[2025-02-11] When Cortical Neurons Talk Sideways: Beyond Feedforward Visual Processing

Laurent Perrinet - https://laurentperrinet.github.io Séminaire Neuromathématiques, Collège de France

Hi, thanks for the introduction! I am Laurent Perrinet, a researcher in computational neuroscience and currently a research director at CNRS at the Institute of Neuroscience of la Timone in Marseille. Thank you for inviting me to participate in this “NeuroMathematics” seminar at the intersection of mathematics and neuroscience.

As an engineer by training, I could have pursued a career in aeronautics rather than becoming a neuroscientist. It is thanks to my mathematics professor Manuel Samuelides that I discovered the beauty of neural networks at the end of my engineering studies. This developped a curiosity, and thanks to him, I was also able to study in a mastere of cognitive sciences (now called CogMaster) in 1998. This is where I particularly want to acknowledge Jean Petitot - for his course I discovered how natural image statistics could link to principles in the central nervous system. This was a vivid revelation, and I’m grateful for his guidance in my academic path. Today’s seminar represents a return to these roots, as I’ll present my research progress since my mastere thesis on this very topic.

Today, I will address our current knowledge about horizontal connectivity rules in V1. Why is this important? As a matter of fact, one main function of sensory systems, such as the pivotal role of the primary visual cortex for vision, is to bind together the different visual features to help ultimately build a global perception.

Paysage catalan (Le Chasseur), Joan Miró (1924)

Trames (Étienne Rey)

Contour detection and the Association Field

Contour detection and the Association Field

Natural Images : Edges are on a common circle

A significant contribution to understanding the association field came from studying edge co-occurrences in natural images by Sigman et al. (2001). They quantified the probability density function of edge co-occurrences based on their relative positions and orientations. The figure demonstrates this by showing the spatial distribution patterns for edges relative to a reference edge at different orientations. For iso-oriented edges (a), the co-occurrence pattern shows clear structure. As the relative orientation increases through 22.5° (b), 45° (c), 67.5° (d), to 90° (e), distinct spatial patterns emerge.

A key finding was that for any given relative orientation between edges, the angle of maximal interaction occurs at the bisector between the orientations. This suggests that co-occurring edges tend to lie on a common circle - a property known as cocircularity. Panel (f) illustrates this geometrical principle: given two edges at angles w (red, 20°) and c (blue, 40°), the cocircularity solutions (green lines at 30° and 120°) represent the possible orientations of connecting circular arcs. This mathematical relationship provides insights into how the visual system might leverage statistical regularities in natural scenes for contour integration. We will go back into the details of this a bit further in the talk.

This association field concept provided a compelling framework for understanding how the visual system may implement contour integration through neural connectivity patterns. but before going there we should go back to the basic anatomy of the visual cortex.

Contour detection and the Association Field

Dynamics of vision

Human Visual system ([Grimaldi *et al* 2022](https://laurentperrinet.github.io/publication/grimaldi-22-polychronies/)) — Human Visual system (Grimaldi *et al* 2022)

Anatomy of the Human Visual system

Thalamic, short- & long-range lateral, inter-areal

A key feature of primary visual cortex is its layered organization, which is shared across cortical areas. The main thalamic input arrives in layer 4, which connects to a dense network of vertical connections across layers. These columns can then communicate via horizontal connections within layers. Hubel and Wiesel also proposed the ice-cube model that every point in the visual field produces a response in a 2 mm x 2 mm area of the cortex. Such an area can contain two complete groups of ocular dominance columns, 16 blobs and interblobs that may contain more than two times all of the orientations possible across 180 degrees. This region of the cortex, which Hubel and Wiesel called a hypercolumn (or, more generally, a cortical module) seems both necessary and sufficient for analyzing the image of a point in visual space. Because the cortex is a continuous cellular layer and because it is very hard to establish the boundaries of these modules physically, their existence from a functional standpoint is still the subject of debate. https://thebrain.mcgill.ca/flash/a/a_02/a_02_cl/a_02_cl_vis/a_02_cl_vis.html

Figure 9.2. Hypercolumn Diagram. Ocular dominance columns are segregated into left and right eye inputs. Orientation columns are neurons that get excited at different orientations and a cluster of these is called a pinwheel. Blobs are color selective and for every pinwheel there is a blob. (Credit: McGill: The Brain from Top to Bottom, Figure of hypercolumns, Copyleft https://copyleft.org/, https://thebrain.mcgill.ca/flash/a/a_02/a_02_cl/a_02_cl_vis/a_02_cl_vis.html. No modifications.)

