PMCCPMCCPMCC

Search tips
Search criteria 

Advanced

 
Logo of nihpaAbout Author manuscriptsSubmit a manuscriptHHS Public Access; Author Manuscript; Accepted for publication in peer reviewed journal;
 
Neuron. Author manuscript; available in PMC 2013 January 12.
Published in final edited form as:
PMCID: PMC3263974
NIHMSID: NIHMS342630

Reciprocal inhibition of inhibition: A circuit motif for flexible categorization in stimulus selection

Abstract

As a precursor to the selection of a stimulus for gaze and attention, a midbrain network categorizes stimuli into “strongest” and “others.” The categorization tracks flexibly, in real-time, the absolute strength of the strongest stimulus. In this study, we take a first principles approach to computations that are essential for such categorization. We demonstrate that classical feedforward lateral inhibition cannot produce flexible categorization. However, circuits in which the strength of lateral inhibition varies with the relative strength of competing stimuli categorize successfully. One particular implementation - reciprocal inhibition of feedforward lateral inhibition – is structurally the simplest, and it outperforms others in flexibly categorizing rapidly and reliably. Strong predictions of this anatomically supported circuit model are validated by neural responses measured in the owl midbrain. The results demonstrate the extraordinary power of a remarkably simple, neurally grounded circuit motif in producing flexible categorization, a computation fundamental to attention, perception, and decision-making.

INTRODUCTION

The segregation of continuously varying stimuli into discrete, behaviorally relevant groups, a process referred to as “categorization,” is central to perception, stimulus identification and decision-making (Freedman and Assad, 2006; Freedman et al., 2001; Leopold and Logothetis, 1999; Niessing and Friedrich, 2010). In some cases, the boundary between categories is fixed (Prather et al., 2009). In most cases, however, the boundary needs to adjust according to context, a process referred to as flexible categorization. Recent research suggests that such flexible categorization also contributes to competitive stimulus selection for gaze and attention (Mysore and Knudsen, 2011b). A midbrain network that plays an essential role in gaze and attention (Cavanaugh and Wurtz, 2004; Lovejoy and Krauzlis, 2010; McPeek and Keller, 2004; Muller et al., 2005) segregates stimuli into “strongest” and “all others” (Mysore and Knudsen, 2011a). The midbrain network includes the optic tectum (called the superior colliculus in mammals) as well as several nuclei in the midbrain tegmentum, referred to as the isthmic nuclei (Knudsen, 2011). Categorization by this network tracks the location of the strongest stimulus in real-time, as a precursor to the selection of the next target for gaze and attention. Despite the importance of flexible categorization to a broad range of functions, how the brain implements it is not known.

Categorization by the midbrain network arises from special response properties of a subset of neurons located in the intermediate and deep layers of the owl optic tectum (OTid) (Mysore et al., 2011; Mysore and Knudsen, 2011a). These neurons display “switch-like” responses, firing at a high rate when the stimulus inside their classical receptive field (RF) is the strongest (highest intensity or speed), but switching abruptly to a lower firing rate when a distant, competing stimulus becomes the strongest. This switch-like property causes the encoding of categories by the OTid to be explicit: the category can be read out directly from the population activity pattern without any further transformations beyond simple linear operations, such as averaging (Gollisch and Meister, 2010). In addition, if the strength of the stimulus inside the RF is increased, a switch-like neuron requires a correspondingly stronger competing stimulus to suppress its responses. This property causes the category boundary to be flexible, enabling network responses to reliably identify the strongest stimulus at each moment in time. Explicit and flexible categorization by this network dramatically improves the discriminability of the strongest stimulus among multiple competing stimuli of similar strength (Mysore et al., 2011; Mysore and Knudsen, 2011a).

The properties of stimulus categorization exhibited by neurons in the owl OTid account well for behavioral deficits in monkeys following the inactivation of the intermediate and deep layers of the superior colliculus (Lovejoy and Krauzlis, 2010; McPeek and Keller, 2004; Nummela and Krauzlis, 2010). In monkeys performing stimulus selection tasks, focal inactivation of the portion of the superior colliculus representing the target stimulus causes an impairment in their ability to select an oddball target or a spatially-cued target among distracters, an impairment that increases dramatically as the distracting stimuli become more similar to the target stimulus. These studies indicate that the midbrain network performs computations that are essential for reliable competitive stimulus selection, especially when competing stimuli are of similar strength.

A neural computation that is fundamental to stimulus competition in the OTid is the suppression of responses to an RF stimulus by stimuli located outside the RF. Such “surround suppression” is observed in many brain areas across many species (Allman et al., 1985). Unlike interactions that occur among stimuli within the RF (such as cross-orientation suppression in the visual cortex (Freeman et al., 2002)), surround suppression is thought to be mediated by lateral inhibition and, often, by feedforward lateral inhibition (Blakemore and Tobin, 1972; Bolzon et al., 2009; Cisek and Kalaska, 2010; Hartline et al., 1956; Kuffler, 1953; Olsen et al., 2010; Yang and Wu, 1991). Anatomical evidence from the avian midbrain network supports lateral inhibition as underlying global suppression in the OTid as well (Fig 1; (Wang et al., 2004)). Specifically, a midbrain GABAergic nucleus, the nucleus isthmi pars magnocellularis (Imc), receives focal input from neurons with dendrites in the retinorecipient layers of the optic tectum, and sends broad projections to neurons in the multimodal and motor layers of the optic tectum, the OTid.

Figure 1
Feedforward lateral inhibition to the OTid

Using this basic feedforward lateral inhibitory circuit as a starting point, we employ a first principles approach to address neural computations that underlie flexible categorization in the OTid. We show that feedforward lateral inhibition, a circuit motif at the heart of most models of selection for attention or action (Cisek and Kalaska, 2010; Lee et al., 1999), cannot account for categorization that is flexible. However, a simple modification – introducing reciprocal inhibition between feedforward lateral inhibitory channels – successfully achieves flexible categorization. The key additional computation that achieves adaptive boundary flexibility in categorization is lateral inhibition that is dependent on relative stimulus strength. Reciprocal inhibition of feedforward lateral inhibition emerges as an extremely simple yet powerful motif that implements this computation rapidly and reliably, thereby producing flexible categorization, a process central to attention, perception, and decision-making (Freedman and Assad, 2011).

RESULTS

To identify neural circuits that might achieve categorization, we begin by first capturing basic properties of neuronal responses to single and multiple, competing stimuli. To this end, we use standard mathematical equations that account accurately for experimental results and that have been employed widely in the literature.

