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.

Figure 1. Feedforward lateral inhibition to the OTid

(A) Key elements of the midbrain network: the superficial layers of the optic tectum (layers 1-9), the intermediate and deep layers of the optic tectum (OTid), and a GABAergic nucleus in the midbrain tegmentum, the Imc. Space-specific excitation due to a stimulus (black arrow) drives neurons both in the OTid and the Imc. The Imc neurons send long-range, lateral inhibitory projections to OTid neurons coding for all other spatial locations (red line). For clarity the projections of only one Imc neuron are shown. Consequently, unlike in sensory normalization circuits, in this circuit a stimulus does not contribute to the suppression of its own responses. (B) Schematic of a feedforward lateral inhibitory circuit (circuit #1) with long-range inhibitory projections from the Imc (red ovals, “inhibitory units”) to the OTid (black circles, “output units”). Black arrows indicate excitatory connections. Two out of many spatial channels in the space map are shown. One channel (#1) represents the RF stimulus, the other (#2) represents the competing stimulus.

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

(A) Strength-response function of a “typical” OTid unit. Curve represents the average loom speed-response function for a single looming stimulus centered in the RF, computed from 61 OTid neurons. (B) Distant competitor produces divisive suppression of spatial tuning curves in the OTid. Left panel: Stimulus protocol. Dashed oval indicates RF, black dot indicates RF stimulus, and gray dot indicates competitor located outside the RF, 30° away. The sizes of the dots represent the strengths (loom speeds) of the stimuli. Note that in experiments, both stimuli were always full contrast dark dots on a light background. Right panel: Black curve: azimuthal tuning curve with RF stimulus alone. Red curve: tuning curve measured in the presence of a competitor, which produces divisive response suppression. (C) Effect of divisive influences on stimulus strength-response functions. Red arrow indicates increasing strength of inhibition. Dashed vertical lines: half-max response strengths. Left panel. Effect of input division. Right panel. Effect of output division. (D) Competitor strength-response profiles (CRPs) measured with competing looming visual stimuli. Left panel: Stimulus protocol. Dashed oval indicates RF, black dot indicates RF stimulus, and gray dot indicates competitor located outside the RF, 30° away. Strength of the RF stimulus is held constant, while that of the competitor is systematically varied. The sizes of the dots represent the strengths (loom speeds) of the stimuli. Middle panel: A CRP showing a gradual, nonlinear increase in response suppression with increasing strength of the competitor. RF stimulus strength = 7.2°/s. Horizontal dotted line indicates responses to RF stimulus alone. Vertical dashed lines indicate the transition range (7.5°/s). Right panel: A switch-like CRP showing an abrupt increase in response suppression with increasing strength of the competitor over a narrow range. RF stimulus strength = 8°/s. Horizontal dotted line indicates responses to RF stimulus alone. Vertical dashed lines indicate the transition range (0.4°/s). Red dot and arrow indicate the switch-value (7.2°/s). (E) Adaptive shift in the switch-values of CRPs with changes in RF stimulus strength. Left panel: Stimulus protocol. The two sets of stimuli were presented in an interleaved fashion. Right panel: The CRP measured with the stronger RF stimulus (blue) is shifted rightward with respect to the CRP measured with the weaker RF stimulus (magenta). [a,b,d,e: Adapted from Mysore et al., 2010 and 2011]. See also Figure S1.

