the applicability of these models in various scenarios.
In short, the spike-triggered covariance analysis is based on comparing the stimulus variance of the complete stimulus set of Gaussian white noise to the variance of the stimulus subset that elicited spikes (“spike-triggered stimulus ensemble”). Typically, the stimulus variance differs between these two ensembles along such stimulus dimensions to which the neuron is sensitive. In other words, the filters of the multi-filter LN model define the only special dimensions of the stimulus space; for all other, orthogonal stimulus dimensions, the original symmetry of the stimulus distribution is preserved, and the stimulus variance stays constant. The stimulus dimensions that do experience a change in variance can be determined from a principal component analysis of the spike-triggered stimulus ensemble. An example for this is given in the presentation of a specific data fitting technique below.
These relevant stimulus features can be selected as the filters of the multi-filter LN model. Once the filters are obtained from the spike-triggered covariance analysis, one may aim at assessing the nonlinearity from the data by measuring how the instantaneous firing rate (or the spike probability) depends on the momentary outputs of the filters. Depending on the amount of available data, however, this is feasible only for a small number of filters. A more detailed account of the spike-triggered covariance methodology can be found in Schwartz et al. .
2.3 2.3 Alternatives to spike-triggered analyses
There have been a number of recent developments regarding alternatives to the spike-triggered analysis techniques for obtaining LN models and variants thereof. In particular, information theory provides a framework for extracting filters that capture maximal information about the neural response (Paninski 2003; Sharpee et al. 2004). Information theory can furthermore be used to combine spike-triggered average and spike-triggered covariance analyses into a single conjoint analysis (Pillow and Simoncelli 2006).
Another set of successful techniques is based on maximum-likelihood approaches (Paninski et al. 2004). This method has also proved quite useful for incorporating additional processing modules, such as neuronal refractoriness and after-spike currents (Paninski et al. 2004; Pillow et al. 2005). One advantage of these alternative spike-triggered methods is that they can be readily applied to more complex stimulus conditions, such as natural stimuli. These typically contain higher order correlations that distort the filters obtained from spike-triggered analyses, which necessitates significant correction procedures (Theunissen et al. 2001; Felsen et al. 2005; Touryan et al. 2005).
3 3 LN models of retinal ganglion cell responses
3.1 3.1 Single-filter models
The neural network of the retina has a long tradition as a system of investigation that combines excellent experimental accessibility and computational rigor in the applied models (Spekreijse 1969; Marmarelis and Naka 1972; Levick et al. 1983; Victor 1987; Sakai 1992; Meister and Berry 1999; Keat et al. 2001; van Hateren et al. 2002; Pillow et al. 2005). One of its principal advantages for studying neuronal network function is the fact that its inputs and outputs are under very good experimental control. The retina can be optically stimulated by projecting images onto its photoreceptor layer. The output of the retina are the spike trains of ganglion cells, whose axons form the optic nerve and transmit all visual information that is accessible to the rest of the brain. These output spike trains can be efficiently and reliably recorded from isolated pieces of retina placed on multi-electrode arrays (Meister et al. 1994; Segev et al. 2004).
LN models and spike-triggered analyses have long been established as standard tools for analyzing responses of retinal ganglion cells. Examples for spike-triggered averages of two ganglion cells are shown in Fig. 1c and d, for a purely temporal stimulus as well as a spatiotemporal stimulus with one spatial dimension, respectively. In the first case, the stimulus is a spatially homogeneous flicker; in the second case, it consists of flickering stripes. All light intensity values, for the full-field illumination as well as for individual stripes, were drawn independently from a Gaussian distribution around some intermediate gray illumination level.
The filters obtained from this spike-triggered average analysis can be used to characterize the response types of the neurons. For both cells shown here, the filters display a negative part close to time zero; on average, the light intensity decreased shortly before the spike occurred. This fact is generally used to classify the cells as OFF-type (Segev et al. 2006). But both cells also show pronounced ON characteristics preceding the OFF part of the filters, giving the filters a strongly biphasic (or triphasic) shape. In fact, the two cells, like many similar ones, respond with bursts of spikes to both step increases and decreases in light intensity, which gives them a signature of ON-OFF cells (Burkhardt et al. 1998). ON and OFF responses in the retina are mediated by activation of ON and OFF bipolar cells that respond to light intensity increases and decreases, respectively. ON-OFF ganglion cells appear to receive inputs through both these pathways (Werblin and Dowling 1969; de Monasterio 1978; Burkhardt et al. 1998; Greschner et al. 2006).
