Visual Pinwheel Center Act as Geometric Saliency Detector

Authors: haixin  zhong
mingyi  huang
wei p dai
haoyu  wang
anna wang roe
yuguo  yu
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-15 13:50:03

Abstract: During natural evolution, the primary visual cortex (V1) of lower mammals typically forms salt-and-pepper organizations, while higher mammals and primates develop pinwheel structures with distinct topological properties. Despite the general belief that V1 neurons primarily serve as edge detectors, the functional advantages of pinwheel structures over salt-and-peppers are not well recognized. To this end, we propose a two-dimensional self-evolving spiking neural network that integrates Hebbian-like plasticity and empirical morphological data. Through extensive exposure to image data, our network evolves from salt-and-peppers to pinwheel structures, with neurons becoming localized bandpass filters responsive to various orientations. This transformation is accompanied by an increase in visual field overlap. Our findings indicate that neurons in pinwheel centers (PCs) respond more effectively to complex spatial textures in natural images, exhibiting stronger and quicker responses than those in salt-and-pepper organizations. PCs act as first-order stage processors with heightened sensitivity and reduced latency to intricate contours, while adjacent iso-orientation domains serve as second-order stage processors that refine edge representations for clearer perception. This study presents the first theoretical evidence that pinwheel structures function as crucial detectors of spatial contour saliency in the visual cortex.