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

(A) Schematic of the circuit. Recording icon indicates the unit for which responses are shown in B and C. (B) Effect of varying the parameters of the inhibitory response function on the CRP transition range of output unit 1. Red line and arrow indicate the range of values of a parameter over which CRPs are switch-like (transition ranges ≤ 4°/s). The value of the indicated parameter was varied systematically while the other parameters were held constant at the following values: k=10, S₅₀=8, m=5, c=15, d_in = 1 and d_out=0.05. (C) The switch-like CRP produced at output unit 1 when the values of the parameters of the inhibitory response function were set at: k=10, S₅₀=8, m=5, c=15, d_in = 1 and d_out =0.05. (D) CRP shift ratio (shift in CRP switch-value / change in the strength of the RF stimulus) as a function of the input and output division factors (d_in and d_out). The change in the RF stimulus strength was 6 °/s (see text). The black shaded values indicate (d_in, d_out) pairs that yielded CRPs for which at least one of them had a maximum change in response smaller than the threshold of 3.9 °/s (Experimental Procedures) or for which one of the CRPs was not switch-like (transition range > 4°/s). Largest value of CRP shift ratio = 0.03, obtained at the (d_in,d_out) pair = (1.5, 0), indicated by the boxed letter E. The resulting CRPs are shown in panel E. (E) The two CRPs corresponding to the (d_in,d_out) pair that yielded the largest CRP shift ratio (d_in = 1.5, d_out = 0). See also Figure S2.

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

(A) Conventions as in Figure 1B. Compared to model circuit #1, this circuit has one modification: reciprocal inhibition among the feedforward lateral inhibitory elements. (B) Conventions as in Figure 1A. Schematic of key elements of the midbrain network showing Imc axon branches that, in addition to projecting broadly back to the OT (Fig. 1A), terminate within the Imc as well (Wang et al., 2004). Whether these terminals form functional inhibitory synapses onto other Imc neurons is yet to be tested. See also Figure S3.

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

(A) Schematic of circuit #2. Recording icon indicates the unit for which responses are shown in B and C. (B) Effect of varying the parameters of the inhibitory response function on the CRP transition range of output unit 1. Red line and arrow indicate the range of values of a parameter over which CRPs are switch-like (transition ranges ≤ 4°/s). The value of the indicated parameter was varied systematically while the other parameters were held at the following fixed values: k=10, S₅₀=8, m=5, c=15, d_in= 1 and d_out=0.05; same values as in Figure 3B. The reciprocal inhibition factors r_in and r_out were chosen to be 0.84 and 0.01, respectively (see Results; Fig. S3C,D). (C) The switch-like CRP produced after choosing the values of the parameters of the inhibitory response function to be k=10, S₅₀=8, m=5, c=15, d_in = 1 and d_out =0.05; same as in Figure 3C. (D) CRP shift ratio (shift in CRP switch-value / change in the strength of the RF stimulus) as a function of the input and output divisive normalization factors (d_in and d_out; ranges of d_in and d_out same as in Fig. 3D). Reciprocal inhibition parameters r_in and r_out were set at 0.84 and 0.01 (Results). All other parameter values were same as in Figure 3D. The change in the RF stimulus strength was 6 °/s. The black shaded values indicate (d_in,d_out) pairs that yielded CRPs for which at least one of them had a maximum change in response less than the threshold of 3.9 °/s (Experimental Procedures), or for which one of the CRPs was not switch-like (transition range > 4°/s). Largest value of CRP shift ratio =0.88, obtained for the (d_in,d_out) pair = (0,0.06), indicated by the boxed letter E. The resulting CRPs are shown in panel e. Note, however, that a large range of nonzero d_in and d_out values yielded CRP shift ratios very close to the maximal value (Fig. S4A). (E) The two CRPs corresponding to the (d_in,d_out) pair that yielded the largest CRP shift ratio (d_in = 0, d_out = 0.06). Inset: normalized CRPs. See also Figure S4.

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

(A-H) Simulated data calculated for output unit 1 in circuit #2 (Fig. 4A). (A-E) Simulated loom speed-response functions obtained from model circuit #2 using different sets of parameter values to illustrate the range of effects of a fixed-strength competing stimulus described in the Results. The values of the parameters used to generate these curves are listed in Table S1. In black: Target-alone response profiles; in orange: Target-with-competitor response profiles. Dashed vertical lines indicate the location of the half-max response speeds. Solid horizontal lines indicate the dynamic ranges. (F) Scatter plot of dynamic ranges. Diagonal line: line of equality. (G) Distribution of the difference between the half-max response speed (L50) of the Target-with-competitor response profile and the strength of the competitor. For panels F-G, 135 pairs of Target-alone response profile and Target-with-competitor response profile were generated by varying the half-max response speed (S₅₀) of the output units systematically over 9 values (from 4°/s through 12°/s) and by varying the half-max response speed of the inhibitory units (from 7°/s through 9°/s) while holding all other parameter values constant. The strength of the competitor was always 8 °/s. (H-O) Experimental results based on measurements of loom-speed response functions from 51/71 OTid units for which (i) both response profiles (Target-alone response profile and Target-with-competitor response profile) were correlated with the strength of RF stimulus, (ii) were well fit by sigmoids (r²>0.8), and (iii) there was an effect of the competitor (Experimental Procedures). Median strength of competitor = 8°/s. (H-L) Loom speed-response functions measured without (black) and with (orange) a competitor stimulus located 30° to the side of the RF. The strength of the competitor in each case is indicated by the position of the black triangle on the x-axis. Dashed vertical lines indicate the location of the half-max response speeds (L₅₀). Solid horizontal lines indicate the dynamic ranges. (M) Scatter plot of dynamic ranges of Target-with-competitor response profiles (TcRPs) vs. those of Target-alone response profiles (TaRPs), showing several neurons that had narrower dynamic ranges in the presence of the competitor (Experimental Procedures). Diagonal line: line of equality. Similar to model predictions in Figure 6F. (N) Scatter plot of the amount of suppression (spikes/s) of responses to the strongest RF stimulus (22°/s) versus suppression to the weakest RF stimulus (0 °/s) showing several neurons with greater suppression of the responses to the weakest RF stimulus. Diagonal line: line of equality. (O1) Scatter plot of half-max response speeds (L₅₀) of Target-with-competitor response profiles (TcRPs) vs. those of Target-alone response profiles (TaRPs) showing that TcRPs are typically right-shifted with respect to TaRPs. Diagonal line: line of equality. (O2) Plot of the difference between the half-max response speed (L₅₀) of the Target-with-competitor response profile and the strength of the competitor showing that the distribution is centered around zero (p>0.05, t-test against 0). Similar to model predictions in Figure 6G. See also Table S1.

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

(A-D) Alternatives to model circuit #2 (Fig. 4) that also implement adaptive, competitive lateral inhibition. (A) Feedback lateral inhibition by output units; inset, equivalent circuit. (B) Same as A, but with additional recurrent excitatory loops to output units. (C) Feedback lateral inhibition to input units. (D) Same as C, but with additional recurrent excitatory loops to input units. (E-F) Comparison of settling times for circuit #2 (orange) and the next simplest circuit, #3, shown in A (green). E. Examples of time courses of responses to an RF stimulus strength of 9 °/s and a competitor stimulus strength of 8 °/s. Top panel: circuit #2. Bottom panel: circuit #3. Dashed lines: settling time (Experimental Procedures). F. Settling time as a function of relative stimulus strength (RF stimulus strength – competitor strength). Competitor strength was held constant at 8 °/s. Orange: circuit #2, Green: circuit #3. (G-H) Comparison of the reliability of the two circuits. G. Monte Carlo simulation was used to generate 100 estimates of the Fano factor (a metric that is inversely related to response consistency; Experimental Procedures) for the steady state response (at t= 100 time steps) with an RF stimulus strength of 9°/s and a competitor strength of 8 °/s (Experimental Procedures). Orange: circuit #2, green: circuit #3. H. Ratio of the mean Fano factor yielded by circuit #2 to that yielded by circuit #3, as a function of the relative stimulus strength (RF stimulus strength – competitor strength). Competitor strength was held constant at 8 °/s. Filled circles indicate p<0.05 (pair-wise ranksum test between Fano factor distributions at each value of relative stimulus strength followed by Holm-Bonferroni correction for multiple comparisons); open circle indicates p>0.05. For the simulations in panels E-H, the parameter values of the circuit #2 were the same as in Figure 5E. The parameters for circuit #3 were chosen to be optimal (minimum model error; Fig. S5 and Experimental Procedures): d_in = 1.1, d_out = 0.005, m=1, h=21, S₅₀ = 9, and k=5. See also Figure S5.