Functional architecture of M1 cells encoding movement direction.

In this paper we propose a neurogeometrical model of the behaviour of cells of the arm area of the primary motor cortex (M1). We will mathematically express as a fiber bundle the hypercolumnar organization of this cortical area, first modelled by Georgopoulos (Georgopoulos et al., 1982; Georgopoulos, 2015). On this structure, we will consider the selective tuning of M1 neurons of kinematic variables of positions and directions of movement. We will then extend this model to encode the notion of fragments introduced by Hatsopoulos et al. (2007) which describes the selectivity of neurons to movement direction varying in time. This leads to consider a higher dimensional geometrical structure where fragments are represented as integral curves. A comparison with the curves obtained through numerical simulations and experimental data will be presented. Moreover, neural activity shows coherent behaviours represented in terms of movement trajectories pointing to a specific pattern of movement decomposition Kadmon Harpaz et al. (2019). Here, we will recover this pattern through a spectral clustering algorithm in the subriemannian structure we introduced, and compare our results with the neurophysiological one of Kadmon Harpaz et al. (2019).

LGN-CNN: A biologically inspired CNN architecture.

In this paper we introduce a biologically inspired Convolutional Neural Network (CNN) architecture called LGN-CNN that has a first convolutional layer composed of a single filter that mimics the role of the Lateral Geniculate Nucleus (LGN). The first layer of the neural network shows a rotational symmetric pattern justified by the structure of the net itself that turns up to be an approximation of a Laplacian of Gaussian (LoG). The latter function is in turn a good approximation of the receptive field profiles (RFPs) of the cells in the LGN. The analogy with the visual system is established, emerging directly from the architecture of the neural network. A proof of rotation invariance of the first layer is given on a fixed LGN-CNN architecture and the computational results are shown. Thus, contrast invariance capability of the LGN-CNN is investigated and a comparison between the Retinex effects of the first layer of LGN-CNN and the Retinex effects of a LoG is provided on different images. A statistical study is done on the filters of the second convolutional layer with respect to biological data. In conclusion, the model we have introduced approximates well the RFPs of both LGN and V1 attaining similar behavior as regards long range connections of LGN cells that show Retinex effects.

Emergence of Lie Symmetries in Functional Architectures Learned by CNNs.

In this paper we study the spontaneous development of symmetries in the early layers of a Convolutional Neural Network (CNN) during learning on natural images. Our architecture is built in such a way to mimic some properties of the early stages of biological visual systems. In particular, it contains a pre-filtering step ℓ0 defined in analogy with the Lateral Geniculate Nucleus (LGN). Moreover, the first convolutional layer is equipped with lateral connections defined as a propagation driven by a learned connectivity kernel, in analogy with the horizontal connectivity of the primary visual cortex (V1). We first show that the ℓ0 filter evolves during the training to reach a radially symmetric pattern well approximated by a Laplacian of Gaussian (LoG), which is a well-known model of the receptive profiles of LGN cells. In line with previous works on CNNs, the learned convolutional filters in the first layer can be approximated by Gabor functions, in agreement with well-established models for the receptive profiles of V1 simple cells. Here, we focus on the geometric properties of the learned lateral connectivity kernel of this layer, showing the emergence of orientation selectivity w.r.t. the tuning of the learned filters. We also examine the short-range connectivity and association fields induced by this connectivity kernel, and show qualitative and quantitative comparisons with known group-based models of V1 horizontal connections. These geometric properties arise spontaneously during the training of the CNN architecture, analogously to the emergence of symmetries in visual systems thanks to brain plasticity driven by external stimuli.

Correction to: From receptive profiles to a metric model of V1.

The authors would like to note an omission, in the published paper, of the Matlab code initially included as Electronic Supplementary Material. Therefore, we hereby re-submit the code in question.

From receptive profiles to a metric model of V1.

In this work we show how to construct connectivity kernels induced by the receptive profiles of simple cells of the primary visual cortex (V1). These kernels are directly defined by the shape of such profiles: this provides a metric model for the functional architecture of V1, whose global geometry is determined by the reciprocal interactions between local elements. Our construction adapts to any bank of filters chosen to represent a set of receptive profiles, since it does not require any structure on the parameterization of the family. The connectivity kernel that we define carries a geometrical structure consistent with the well-known properties of long-range horizontal connections in V1, and it is compatible with the perceptual rules synthesized by the concept of association field. These characteristics are still present when the kernel is constructed from a bank of filters arising from an unsupervised learning algorithm.

Curvature Integration in a 5D Kernel for Extracting Vessel Connections in Retinal Images.

Tree-like structures, such as retinal images, are widely studied in computer-aided diagnosis systems for large-scale screening programs. Despite several segmentation and tracking methods proposed in the literature, there still exist several limitations specifically when two or more curvilinear structures cross or bifurcate, or in the presence of interrupted lines or highly curved blood vessels. In this paper, we propose a novel approach based on multi-orientation scores augmented with a contextual affinity matrix, which both are inspired by the geometry of the primary visual cortex (V1) and their contextual connections. The connectivity is described with a 5D kernel obtained as the fundamental solution of the Fokker-Planck equation modeling the cortical connectivity in the lifted space of positions, orientations, curvatures, and intensity. It is further used in a self-tuning spectral clustering step to identify the main perceptual units in the stimuli. The proposed method has been validated on several easy as well as challenging structures in a set of artificial images and actual retinal patches. Supported by quantitative and qualitative results, the method is capable of overcoming the limitations of current state-of-the-art techniques.

Local and Global Gestalt Laws: A Neurally Based Spectral Approach.

This letter presents a mathematical model of figure-ground articulation that takes into account both local and global gestalt laws and is compatible with the functional architecture of the primary visual cortex (V1). The local gestalt law of good continuation is described by means of suitable connectivity kernels that are derived from Lie group theory and quantitatively compared with long-range connectivity in V1. Global gestalt constraints are then introduced in terms of spectral analysis of a connectivity matrix derived from these kernels. This analysis performs grouping of local features and individuates perceptual units with the highest salience. Numerical simulations are performed, and results are obtained by applying the technique to a number of stimuli.

Cortical spatiotemporal dimensionality reduction for visual grouping.

The visual systems of many mammals, including humans, are able to integrate the geometric information of visual stimuli and perform cognitive tasks at the first stages of the cortical processing. This is thought to be the result of a combination of mechanisms, which include feature extraction at the single cell level and geometric processing by means of cell connectivity. We present a geometric model of such connectivities in the space of detected features associated with spatiotemporal visual stimuli and show how they can be used to obtain low-level object segmentation. The main idea is to define a spectral clustering procedure with anisotropic affinities over data sets consisting of embeddings of the visual stimuli into higher-dimensional spaces. Neural plausibility of the proposed arguments will be discussed.

The constitution of visual perceptual units in the functional architecture of V1.

In this paper we show that the emergence of perceptual units in V1 can be explained in terms of a physical mechanism of simmetry breaking of the mean field neural equation. We consider a mean field neural model which takes into account the functional architecture of the visual cortex modeled as a group of rotations and translations equipped with a degenerate metric. The model generalizes well known results of Bressloff and Cowan which, in absence of input, accounts for hallucination patterns. The main result of our study consists in showing that in presence of a visual input, the stable eigenmodes of the linearized operator represent perceptual units of the visual stimulus. The result is strictly related to dimensionality reduction and clustering problems.

An uncertainty principle underlying the functional architecture of V1.

We present a model of the morphology of orientation maps in V1 based on the uncertainty principle of the SE(2) group. Starting from the symmetries of the cortex, suitable harmonic analysis instruments are used to obtain coherent states in the Fourier domain as minimizers of the uncertainty. Cortical activities related to orientation maps are then obtained by projection on a suitable cortical Fourier basis.

A model of natural image edge co-occurrence in the rototranslation group.

In this paper, we propose to model the edge information contained in natural scenes as points in the 3D space of positions and orientations. This space is equipped with a strong geometrical structure and it is identified as the rototranslation group. In this space, we compute a histogram of co-occurrence of edges from a database of natural images and show that it can be interpreted as a probability density function, expressed by the fundamental solution of a suitable Fokker-Planck equation defined in the 3D structured space. Both estimated statistics and model predictions are reconsidered and compared with the partial gestalt association fields proposed by D. J. Field, A. Hayes, and R. F. Hess (1993). Finally, parametric identification allows to estimate the variance of the co-occurrence random process in natural images.

Functional geometry of the horizontal connectivity in the primary visual cortex.

We present a geometrical model of the functional architecture of the primary visual cortex. In particular we describe the geometric structure of connections found both in neurophysiological and psychophysical experiments, modeling both co-axial and trans-axial excitatory connections. The model shows what could be the deep structure for both boundary and figure completion and for morphological structures such as the medial axis of a shape.

The symplectic structure of the primary visual cortex.

We propose to model the functional architecture of the primary visual cortex V1 as a principal fiber bundle where the two-dimensional retinal plane is the base manifold and the secondary variables of orientation and scale constitute the vertical fibers over each point as a rotation-dilation group. The total space is endowed with a natural symplectic structure neurally implemented by long range horizontal connections. The model shows what could be the deep structure for both boundary and figure completion and for morphological structures, such as the medial axis of a shape.

From neural oscillations to variational problems in the visual cortex.

Aim of this study is to provide a formal link between connectionist neural models and variational psycophysical ones. We show that the solution of phase difference equation of weakly connected neural oscillators gamma-converges as the dimension of the grid tends to 0, to the gradient flow relative to the Mumford-Shah functional in a Riemannian space. The Riemannian metric is directly induced by the pattern of neural connections. Next, we embed the energy functional in the specific geometry of the functional space of the primary visual cortex, that is described in terms of a subRiemannian Heisenberg space. Namely, we introduce the Mumford-Shah functional with the Heisenberg metric and discuss the applicability of our main gamma-convergence result to subRiemannian spaces.