- Physical Review Journals
All JournalsPhysics Magazine

PRX Life

Reuse & Permissions

It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

Export citation

Export citation

Choose format for download:

Download Citation
  • Open Access

Stochastic Synaptic Dynamics Under Learning

Jakob Stubenrauch1,2,*, Naomi Auer3, Richard Kempter1,3,4, and Benjamin Lindner1,2

  • *Contact author: jakob.stubenrauch@rwth-aachen.de
PDF

PRX Life 4, 023006 – Published 13 April, 2026

DOI: https://doi.org/10.1103/r21z-29nv

Abstract

Learning is based on synaptic plasticity, which affects and is driven by neural activity. Because pre- and postsynaptic spiking activity is shaped by randomness, the synaptic weights follow a stochastic process, requiring a probabilistic framework to capture the noisy synaptic dynamics. We consider a paradigmatic supervised learning example: a presynaptic neural population impinging in a sequence of episodes on a recurrent network of integrate-and-fire neurons through synapses undergoing spike-timing-dependent plasticity (STDP) with additive potentiation and multiplicative depression. We first analytically compute the drift and diffusion coefficients for a single synapse within a single episode (microscopic dynamics), mapping the true jump process to a Langevin and the associated Fokker-Planck equations. Leveraging recent analytical tools, we include spike-time-resolving cross-correlations between pre- and postsynaptic spikes, which corrects substantial deviations seen in standard theories purely based on firing rates. We then apply this microdynamical description to the network setup in which hetero-associations are trained over one-shot episodes into a feed-forward matrix of STDP synapses connecting to neurons of the recurrent network (macroscopic dynamics). By mapping statistically distinct synaptic populations to instances of the single-synapse process above, we self-consistently determine the joint neural and synaptic dynamics and, ultimately, the time course of memory degradation and the memory capacity. We demonstrate that specifically in the relevant case of sparse coding, our theory can quantitatively capture memory capacities which are strongly overestimated if spike-time–resolving cross-correlations are ignored. On the single-synapse level, our theory predicts a transition between acceleration and deceleration of the synaptic weight, and a possibly nonmonotonic dependence of the finite-time diffusion coefficient on the synaptic weight. On the memory level, we showcase two complementary roles of homeostatic plasticity: namely, it is both responsible for forgetting and a requirement for the ability to learn. Additionally, we show how strength of plasticity and duration of exposure to stimuli must be scaled to evade a tradeoff dilemma between memory capacity and early accuracy. We conclude with a discussion of the many directions in which our framework can be extended.

View figure in article

Physics Subject Headings (PhySH)

Article Text

References (72)

