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Register a new userThe mean-field equation of a leaky integrate-and-fire neural network: measure solutions and steady states
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Neural network dynamics emerge from the interaction of spiking cells. One way to formulate the problem is through a theoretical framework inspired by ideas coming from statistical physics, the so-called mean-field theory. In this document, we investigate different issues related to the mean-field description of an excitatory network made up of leaky integrate-and-fire neurons. The description is written in the form a nonlinear partial differential equation which is known to blow up in finite time when the network is strongly connected. We prove that in a moderate coupling regime the equation is globally well-posed in the space of measures, and that there exist stationary solutions. In the case of weak connectivity we also demonstrate the uniqueness of the steady state and its global exponential stability. The method to show those mathematical results relies on a contraction argument of Doeblins type in the linear case, which corresponds to a population of non-interacting units.
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We show that the stochastic Morris-Lecar neuron, in a neighborhood of its stable point, can be approximated by a two-dimensional Ornstein-Uhlenbeck (OU) modulation of a constant circular motion. The associated radial OU process is an example of a leaky integrate-and-fire (LIF) model prior to firing. A new model constructed from a radial OU process together with a simple firing mechanism based on detailed Morris-Lecar firing statistics reproduces the Morris-Lecar Interspike Interval (ISI) distribution, and has the computational advantages of a LIF. The result justifies the large amount of attention paid to the LIF models.
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