Emergent Orientation Maps —— Mechanisms, Coding Efficiency and Robustness
Authors: Haixin Zhong
Haoyu Wang
Wei P Dai
Yuchao Huang
Mingyi Huang
Rubin Wang
Anna Wang Roe
yuguo yu
Venue: ICLR 2025 Spotlight (2025)
Keywords: Vision
Energy Efficient Coding
Neural Network
Sensory Coding
Spiking Mechanisms
Primary area: applications to neuroscience & cognitive science

Abstract: Extensive experimental studies have shown that in lower mammals, neuronal orientation preference in the primary visual cortex is organized in disordered "salt-and-pepper" organizations. In contrast, higher-order mammals display a continuous variation in orientation preference, forming pinwheel-like structures. Despite these observations, the spiking mechanisms underlying the emergence of these distinct topological structures and their functional roles in visual processing remain poorly understood. To address this, we developed a self-evolving spiking neural network model with Hebbian plasticity, trained using physiological parameters characteristic of rodents, cats, and primates, including retinotopy, neuronal morphology, and connectivity patterns. Our results identify critical factors, such as the degree of input visual field overlap, neuronal connection range, and the balance between localized connectivity and long-range competition, that determine the emergence of either salt-and-pepper or pinwheel-like topologies. Furthermore, we demonstrate that pinwheel structures exhibit lower wiring costs and enhanced sparse coding capabilities compared to salt-and-pepper organizations. They also maintain greater coding robustness against noise in naturalistic visual stimuli. These findings suggest that such topological structures confer significant computational advantages in visual processing and highlight their potential application in the design of brain-inspired deep learning networks and algorithms.

Published as a conference paper at ICLR 2025
EMERGENT ORIENTATION MAPS CODING EFFICIENCY AND ROBUSTNESS
MECHANISMS,
Haixin Zhongl.2
Haoyu Wang? ha Sudan
Wei P. Dai
hxzhonge fudan
fudan
Yuchao Huang 5.6
Mingyi Huang! myhtang20 fudan
Rubin Wang"
hyc2lemails..tinghh
rbwang@163. com
Anna Wang Roe
Yuguo Yu 12,3,44, yuyuguo8 fudan
annawangezju
1. Research Institute of Intelligent Complex Systems, Fudan University.
2. State Key Laboratory of Medical Neurobiology and MOE Frontiers Center for Brain Science,
Institutes of Brain Science, Fudan University.
Institute of Science and Technology for Brain-Inspired Intelligence, Fudan University.
4. Shanghai Artificial Intelligence I Laboratory.
5. IDG/McGovern Institute for Brain Research, School of Medicine, Tsinghua University.
6. Peingnu Joint Center for Life Sciences.
7. Institute for Cognitive Neurodynamics, East China University of Science and Technology. Integration, School of Brain Science and Brain Medicine, Key Laboratory of Biomedical Engineering of Ministry of Education, College of Biomedical Engineering and Instrument Science, Zhejiang University.
8. MOE Frontier Science Center for Brain Science and
* Corresponding author.
Note: Append .ddu. en to author email addresses.
ABSTRACT
Extensive experimental studies have shown that in lower mammals, neuronal orientation preference in the primary visual cortex is organized in disordered "salt- and-pepper" organizations. In contrast, higher-order mammals display continuous variation in orientation preference, forming pinw heel-eel-like structures. Despite these observations, the spiking mechanisms underlying the emergence of these distinct topological structures and their functional roles in visual processing remain poorly understood. To address this, we developed a self-evolving spiking neural network model with Hebbian plasticity, trained using physiological parameters characteristic of rodents, cats, and primates, including retinotopy, neuronal morphology, and connectivity patterns. Our results identify critical factors, such the degree of input visual field overlap, neuronal connection range, and the balance between localized connectivity and rong-range competition, that determine the emergence of either salt-and-pe -nt-per or pinwheel-like topologies. Furthermore, we demonstrate that pinwheel structures exhibit lower wiring costs and enhanced sparse coding capabilities compared to salt-and-pepper organizations. They also maintain greater coding robustness against noise in naturalistic visual stimuli. These findings suggest that such topological structures confer significant computational ad advattages in visual processing and highlight their potential application the design of brain- inspired deep learning networks and algorithms.
INTRODUCTION
In the primary visual cortex (V1) of 1962) 1974; Blaaddel & Salama 1986 Bonhoeffer & Grinvald 1991) revealed that neurons are or- ganized into distinct spatial clusters, forming "pinwheel" structures centered around singularities
mammals, seminal works (Hubel & Wiesel

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within the orientation preference map. This organization contrasts with that of some mammals, such as rodents, which exhibit "salt-and- and-perper" organizations, characterized by randomly interspersed neurons with varying orientation preferences, or "mini-columns" that show only weak spatial clus- tering, revealed by studies(Girnan et al., 1999) Ringach et al. 2016 Bargmann & Newsome (0144) Kaschube et al. 2010). Despite these structural differences, the spiking mechanisms that give rise these species-specific patterns, and their roles in visual processing, remain poorly understood.
Existing computational models, such as self-organizing maps (Kohonen 1982), on-off theoretical
models (iiller 1994, Jang et al., 2020; Song et al. 2021] Najañan et al. 2022), and various artificial
neural networks (Maggalit al.
2023 Chizhov & Graham 2021) have provided insights into the
generation of orientation preference maps through mathematical abstractions or firing rate models. However, these approaches often lack the biological realism of spiking neural networks (SNNs), which better capture the intricate temporal dynamics of real neuronal systems. Furthermore, many existing SNNs (Lufkin et al. 2022 Chizhov & Graham, 202T Srinivasa & Jiang (013) struggle to reproduce the emergent properties of orientation preference maps over time, as they often overlook key anatomical and physiological factors, such as shared visual inputs (Zhang et al. 2018 Jang el al.
2020; Najafian al. 2022), cortical size (Meng et al. 2012). localized neuronal densities (Weigand et al. 2017), and genetically encoded connectivity patterns Blauch et al. 2022).
In response to these challenges, research contributes the following:
We propose retina-V1 realistic self-evolving spiking neural network (SESNN) that in- tegrates comprehensive data on retinotopy, neuronal morphology, and connectivity from rats, cats and primates. This model is capable of dynamically simulating cortical structure formation across species using Hebbian-like learning rules and natural scene inputs. By leveraging visual field overlap and neuronal density, the SESNN accurately distinguishes between salt-and-pepper and pinwheel cortical patterns.
Pinwheel structures arise from local synaptic plasticity rules in spiking neural networks, where lateral connections are strengthened between neurons with similar stimulus responses and weakened between those with dissimilar responses. This selective connectivity forms iso-orientation domains with consistent orientation tuning. Near pinwheel centers, connec- tions become more homogeneous, while neurons maintain diverse orientation preferences and dynamic firing patterns. This balance between connectivity and neural diversity en- ables efficient, sparse coding of both simple and complex visual stimuli, optimizing visual information processing.
We further demonstrate that pinwheel organizations exhibit superior information-com ef- ficiency, increased coding sparsity, and greater robustness to noise compared to salt-and- pepper patterns. These properties enhance the detection and coding of varied orientation features, indicating that the spatial structure of pinwheels promotes highly efficient and reliable visual processing.
2 RETINA-VI TOPOLOGICAL SELF-EVOLVING SPIKING NEURAL NETWORK
Table SESNN pinwheels vs. macaque pinwheels. (mean + SD)
Metric
E-I baseline
SESNN (macaque)
Macaque
Pinwheel density (pinwheels/A33 Nearest-neighhor pinwheel distance (NNPD) (mm)
2.941
3.175 + 0.397
3.327
N/A
0.277 + 0.043
0.242
Hypercolumn size (mm)
N/A
0.839 + 0.054
0.760
2.1 STIMULUS TRAINING SETS
We use 20 original whitened images which reduce pairwise correlations among pixels, mimicking the processing of the lateral geniculate nucleus (Dan al. (1996); Zylberberg et al. (201))) 1)11] Paul D King et (2013)) To ensure these images encompassed orientation details, each image is rotated clockwise every 90-degree clockwise rotation and subsequent flip, producing variations per
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Salf-evalvin spiking neural network
Spotial saf-organization
plvals
Image net: $12+5522-180
overtpp
Rancomness
SESNN Pinwheels
Foodlowward connection, FP
Butch slzu: 100
100
connectioon W!
