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The elapsed-time model describes the behavior of interconnected neuronsthrough the time since their last spike. It is an age-structured non-linearequation in which age corresponds to the elapsed time since the last discharge,and models many interesting dynamics depending on the type of interactionsbetween neurons. We investigate the linearized stability of this equation byconsidering a discrete delay, which accounts for the possibility of a synapticdelay due to the time needed to transmit a nerve impulse from one neuron to therest of the ensemble. We state a stability criterion that allows to determineif a steady state is linearly stable or unstable depending on the delay and theinteraction between neurons. Our approach relies on the study of the asymptoticbehavior of related Volterra-type integral equations in terms of theirs Laplacetransforms. The analysis is complemented with numerical simulationsillustrating the change of stability of a steady state in terms of the delayand the intensity of interconnections.

On the local stability of the elapsed-time model in terms of the transmission delay and interconnection strength