  1. L. F. Abbott and S. B. Nelson, Synaptic plasticity: Taming the beast, Nat. Neurosci. 3, 1178 (2000).
  2. J. Sjöström and W. Gerstner, Spike-timing dependent plasticity, Scholarpedia 5, 1362 (2010).
  3. W. Gerstner, R. Kempter, J. L. van Hemmen, and H. Wagner, A neuronal learning rule for sub-millisecond temporal coding, Nature (London) 383, 76 (1996).
  4. H. Markram, J. Lübke, 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).
  5. G. Bi and M. Poo, Synaptic modifications in cultured hippocampal neurons: Dependence on spike timing, synaptic strength, and postsynaptic cell type, J. Neurosci. 18, 10464 (1998).
  6. G. Bi and M. Poo, Synaptic modification by correlated activity: Hebb's postulate revisited, Annu. Rev. Neurosci. 24, 139 (2001).
  7. N. Yasumatsu, M. Matsuzaki, T. Miyazaki, J. Noguchi, and H. Kasai, Principles of long-term dynamics of dendritic spines, J. Neurosci. 28, 13592 (2008).
  8. R. Rosenbaum, J. E. Rubin, and B. Doiron, Short term synaptic depression imposes a frequency dependent filter on synaptic information transfer, PLoS Comput. Biol. 8, e1002557 (2012).
  9. R. Rosenbaum, J. E. Rubin, and B. Doiron, Short-term synaptic depression and stochastic vesicle dynamics reduce and shape neuronal correlations, J. Neurophysiol. 109, 475 (2013).
  10. A. Statman, M. Kaufman, A. Minerbi, N. E. Ziv, and N. Brenner, Synaptic size dynamics as an effectively stochastic process, PLoS Comput. Biol. 10, e1003846 (2014).
  11. R. Kempter, W. Gerstner, and J. L. van Hemmen, Hebbian learning and spiking neurons, Phys. Rev. E 59, 4498 (1999).
  12. M. C. W. van Rossum, G. Q. Bi, and G. G. Turrigiano, Stable Hebbian learning from spike timing-dependent plasticity, J. Neurosci. 20, 8812 (2000).
  13. A. N. Burkitt, H. Meffin, and D. B. Grayden, Spike-timing-dependent plasticity: The relationship to rate-based learning for models with weight dynamics determined by a stable fixed point, Neural Comput. 16, 885 (2004).
  14. H. Meffin, J. Besson, A. N. Burkitt, and D. B. Grayden, Learning the structure of correlated synaptic subgroups using stable and competitive spike-timing-dependent plasticity, Phys. Rev. E 73, 041911 (2006).
  15. M. Gilson, A. N. Burkitt, D. B. Grayden, D. A. Thomas, and J. L. van Hemmen, Emergence of network structure due to spike-timing-dependent plasticity in recurrent neuronal networks. I. Input selectivity–strengthening correlated input pathways, Biol. Cybern. 101, 81 (2009).
  16. N. Frémaux and W. Gerstner, Neuromodulated spike-timing-dependent plasticity, and theory of three-factor learning rules, Front. Neural Circuits 9, 85 (2016).
  17. G. K. Ocker and B. Doiron, Training and spontaneous reinforcement of neuronal assemblies by spike timing plasticity, Cereb. Cortex 29, 937 (2019).
  18. A. E. Akil, R. Rosenbaum, and K. Josić, Balanced networks under spike-time dependent plasticity, PLoS Comput. Biol. 17, e1008958 (2021).
  19. B. Sosis and J. E. Rubin, Distinct dopaminergic spike-timing-dependent plasticity rules are suited to different functional roles, bioRxiv (2024).
  20. T. Keck, T. Toyoizumi, L. Chen, B. Doiron, D. E. Feldman, K. Fox, W. Gerstner, P. G. Haydon, M. Hübener, H.-K. Lee, J. E. Lisman, T. Rose, F. Sengpiel, D. Stellwagen, M. P. Stryker, G. G. Turrigiano, and M. C. van Rossum, Integrating Hebbian and homeostatic plasticity: The current state of the field and future research directions, Phil. Trans. R. Soc. B 372, 20160158 (2017).
  21. B. Lindner, B. Doiron, and A. Longtin, Theory of oscillatory firing induced by spatially correlated noise and delayed inhibitory feedback, Phys. Rev. E 72, 061919 (2005).
  22. D. Grytskyy, T. Tetzlaff, M. Diesmann, and M. Helias, A unified view on weakly correlated recurrent networks, Front. Comput. Neurosci. 7, 131 (2013).
  23. M. Layer, M. Helias, and D. Dahmen, Effect of synaptic heterogeneity on neuronal coordination, PRX Life 2, 013013 (2024).
  24. W. Gerstner, Coding properties of spiking neurons: Reverse and cross-correlations, Neural Netw. 14, 599 (2001).
  25. G. K. Ocker, A. Litwin-Kumar, and B. Doiron, Self-organization of microcircuits in networks of spiking neurons with plastic synapses, PLoS Comput. Biol. 11, e1004458 (2015).