Manaque pinwhoels
modal (Baseline model)
FFRHOW &
panait roo
HHO
CM
Excitatory and inhibitory nourons
CM2
Parlodic boundary condition
groups
connection
Exnerron
Expiatory connestion
HD ruler: APP Wy
I-neuron
Inhiitory connection
CM nule: syl-o, Wi-h and Wi-l
Figure 1: Architecture of proposed SESNN model. a. The image set with 160 whitened natural images. b. Stimulus images are randomly selected from the image and presented to E neurons for 100 ms. c. The connections between excitatory (E) and inhibitory (I) neurons follow HO and CM rules. Recurrent connection among neurons. The spatial arrangement and periodic boundary conditions of E- and I- neurons. E- and I- neurons share the same spatial coordinates, and the boundaries are linked together according to the arrows shown in the diagram. Arrows of the same color represent identical connections. e. Spatial self-organization within the neuronal population post-training and the comparison among the SESNN model's orientation preference map, macaque's 's orientation preference map and SNN-based model (baseline) (Srinivasa & Jiang 2013). The color bars represent orientation preferences, and the scale bar corresponds to mm on the V1 cortical surface. This comparative analysis is detailed in Table
original image. This method generates a total of 160 images comprising our training dataset (refer to Fig. Tι).
In each trial, all E-neurons process visual input from 100 random patches, each presented for 100 ms(see Fig. [b.. Each E-neuron perceives a small patch 16x16. The visual fields the E-neurons modeled overlapping on the retina (see Fig, Ta). with the total visual input dimension being [166 z E] pixels on each side (n is neuron number, visual overlap metric e). To address the visual overlap among adjacent neurons, which adheres to periodic boundary conditions, the input design features a wraparound effect to ensure continuity and uniformity throughout the neural network.
2.2 NEURAL MODEL AND THE PLASTICITY RULES
The SESNN model includes 4900 E- and 1225 neurons, forming recurrent network with an E-I ratio of 4:1 (Alreja et al. (2022); Tao al. (0004): Pfeffer et (013)). E-neurons are stimulated by natural images and receive background noise from other brain areas, while I-neurons indirectly process natural image stimuli through their interactions with E-neurons. The spiking dynamics of the
network are simulated by the classical leaky Integrate-and-fff neurons (Stevens et al. (2013)). which
also feature refractory periods of 3 ms and adaptive firing thresholds (Földiák 1990)). ). The detailed description of the neuronal model can be found in the appendix (A.
The learning process utilizes two primary forms of Hebbian-like learning rules: Hebbian Oja's variant (HO) and Correlation Measuring (CM) (Oja (1922): Zylberberg et al. (2011); Paul D. King et al. (2013)). Consistent with experimental evidence Holmgren al. (2003); Hofer et al. (011)). pyramidal neurons exhibit weaker synaptic strengths compared to other types, crucial for avoiding over-excitation and ensuring neural balance. To accurately model these self-evolving dynamics within our SESNN, implement E-E connections governed by the HO rule (Oja (1982)), with normalization factor (the second term in Eq. [[) to regulate synaptic weight growth within to range. In contrast, lateral connections between excitatory and inhibitory neurons governed by the CM
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rule. which inherently allows for stronger synaptic weights without normalization (Zylberberg et al. (201T); Paul D. King et al. (2013), enabling these connections to better capture correlations between input patterns and neuronal responses. This distinction in learning rules is crucial for maintaining functional balance and enhancing representational efficiency within the network. Specifically, the HO rule learns the principal component of input patterns, allowing E-E connections to align with the largest variance in the input space. While this supports self celf-organized structure formation, such as oriented receptive fields (RFs), it can lead redundancy across excitatory and inhibitory neurons. By comparison, the CM rule captures correlations between input activity and a neuron's 's response, ensuring that predictable components of excitatory activity are expressed in inhibitory neurons. These components can then be suppressed through E-I-E loops, mechanism that supports predictive coding by minimizing redundancy and promoting sparse, efficient neural representations. Both HO and CM rules support long-term potentiation and depression, functioning similarly to other rate-based learning rules that do not require precise spike timing. We selected the HO and CM rules for their minimal tuning requirements and their ability to stabilize recurrent excitation.
Following every 100 ms of stimulus presentation, synaptic weight changes AW within the network computed:
HO : AW x yr - y b.
CM (K-KK) x yit - (wi) (az) (I+ W. w.(K-K))
where x, refer to the spike rate of presynaptic (j) and postsynaptic neurons (i) and K, K determine the type of neurons (E or I), the operator (-) indicates the lifetime average value. Following each 100 ms stimulus presentation, we compute the network's 's neuronal instantaneous spike rates as exponential moving averages, which accumulate spikes over time (see Eq. B). Exponential moving averages are utilized to track recent neuronal activity levels. Concurrently, lifetime average values are also calculated using exponential moving averages, which are crucial for maintaining homeostatic stability. This method helps stabilize the neural network by adjusting neuronal properties synaptic strengths to sustain consistent activity levels over time.
2,(() = (1 - (tx, + (t).
where Ç ç- 1-1 to indicating that the 10 ms is temporal window of the moving average weighted with exponential decay. The initialization of 2, is O. The exponential moving average is calculated dynamically and updated along with synaptic weights. 23 2(K*) (t) represents the spike output from neuron j at time f, and 0 otherwise.
(z (x,) + イ.
where { = =-1--1 It is dynamically updated to ensure the sum of the weights remains constant over time. Eq. captures the short-term firing rate of neurons by tracking recent activity through short time window, while Eq. X commputes the long-term average to monitor activity trends and maintain homeostatic stability, together ensuring the network's k adaptability and balanced dynamics over multiple timescales. I, represents the average firing rate of neuron j in one trial. The hyperparameters of the SESNN model are as follows: the learning rates are //F = 0.2 (image to E-neuron). THE = 0.01 (E- to E-neuron), Tha 0.7 (I- to E-neuron), ?/1 1.5 (I- I-neuron), and M/ 0.7 (E- to I-neuron), while the neural connectivity parameters are mmax, E = 1.0 (E- max weight) and (mmax. = 0.5 (I- max weight). These learning rate settings are crucial for stabilizing the training of the neural network. Specifically, setting a slower learning rate for E-E connections than for others helps prevent over- excitation among E-neurons. This approach is consistent with empirical findings (Hofer et al. 2011 Holmgren et al., 2003; Sato et al. 2016).
2.3 EXPERIMENTAL DATA-JUSTIFIED NEURAL CONNECTIVITY IN A 2D CORTICAL AREA
E- and I- neurons are arranged symmetrically on a two-dimensional lattice within SESNN, The initial connection weights between neurons modeled by Gaussian function of Fuclidean distance, which can be expressed as:
[d(;,))
WK-K'' (i.j) = ag x exp
(5)
20R² ?
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where d(i,j) represents the Euclidean distance from neuron to neuron in grid, a determines the maximum connection weight, which is set to OEE = 1, agi = 1, GIE = 0.5, au = 0.5, and 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 synaptic connection range, denoted by o, are set to
OEE = 3.5, σEI 2.9, OTE = 2.6, on 2.1. These values are based on anatomical data indicating that the axon lengths of E- and I-neurons are approximately 200 jam and 100 pm, respectively, while the dendrite lengths are around 150 ym for E-neurons and 75 jmm for I-neurons in the V1 (Tao et al. (2004): [Stepanyants et al. (2009); Amatrudo et al. (2012)). This careful selection of parameters and neuron distribution ensures our model's biological fidelity, particularly in representing synaptic connectivity and neuronal density. We discard any connection strengths below a threshold 0.01 to maintain computational efficiency and biological plausibiliity
Given that the majority of RFs are concentrated in the area centralis Van Essen et al. (1844); Born cet al. 2015), which projects to a disproportionately large area of VI, we focus our analysis solely on RF's within this central visual field (see Table 2). To control for variables, we use a constant RF size near the central vision, where eccentricity is close to zero.
