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Limits and Dynamics of Stochastic Neuronal Networks with Random Heterogeneous Delays

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Abstract

Realistic networks display heterogeneous transmission delays. We analyze here the limits of large stochastic multi-populations networks with stochastic coupling and random interconnection delays. We show that depending on the nature of the delays distributions, a quenched or averaged propagation of chaos takes place in these networks, and that the network equations converge towards a delayed McKean-Vlasov equation with distributed delays. Our approach is mostly fitted to neuroscience applications. We instantiate in particular a classical neuronal model, the Wilson and Cowan system, and show that the obtained limit equations have Gaussian solutions whose mean and standard deviation satisfy a closed set of coupled delay differential equations in which the distribution of delays and the noise levels appear as parameters. This allows to uncover precisely the effects of noise, delays and coupling on the dynamics of such heterogeneous networks, in particular their role in the emergence of synchronized oscillations. We show in several examples that not only the averaged delay, but also the dispersion, govern the dynamics of such networks.

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Notes

  1. The degree of freedom given by the choice of m-dimensional Brownian motions allows to take into account different sources of independent sources of noise, typically in conductance-based neurons thermal noise affecting the voltage and channel noise affecting ionic concentrations.

  2. Note that this model has the disadvantage to possibly change the excitatory or inhibitory nature of the connection from neuron i to neuron j when the sign of w ij changes. Other more biologically relevant problems involve enforcing the synaptic weight not to change sign by modeling it as the solution of particular stochastic differential equation that does not change sign, such as the Cox-Ingersoll-Ross process, and by fit our framework by formally adding the P synaptic weights of neuron i in the state variable X i of neuron i.

  3. McKean-Vlasov is the generic term in the mathematical literature on statistical physics for describing the type of equation we obtain. It is differs from the well-know to physicists Vlasov equation, which describes the evolution of the distribution function of plasma consisting of charged particles with long-range interactions, in particular in that our equation includes a diffusion term (Brownian motion) and is written as a stochastic equation, whereas usual the Vlasov equation is written as a nonlinear PDE closer from what we call the McKean-Vlasov-Fokker-Planck equation.

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Acknowledgements

The author warmly thanks Tanguy Cabana for very interesting discussions and suggestions.

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Authors and Affiliations

  1. The Mathematical Neuroscience Laboratory, CIRB and INRIA Bang Laboratory, Collège de France, 11, place Marcelin Berthelot, 75005, Paris, France

    Jonathan Touboul

  2. CNRS UMR 7241m INSERM U1050, MEMOLIFE Laboratory of excellence and Paris Sciences Lettres PSL*, Université Pierre et Marie Curie ED 158, Paris, France

    Jonathan Touboul

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Touboul, J. Limits and Dynamics of Stochastic Neuronal Networks with Random Heterogeneous Delays. J Stat Phys 149, 569–597 (2012). https://doi.org/10.1007/s10955-012-0607-6

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