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  • Open Access

Impact of Dendritic Nonlinearities on the Computational Capabilities of Neurons

Clarissa Lauditi1, Enrico M. Malatesta2, Fabrizio Pittorino3,*, Carlo Baldassi2, Nicolas Brunel2,4, and Riccardo Zecchina2

  • 1Department of Applied Math, John A. Paulson School of Engineering and Applied Sciences, Harvard University, 02138 Cambridge, Massachusetts, USA
  • 2Department of Computing Sciences and Bocconi Institute for Data Science and Analytics (BIDSA), Bocconi University, 20136 Milano, Italy
  • 3Department of Electronics, Information and Bioengineering, Politecnico di Milano, 20133 Milano, Italy
  • 4Departments of Neurobiology and Physics, Duke University, Durham, North Carolina, USA

  • *Contact author: fabrizio.pittorino@polimi.it
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PRX Life 3, 033003 – Published 8 July, 2025

DOI: https://doi.org/10.1103/d7f1-xc8q

Abstract

How neurons integrate the myriad synaptic inputs scattered across their dendrites is a fundamental question in neuroscience. Multiple neurophysiological experiments have shown that dendritic nonlinearities can have a strong influence on synaptic input integration. These nonlinearities have motivated mathematical descriptions of single neurons as a two-layer computational units, which have been shown to increase substantially the computational abilities of neurons, compared to linear dendritic integration. However, current analytical studies are restricted to neurons with unconstrained synaptic weights and unplausible dendritic nonlinearities. Here we introduce a two-layer model with sign-constrained synaptic weights and a biologically plausible form of dendritic nonlinearity and investigate its properties using both statistical physics methods and numerical simulations. We find that the dendritic nonlinearity enhances both the number of possible learned input-output associations and the learning velocity. We characterize how capacity and learning speed depend on the implemented nonlinearity and the levels of dendritic and somatic inhibition. We calculate analytically the distribution of synaptic weights in networks close to maximal capacity and find that the dendritic nonlinearity increases the fraction of zero-weight (“silent” or “potential”) synapses, compared with the standard perceptron model, when no or weak robustness constraints are present, while the opposite occurs with strong robustness constraints. We test our model on standard real-world benchmark datasets and observe empirically that the nonlinearity provides an enhancement in generalization performance and that it enables to capture more complex input-output relations, compared to the perceptron model.

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References (68)