The anatomical data for neural connectivity is based on the findings of (Tao et al. 2004 Stepanyants
ct al.
|009] Amatrudo al. 2012). The plasticity rule settings are based on the findings of (Holmgren
ef al.
2003 Hofer et al. 2011 1). For retinotopic data, we refer to (Srinivasan et al. 2015 | Tehovnik
& Slocum|
2007| (ccool et al. 2013: Veit et al., [2044] Engelmann & Peichl| 1996 Weigand et al.
2017
Law et al.
1988] Huberman et al., [2066] Niell & Stryker 2008 van Beest et al.) [02]] Foik
al.
2020). The anatomical data on orientation preference maps is sourced from (Naaaaaan al.
2022).
The details of the data usage and visual input overlap calculation can be found in the appendix
A.2
A.3
A.4
Table 2: Comparative anatomical data of the retina and V1 across species.
VI neurons density (neurons/mm²)
VI RF size
Peak CMF
Pinwheel numbers
Species
(deg)
(mm/deg)
(per mm?)
Macaque
243,000
0.2b
18.18
Cat
99.200
1.0
1.90
Mouse
86,600
4.0F
0.03
N/A
Srinivasan et al. (2015)
Tehovnik & Slocum (2007)
Scholl et al.
2013
Tusa et al. (1978)
Niell & Stryker (2008)
van Beest et al. 20021)
SESNN REVEALS THAT VISUAL OVERLAP BETWEEN NEURONS IS CRITICAL FOR V1 ORIENTATION MAP FORMATION
For quantitative analysis of orientation maps, we utilize metrics including pinwheel density (pinwheels/mm' ). pinwheel counts (Kaschube et al. 2010); Stevens et al. (2013) (Fig. (b)) NNPD (mm) (Müller et (2000); Kaschube et al. (2010) (Fig.) 2t., and hypercolumn size (mm) (Stevens) cet al. (2013)) (Fig. 1D.. Pinwheel density is defined as the number of pinwheels per VI surface area mm NNPD refers to the shortest Euclidean distance between pinwheel centers. Pinwheel centers are identifiable via 2D Fast Fourier Transform (Kaschube et al. (2010)). located where the real and imaginary components of the transform intersect at zero (Stevens al. (2013)). A hypercolumn is defined as region with periodicity measured by 2D fast Fourier transform. It's noteworthy that pinwheel density is excluded from the Fig. 2 analysis, because, regardless of hypercolumn size, A 31), conforming to topological constraints (Stevens et al. (2013); Kaschube et al. (2010)). This paper does not account for left- and right-eye dominance columns, the hypercolumn size defined the full 180 cycle of repeating column spacing (A) (mm).
the observed pinwheel density consistently approaches
Our SESNN model effectively generates various orientation selectivity patterns, from aalt-and-pepeer to pinwheel structures. We modify the overlap parameter E to range from 10 to 15 pixels (see Fig. 2 This visualization highlights how variations in the degree of overlap in visual stimuli can influence
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maps
Ovatlip
Ovatisp
Overlap
Number of pinereess
200
0.4
NND (mm))
overlop
10
12
13
15
16
0.1
13
15
Overlapping visual Nells
Visual input ovartap (pteab)
Overlop
Hypae aoperooaannnn size (mo))
Aratomiral dees conday
100
0.97
E-nouron stoual fields
20-00.00 0,077' $2.97
0.8
Moco
Macaque
Car
0.4
15
15
50
60
70
50
100
overioo (C) (p0kot))
Law ovarlap
High overap
Figure 2: The overlap of visual fields among adjacent neurons contributes to the formation of pinwheel structures. a. The diverse overlapping extent of visual fields neurons forms various iso-orientation domains of species in the orientation preference maps when perceiving natural images. The increasing overlap is consistent with the transition from lower to higher mammals (see the animal sketches). b. The correlation between visual overlap and pinwheel counts. c. The relationship between visual overlap and NNPD. d. The hypercolumn size varies with different visual overlaps. e. Comparing the SESNN model overlap with actual anatomical data overlap percentages in different species (mice, cats, and macaques). Color scheme: orientation preference. All data: mean # SD.
the density and arrangement of pinwheel structures in the visual cortex. Notably, setting the overlap
to E2 = 9 pixels results in the model producing salt-and-pepper organizations (see the first panel of Fig. [). rather than the anticipated pinwheel structures. Thus, it is evident that above the threshold of overlap (in our scenarios, Whresholde = 10 pixels), pinwheel structures emerge. Specifically, we find that a greater overlap (e.g., 22 15 pixels) fosters the formation of stronger local clusters, resulting in larger hypercolumn sizes (Fig. 21), a reduced number of pinwheels (Fig. 22). and extended NNPD (Fig. 2b. compared with the scenario of smaller overlap (e.g.. E 12 pixels). These findings suggest that visual input overlap critically determines the species-specific organization patterns of V1. The overlap degrees (E) in visual fields vary across species (including macaques, cats, rodents, etc.) (Fig. 2 to accommodate the size of iso-orientation domains in the visual cortex representation as observed by the study of (Najafian et al., 2022). When the overlap of visual stimuli is large, multiple neurons receive similar visual inputs. This causes these neurons form stronger local connections through Hebbian plasticity. Neurons with similar RFs tend to develop highly consistent local connections, and their orientation preferences gradually align, creating region of locally continuous orientation preferences. Additionally, Neuronal density is another factor that impacts the size of hypercolumns. Although this has been reported previously Philips et al. (017)), we have included additional details in Appendix A.3.2 for completeness.
3.1 THE EMERGENCE OF THE PINWHEEL STRUCTURE AND MECHANISM
We investigate the development of orientation preference maps in VI of animals eye-opening using the SESNN model, where the initial visual experience of natural stimuli are retinotopically mapped to V1 neurons (Fig. 3h) (Stevens al. (2013)). Our comparative analysis with empirical ferret data (teeenns et al. 2013); Chapman et al. 1996); Chapman & Stryker (1993)) shows high correlation 0.98, Pearson correlation) for orientation preference maps observed from postnatal day 33 (P33)
to P41 when they reach high stability (Fig. B3)) By normalizing the time scale, we are able to align the SESNN model's developmental trajectory with ferret empirical data. To quantitatively assess how the stability change over time, we use the orientation stability index (SI) (Chapman et al. (199)). calculated as follows:
SI = 1
4
(Fs
O2) mod T
(6)

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Orientation map development
Trid 11
The 50
Trie - 100
Trill 150
Trai 1600
Inal 3000 2700 plowhooksW
time
vs. Ferret map development
Information oose officency during development
species SE3NN
0.8
0.8
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0.4
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0.4
Winng tsed capacity
ICE
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08
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Time (Namrddood)
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100 meurrnn funed
E-neurons connect 53-degree 100 neuron
ICD neuran tuned 169 degrees
E-neurons connect to 100-degree neuron
53 degrees
8.
Meen
Convection drengh
0.8
Mean wher
0.
04
0.4
0.2
0.2
50
185
90
130
190
connections
Connadad E-neuroma oniertationn proferance
connections
Conneccad E-nsurona crierattion profenance
neuron tuned degrees
E-nourons connect 121-dogrea neuron
PC neuron
E-neurons neurron
Mexn befone trainigg
before traning trokiing
0.
Coneccionn Ceneettoantttt struggh
0.8
0.8
04
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Conraced E-meurraaa unteradioo prifenecce
18C
Homageneaus Interd
Connactd crienbaainn proferance
135
Figure 3: Orientation preference map development in SESNN model. a. The temporal evolution of the SESNN model's orientation preference maps, presented in HSV format. The final trial orientation map displays measured density per hypercolumn area (A²). The correlation between orientation preference maps from the SESNN model and observed orientation preference maps in the ferret over a normalized time scale. e. The changes in three different metrics wwiling cost, information capacity, and information-com cof.iinnncotttfffiiiineey during orientation map development. d. Comparative analysis of pinwheel densities relative to hypercolumn areas for ferrets, galagos, and tree shrews (Kaschube et al. (2010): Stevens et al. (2013)) alongside the SESNN model. e-h. Subplots with insets of pinwheel structures identify focal neurons and depict changes in connection strength among E-neurons with various orientation tunings before and after training (dashed and solid lines: mean). e-g. Display heterogeneous connection weights that bias toward iso-orientation domains. h. The subplot shows that the homogeneous connection weights contributed equally from nearby iso-orientation domains. The thickness of the arrows represents connection strength. All shaded areas: SD.