Main paper: 
 introduction: 
 The seminal work of Hubel and Wiesel revealed orientation-selective columns in the visual cortex of higher mammals [1, 2] . In higher mammals' primary visual cortex (V1), neurons cluster into "pinwheel" structures around singularities [3] , unlike in some mammals like rodents, which display "salt-and-pepper" organizations [4] or mini-columns [5] . While there are established theories and experiments for studying the formation of topological organization maps in the visual cortex [6, 7, 8, 9, 10, 11] , the functional significance of pinwheel-like columnar organization remains an unresolved question and is even debated [12, 13] . Sophisticated visual analyses, such as image pattern extraction [14] , pattern symmetry [15] , material properties [16] , and textures [17] , are crucial for understanding complex visual inputs. Imaging and electrophysiological studies have shown that iso-orientation domains (IODs) undergo crossorientation suppression [18] , reducing a neuron's response to its preferred orientation when another orientation is also present in the stimulus [13, 19, 20] . This indicates IODs encoding the linear oriented stimuli, which is crucial for detecting edges and contours [21, 22] . Cross-orientation suppression is believed to facilitate the detection of local discontinuities, such as orientation discontinuities [23, 24, 25] , leading to perceptual "pop-out" effects and the perception of illusory contours [24, 26, 27] . In contrast, neurons at pinwheel centers (PCs) exhibit greater selectivity for cross-orientation stimuli [12, 13] . This indicates that PCs respond more effectively to multi-orientation patterns, such as pattern symmetry than IODs [12] . This indicates PCs may contribute to encode more complex contour features. However, PCs are less selective but have longer response latency than IODs for stimulus orientation in the hierarchy process within OPMs when it comes to a single stimulus orientation [13, 19, 28] . Some studies indicate that colors [29] , textures [30] , darks and lights [31] , luminance [32] , and mirror symmetry [15] play a role in salient to visual processing. Despite these insights, the functional implications of how neurons within IODs and PCs of pinwheels process complex contour stimuli-potentially affecting stimulus salience for both IODs and PCs-from bottom-up visual inputs remain poorly understood, particularly from a temporal-spatial neural dynamics standpoint.In response to these challenges, our research contributes the following:• We propose a novel 2D self-evolving spiking neural network (SESNN) model that investigates the spiking mechanisms behind orientation preference maps (OPMs), spanning from salt-and-pepper organizations in mice to pinwheel structures in cats and macaques. The SESNN uniquely produces sparse codes through local synaptic plasticity during natural scene learning, establishing a new benchmark for neural coding strategies.• PCs act as first-stage processors, detecting natural images and initiating spiking waves to neighboring IODs, which then process as second-stage neurons. This indicates that early processing involves complex contours, not just edge detection.• PCs react faster to a variety of orientation features than IODs, indicating their function in detecting complex orientations and serving as geometric saliency detectors. This suggests PCs have an evolutionary advantage due to self-organized pinwheel structures, which improves their ability to process complex contours.2 Results 
visual overlap underlying pinwheels emergence: 
 Our SESNN model generates diverse OPMs, from salt-and-peppers to pinwheel structures, by adjusting the visual overlap metric ε. This metric, crucial for the variety of visual topologies across species, is shown in Fig. 1a to produce pinwheel structures at high overlap, akin to those in cats and macaques, while low overlap results in salt-and-pepper organizations, typical of mice or rats. High overlap also enables cortical neurons to sample natural scenes more frequently, aiding in generating high-resolution images during decoding.Fig. 1a shows how visual input overlap levels from 10 to 15 pixels affect pinwheel structure representation in the model. The top panel illustrates a higher overlap (15 pixels), and the middle, a lower overlap (12 pixels ). This comparison reveals the impact of stimulus overlap on pinwheel density and layout in the visual cortex. Below the threshold (10 pixels in our case), salt-and-pepper patterns form, as the bottom panel indicates. Thus, 9 pixels of overlap are excluded from pinwheel analysis, as shown in Fig. 1b-d .We quantitatively analyze the OPMs shown in Fig. 1a with several metrics [7, 33] :Pinwheel counts, defined as the number of PCs, can be measured by 2D FFT [34] which are located at the intersection of the real and imaginary components that equal 0 [33] . It exhibits a decreasing trend as the visual input overlap increases (illustrated in Fig. 1b ), suggesting that a greater overlap in the visual field may lead to a reduction in the number of discrete pinwheel structures. The nearest-neighbor pinwheel distance (NNPD) in millimeter (mm) unit is defined as the distance between the two nearest PCs. The increasing trend of visual input overlap expands the distance between neighboring pinwheels (Fig. 1c ).The size of hypercolumns (mm) is defined with periodicity measured by 2D FFT and also increases with the visual input overlap (shown in Fig. 1d ). This paper does not account for left-and right-eye dominance columns, so the hypercolumn size is defined as the full 180°cycle of repeating column spacing (Λ) (mm).It's noteworthy that pinwheel density is not included as a metric in our analysis. This omission is because the observed pinwheel density, irrespective of the hypercolumn size, approaches π pinwheels/Λ 2 , conforming to topological constraints [33, 34] .Our findings emphasize the importance of overlap degrees. Greater overlap (e.g., ε 1 = 15 pixels) fosters stronger local clustering, leading to larger hypercolumn sizes, fewer pinwheels, and longer NNPDs, versus lower overlap (e.g., ε 2 = 12 pixels). Minimal overlap (e.g., ε 3 = 9 pixels), yields weak clustering, resembling salt-and-pepper organizations (Fig. 1a ). This suggests that shared input among V1 neurons significantly influences OPM and salt-and-pepper formation. A strong positive correlation (R 2 = 0.99) between the SESNN model and species' visual RF overlaps (mouse, cat, macaque) is observed in Fig. 1e . This relationship highlights the overlap index's key role in spatial organization within orientation maps. The model's predictions on IOD sizes and visual field extent (Fig. 1f ) align with empirical data [7] , confirming the SESNN model's robustness in simulating neuroanatomical organization and the biological development of orientation maps.V1 neurons stimulated by natural images primarily fire within pinwheel structures, particularly within and around PCs (Fig. 2a-b ). This pattern is especially pronounced in higher mammals with large IODs, such as macaques and cats. We define the response onset latency as 1 ms for the initial discharge from pinwheel structures, with subsequent firings occurring at 2 ms, based on a 1 ms time unit. Stimulated by natural images, the discharges start at the PCs and exhibit pronounced diffusion within the IODs sequentially, depending on their distance from the center, as suggested in Fig. 2c .In this section, we investigate whether pinwheel structures respond distinctly to salient features in input images. The ground truth boundary from the BSDS 500 dataset [35] used as binary input represents geometric complexity (edges and curves) (Fig. 3a ). The complexity is measured by calculating the local pixel entropy using sliding windows, with a 15×15 pixel neighborhood to assess pixel value dispersion in the binary images. The computation adheres to the following equation:EQUATIONwhere H(i, j) denotes the entropy at pixel position (i, j) in the entropy map, L the count of distinct gray levels within the local neighborhood around pixel (i, j), and m k the kth gray level within this specified neighborhood. A large entropy value reflects great unpredictability or complexity in the pixel values, signifying a highly variable pixel value distribution. Conversely, a low entropy value indicates a high degree of predictability, less variation, and reduced complexity in the contours of pixel values. In addition, the saliency map of images is generated based on the classical methodology [36] .Furthermore, we propose a bimodal ratio analysis to compute the orientation bimodal ratio (OBR) to indicate a neuron's orientation tuning curve as either unimodal (single peak) or perfectly bimodal (two peaks of equal strength). This analysis focuses on identifying the peaks in the orientation tuning curve and quantifying their relative strengths. where R 1 and R 2 represent the normalized firing rates corresponding to the strengths of the two most pronounced peaks in the orientation tuning curve. The OBR ranges from 0, denoting unimodality, to 1, indicating perfect bimodality in the neuron's orientation tuning.EQUATIONA positive correlation is observed between the saliency map and the geometrical complexity of the BSDS500 dataset (Fig. 3b ), demonstrating that higher geometrical complexity correlates with increased saliency. Significantly, in response to the stimulus shown in the BSDS500 image (Fig. 3a ), pinwheel structures primarily activate in areas of high contour complexity (regions with the highest saliency in this binary image), which is a response pattern have not been observed in salt-and-peppers (Fig. 3c ).To confirm the disparity in contour complexity responses between pinwheel and salt-and-pepper organizations, we design a star-like binary input (depicted in Fig. 4a ), including four identical entities to negate the impact of neuronal positioning within the SESNN model. This approach reaffirms the saliency-complexity correlation (Fig. 4b ) and the priority of pinwheel activation over salt-and-peppers in response to heightened complexity (Fig. 4c ).Findings show that PCs exhibit enhanced saliency detection and significantly faster response times than IODs, indicating that PCs respond more quickly and sensitively to geometrically complex stimuli, while IODs are slower and react to simpler geometrical stimuli (see Fig. 4d ). Both saliency and latency measurements are normalized to a 0-1 scale for comparison.The enhanced saliency detection of PCs is due to the complex orientation preference in RFs. As addressed in Fig. 4e , the ordinate represents the OBR, reflecting that neurons near PC generally exhibit bimodal orientation tuning curves with near-equal peak strengths while there is a primary peak and a secondary peak at a relatively far position from PC (x = 2). And the secondary peak is nearly absent at the IODs level (x = 3), leading to an OBR close to 0. Salt-and-peppers, however, show less variation, maintaining a consistent OBR.In conclusion, PCs demonstrate selectivity for more intricate orientations. This is experimentally supported by [13, 37] , who suggest that PCs are particularly sensitive to specific geometric configurations, such as T junctions. Characterized by multiple orientations and an OBR nearing 1 (Fig. 4e ), these neurons tend to initiate action potentials in response to complex orientations. Consequently, this leads to pinwheels being the first to respond. In contrast, neurons in salt-and-peppers do not exhibit a similar responsiveness to complex orientations as observed in pinwheels. 
the architecture of sesnn model: 
 Our SESNN model is a two-dimensional network of excitatory (E-) and inhibitory (I-) leaky integrateand-fire neurons (LIF) (5) , stimulated by whitened natural images to mimic the LGN's functions of contrast normalization and edge enhancement without complex modeling [42, 43, 44] . We use 160 whitened natural images as the training dataset, normalized to zero mean and uniform variance, derived from 20 base images (512×512 pixels) [44, 45] . To capture orientation details, each of the base images undergoes a 90-degree clockwise rotation and flip, creating 8 variations per original.The configuration features E-and I-neurons in recurrent networks with periodic boundary conditions (PBC), simulating a continuous 2D cortical surface. Under natural image stimuli, the SESNN forms single neuron receptive fields and population-level pinwheel structures in the OPM (Fig. 5e-f ). To validate the model, we compare its evolution from randomness to organized states against biological data from macaque pinwheel structures and a baseline model [41] , using metrics such as pinwheel density (pinwheels/ Λ 2 ), NNPD (mm), and hypercolumn size (mm) [7, 33, 34] (Fig. 5f and Table 1 ). In each trial, E-neuron processes 100 different 16×16 patches for 100 milliseconds each, randomly selected from the training dataset to serve as the receptive fields (RFs) (see Fig. 5a middle and right panels). It is assumed that these visual inputs overlap on the retina (Fig. 5a , middle panel and its inset). To reflect biological conditions, we perform a statistical analysis based on data from cats, macaques, and mice (Table 2 ), calculating average overlaps of 99.84% for cats, 99.70% for macaques, and 96.08% for mice using (Eq. 3). These results closely align with our SESNN model's configurations (refer to Fig. 1e ). Receptive field (RF) size in V1 is more related to resolution than orientation map formation. In macaque V1, RF size increases more than tenfold from fovea to periphery, while orientation map properties show little variation [46, 47] . Our study does not focus on RF size variations across the retina, as we expect minimal effects from these shifts across species, provided the overlap remains constant. Since the fovea is key for detailed visual information, we use V1 RFs in the area centralis to modeling.We propose the visual input overlap metric ε percentage , which is defined as follows:EQUATIONwhere S unit represents the unit cortical area, RF size denotes the size of the RF in V1, ρ V1 represents the density of neurons in V1, L unit denotes the unit length of the RF and M refers to the cortical magnification factor (CMF). We consider only an effective cortical layer composed of output neurons. This is because the apparent overlap within a vertical cortical column primarily contributes to intermediary processing stages for the same input. Therefore, such overlaps should not be conflated with overlaps in the input space.E-neurons receive stimuli from natural images as well as noise N (0, 0.04) from other brain areas (noise term). I-neurons indirectly receive natural image stimuli by adjusting E-neurons. The neural ∼ 0.02 [53] spiking dynamics are modeled using biologically inspired LIF neurons, incorporating refractory periods and adaptive firing thresholds [54] . The neural dynamics are iteratively formulated as follows:EQUATIONEQUATIONEQUATIONwhere i = 1, 2, . . . , N th (the neuron IDs of E-neurons and I-neurons).In neural dynamics equation, u(K) i (t)denotes the membrane potential of neuron i at time t, applicable to neurons of class K, which includes E-and I-neuron groups. The membrane time constant, symbolized by τ in the resistor-capacitor circuit, governs the decay rate of the membrane potential in individual neurons. Notably, inhibitory neurons are configured to fire more rapidly than excitatory neurons [43, 55] . This setup reduces reconstruction error and hastens system convergence, leading to a more efficient and accurate representation of input stimuli.The learning rules consist of the Hebbian Oja's variant (HO) [42] for input weight adjustments and the Correlation Measuring (CM) rules [43, 44] for intra-network weight changes (Fig. 5a-c ). These facilitate adaptive synaptic weight adjustment based on firing pattern correlations, emulating a key learning mechanism in biological neural networks.The formula for these adjustments is given by: HO : ∆WEQUATIONCM : ∆WEQUATIONwhere x, y denote the spike rates of presynaptic and postsynaptic neurons, respectively, with ⟨•⟩ denotes the lifetime average. After each stimulus presentation of 100 ms, we calculate the network's neuronal instantaneous spike rates using exponential moving averages (EMAs), which aggregate spikes over time to reflect recent activity (see section A.1). Lifetime averages, also computed as EMAs, are crucial for homeostatic stability, helping to modulate neuronal properties or synaptic strengths for consistent activity. See the appendix for the hyperparameters.Our SESNN model reflects experimental findings [56, 57] by representing V1 pyramidal neurons with weaker synaptic strengths, essential for preventing over-excitation and maintaining neural balance. We apply the HO rule [42] to E-E connections with a normalization factor to keep synaptic weights between 0 and 1, while stronger lateral E-I connections under the CM rule lack this normalization [44, 43] . Post-training synaptic strengths are depicted in Fig. 5d . Stabilizing neural network training requires careful learning rate adjustment. A slower rate for E-E connections compared to others is crucial to prevent E-neuron over-excitation, aligning with empirical data [56, 57, 58] .The HO and CM rules facilitate LTP and LTD mechanisms, common in rate learning rules that do not require precise spike timing. We selected these rules for their ease of tuning and ability to stabilize recurrent excitation. 
related works: 