  1. M. Minsky and S. A. Papert, Perceptrons—Expanded Edition (MIT Press, Cambridge, MA, 1988).
  2. T. M. Cover, Geometrical and statistical properties of systems of linear inequalities with applications in pattern recognition., IEEE Trans EC-14, 326 (1965).
  3. E. Gardner, The space of interactions in neural network models, J. Phys. A: Math. Gen. 21, 257 (1988).
  4. A. Engel and C. Van den Broeck, Statistical Mechanics of Learning (Cambridge University Press, Cambridge, UK, 2001).
  5. M. London and M. Häusser, Dendritic computation, Annu. Rev. Neurosci. 28, 503 (2005).
  6. M. Larkum, A cellular mechanism for cortical associations: An organizing principle for the cerebral cortex, Trends Neurosci. 36, 141 (2013).
  7. G. Major, M. E. Larkum, and J. Schiller, Active properties of neocortical pyramidal neuron dendrites, Annu. Rev. Neurosci. 36, 1 (2013).
  8. J. Schiller, G. Major, H. J. Koester, and Y. Schiller, NMDA spikes in basal dendrites of cortical pyramidal neurons, Nature (Lond.) 404, 285 (2000).
  9. T. Branco and M. Hăusser, Synaptic integration gradients in single cortical pyramidal cell dendrites, Neuron 69, 885 (2011).
  10. M. E. Larkum, J. J. Zhu, and B. Sakmann, A new cellular mechanism for coupling inputs arriving at different cortical layers, Nature (Lond.) 398, 338 (1999).
  11. M. E. Larkum, W. Senn, and H.-R. Luscher, Top-down dendritic input increases the gain of layer 5 pyramidal neurons, Cereb. Cortex 14, 1059 (2004).
  12. T. Nevian, M. E. Larkum, A. Polsky, and J. Schiller, Properties of basal dendrites of layer 5 pyramidal neurons: A direct patch-clamp recording study, Nat Neurosci. 10, 206 (2007).
  13. A. Polsky, B. W. Mel, and J. Schiller, Computational subunits in thin dendrites of pyramidal cells, Nat. Neurosci. 7, 621 (2004).
  14. P. Poirazi, T. Brannon, and B. W. Mel, Pyramidal neuron as two-layer neural network, Neuron 37, 989 (2003).
  15. N. Brunel, V. Hakim, and M. J. Richardson, Single neuron dynamics and computation, Curr. Opin. Neurobiol. 25, 149 (2014).
  16. B. B. Ujfalussy, J. K. Makara, M. Lengyel, and T. Branco, Global and multiplexed dendritic domputations under in-vivo-like conditions, Neuron 100, 579 (2018).
  17. M. E. Larkum, T. Nevian, M. Sandler, A. Polsky, and J. Schiller, Synaptic integration in tuft dendrites of layer 5 pyramidal neurons: A new unifying principle, Science 325, 756 (2009).
  18. D. Beniaguev, I. Segev, and M. London, Single cortical neurons as deep artificial neural networks, Neuron 109, 2727 (2021).
  19. M. Pagkalos, R. Makarov, and P. Poirazi, Leveraging dendritic properties to advance machine learning and neuro-inspired computing, Curr. Opin. Neurobiol. 85, 102853 (2024).
  20. S. Sardi, R. Vardi, A. Sheinin, A. Goldental, and I. Kanter, New types of experiments reveal that a neuron functions as multiple independent threshold units, Sci. Rep. 7, 18036 (2017).
  21. G. J. Mitchison and R. M. Durbin, Bounds on the learning capacity of some multi-layer networks, Biol. Cybern. 60, 345 (1989).
  22. E. Barkai, D. Hansel, and H. Sompolinsky, Broken symmetries in multilayered perceptrons, Phys. Rev. A 45, 4146 (1992).
  23. R. Monasson and R. Zecchina, Weight space structure and internal representations: A direct approach to learning and generalization in multilayer neural networks, Phys. Rev. Lett. 75, 2432 (1995).
  24. C. Baldassi, E. M. Malatesta, and R. Zecchina, Properties of the geometry of solutions and capacity of multilayer neural networks with rectified linear unit activations, Phys. Rev. Lett. 123, 170602 (2019).
  25. J. A. Zavatone-Veth and C. Pehlevan, Activation function dependence of the storage capacity of treelike neural networks, Phys. Rev. E 103, L020301 (2021).
  26. C. Baldassi, C. Borgs, J. T. Chayes, A. Ingrosso, C. Lucibello, L. Saglietti, and R. Zecchina, Unreasonable effectiveness of learning neural networks: From accessible states and robust ensembles to basic algorithmic schemes, Proc. Natl. Acad. Sci. USA 113, E7655 (2016).
  27. C. Baldassi, F. Pittorino, and R. Zecchina, Shaping the learning landscape in neural networks around wide flat minima, Proc. Natl. Acad. Sci. USA 117, 161 (2020).