The SI is computed by iterating over all n data points, where indexes individual pixels, F; represents the final orientation preference value of the neurons, and O denotes the orientation preference at prior developmental stage. The higher SI at later development stage of the orientation preference maps implying that maps remain unchanged during their continued exposure to varying natural stimuli (Fig. (p). Furthermore, we also investigated whether the model's performance would change trained with a new set of images after stabilization. Our results show that while some detailed aspects of the pinwheel structure altered slightly. the overall locations of the pinwheel centers and the domains remained largely unchanged, indicating strong stability. The changes in neuron tuning toward specific orientations were also observed (detailed in Appendix A.5) Additionally, examine
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the relationship between dimensionless pinwheel densities (p) and hypercolumn areas (A2) across the SESNN model, ferrets, galagos, and tree shrews, comparing findings with empirical data (Fig. Bi)..
Considering the trained connections Waj. we follow the hypothesis that the information capacity can be depicted by the entropy of weight distribution in the complete connection matrix (distribution of all the trained connection strength) and the energy cost of synaptic transmission is inversely proportional to the connection strength. We have the information-cost efficiency (ICE) defined as:
H ICE C H
Σ Palog (Pt):
Σ W
where H represents the information encoding capacity, and C' represents the energy cost of synaptic transmission, P& the discrete probability of the distribution, that is, the normalized count of W in k-bins of the same size (0.01). Here, the calculation of wiring costs is based on Eq. / from Bertens Lee (022). In this approach, the network wiring cost is inversely proportional to the summed synaptic connection weights between pairs of neurons, indicating that stronger synaptic weights not only facilitate efficient communication but also contribute to lower wiring costs. In Fig. 3 we observed decreasing wiring costs during the development, which is consistent with previous research (Koulakov & Chklovski 200T). Meanwhile, the growing information capacity and the ICE suggest that the emergence of the pinwheel structure effectively maximizes coding capacity under the constraint of energy cost.
To further understand the circuit mechanisms underlying the emergence of pinwheel structure, examine the connections between E-neurons for ones at the iso-orientation domains (e.g.. Fig. 3pgg) and ones at the pinwheel centers (e.g., Fig. 3)) All connections of E-neurons are initialized with caussian-distributee strengths (see Eq. 5 and distributed uniformly across neurons of different orientation preferences (determined after stabilization), as indicated by the gray dashed curves. After the orientation preference maps stabilized, the connectivity patterns evolved significantly, leading to selectively stronger connections to neurons with similar orientation preferences. This clearly demonstrated by the red solid curves and the varying thickness of the arrows. The temporal in-variance of the connection distribution at the pinwheel center over time (except for an increase in overall strength), therefore, clearly demonstrated that the iso-orientation domains must take form first, as also visually verified in early trials of our model (Fig. 3p. This local continuity of iso-orientation domains is consistent with the demand of Hebbian-likk learning rule (Eq. I together with the overlap between RFs. The domain size is determined by the extent of E-connections, while swirling patterns arise from the varied local orientations in stimuli, enabling animals to perceive different orientations within the same visual field while maintaining local continuity. In -et--eed- organizations, reduced overlap disrupts this continuity. Thus, visual stimulus overlap shapes orientation preference maps in V1: greater overlap fosters pinwheel structures, while reduced overlap leads to salt-and-pepper patterns.
3.2 CODING EFFICIENCY AND ROBUSTNESS EMERGENCE WITHIN PINWHEELS
To assess the response reliability of neurons under stochastic conditions- for trustworthy and robust neural information processing- we analyze neuron reliability within orientation preference maps under noisy environments (N (N(0, of) We utilize a spike-based measurement approach to quantify this reliability. Specifically, for each neuron, we examine spike time sequences generated under identical stimulus conditions with added noise, calculating the mean cross-correlation between these sequences. We then determine the neuron's 's reliability by identifying the maximum in the average cross-correlation the mean of each train to zero, we systematically calculate the cross-correlation coefficient between each pair of sequences from these n trains:
For each neuron, we analyze n spike trains of duration T. After normalizing
(zy(m) E [IT+m!T],
(8)
where m denotes lag, the range of which can extend over the length of the time sequence T. and denote two different spike trains. Through this procedure, the average correlation coefficient of the same neuron can be determined:
Σee Rey (m)
(my(m))
(9)
C
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Original image
Natural image
Sallaan--eppper reliabllty
Pirwheels relibillty
d,
Raspurse rubablty
Ressoonee elatiity
0.8
2.8
0.5
0.6
0.4
0.2
0.1
0.2
0.3
0.1
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of Goussian nelce increases
0 of Gaussian nosee
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Information.caat efficnncy comparisons
Sparsoncos with varying hypercolumn s22
1.2
1.2
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Informaioon caperdyy
Populaton spersanece
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0.8
0.8
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0.4
0.2
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0.4
0.1
0.2
0,3
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Pineheeto
Pinnheale Salt and papplirs
Fineteeai Bati peddpppontt
hopercalummn SEow (nnrrmal
Figure 4: Population response analysis investigates neurons across different orientation preference maps. The SESNN model is trained using natural image inputs that include background distur- bances characterized by Gaussian noise (a, E [0, 0.3)). The response reliability of all neurons in the salt-and-peppers gradually decreases as a, increases. c-d. The response reliability about noise level decreases distinctly for neurons within the pinwheel center and surrounding areas across different hypercolumn sizes. Insets illustrate these patterns, with color bars indicating orientation preferences. A dashed blue line at 0.5 acts as a benchmark for comparing the responses of neuron types under specific noise conditions. e. Comparisons of three metrics-wirine capacity, and ICE across between pinwheels and aalt-and-peppers f. The averaged population re- sponse sparseness of neurons increases for pinwheels with larger hypercolumn sizes. (Data presented as mean SD, statistical significance denoted s*#p<0.001, $#$*p<0.0000, Welch's 's r-test).
cost, information
We posit that the maximum lag for similar time sequences generated by the same neuron cannot exceed half the length of the time sequence. Consequently, the reliability is obtained as follows:
Reliability
max S
(zy(m).
(10)
To assess the response reliability of the SESNN model, we use a corse-corrupted natural image as the input. This involves superimposing Gaussian white noise onto the image. The model is then stimulated with this modie-eo image, with the same variance noise applied in 10 trials.
Our findings show that neurons in orientation maps with pinwheel structures are more reliable than those salt-and--epper or mini-column configu anfiiuraations (Fig. (D-d). Pinwheel structures in higher mammals (Fig. 4d enhance signal processing reliability against noise and visual perception. In Fig. Ap. we compare wiring cost, information capacity, and ICE across pinwheels and salt-and- salt-and-peppers. Pinwheels exhibit lower wiring costs and higher information capacity, making them more efficient, as indicated by the ICE metric (Crumiller et al. 2013). Additionally, we calculate the population sparseness /Vinje & Gallant (000) for different orientation map sizes using 100 natural image patches within a 100 ms window (Fig. (4). Population sparseness measures the distribution of neuron responses to single patch, depicted below:
Σ
(ad)
(ax
Population sparseness
(11)
where a; refers to the mean spike rate of a neuron out of a total m neurons in a time window.
The model's findings reveal pattern in which population sparseness intensifies with the enlargement of hypercolumn sizes, spanning from small to large across various species. This pattern is illustrated in Fig. 4. which displays the progressively rising average population sparseness about hypercolumn size

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across 100 patches. Our model's findings consistent with empirical data (population sparseness: SESNN pinwheels ~ 0.74 vs. ~ 0.67 macaque pinwheels (Zandvakili & Kohn 2015), SESNN salt-and salt-and-pepper pepper organizations 0.25 vs. ~ 0.36 mice (Froudarakis et al. 2014).