From: https://pressbooks.umn.edu/sensationandperception/chapter/columns-and-hypercolumns-in-v1/

Thalamic, short- & long-range lateral, inter-areal

Anatomy of the Primary Visual Cortex

V1 is central to these pathways and shows distinctive anatomical and functional properties along with a complex topographical organization.

This figure from Kaschube et al. (2010) illustrates the organization of orientation preference maps in primary visual cortex (V1). Individual V1 neurons exhibit selective responses to oriented visual stimuli (as denoted by varying hues Colors code preferred ORs as indicated by the bars in (C)), with their spatial arrangement following highly structured patterns across the cortical surface. Panel B shows Synthetic orientation-maps of equal column spacing Λ but widely different pinwheel densities ρ. Left to right: solutions of different models: (13–16).. (C) High (blue frame) and low (orange frame) pinwheel density regions in tree shrew visual cortex. (D to F), Optically recorded orientation-maps in tree shrew (D), galago (E), and ferret (F) visual cortex. Regions shown in (C) are marked in (D). White arrows in (F) mark selected pinwheel centers. Framed regions in (C) and (F) are magnified. In many mammals including cats, monkeys and ferrets, orientation preference is organized in a quasi-periodic manner, forming what are known as orientation preference maps. These maps show remarkable consistency in their geometric properties across species, particularly in the spatial organization of pinwheel centers where orientation preferences converge.

However, this organization shows important species-specific variations. Most notably, while primates and carnivores display orderly orientation maps with smooth transitions between preferred orientations, rodents lack such maps and instead show a “salt-and-pepper” arrangement where neighboring neurons have seemingly random orientation preferences. This organizational diversity raises interesting questions about the computational advantages of these different architectures and their relationship to visual processing requirements and behavioral needs across species.

Horizontal connectivity links different hypercolumns

Contour detection and the Association Field

The like-to-like hypothesis

The resemblance between what was shown by Bosking and the structure of the association field that we saw above is such that it is tempting to align both and state that the function of horizontal connections is to bind neurons with a selectivity to similar orientations* over long distances. This like-to-like hypothesis has been influential in understanding horizontal connectivity patterns.

However, we should be cautious about overstating these relationships. While horizontal connections show some orientation specificity, recent evidence indicates the connectivity patterns are more complex and heterogeneous than initially proposed. The functional role of this diverse connectivity remains an active area of investigation.

During the remainder of this talk, I will try to shed light on our current knowledege on horizontal connectivities.

Supplementary: the HMAX model

[Serre and Poggio, 2007]

Supplementary: Convolutional Neural Nets (CNN)

[[Boutin *et al*, 2021](https://laurentperrinet.github.io/publication/boutin-franciosini-chavane-ruffier-perrinet-20/)] — [Boutin *et al*, 2021]

Supplementary: Orientation selectivity in V1

[Hubel & Wiesel, 1962]

Supplementary: Marr’s three levels of analysis

Marr, 1982

cut in different levels: Marr (+ Poggio)
arbitrary, but useful division of labor= computational / algorithm / hardware
here:
anatomy
algorithm / model
function

First: What is the anatomy of horizontal connections?

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[Marr, 1982]]

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Challenging the like-to-like hypothesis

[[Chavane, LP and Rankin, 2022]](https://laurentperrinet.github.io/publication/chavane-22/) — [Chavane, LP and Rankin, 2022]

Challenging the like-to-like hypothesis

Revisiting Horizontal Connectivity Rules in V1: From like-to-like towards like-to-All [[Chavane, LP and Rankin, 2022]](https://laurentperrinet.github.io/publication/chavane-22/) — Revisiting Horizontal Connectivity Rules in V1: From like-to-like towards like-to-All [Chavane, LP and Rankin, 2022]

This figure illustrates different hypothetical connectivity rules for horizontal connections in V1. The target neuron (large circle on left) has a specific orientation preference indicated by its color. Following the classical like-to-like hypothesis (shown in panel A), this neuron would preferentially connect to other neurons with matching orientation preference (similar colors) across multiple hypercolumns, as indicated by the vertical red arrows. The radial spread of connections spans approximately three hypercolumns, consistent with anatomical observations. Each hypercolumn contains a complete set of orientation preferences, represented by the different colored neurons.