Responses of the OTid to single stimuli are nonlinear

OTid neurons respond nonlinearly to increasing strengths of a single stimulus inside their RFs. Strong stimuli (high contrast, high sound level, fast motion, etc) drive neurons to saturation. These nonlinear responses are well fit by sigmoidal functions (Mysore et al., 2010, 2011). In this study, looming visual stimuli (expanding dots) were used to drive neural responses. A standard sigmoidal equation, the hyperbolic ratio function (Naka and Rushton, 1966), describes OTid responses to an RF stimulus of loom speed l:

equation M1
(1)

The parameters are a, the minimum response, b, the maximum change in response, L50, the loom speed that yields a half-maximum response, and n, a factor that controls response saturation. The mechanisms that underlie response saturation to single stimuli are distinct from those that mediate global surround suppression, the focus of this study (Freeman et al., 2002; Mysore et al., 2010). Therefore, without loss of generality, we focus on the lateral inhibition for surround suppression while using the sigmoidal function as a description of OTid responses to single stimuli.

For subsequent simulations, the best sigmoidal fit to the experimentally measured, average loom speed-response function from 61 OTid neurons (Fig. 2A) was used as the response function of a “typical” OTid unit:

equation M2
(2)

Here, the first term (5.3) represents the contribution of the contrast of a stationary dot (loom speed = 0 °/s) to the response: the average response to a loom speed of 0 °/s at full contrast was 5.3 sp/s. As this contribution due to stimulus contrast is small, we made the simplifying assumption that the dependence of the response on the contrast of a stationary dot was linear. Because all responses were simulated for full contrast stimuli (contrast=1), the contrast-related term was simply a constant, 5.3.

Figure 2
Neuronal response properties and signatures of explicit, flexible categorization in the owl OTid

Lateral inhibition is divisive and scales nonlinearly with competitor strength

Responses to RF stimuli are divisively suppressed by a competing stimulus located outside the RF (Fig. 2B; Mysore et al., 2010). We captured this divisive effect of lateral inhibition by introducing both input and output divisive influences in a manner similar to previously published reports (eqn. 3; (Olsen et al., 2010)).

equation M3
(3)

Here, sin and sout are suppressive factors that produce input and output division, respectively (Supplemental Experimental Procedures).

Experiments in the OTid have demonstrated that response inhibition increases as the strength of the competitor increases, and does so, typically, in a nonlinear (sigmoidal) manner (Fig. 2D, (Mysore et al., 2011)). To incorporate competitor strength-dependent inhibition, the suppression factors sin and sout were taken to be proportional to the activity I of the inhibitory units driven by stimuli located outside the RF:

sindinIsoutdoutI
(4)

where din and dout were proportionality constants, and I was the inhibitory activity driven by the competitor. Recordings of Imc responses to single looming stimuli have shown that they are well-fit by sigmoidal functions (unpublished results). Consequently, inhibitory activity as a function of the loom speed of the competitor stimulus was modeled as having the same form as equation 1:

equation M5
(5)

The free parameters were m, the minimum response, h, the maximum change in response, S50, the loom speed that yields a half-maximum response, and k, a factor that controls response saturation. The effect of changing the values of each of these parameters on I is illustrated in Figure S1. A linear dependence between the input and output divisive factors (sin and sout) and the inhibitory activity (I) was assumed in equation 4 for simplicity. This formulation minimized the number of free parameters in the model, while still allowing for nonlinear competitor strength-dependent response suppression, due to the nonlinearity of I.

Response properties essential to explicit and flexible categorization in the OTid

We now describe the special response properties underlying “strongest” vs. “other” categorization that need to be accounted for by the model. These were revealed in experiments in which a looming stimulus of fixed speed was presented inside the RF, while a second, competing stimulus of variable speed was presented far outside the RF, about 30° away. The resulting responses are referred to as the “competitor strength-response profile”, or CRP (Mysore et al., 2011).

Explicit encoding of categories

Essential to the explicit representation of categories in the OTid is the abrupt, “switch-like” increase in response suppression, observed in about 30% of OTid neurons, as the strength of a competing stimulus is increased (Fig. 2D, right panel). The abruptness of the transition is quantified as the range of competitor strengths over which CRP responses drop from 10% to 90% of the maximum change in response, and is referred to as the “transition range.” Switch-like CRPs were defined as those for which the transition range was very narrow: ≤ 4 °/s, equivalent to ≤ 1/5th the full range of loom speeds tested. Population activity patterns that include switch-like responses (along with non switch-like responses) explicitly categorize stimuli into two categories, “strongest” and “others,” as determined by cross correlational analysis (Mysore and Knudsen, 2011a). Conversely, excluding the top 20% of the neurons with the most abrupt response transitions (switch-like responses) from the population analysis, eliminates categorization by the population activity.

Switch-like responses are not winner-take-all: the responses to the RF stimulus when it is weaker than the competitor, i.e., when it is the “losing” stimulus, are not driven to zero. Rather, the responses scale with the absolute strength of the losing RF stimulus (Fig. 2E, right panel, magenta versus blue data on the right side; Fig. S1E-I) (Mysore et al., 2011).

Flexibility of categorization

The flexibility of categorization in the OTid requires that the boundary between categories track dynamically the strength of the strongest stimulus. For switch-like CRPs, the strength of the competitor that caused responses drop from a high to a low level (Fig. 2D, red dot), called the “switch-value”, equaled, on average, the strength of the RF stimulus, and was therefore indicative of the categorization boundary. Moreover, when two CRPs were measured for a unit using two different RF stimulus strengths, the switch-value shifted with the strength of the RF stimulus (Fig. 2E), and across all tested switch-like units, the average shift in the switch-value was equal to the change in the strength of the RF stimulus.

Population activity patterns constructed using these CRP responses exhibited an appropriately shifting category boundary with RF stimulus strength (Mysore and Knudsen, 2011a). Conversely, when switch-like responses were removed from the population, flexible categorization did not occur. Thus, switch-like responses and adaptive shifts in switch-value with changes in RF stimulus strength are, respectively, the signatures of the explicit and flexible categorization in the OTid.

Results from model circuit #1: Feedforward lateral inhibition

We asked whether a feedforward lateral inhibitory circuit could produce the two response signatures critical for categorization in the OTid. This circuit architecture served as a good starting point because it is anatomically supported in the midbrain network and similar architectures have been used to model sensory processing of multiple stimuli as well as the selection of stimuli for attention and action in many different brain structures.