4 RELATED WORKS
Traditional models. Traditional models such as self-organizing maps Kohonen, 1982, ON-OFF
theoretical models (Miller, 1994| Dang et al., 2020) [Song et al. 202T, Najafan et al., 2022). . and
artificial neural networks (Margalit et al. [023 Chizhov & Graham (021) have explored orientation
map mechanisms but often lack the complexity found in spiking neural networks. Recent spiking
neural networks (Lufkin et al. 2022 Keane& & Gong 2015 Chizhov & Graham 2021) have struggled with static inputs and connections, failing to accurately capture neural dynamics. Most importantly, no previous spiking neuronal networks with plasticity have actually successfully reproduced pinwheel structures. Line discontinuity instead discrete point discontinuity (as in real pinwheel and our model) plague the functional map (Srinivasa & Jiang] 2013). Thus, major improvement, our model's quantitative match with experiment results greatly enhanced the explaining power of its mechanisms.
Visual overlap among neighhoring neurons. Numerous studies have explored the variability in V1 neuronal organization and its impact on afferent segregation, with factors like visual sampling density playing a significant role (Baden et al.] 20166] Román Rosón al. 2019; Srinivasan et al. 2015
Najatian et al., 2022 Jin et al. [201 I] Lee et al. 2016; Kremkow et al. 2016 D Jang et al. 2020 Lien
& Scanziani| 2013). While these studies have highlighted the importance of anatomical factors such as visual overlap and neuronal density in distinguishing pinwheel structures from salt-and and-pepper organizations, they primarily rely on phenomenological static models, lacking detailed mechanisms for how these factors influence functional organization. Our study provides a clear mechanism by demonstrating that visual input overlap is crucial in forming pinwheel structures. Through our spiking neural network model, we show that increased overlap between neighboring neurons enhances local connectivity and functional clustering in iso-orientation domains, leading to the emergence of pinwheels. In contrast, reduced overlap weakens these local connections, resulting in the disorganized salt-and- pepper patterns. This finding establishes a direct link between the structure of input overlap and the functional organization of VI, making the connection between structure and function explicit.
Wiring cost minimization and sparse coding strategy. Research suggests that wiring minimization optimizes orientation maps (Koulakov & Chklovskii] 2001 Petitot 2017: Dai et al. 2018). Our study expands on this by not only examining the wiring costs but also the information capacity, thereby finding that the ICE of pinwheel structures is much improved compared to aal--and-pepper organizations, Studies also suggest that sparseness in neural coding enhances precision and reduces redundancy (Zhou & Yu 2018 Yu et al., 2014 Olshausen & Field 1997| Rolls & Tovee 1995] Vinje & Gallant (000). We found that pinwheel structures strike a balance between reliability and coding sparsity. Lateral connections in pinwheels strengthen synapses among neurons with similar orientations and weaken those with dissimilar orientations. Thus, by enhancing functional clustering in the iso-orientation domains, these neurons boost reliability with some redundancy but still effectively implement sparse coding by suppressing nearby iso-orientation domains. While near pinwheel centers, where connections are less selective, neurons display varied response patterns instead of clear orientation preferences, therefore implements sparse coding of high-order pattern in addition to plain feature space of single orientations.
CONCLUSION
The SESNN model integrates anatomical and physiological factors like visual overlap, neuronal den- sity, and synaptic plasticity, providing more realistic, dynamic representation of spiking mechanisms in VI, especially in pinwheel formation. Our study identifies visual overlap, neuron density, and local plasticity key factors driving the transition from salt-and-pepper to pinwheel topologies, linking structural features to functional performance- -a significant advancement in understanding visual coding efficiency. By addressing wiring cost minimization and information capacity, our findings reveal optimal cortical network strategies for sparse and robust visual encoding. These insights can inform the design of more efficient brain-inspired computational models.
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ACKNOWLEDGMENTS
We extend sincere gratitude for the support provided by the Science and Technology Innova- tion 2030 Brain Science and Brain-Inspired Intelligence Project (2021ZD020131)) as well as the National Natural Science Foundation of China (U20A20221, 12201 125, 120721 13). We also acknowledge funding from the Shanghai Municipal Science and Technology Committee Shanghai Outstanding Academic Leaders Plan (21XD1400400) the Yang Fan Plan (22YF1403300) and the China Postdoctoral Science Foundation (2023M740724).
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A APPENDIX
A.1 DETAILED NEURAL MODEL
The spiking neuronal dynamics are iteratively formulated as follows:
u (1) (t + u, (K) (t)e ,.k0 +kk() Σ FF [e image)
Xj
(12)
(((KK)) (K') Bi 3lktK'), 1/ 22 (tk)(() noise,
K
1, if an E-neuron ID,
hk(11
(13)
0, if is an I-neuron ID,
AO(K) p( (K*) 1) -pp (K)
(14)
where K = Excitatory (E) neurons or Inhibitory (I) neurons: i= 1,11,, NN (numbers of E-neurons and I-neurons).
In Eq. 12 u (K) (t) represents the membrane potential of neuron i at time 1. applicable to neurons of class K. The decay rate of the membrane potential is governed by the resistor-capacitor circuit time constant set (E) 10 ms for E-neurons and 70) 5 ms for I-neurons, reflecting the faster firing rate of I-neurons compared to E-neurons (Paul D. King et al. 2013);; Thomson Lamy (2007)) This configuration reduces reconstruction error and speeds up system convergence, enhancing the representation of input stimuli. The feedforward (FF) connection, Fp!EE-image) connects pixel Xm from a whitened image patch (normalized to zero mean and unit variance) to E-neuron i. Connection weight W from neuron of class K* to neuron of class K includes a sign /3 (((KKK)) 'ndicatinn
1j excitatory (+1) or inhibitory (-1) connections. 2j (t) H (u. (KK) (t) 0 o( where (x) is the
(K')
Heaviside step function, defined as H = 1 if x > O and O otherwise. Here, 2, 2((²) (t) represents
the spike output from neuron time which triggers reset of the membrane potential to 0 after reaching the spike threshold 0 (initially set at 2). The firing threshold 0 adapts based on the
discrepancy between the current and target firing rates p; (t) and (K) (n(t)) 2, p( 4) (see
Eq. 14 (Földiák 1990)). Neurons receive Gaussian white noise (00 0.04) from other brain areas (noise term). For computational efficiency, the time step 77 is set ms.
A.2 ANATOMICAL DATA INTEGRATION
NEURAL CONNECTION DATA
The experimental subjects include six adult cats with unknown genders, with data sourced from research by Armen Stepanyants et ateepannnsss et al. 2009); and eight macaques, aged 5-11 years, including six males and two females, with data sourced from research by Joseph Amatrudo et al. all(Aamarudoo et al. 2012).
NEURONAL SYNAPTIC PLASTICITY
The subjects are rats aged 14- 16 days, with unknown gender and quantity, with data sourced from research by Holmgren et al. Holmmgen al. (200);; transgenic mice, with unknown quantity and gender, with data sourced from research by Hofer et al. (Hofer et al. 2011).
RETINAL-V V TOPOLOGICAL PROJECTION DATA
Receptive field (RF) data: Primary visual cortex (V1) neuron counts for macaques, cats, tree shrews, ferrets, mice, rats, and gray squirrels respectively come from Tehovnik et al. (Tehovnik & Lee 1993) (subjects: macaques, unknown gender and age), Scholl et al. (Scholl et al. (013) (subjects: cats, unknown gender and age), Veit al. (eii et al.) 2014) (subjects: 9 male and 7 female tree shrews, aged 3-8 years), Huberman et al. (Huberman et al., (006) (subjects: ferrets, unknown gender and age), Niell et al. (Niell & Stryker, 2008) (subjects: mice, aged 2-6 months, unknown gender), Foik
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al. (Foik al. (sbbeees:: 21 rats, unknown gender and age), and Hall al. (Hall al. 1971
(subjects: 17 gray squirrels, unknown gender and age). VI neuronal density: neuronal density data for macaques, cats, mice, rats, and gray squirrels come from Srinivasana et al. (Srinivasan et al. 2015) (subjects: unknown gender and age); tree shrew, ferret, and gray squirrel density data respectively come from Weigand al. (Weigand et al. 2017).