It has recently been shown that the characterization of ON-type and OFF-type filters is not completely static. This became apparent by the following experiment (Geffen et al. 2007): ganglion cells of the salamander retina were stimulated by flickering light in their receptive field center. Under stationary stimulus conditions, the reverse correlation revealed typical OFF-type filters for many neurons. For some of these, the filter characteristics changed, however, when a sudden shift of a visual pattern occurred in the periphery—similar to the global image shifts that accompany saccadic eye movements. In the ensuing about 100 ms after this shift, some of these ganglion cells yielded filters typical for ON-type cells; this means that, temporarily, the filter shapes were nearly inverted as compared to stationary conditions. As we will see below, these intriguing findings can be explained by specific filter models that capture contributions from the ON and OFF pathways in separate filters.
3.2 3.2 Multi-filter models
The dynamic changes between ON and OFF characteristics of ganglion cells motivated a model with explicit input from ON and OFF bipolar cells. Experimental support that this circuit structure is relevant for the observed phenomena came from pharmacological tests. To investigate the involvement of ON bipolar cells, the drug 2-amino-4-phosphono-butyrate (APB) can be applied to the retina. APB is known to block the synaptic input from photoreceptors to ON bipolar cells (Slaughter and Miller 1981; Yang 2004). Indeed, the effect of the drug was to abolish the occurrence of the ON characteristics after the peripheral shift (Geffen et al. 2007).
Separate inputs from ON and OFF pathways into specific ganglion cells have also been suggested by a generic investigation of multi-filter LN models under spatially homogeneous flicker stimulation (Fairhall et al. 2006). In this study, the modeling goal was not to match a specific circuitry, but to find good quantitative descriptions of the ganglion cell responses and to classify the cells according to the number and shapes of filters obtained. The approach was to apply a spike-triggered covariance analysis, and the resulting models capture the ganglion cell responses remarkably well; using tools from information theory, this study found that the models generally account for more than 80% of the information that is transmitted by the instantaneous firing rate.
Of course, the spike-triggered analysis does not automatically lead to an understanding of which features of the neuronal circuitry correspond to the obtained filters and nonlinearities in the model. In some cases, however, certain features of the resulting model structure can be explained in terms of the biological substrate. For some ganglion cells, for example, the obtained two-filter LN model can be understood as resulting from threshold-based spike generation mechanisms (Fairhall et al. 2006). In other cases—and more importantly for our present purpose—the two filters arise from a confluence of ON and OFF inputs (Fairhall et al. 2006; Geffen et al. 2007).
4 4 Spike timing at stimulus onsets
Most modeling approaches that we have discussed so far aim at capturing the (time-dependent) firing rate of a neuron under continuous, stationary stimulus conditions. Another fundamental stimulus paradigm is given by the sudden appearance of a visual image. In natural vision, such sudden stimulus onsets are caused by saccades, i.e. rapid shifts of the direction of gaze (Land 1999). The prominent temporal structure that saccades enforce on the natural stream of visual signals falling onto the eye makes the study of neuronal responses to stimulus onsets of obvious relevance.
Even for the simplest stimulus onsets—step increases and decreases of the light intensity with no spatial structure—one finds intriguing phenomena in the timing of spike events elicited in ON-OFF ganglion cells. In the turtle retina, specific ON-OFF ganglion cells have been shown to display peculiar spike patterns to steps in light intensity (Greschner et al. 2006; Thiel et al. 2006). Whereas the first spike after the change in light intensity was monotonically shifted to earlier times with increasing size of the intensity step, the timing of a second spike event had a non-monotonic dependence on step size, with the shortest timing occurring for intermediate changes in light intensity. To explain these response characteristics, models were employed that combine parallel ON and OFF pathways with feedback components and gain control. Both in the form of a phenomenological cascade model (Greschner et al. 2006) as well as in the form of a biophysical model of the retina network (Thiel et al. 2006), this allowed an accurate reproduction of the encountered response phenomena.
Most interestingly for the present discussion, many cells reliably responded with a burst of spikes to all spatial phases of the grating. This included responses to stimuli that were completely reversed in polarity so that bright and dark regions of the image were exchanged. Moreover, the latency of the response shifted systematically with the spatial phase of the grating. Early responses were observed when dark bars of the grating fell onto the neuron’s spatial receptive field; bright bars caused late responses. This relation between spatial phase of the stimulus and response latency can be summarized in a tuning curve (Fig. 3b) and compared to the corresponding tuning in spike count. For most recorded neurons, the latency was much more strongly tuned and consequently contained more information about the spatial phase of the stimulus. Moreover, this information is available already with the first spike, thus providing a potential signal for very rapid visual processing (Potter and Levy 1969; Thorpe et al. 2001; Kirchner and Thorpe 2006).