 Functional roles of pinwheel structure can be revealed by SESNN model The classical selforganizing map model [8] and other computational approaches like on-off models [6, 7, 9, 59 ] and related ANNs [10, 60, 61] lack the dynamic and temporal fidelity needed to realistically simulate the emergence of pinwheel structures in the visual cortex. To address these shortcomings, we propose the novel SESNN model, integrating retinotopy data [48, 49, 51, 52] , detailed morphological data [62, 63, 64] , and CMF [50, 51, 53] to enhance biological fidelity. The SESNN model effectively simulates macaque cortical organization and pinwheel development within OPMs (Fig. 5f ). Furthermore, our investigations reveal that the degree of overlap-reflecting similar feedforward inputs from identical RGCs to neighboring neurons-positively correlates with the retino-cortical mapping ratio [6] , aiding in distinguishing between different V1 organizational patterns.PCs and IODs in neural processing hierarchies Our findings show that PCs and IODs exhibit distinct neural activity waves, leading to varied responses to contour complexity from spatial-temporal dynamics: PCs react first to complex contours, having more multi-orientation selective neurons (Fig. 4e , x = 1) before activity spreads to IODs, which process simpler edges (Fig. 4d and e ). PCs display a stronger correlation with contour saliency, indicating a heightened role in processing visual stimuli over IODs (Figs. 3b and 4b ). In rodents with salt-and-pepper organizations, contour saliency is less pronounced (Figs. 3c and 4c ). While PCs are thought to indicate higher-order processing due to delayed response [13, 28] , this is likely due to the nature of the stimuli. Studies reveal IODs show cross-orientation suppression under complex stimuli [12] , unlike PCs with broader tuning. The SESNN model illustrates a preference for complex stimuli in PCs and simple stimuli in IODs, with activity cascading from PCs to IODs upon encountering complex contours (Fig. 4d and e ).The SESNN model reveals PCs have broader orientation tuning and less selectivity for complex contours, unlike IODs, which show sharper tuning and crossorientation suppression, preferring simpler edges (x = 1 in Fig. 4e ) [12, 13, 19, 58, 65, 66, 67] . PCs' excitation leads to reduced cross-orientation suppression. With binary input, PCs correlate more positively with contour complexity than IODs (Figs. 3b and 4b ), making them more salient in processing visual stimuli. This differs from rodents with salt-and-pepper organizations that lack distinct contour complexity saliency (Figs. 3c and 4c ). Prior studies [12, 13, 28] suggest PCs have delayed response latency, indicative of higher-order processing. This arises from using drifting grating stimuli that activate IODs more readily. Koch et al. [12] note that IODs show cross-orientation suppression under complex stimuli, narrowing their tuning, unlike PCs. However, these studies omit temporal neural data within pinwheel structures. The SESNN model supports physiological findings that IODs and PCs favor single and complex orientation stimuli, respectively. 
conclusion and limitations: 
 The advantages of pinwheel structures in visual representation and encoding are not fully understood.To address this, we develop a two-dimensional self-evolving spiking neural network (SESNN) that incorporates Hebbian-like plasticity and empirical morphological data. This model evolves to function as localized, bandpass filters, enhancing its responsiveness to a range of orientations and complex spatial textures in natural images. Our findings reveal that neurons within pinwheel structures respond more effectively to these textures, with stronger and quicker reactions than those in salt-and-pepper configurations. Specifically, PCs act as first-order stage processors with heightened sensitivity and reduced response latency to intricate contours, while IODs function as second-stage processors, refining edge representation for greater clarity. This advanced processing capability of pinwheel structures, particularly in detecting spatial contour saliency, not only deepens our understanding of visual processing in higher mammals but may also inform new strategies for visual saliency algorithms in computational models.Using sliding windows, local entropy assesses variation and complexity in spatial distributions by capturing local intensity changes, indirectly reflecting geometric complexity through edges, corners, and patterns. Since this method cannot directly measure geometric shapes, we verify the use of the Ramer-Douglas-Peucker algorithm to approximate and directly measure geometric structures (refer to section A.5) [68] . This algorithm simplifies shape contours by reducing vertices while preserving the overall form. The resulting polygon will allow us to calculate the distribution of edge lengths and angles, with geometric entropy defined as the sum of these entropy values. In future studies, we will utilize the Ramer-Douglas-Peucker algorithm to enhance our geometric analysis by identifying and measuring the complexity of specific structural features, such as junctions, sharp corners, and textures, which are essential in complex visual scenes. 
broader impacts: 
 Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?Answer: [NA] Justification: Our work mainly focus on explaning the biological mechanisms underline pinwheel structure in the visual system, thus has no societal impact.Guidelines:• The answer NA means that there is no societal impact of the work performed.• If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact. • Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.• The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster. • The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology. • If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML). 
safeguards: 
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 Appendix: We compute the network's neuronal instantaneous spike rates as exponential moving averages (EMAs), which accumulate spikes over time (see Eq. 9). EMAs are utilized to track recent neuronal activity levels. Concurrently, lifetime average values are also calculated using EMAs, which are crucial for maintaining homeostatic stability. This method helps stabilize the neural network by adjusting neuronal properties or synaptic strengths to sustain consistent activity levels over time.EQUATIONwhere ζ = 1 -e -1 10 , indicating that the 10 ms is a temporal window of the moving average weighted with exponential decay. The initialization of x j is 0. The exponential moving average is calculated dynamically and updated along with synaptic weights.EQUATIONwhere ξ = 1 -e -1 . It is dynamically updated to ensure the sum of the weights remains constant over time.A.2 Detailed parameters and connectivity settings for SESNN Detailed neural dynamics: The FF connection, labeled FF(image→E) ij, links pixel X j of the whitened image patch to E-neuron i. W (K * →K) ij signifies the synaptic weight from neuron j of neuron class K * to neuron i of neuron class K, with its sign determined by the connection type, described as β(K * →K) ij(the neuron receives excitatory connections, set as +1; conversely, the neuron receives inhibitory connections, the sign is set as -1). z (K * ) j (t) indicates the spike output of neuron j at time t .Upon reaching the spike threshold θ (initialized as 2), a spike is emitted, z (K * ) j (t) is set to 1, then the membrane potential is reset to 0 mV, remaining so until the refractory period (3 ms) concludes. Within V1, homeostatic plasticity [33, 54] ensures neural activity stability by dynamically adjusting the firing threshold θ. This adjustment is based on the deviation of the current firing rate p i (t) from the target rates p (K) i (p (E) = 2, p (I) = 4), as outlined in Eq. 6 [54] . We assign τ (E) = 10 ms for E-neurons and τ (I) = 5 ms for I-neurons. To enhance computational efficiency, we set the time step to 1 ms.Hyperparameters: For the synaptic plasticity,learning rates are η FF = 0.2 (image to E-neuron), η EE = 0.01 (E-to E-neuron), η EI = 0.7 (I-to E-neuron), η II = 1.5 (I-to I-neuron), and η IE = 0.7 (Eto I-neuron), while the neural connectivity parameters are α max,E = 1.0 (E-max weight), α max,I = 0.5 (I-max weight), σ EE = 3.5 (E-E coupling range), σ EI = 2.9 (E-I coupling range), σ IE = 2.6 (I-E coupling range), and σ II = 2.1 (I-I coupling range).Neural connectivity within 2D cortical area: E-and I-neurons are arranged symmetrically on a two-dimensional lattice, as illustrated in Fig. 5b . PBCs are employed to mimic the large number of neurons in the actual V1 cortical surface. Specifically, neurons at the boundary are connected to neurons at corresponding symmetric positions on the opposite boundary. The initial connection weights between neurons are modeled by a Gaussian function of their distance (see Fig. 5c ), which can be expressed as:EQUATION)In this equation, d(i, j) represents the Euclidean distance from neuron i to neuron j in a grid, α determines the maximum connection weight, which is set to α EE = 1, α EI = 1, α IE = 0.5, α II = 0.5, and σ governs the rate at which the weight decays with distance. The synaptic types predominantly determine the parameters for this connection weight distribution function. To accurately replicate the neuronal architecture of V1 in macaques. The connectivity radiuses, denoted by σ, are set to σ EE = 3.5, σ EI = 2.9, σ IE = 2.6, σ II = 2.1. These values are based on anatomical data indicating that the axon length scales of E-and I-neurons are approximately 200 µm and 100 µm, respectively, while the dendrite length scales are around 150 µm for E-neurons and 75 µm for I-neurons in the V1 [62, 63, 64] . We prune any connection strengths below a threshold of 0.01 to maintain computational efficiency and biological plausibility.Neural connection dataThe experimental subjects include six adult cats with unknown genders, with data sourced from research by Armen Stepanyants et al. [63] ; and eight macaques, aged 5-11 years, including six males and two females, with data sourced from research by Joseph Amatrudo et al. [64] .The subjects are rats aged 14-16 days, with unknown gender and quantity, with data sourced from research by Holmgren et al. [56] ; transgenic mice, with unknown quantity and gender, with data sourced from research by Hofer et al. [57] .Receptive field data: V1 neuron counts for macaques, cats, tree shrews, ferrets, mice, rats, and gray squirrels respectively come from Cortical magnification factor data for macaques, cats, tree shrews, ferrets, mice, rats, and gray squirrels are sourced from Tehovnik et al. [69] (subjects: 3 macaques, unknown gender and age), Veit et al. [50] (subjects: cats, unknown gender and age), Bosking et al. [46] (subjects: tree shrews, unknown gender and age), Rockland et al. [74] (subjects: 9 ferrets, female, unknown age), Beest et al. [53] (subjects: 28 mice, 11 males and 17 females, ages 2-14 months), Keller et al. [75] (subjects: male rats, age 3 months), and Hall et al. [72] (subjects: 17 gray squirrels, unknown gender and age).Additionally, the anatomical data concerning inter-ocular distances are obtained from Najafian et al. [7] .A.4 Unveiling species-specific factors distinguishing pinwheels and salt-and-peppers A.4.1 Anatomical data suggests RFs density underlying V1 organizations We analyzed anatomical data from seven species, including primates (e.g., macaques) and nonprimates (e.g., mice, rats, cats, tree shrews, gray squirrels, and ferrets), as detailed in Table 3 . We first find that V1 RFD (ρ RF ) acts as a linear classifier (y = 4.42 × 10 4 x), effectively distinguishing species with pinwheel structures from those with salt-and-pepper organizations. In this classifier, species like macaques, cats, tree shrews, and ferrets, which have higher RFD, are associated with pinwheel structures (light red area in Fig. 6 ) and exceed the classification threshold. In contrast, species with lower RFD, such as mice, rats, and gray squirrels, are linked to salt-and-pepper organizations (light blue area in Fig. 6 ). Thus, V1 RFD serves as a predictive metric for V1 organizational patterns across species. The ρ RF is calculated as follows:EQUATION)where n denotes the total number of neurons in V1, s r indicates the retinal surface area, s V1 corresponds to the V1 2D surface area, and ρ V1 signifies the neuronal density within V1. The variable ε quantifies the degree of visual input overlap among adjacent neurons, and s RF represents the RF size. Concerning the Eq. 12 and anatomical data (Table 3 ), the two main factors influencing RFD ρ RF are the overlap ε of visual inputs between adjacent RFs and V1 neuronal density ρ V 1 . We discuss the overlap in the main text. Neuronal density is discussed in the following sections. The anatomical data in Table 3d for seven species show variability in V1 neuronal density (ρ V1 ), which influences inter-neuronal spacing and connection strength. We explore how V1 cortical orientation patterns form by adjusting the lateral connection range, impacting axon reach among Eand I-neurons, as depicted in Fig. 7 . We modulate axonal arborization through parameter σ to adjust the connection range, allowing us to simulate neuronal connections in areas with varying densities. This setup enables the SESNN model to predict changes in cortical patterns (Fig. 7 ). Our observations indicate that increasing axon lengths, thereby extending the connection range, enlarges hypercolumn sizes within pinwheel structures (Fig. 7d ), reduces the overall number of pinwheels (Fig. 7b ), and increases NNPD (Fig. 7c ). These findings underscore the critical role of neural synaptic connection range in organizing orientation maps.To address the limitations of using local pixel entropy (LPE) with sliding windows alone to capture complex geometric properties, we conduct a new analysis comparing the maximum values of LPE with local geometrical entropy (LGE) across various shapes. These shapes include lines, angles, and junctions (L-, T-, X-junctions), as well as jagged edges. Both LPE and LGE values were normalized to the range [0,1] for consistency.Let P = {v 1 , v 2 , . . . , v n } be a polygon with vertices v i = (x i , y i ), where i = 1, 2, . . . , n. The edges of the polygon are the line segments between consecutive vertices, denoted as e i = ∥v i+1 -v i ∥, where ∥ • ∥ represents the Euclidean distance. The angle θ i between two consecutive edges e i and e i+1 can be computed using the dot product:EQUATIONWith the set of edge lengths {e 1 , e 2 , . . . , e n } and angles {θ 1 , θ 2 , . . . , θ n }, we calculate the entropy for both distributions. The entropy H of a discrete distribution X with probability mass function p(x) is given by:EQUATIONFor the edge lengths and angles, the probability mass function is estimated by normalizing the frequency of occurrence of each unique edge length and angle in the polygon:EQUATIONEQUATIONTo enhance the sensitivity of geometrical entropy to structural complexity, particularly in differentiating shapes that have similar edge lengths and angles but different structural arrangements, we introduce a scaling factor based on the logarithm of the number of vertices n. The defined geometrical entropy (GE) with the scaling factor is thus defined as:EQUATIONThis modification allows GE to capture additional complexity arising from intersections and the global arrangement of vertices, providing a more comprehensive assessment of the shape's structural intricacies.Our results, summarized in Table 4 , show that while LPE can reflect the complexity of certain patterns, it does not fully capture the geometric variations seen in more intricate shapes. For instance, the LPE values for line structures remain relatively low compared to those for jagged edges, which have the highest LPE and LGE values due to their high structural complexity. This comparison highlights the added value of incorporating LGE to better characterize local geometric structures, providing a more nuanced measure of complexity that includes both intensity distribution and spatial organization. Understanding the tuning of PCs in V1 to edges, corners, and junctions is essential. In Fig. 4e , we show that PCs exhibit broader orientation tuning curves than IODs when using star-like patterns as stimuli, potentially enabling the detection of T-junctions and corners, as demonstrated by Ming Li et al. [13] and Erin Koch et al. [12] . We further examine the distribution of PCs' tuning curves using gratings as inputs, specifically analyzing acute angles formed by the primary and secondary peaks (Fig. 8a ). This analysis reveals that PCs are more frequently associated with larger acute angles, closer to orthogonal (90°), suggesting a preference for orthogonal junctions. However, this result does not differentiate between L-and T-junctions based solely on angle. We propose that such high-order feature extraction be deferred to higher visual cortices, like V2 and V4, which are involved in texture detection, as noted by Tianye Wang et al. [78] and Anna W. Roe et al. [79] .We present a mechanism of multiple orientation tuning that is essential for processing complexity. Our analysis of PCs' preferred acute angles (Fig. 8a ) suggests that their broad tuning enables the detection of complex junctions, such as T-and L-junctions, likely due to variations in local connectivity within and between IODs.To test this, we conduct an ablation study by disrupting local connectivity and shuffling the spatial arrangement of orientation-tuned RFs in the pinwheel orientation map, while keeping other properties constant (Fig. 8b ). The control group (red) maintains higher complexity over time, whereas shuffling connections-especially both feedforward and lateral-resulted in a decline in complexity. This highlights the importance of structured connectivity in preserving complex neural responses in V1 and supports the conclusion that structured connectivity underlies enhanced saliency detection by pinwheels.A.8 Computing infrastructure The simulations and analyses in this study are performed on a high-performance computing infrastructure to ensure efficient processing of large datasets and complex models. The system is powered by an Intel® Xeon® Gold 6348 CPU running at 2.60 GHz and an NVIDIA A100 GPU, providing robust computational power for intensive tasks. The system includes 512 GB of memory, which supports handling memory-intensive applications and large-scale simulations. The operating system used is Ubuntu 20.04.6 LTS, known for its stability and compatibility with software. The simulations are conducted using MATLAB R2023a and Python 3.9, both of which are widely used in scientific computing and neural modeling, enabling effective implementation and analysis of the models presented in this study.Answer Guidelines:• The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.• If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics. • The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction). 