CORTICAL MAGNIFICATION FACTOR
Cortical magnification factor (CMF) data for macaques, cats, tree shrews, ferrets, mice, rats, and gray squirrels are sourced from Tehovnik et al. (Tehovnik & Lee 1993) (subjects: macaques, unknown gender and age), Veit et al. (Veit et al. 2014) (subjects: cats, unknown gender and age), Bosking et al. (Bosking et al. 1997) (subjects: tree shrews, unknown gender and age), Rockland et al. (Law et al.) 1988) (subjects: 9 ferrets, female, unknown age), Beest al. (van Beest et al. (202T) (subjects: 28 mice 11 males and 17 females, ages 2-14 months), Keller et al. (Keller et al. 20000) (subjects: male rats, age 3 months), and Hall et al. (Hall et al. 1971) (subjects: 17 gray squirelss unknown gender and age).
A.3 UNVEILING SPECIES-SPECIFIC FACTORS DISTINGUISHING PINWHEELS AND SALT-AND-PEPPERS
A.3.1 ANATOMICAL DATA SUGGESTS RFs DENSITY UNDERLYING VI ORGANIZATIONS
Table 3: Comparative anatomical data the retina and V1 across species.
d. V1 neurons density (ncurons/mm")
e. V1 RF
f. RFs density ((c) x (d)/(b) (RFs/mm")
a. Species (mean)
b. Retina (mm²)
VI size (mm²)
size in area centralis (deg)
Macaque
636
1,090
243,000
0.2
416,462.26
Cat
510*
380
99,200*
1.0d
73,913.73
Tree shrew
122
73.,a
192,800
2.09
115,363.93
Ferret
8
78h
95,813/
3.0*
90,041.13
Mouse
15
2.5t
86,600
4.07
14,433.33
Rat
m2n
7.1
90,800*
3.0¹
12,397.69
Gray squirrel
a05
32"
84,213
2.0
13,145.44
a
Jang et al. (2020)
Srinivasan et al. (0155
Tehovnik & Slocum (2077)
d
Scholl et al. (201 3 Engellann & Peichl 1996) Weigand et al. (2017)
9
Veit et al. (2014)
Law et al. (1988) Huberman et al. (2006)
Niell & Stryker
2008)
Hughes 1979
Foik et al. (2020)
Hall et al.
1971
We analyzed anatomical data from seven species, including primates (e-g., macaques) and non- primates (e.g., mice, rats, cats, tree shrews, gray squirrels, and ferrets), detailed in Table We first find that RFs density (PRF) acts as linear classifier (y = 4.42 10*x), effectively distinguishing species with pinwheel structures from those with allt-and-pepper organizations. In this classifier, species like macaques, cats, tree shrews, and ferrets, which have higher RFs density, are associated with pinwheel structures (light red area in Fig. 5 and exceed the classification threshold. In contrast, species with lower RFs density, such as mice, rats, and gray squirrels, are linked to salt-and-pepper organizations (light blue area in Fig. 5). Thus, V1 RFs density serves predictive metric for V1 organizational patterns across species, The PRF is calculated as follows:
n'
PRF
(15)
[(RRF - E) (vn-1) sk
where ní denotes the total number of neurons in V1, s/ indicates the retinal surface area. We have n' = Sv1 x Pv1. where Sy1 corresponds to the V1 2D surface area, and fv1 signifies the neuronal density within V1. The variable quantifies the degree of visual input overlap among adjacent neurons, n denotes the total number of neurons, and sgr represents the RF size in the self-evolving spiking neural network (SESNN).
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RI's density dessifier
Species dessiiie V1 neurone-reling
Salt and peoper cogancations
iP Pinwheel aiructures
TBLs
Number of V1 nouron
10
CM
Spacies
Pinaheel structures
Cray scurrel
Tree shrew
10
y 7-4-2--10'
Ferret
popper organizations
y= 4,42x10
10"
10"
10"
16
10a
RF density (RFaimm")
Retina size (mm))
Figure 5: A linear classifier based on RFs density (y = 4.42 x 10 effectively differentiates species with salt-and-pepper organizations (rats, mice, gray squirrels) from those with pinwheel structures (macaques, ferrets, cats, tree shrews). a. This classifier reflects variations in VI organizations across species. plot categorizing species by the ratio of V1 neuron number retina size acts divider, implying a critical ratio for the formation of pinwheel structures.
A.3.2 SESNN REVEALS THE INFLUENCE NEURONAL CONNECTION RANGE VI ORGANIZATIONS
Synaplic connection
Synaptic connection range
Pinwheelss
Pinwhests
range
Increase the range
299
Ar = 02
mm
mm
Number pindhasls
200
0.35
Hypencolumee dzz (mm)
150
0.3
100
NNPD
0.2
0.8
50
0.15
0.4
26 3.2
2.0 23
26
29 3.2
Synapdo connection range (9)
Synaptc connection range
Synaptic
Figure 6: Synaptic connection range within VI contributes to the formation of pinwheel structures. a. Modifying the synaptic connection range reshapes the dimensions pinwheel structures. b-d. The relationship between the synaptic connection range (α) and the number of pinwheels, nearest- neighbor pinwheel distance (NNPD) (mm), and hypercolumn size (mm). The scale bar: 1 mm in V1 cortical surface. Color scheme: orientation preference. Lines: mean. Shaded area: SD.
The anatomical data Table 3 for seven species show variability in V1 neuronal density (pv1). 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 E- and I-neurons, as depicted in Fig. [ We modulate axonal arborization through parameter 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. 6). Our observations indicate that increasing axon lengths, thereby extending the connection range, enlarges hypercolumn sizes within pinwheel structures (Fig. (d)), reduces the overall number of pinwheels (Fig. (6p). and increases NNPD (Fig. (b)) These findings underscore the critical role of neural synaptic connection range in organizing orientation maps.
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A.4 QUANTITATIVE DATA CORTICAL PINWHEEL STRUCTURES
The visual input overlap is quantified as follows:
VPVi Suii Linihh
Eratio
(16)
v VPviSunit
RFF
where Sunit represents the unit cortical area (mm²), S denotes the size of the RF in VI with unit in degrees, e' denotes the visual overlap degree (in degrees) between neighboring neurons, and pvi represents the density of neurons in V1. We consider only an effective cortical layer composed of output neurons. This 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. M refers to the CMF. The trend shown in Fig. Z] indicates increasing overlap for mice, macaques, and cats, which is consistent with the overlap settings of our model. To express the visual input overlap ratio as a percentage, we multiply the ratio by 100%. The escalation in overlap corresponds the expansion of iso-orientation domains and pinwheels, presented by our model's predictions and anatomically detailed by Najafian et al. (Njaffan
al. 2022).
A.5 STABILITY AND ADAPTABILITY OF PINWHEEL STRUCTURES
Stabilired Orientation Map
liaggll Trained Map
0.8
92
n.S
af Protents
70
BC
0.6
0.2
0.15
Painsise Casire Similntyy
40
15
Stakittedd Map
Imageo Trained Map
New bmages Trained Map
Stabiiiod Map
Stabiited Map
Imagro Treened Map
Figure 7: Plasticity of the pinwheel structure in response natural images. (a) Stabilized orientation map and orientation map after training with new natural images. (b) Number of pinwheels. (c) Hypercolumn size. (d) NNPD. (e) Pairwise cosine similarity between neuronal representations before and after introducing new environments.
We conducted additional experiments to examine the plasticity of the pinwheel structure in response to new visual environments. new set of natural images, including forests, large stones, tree branches and leaves, reeds, stones of various sizes, tree trunks, leaves, and grasses, was introduced after the pinwheel structure had stabilized. After further training with these new images, we observed slight modifications in some finer aspects of the pinwheel structure; however, the overall structure, including the locations of the pinwheel centers, remained largely stable (Fig. 4.
To quantitatively assess these changes, employed three different metrics: (i) the number of pinwheels, (ii) hypercolumn size, and (iii) NNPD. These metries were used to compare the pinwheel structure before and after introducing the new images (Fig. 7 c, d). Additionally, we utilized pairwise cosine similarity to measure the representation similarity of the neuronal spike train population (Fig. (7
Although we did not observe significant changes in the morphology of the pinwheels, some detailed aspects of the pinwheel structure did change. Specifically, while the locations of the pinwheel centers
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and the overall domains remained nearly unchanged, some neurons exhibited tuning adjustments towards specific orientations. Furthermore, the representation of neurons to the same stimulus remained overall stable, with minor variations before and after the introduction of new environments, suggesting that the pinwheel structure is highly adaptable (Fig. 7-
These results indicate that the model's organizational structure is deeply rooted in the statistical features of natural images and demonstrates a strong capacity for adaptation to new environments while preserving its core structural properties.
