Fig 1.
Species-specific existence of long-range horizontal connectivity in V1.
(A). Illustration of long-range horizontal connections observed in the primary visual cortex (V1) of tree shrews (following Bosking 1997). Red dots represent synaptic boutons of lateral connections with retrograde labeling by injection. (B). Illustration of distribution of the connection lengths of lateral connections in various species (following Seeman 2018 and Bosking 1997). LRCs are not found in species with a small V1 area, such as mice. (C). Illustrations of retino-cortical projections in species with different magnification factors. According to the size of the cortex, the same object in the visual space can cover different sizes of the spatial area in the cortical space. (D). LRCs may act as shortcuts for inter-neural communications in a large V1 network but not in a small network, where local connections suffice.
Fig 2.
Integration of global information by LRCs in a large network.
(A). Design of a visual stimulus containing “global” information in the form of positions of dots. (B). A simplified three-layer network model of the early visual pathway. (C). Two distinct sizes of the model network used in the simulations. (D). LRC ratio, i.e., the number of LRCs relative to the total number of connections, varying from 0% to 100%, while the total number of connections remains constant. The connection length in the simulation was sampled from observed statistics in biological data [5]. (E)-(F). The classification accuracy for the “position” dataset under variations of the LRC ratio in large and small networks. Thin lines indicate the results for different conditions of the center-dot distance in the stimulus (See S2A Fig for details). Bold lines indicate performance averaged over all stimulus conditions. (G)-(H). Visualization of the connectivity projection from the processing layer to a single readout unit in a large and a small network. The gray level of each pixel represents the number of connections linked to a readout neuron, and the total grayscale area indicates the effective “recognition” range of the readout. (I)-(J). Illustration of the average path length L of the network. Note that a wide effective range of lateral connections, leading to a small L, is necessary to identify global positional information in the visual stimulus. (K). Similarity between the modulation of the value of 1/L and of the classification accuracy of the network during the rewiring process when varying the LRC ratio. (L). A strong correlation is observed between 1/L and the classification accuracy of the network regardless of the network size and/or the stimulus condition. For each value of 1/L on the x axis, transparent dots indicate the results for different conditions of the center-dot distance in the stimulus and bold dots indicate the result averaged over all stimulus conditions. Error bars represent the confidence interval for 20 trials.
Fig 3.
Local connections are required to integrate local information in a large network.
(A). Design of a visual stimulus containing “local” or “shape” information with an MNIST digit located in the center. (B). Classification accuracy for the “shape” dataset with variations of the LRC ratio and of the network size. For both large and small networks, the performance decreased as the LRC ratio was increased. Thin lines indicate the individual network performance outcomes under variations of the stimulus parameters, and bold lines indicate the averaged performance (See S2C Fig for details). (C). Illustration of the clustering coefficient C of the network. Note that a high connection density, leading to high C, is necessary to recognize the detailed shapes of the digits. (D). Similarity between the modulation of the value of C and of the classification accuracy of the network during the rewiring process when varying the LRC ratio. (E). A strong correlation is observed between C and the classification accuracy of the network regardless of the network size and/or the stimulus condition. For each value of C on the x axis, transparent dots indicate the results for different combinations of digits in the stimulus and bold dots indicate the result averaged over all stimulus conditions. Error bars represent the confidence interval for 20 trials.
Fig 4.
LRCs organize a small-world network to enable the recognition of various visual features.
(A). Design of visual stimulus containing both “global” and “local” information with an MNIST digit in the center and a dot located in one of the quadrants. (B). Classification accuracy with variations of the LRC ratio. The large network (left) showed the best performance with a combination of sparse (10%) LRCs and dense local connections, while the small network (right) achieved its best result with local connections only. Arrows indicate the condition of the best performance. Thin lines indicate the individual network performance under variations of the stimulus parameters, and bold lines indicated the averaged performance (See S3 Fig for details). (C). Similarity and differences in the modulation of L and C during LRC ratio variations in large and small networks. Note that L changes significantly due to the modulation of the LRC ratio in a large network, but not in a small network, while C changes similarly in both small and large networks. (D). Similarity between the modulation of the small-world coefficient (SW) and the classification accuracy of the network during the rewiring process when varying the LRC ratio. (E). A strong correlation is observed between SW and the classification accuracy of the network regardless of the network size and/or the stimulus condition. For each value of SW on the x axis, transparent dots indicate the results of different combination of digits and dot positions in the stimulus, and bold dots indicate the result averaged over all stimulus conditions. Error bars represent the confidence interval for 20 trials.
Fig 5.
Small-world coefficient of the network predicts the size-dependent effect of LRCs for visual encoding.
(A). A hypothetical scenario for the network-size-dependent existence of LRCs. The species-specific emergence of LRCs is explained by the size-dependent enhancement of small-world coefficient (ΔSW) with LRCs. (B). To validate the hypothesis, the small-world coefficient (SW) and the performance of the network were examined while the network size and the LRC ratio varied. (C). Change in the SW of the network as a function of the LRC ratio with different sizes of networks. Arrows indicate the maximum value point and dashed lines indicate the value of SW with 0% LRCs. ΔSW is defined as the difference between the two values, as illustrated. (D). Modulation of the normalized classification accuracy (ΔAccu) as a function of the LRC ratio with different sizes of networks. ΔAccu is defined as the difference between the maximum performance and that with 0% LRCs. (E)-(F). Changes of ΔSW and ΔAccu as a function of the network size. Note that both ΔSW and ΔAccu become positive only when the network size exceeds a certain threshold. Thin lines indicate the performance under the variation of local and global parameters of the stimulus (See S3 Fig for details). Bold lines indicated the averaged performance during variation of the stimulus. Error bars represent the confidence interval for 20 trials.
Fig 6.
Emergence of LRCs for size-dependent optimization of the performance and wiring cost.
(A). Networks of two different sizes were trained and pruned from random initial wiring conditions to find the optimal condition that maximized the classification performance at the minimal wiring cost. (B). The classification accuracy increased and the total wiring length decreased in both networks, as intended by the new loss function that consists of classification error and length-penalty terms. (C). Distribution of the lateral connection length in a large network after training, with (red) and without (black) a length penalty. (D). A certain portion of LRCs survived after training even with a length penalty. Note that the LRC ratio quickly decreases initially but then converges asymptotically to a constant value. (E). The maximum length of surviving connections also converges to a certain value above the LRC threshold (10 units). (F). Distribution of the lateral connection length in a small network after training, with (blue) and without (black) a length penalty. (G)-(H). LRCs scarcely survive after training in a small network. The LRC ratio and the maximum length of connections after training are significantly lower than those of a large network (insets). Error bars represent the standard deviation for 20 trials.