This first schematic (noted A) represents one of the like-to-like connectivity rules, where horizontal connections strictly follow orientation similarity.

Challenging the like-to-like hypothesis

Let’s first shows some functional evidence.

This video shows voltage-sensitive dye imaging (VSDI) data from cat primary visual cortex (area 17) in response to a local oriented grating stimulus. The visualization reveals two key aspects:

The broader activation pattern shown by overall fluorescence changes (gray)
The more restricted orientation-selective response pattern (colored regions)

Two contours are overlaid: a red line marking the boundary of significant activation, and a white line delineating regions with statistically significant orientation selectivity. The orientation selectivity is encoded by color hue.

The bottom plots quantify the spatiotemporal dynamics by showing:

Left: The total activated cortical area over time
Right: The extent of orientation-selective regions over time

Together, these measurements demonstrate how orientation-selective signals propagate laterally beyond the classical feedforward input zone through horizontal connections, while maintaining some degree of feature selectivity.

Challenging the like-to-like hypothesis

This figure shows spatial and temporal dynamics of orientation selectivity in cat V1 analyzed from voltage-sensitive dye imaging data. Panel A displays a cortical orientation map averaged over the final 145ms of the response, where hue indicates preferred orientation and brightness shows orientation tuning strength. The dotted red line delineates the expected retinotopic boundary of feedforward input based on Albus (2004).

The inset quantitatively compares the spatial extent of:

Total cortical activation (grey contour)
Orientation-selective activation (black contour)
Theoretical feedforward input boundary (red contour)

This data demonstrates that orientation-selective responses propagate laterally beyond the classical feedforward input zone through horizontal connections, while maintaining some degree of feature selectivity. The systematic comparison between total activation and selective activation provides direct evidence for how horizontal connectivity shapes the spatiotemporal dynamics of orientation processing in V1.

Challenging the like-to-like hypothesis

Panel C displays intracellular recordings of subthreshold responses visualized as a visuotopic orientation polar map. The color hue represents preferred orientation while brightness indicates the strength of orientation tuning in the membrane potential. White contours outline regions showing statistically significant responses based on both amplitude and orientation selectivity criteria. The middle plots show averaged subthreshold responses to four different oriented stimuli (color-coded) at specific recording locations (marked by circle, triangle and square symbols), with scale bars indicating 50 ms and 1 mV. On the right, normalized orientation tuning curves are shown, computed by integrating responses within a fixed temporal window (shaded region in middle panel). The black circle marks the spontaneous activity level for the depolarizing integral measurement.

These shows a direct functional evidence for a diversity of tuning profile in th horizontal connectivity.

Challenging the like-to-like hypothesis

[[Voges and LP, 2012]](https://laurentperrinet.github.io/publication/voges-12/) — [Voges and LP, 2012]

Challenging the like-to-like hypothesis

Figure 4 illustrates a neural field model that bridges anatomical structure with functional observations in V1, as developed by Rankin and Chavane (2017).

Panel A depicts radial connectivity profiles with Gaussian-decaying inhibition and distance-dependent excitation that peaks periodically at multiples of distance L. The Ring Width (RW) parameter controls the spread of these excitatory peaks.

Panel B shows how local orientation preference maps influence lateral connectivity patterns under different orientation bias (BR) values in the recurrent connections.

Panel C quantifies the orientation tuning that emerges from these connectivity patterns. While orientations are uniformly represented globally, the local excitatory component shows strong bias around -60°. As BR increases above 0.5, the lateral connection orientation bias strengthens, reaching values around k=1 (consistent with Buzás et al. 2006).

Challenging the like-to-like hypothesis

Panel F maps the normalized selective area (relative to the feedforward footprint) across Ring Width (RW) and orientation bias (BR) parameters. White contours delineate anatomically plausible ranges where k values fall between 0.7-1.2, consistent with experimental measurements. The green region indicates parameter combinations that additionally satisfy constraints on both orientation preference and the observed radial decay of selectivity.

The neural field model effectively connects anatomical connectivity patterns with functional observations of orientation selectivity propagation in V1. The resulting connectivity structure exhibits similarities with “association field” patterns, suggesting potential optimization for encoding natural image statistics. This framework provides a quantitative basis for investigating computational principles underlying horizontal connectivity in visual cortex.

Challenging the like-to-like hypothesis

Modelling the Association field

Edge co-occurences in natural images

Panel A illustrates the groundbreaking approach developed by Geisler et al. (2001) for analyzing edge statistics in natural images. The method involves:

Detecting oriented edge elements in natural images (shown as red segments)
For each edge pair, measuring:
- Their relative orientation difference (𝜃)
- The relative position angle (𝜙)
- Center-to-center distance (d)
- Azimuth difference (φ)
- Co-circularity parameter ψ = φ - θ/2

This quantitative analysis reveals two key distributions:

A strong bias for parallel edge arrangements, evident in the orientation difference histogram
A marked preference for co-circular alignments, shown in the relative position histogram

These statistics vary significantly across image databases. For example, images containing animals exhibit enhanced co-circularity compared to general natural scenes. This suggests that rather than implementing a single fixed association field, the visual system may need to handle diverse statistical regularities present in natural inputs.

The next section will examine how these statistical regularities inform computational models of the association field.

Edge co-occurences in natural images

[Geisler, 2001]

Edge co-occurences in natural images

Edge co-occurrences can account for rapid categorization of natural versus animal images [[LP and Bednar, 2015]](https://laurentperrinet.github.io/publication/perrinet-bednar-15/) — Edge co-occurrences can account for rapid categorization of natural versus animal images [LP and Bednar, 2015]

The probability distribution function p(ψ,θ) represents the distribution of the different geometrical arrangements of edges’ angles, which we call a “chevron map”. We show here the histogram for non-animal natural images, illustrating the preference for co-linear edge configurations. For each chevron configuration, deeper and deeper red circles indicate configurations that are more and more likely with respect to a uniform prior, with an average maximum of about 3 times more likely, and deeper and deeper blue circles indicate configurations less likely than a flat prior (with a minimum of about 0.8 times as likely). Conveniently, this “chevron map” shows in one graph that non-animal natural images have on average a preference for co-linear and parallel edges, (the horizontal middle axis) and orthogonal angles (the top and bottom rows), along with a slight preference for co-circular configurations (for ψ =0 and ψ = ± π/2, just above and below the central row).

Edge co-occurences in natural images

This figure shows classification performance across image categories using different statistical features. We used an SVM classifier with three feature sets: first-order orientation statistics (FO), the reduced 2D “chevron map” (CM), and full 4D second-order statistics (SO). The classification accuracy (F1 score) was tested for distinguishing between image categories. Results show strong performance in separating man-made from natural images, as expected. More notably, the classifier achieved ~80% accuracy in discriminating animal vs non-animal natural images, matching human performance levels reported by Serre et al. This suggests that relatively simple edge co-occurrence statistics contain sufficient information for basic image categorization tasks, without requiring higher-level semantic processing.

We also found that our model made the same errors as humans do: if an image without an animal contains more co-circular edges, it is more likely to be falsely categorized as containing an animal.

Edge co-occurences in natural images

Can we explain the diversity ?

Indeed, this diversity is revealed in the anatomical data: V1 horizontal connectivity exhibits more complexity than suggested by the classical like-to-like hypothesis. While orientation-specific connections exist, they coexist with non-selective connections that link neurons irrespective of their tuning preferences. This diversity likely serves multiple computational functions:

Specific connections could support contour integration and feature binding
Non-selective connections may enable broad contextual modulation
Mixed connectivity patterns could help maintain network stability while preserving functional specificity

This anatomical heterogeneity aligns with V1’s role in both specialized feature detection and broader contextual processing. Understanding how these distinct connectivity patterns interact remains an active area of research in visual neuroscience.

Predictive processing

To bridge the gap between anatomical observations and functional requirements of visual processing, We added two key ingredients in the sparse deep predictive coding (SDPC) model :

Sparse connectivity patterns:
- Enforcing regularization of the activity map using L1 penalty
- Activity computed via recurrent local connectivity
- Similar to biological observations
Feedback from efferent layers:
- Predicts activity of afferent layer
- Only residual prediction error is processed
- Defines long-range inter-areal connectivity
- Specific influence demonstrated in Neural Computation paper

By defining a cost on minimizing the prediction error in each layer, everything stays derivable, such that we can use a classical gradient descent. These additions should allow us to better understand how feedback shapes visual processing in biological neural networks.

Predictive processing

More specifically in the context of our focus today, we can look at the co-occurence

llustration of the procedure to generate interaction map. In this illustrative example we consider a V1 representation with only 4 feature maps (represented in the upper-left box). Step 1 is to extract a neighborhood (of size 3x3 in the illustration only) around the most strongly activated neuron (represented with a red square in the illustration) for a given central preferred orientation (denoted ✓ c ). Step 2 is to normalize the neural activity in the extracted neighborhood using the marginal activity (see Eq.8). Step 3 is to compute the resulting orientation and activity at every position of the neighborhood using a circular mean (see Eq. 11 and Eq. 12 respectively). To keep a concise figure we have illustrated the computation of the central edge of the interaction map only. For simplification, the illustration shows only 1 neighborhood extraction whereas the interaction maps shown in the paper are computed by averaging neighborhoods centered on the 10 most strongly activated neurons

Predictive processing

Predictive processing with pooling

[[Boutin *et al*, 2022](https://laurentperrinet.github.io/publication/franciosini-21/)] — [Boutin *et al*, 2022]

Challenging the like-to-like hypothesis

To conclude, our review of horizontal connectivity in V1 reveals patterns more complex than initially theorized. The classical like-to-like hypothesis, while valuable, doesn’t fully capture the diversity of observed connectivity patterns.

Mathematical modeling has proven essential in bridging theory and biology. Our predictive processing framework shows how simple computational principles can explain the emergence of these complex connectivity patterns. The model demonstrates how feedback influences lateral interactions and reproduces key experimental observations.

However, important questions remain unanswered. We need to better understand how precise timing information is encoded in these circuits, how temporal dynamics shape processing, and whether similar principles apply across other cortical areas.

These fundamental questions will guide future experimental and theoretical work as we continue to unravel the computational principles of cortical processing.

[2025-02-11] When Cortical Neurons Talk Sideways: Beyond Feedforward Visual Processing

Laurent Perrinet - https://laurentperrinet.github.io Séminaire Neuromathématiques, Collège de France

Dynamics of vision

[[Marr, 1982](https://outde.xyz/2020-01-12/overappreciated-arguments-marrs-three-levels.html)] — [Marr, 1982]

Dynamics of vision

Visual latencies ([see review](https://laurentperrinet.github.io/publication/grimaldi-22-polychronies/)). — Visual latencies (see review).

Dynamics of vision

Sensorimotor delays ([Perrinet & Friston 2014](https://laurentperrinet.github.io/publication/perrinet-adams-friston-14/)) — Sensorimotor delays (Perrinet & Friston 2014)

Dynamics of vision

Diagonal markov model ([Khoei *et al*, 2017](https://laurentperrinet.github.io/publication/khoei-masson-perrinet-17/)). — Diagonal markov model (Khoei *et al*, 2017).

Dynamics of vision

Flash-lag effect: MBP (Khoei et al, 2017)

Dynamics of vision

Dynamics of vision: Neural modeling

Spiking Neural Networks: Spiking motifs

[Grimaldi *et al*, 2023, [Precise Spiking Motifs](https://laurentperrinet.github.io/publication/grimaldi-22-polychronies/)] — [Grimaldi *et al*, 2023, Precise Spiking Motifs]

Spiking Neural Networks: Spiking motifs

Review on [Precise Spiking Motifs](https://laurentperrinet.github.io/publication/grimaldi-22-polychronies/). — Review on Precise Spiking Motifs.

Spiking Neural Networks: Spiking motifs

Spiking Neural Networks: HD-SNN

[Grimaldi & LP (2023) Biol Cybernetics](https://laurentperrinet.github.io/publication/grimaldi-23-bc/) — Grimaldi & LP (2023) Biol Cybernetics

Spiking Neural Networks: HD-SNN

2 MINUTE

One advantage of this network is that it is differentiable, enabling us to apply classical machine learning methods, notably supervised learning. We then see the emergence of different convolution kernels, and here I represent a subset of its kernels for different directions, as denoted by the red arrows on the left of the graph. It shows the kernels obtained on the spatial representation according to the different columns, and each row represents the different delays from a delay of one on the right to a delay of 12 time steps on the left. Detectors that follow the motion emerge. For example, for the top line from top to bottom. These kernels integrate both positive neurons in red and negative polarity inputs in blue. Such spatio-temporal filtering is observed in neurobiology, but to my knowledge had never been observed in a model of spiking neurons trained under natural conditions.