In the following simulations, we present the results from the perspective of output unit 1 (Fig. 1B, black circle) and the inhibitory unit that suppresses it, inhibitory unit 2 (Fig. 1B, red oval). Because the connections and weights are symmetrical, the results would apply to neurons representing additional spatial channels in the output or inhibitory unit layers.

Feedforward lateral inhibition alone can produce switch-like CRPs

To test if this circuit model can produce switch-like CRPs at the output (OTid) units, responses were simulated with the strength of the RF stimulus held constant at 8 °/s and the strength of the competitor stimulus increased systematically from 0 to 22 °/s. We expected that any parameter that affected the steepness of the inhibitory response function would, in turn, affect the steepness of the CRP. Therefore, increasing the saturation parameter k (Fig. S1A) and decreasing the half-max parameter S50 (Fig. S1b), both of which make the inhibitory response function steeper, should yield CRPs with transition ranges narrower than 4°/s.

Consistent with these expectations, large enough values of k (Fig. 3B, top left panel) and small enough values of S50 over a wide range (Fig. 3B, top right panel) successfully produced switch-like CRPs at the output units. Based on these results we chose the parameter values of the inhibitory response function for subsequent simulations to be k=10, S50=8, m=5, and h=15. The resulting switch-like CRP is shown in Figure 3C.

Figure 3
Feedforward lateral inhibition (circuit #1) can produce switch-like CRPs, but not adaptive shifts in CRP switch-value

Feedforward lateral inhibition cannot produce adaptive shifts of CRPs

Next, we tested if this circuit model can produce adaptive shifts in the CRP switch-value. We simulated two CRPs with RF stimulus strengths of 8 °/s and 14 °/s, respectively, and asked if any combination of input and output divisive inhibition (din and dout, respectively; eqns. 2 and 3) could appropriately shift the CRP switch-value.

The ranges of din and dout tested, [0, 3] and [0, 0.24], respectively, were chosen such that the smallest value produced no modulation of the RF stimulus-response function and the largest value produced 90% of the maximum possible modulation (Fig. S2A,B). All the parameters of the inhibitory response function were maintained at the previous values, chosen to yield switch-like CRPs. For each pair of din and dout values, we computed the switch-values for the two CRPs and calculated as the “CRP shift ratio”, the ratio of the shift in the switch-value to the change in the RF stimulus speed; a ratio of 1 represents a perfectly adaptive shift.

The plot of model CRP shift ratios as a function of din and dout demonstrated that this circuit produced almost no shift in CRP switch-values in response to an increase in the strength of the RF stimulus (Fig. 3D and Fig. S2C). The maximum shift ratio produced was 0.03 (din=1.5, dout=0), and the two CRPs corresponding to this shift ratio are shown in Figure 3E.

To understand why this circuit cannot produce adaptive shifts in the CRP switch-value, we compared the patterns of inhibition in the two CRP measurement conditions. Since the activity of the inhibitory neuron (I) depended only on the strength of the competitor, and not on the strength of the RF stimulus (I, sin and sout (eqn. 4)), the pattern of inhibition was identical in both cases (Fig. S2E; identical magenta and blue lines). Therefore, the only difference between the two CRPs measured at the output unit was the upward (without a rightward) shift (Fig. 3E, blue curve relative to magenta curve), reflecting the increased excitatory drive caused by the stronger RF stimulus (l in eqn. 3). The simulations for Figure 3 explored a large portion, but not the entire space of parameter values. Nonetheless, it is clear from the above observation that no possible combination of parameters for this circuit can produce adaptive, rightward shifts in the CRP when the strength of the RF stimulus is changed.

Thus, feedforward lateral inhibition, as modeled with widely used divisive normalization (eqn. 5), while able to produce switch-like CRPs, is unable to produce adaptive shifts in the CRP switch-value. Therefore, it is unable to account for flexible categorization.

Model circuit #2: Reciprocal inhibition of feedforward lateral inhibition

The above results suggest that, in order for CRP switch-values to shift adaptively with changes in the strength of the RF stimulus, the strength of inhibition must depend on the relative strengths of the competitor and RF stimuli, rather than just on the strength of the competitor alone. In other words, the term I in equation 4 must depend on relative stimulus strength.

From a circuit perspective, the simplest modification to achieve this goal is to have the inhibitory units inhibit each other (reciprocal inhibitory connections; Fig. 4A). Indeed, structural support for such a circuit motif in the Imc has been found in an anatomical study (Wang et al., 2004). The study showed that in addition to projecting to the OTid, Imc axonal branches also terminate within the Imc itself (Fig. 4B). Such reciprocal connections will cause the inhibitory units representing each location to inhibit the inhibitory units representing all other locations. As a result, the activity of each inhibitory unit should depend on the strength of its excitatory drive relative to the excitatory drive to other inhibitory units.

Figure 4
Model circuit #2: Reciprocal inhibition among feedforward lateral inhibitory elements

To model the reciprocal connectivity, each inhibitory unit was first modeled as being affected by a combination of input and output divisive inhibition (along with an implicit subtractive component; (eqn. 6)). This formulation was general as it allowed for the inhibition onto inhibitory units to be any arbitrary combination of the commonly observed forms of inhibition in the literature.

equation M6
(6)

Here, I(t) is the inhibitory activity at computational time step t. iin(t) and iout(t) were the input and output divisive factors at time-step t, modeled as being proportional to the activity of the inhibitory units at the previous time-step (compare to eqn. 4):

iin(t) = rin  ·  I(t − 1),  iout(t) = rout  ·  I(t − 1)
(7)

rin and rout are proportionality constants. In this formulation, transmission and synaptic delays were assumed to be equal to one computational time-step, for simplicity. These equations were applied iteratively until there was no further change in the inhibitory activity, i.e., I(t) = I(t+1). The resulting steady-state activity of the inhibitory units was referred to as Iss. Consequently, at steady state, the input and output divisive factors in equation 7 reduce to

iinrinIss, ioutroutIss
(8)

The single stimulus response functions of the inhibitory and excitatory units were unchanged from before.