A.6 ORIENTATION MAP FORMATION FROM OUR PROPOSED SPIKING NETWORK MODEL
A.6.1
HO RULE LEARNS VI RF
The HO rule learns RFs from visual images, described by the following formula:
AFF AFF(Ee-mmar (me-Tmeo)) im
x y:
y F km
(Ee-Image)
(17)
where Zm and y: represent the gray value of pixel and instantaneous firing rate of post-synaptic neuron i. This rule ensures that weights converge when:
(y:Tm) (y?)
STA, (v
FF
(18)
where the operator (-) denotes the moving average value (refer to Eq. which corresponds the normalized spike-triggered average (STA). The HO rule stabilizes learning by constraining synaptic growth, resulting in biologically realistic V1-like RFs.
A.6.2 RETINOTOPIC TOPOGRAPHIC CONSTRAINTS
Each neuron encodes only small portion of the visual field (Fig. D.. with the small patch E R #' xw" smaller than the input image I E IR #x To represent the entire visual stimulus, the population of neurons collectively tiles the input space through their overlapping visual fields. This organization mirrors the retinotopic mapping observed biologically, where neighboring neurons in the cortex encode adjacent regions of the visual scene, preserving spatial relationships from the retina.
A.6.3 HIGH-OVERLAP STRENGTHENS LOCAL E-E CONNECTIONS FOR ORIENTATION MAP CONTINUITY
E-E connections align adjacent neurons' firing activities, promoting similar orientation preferences. This requires a high overlap (Fig. 2) where correlated feedforward input initially strengthens local excitatory connections, ensuring continuous orientation preference and forming iso-orientation domains with similar orientation preferences (Fig. 3--
A.6.4
CM RULE VIA E-I-E LOOPS ENSURES ORIENTATION DIVERSITY AND MAP COVERAGE
E-I, I-F, and I-I connections are updated through the CM rule:
AW (K-K K<) x yıæ, - (z,) (1+ w.(k-k)))
(19)
where zy and y represent the instantancous firing rates of pre- and post-synaptic neuron j and At equilibrium, the CM rule converges to:
We (Ui2,) (2)) (wi)(e)))
(20)
We have:
We (wary) - (wx) (xr)) Cov(us,
(21)
Covariance shapes neural connectivity based on activity covariance, where synaptic weights are strengthened between correlated neurons and weakened between uncorrelated ones. This mechanism
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ensures that inhibitory neurons regulate overall activity between neighboring E-neurons through the E-I-E loops (lateral inhibition), suppressing predictable responses to maintain sparse coding. This also prevents excitatory neurons from excitatory connections overly responding to the same orientation, enabling preference for other orientations and ensuring full coverage. Over larger connection ranges, highly correlated feedforward correlations disrupted and inhibitory neurons drive diverse responses, resulting in distinct domains with varying preferred orientations, forming pinwheel structures within the functional orientation map.
The exact size of the domain, as explained earlier, is determined by the extent of the E-connections; while their swirling patterss are instead demanded by the rich local orientations each stimulus such that the animal would be able to perceive different orientations in the same local visual field but still satisfying most of the continuity locally. Note that salt-and-pepper organizations, the mechanisms are disrupted due to reduced overlap. The local continuity of iso-orientation domains is consistent with the HO rule and the overlap between RFs. The domain size determined by the extent of E-connections, while swirling patterns arise from the diverse local orientations in stimuli. This allows animals to perceive varying orientations within the same local visual field while maintaining local continuity. In salt-and- aatt-and-epper pepper organizations, reduced overlap disrupts these mechanisms. Hence, visual stimulus overlap critically shapes orientation preference maps in VI: greater overlap promotes pinwheel structures, while reduced overlap leads to aal-annd-pepper patterns.
A.7 INTEGRATE INTO MODERN DEEP LEARNING ALGORITHMS
To demonstrate the practical relevance of model, we explore its capability front-end encoder for deep spiking neural networks. Specifically, we design five-layer convolutional and pooling-based SNN to classify images from the FaSSionMNIST dataset. The classification accuracy achieved using our model's ' spike sequences is compared with that of Poisson spike sequences, as summarized in Table
Table 4: Pinwheels vs. Poisson encoder accuracy (FashionMNIST)).
Dataset class
Pinwheels encoder accuracy(%)
Poisson encoder accuracy(%)
T-shirt top
86.80
85.40
Trouser
97.30
96,40
Pullover
83.40
83.80
Dress
91.10
90.30
Coat
83.60
81.60
Sandal
96.20
96,80
Shirt
73.20
67.80
Sneaker
95,40
95.40
Bag
97.80
97.00
Ankle boot
96.10
94.20
Overall
90.09
88.87
From these results, we observe that, compared to the randomness inherent in Poisson- based encoding, the functional organization of pinwheels produces spike sequences with more structured spatiotempo- ral patterns. This organization encodes richer information, contributing to improved classification accuracy across most categories.
Additionally, we evaluate the noise robustness of our model by introducing Gaussian noise to the input images. The results are shown in Table 5
Compared the Poisson encoder, the functional organization inherent in pinwheel encoding provides better resistance to noise, as it retains meaningful features even noisy environments. Morcover, the salt-and- pepper encoding also shows degree of robustness, albeit slightly less effective than pinwheel encoding under higher noise levels.
These findings highlight that the spatiotemporal regularity of the pinwheel maps significantly benefits both classification performance and noise resistance, suggesting their potential utility in enhancing the robustness of modern deep learning frameworks.
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Table 5: Noise robustness: Pinwheels Poisson Salt-and-peppers encoders.
Noise level
Pinwheels encoder
Poisson encoder
Salt-and- encoder accuracy (%)
accuracy (%)
accuracy (%)
0.0
89.47
93.16 14
87.92
0.1
88.98
92.14
87.21
0.2
88.41
82.81
86.18
0.3
88.11
62.79
84.51
0.4
86.22
36.61
80.52
0.5
82.91
18.79
67.39
0.6
70.88
12.03
38.37
0.7
43.60
10.89
19.95
0.8
21.20
10.50
13,00
0.9
12.51
10.20
10.71
A.8
COMPUTING INFRASTRUCTURE
Simulations are executed on a high-performance system featuring an Intel Xeon@ Gold 6348 CPU (2.60 GHz). an NVIDIA@ A100 GPU, and 512 GB of memory. The workflow was managed on Ubuntu 20.04.6 LTS, with computational tasks implemented in MATLAB R2023a and Python 3.9.
Table 6:
Computing infrastructure.
CPU
Intel Xeon(R) Gold 6348 CPU @ 2.60GHz
GPU
A100
Memory
512 GB
Operating system
Ubuntu 20.04.6 LTS
Simulation platform
MATLAB R2023a and Python 3,9
A.9
ACRONYM
Table 7 summarizes essential acronyms and their full terms, spanning neural structures (e.g.. VI), computational models (SESNN, SNN), learning rules (HO, CM), and analytical metrics (NNPD, ICE, SI).
Table 7: List of acronyms and their corresponding terms.