  26. N. Brunel, F. S. Chance, N. Fourcaud, and L. F. Abbott, Effects of synaptic noise and filtering on the frequency response of spiking neurons, Phys. Rev. Lett. 86, 2186 (2001).
  27. B. Lindner and L. Schimansky-Geier, Transmission of noise coded versus additive signals through a neuronal ensemble, Phys. Rev. Lett. 86, 2934 (2001).
  28. M. J. E. Richardson, Spike-train spectra and network response functions for non-linear integrate-and-fire neurons, Biol. Cybern. 99, 381 (2008).
  29. M. J. E. Richardson and R. Swarbrick, Firing-rate response of a neuron receiving excitatory and inhibitory synaptic shot noise, Phys. Rev. Lett. 105, 178102 (2010).
  30. F. Droste and B. Lindner, Exact analytical results for integrate-and-fire neurons driven by excitatory shot noise, J. Comput. Neurosci. 43, 81 (2017).
  31. F. Droste and B. Lindner, Exact results for power spectrum and susceptibility of a leaky integrate-and-fire neuron with two-state noise, Phys. Rev. E 95, 012411 (2017).
  32. J. Stubenrauch and B. Lindner, Furutsu-Novikov–like cross-correlation–response relations for systems driven by shot noise, Phys. Rev. X 14, 041047 (2024).
  33. R. Gütig, R. Aharonov, S. Rotter, and H. Sompolinsky, Learning input correlations through nonlinear temporally asymmetric Hebbian plasticity, J. Neurosci. 23, 3697 (2003).
  34. Y. F. K. Kossio, S. Goedeke, C. Klos, and R.-M. Memmesheimer, Drifting assemblies for persistent memory: Neuron transitions and unsupervised compensation, Proc. Natl. Acad. Sci. USA 118, e2023832118 (2021).
  35. A. Morrison, M. Diesmann, and W. Gerstner, Phenomenological models of synaptic plasticity based on spike timing, Biol. Cybern. 98, 459 (2008).
  36. P. J. Sjöström, G. G. Turrigiano, and S. B. Nelson, Rate, timing, and cooperativity jointly determine cortical synaptic plasticity, Neuron 32, 1149 (2001).
  37. J.-P. Pfister and W. Gerstner, Triplets of spikes in a model of spike timing-dependent plasticity, J. Neurosci. 26, 9673 (2006).
  38. W. Gerstner, R. Kempter, J. L. van Hemmen, and H. Wagner, Hebbian learning of pulse timing in the barn owl auditory system, in Pulsed Neural Networks, edited by W. Maass and C. M. Bishop (MIT Press, Cambridge, MA, 1998), pp. 353–377.
  39. R. Kempter, W. Gerstner, and J. L. v. Hemmen, Intrinsic stabilization of output rates by spike-based Hebbian learning, Neural Comput. 13, 2709 (2001).
  40. C. Clopath, L. Büsing, E. Vasilaki, and W. Gerstner, Connectivity reflects coding: A model of voltage-based STDP with homeostasis, Nat. Neurosci. 13, 344 (2010).
  41. F. Zenke, E. J. Agnes, and W. Gerstner, Diverse synaptic plasticity mechanisms orchestrated to form and retrieve memories in spiking neural networks, Nat. Commun. 6, 6922 (2015).
  42. A. Rauch, G. La Camera, H.-R. Lüscher, W. Senn, and S. Fusi, Neocortical pyramidal cells respond as integrate-and-fire neurons to in vivo-like input currents, J. Neurophysiol. 90, 1598 (2003).
  43. N. Fourcaud and N. Brunel, Dynamics of the firing probability of noisy integrate-and-fire neurons, Neural Comput. 14, 2057 (2002).
  44. A. Roxin, N. Brunel, D. Hansel, G. Mongillo, and C. van Vreeswijk, On the distribution of firing rates in networks of cortical neurons, J. Neurosci. 31, 16217 (2011).
  45. N. Brunel, Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons, J. Comput. Neurosci. 8, 183 (2000).
  46. M. Beierlein, J. R. Gibson, and B. W. Connors, Two dynamically distinct inhibitory networks in layer 4 of the neocortex, J. Neurophysiol. 90, 2987 (2003).
  47. D. Bernardi, G. Doron, M. Brecht, and B. Lindner, A network model of the barrel cortex combined with a differentiator detector reproduces features of the behavioral response to single-neuron stimulation, PLoS Comput. Biol. 17, e1007831 (2021).
  48. F. Zenke and W. Gerstner, Hebbian plasticity requires compensatory processes on multiple timescales, Phil. Trans. R. Soc. B 372, 20160259 (2017).
  49. J. N. Bourne and K. M. Harris, Coordination of size and number of excitatory and inhibitory synapses results in a balanced structural plasticity along mature hippocampal CA1 dendrites during LTP, Hippocampus 21, 354 (2011).
