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Global modules robustly emerge from local interactions and smooth gradients

Nature volume 640, pages 155–164 (2025)Cite this article

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Abstract

Modular structure and function are ubiquitous in biology, from the organization of animal brains and bodies to the scale of ecosystems. However, the mechanisms of modularity emergence from non-modular precursors remain unclear. Here we introduce the principle of peak selection, a process by which purely local interactions and smooth gradients can drive the self-organization of discrete global modules. The process combines strengths of the positional and Turing pattern-formation mechanisms into a model for morphogenesis. Applied to the grid-cell system of the brain, peak selection results in the self-organization of functionally distinct modules with discretely spaced spatial periods. Applied to ecological systems, it results in discrete multispecies niches and synchronous spawning across geographically distributed coral colonies. The process exhibits self-scaling with system size and ‘topological robustness’1, which renders module emergence and module properties insensitive to most parameters. Peak selection ameliorates the fine-tuning requirement for continuous attractor dynamics in single grid-cell modules and it makes a detail-independent prediction that grid module period ratios should approximate adjacent integer ratios, providing a highly accurate match to the available data. Predictions for grid cells at the transcriptional, connectomic and physiological levels promise to elucidate the interplay of molecules, connectivity and function emergence in brains.

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Fig. 1: Positional versus pattern-forming mechanisms for structure formation and our hypothesis.
Fig. 2: Two local interactions with graded and fixed widths, respectively, lead to global module emergence.
Fig. 3: Multiscale linear instability, theory of topological peak selection and parameter-insensitive period ratio prediction.
Fig. 4: Emergent self-scaling of modules with system size.
Fig. 5: Enhanced robustness to weight heterogeneity, noise and activity perturbation by peak selection.
Fig. 6: Self-organization of ecological niches and synchronous coral spawning through peak selection.

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Data availability

Data were collected by running the codes available at https://github.com/FieteLab.

Code availability

Codes used to run the model and analyse data are available at https://github.com/FieteLab.

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Acknowledgements

We are grateful to M. Kardar for insightful discussions and to J. Widloski, A. Boopathy and L. Dong for comments on the manuscript. This work has been supported by ONR award N00014-19-1-2584, NSF-CISE award IIS-2151077 under the Robust Intelligence programme, ARO-MURI award W911NF-23-1-0277, Simons Foundation SCGB programme 1181110 and the MathWorks Science Fellowship.

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Author notes

  1. These authors contributed equally: Mikail Khona, Sarthak Chandra

Authors and Affiliations

  1. Department of Brain and Cognitive Sciences, MIT, Cambridge, MA, USA

    Mikail Khona, Sarthak Chandra & Ila Fiete

  2. McGovern Institute, MIT, Cambridge, MA, USA

    Mikail Khona, Sarthak Chandra & Ila Fiete

  3. Department of Physics, MIT, Cambridge, MA, USA

    Mikail Khona

Authors
  1. Mikail Khona
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  2. Sarthak Chandra
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Contributions

Conceptualization and writing were by I.F., M.K. and S.C. Funding was acquired by I.F. Coding and analysis were undertaken by M.K. and S.C.

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Correspondence to Sarthak Chandra or Ila Fiete.

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Nature thanks Matthew Nolan and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

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Extended data figures and tables

Extended Data Fig. 1 Naive addition of gradient in CAN does not form modularity.

(a-c) Naive merger of the two mechanisms by smoothly scaling the width of the pattern-forming lateral interaction in the grid cell CAN model18 does not generate global modularity in 2-dimensional (b) or 1-dimensional (c) grid models: the result is one smoothly varying periodic pattern.

Extended Data Fig. 2 Generalized peak-selection mechanism leads to modularity emergence.

(a) Energy landscape (Lyapunov function) for dynamics of the abstract state variable x consisting of a rugged multi-minimum function and a smooth, broad single-minimum function with minimum located at x*. (b) As a parameter θ is varied, x* varies as g(θ), where g is some monotonic function. (c) The resulting fixed points \(\bar{x}\), as a function of the smoothly varied θ, form sets with a constant value, followed by an abrupt jump to a new set of values, and so on in a series of discrete steps, defining a set of discrete modules. (See SI Sec. I for simulation details).

Supplementary information

Supplementary Information

Supplementary text, Figs. 1–17 and references.

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Supplementary Video

Peak selection process in Fourier space for a graded kernel and a fixed-scale kernel. Although the continuous varying pattern-forming kernel (in red) has a smoothly varying maxima, when it is added to a fixed-scale kernel (in blue), the composite interaction (in purple) has a maximum that changes in discrete jumps (denoted by black circle). Increasing time in the animation corresponds varying dorsoventral position. The discrete changes in the location of the maximum, despite continuous gradients, result in the formation of grid-cell modules in a single attractor sheet with smoothly varying continuous biophysical gradients.

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Khona, M., Chandra, S. & Fiete, I. Global modules robustly emerge from local interactions and smooth gradients. Nature 640, 155–164 (2025). https://doi.org/10.1038/s41586-024-08541-3

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