Before exploring the effect of reciprocal inhibition on output unit activity, we first analyzed its effect on the steady state inhibitory activity. We plotted Iss for inhibitory unit #2 during a CRP measurement protocol with a RF stimulus of strength 8°/s (Fig. S3A,B). Compared to the inhibitory activity obtained with circuit #1 (Fig. S2E), the steady state inhibitory activity obtained with circuit #2 (Fig. S3B) displayed two key differences. First, the rate of increase in inhibitory activity with increasing competitor strength was steeper in the presence of reciprocal inhibition (Fig. S3B, solid magenta versus dashed magenta lines). This increase in steepness reflected, as expected, an iterative amplification of the difference in activity between the two inhibitory units, due to the inhibitory feedback motif. Second, when another CRP was obtained with a different RF stimulus (14 °/s), the steady state inhibitory activity of inhibitory unit #2 was conspicuously shifted to the right (Fig. S3B, solid blue vs. solid magenta), in contrast to the results from the feedforward circuit (Fig. S2E). This rightward shift in the steady state inhibitory activity predicted that, following a change in the strength of the RF stimulus, output unit CRPs would also shift adaptively. We test this prediction below. For subsequent simulations, we chose the reciprocal inhibitory parameter values as follows: rin =0.84 (which yielded the maximum rightward shift of the inhibitory activity; Fig. S3C) and rout = 0.01 (Fig. S3D).

Results from model circuit #2

We asked whether a circuit with reciprocal inhibition of feedforward lateral inhibition could produce the two response signatures critical for categorization in the OTid.

Reciprocal inhibition of feedforward lateral inhibition can produce switch-like CRPs

To test if this circuit model can produce switch-like CRPs at the output (OTid) units, we simulated output unit CRPs and, as before (Fig. 3B), plotted their transition ranges as a function of each of the parameters of the inhibitory response function (Fig. 5B). For these plots, the values of din and dout and the values of the fixed inhibitory parameters were chosen to be the same as those used previously in testing circuit #1 (Fig. 3B).

Figure 5
Reciprocal inhibition among feedforward lateral inhibitory elements (circuit #2) can produce switch-like CRPs and adaptive shifts in CRP switch-value

We found that large enough values of the saturation parameter k and small enough values of the half-max response loom speed (S50) yielded switch-like CRPs (Fig. 5B). An example of a switch-like CRP, obtained with the same values of the inhibitory parameters used for circuit #1 (Fig. 3C), is shown in Figure 5C. As expected by the steeper inhibitory response function (Fig. S3B, solid versus dashed magenta), this CRP at the output unit is also steeper (compare with Fig. 3C).

Reciprocal inhibition of feedforward lateral inhibition can produce adaptive shifts of CRPs

Next, we tested if this circuit can produce adaptive shifts in the CRP switch-value. As before, we asked if any combination of input and output divisive inhibition (din and dout, respectively; eqns. 2 and 3) could produce a shift in the CRP switch-value with a 6°/s increase in the RF stimulus strength. The strength of reciprocal inhibition was unchanged from before (rin=0.84 and rout=0.01).

A plot of model CRP shift ratios (ratio of switch-value shift to change in RF stimulus strength) as a function of din and dout shows that a large set of (din, dout) values successfully produced adaptive shifts in the switch-value (shift ratio near 1; Fig. 5D and Fig. S4A). The ranges of din and dout tested were the same as those used for testing the feedforward circuit (Fig. 3D). The largest CRP shift ratio (for this particular set of rin and rout values) was 0.88, nearly identical to the average value of shift ratios observed experimentally (0.90±0.16; (Mysore et al., 2011). The two CRPs that yielded this shift ratio are shown in Figure 5E. In addition to displaying a rightward shift, the CRP computed with the stronger RF stimulus (Fig. 5E, blue curve) was scaled upwards with respect to the CRP that was computed with the weaker RF stimulus (Fig. 5E, magenta curve), consistent with experimental results (Fig. 2E; (Mysore et al., 2011)). However, when divisive inhibition was exceptionally strong, the scaling of the responses to the losing RF stimulus was eliminated, resulting in winner-take-all responses.

The strengths of reciprocal inhibition and the values of the parameters of the inhibitory response functions chosen to demonstrate these rightward shifts were not special. Wide ranges of values for these parameters produced adaptive shifts in the CRP switch-value (Fig. S4).

Thus, reciprocal inhibition between feedforward lateral inhibitory units can produce switch-like CRPs and adaptive shifts in the switch-value in response to changes in RF stimulus strength, thereby creating an explicit, and flexible categorical representation of stimuli based on relative stimulus strength.

Model predictions

Thus far, we have demonstrated that model circuit #2, involving the reciprocal inhibition of lateral inhibition motif, successfully accounts for experimentally measured CRP properties. To further evaluate the validity of this circuit, we used it to predict output unit activity in a new, two-stimulus paradigm, one that had not been previously tested experimentally. In this paradigm, the responses to a receptive field stimulus of increasing strength were obtained both without a competitor, and with a competitor of fixed strength. The resulting profiles of output unit activity are called, respectively, the “Target-alone response profile” and the “Target-with-competitor response profile.” Comparing these profiles allowed us to assess the effect of a fixed competitor strength on the classic, strength-response profile.

We show next that model circuit #2 predicts a wide range of shapes for Target-with-competitor response profiles, the bulk of which are not predicted by model circuit #1. We also demonstrate with new experimental results that neuronal responses to this two-stimulus paradigm are fully in line with the predictions of model circuit #2, but not model circuit #1.

The feedforward lateral inhibitory circuit (circuit #1; Fig. 1B) produced Target-with-competitor response profiles that reflected, essentially, various combinations of purely input divisive (Fig. 2C, left panel) and purely output divisive (Fig. 2C, right panel) influences caused by the competitor stimulus. The full range of effects are summarized in Table 1 (middle column) and are typical of structures that process sensory information (Cavanaugh et al., 2002; Olsen et al., 2010; Williford and Maunsell, 2006).

Table 1
Comparison of the Properties of Target-with-Competitor versus Target-Alone Response Profiles

Addition of the reciprocal inhibition motif to the feedforward lateral inhibitory circuit (circuit #2; Fig. 4B) yielded not only Target-with-competitor response profiles that were similar to those described above (Fig. 6A,B) but also, importantly, profiles that were qualitatively distinctive in each of three respects (Fig. 6C-E; and Fig. 6F,G), as summarized in Table 1 (last column). These kinds of effects are not typically observed in structures that process sensory features. Thus, the two circuit models make predictions that are qualitatively different.