  50. C. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry, and the Natural Sciences, Proceedings in Life Sciences (Springer, Berlin, 1985).
  51. M. Layer, J. Senk, S. Essink, A. van Meegen, H. Bos, and M. Helias, NNMT: Mean-field based analysis tools for neuronal network models, Front. Neuroinform. 16, 835657 (2022).
  52. M. R. Rahimi Tabar, Analysis and Data-Based Reconstruction of Complex Nonlinear Dynamical Systems, 2019th ed., Understanding Complex Systems (Springer Nature, Cham, 2019).
  53. M. Ragwitz and H. Kantz, Indispensable finite time corrections for Fokker-Planck equations from time series data, Phys. Rev. Lett. 87, 254501 (2001).
  54. R. D. Vilela and B. Lindner, Are the input parameters of white-noise-driven integrate & fire neurons uniquely determined by rate and CV? J. Theor. Biol. 257, 90 (2009).
  55. D. J. Amit and S. Fusi, Learning in neural networks with material synapses, Neural Comput. 6, 957 (1994).
  56. N. Auer, L. Chen, J. Stubenrauch, B. Lindner, and R. Kempter, Population sparseness determines strength of Hebbian plasticity for maximal memory lifetime in associative networks, bioRxiv (2025).
  57. G. Palm, Neural associative memories and sparse coding, Neural Netw. 37, 165 (2013).
  58. R. Heitz and J. Schall, Neural mechanisms of speed-accuracy tradeoff, Neuron 76, 616 (2012).
  59. B. J. Bhasin, J. L. Raymond, and M. S. Goldman, Synaptic weight dynamics underlying memory consolidation: Implications for learning rules, circuit organization, and circuit function, Proc. Natl. Acad. Sci. USA 121, e2406010121 (2024).
  60. J. J. Hopfield, Neural networks and physical systems with emergent collective computational abilities. Proc. Natl. Acad. Sci. USA 79, 2554 (1982).
  61. J. P. Nadal, G. Toulouse, J. P. Changeux, and S. Dehaene, Networks of formal neurons and memory palimpsests, Europhys. Lett. 1, 535 (1986).
  62. J. T. Buckingham, Delicate Nets, Faint recollections: A study of partially connected associative network memories, Ph.D. thesis, University of Edinburgh, 1992.
  63. D. Tyulmankov, C. Fang, A. Vadaparty, and G. R. Yang, Biological learning in key-value memory networks, Adv. Neural Inf. Process. Syst. 34, 22247 (2021).
  64. G. Iatropoulos, J. Brea, and W. Gerstner, Kernel memory networks: A unifying framework for memory modeling, Adv. Neural Inf. Process. Syst. 35, 35326 (2022).
  65. M. W. H. Remme, U. Bergmann, D. Alevi, S. Schreiber, H. Sprekeler, and R. Kempter, Hebbian plasticity in parallel synaptic pathways: A circuit mechanism for systems memory consolidation, PLoS Comput. Biol. 17, e1009681 (2021).
  66. B. Dummer, S. Wieland, and B. Lindner, Self-consistent determination of the spike-train power spectrum in a neural network with sparse connectivity, Front. Comput. Neurosci. 8, 104 (2014).
  67. A. van Meegen and H. Sompolinsky, Coding schemes in neural networks learning classification tasks, Nat. Commun. 16, 3354 (2025).
  68. D. G. Clark and L. F. Abbott, Theory of coupled neuronal-synaptic dynamics, Phys. Rev. X 14, 021001 (2024).
  69. J. Stubenrauch, N. Auer, R. Kempter, and B. Lindner, Stochastic synaptic dynamics under learning—Scripts for simulation and illustration, Zenodo (2026), https://doi.org/10.5281/zenodo.19019985.
  70. S. Behnel, R. Bradshaw, C. Citro, L. Dalcin, D. Seljebotn, and K. Smith, Cython: The best of both worlds, Comput. Sci. Eng. 13, 31 (2011).
  71. B. Lindner, Coherence and Stochastic Resonance in Nonlinear Dynamical Systems (Logos Verlag, Berlin, 2002).
  72. T. Schwalger, M. Deger, and W. Gerstner, Towards a theory of cortical columns: From spiking neurons to interacting neural populations of finite size, PLoS Comput. Biol. 13, e1005507 (2017).

Outline

Information

PRX Life 4, 023006– Published 13 April, 2026
  • Received 5 September 2025
  • Accepted 9 March 2026
Crossmark - Check for updates button
Creative Commons Logo

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.

Published by the American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation