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We study a stochastic process describing the continuous time evolution of themembrane potentials of finite system of neurons in the absence of externalstimuli. The values of the membrane potentials evolve under the effect of {\itchemical synapses}, {\it electrical synapses} and a \textit{leak current}. Theevolution of the process can be informally described as follows. Each neuronspikes randomly following a point process with rate depending on its membranepotential. When a neuron spikes, its membrane potential is immediately reset toa resting value. Simultaneously, the membrane potential of the neurons whichare influenced by it receive an additional positive value. Furthermore, betweenconsecutive spikes, the system follows a deterministic motion due both toelectrical synapses and the leak current. Electrical synapses push the systemtowards its average potential, while the leak current attracts the membranepotential of each neuron to the resting value.  We show that in the absence leakage the process converges exponentially fastto an unique invariant measure, whenever the initial configuration is non null.More interesting, when leakage is present, we proved the system stops spikingafter a finite amount of time almost surely. This implies that the uniqueinvariant measure is supported only by the null configuration.

A model for neural activity in the absence of external stimuli