A Formalism for Evaluating Analytically the Cross-Correlation Structure of a Firing-Rate Network Model

Fasoli, Diego; Faugeras, Olivier; Panzeri, Stefano

doi:10.1186/s13408-015-0020-y

Cited by 9 publications

(42 citation statements)

References 79 publications

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“…In more detail, we consider a widely used rate model to describe the dynamics of single neurons [ 30 , 35 , 38 , 40 , 48 , 53 ]: where N ≥ 4 represents the number of neurons in the network. The function V i ( t ) is the membrane potential of the i th neuron, while τ i is its membrane time constant.…”

Section: Methodsmentioning

confidence: 99%

“…This approximation becomes exact in the limit of networks with an infinite number of neurons, the so-called thermodynamic limit . For finite-size networks, however, the mean-field theory provides only an approximation of the real behavior of the system, and therefore may neglect important phenomena, such as qualitative differences in the transitions between static regimes and chaos [ 39 ], or in the degree or nature of correlations among neurons [ 40 ]. Clearly, these macroscopic differences in the dynamical and statistical behavior of finite and infinite-size networks may have important consequences on the information processing capability of the system, as potential oversimplifications in the mean-field approximation may hide important neural processes that are fundamental for the comprehension of neural circuits of finite size.…”

Section: Introductionmentioning

confidence: 99%

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The Complexity of Dynamics in Small Neural Circuits

2016

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Mean-field approximations are a powerful tool for studying large neural networks. However, they do not describe well the behavior of networks composed of a small number of neurons. In this case, major differences between the mean-field approximation and the real behavior of the network can arise. Yet, many interesting problems in neuroscience involve the study of mesoscopic networks composed of a few tens of neurons. Nonetheless, mathematical methods that correctly describe networks of small size are still rare, and this prevents us to make progress in understanding neural dynamics at these intermediate scales. Here we develop a novel systematic analysis of the dynamics of arbitrarily small networks composed of homogeneous populations of excitatory and inhibitory firing-rate neurons. We study the local bifurcations of their neural activity with an approach that is largely analytically tractable, and we numerically determine the global bifurcations. We find that for strong inhibition these networks give rise to very complex dynamics, caused by the formation of multiple branching solutions of the neural dynamics equations that emerge through spontaneous symmetry-breaking. This qualitative change of the neural dynamics is a finite-size effect of the network, that reveals qualitative and previously unexplored differences between mesoscopic cortical circuits and their mean-field approximation. The most important consequence of spontaneous symmetry-breaking is the ability of mesoscopic networks to regulate their degree of functional heterogeneity, which is thought to help reducing the detrimental effect of noise correlations on cortical information processing.

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Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Complexity of Dynamics in Small Neural Circuits

2016

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“…The middle panels show the same comparison for the crosscorrelations between neurons 0 and 1 (the red curves are described by Eqs (13),. left, and(14), right). The bottom panels show examples of highly correlated activity (synchronous states) between the membrane potentials (for I " 0, left) and between the firing rates (for I " 2, right).…”

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confidence: 98%

Pattern Storage, Bifurcations, and Groupwise Correlation Structure of an Exactly Solvable Asymmetric Neural Network Model

2018

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Despite their biological plausibility, neural network models with asymmetric weights are rarely solved analytically, and closed-form solutions are available only in some limiting cases or in some mean-field approximations. We found exact analytical solutions of an asymmetric spin model of neural networks with arbitrary size without resorting to any approximation, and we comprehensively studied its dynamical and statistical properties. The network had discrete time evolution equations and binary firing rates, and it could be driven by noise with any distribution. We found analytical expressions of the conditional and stationary joint probability distributions of the membrane potentials and the firing rates. By manipulating the conditional probability distribution of the firing rates, we extend to stochastic networks the associating learning rule previously introduced by Personnaz and coworkers. The new learning rule allowed the safe storage, under the presence of noise, of point and cyclic attractors, with useful implications for content-addressable memories. Furthermore, we studied the bifurcation structure of the network dynamics in the zero-noise limit. We analytically derived examples of the codimension 1 and codimension 2 bifurcation diagrams of the network, which describe how the neuronal dynamics changes with the external stimuli. This showed that the network may undergo transitions among multistable regimes, oscillatory behavior elicited by asymmetric synaptic connections, and various forms of spontaneous symmetry breaking. We also calculated analytically groupwise correlations of neural activity in the network in the stationary regime. This revealed neuronal regimes where, statistically, the membrane potentials and the firing rates are either synchronous or asynchronous. Our results are valid for networks with any number of neurons, although our equations can be realistically solved only for small networks. For completeness, we also derived the network equations in the thermodynamic limit of infinite network size and we analytically studied their local bifurcations. All the analytical results were extensively validated by numerical simulations.

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“…In the present section, we assume that the population has all to all connectivity as in [20,21]. For discussions on this topic see [22][23][24].…”

Section: Excitatory Synapses Uniform Connectivitymentioning

confidence: 99%

Mean Field Analysis of Large-Scale Interacting Populations of Stochastic Conductance-Based Spiking Neurons Using the Klimontovich Method

2017

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We investigate the dynamics of large-scale interacting neural populations, composed of conductance based, spiking model neurons with modifiable synaptic connection strengths, which are possibly also subjected to external noisy currents. The network dynamics is controlled by a set of neural population probability distributions (PPD) which are constructed along the same lines as in the Klimontovich approach to the kinetic theory of plasmas. An exact non-closed, nonlinear, system of integro-partial differential equations is derived for the PPDs. As is customary, a closing procedure leads to a mean field limit. The equations we have obtained are of the same type as those which have been recently derived using rigorous techniques of probability theory. The numerical solutions of these so called McKean-Vlasov-Fokker-Planck equations, which are only valid in the limit of infinite size networks, actually shows that the statistical measures as obtained from PPDs are in good agreement with those obtained through direct integration of the stochastic dynamical system for large but finite size networks. Although numerical solutions have been obtained for networks of Fitzhugh-Nagumo model neurons, which are often used to approximate Hodgkin-Huxley model neurons, the theory can be readily applied to networks of general conductance-based model neurons of arbitrary dimension.Classifications (2016).

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A Formalism for Evaluating Analytically the Cross-Correlation Structure of a Firing-Rate Network Model

Cited by 9 publications

References 79 publications

The Complexity of Dynamics in Small Neural Circuits

The Complexity of Dynamics in Small Neural Circuits

Pattern Storage, Bifurcations, and Groupwise Correlation Structure of an Exactly Solvable Asymmetric Neural Network Model

Mean Field Analysis of Large-Scale Interacting Populations of Stochastic Conductance-Based Spiking Neurons Using the Klimontovich Method

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