Mathematical studies of the dynamics of finite-size binary neural networks: A review of recent progress

Diego Fasoli; Stefano Panzeri; Diego Fasoli; Stefano Panzeri

doi:10.3934/mbe.2019404

Mathematical Biosciences and Engineering

2019, Volume 16, Issue 6: 8025-8059. doi: 10.3934/mbe.2019404

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Review Special Issues

Mathematical studies of the dynamics of finite-size binary neural networks: A review of recent progress

Diego Fasoli ^,,
Stefano Panzeri

Laboratory of Neural Computation, Center for Neuroscience and Cognitive Systems@UniTn, Istituto Italiano di Tecnologia, 38068 Rovereto, Italy

Received: 30 April 2019 Accepted: 23 August 2019 Published: 04 September 2019

Several mathematical approaches to studying analytically the dynamics of neural networks rely on mean-field approximations, which are rigorously applicable only to networks of infinite size. However, all existing real biological networks have finite size, and many of them, such as microscopic circuits in invertebrates, are composed only of a few tens of neurons. Thus, it is important to be able to extend to small-size networks our ability to study analytically neural dynamics. Analytical solutions of the dynamics of small-size neural networks have remained elusive for many decades, because the powerful methods of statistical analysis, such as the central limit theorem and the law of large numbers, do not apply to small networks. In this article, we critically review recent progress on the study of the dynamics of small networks composed of binary neurons. In particular, we review the mathematical techniques we developed for studying the bifurcations of the network dynamics, the dualism between neural activity and membrane potentials, cross-neuron correlations, and pattern storage in stochastic networks. Then, we compare our results with existing mathematical techniques for studying networks composed of a finite number of neurons. Finally, we highlight key challenges that remain open, future directions for further progress, and possible implications of our results for neuroscience.
- binary rate networks,
- threshold units,
- discontinuous maps,
- bifurcation theory,
- cross-neuron correlations,
- patterns storage,
- learning rules,
- quenched disorder
Citation: Diego Fasoli, Stefano Panzeri. Mathematical studies of the dynamics of finite-size binary neural networks: A review of recent progress[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 8025-8059. doi: 10.3934/mbe.2019404

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Abstract

Several mathematical approaches to studying analytically the dynamics of neural networks rely on mean-field approximations, which are rigorously applicable only to networks of infinite size. However, all existing real biological networks have finite size, and many of them, such as microscopic circuits in invertebrates, are composed only of a few tens of neurons. Thus, it is important to be able to extend to small-size networks our ability to study analytically neural dynamics. Analytical solutions of the dynamics of small-size neural networks have remained elusive for many decades, because the powerful methods of statistical analysis, such as the central limit theorem and the law of large numbers, do not apply to small networks. In this article, we critically review recent progress on the study of the dynamics of small networks composed of binary neurons. In particular, we review the mathematical techniques we developed for studying the bifurcations of the network dynamics, the dualism between neural activity and membrane potentials, cross-neuron correlations, and pattern storage in stochastic networks. Then, we compare our results with existing mathematical techniques for studying networks composed of a finite number of neurons. Finally, we highlight key challenges that remain open, future directions for further progress, and possible implications of our results for neuroscience.

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