To test whether the multilobed structure of Imc RFs was an artifact of our experimental or analytical methods, we performed three controls. First, we tested whether errors in spike sorting might have caused multiple units with single-lobed RFs to be misidentified as a single unit with a multilobed RF. To this end, we applied an additional separability criterion to our sorted units. We tested the statistical separability of the waveforms of each sorted unit with those of any other unit as well as with outlier waveforms recorded at the same site, and retained only those units that were well separated ( STAR Methods ). We found that the majority of the sorted units (114/116) satisfied the separability criterion as well (p < 0.05; Figure 1 I), ruling out multiunit contamination as a source of error. Second, we examined whether the spatial sampling resolution used for RF measurement, as well as neuronal response variability, might have caused the erroneous identification of single-lobed RFs as being multilobed ( Figure S2 G). Using experimentally grounded simulations, we mapped out the values of sampling step size and response Fano factor that yielded a multilobe misidentification rate of 5% or greater ( Figure 1 J, red zone; STAR Methods ). By comparing with experimental data, we found that the values of these parameters from each recorded unit fell outside the 5% misidentification zone. As a final control, because it is well established that OT RFs have single spatial response fields, we measured visual RFs of OT neurons. Our methods correctly identified all of the measured OT RFs as being single-lobed ( Figures 1 K and S2 H). Together, these results confirmed the veracity of our conclusion that the Imc contains predominantly “multilobe” neurons (68%; 78/114; Figure 1 L).

We found that individual Imc neurons possessed visual RFs with multiple, distinct response fields or lobes ( Figures 1 A–1H, S2 A, and S2B). The number of lobes in each RF was estimated in an unbiased manner using a two-step process ( STAR Methods ): (1) a nonlinear clustering method () to fit different numbers of clusters to the spatial map of firing rates followed by (2) a model selection method () to robustly select the optimal number of clusters in the data ( Figures 1 C, 1G, and S2 C–S2F). We found that about two-thirds of Imc neurons had multilobed RFs (80/116; see also Figure 1 L).

We measured the visuospatial RFs of Imc neurons using extracellular recordings ( STAR Methods ). Individual Imc units were identified by spike sorting single and multiunit data; only those units deemed to be of “high quality” were included in the analysis ( STAR Methods ). Consistent with published data, Imc neurons have high firing rates (median, 76.5 Hz []; Figures 1 A, 1B, 1E, and 1F ).

(I) Plot of p values (logarithmic scale) obtained from separability testing for each sorted unit; one-way ANOVA followed by correction for multiple comparisons ( STAR Methods ). p value < 0.05 (blue data): units that are deemed “well separated” from co-recorded units as well as outliers (n = 114). Red data: units not well separated form cohort.

(C) Rate map in (B) re-plotted as distribution of points in a 2D plane and subjected to spatial clustering ( STAR Methods ). Shown are the best single (top left), best two (top right), and best three clusters (bottom left) fitted to the data using the density peaks clustering method () ( Figure S2 C; STAR Methods ). Bottom right: plot of GAP statistic, a robust model selection metric, against the number of clusters (k) fitted to data () ( STAR Methods ). Red point: statistically optimal number of clusters (k∗), identified as the smallest k for which GAP exceeds zero; here k∗ = 2 ( STAR Methods ) ().

(A) 2D visual receptive field (RF) of Imc neuron: raster plot of neuron’s responses to visual stimulus presented at different spatial locations. Inset top: gray line, stimulus onset; red lines, time window used to calculate firing rate; evoked firing rates in Imc were high (median, 76.5 Hz; n = 114 neurons). Inset bottom: average spike waveform for neuron in (A); identified as high-quality unit ( STAR Methods ); mean (black) ± SD (gray).

These results demonstrated that whereas azimuthal space is encoded in a topographic manner along the rostrocaudal extent of the Imc, elevational space is encoded by RFs with multiple, arbitrarily spaced, and widely distributed lobes of varying number and size ( Figures S3 E–S3J), with a maximum of three RF lobes per neuron ( Figure 1 L).

The encoding of elevation by Imc neurons was strikingly different. RF lobes of individual multilobed neurons were spaced arbitrarily in elevation ( Figure 2 A: large range of red data). Additionally, RF lobes of multilobed Imc neurons were distributed widely across elevational space: for each multilobed neuron ( Figure 2 A, inset: large median of data), across neurons recorded at a given site ( Figure 2 B, red), and across sites recorded along both dorsoventral and rostrocaudal axes ( Figures S3 A–S3D). There was also no systematic relationship between encoded elevations and distance along either principal axis ( Figures S3 A and S3B).