  28. C. Lucibello, F. Pittorino, G. Perugini, and R. Zecchina, Deep learning via message passing algorithms based on belief propagation, Mach. Learn.: Sci. Technol. 3, 035005 (2022).
  29. B. L. Annesi, C. Lauditi, C. Lucibello, E. M. Malatesta, G. Perugini, F. Pittorino, and L. Saglietti, Star-shaped space of solutions of the spherical negative perceptron, Phys. Rev. Lett. 131, 227301 (2023).
  30. K. Hornik, Approximation capabilities of multilayer feedforward networks, Neural Netw. 4, 251 (1991).
  31. N. Brunel, V. Hakim, P. Isope, J.-P. Nadal, and B. Barbour, Optimal information storage and the distribution of synaptic weights: Perceptron versus purkinje cell, Neuron 43, 745 (2004).
  32. N. Brunel, Is cortical connectivity optimized for storing information? Nat. Neurosci. 19, 749 (2016).
  33. V. Braitenberg and A. Schütz, Anatomy of the Cortex (Springer-Verlag, Berlin, 1991).
  34. D. M. Iascone, Y. Li, U. l, M. Doron, H. Chen, V. Andreu, F. Goudy, H. Blockus, L. F. Abbott, I. Segev, H. Peng, and F. Polleux, Whole-neuron synaptic mapping reveals spatially precise excitatory/inhibitory balance limiting dendritic and somatic spiking, Neuron 106, 566 (2020).
  35. G. N. Elston, R. Benavides-Piccione, A. Elston, P. R. Manger, and J. Defelipe, Pyramidal cells in prefrontal cortex of primates: Marked differences in neuronal structure among species, Front. Neuroanat. 5, 2 (2011).
  36. E. Gardner and B. Derrida, Optimal storage properties of neural network models, J. Phys. A: Math. Gen. 21, 271 (1988).
  37. D. Amit, K. Y. Wong, and C. Campbell, Perceptron learning with sign-constrained weights, J. Phys. A: Math. Gen. 22, 2039 (1989).
  38. M. Stojnic, Capacity of the treelike sign perceptrons neural networks with one hidden layer–RDT based upper bounds, arXiv:2312.08244.
  39. Y. LeCun, L. D. Jackel, L. Bottou, A. Brunot, C. Cortes, J. S. Denker, H. Drucker, I. Guyon, U. A. Muller, E. Sackinger, P. Simard, and V. Vapnik, Comparison of learning algorithms for handwritten digit recognition, in International Conference on Artificial Neural Networks, edited by F. Fogelman and P. Gallinari (EC2 & Cie, Paris, 1995), pp. 53–60.
  40. H. Xiao, K. Rasul, and R. Vollgraf, Fashion-MNIST: A novel image dataset for benchmarking machine learning algorithms, arXiv:1708.07747.
  41. A. Krizhevsky, Learning Multiple Layers of Features from Tiny Images (Canadian Institute for Advanced Research, Toronto, 2009).
  42. Code available at https://gitlab.com/bocconi-artlab/biologicalneuron.
  43. E. M. Malatesta, High-dimensional manifold of solutions in neural networks: Insights from statistical physics, arXiv:2309.09240.
  44. M. Mezard, G. Parisi, and M. Virasoro, Spin Glass Theory and Beyond: An Introduction to the Replica Method and Its Applications, Vol. 9 (World Scientific, Singapore, 1987).
  45. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory (Springer Science & Business Media, New York, 2013).
  46. C. Clopath, J.-P. Nadal, and N. Brunel, Storage of correlated patterns in standard and bistable Purkinje cell models, PLoS Comput. Biol. 8, e1002448 (2012).
  47. G. Parisi, Infinite number of order parameters for spin-glasses, Phys. Rev. Lett. 43, 1754 (1979).
  48. A. Engel, H. M. Köhler, F. Tschepke, H. Vollmayr, and A. Zippelius, Storage capacity and learning algorithms for two-layer neural networks, Phys. Rev. A 45, 7590 (1992).
  49. C. Baldassi, E. M. Malatesta, G. Perugini, and R. Zecchina, Typical and atypical solutions in nonconvex neural networks with discrete and continuous weights, Phys. Rev. E 108, 024310 (2023).
  50. B. L. Annesi, E. M. Malatesta, and F. Zamponi, Exact full-RSB SAT/UNSAT transition in infinitely wide two-layer neural networks, SciPost Phys. 18, 118 (2025).
  51. C. Baldassi, A. Ingrosso, C. Lucibello, L. Saglietti, and R. Zecchina, Subdominant dense clusters allow for simple learning and high computational performance in neural networks with discrete synapses, Phys. Rev. Lett. 115, 128101 (2015).
  52. C. Baldassi, C. Lauditi, E. M. Malatesta, G. Perugini, and R. Zecchina, Unveiling the structure of wide flat minima in neural networks, Phys. Rev. Lett. 127, 278301 (2021).