Acronym
Term
VI
Primary Visual Cortex
SESNN
Self-Evolving Spiking Neural Network
SNN
Spiking Neural Network
HO
Hebbian Oja's 's variant
CM
Correlation Measuring
NNPD
Nearest-neighbor pinwheel distance
ICE
Information-C25 Efficiency
CMF
Cortical Magnification Factor
E-1
Excitatory-Inhiitory
FF
Feedforward
RF
Receptive Field
SI
Stubility index
24


Reviews:
[{"review_id": "mADRenWLVe", "reviewer": "Reviewer_seTg", "summary": "The paper investigates how the pattern of orientation map emerges from a recurrent spiking network model with synaptic plasticity, and concludes that the feedforward input overlap is crucial for forming different map patterns, i.e., salt-and-pepper vs. pinwheel structures.", "questions": "1. What is the $z_j(t)$ in Eq. 3?\n\n2. Could the authors comment on the differences between the current model with Hansel & Vreeswijk, J. Neurosci., 2012?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "1. Biological plausibility of the recurrent spiking network model with plasticity. The model is well supported by neurobiological experiments and is quite similar to the conventional recurrent spiking network used in neuroscience studies.", "weaknesses": "1. Although the authors claim they provide a spiking network mechanism of orientation map, the spiking dynamics seem unnecessary, because none of the results rely on spike timing information. All learning rules are based on the firing rate of neurons only. This makes the spiking network model look like a \"strawman\" a bit. I envision the spiking network model with spike-time-dependent plasticity rule might have different mechanisms from the rate-based Hebbian rules. Anyhow, I think the author can provide some explanations about how much we could gain about the orientation map formation from the proposed spiking network model. \n\n2. I am still debating about the contribution of this work to ICLR society. That is, how much we could gain from this study to develop the next-generation AI algorithm? Or how the results in the paper can be incorporated into the modern deep learning algorithms? Another thing is although the model is biologically plausible, all learning rules and network model architecture are not new.\n\n3. It is uncommon to see the title of the paper PDF differ from the title shown in the OpenReview. This makes me concerned that the paper might not be well prepared, at least in the submission stage.\n\n### Minor\n1. When comparing output orientation map size among different species, the model should scale with the different neuron density, RF size, and magnification factor (in Table 2) across species. For example, the patch size that each neuron perceives is a constant (16 pixel), while for different species it should be scaled with the following term:\n\n    V1 RF size (deg) * magnification factor (mm/deg) * sqrt(neuron density) (neuron / mm)\n\n(so the outcome is the receptive field size scaled with neuron number)\n\nAlso, the recurrent wiring size scaled with neuron number in Eq. (5), presumably conservative about 100 ~ 200 um across species, should be scaled with sqrt(neuron density) (neuron / mm) in different species.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Hglq7KIsm7", "reviewer": "Reviewer_zfyC", "summary": "Authors study functional organization of the primary visual cortex across animal species. Depending on the species, primary visual cortex is organized in a continuum going between disordered salt-and-pepper and ordered pinwheel structures. Authors model these functional organizations with an E-I spiking neural network that is equipped with two types of Hebbian-like plasticity rules. An important finding is that  the degree of synaptic organization depends crucially on the overlap of feedforward inputs incoming to the network. They also find that once learned, the pinwheel organisation is stable and promotes efficient processing of visual information.", "questions": "1) Seen that Eq. 4 is the one actually used for modelling, it is not clear what is the purpose of Eq. 3? Also, it is unclear what the variable z in Eq. 3 stands for.\n\n2) Why is the \"neural connectivity parameter\" (max weight) higher for E compared to I neurons? At the same time, authors report that E-I connectivity should be stronger compared to E-E connectivity.\n\n3)  In lines 173-174, authors comment on their choice of hyperparameters and state that \"their approach is consistent with empirical findings\". It remains unclear how specific hyperparameters of their model relate to empirical findings. Could authors be more specific?\n\n4) How is the coefficient in Eq. 8 different from a well-known measure of the cross-correlogram on neural spike trains (Bair et al., J.Neurosci. 2001)?\n\n5) A recent study showed that efficient encoding of uncorrelated stimulus features with E-I spiking networks does not require E-E connectivity (Koren et al., eLife 2024). However, your study seems to suggest the necessity of E-E connectivity. Could authors comment on this discrepancy?\n\n6) One of the take-home messages of the paper seems to be that the pinwheel structure is more efficient than the salt-and-pepper structure. However, salt-and-pepper organization can be observed in biological brains, and it seems unlikely that it would survive evolution if it was inefficient. Moreover, authors show that salt-and-pepper type of organisation arises when the overlap of inputs to E neurons is small, which is the case in some animal species. All together, it seems to me that there is no single \"most efficient\" type of structure, but that there are multiple efficient solutions depending on a specific model parameter. Rather than making a simplistic (and not quite correct ?) conclusion and point to one and only best solution, it would seem to me more appealing if these results are interpreted more carefully to give us better insights about biology.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "Authors presented a rather convincing paper with a clear and timely scope. The paper is technically correct and rigorous in methods. It brings potentially important and new insights into why biological neural networks organise in a continuum of salt-and-pepper and pinwheel structures.", "weaknesses": "Major weakness of this paper are the presentation, the lack of justification and elaboration on modelling and parameter choices, and insufficient comment on results. While the presentation is rather transparent, the clarity of the text can be substantially improved. Also, the text contains a number of typos.\n\n1) I recommend to omit all abbreviations besides the very common ones, such as  V1, E-I and potentially FF. The acronym of the network, SESNN, is also an exception and is useful to have. All other acronyms seem counterproductive. While authors have compiled a list of acronyms in the Appendix A.7, frequent use of diverse acronyms throughout the text impedes readers that are less familiar with the topic of self-organising maps to appreciate the results.  \n\n2) On many places, the text is rather difficult to follow, with long and unnecessarily complicated sentences. In some cases, sentences are not grammatically correct. See for example line 275, Figure Caption 3a and h, lines 368-371. On page 9, there are references to Figure 6, but likely authors want to refer to their Figure 4. In line 262, authors refer to the third panel of fig 2a, but it might rather be the first panel?\n\n3) It is not clarified enough what is the difference in effects between the Hebbian Oja's and Correlation Measuring type of learning. The only reasoning I gathered is that Hebbian Oja plasticity rule has the desired property of normalization that prevents too strong E-E connectivity.  What is the drawback of using Hebbian Oja's plasticity for E-I and I-I connectivity? Authors should go deeper into explaining this, as it seems to be crucial for the results they obtain.\n\n4) In Eq. 7, authors define the energy cost of synaptic transmission as inversely proportional to the connection strength. While this is an interesting choice, it is not intuitive why a quantity inversely proportional to the connection strength is used as the metabolic cost. Also, it is not well justified why is the information capacity computed as the entropy of the weight distribution. Could authors elaborate on that and provide references if applicable?\n\n5) The measure of reliability in Eq. 10 is based on the encoding capability of single neurons. Today, however, there is ample evidence suggesting that in the cortex, signal processing is better captured on the level of neural populations.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "uKQ2Lc8UZ3", "reviewer": "Reviewer_GMwp", "summary": "Authors propose a comprehensive model of the primary visual cortex encompassing anatomical factors, such as natural image inputs, precise and realistic representation of neuronal responses, as well as spiking mechanisms in order to understand the emergence of pinwheel structures. They successfully show that the model can reproduce the emergence of different orientation maps in the visual cortex and that it is compatible with the maps of different species as a function of different factors such a neuron density or RF overlap.", "questions": "It has been proven multiple times that the anatomy of V1 is efficient for processing visual information. How do you think your model could be used to understand the emergence of orientation maps in the visual cortex? Is the tiling of receptive fields a possible factor ? (see eg https://doi.org/10.1016/j.neuron.2016.07.015 )\n\nThe model uses 1/ static images, 2/ images represented on a Cartesian grid. Extensions using dynamic images or images represented on a polar grid could be interesting. The evaluation of the model is limited to the comparison of the orientation maps with experimental data. A more quantitative evaluation of the model's performance on other tasks could provide a more comprehensive assessment of its capabilities.", "limitations": "", "rating": 8, "confidence": 5, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "The paper is clearly written and presents a novel approach to analyzing the emergence of orientation maps in the visual cortex. The proposed model is well-motivated and supported by mathematical formalization. The authors provide a detailed explanation of the model and its components, which enhances the understanding of the proposed method.", "weaknesses": "Stating in the abstract that the subject in \"largely unexplored\" is a strong claim that needs to be supported by a more extensive literature review. The paper could benefit from a more in-depth discussion of related work to provide context for the proposed approach. Your extensive bibliography provides already a good illustration of the number of works on the subject, and more for instance on the emergence of orientation maps in the visual cortex from haphazard wiring, or from the emergence of both types of maps using pooling mechanisms in a sparse deep predictive coding network, could be useful. More generally, it would be useful to highlight the difference of your work with [Stevens et al., 2013].", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Spotlight)