To investigate organizing principles underlying spatial encoding by Imc neurons, we analyzed the properties of the measured visual RFs along the two major anatomical axes of the Imc ( Figure S1 A). The azimuthal centers of RF lobes were nearly identical for lobes of individual multilobed neurons ( Figure 2 A, blue data; STAR Methods ), across neurons recorded at a given site ( Figure 2 B, blue data), and across sites recorded along the dorsoventral axis of the Imc ( Figure 2 C; STAR Methods ). However, azimuthal encoding varied systematically along the rostrocaudal axis of the Imc: centers of RF lobes encoded progressively more peripheral azimuths as the recording electrode was moved from rostral to caudal portions of the Imc ( Figure 2 D; ()).

(D) Plot of average azimuthal “center” of a dorsoventral penetration against the rostrocaudal position of electrode in the Imc in that recording session ( STAR Methods ). Colors: different recording sessions; Spearman correlation = 1 in each case. See also Figure S3

(C) Plot of average azimuthal center of a recording site against the dorsoventral position of the site within the Imc ( STAR Methods ); colors: different penetrations. Inset: data re-plotted as histogram of pairwise differences in the azimuthal centers of recording sites along a dorsoventral penetration (p = 0.18, one-tailed rank sum test).

(B) Histograms of maximum distances between centers of RF lobes of multilobed neurons sorted from individual recording sites ( STAR Methods ); conventions as in (A); p = 0.65 for azimuth; one-tailed rank sum test, p < 0.05 for elevation; one-tailed t test.

(A) Histograms of pairwise distance between centers of RF lobes of individual multilobed neurons ( STAR Methods ). Blue: azimuthal distance; red: elevational distance; marked range: 5th to 95th percentile range of red data; large range indicates arbitrary spacing of RF lobes. Arrows: median values;: median significantly different from 0 (p = 0.17, azimuth; p < 0.05, elevation; one-tailed rank sum tests). Inset: histogram of maximum elevational distance between centers of RF lobes of an individual multilobed neuron. Arrow: median value; significantly different from 0 (p < 0.05, one-tailed rank sum test); large median indicates widely distributed RF lobes.

RF Lobes of Multilobe Imc Neurons Are Distributed along Elevation but Not Azimuth, and RFs Are Organized Topographically in Azimuth, but Not Elevation

Figure 2 RF Lobes of Multilobe Imc Neurons Are Distributed along Elevation but Not Azimuth, and RFs Are Organized Topographically in Azimuth, but Not Elevation

These results indicated that multilobed encoding by Imc neurons is consistent with the need for the Imc-OT circuit to achieve stimulus selection at all possible elevational location pairs in the face of a scarcity of Imc neurons encoding elevation ( Figures 3 B and 3C).

In contrast, along the azimuth, Nis greater than or equal to L. The OTid encodes azimuths ranging typically from −15° to 60° at a spatial resolution of at best 1° ( STAR Methods ). As a result, the number of distinct azimuthal locations encoded by OTid is at most 75 (L≤ 75). On the other hand, we estimated that there are at least 84 neurons involved in encoding these distinct azimuths, N≥ 84 ( STAR Methods ). Thus, there are more Imc neurons than there are encoded azimuthal locations (N≥ L), an observation that is consistent with the absence of multilobed RFs along the azimuth.

To examine the biological applicability of this insight, we estimated L and N in the owl brain. For a given azimuth, the OTid encodes elevations ranging typically from −60° to +60°, and does so at a spatial resolution of at least 3° ( STAR Methods ). Consequently, the number of distinct elevational locations encoded by the OTid at a given azimuth (L) is at least 120°/3° = 40 (L≥ 40). Next, we estimated the number of Imc neurons encoding these elevational locations (N). Because visual azimuth is organized topographically along Imc’s rostrocaudal axis ( Figure 2 D), transverse sections of the Imc provide snapshots of Imc tissue encoding all elevations at a given azimuth ( Figures 3 A and 3B ). We obtained histological sections perpendicular to the rostrocaudal axis of the Imc and performed Nissl staining to visualize cell bodies ( STAR Methods ). Counts of the number of Nissl-stained somata () showed that the majority of sections (75%) had fewer than 28 neurons per section (N Figures 3 B and 3C). Thus, Nis typically much smaller than L(median N/L< 26/40 = 0.65).