Again, the responses are intimately connected to the convergence of ON and OFF inputs; when APB was used to block ON inputs, the observed response phenomena disappeared, and the neurons behaved like pure OFF-type cells (Gollisch and Meister 2008). In the following, we will first discuss a model structure that captures these latency-tuning effects and subsequently elaborate on how the model parameters are obtained from electrophysiological data.
5 5 Modeling first-spike latencies for ON-OFF ganglion cells
The following model approach is aimed at capturing specifically the first spike latency after the onset of a flashed stimulus. The potential for rapid information transmission by latencies warrants special efforts to model this response feature. As pharmacological experiments indicated the necessity of signals from ON and OFF bipolar cells, a key aspect of the modeling will be the use of parallel ON and OFF pathways that correspond to separate stimulus filters.
This model fails to explain the measured responses (Fig. 4b). The primary reason is simply that the model produces no spikes at all for several of the stimuli; if one grating leads to a strong positive activation a(t), then the inverted grating results in a negative a(t).
As anticipated, this model now produces spikes for all stimuli, but the tuning of the latency curve is not well reproduced quantitatively (Fig. 4d). From the perspective of the neuronal circuitry, a flaw of this model version is that it takes into account the partition of bipolar cells into ON and OFF type, but not their smaller receptive field sizes as compared to ganglion cells; both the ON and the OFF field are still integrated linearly over space.
6 6 Obtaining the filters for an ON-OFF multi-pathway model
6.1 6.1 ON and OFF filters for spatially homogeneous stimulation
To obtain the ON and OFF filters, we need to separate the receptive field into contributions from these two pathways. To explain this procedure, we will first consider the case of spatially homogeneous stimuli where only the temporal stimulus dimension needs to be considered. Several studies (Fairhall et al. 2006; Greschner et al. 2006; Geffen et al. 2007; Gollisch and Meister 2008) have pursued this separation with variants of the same basic technique, which we will also follow here. It makes use of the fact that the ON and OFF pathways are sensitive to stimuli that are nearly inverted with respect to each other. It follows that typically one of the pathways can be excited, not both simultaneously. This allows a classification of the spikes according to the pathway that was responsible for providing excitation.
As in the computation of the spike-triggered average (Fig. 1c), the analysis is based on an experiment with flickering illumination and begins with collecting the stimulus segments that led to spikes, the spike-triggered stimulus ensemble. The light intensities are drawn from a Gaussian distribution and, for simplicity, normalized to zero mean and unit variance.
The spectrum of eigenvalues could now be analyzed statistically to find those components that significantly differ from unity, for example by computing the distribution of eigenvalues for temporally shuffled spike trains (Rust et al. 2005; Schwartz et al. 2006). This allows a formal analysis of how many filters should be included in the multi-filter LN model. Here, however, we are only interested in finding those (one or two) stimulus dimensions that let us best distinguish between contributions from the ON and OFF pathways. Therefore, we simply focus on the highest and lowest eigenvalue of the spectrum, which furthermore allows us to easily automate the analysis.
Clearly, the largest eigenvalue sticks out from the rest. This is typical for the analyzed neurons with ON-OFF response characteristics. The large eigenvalue corresponds to the fact that two nearly opposing pathways contribute to the response, which makes the variance of the spike-triggered stimulus ensemble along this direction particularly large. Thus, if no such eigenvalue emerges from the analysis, it is unlikely that both ON and OFF pathways contribute strongly. The lowest eigenvalue also deviates substantially from unity and is thus a candidate for denoting a relevant stimulus structure. Because its value is smaller than unity, the spike-triggered stimulus ensemble is compressed along this stimulus component. In specific contexts, this has been associated with suppressive response pathways (Schwartz et al. 2002), but it can arise from various sources, such as the dynamics of spike generation (Fairhall et al. 2006).
Most strikingly, for the present case, the firing rate for PC1 is “U-shaped”, which means that large positive projections and large negative projections both caused the cell to fire. This phenomenon becomes more evident when we take a look at the projections of all spike-triggered stimulus segments on both PC1 and PC2. When these projection values are displayed in a scatter plot, as in Fig. 5d, two clouds of data points become apparent. For almost all spikes, the projection onto PC2 was negative, but the projection onto PC1 could have either large positive or large negative values.