Decision: The paper proposes a novel self-evolving spiking neural network (SESNN) that develops either salt-and-pepper or pinwheel organization reminiscent of primary visual cortex. The author propose that pinwheel centers respond more effectively to complex patterns than salt-and-pepper populations. The reviewers recognize the work as novel and innovative, especially the fact that the model replicates the pinwheel structure of monkeys and that it illustrates a functional difference between pinwheel and salt-and-pepper organization. However, the reviewers also criticized the entropy-based measure used for evaluation and this criticism could not be fully resolved during the author-reviewer discussion. Overall, the positives outweigh the negatives and I suggest the paper for acceptance. However, I ask the authors to tune down their claims that rest on the contested metric and add a discussion of its limitations to the final version of the paper.

Reviews: review: paper_summary: paper_summary: This paper uses a self-evolving spiking neural network model to investigate why some visual systems develop pinwheel structures while others have salt-and-pepper organization of orientation tuning. The simulation shows that the organization depends on the amount of receptive field overlap between neighbouring neurons. Experiments on the trained network show that pinwheel centers respond more efficiently to complex edge structure in images than salt-and-pepper neurons.
strength_weakness: strength_weakness: - The experiments seem to use binary edge map images only, if I’ve understood them correctly. No visual system evolved to process these kinds of binary images, so it’s unclear what conclusions we should draw from these experiments. It would be much more useful to show experimental results for real images.
- Entropy in a local region of the edge map is used as a proxy for geometric complexity, but this just measures the proportion of pixels in a local part of the edge map which are white vs. black.  A little patch of random noise (50% white and 50% black pixels) would have maximum entropy but no geometric information. From the experiments, it’s hard to tell what aspect of the high-entropy regions the pinwheel centers are more responsive to – would they respond highly to random noise? Or are they responding to edge intersections? Corners? Textures? Given that the input to the model is a binary edge map, there are many ways the edges could be characterized to better understand the tuning.
questions: questions: Do the pinwheel centers show faster responses to regions of the image that are ground-truth boundary vs. other regions of the image? (If a BSDS image is input to the model, do the pinwheel centers respond differently to the pixels which are 1 in the edge map than those which are 0?) This might show an advantage for these cells in boundary detection or figure-ground analysis.

Are pinwheel centers tuned for particular types of edge structure (for example, giving a different response for T-junctions than other types of junctions)?
limitations: limitations: Not applicable
paper_summary: paper_summary: The authors present a comprehensive model of the primary visual cortex adapted for various mammalian species, demonstrating its ability to reproduce orientation maps and compatibility with experimental data across different factors such as neuron density or RF overlap. Importantly, they provide evidence in their model that pinwheel centers act as saliency detectors.
strength_weakness: strength_weakness: Given the link with neuroscience, more links with behavioural data would be beneficial: are animals without PCs less efficient in detecting saliency? Is the density of PCs compatible with the "resolution" of the saliency map? In general, this feature of PCs as detecting features containing multiple orientations should be more broadly tested, for instance by using existing results on textures with different orientation bandwidths. such work could help better undertand the underlying principles which give rise to that particular feature.  (The link to https://github.com/HenryGithub1?tab=following (and to its followers) may reveal the authorship, which may be a problem for NeurIPS. Use anonymous links instead. There is a strong overlap with submission 16614, most certainly from the same authors.)
questions: questions: In the computation of entropy, why use luminance values instead of that of the (whitened) images used during the training? Also the numbers used in the paper and synthesized in table 2 are given following previous papers, bt it is not said at what eccentricity they are computed. Could you please provide this information? How do you justify that the emergence is uniform while retinotopic space is not (certainly thanks to the fact that RF size is function of eccentricity, as well as the RF density, but this is not discussed in the paper as far as I could read)?
limitations: limitations: The main result is figure 3, and to better the mechanisms leading to this result, an ablation study would be an asset to the paper. Minor: in Figure 4c, your error bar escapes the imit of valid values (normalized entropy higher than 1). Use quantile regression to estimate the 95% confidence interval?
paper_summary: paper_summary: This paper introduced a novel spiking neural network (SNN) to investigate the functional roles of pinwheel structures in the primary visual cortex of higher mammals and primates. By adjusting a visual RF overlapping parameter, their model can produce the salt-and-pepper and pinwheel organizations observed in lower (rodent) and higher (macaque and cat) mammals. Their results suggest neurons in pinwheel centers are more responsive towards complex geometry and spatial textures than those in the iso-orientation domains.
strength_weakness: strength_weakness: - The paper lacks a comparison with previous work or verification with experimental data. For instance, only one metric from the baseline model is reported in Table 1. Please see points 2 and 3 in “Questions”. 
- The SESNN model shows a rippling effect/pattern (Figure 2) and a heightened response with high contour complexity (Figure 3) in pinwheel centers, it is unclear whether or not this is indeed the case in the visual cortex without validation/comparison with experimental data.
questions: questions: - One of the main claims is the differences in neuronal response time with respect to the complexity of the visual scene between salt-and-pepper and pinwheel organizations. Based on Figure 3c, other than the initial time point (1 ms), the two structures seem to have similar response ranges and a downward trend as latency increases. Can the author comment on that?
- In Table 1, the authors compare a baseline model and the SESNN model against the experimental data obtained from macaques. Can the authors clarify why the NNPD and hypercolumn size are “N/A” for the baseline model? Without a proper baseline, it is difficult to evaluate how well the proposed model is performing in this specific task. 
  - Moreover, can the authors clarify why the pinwheel density is omitted as a metric in Section 2.1, though it is included in Table 1?
  - Finally, I believe a brief description of the baseline model, and how it differs from SESNN, should be added to the main text.
- I believe the analysis would be more complete if the authors could add the pinwheel data from rodents and cats in Table 1 to compare the performance of the proposed model against the baseline.
limitations: limitations: I strongly suggest the authors discuss the limitations of this work and potential future research direction in the paper.
paper_summary: paper_summary: This work aims to explain the origin and functional benefits of pinwheel structures in V1 compared to salt-and-pepper configurations. They use a two-dimensional self-evolving spiking neural network (SESNN) model with Hebbian-like plasticity and empirical morphological data to simulate the evolution from salt-and-pepper clusters to pinwheel structures. Their findings reveal that neurons at pinwheel centers exhibit heightened sensitivity and quicker responses to complex spatial textures in natural images, acting as primary processors for intricate contours, while adjacent iso-orientation domains refine edge representations.
strength_weakness: strength_weakness: - More analyses to support the main claim in needed (see limitations)
- The use of different learning rules for E>E vs E>I is justified computationally (for stability) but not discussed in terms of bio-plausibility.
questions: questions: - In Related works line 228, when referring to ANN models explaining pinwheels, the lack of temporal processing in these models were mentioned but their ability in connecting these maps to the overall functionality of the network is not discussed. Since the main claim of this paper is about functionality of PCs, I wonder what authors think about the functional role previous work attributed to PCs.
limitations: limitations: - In the setup, It seems that the difference between receptive filed sizes of monkeys, cats and mice was ignored. In mice receptive fields for V1 are generally much larger than monkeys (>20x, see ref below for example). 

- The claim about functionality of pinwheels requires more support: the question whether the differences in responses of PCs and IODs in terms of response latency, etc have a functional role could be verified by using SESNN as a frontend for a simple DNN performing object recognition, motion detection, etc. At this point, it remains a different in response properties.

- Code is not shared at this version.

Van den Bergh G, Zhang B, Arckens L, Chino YM. Receptive-field properties of V1 and V2 neurons in mice and macaque monkeys. J Comp Neurol. 2010 Jun 1;518(11):2051-70. doi: 10.1002/cne.22321. PMID: 20394058; PMCID: PMC2881339.

Comments: title: Response to rebuttal
Response to Official Comment 2
Response to Official Comment 1
Response to rebuttal
comment: We are still awaiting feedback on our rebuttal from Reviewer TFr2. We have carefully addressed all the concerns and sincerely hope to receive their insights soon, as it would greatly assist us in improving our work. If our clarifications address your concerns satisfactorily, we would greatly appreciate it if you could consider raising the score.
Thank you for the clarification. I apologize for any misunderstanding regarding the submission guidelines. Our intention was to help the reviewers better understand our work.
**Related Works**

**Functional Roles of Pinwheel Structures Revealed by the SESNN Model**: Traditional models, such as the self-organizing map (Kohonen, 1982) and computational approaches like on-off models (Miller, 1994; Jang et al., 2020; Song et al., 2021; Najafian et al., 2022) and related ANNs (Margalit et al., 2023; Chizhov and Zaitsev, 2021; Lufkin et al., 2022), lack the dynamic and temporal fidelity needed to realistically simulate the emergence of pinwheel structures in the visual cortex. To address these limitations, we propose the SESNN model, integrating retinotopy data (Srinivasan et al., 2015; Scholl et al., 2013; Tehovnik et al., 2007; Niell and Stryker, 2008), detailed morphological data (Tao et al., 2004; Stepanyants et al., 2009; Amatrudo et al., 2012), and CMF (Veit et al., 2014; Tehovnik et al., 2007; Van Beest et al., 2021) to enhance biological fidelity. The proposed self-evolving spiking neural network (SESNN) model effectively simulates macaque cortical organization and pinwheel development within the orientation preference map (OPM). Additionally, our findings reveal that overlap degree, reflecting similar feedforward inputs from identical RGCs to neighboring neurons, positively correlates with the retino-cortical mapping ratio (Jang et al., 2020), distinguishing different V1 organizational patterns.

**PCs and IODs in Neural Processing Hierarchies**: Our results show that pinwheel centers (PCs) and iso-orientation domains (IODs) exhibit distinct neural activity waves, leading to varied responses to contour complexity in spatial-temporal dynamics. PCs, with broader multi-orientation selectivity, respond first to complex contours, followed by IODs, which process simpler edges. PCs correlate strongly with contour saliency, indicating a more prominent role in visual processing than IODs. In contrast, rodents with salt-and-pepper organizations exhibit less pronounced contour saliency. While PCs are associated with higher-order processing due to delayed responses (Li et al., 2019; Song et al., 2020), this delay is likely stimulus-dependent. Previous studies show IODs experience cross-orientation suppression under complex stimuli (Koch et al., 2016), whereas PCs, with their broader tuning, are less affected.

**PCs as Geometric Saliency Detectors**: The SESNN model reveals that PCs have broader orientation tuning and less selectivity for complex contours compared to IODs, which exhibit sharper tuning and cross-orientation suppression, favoring simpler edges (Koch et al., 2016; Li et al., 2019; Ferster and Miller, 1996; Sato et al., 2016; Nauhaus et al., 2008; Blakemore and Tobin, 1972; Bonds, 1989). PCs' excitation reduces cross-orientation suppression, making them more responsive to contour complexity, especially with binary input. This contrasts with rodents having salt-and-pepper organizations, where distinct contour complexity saliency is lacking. Prior studies (Koch et al., 2016; Li et al., 2019; Song et al., 2020) indicate that PCs have delayed response latency, suggesting higher-order processing. This is attributed to the use of drifting grating stimuli that preferentially activate IODs. However, these studies omit temporal neural data within pinwheel structures. The SESNN model corroborates physiological findings that IODs and PCs prefer simple and complex orientation stimuli, respectively.
We are currently awaiting feedback on our rebuttal from Reviewer Boae. We have thoroughly addressed all the concerns and eagerly anticipate their insights, which would significantly contribute to enhancing our work.
Thank you for acknowledging our effort. As for quantifying the difference between downward trends in pinwheels and salt-and-pepper, we consider statistical significance between every other 2 ms for both pinwheel and salt-and-pepper structures to show the downward trend is only significant among pinwheels. (Since recorded neurons are not guaranteed to be in the same pinwheel, the statistical significance of a direct comparison of slopes is unavailable.):
| Latency-Gap(ms) | p for Pinwheel | p for Salt & Pepper|
|-------------|---|---|
|1-3  |**0.00002**|0.25352|
|3-5  |**0.01208**|0.73593|
|5-6  |0.64746|0.16130|
|7-9  |0.95521|0.99757|

The statistical significance of pinwheels vs salt-and-pepper at all time points:
| Latency(ms) | 1 | 2 | 3 | 4 | 5 | 6| 7| 8| 9| 10|
|-------------|---|---|---|---|---|---|---|--|--|---|
| p (Pinwheel vs Salt & Pepper)    |**0.0001**|**0.0180**|**0.0034**|**0.0173**|0.2146|0.1233|**0.0490**|**0.0004**|**0.0274**|0.5049|
I thank the authors for their thorough responses and they have addressed most of my main concerns. I have therefore updated the score. 

As to "Neuronal Response Time to Complexity of Visual Scene”, it is not visually obvious that the response time downward trend is stronger in the pinwheel structure. I suggest the authors quantify the downward trends in Figure 3c between pinwheels vs salt-and-pepper, and show their statistical significance at all time points.
Thank you for your interest in our work. We appreciate you taking the time to review it.
**Table 2: Standard deviation values of local pixel entropy and directly calculated global GE can both identify noise from various shapes (lines, angles, L-, T-, X- junctions, and jagged edges).** 

|**Various shapes** |**$\sigma$ of local pixel entropy** | **Global geometrical entropy (bits)** |
|------------------|----------------------------|------------------------------------|
|White Noise       |4.34e-04                     |NaN                                 |
|line 1            |0.13                         |  0                                 |
|line 2            |0.13                         | 0                                  |
|angle 1           |0.17                         | 1                                  |
|angle 2           |0.16                         | 1                                  |
|angle 3           |0.17                         | 1                                  |
| L-junction       |0.13                         | 1                                  |
| T-junction       |0.14                         | 2.37                               |
| X-junction       |0.16                         | 2.73                               |
| jagged edges     |0.21                         | 4.65                               |

We appreciate the reviewer's careful and instructive comments and plan to incorporate GE in revisions and future work. This will enhance the robustness of our analysis and clarify how pinwheel centers can sensitively respond to the complexity of natural images.

**Model's response to binary edge maps**: We used binary edge maps to isolate geometric complexity by eliminating luminance and contrast. This allowed us to focus on detecting complex features inherent in natural scenes. When images are binarized, texture complexity often results in variations in local geometric edges and shapes. These variations can be captured by the measure of local pixel entropy with sliding window, which reflects the local intensity distribution's complexity.

**The natural images response pattern for salt-and-peppers**: Due to the restrictions on uploading figures, we have instead provided the data in a tabular format (**table 3**). The latency for salt-and-pepper neurons on boundaries is significantly lower than that for neurons in other areas (P < 0.0001) (Corresponding figure is available in anonymous GitHub repository).

**Table 3: Latency for salt-and-pepper neurons on boundaries and neurons in other areas.** 

| **Salt-and-peppers' neuron type**    | **Mean latency (ms)** | **Standard deviation (ms)** |
|--------------------------------|---------------------|--------------------------------|
| Neurons on boundaries          | **2.832**          | 2.998                           |
| Neurons in other areas         | 5.137              | 4.626                           |

**Filtered images**: The BSDS 500 images we used are whitened, aligning with our model's training process (Olshausen and Field, 1996, *Nature*) (citation 36 in our paper). Whitening reduces nearby pixel correlation by down-weighting low-frequency content, which typically dominates and causes correlations. This process also enhances important image features, such as edges and contours, to preserve structural details. Additionally, whitening attenuates high frequencies, which often correspond to noise, thereby reducing noise from being introduced into the image.

**PCs tuning for particular types of edge structure**: Figure 4e in our paper shows that PCs have broader orientation tuning curves than iso-orientation domains, which may enable them to detect T-junctions and corners, as demonstrated by Ming Li et al. (2019, *Sci.Adv.*) and Erin Koch et al. (2016,*Nat. Commun.*). We analyzed the acute angles formed by the primary and secondary peaks in the tuning curves (**Table 4**), revealing that PCs prefer larger acute angles, closer to orthogonal (90°), indicating a preference for orthogonal junctions. While this result doesn't distinguish between "L" and "T" junctions beyond their angle, we suggest that higher visual cortices like V2 and V4 handle such distinctions, as shown by Tianye Wang et al. (2024, *Nat. Commun.*) and Anna W. Roe et al. (2012, *Neuron*).

**Table 4: Probability distribution of preferred adjusted acute angles in pinwheel centers.** (Available in anonymous GitHub)

| Adjusted acute angle range (degrees) | Probability (%) |
|-----------------------|-----------------|
| 0 - 9                 | 0               |
| 9 - 18                | 0               |
| 18 - 27               | 1.30            |
| 27 - 36               | 0               |
| 36 - 45               | 3.90            |
| 45 - 54               | 6.49            |
| 54 - 63               | 5.19            |
| 63 - 72               | 14.29           |
| 72 - 81               | 22.08           |
| 81 - 90               | 46.75           |

**We would like to express our sincere gratitude for your time and effort in reviewing our paper.**
Thank you for your insightful comments regarding the distinction between geometric complexity and texture complexity. We appreciate the opportunity to clarify our approach and address your concerns:

**How to distinguish geometric complexity and texture**:

1. **Local pixel entropy with sliding windows (LPESW)**: Global pixel entropy, indeed, does not measure geometric complexity within a window. Instead, we used MATLAB's `entropyfilt` function, which calculates local entropy for each pixel's neighborhood using sliding windows. This LPESW method evaluates the variation and complexity of pixel spatial distributions. While it doesn’t directly measure geometric shapes, it reflects their complexity through local intensity variations. By sliding a small window across the binary images, the local entropy measure captures the spatial distribution of entropy within these windows. This spatial distribution inherently includes geometric information, allowing it to detect local changes in intensity that characterize edges, corners, and other complex patterns.
2. **Verification with additional experiments by LPESW**: To verify our paper's approach, we conducted new experiments using various shapes, including lines, angles, L-, T-, and X-junctions (geometric complexity), as well as jagged edges (texture). The maximum entropy values obtained were 0.52 (lines), 0.72 (angles), 0.73 (junctions), and 0.94 bits (jagged edges), respectively. These results confirm that our LPESW method is sensitive to the complexity of geometric structures. (Available in the previously mentioned anonymous GitHub repository.)
3. **Comparison with geometrical entropy (GE)**: To further validate LPESW, we first approximate the contours of the shape using the Ramer-Douglas-Peucker algorithm, which simplifies the contour by reducing the number of vertices while preserving the general shape. The resulting vertices form a polygon, which serves as the basis for calculating the distribution of edge lengths and angles. GE is then defined as the sum of entropy values from these two distributions. We have tested the same shapes (as mentioned in point 2) using the GE measure. The results showed a strong correlation with in max values with LPESW (**table 1**) when calcuated with siliding windows, confirming the validity of LPESW for our experiments. (Also available in the previously mentioned anonymous GitHub repository.)
4. **Recognition of white noise**:  While the global GE for the pattern returns a NaN directly for white noise without structures, we understand that pixel entropy cannot differentiate noise from geometric shapes with max values as the reviewer suggested. However, since white noise will lead to consistently large pixel entropy in all sliding windows, geometrical structures would exhibit much variable distribution of pixel entropy across windows. Thus, we can also differentiate with standard deviation ($\sigma$) of local pixel entropy when performing LPESW as in **table 2**.