(B) Zoomed-in image showing individual, Nissl-stained, Imc somata (arrowheads); 22 somata in this section. The zoomed-in image was obtained by stitching 5 images (each taken at 40× magnification) with overlapping fields of view at the edges.

To examine the implications of spatial selection on Imc RF structure and, specifically, of the need for implementing stimulus selection at all possible location pairs, we turned to theory ( STAR Methods ). Briefly, we compared the total number of location pairs at which selection must occur in the OTid, with the number of location pairs in the OTid at which selection can be achieved by a set of Imc neurons. Since multilobed Imc encoding is restricted along the elevation ( Figures 2 A, 2B, and S3 A–S3D), we focused on stimulus selection between all possible pairs of elevations at any azimuth. We proved mathematically that if the number of Imc neurons (N) encoding different elevations at a given azimuth is less than the number of distinct elevational locations (L) encoded by the OTid at that azimuth (N < L), then multilobed Imc RFs are necessary for stimulus selection at all possible location pairs ( STAR Methods ).

The multilobed encoding of elevational space by Imc neurons was puzzling. This was especially so because neurons that provide input to the Imc (OT), as well those that receive Imc’s output (OTid), all tile sensory space with single-lobed spatial RFs organized topographically in both elevation and azimuth ( Figure 1 K) (). Might the implementation of stimulus selection across space, a main function of the Imc (), impose demands on the spatial coding properties of Imc neurons that can explain multilobed RFs?

Model Predicts Combinatorially Optimized Inhibition for Selection at All Location Pairs

Figure 4 Example Optimal Solution from Model Illustrates Stimulus Selection at All Location Pairs with an Under-Complete Set of Neurons Show full caption ∗) needed by model to solve selection across all locations for different numbers of locations (L) ( max : maximum number of RF lobes allowed for each neuron ( (A) Summary plot showing the fewest number of neurons (N) needed by model to solve selection across all locations for different numbers of locations (L) ( Figure S5 A; STAR Methods ). k: maximum number of RF lobes allowed for each neuron ( STAR Methods ). (B–D) Illustration of selection at all possible location pairs by an optimal model solution for L = 5 locations (numbered a–e) and N = 4 neurons (numbered #1–#4). (B) The four RFs in the optimal solution. Shaded areas: RF of neuron; two neurons have multilobed RFs (#1, two lobes; #3, three lobes). a and S b at location pair ab (extreme left). S a and S b are of equal priority (1 unit for simplicity). Top row: information flow through the model OT 10 -Imc-OTid circuit triggered by S a . First column: activation of OT 10 space map. Second column: activation of individual Imc neurons. Third column: suppression pattern generated by each activated Imc neuron (spatial inverse of the neuron’s RF; consistent with published anatomical results; Wang et al., 2004 Wang Y.

Major D.E.

Karten H.J. Morphology and connections of nucleus isthmi pars magnocellularis in chicks (Gallus gallus). a location b. Dark-gray shading: “activated” neuron (#2); defined as a neuron driven by S a but that does not send inhibition to location b. Red shading: “recruited” neuron (#3); defined as activated neuron that sends inhibition to location b, thereby involved in selection for location pair ab. Bottom row: same as top row, but for stimulus S b . (C) Optimal solution in (B) implements selection between stimuli Sand Sat location pair ab (extreme left). Sand Sare of equal priority (1 unit for simplicity). Top row: information flow through the model OT-Imc-OTid circuit triggered by S. First column: activation of OTspace map. Second column: activation of individual Imc neurons. Third column: suppression pattern generated by each activated Imc neuron (spatial inverse of the neuron’s RF; consistent with published anatomical results; Figures S1 B–S1E []). Fourth column: combined pattern of suppression in the OTid. Dark colors: 2 units of suppression; light colors: 1 unit ( STAR Methods ). Curved arrow: net suppression driven by Slocation b. Dark-gray shading: “activated” neuron (#2); defined as a neuron driven by Sbut that does not send inhibition to location b. Red shading: “recruited” neuron (#3); defined as activated neuron that sends inhibition to location b, thereby involved in selection for location pair ab. Bottom row: same as top row, but for stimulus S (D) Selection matrix summarizing implementation of selection at all location pairs by optimal model solution in (A). Columns: ten possible location pairs; rows: the four neurons. In each column: dark-gray, activated neurons; red, recruited neurons; blank, neurons not activated by either stimulus. (E) Example optimal solution for (L, k max , N∗) = (40, 3, 27). Black pixels: locations inside neurons’ RF; white pixels, locations outside neurons’ RF. To explore how an under-complete set of Imc neurons might implement selection at all possible location pairs, we turned to computational modeling. We set up stimulus selection across spatial locations as an optimization problem with L locations (elevations at a given azimuth), and N model neurons encoding those elevations (N < L; Figure S4 STAR Methods ). We imbued all model neurons with Imc-like spatially inverting connectivity with the OT ( Figures S1 and S4 ). The spatial RFs of these model Imc neurons were represented, for simplicity, using ones and zeros, with ones corresponding to locations inside the RF, and zeros, outside ( Figure 4 B; also see Figure S4 for validity of model even when this assumption is relaxed).