Another illustrative way of displaying this information is achieved by plotting the instantaneous firing rate as a function of both projection values, as in Fig. 5e. Here, the data are combined into bins with similar projections onto PC1 and PC2, respectively. For each bin, the firing rate is calculated as the average rate during the final stimulus frame of all stimulus segments in that bin. In contrast to the scatter plot of Fig. 5d, this form of display takes into account that, because of the Gaussian distribution of stimulus values, many more stimuli are presented near the center of the plots, where the projection values are close to zero, than in the periphery.
The scatter plot in Fig. 5d and the display of the firing rate in Fig. 5e show that the spike-triggered stimulus ensemble can be separated into two clusters. These two-dimensional displays reinforce the notion that two fundamentally different types of stimuli elicit spikes. Both PC1 and PC2 influence the shapes of the clusters, and it is likely that further stimulus components (corresponding to further eigenvalues of the spectrum shown in Fig. 5a) also contribute to separating the clusters.
Different techniques have been utilized to separate the clusters, such as a formal multi-dimensional cluster analysis (Geffen et al. 2007), a classification of the spike-triggered stimulus segments depending on whether they show an average intensity increase or decrease in a short window prior to the spike (Greschner et al. 2006), or a separation along the zero axis of the first principal component PC1 (Fairhall et al. 2006; Gollisch and Meister 2008). Here, we follow the latter approach, which yields a good separation of the clusters in many cases, owing to the pronounced U-shape of the firing rate dependence on the PC1 projection in Fig. 5c, where the firing rate drops down to zero when the projection is zero. We thus assign the stimulus segments to clusters depending on whether the projection onto PC1 was positive or negative. This approach allows us to easily automate this step in the analysis explained below. We then calculate the spike-triggered average for each cluster separately (Fig. 5f). Their shapes are not constrained to the space spanned by PC1 and PC2; the calculation is performed in the original full stimulus space. This takes into account that the two clusters may also differ along further stimulus dimensions. The reduction to the two dimensions PC1 and PC2 merely serves for separating the clusters.
The shapes of the two obtained filters, shown in Fig. 5f, can be interpreted as representing processing through ON and OFF bipolar cells, respectively. The strong biphasic nature of both these filters results from the fact that the spatially homogeneous stimulus not only excites receptive field centers of bipolar cells and ganglion cells, but also the inhibitory surround. The filtering characteristics of this surround are typically temporally delayed and inverted with respect to the center (ON-center cells have an OFF surround and vice versa). Their superposition thus yields the biphasic filter shape under activation of the whole space.
Of particular importance is the observation that the OFF filter has “faster kinetics”, i.e., its peaks are closer to time zero as compared to the ON filter. This means that activation of the OFF filter affects spike probability with a shorter latency—an observation that is of obvious importance for explaining the differences in latency for the flashed gratings. This was consistently observed in all cells in the salamander retina where the separation of ON and OFF contributions was possible. The likely cause is a delay in the processing of ON stimuli that results from the involvement of metabotropic receptors at the synapse between photoreceptors and ON bipolar cells (Ashmore and Copenhagen 1980; Yang 2004). Now that we have separated contributions from the ON and OFF pathway for spatially homogeneous stimuli and obtained two biologically plausible filters, let us consider the case where the stimulus includes spatial structure.
6.2 6.2 Spatially local ON and OFF filters
As for the case of spatially homogeneous stimulation, the relevant stimulus structures are again obtained from a principal component analysis. The analysis is shown in Fig. 6b–e, for those three stripes that lie in the center of the spatial receptive field of the sample neuron. Each eigenvalue spectrum (Fig. 6b) displays one eigenvalue that is much larger than unity. Most of the other eigenvalues are close to unity so that other relevant stimulus structures appear to be largely covered by noise. Consequently, as shown in Fig. 6c, the principal component corresponding to the largest eigenvalue, PC1, has a similar shape as previously, whereas the principal component corresponding to the minimal eigenvalue, PC2, is often dominated by noise. As we had seen in the previous section, however, a single stimulus component can suffice to separate ON and OFF contributions.
Indeed, a plot of the instantaneous firing rate in the space of PC1 and PC2 (Fig. 6d) reveals that spikes appear primarily when the projection onto PC1 is either strongly positive or strongly negative. Thus, we can separate stimulus segments activating the ON and OFF pathway, respectively, by selecting for positive or negative projection onto PC1. The resulting spike-triggered averages for each cluster, shown in Fig. 6e, display similar differences in kinetics as for the case of spatially homogeneous stimulation (Fig. 5f); for each stimulus stripe, the peaks of the OFF filters are closer to time zero than those of the ON filters. Also, all filters show a mild, but systematic biphasic structure, evident by the slow tail of opposite polarity as compared to the main peak. The biphasic nature of the filter is less pronounced than in the case of spatially homogeneous stimulation; the inhibitory surround that is responsible for the delayed inverted peak in the filter is still activated for individual stripes of the stimulus, but proportionally less so as compared to the spatially homogeneous stimulation.
Note that it is important to revert to the original stimulus segments for calculating the spike-triggered averages separately for the two clusters. The fact that the obtained ON and OFF filters are not exact inversions of each other, but indeed show systematic differences in their timing, underscores the importance of stimulus structures beyond the first principal component. Note also that the sets of ON filters and OFF filters are very similar across different stripes despite the fact that these were analyzed independently. This supports the reliability of the method. The actual test for the performance of the obtained model, however, is how closely it fits the data of the latency tuning curve (Fig. 4f).
7 7 Discussion
Neuronal models that are based on a single linear filter in the first stage of processing have a long and successful history, in the form of the widely used LN model (Hunter and Korenberg 1986; Chichilnisky 2001; Baccus and Meister 2002) as well as in combination with more complex mechanisms for processing and spike generation after the filtering stage (Keat et al. 2001; Pillow et al. 2005; Gollisch 2006). When parallel processing pathways are relevant for the function of a neuron, these single-filter models may be too simplistic. The natural extension is to use multiple parallel filters that represent these pathways. This more complex model structure, however, naturally brings about a more demanding task of extracting the model parameters from experimental data. Several earlier investigations have shown how generic multi-filter models can be obtained based on spike-triggered covariance analysis (de Ruyter van Steveninck and Bialek 1988; Schwartz et al. 2006) or on information theoretic approaches (Paninski 2003; Sharpee et al. 2004; Pillow and Simoncelli 2006).
In the examples presented here, the goal was to find filters that correspond to the synaptic inputs from a pool of bipolar cells, including both ON-type and OFF-type bipolar cells. One particular challenge for separating contributions from the ON and OFF pathways is that their preferred stimuli are nearly inverted with respect to each other. Therefore, they cannot naturally emerge as separate filters from a spike-triggered covariance analysis, for which the resulting filters are by design orthogonal to each other. Nevertheless, this covariance analysis serves as a good starting point because it singles out stimulus components for which the variance in the spike-triggered stimulus ensemble is particularly large. Such a stimulus component is a good candidate for providing a separation of clusters with nearly inverted stimulus characteristics.
One particular goal for focusing on separating ON- and OFF-pathway contributions is to use the resulting modeling framework as a data-analysis tool. The parameters of the model, such as the shapes of the filters and the relative strength of ON- and OFF-pathway contributions are here obtained for a specific stimulus context and will likely vary with this context. In the presented example, this context is given by the mean light intensity and the variance of the flickering light stimulus. Under different stimulus conditions, one may obtain different values for the model parameters, which could be used, for example, to investigate adaptation phenomena, similar to applications of the LN model (Chander and Chichilnisky 2001; Kim and Rieke 2001; Baccus and Meister 2002). In the discussed work of Geffen et al. , the multi-filter model was used in this way to study the effect of saccadic stimulus shifts, which revealed that the weights of the ON- and OFF-pathways transiently change after the saccade. For a better mechanistic understanding of these adaptive and contextual effects on the gating of these pathways, future extensions of the multi-pathway model may aim at incorporating how these pathway weights are determined by the stimulus context.
The applied model structure can be viewed as a hybrid between a purely phenomenological and a biologically inspired approach; based on the descriptive LN model, the use of parallel, spatially localized ON and OFF filters aims at capturing properties of the neuronal circuit that are thought to be fundamental for the investigated phenomena. In the discussed examples, the involvement of ON and OFF pathways was corroborated by experiments under pharmacological perturbation of the circuitry.
Generally speaking, however, it should be noted that failure of the single-filter LN model does not always mean that a multi-filter model is required. In fact, additional dynamics that follow after stimulus integration, such as spike generation dynamics (Aguera y Arcas and Fairhall 2003; Fairhall et al. 2006) and spike time jitter (Aldworth et al. 2005; Dimitrov and Gedeon 2006; Gollisch 2006) can lead to the appearance of multiple relevant filters in a spike-triggered covariance analysis. Although the multi-filter models may then still provide accurate descriptions of the neuronal responses, extensions of the model cascade with explicit spike generation dynamics (Keat et al. 2001; Pillow et al. 2005) or additional filtering stages (Victor and Shapley 1979; Korenberg and Hunter 1986; Sakai 1992) may provide a closer match to the biological processes. In the retina, for example, other successful approaches include Wiener series modeling (Marmarelis and Naka 1972) and LNL cascades (Spekreijse 1969; Victor and Shapley 1979). In fact, by analyzing the structures of first- and second-order Wiener kernels, one may estimate whether within the realm of LNL cascades, linear filtering acts primarily before the nonlinear transformation, after it, or both (Victor and Shapley 1979, 1980; Korenberg and Hunter 1986; Korenberg et al. 1989).
For the ganglion-cell responses analyzed here, such an analysis supports the importance of linear filtering that precedes the nonlinearity. For some cells, the second-order Wiener kernel indicates additional filtering that follows after the nonlinearity, which may correspond to feedback dynamics resulting from adaptation or gain control. These dynamics are not included in the model structure discussed here, which instead focuses on capturing a specific aspect of the retinal circuitry, the convergence of ON and OFF pathways. For a more general model of the cells’ response characteristics, additional dynamics should also be considered.
Generalizing the model structure in such a way is straightforward; the parallel filters can act as a front end to existing modules for gain control (Victor 1987; Berry and Meister 1998; Berry et al. 1999; Pillow et al. 2005) or additional filtering (Spekreijse 1969; Victor and Shapley 1979), which would act on the activation function that results from combining the spatially localized ON and OFF filter contributions. Fitting the complete model structure to experimental data becomes, of course, increasingly challenging with increasing number of model parameters. How well it works will depend on the specific model extension and the amount of available data. A promising approach here seems to be to resort to maximum-likelihood estimation techniques, for which the parallel filters can be initialized by the shapes obtained from the separation procedure described here. This approach is also amenable to various desirable model extensions discussed below.
7.1 7.1 Shortcomings and extensions
One shortcoming of the current approach is the ad-hoc definition of the spatial subfields. For ease of analysis, the subfields are modeled as rectangular and non-overlapping, whereas actual bipolar cells are better described by a smooth center-surround structure (Dacey et al. 2000; Baccus et al. 2008) that suggests, for example, a “difference-of-Gaussians” model. To fit such a more elaborate model to data will require stimulation with finer spatial structures and consequently more experimental time for data acquisition.
A simplification in the model comes from the fixed half-wave rectification that follows after each filter. The shape of this nonlinearity is motivated by findings that support rectification of synaptic inputs from bipolar cells to ganglion cells (Victor and Shapley 1979; Demb et al. 2001). For the case of spatially homogeneous stimulation, the shape of the nonlinear transformations could also be obtained from the experimental data by analyzing the relationship between the spikes from an individual spike-triggered stimulus cluster (Fig. 5d) and the output of the corresponding ON or OFF filter. For the case of flickering stripes, however, this approach is not suited because of the larger number of filters whose outputs simultaneously affect the firing rate. The relation between the firing rate and an individual filter (i.e., the marginal spike probability that depends only on a single filter output) is distorted by the large number of spikes that are generated primarily by activation from neighboring stripes. One may also consider a full multi-dimensional exploration of the nonlinearity by sampling the spike probability as depending on the joint outputs of all spatially local ON and OFF filters. However, given that around six to ten filters are typically required to span the receptive field center, this analysis is currently precluded by the large amounts of data that would be required for sufficient sampling. As a suitable alternative, a parameterization of the nonlinear transformation could be applied and included in a maximum-likelihood fitting procedure. Initial explorations suggest that threshold-linear or threshold-quadratic nonlinearities with positive thresholds may lead to an improved model version (Fig. 7c).
Finally, the spike-generation part of the model is currently limited to predicting the first spike in response to a stimulus. For a full account of the neuronal response, including the prediction of the time-dependent firing rate, more details are needed in the final model stage. In particular, effects of refractory period, adaptation, and contrast gain control need to be considered. Obtaining such a complete model description for spatiotemporal stimulation of ON-OFF-type neurons from experimental data will be a formidable, yet worthwhile task.
This work was supported by grants from the National Eye Institute (M.M.) and the Human Frontier Science Program Organization (T.G.) and by the Max Planck Society.
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