Abstract
Since the pioneering works of Lapicque [17] and of Hodgkin and Huxley [16], several types of models have been addressed to describe the evolution in time of the potential of the membrane of a neuron. In this note, we investigate a connected version of N neurons obeying the leaky integrate and fire model, previously introduced in [1–3,6,7,15,18,19,22]. As a main feature, neurons interact with one another in a mean field instantaneous way. Due to the instantaneity of the interactions, singularities may emerge in a finite time. For instance, the solution of the corresponding Fokker-Planck equation describing the collective behavior of the potentials of the neurons in the limit N ⟶ ∞ may degenerate and cease to exist in any standard sense after a finite time. Here we focus out on a variant of this model when the interactions between the neurons are also subjected to random synaptic weights. As a typical instance, we address the case when the connection graph is the realization of an Erdös-Renyi graph. After a brief introduction of the model, we collect several theoretical results on the behavior of the solution. In a last step, we provide an algorithm for simulating a network of this type with a possibly large value of N.
Highlights
We address a network of N connected neurons that evolve according to a variant of the above stochastic LIF model
We describe the discrete system of neurons through the evolution in time of the membrane potentials of each of the neurons: The membrane potential of neuron i is described by a process Xi = (Xti)t≥0
Because of the normalization factor α/N in the interaction term and by independence of the various sources of noise, we may expect from standard results on large particle systems with mean field interaction, see for instance [24], that neurons become independent asymptotically, each of them solving in the limit a nonlinear stochastic differential equation with interaction with the common theoretical distribution of the network
Summary
One of the first models for neurons was introduced by Louis Lapicque in 1907, see [17]. Because of the normalization factor α/N in the interaction term and by independence of the various sources of noise, we may expect from standard results on large particle systems with mean field interaction, see for instance [24], that neurons become independent asymptotically, each of them solving in the limit a nonlinear stochastic differential equation with interaction with the common theoretical distribution of the network. A blow-up occurs when a large proportion of neurons in the finite network spike at the same time; in the limit regime, a blow up occurs when the time derivative of the mean field term in (2) is infinite It is proven in [1, 6] that a blow up appears if the mass of the initial distribution is too concentrated near the firing threshold XF and, that no blow up appears if the initial mass is sufficiently far away from XF. It is worth noticing that another form of inhomogeneous connection is addressed in [20]: Therein, a finite network of distinct infinite populations is studied; inhomogeneous weights are used to describe the connections between populations
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