  53. P. Isope and B. Barbour, Properties of unitary granule cell: Purkinje cell synapses in adult rat cerebellar slices, J. Neurosci. 22, 9668 (2002).
  54. S. Song, P. J. Sjöström, M. Reigl, S. Nelson, and D. B. Chklovskii, Highly nonrandom features of synaptic connectivity in local cortical circuits, PLoS Biol. 3, e68 (2005).
  55. H. Markram, J. Lubke, M. Frotscher, A. Roth, and B. Sakmann, Physiology and anatomy of synaptic connections between thick tufted pyramidal neurones in the developing rat neocortex, J. Physiol. 500, 409 (1997).
  56. P. J. Sjöström, G. G. Turrigiano, and S. Nelson, Rate, timing, and cooperativity jointly determine cortical synaptic plasticity, Neuron 32, 1149 (2001).
  57. S. Lefort, C. Tomm, J. C. Floyd Sarria, and C. C. Petersen, The excitatory neuronal network of the C2 barrel column in mouse primary somatosensory cortex, Neuron 61, 301 (2009).
  58. L. Campagnola, S. C. Seeman, T. Chartrand, L. Kim, A. Hoggarth, C. Gamlin, S. Ito, J. Trinh, P. Davoudian, C. Radaelli, M.-H. Kim, T. Hage, T. Braun, L. Alfiler, J. Andrade, P. Bohn, R. Dalley, A. Henry, S. Kebede, A. Mukora, D. Sandman, G. Williams, R. Larsen, C. Teeter, T. L. Daigle, K. Berry, N. Dotson, R. Enstrom, M. Gorham, M. Hupp, S. D. Lee, K. Ngo, P. R. Nicovich, L. Potekhina, S. Ransford, A. Gary, J. Goldy, D. McMillen, T. Pham, M. Tieu, L. Siverts, M. Walker, C. Farrell, M. Schroedter, C. Slaughterbeck, C. Cobb, R. Ellenbogen, R. P. Gwinn, C. D. Keene, A. L. Ko, J. G. Ojemann, D. L. Silbergeld, D. Carey, T. Casper, K. Crichton, M. Clark, N. Dee, L. Ellingwood, J. Gloe, M. Kroll, J. Sulc, H. Tung, K. Wadhwani, K. Brouner, T. Egdorf, M. Maxwell, M. McGraw, C. A. Pom, A. Ruiz, J. Bomben, D. Feng, N. Hejazinia, S. Shi, A. Szafer, W. Wakeman, J. Phillips, A. Bernard, L. Esposito, F. D. D'Orazi, S. Sunkin, K. Smith, B. Tasic, A. Arkhipov, S. Sorensen, E. Lein, C. Koch, G. Murphy, H. Zeng, and T. Jarsky, Local connectivity and synaptic dynamics in mouse and human neocortex, Science 375, eabj5861 (2022).
  59. F. Pittorino, C. Lucibello, C. Feinauer, G. Perugini, C. Baldassi, E. Demyanenko, and R. Zecchina, Entropic gradient descent algorithms and wide flat minima, in International Conference on Learning Representations (ICLR, Vienna, Austria, 2021).
  60. A. A. Faisal, L. P. Selen, and D. M. Wolpert, Noise in the nervous system, Nat. Rev. Neurosci. 9, 292 (2008).
  61. F. Pittorino, A. Ferraro, G. Perugini, C. Feinauer, C. Baldassi, and R. Zecchina, Deep networks on toroids: Removing symmetries reveals the structure of flat regions in the landscape geometry, in Proceedings of the 39th International Conference on Machine Learning, Proceedings of Machine Learning Research, Vol. 162, edited by K. Chaudhuri, S. Jegelka, L. Song, C. Szepesvari, G. Niu, and S. Sabato (PMLR, Baltimore, MD, 2022), pp. 17759–17781.
  62. A. Gidon, T. A. Zolnik, P. Fidzinski, F. Bolduan, A. Papoutsi, P. Poirazi, M. Holtkamp, I. Vida, and M. E. Larkum, Dendritic action potentials and computation in human layer 2/3 cortical neurons, Science 367, 83 (2020).
  63. J. H. Kirchner and J. Gjorgjieva, Emergence of synaptic organization and computation in dendrites, Neuroforum 28, 21 (2022).
  64. J. Chapeton, T. Fares, D. LaSota, and A. Stepanyants, Efficient associative memory storage in cortical circuits of inhibitory and excitatory neurons, Proc. Natl. Acad. Sci. USA 109, E3614 (2012).
  65. R. Rubin, L. F. Abbott, and H. Sompolinsky, Balanced excitation and inhibition are required for high-capacity, noise-robust neuronal selectivity, Proc. Natl. Acad. Sci. USA 114, E9366 (2017).
  66. R. Gütig and H. Sompolinsky, The tempotron: A neuron that learns spike timing-based decisions, Nat. Neurosci. 9, 420 (2006).
  67. F. Rosenblatt, Principles of Neurodynamics: Perceptrons and the Theory of Brain Mechanisms (Spartan Books, Washington, DC, 1962).
  68. F. S. Werblin and J. E. Dowling, Organization of the retina of the mudpuppy necturus maculosus. ii. intracellular recordings, J. Neurophysiol. 32, 339 (1969).

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PRX Life 3, 033003– Published 8 July, 2025
  • Received 9 January 2025
  • Accepted 16 June 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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