Systematic Fluctuation Expansion for Neural Network Activity Equations
Open accessAbstract
Population rate or activity equations are the foundation of a common approach to modeling for neural networks. These equations provide mean field dynamics for the firing rate or activity of neurons within a network given some connectivity. The shortcoming of these equations is that they take into account only the average firing rate, while leaving out higher-order statistics like correlations between firing. A stochastic theory of neural networks that includes statistics at all orders was recently formulated. We describe how this theory yields a systematic extension to population rate equations by introducing equations for correlations and appropriate coupling terms. Each level of the approximation yields closed equations; they depend only on the mean and specific correlations of interest, without an ad hoc criterion for doing so. We show in an example of an all-to-all connected network how our system of generalized activity equations captures phenomena missed by the mean field rate equations alone.
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Evidence suggests that neural fluctuations involve multiplicative rather than merely additive noise components, where the variance of the noise depends on the system's state.
“Evidence suggests that neural fluctuations involve multiplicative rather than merely additive noise components, where the variance of the noise depends on the system’s state (Buice et al. 2010; Roberts et al. 2019).”
From The myth of the Bayesian brain (2025) · cites this paper as Buice et al. 2010
In §5 (page not provided), the paper adds multiplicative Gaussian noise to a neural-network model, with a transition rate that depends on activity. But it calls this added transition artificial; it does not establish that neural fluctuations generally involve multiplicative rather than merely additive noise. (LLM, conf=0.85)
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References 77 · Cited by 164
- Authors
- Buice, M. A., Cowan, J. D., & Chow, C. C.
- Affiliations
- National Institutes of Health (US)
- University of Chicago (US)
- Venue
- Neural Computation
- Publisher
- MIT Press - Journals
- Published
- 2010-02
- Volume
- 22
- Issue
- 2
- Pages
- 377-426
- ISSN
- 0899-7667 (ISSN-L)
- 1530-888X
- PMC
- PMC2805768
- OpenAlex
- W2115228179
- Cited by
- 164
- References
- 77
- Citations by year
- 2026: 5 · 2025: 8 · 2024: 7 · 2023: 6 · 2022: 8 · 2021: 7 · 2020: 6 · 2019: 8 · 2018: 3 · 2017: 15 · 2016: 9 · 2015: 13 · 2014: 11 · 2013: 26 · 2012: 17
- FWCI
- 5.02
- Type
- journal article
- Language
- en
- Open access
- green
- www.ncbi.nlm.nih.gov
- Locations
- Neural Computation (publishedVersion)
- PubMed (publishedVersion)
- PubMed Central (acceptedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Institutes of Health
- National Institute of Diabetes and Digestive and Kidney Diseases (Z01DK013024)
- Topics
- Life Sciences › Neuroscience › Cognitive Neuroscience › Neural dynamics and brain function
- Physical Sciences › Computer Science › Artificial Intelligence › Neural Networks and Applications
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › stochastic dynamics and bifurcation
- Keywords
- mean-field theory, stochastic neural networks, higher-order statistics, two-particle correlations, closed-form expressions
- MeSH terms
- Action Potentials, Algorithms, Animals, Artificial Intelligence, Brain, Humans, Mathematical Computing, Nerve Net, Neurons, Neural Networks, Computer, Mathematical Concepts
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