Figure 6
Comparison of model predictions with experimental data from the owl OTid

Testing model predictions

We tested these strong predictions of the models experimentally in the barn owl OTid. For each OTid neuron, we measured Target-alone response profiles by varying the strength (loom speed) of a stimulus presented at the center of the RF (n=71 neurons). Randomly interleaved with these were the Target-with-competitor response profiles, measured with a second, simultaneously presented competitor stimulus, located far outside the RF (30° away). The responses were fit with sigmoidal functions, and various parameters of the fit were estimated and compared to predictions.

The range of effects on loom speed-response profiles observed in the OTid due to the presence of a competitor stimulus matched those predicted by model circuit #2, and exceeded those predicted by model circuit #1. In addition to Target-with-competitor response profiles that reflected pure feedforward input or output division (Figs. 6H and I compared with Figs. 6A,B), we found Target-with-competitor response profiles that exhibited smaller dynamic ranges (Fig. 6J,K), more suppression of the responses to the weakest than the strongest RF stimulus (Figs. 6J,L), as well as right-shifted (Figs. 6H,J-L) or unshifted (Fig. 6I) half-max response strengths. These representative results were confirmed by population analyses (Figs. 6M-6O).

The correspondence between the predictions made by model circuit #2 and the experimental results supports the validity of the reciprocal inhibition of feedforward lateral inhibition model.

Advantages of the reciprocal inhibition of feedforward lateral inhibition circuit over alternative circuit models

Reciprocal inhibition among feedforward lateral inhibitory units is only one of many circuit architectures for producing competitive inhibition that adapts to the relative strengths of drive to competing stimulus channels. Alternative circuits that accomplish the same goal include feedback inhibition among output units (Fig. 7A; circuit #3), feedback inhibition from output units to input units (Fig. 7C), and each of these circuits with an additional recurrent excitatory loop (Figs. 7B and D, respectively). However, reciprocal inhibition of feedforward lateral inhibition accomplishes the computational goal with structural efficiency: by interconnecting the inhibitory units themselves, the inhibitory drive from each competing channel is modulated directly by the inhibitory drives from all other channels. With this motif, the feedback loop involves only the inhibitory units and the two synapses that connect them. In contrast, all other feedback architectures involve additional units and synapses in the feedback loop.

Figure 7
Comparison of the performance of alternative circuit implementations of feedback lateral inhibition

We studied the consequences of structural complexity of the feedback motif on the ability of the model to compute steady state responses to competing stimuli rapidly and reliably. We compared the performance of the reciprocal inhibition of feedforward lateral inhibition motif (Fig. 4; circuit #2) with that of the next most structurally simple motif: feedback lateral inhibition by output units (Fig. 7A, circuit #3). The parameter values for the circuit #2 model were chosen to be the same as those in Figure 5E. The parameter values for the circuit #3 model were chosen such that the circuit yielded output unit responses at steady state that were nearly identical to those from the circuit #2 model (Fig. S5A-D). The quality of the match between the responses of the two circuits was particularly sensitive to the values of the parameters for circuit #3, with the best match occurring over a narrow range of values (Fig. S5E-J).

We measured calculation speed as the “settling time”, defined as the first time step after which responses did not change any further (Experimental Procedures). The time courses of the responses from the two models, calculated for an RF stimulus of strength of 9°/s and a competitor strength of 8 °/s (relative strength = 1°/s), demonstrated that circuit #2 settled faster than circuit #3 (Fig. 7E). This finding held true for all relative stimulus strength values (Fig. 7F). Both models exhibited longer settling times as the relative strength between the competing stimuli decreased, consistent with the experimental observation that difficult discriminations take longer to resolve (Gold and Shadlen, 2007).

We assessed the reliability of the calculation as the consistency of the steady state response. Gaussian noise was introduced into the calculation of the response for each unit at each time-step. Consistency was quantified by calculating the Fano factor (Experimental Procedures), a metric that is inversely related to response consistency. The distribution of Fano factors at steady state was estimated using Monte Carlo analyses (Experimental Procedures).

Comparison of the Fano factors from the two models for an RF stimulus of strength of 9°/s and a competitor strength of 8 °/s (relative strength = 1°/s) showed that circuit #2 produced less variability (smaller Fano factor) than circuit #3 (Fig. 7G; Average Fano factors were 0.71±0.01 and 0.78±0.01, respectively; p<10-4, ranksum test). Circuit #2 exhibited superior reliability for all values of the RF stimulus from 1 to 9 °/s (competitor = 8 °/s), with the reduction in Fano factor being substantial (approximately 75%) when the RF stimulus was weaker or as strong as the competitor (Fig. 7H). When the RF stimulus was substantially stronger than the competitor (RF stimulus strengths = 11-16 °/s), the Fano factors yielded by circuit #2 were slightly greater than those yielded by circuit #3 (Fig. 7H). Considered together, these effects show that for all values of relative stimulus strength, the discriminability between the responses to the stronger (winning) stimulus and the weaker (losing) stimulus is substantially greater for the circuit #2 model that contained the inhibition of inhibition motif.

Thus, the structural simplicity of the reciprocal inhibition of feedforward lateral inhibition motif enabled both faster and more reliable categorization of competing stimuli than the next most structurally simple implementation of this competitive rule.

DISCUSSION

Although flexible categorization has been studied extensively in systems and cognitive neuroscience, how neural circuits might implement it has been unclear. Our goal was to provide an intuitive, circuit level account of the key computations involved in creating an explicit and flexible categorization of stimuli while being agnostic to their biophysical implementation. Through a first principles approach, we showed that while classical feedforward lateral inhibition, implemented with sufficiently steep inhibitory stimulus-response functions, can successfully produce categorical responses, it cannot adjust the category boundary flexibly in response to changes in the absolute strengths of competing stimuli. In contrast, relative strength-dependent lateral inhibition (feedback inhibition) achieves both explicit and flexible categorization. While many different circuits can implement relative strength-dependent inhibition, reciprocal inhibition among the feedforward laterally inhibitory units is structurally the simplest, involving the fewest possible units and synapses within the feedback loop, and it categorizes stimuli faster and more reliably than the next most simple circuit. The superior performance of this motif suggests that it may occur in networks that are engaged in flexible categorization, identification or decision-making, particularly when speed or reliability are important.

Reciprocal inhibition of inhibitory elements is a circuit motif that has been observed in several other brain areas such as the thalamic reticular nucleus (Deleuze and Huguenard, 2006), the neocortex (Pangratz-Fuehrer and Hestrin, 2011), and the hippocampus (Picardo et al., 2011), but for which a clear function has not been ascribed. Our analysis indicates that the primary power of this circuit motif is in both enhancing and providing flexibility to the comparison of information across channels.

Normalization models of sensory processing

The feedforward lateral inhibition motif, which served as the core of the model in this study, has been employed widely in models of sensory information processing and attentional modulation of sensory representations. One of these models was of olfactory processing in the fly antennal lobe (Olsen et al., 2010). In that study, the effect of increasing the strength of a competing stimulus on the representation of an RF stimulus (which we call a CRP) was tested with RF stimuli of different strengths. Consistent with our results, they report no adaptive shifts in CRPs with changing RF stimulus strengths from this feedforward lateral inhibitory circuit.

In one model of information processing in the primate visual cortex (V1), nonlinear properties of response normalization, consistent with input divisive normalization, were accounted for with feedback inhibition (Carandini et al., 1997). Our study and others (Olsen et al., 2010; Pouille et al., 2009) have demonstrated, however, that a feedforward circuit is sufficient to achieve input divisive normalization. The necessity for feedback inhibition in that study was not explored. Our results, and those from a recent model of V1 (Ayaz and Chance, 2009), indicate that feedback inhibition enhances the nonlinearity of competitive response profiles. In addition, our results indicate that feedback inhibition is required for adaptive shifts of CRPs of the kind observed in V1 (Carandini et al., 1997). In sensory processing, then, feedback lateral inhibition causes normalization that adjusts adaptively according to relative stimulus strengths, and reciprocal inhibition of feedforward lateral inhibition could be an efficient circuit motif to implement such a flexible normalization rule.

Other models of sensory normalization, particularly those simulating interactions of stimuli within the RF (like cross-orientation suppression in V1 or biased stimulus competition for attention), typically invoke mechanisms that are distinct from those that affect responses outside of the RF, explored in this study (Busse et al., 2009; Carandini et al., 2002; Freeman et al., 2002; Lee and Maunsell, 2009; Ohshiro et al., 2011; Reynolds et al., 1999; Reynolds and Heeger, 2009).

Comparison with models of selection for action and attention

Different kinds of models have been proposed to explain the major steps in stimulus selection for action or attention (Cisek and Kalaska, 2010; Itti and Koch, 2001; Lee et al., 1999), with one step being a winner-take-all operation (Edwards, 1991; Hahnloser et al., 1999; Koch and Ullman, 1985), which we have shown to be a special case of flexible categorization. However, these models were strictly computational, with no explicit correspondence between component computations and neural circuitry.

The patterns of connections within the midbrain network facilitate the inference of component computations from neural structure. The striking anatomy of the GABAergic Imc circuit (Fig. 4B) has inspired the proposal that it participates in a winner-take-all selection of the highest priority stimulus (Marin et al., 2007; Sereno and Ulinski, 1987). A recent model of this network (Lai et al., 2010) invoked connections between the optic tectum, the Imc, and a cholinergic nucleus in the isthmic complex (Asadollahi et al., 2010) to attempt to explain winner-take-all responses. Although the circuit model used was a variation of the model in Fig. 7B, one of the circuits that can produce explicit and flexible categorization (Fig. S5A-D; Koch and Ullman, 1985), their particular choice of parameter values did not yield flexible categorization.

Our study suggests that the Imc circuit can, by itself, mediate categorization in the midbrain network. We propose a simpler and faster circuit motif for implementing flexible categorization and, possibly, winner-take-all decisions: reciprocal inhibition of feedforward lateral inhibition within the Imc. Anatomical support for such a motif has been found in a study of the projection patterns of Imc neurons (Fig. 4B; (Wang et al., 2004). Future experiments will be needed to determine the contribution of the Imc to categorization in controlling gaze and attention.

Generality of the reciprocal inhibition of feedforward lateral inhibition motif

The computations explored in this study that account for explicit and flexible categorization of relative stimulus strengths in the midbrain network, may generalize to other examples of categorical decisions and, therefore, to other brain areas (Wang, 2008). Classification of direction of stimulus motion with respect to a flexible reference (Freedman and Assad, 2006), of speed of stimulus motion with respect to a flexible reference (Ferrera et al., 2009), of odor based on relative odor strengths in a mixture (Niessing and Friedrich, 2010), and of tactile stimulus frequency relative to a prior sample frequency (Machens et al., 2005) can each be thought of as decisions based on such categorization. Indeed, the model that was proposed to account for neural responses in the monkey prefrontal cortex during the discrimination of tactile stimulus frequency relative to a prior sample frequency employed feedback inhibition (Machens et al., 2005). In this task, the decision of whether the test frequency was higher or lower than the sample frequency can be thought of as a form of flexible categorization, in which the comparison of stimulus representations occurs over time rather than space. Like other models of decision, the model that was proposed was purely computational and without neural correlates, and the specific computational contributions of the different circuit elements to the decision were not explored.

Recently, parallels between such potentially abstract decision-making processes and competitive stimulus selection have been recognized (Cisek and Kalaska, 2010; Freedman and Assad, 2011). We propose that reciprocal inhibition of feedforward lateral inhibition, working in various brain areas, could serve as a highly efficient motif for flexible categorization for decisions as well as for flexible normalization.

EXPERIMENTAL PROCEDURES

Computational modeling

Data analysis

The transition range of a CRP was defined as the range of competitor strengths over which responses dropped from 90% to 10% of the total range of responses. Switch-like CRPs were defined as those for which the CRP transition range was ≤ 4°/s (Mysore et al., 2011).

The switch-value of a switch-like CRP was defined as the strength of the competitor stimulus at which responses to the paired stimuli changed from high to low values. It was estimated as the competitor strength that yielded the half-maximum response.

CRP height was measured as the difference between the maximum and minimum response over the standard range of competitor loom speeds (0 to 22°/s). For experimentally measured CRPs, maximum and minimum responses were estimated from the best sigmoidal fit to the data. Experimental results (Mysore et al., 2011) indicate that only ~70% of CRPs measured in the OTid are significantly correlated with the strength of the competitor stimulus (“correlated CRPs”); for the remaining CRPs, the maximum change in response with competitor strength (“CRP height”, Experimental Procedures) is not sufficiently large to yield a significant correlation. The smallest value of CRP height for correlated CRPs, estimated as the 5th percentile value of the distribution of heights for such CRPs, was 3.9 sp/s (n=107). To translate this constraint to our model, simulated CRPs with heights smaller than the 3.9 sp/s were considered not-correlated and were excluded from subsequent analysis.

