Unfolding an electronic integrate-and-fire circuit

Abstract

Many physical and biological phenomena involve accumulation and discharge processes that can occur on significantly different time scales. Models of these processes have contributed to understand excitability self-sustained oscillations and synchronization in arrays of oscillators. Integrate-and-fire (I+F) models are popular minimal fill-and-flush mathematical models. They are used in neuroscience to study spiking and phase locking in single neuron membranes, large scale neural networks, and in a variety of applications in physics and electrical engineering. We show here how the classical first-order I+F model fits into the theory of nonlinear oscillators of van der Pol type by demonstrating that a particular second-order oscillator having small parameters converges in a singular perturbation limit to the I+F model. In this sense, our study provides a novel unfolding of such models and it identifies a constructible electronic circuit that is closely related to I+F.

Growth Dynamics Explain the Development of Spatiotemporal Burst Activity of Young Cultured Neuronal Networks in Detail

“…This paper is the third in a series of three on simulation models of cultured networks. Our two previous studies [26], [27] have shown that random recurrent network activity models generate intra- and inter-bursting patterns similar to experimental data. The networks were noise or pacemaker-driven and had Izhikevich-neuronal elements with only short-term plastic (STP) synapses (so, no long-term potentiation, LTP, or depression, LTD, was included).…”

The faithful copy neuron

“…This has proven to be a remarkably good approximation to the Hodgkin-Huxley model if spike duration times are short compared to interspike times (Knight 1972;Dayan and Abbott 2001;Carrillo and Hoppensteadt 2010). Equation (16) is to be solved under the condition that x is reset to rest, x = 0, on firing a spike, x = 1, a non-linear framework.…”

Spiking Neurons and the First Passage Problem

“…with firing threshold x = 1 and resting state x = 0, has proven a trustworthy approximation when the input timescale is large compared to the duration of the action potential (Knight, 1972;Keener & Sneyd, 1998;Dayan & Abbott, 2001;Carrillo & Hoppensteadt, 2010). Equation 2.7 models the activity of the membrane by an RC circuit where γ represents leakage conductance and s is the input current.…”