The goal of the optimization was to identify the spatial RF structures of these N neurons (i.e., the numbers of their RF lobes and their spatial locations), such that when two stimuli of equal priority were placed at any pair of locations, they suppressed each other equally. This necessary and sufficient condition for implementing selection at all location pairs was captured by a specially constructed cost function whose value decreased as the number of location pairs at which the above condition was satisfied increased. The cost function took the lowest possible value of –L(L − 1) if and only if the condition was satisfied at all location pairs ( STAR Methods ). Any set of Imc RFs that achieved this minimum value, i.e., that achieved selection for all location pairs, was called an “optimal solution.”

max . The values of k max tested were 1, 3, and 10, corresponding to key experimentally relevant values: k max = 1 only permitted model neurons with (traditional) single-lobed RFs as potential solutions to the optimization problem; k max = 3 permitted up to three lobed RFs, in line with the experimental data ( max = 10 allowed up to 10 lobes per RF, representing the largest number of typical Imc RF lobes that one can fit within the encoded elevational space ( max (maximum number of RF lobes allowed per neuron), and N (number of Imc neurons). For each triplet of (L, k max, N), we ran the optimization problem 1,000 times (Monte Carlo simulation), each time with a different, randomly chosen initial condition, to explore the space of potential optimal solutions. For each value of L, we varied the number of neurons in the model from N = 1 to N = L. In addition, in each case, we examined the impact of single as well as multilobed RFs on the existence and nature of optimal solutions. We did so by including a constraint that specified the maximum number of lobes allowed in a model neuron’s RF, denoted by k. The values of ktested were 1, 3, and 10, corresponding to key experimentally relevant values: k= 1 only permitted model neurons with (traditional) single-lobed RFs as potential solutions to the optimization problem; k= 3 permitted up to three lobed RFs, in line with the experimental data ( Figure 1 L); and k= 10 allowed up to 10 lobes per RF, representing the largest number of typical Imc RF lobes that one can fit within the encoded elevational space ( STAR Methods ). Therefore, the main parameters in the optimization problem were L (number of locations), k(maximum number of RF lobes allowed per neuron), and N (number of Imc neurons). For each triplet of (L, kN), we ran the optimization problem 1,000 times (Monte Carlo simulation), each time with a different, randomly chosen initial condition, to explore the space of potential optimal solutions.

∗, was 4 (∗). We found that as L increased, neuronal savings increased (∗ = 27 neurons to solve selection at all location pairs (savings of 13 neurons or 32%). Neuronal savings also increased as a function of k max , the maximum number of RF lobes allowed per neuron ( max = 1), N∗ was always equal to L, and there were no neuronal savings ( We found that L = 5 was the smallest number of locations for which selection could be solved at all location pairs with fewer than 5 neurons ( Figure 4 B). The fewest number of neurons needed by the model in this case, called N, was 4 ( Figure S5 A; STAR Methods ). Therefore, the maximum “savings” in the number of Imc-like neurons for L = 5 locations was 1 (=L − N). We found that as L increased, neuronal savings increased ( Figure 4 A; orange data), with L = 40 locations requiring just N= 27 neurons to solve selection at all location pairs (savings of 13 neurons or 32%). Neuronal savings also increased as a function of k, the maximum number of RF lobes allowed per neuron ( Figure 4 A; black versus orange data). Notably, when only single-lobed RFs were allowed in the model (k= 1), Nwas always equal to L, and there were no neuronal savings ( Figure 4 A; blue data). Thus, consistent with our theoretical prediction, selection at all possible location pairs could not be achieved with fewer than L neurons if all neurons only had single-lobed RFs.

max = 3 lobes, and N = 4 neurons ( max = 3 lobes, and N = 27 (N∗) neurons. The primary motivation for our optimization-based modeling approach was to gain insight into the computational logic underlying successful stimulus selection at all possible location pairs when neurons are scarce. An example optimal solution obtained when L = 5 locations, k= 3 lobes, and N = 4 neurons ( Figure 4 B), illustrates how fewer than L inhibitory neurons can successfully achieve selection at all location pairs ( Figures 4 C and 4D). Figure 4 E shows another example optimal model solution, obtained when L = 40 locations, k= 3 lobes, and N = 27 (N) neurons.

a ( a was shared with the lobes of neuron B, but the lower lobe of M a was not. Similarly, there also existed another neuron C (for instance, neuron #3 in a was shared with the lobes of neuron C, but not the upper lobe. (Here, the neurons B and C could be either single-lobed or multilobed.) This property was quantified using a binary score: briefly, each optimal solution was assigned a score of 1 if every multilobed neuron in that solution satisfied the optimized lobe overlap property (as illustrated in Second, the RFs of the neurons in optimal solutions collectively exhibited the “optimized lobe overlap” property: every multilobed neuron shared each of its lobes, but not all, with another neuron ( Figures 5 B–5D). To visualize this property, consider a two-lobed neuron M Figure 5 B: for instance, neuron #1 in Figure 4 B). There necessarily existed another neuron B (for instance, neuron #2 in Figure 4 B) in the solution such that the upper lobe of Mwas shared with the lobes of neuron B, but the lower lobe of Mwas not. Similarly, there also existed another neuron C (for instance, neuron #3 in Figure 4 B) in the optimal model solution such that the lower lobe of Mwas shared with the lobes of neuron C, but not the upper lobe. (Here, the neurons B and C could be either single-lobed or multilobed.) This property was quantified using a binary score: briefly, each optimal solution was assigned a score of 1 if every multilobed neuron in that solution satisfied the optimized lobe overlap property (as illustrated in Figure 5 B), and 0 otherwise. We found that every optimal solution obtained had a score of 1 ( Figure 5 C). Conceptually, the optimized lobe overlap property is necessary because selection needs to be solved also when two stimuli are placed at the locations encoded by different lobes of an individual multilobed neuron ( Figure 5 D). Consequently, this imposes a severe constraint on the relative organization of RF lobes across neurons in optimal solutions—one that causes structured non-orthogonality of the RFs.

Third, neurons in optimal solutions used a “combinatorial inhibition” strategy to achieve stimulus selection at all location pairs ( Figures 5 E–5J). The combinatorial nature was quantified via a pair of necessary and sufficient conditions, namely that assorted subsets of neurons were selectively recruited to solve stimulus selection for individual location pairs, with the subsets corresponding to different location pairs intersecting extensively.

An optimal solution was said to exhibit the extensive intersection feature if neural subsets recruited to solve selection even for two location pairs in distant portions of space shared common neurons ( Figures 5 H and 5I). This feature was quantified by first identifying “doublets”: two location pairs such that the locations within each pair were nearby locations, but such that the two pairs themselves occupied distant portions of space. Then we checked whether the neural subsets recruited to solve selection for at least one such doublet involved a common neuron, and scored the solution as 1 if they did ( STAR Methods ). We found that optimal solutions obtained from all runs exhibited this feature ( Figure 5 I; see also Figure S6 ), demonstrating the extensively intersecting nature of optimal solutions. Together, the above results indicated that combinatorial inhibition was a signature property of optimal solutions.

Conceptually, combinatorial patterns of inhibition are a consequence of the RF lobes of model Imc neurons being widely distributed and arbitrarily spaced in the optimal solutions ( Figures 5 J, S6 B, and S6D): restricting RF lobes to only nearby locations substantially limits the number of available RF configurations, potentially precluding optimal solutions. Stated equivalently, the combinatorial inhibition strategy arises because of the combinatorial coding of space by the model Imc neurons: individual neurons do not always encode only for neighboring locations ( Figures 4 B, 4E, S6 B, and S6D), and conversely, nearby locations are not always encoded by “nearby” neurons (no matter what permutation of neurons is considered; Figures S5 B, S6 B, and S6D). Therefore, when two stimuli are presented, two groups of the inhibitory Imc neurons are activated in a non-ordered fashion, resulting in a combinatorial pattern of inhibition.

Taken together, the model revealed that selection at all possible location pairs when N < L, as is the case with Imc’s elevational coding, necessitates a combinatorially optimized inhibition strategy by multilobed neurons.