**Table 1: Relationship between maximum values of local pixel entropy (both normalized to [0,1]) and local geometrical entropy ($r^{2} = 0.85$) for various shapes (lines, angles, L-, T-, X- junctions, and jagged edges).** 

|**Various shapes** |**Max local pixel entropy** | **Max local geometrical entropy** |
|-------------------|----------------------------|------------------------------------|
|line 1            |0.56                         | 0.43                               |
|line 2            |0.56                         | 0.43                               |
|angle 1           |0.81                         | 0.87                               |
|angle 2           |0.79                         | 0.86                               |
|angle 3           |0.77                         | 0.87                               |
| L-junction       |0.78                         | 0.74                               |
| T-junction       |0.78                         | 0.64                               |
| X-junction       |0.78                         | 0.84                               |
| jagged edges     |1.00                         | 1.00                               |

**(please see the next official comment)**
Thank you for clarifications. I raised the score.
Please note that you should not do this, as was explicitly pointed out by the instructions sent out yesterday:

> The deadline for submitting the rebuttal has now passed. The reviewers will now read the rebuttals you posted. When relevant, they will ask for clarification questions. **The discussion phase is meant to clarify these questions, rather than to provide further comments regarding the reviews.**

I will leave it to the reviewers to decide whether they want to take this comment into account or not.
We are now providing this additional information to ensure a comprehensive understanding of our study and to address your valuable feedback more thoroughly.

**About behavioral data**: Linking our model with behavioral data could further strengthen our findings. Currently, no studies show that animals (like rodents) without PCs are less efficient in detecting saliency. However, electrophysiological studies suggest that PCs have delayed response latency, indicative of higher-order processing [citations 6, 7, 21 in our paper]. This arises from using drifting grating stimuli that activate IODs more readily. Koch et al. [citation 6 in our paper] note that IODs exhibit cross-orientation suppression under complex stimuli, narrowing their tuning, unlike PCs. This study, however, omits temporal neural data within pinwheel structures. The SESNN model aligns with physiological findings, showing that IODs and PCs favor single and complex orientation stimuli, respectively.

**About the textures with different orientation bandwidths**: Understanding the tuning of PCs in V1 to various edges, corners, and junctions is crucial. In Fig. 4e of our paper, we show that PCs exhibit broader orientation tuning curves than iso-orientation domains, which may allow them to detect T-junctions and corners, as demonstrated by Ming Li et al. (2019, *Science Advances*) and Erin Koch et al. (2016, *Nature Communications*). Your insightful comment prompted us to further examine the distribution of PCs' tuning curves. We have analyzed the acute angles formed by the primary and secondary peaks in the orientation tuning curve (**Table 1**). **Table 1** shows that PCs are more frequently associated with larger acute angles (closer to orthogonal, 90°), indicating a preference for orthogonal junctions. However, this result does not distinguish between "L" and "T" junctions beyond their angle. We suggest deferring such high-order feature extraction to higher visual cortices like V2 and V4, which are involved in texture detection, as discussed by Tianye Wang et al. (2024, *Nature Communications*) and Anna W. Roe et al. (2012, *Neuron*).

**Table 1: Probability distribution of preferred adjusted acute angles in pinwheel centers.** (Corresponding figure is available in anonymous GitHub repository)

| Adjusted acute angle range (degrees) | Probability (%) |
|-----------------------|-----------------|
| 0 - 9                 | 0               |
| 9 - 18                | 0               |
| 18 - 27               | 1.30            |
| 27 - 36               | 0               |
| 36 - 45               | 3.90            |
| 45 - 54               | 6.49            |
| 54 - 63               | 5.19            |
| 63 - 72               | 14.29           |
| 72 - 81               | 22.08           |
| 81 - 90               | 46.75           |

**Ablation study**: In our paper (Fig. 4e), we present a mechanism of multiple orientation tuning that is crucial for processing complexity. Our experiment (**Table 1**) on PCs' preferred acute angles suggests that their broad tuning enables detection of complex junctions like T- and L-junctions, likely due to differences in local connectivity within and between iso-orientation domains.

We appreciate the reviewer's feedback and conducted an ablation study by disrupting local connectivity and shuffling the spatial positions of orientation-tuning receptive fields in the pinwheel orientation map, while keeping other properties constant (**Table 2**). This supports the conclusion that structured connectivity, as shown in Fig. 4e, underlies the enhanced saliency detection by pinwheels.

**Table 2: Ablation study on local connectivity and orientation-tuning receptive fields (available in anonymous Github). The table shows normalized entropy values (mean ± std).**

| Latency (ms) | 1| 2| 3| 4| 5| 6| 7|
|-|-|-|-|-|-|-|-|
|Pinwheels (control group) | 0.915 ± 0.049 | 0.950 ± 0.170 | 0.899 ± 0.259 | 0.468 ± 0.214 | 0.462 ± 0.202 | 0.491 ± 0.173 | 0.491 ± 0.235 |
| Interchange feedforward connection (FF)| 0.948 ± 0.070 | 0.897 ± 0.222 | 0.889 ± 0.226 | 0.558 ± 0.208 | 0.503 ± 0.159 | 0.486 ± 0.158 | 0.464 ± 0.253 |
|Shuffle lateral connections| 0.959 ± 0.288 | 0.916 ± 0.208 | 0.909 ± 0.262 | 0.500 ± 0.205 | 0.947 ± 0.270 | 0.874 ± 0.194 | 0.501 ± 0.264 |
|Interchange FF, shuffle lateral connections| 0.866 ± 0.221 | 0.904 ± 0.186 | 0.524 ± 0.181 | 0.907 ± 0.200 | 0.626 ± 0.229 | 0.492 ± 0.184 | 0.738 ± 0.268 |
|Shuffle all FF| 0.893 ± 0.073 | 0.937 ± 0.193 | 0.816 ± 0.093 | 0.783 ± 0.151 | 0.820 ± 0.029 | 0.905 ± 0.126 | 0.837 ± 0.184 |
|Shuffle all connections| 0.849 ± 0.101 | 0.875 ± 0.100 | 0.888 ± 0.136 | 0.821 ± 0.125 | 0.818 ± 0.119 | 0.699 ± 0.140 | 0.767 ± 0.098 |
I understand how the entropy measure is calculated, but it doesn't seem to distinguish between increasing geometric complexity (e.g., more highly-branching junctions) and increasing texture (e.g., more jagged-looking edges). The measure only cares about how many of the pixels in the window are white. The entropy of a straight line and a corner are the same according to this measure, even though the corner is (I assume?) more geometrically complex. It seems like in most cases, the highest-entropy parts of the image would be textured natural background structures like trees or mountains, while the more important foreground objects (like a person) may have lower entropy because they have smoother boundaries.

I also don't really understand what we can take away from the model's response to binary edge maps, given that the cells were trained on natural images. The analysis in Fig.S1(a) looks promising. What is the pattern for salt-and-pepper?

Also, just out of curiousity, why are the images filtered as shown in the supplemental? It looks like they have been bandpass filtered to remove low spatial frequency information.