The dynamic range of either a Target-alone response profile or a Target-with-competitor response profile was defined, analogous to the CRP transition range, as the range of RF stimulus loom speeds over which responses increased from 10% to 90% of the total range of responses. Both the transition and dynamic ranges are directly related to the maximum (normalized) slope of the responses: smaller dynamic range <=> higher maximum (normalized) slope.

For circuits involving inhibitory feedback (Figs. 4A and and7A),7A), in which steady state responses were iteratively computed, the speed at which steady state was achieved was quantified using response settling time. This was defined as the first iteration time step at which the response did not change any further (< 5% change thereafter).

To estimate the reliability of the responses produced by these circuits, Gaussian noise was introduced at each computation of a unit’s response using its input-output function. The standard deviation of the noise of the response was assumed to be proportional to its mean (SD = mean/5). Monte Carlo simulation was used to obtain multiple (n=100) estimates of the steady state response.

Response variability was estimated using the Fano factor, defined as the ratio of the variance of the responses to the mean of the responses to a given stimulus strength. This procedure was repeated 100 times to estimate the distribution of the Fano factor.

The “model error” quantified the mismatch in the responses of output unit 1 in circuit #3 with respect to the responses of output unit 1 in circuit #2. It was computed by simulating the responses with both circuits to four stimulus protocols: Target-alone response profile, Target-with-competitor response profile, CRP1, and CRP2. The absolute values of the differences in responses between the two circuits for all four protocols were added up to yield the model error.

Neurophysiology

Experimental protocol

Experiments to test the model predictions were performed following protocols that have been described previously (Mysore et al., 2010, 2011), and key aspects are listed in the Supplemental Experimental Procedures. Briefly, epoxy-coated tungsten microelectrodes (FHC, 250um, 1-5 MW at 1kHz) were used to record single and multi-units extracellularly in 7 barn owls that typically were tranquilized with a mixture of nitrous oxide and oxygen. Multi-unit spike waveforms were sorted off-line into putative single units. All recordings were made in layers 11-13 of the optic tectum (OTid). Visual looming stimuli were presented on a tangent screen in front of the owl.

Supplementary Material

01

Acknowledgments

This work was supported by funding from the NIH (9R01 EY019179-30, EIK). We thank Daniel Kimmel, Valerio Mante and Alireza Soltani for critically reading the manuscript and for discussions.

Footnotes

AUTHOR CONTRIBUTIONS SPM and EIK designed the research and wrote the manuscript. SPM performed the simulations, experiments, and analyses.

SUPPLEMENTAL INFORMATION Supplemental Information includes 5 figures, one table, supplemental experimental procedures, and supplemental references.

The authors declare that there are no conflicts of interest.

Publisher's Disclaimer: This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

