The Complexity of Dynamics in Small Neural Circuits

Fasoli, Diego; Cattani, Anna; Panzeri, Stefano

doi:10.1371/journal.pcbi.1004992

Cited by 21 publications

(69 citation statements)

References 104 publications

(211 reference statements)

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“…Note that N E,I can be arbitrary, but for illustrative purposes in this section we consider the case N E = N I (rather than the N E \N I = 4 ratio experimentally observed in biological systems [25]), because we found that the network complexity increases with the size of the inhibitory population. Interestingly, the same phenomenon was found to occur also in multi-population networks with graded activation function [9]. Moreover, without further loss of generality, we index the neurons of the excitatory population as i = 0, .…”

Section: Examples Of Network Topologiessupporting

confidence: 56%

“…(3) we provided examples that show how the network dynamics depends on the external stimuli, the network size and the topology of the synaptic connections. Similarly to the case of graded firing-rate network models [3][4][5]9,15,27], this analysis revealed a complex bifurcation structure, encompassing several changes in the degree of multistability of the network, oscillations with stimulus-dependent frequency, and various forms of spontaneous symmetrybreaking of the neural activity.…”

Section: Discussionmentioning

confidence: 82%

“…Neural complexity refers to the wide variety of dynamical behaviors that occur in neural networks [5,9,27]. This set of dynamical behaviors includes variations in the number of stable solutions of neuronal activity, the formation of neural oscillations, spontaneous symmetry-breaking, chaos and much more [1,16].…”

Section: Introductionmentioning

confidence: 99%

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Optimized brute-force algorithms for the bifurcation analysis of a binary neural network model

Fasoli¹,

Panzeri²

2019

Phys. Rev. E

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Bifurcation theory is a powerful tool for studying how the dynamics of a neural network model depends on its underlying neurophysiological parameters. However, bifurcation theory has been developed mostly for smooth dynamical systems and for continuous-time non-smooth models, which prevents us from understanding the changes of dynamics in some widely used classes of artificial neural network models. This article is an attempt to fill this gap, through the introduction of algorithms that perform a semianalytical bifurcation analysis of a spin-glass-like neural network model with binary firing rates and discrete-time evolution. Our approach is based on a numerical brute-force search of the stationary and oscillatory solutions of the spin-glass model, from which we derive analytical expressions of its bifurcation structure by means of the state-to-state transition probability matrix. The algorithms determine how the network parameters affect the degree of multistability, the emergence and the period of the neural oscillations, and the formation of symmetry-breaking in the neural populations. While this technique can be applied to networks with arbitrary (generally asymmetric) connectivity matrices, in particular we introduce a highly efficient algorithm for the bifurcation analysis of sparse networks. We also provide some examples of the obtained bifurcation diagrams and a Python implementation of the algorithms. IntroductionNeural complexity refers to the wide variety of dynamical behaviors that occur in neural networks [5,9,27]. This set of dynamical behaviors includes variations in the number of stable solutions of neuronal activity, the formation of neural oscillations, spontaneous symmetry-breaking, chaos and much more [1,16]. Qualitative changes of neuronal activity, also known as bifurcations, are elicited by variations of the network parameters, such as the strength of the external input to the network, the strength of the synaptic connections between neurons, or other network characteristics.Bifurcation theory is a standard mathematical formalism for studying neural complexity [20]. It allows the construction of a map of neuronal activity, known as bifurcation diagram, that links points or sets in the parameters space to their corresponding network dynamics. In the study of firing-rate network models, bifurcation theory has been applied mostly to graded (smooth) neural networks with analog firing rates (e.g. [3-5, 9, 15, 27]), proving itself as an effective tool for deepening our understanding of network dynamics. The bifurcation analysis of smooth models is based on differential analysis, in particular on the Jacobian matrix of the system. However, the Jacobian matrix is not defined everywhere for artificial neuronal models with discontinuous activation function, such as networks of binary neurons. For this reason, bifurcation theory of smooth dynamical systems cannot be applied directly to these models. On 1 arXiv:1705.05647v1 [q-bio.NC] 16 May 2017 the other hand, while the bifurcation analysis of non-sm...

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Section: Examples Of Network Topologiessupporting

confidence: 56%

Section: Discussionmentioning

confidence: 82%

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Optimized brute-force algorithms for the bifurcation analysis of a binary neural network model

Fasoli¹,

Panzeri²

2019

Phys. Rev. E

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“…For this reason, it is important to be able to study mathematically the dynamics of binary neural network models (or of any network model, see e.g. [13,28,63]), for a wide range of different network sizes.Large-scale networks composed of several thousands of neurons or more, are typically studied by taking advantage of the powerful techniques of statistical mechanics, such as the law of large numbers and the central limit theorem, see e.g. [21,29,33,74].…”

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confidence: 99%

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confidence: 99%

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

Fasoli

Panzeri

2019

Mathematical Biosciences and Engineering

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Traditional mathematical approaches to studying analytically the dynamics of neural networks rely on the mean-field approximation, which is 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 finite-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. Finally, we highlight key challenges that remain open, future directions for further progress, and possible implications of our results for neuroscience. arXiv:1904.12798v1 [q-bio.NC] 29 Apr 2019 is a consequence of their thresholding activation function, which can be considered as the simplest, piecewise-constant, approximation of the non-linear (and typically sigmoidal shaped) graded input-output relationship of biological neurons. Despite their simplicity, as shown both by classic work [1,33,40,50,64], as well as by our work reviewed here [29][30][31], the jump discontinuity of their activation function at the threshold is sufficient to endow binary networks with a complex set of useful emergent dynamical properties and non-linear phenomena, such as attractor dynamics [4], formation of patterns and oscillatory waves [3], chaos [73], and information processing capabilities [2], which are reminiscent of neuronal activity in biological networks.The importance of binary network models is further strengthened by their close relationship with spin networks studied in physics [43,55,56,70]. The temporal evolution of a binary network in the zero-noise limit is isomorphic to the dynamics of kinetic models of spin networks at absolute temperature [34]. This allowed computational neuroscientists to study the behavior of large-size binary networks, by applying the powerful techniques of statistical mechanics already developed for spin models (see e.g. [21]).Sizes of brains and of specialized neural networks within brains change considerably across animal species, ranging from few tens of neurons in invertebrates such as rotifers and nematodes, to billions of neurons in cetaceans and primates [76]. Network size changes also across levels of spatial organizations, ranging from microscopic and mesoscopic levels of organization in cortical micro-columns and columns (including from few tens [58] to few tens of thousands ...

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The Effect of Inhibitory Neurons on a Class of Neural Networks

Neogrády-Kiss

Šimon

2020

Trends in Biomathematics: Modeling Cells, Flows, Epidemics, and the Environment

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The Complexity of Dynamics in Small Neural Circuits

Cited by 21 publications

References 104 publications

Optimized brute-force algorithms for the bifurcation analysis of a binary neural network model

Optimized brute-force algorithms for the bifurcation analysis of a binary neural network model

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

The Effect of Inhibitory Neurons on a Class of Neural Networks

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