[Paper Review] Cortical Divisive Normalization from Wilson-Cowan Neural Dynamics

Jesús Malo, José J. Esteve‐Taboada+1 more|arXiv (Cornell University)|Jun 19, 2019

Neural dynamics and brain function41 references4 citations

TL;DR

This paper derives divisive normalization—a key model of cortical nonlinear processing—from the Wilson-Cowan neural field equations by showing that divisive normalization emerges as the steady-state solution of Wilson-Cowan dynamics. The derived kernel is signal-dependent, explaining why ad-hoc modifications (e.g., high-pass filtering) were needed in prior divisive normalization models, and demonstrates that Wilson-Cowan dynamics can reproduce contrast responses and visual masking, providing a mechanistic foundation for signal-dependent normalization.

ABSTRACT

Divisive Normalization and the Wilson-Cowan equations are influential models of neural interaction and saturation [Carandini and Heeger Nat.Rev.Neurosci. 2012; Wilson and Cowan Kybernetik 1973]. However, they have not been analytically related yet. In this work we show that Divisive Normalization can be obtained from the Wilson-Cowan model. Specifically, assuming that Divisive Normalization is the steady state solution of the Wilson-Cowan differential equation, we find that the kernel that controls neural interactions in Divisive Normalization depends on the Wilson-Cowan kernel but also has a signal-dependent contribution. A standard stability analysis of a Wilson-Cowan model with the parameters obtained from our relation shows that the Divisive Normalization solution is a stable node. This stability demonstrates the consistency of our steady state assumption. The proposed theory provides a physiological foundation (a relation to a dynamical network with fixed wiring among neurons) for the functional suggestions that have been done on the need of signal-dependent Divisive Normalization [e.g. in Coen-Cagli et al., PLoS Comp.Biol. 2012]. Moreover, this theory explains the modifications that had to be introduced ad-hoc in Gaussian kernels of Divisive Normalization in [Martinez et al. Front. Neurosci. 2019] to reproduce contrast responses. The proposed relation implies that the Wilson-Cowan dynamics also reproduces visual masking and subjective image distortion metrics, which up to now had been mainly explained via Divisive Normalization. Finally, this relation allows to apply to Divisive Normalization the methods which up to now had been developed for dynamical systems such as Wilson-Cowan networks.

Motivation & Objective

• To establish a formal analytical link between two influential models of neural interaction: Wilson-Cowan equations and divisive normalization.
• To resolve the open question of whether divisive normalization can be derived from biophysically plausible neural dynamics.
• To explain why previous divisive normalization models required signal-dependent or modified kernels (e.g., high-pass filters) in contrast response modeling.
• To validate the stability of the divisive normalization solution within the Wilson-Cowan framework, supporting its use as a steady-state approximation.
• To demonstrate that Wilson-Cowan dynamics can reproduce visual masking and image distortion, traditionally explained only via divisive normalization.

Proposed method

• Assume that divisive normalization is the steady-state solution of the Wilson-Cowan differential equations.
• Derive the divisive normalization kernel as a function of the Wilson-Cowan interaction kernel, weighted by signal-dependent terms arising from the dynamics.
• Perform a standard stability analysis of the Wilson-Cowan system with parameters derived from the relation, confirming the solution is a stable node.
• Use Euler integration to simulate the Wilson-Cowan dynamics and verify convergence to the divisive normalization solution.
• Compare the derived kernel structure with prior models (e.g., Martinez et al., 2019) to explain the need for ad-hoc kernel modifications.
• Validate the model’s ability to reproduce contrast response curves and visual masking effects using simulated and real data.

Experimental results

Research questions

• RQ1 Can divisive normalization be formally derived as the steady-state solution of Wilson-Cowan neural dynamics?
• RQ2 Why do previous divisive normalization models require signal-dependent or modified kernels (e.g., high-pass filters) to fit contrast response data?
• RQ3 Does the Wilson-Cowan model with the derived kernel structure reproduce visual masking and contrast response curves observed in V1 cortex?
• RQ4 Is the divisive normalization solution stable within the Wilson-Cowan framework, justifying its use as a static approximation?
• RQ5 Can the Wilson-Cowan model serve as a mechanistic basis for divisive normalization, explaining its functional advantages?

Key findings

• The divisive normalization kernel is not fixed but depends on the signal through a nonlinear transformation of the Wilson-Cowan kernel, providing a mechanistic explanation for signal-dependent normalization.
• The derived kernel structure explains why high-pass filtering was required in Gaussian kernels of divisive normalization to match V1 contrast responses (Martinez et al., 2019).
• The Wilson-Cowan model with the derived parameters produces stable fixed points, confirming that the divisive normalization solution is a stable node, supporting the steady-state assumption.
• The model reproduces contrast response curves and visual masking effects, which were previously explained only via divisive normalization, suggesting a broader explanatory power for Wilson-Cowan dynamics.
• The dynamic Wilson-Cowan system converges to the divisive normalization solution in approximately 4–5 ms (with Δt = 10⁻⁵ s), suggesting that static divisive normalization is valid for slowly varying stimuli.
• The approach enables the development of new image quality metrics based on Wilson-Cowan dynamics, extending beyond existing divisive normalization-based metrics.