References

  • Allman J, Miezin F, McGuinness E. Stimulus specific responses from beyond the classical receptive field: neurophysiological mechanisms for local-global comparisons in visual neurons. Annu Rev Neurosci. 1985;8:407–430. [PubMed]
  • Asadollahi A, Mysore SP, Knudsen EI. Stimulus-driven competition in a cholinergic midbrain nucleus. Nat Neurosci. 2010;13:889–895. [PMC free article] [PubMed]
  • Ayaz A, Chance FS. Gain modulation of neuronal responses by subtractive and divisive mechanisms of inhibition. J Neurophysiol. 2009;101:958–968. [PubMed]
  • Blakemore C, Tobin EA. Lateral inhibition between orientation detectors in the cat’s visual cortex. Exp Brain Res. 1972;15:439–440. [PubMed]
  • Bolzon DM, Nordstrom K, O’Carroll DC. Local and large-range inhibition in feature detection. J Neurosci. 2009;29:14143–14150. [PubMed]
  • Busse L, Wade AR, Carandini M. Representation of concurrent stimuli by population activity in visual cortex. Neuron. 2009;64:931–942. [PMC free article] [PubMed]
  • Carandini M, Heeger DJ, Movshon JA. Linearity and normalization in simple cells of the macaque primary visual cortex. J Neurosci. 1997;17:8621–8644. [PubMed]
  • Carandini M, Heeger DJ, Senn W. A synaptic explanation of suppression in visual cortex. J Neurosci. 2002;22:10053–10065. [PubMed]
  • Cavanaugh J, Wurtz RH. Subcortical modulation of attention counters change blindness. J Neurosci. 2004;24:11236–11243. [PubMed]
  • Cavanaugh JR, Bair W, Movshon JA. Nature and interaction of signals from the receptive field center and surround in macaque V1 neurons. J Neurophysiol. 2002;88:2530–2546. [PubMed]
  • Cisek P, Kalaska JF. Neural mechanisms for interacting with a world full of action choices. Annu Rev Neurosci. 2010;33:269–298. [PubMed]
  • Deleuze C, Huguenard JR. Distinct electrical and chemical connectivity maps in the thalamic reticular nucleus: potential roles in synchronization and sensation. J Neurosci. 2006;26:8633–8645. [PubMed]
  • Edwards DH. Mutual inhibition among neural command systems as a possible mechanism for behavioral choice in crayfish. J Neurosci. 1991;11:1210–1223. [PubMed]
  • Ferrera VP, Yanike M, Cassanello C. Frontal eye field neurons signal changes in decision criteria. Nat Neurosci. 2009;12:1458–1462. [PMC free article] [PubMed]
  • Freedman DJ, Assad JA. Experience-dependent representation of visual categories in parietal cortex. Nature. 2006;443:85–88. [PubMed]
  • Freedman DJ, Assad JA. A proposed common neural mechanism for categorization and perceptual decisions. Nat Neurosci. 2011;14:143–146. [PubMed]
  • Freedman DJ, Riesenhuber M, Poggio T, Miller EK. Categorical representation of visual stimuli in the primate prefrontal cortex. Science. 2001;291:312–316. [PubMed]
  • Freeman TC, Durand S, Kiper DC, Carandini M. Suppression without inhibition in visual cortex. Neuron. 2002;35:759–771. [PubMed]
  • Gold JI, Shadlen MN. The neural basis of decision making. Annu Rev Neurosci. 2007;30:535–574. [PubMed]
  • Gollisch T, Meister M. Eye smarter than scientists believed: neural computations in circuits of the retina. Neuron. 2010;65:150–164. [PMC free article] [PubMed]
  • Hahnloser R, Douglas RJ, Mahowald M, Hepp K. Feedback interactions between neuronal pointers and maps for attentional processing. Nat Neurosci. 1999;2:746–752. [PubMed]
  • Hartline HK, Wagner HG, Ratliff F. Inhibition in the eye of Limulus. J Gen Physiol. 1956;39:651–673. [PMC free article] [PubMed]
  • Itti L, Koch C. Computational modelling of visual attention. Nat Rev Neurosci. 2001;2:194–203. [PubMed]
  • Knudsen E. Control form below: The contribution of a midbrain network to spatial attention. European Journal of Neuroscience. 2011;33:1961–1972. [PMC free article] [PubMed]
  • Koch C, Ullman S. Shifts in selective visual attention: towards the underlying neural circuitry. Hum Neurobiol. 1985;4:219–227. [PubMed]
  • Kuffler SW. Discharge patterns and functional organization of mammalian retina. J Neurophysiol. 1953;16:37–68. [PubMed]
  • Lai D, Brandt SF, Luksch H, Wessel R. Recurrent Antitopographic Inhibition Mediates Competitive Stimulus Selection in an Attention Network. J Neurophysiol. 2010;105:793–805. [PubMed]
  • Lee DK, Itti L, Koch C, Braun J. Attention activates winner-take-all competition among visual filters. Nat Neurosci. 1999;2:375–381. [PubMed]
  • Lee J, Maunsell JH. A normalization model of attentional modulation of single unit responses. PLoS One. 2009;4:e4651. [PMC free article] [PubMed]
  • Leopold DA, Logothetis NK. Multistable phenomena: changing views in perception. Trends Cogn Sci. 1999;3:254–264. [PubMed]
  • Lovejoy LP, Krauzlis RJ. Inactivation of primate superior colliculus impairs covert selection of signals for perceptual judgments. Nat Neurosci. 2010;13:261–266. [PMC free article] [PubMed]
  • Machens CK, Romo R, Brody CD. Flexible control of mutual inhibition: a neural model of two-interval discrimination. Science. 2005;307:1121–1124. [PubMed]
  • Marin G, Salas C, Sentis E, Rojas X, Letelier JC, Mpodozis J. A cholinergic gating mechanism controlled by competitive interactions in the optic tectum of the pigeon. J Neurosci. 2007;27:8112–8121. [PubMed]
  • McPeek RM, Keller EL. Deficits in saccade target selection after inactivation of superior colliculus. Nat Neurosci. 2004;7:757–763. [PubMed]
  • Muller JR, Philiastides MG, Newsome WT. Microstimulation of the superior colliculus focuses attention without moving the eyes. Proc Natl Acad Sci U S A. 2005;102:524–529. [PubMed]
  • Mysore SP, Asadollahi A, Knudsen EI. Global inhibition and stimulus competition in the owl optic tectum. J Neurosci. 2010;30:1727–1738. [PMC free article] [PubMed]
  • Mysore SP, Asadollahi A, Knudsen EI. Signaling of the strongest stimulus in the owl optic tectum. J Neurosci. 2011;31:5186–5196. [PMC free article] [PubMed]
  • Mysore SP, Knudsen EI. Flexible categorization of relative stimulus strength by the optic tectum. J Neurosci. 2011a;31:7745–7752. [PMC free article] [PubMed]
  • Mysore SP, Knudsen EI. The role of a midbrain network in competitive stimulus selection. Curr Opin Neurobiol. 2011b;21:653–660. [PMC free article] [PubMed]
  • Naka KI, Rushton WA. S-potentials from luminosity units in the retina of fish (Cyprinidae) J Physiol. 1966;185:587–599. [PubMed]
  • Niessing J, Friedrich RW. Olfactory pattern classification by discrete neuronal network states. Nature. 2010;465:47–52. [PubMed]
  • Nummela SU, Krauzlis RJ. Inactivation of primate superior colliculus biases target choice for smooth pursuit, saccades, and button press responses. J Neurophysiol. 2010;104:1538–1548. [PubMed]
  • Ohshiro T, Angelaki DE, Deangelis GC. A normalization model of multisensory integration. Nat Neurosci. 2011;14:775–782. [PMC free article] [PubMed]
  • Olsen SR, Bhandawat V, Wilson RI. Divisive normalization in olfactory population codes. Neuron. 2010;66:287–299. [PMC free article] [PubMed]
  • Pangratz-Fuehrer S, Hestrin S. Synaptogenesis of electrical and GABAergic synapses of fast-spiking inhibitory neurons in the neocortex. J Neurosci. 2011;31:10767–10775. [PMC free article] [PubMed]
  • Picardo MA, Guigue P, Bonifazi P, Batista-Brito R, Allene C, Ribas A, Fishell G, Baude A, Cossart R. Pioneer GABA Cells Comprise a Subpopulation of Hub Neurons in the Developing Hippocampus. Neuron. 2011;71:695–709. [PMC free article] [PubMed]
  • Pouille F, Marin-Burgin A, Adesnik H, Atallah BV, Scanziani M. Input normalization by global feedforward inhibition expands cortical dynamic range. Nat Neurosci. 2009;12:1577–1585. [PubMed]
  • Prather JF, Nowicki S, Anderson RC, Peters S, Mooney R. Neural correlates of categorical perception in learned vocal communication. Nat Neurosci. 2009;12:221–228. [PMC free article] [PubMed]
  • Reynolds JH, Chelazzi L, Desimone R. Competitive mechanisms subserve attention in macaque areas V2 and V4. J Neurosci. 1999;19:1736–1753. [PubMed]
  • Reynolds JH, Heeger DJ. The normalization model of attention. Neuron. 2009;61:168–185. [PMC free article] [PubMed]
  • Sereno MI, Ulinski PS. Caudal topographic nucleus isthmi and the rostral nontopographic nucleus isthmi in the turtle, Pseudemys scripta. J Comp Neurol. 1987;261:319–346. [PubMed]
  • Wang XJ. Decision making in recurrent neuronal circuits. Neuron. 2008;60:215–234. [PMC free article] [PubMed]
  • Wang Y, Major DE, Karten HJ. Morphology and connections of nucleus isthmi pars magnocellularis in chicks (Gallus gallus) J Comp Neurol. 2004;469:275–297. [PubMed]
  • Williford T, Maunsell JH. Effects of spatial attention on contrast response functions in macaque area V4. J Neurophysiol. 2006;96:40–54. [PubMed]
  • Yang XL, Wu SM. Feedforward lateral inhibition in retinal bipolar cells: input-output relation of the horizontal cell-depolarizing bipolar cell synapse. Proc Natl Acad Sci U S A. 1991;88:3310–3313. [PubMed]