In this paper we present a numerical study of a mathematical model of spiking
neurons introduced by Ferrari et al. in an article entitled Phase transition
forinfinite systems of spiking neurons. In this model we have a countable
number of neurons linked together in a network, each of them having a membrane
potential taking value in the integers, and each of them spiking over time at a
rate which depends on the membrane potential through some rate function $ϕ$.
Beside being affected by a spike each neuron can also be affected by leaking.
At each of these leak times, which occurs for a given neuron at a fixed rate
$γ$, the membrane potential of the neuron concerned is spontaneously reset
to $0$. A wide variety of versions of this model can be considered by choosing
different graph structures for the network and different activation functions.
It was rigorously shown that when the graph structure of the network is the
one-dimensional lattice with a hard threshold for the activation function, this
model presents a phase transition with respect to $γ$, and that it also
presents a metastable behavior. By the latter we mean that in the sub-critical
regime the re-normalized time of extinction converges to an exponential random
variable of mean 1. It has also been proven that in the super-critical regime
the renormalized time of extinction converges in probability to 1. Here, we
investigate numerically a richer class of graph structures and activation
functions. Namely we investigate the case of the two dimensional and the three
dimensional lattices, as well as the case of a linear function and a sigmoid
function for the activation function. We present numerical evidence that the
result of metastability in the sub-critical regime holds for these graphs and
activation functions as well as the convergence in probability to $1$ in the
super-critical regime.
arXiv Quantitative Biology
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