On the firing maps of a general class of forced integrate and fire neurons

Carrillo, Humberto; Ongay, Fernando

doi:10.1016/s0025-5564(01)00060-8

Cited by 10 publications

(19 citation statements)

References 17 publications

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“…For example, they often appear in mathematical neuroscience, where they are viewed as being mathematical simplifications of the HodgkinHuxley theory of neuron membranes, and are used for qualitative analysis and simulation. They have provided insights to how neurons might interact (Izhikevich 2003(Izhikevich , 2007, they have suggested experiments that have revealed neuron behavior 1 (Knight 1972;Winfree 2000), and they have been analyzed and simulated using mathematical methodologies (Carrillo and Ongay 2001;Hoppensteadt and Izhikevich 1997;Hoppensteadt 1997;Keener et al 1981). In addition, they have provided useful ways to conceptualize certain electronic circuits (Kennedy and Chua 1986;Yang et al 2001).…”

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confidence: 99%

“…This class of models uses a differential equation to describe the charging process and an artificial mechanism to produce instantaneous discharge (Carrillo and Ongay 2001;Keener et al 1981). Electrical I+F models usually involve a source of electric current, an energy storage device such as a capacitor or an inductor, and a device that allows current to pass when a threshold voltage is exceeded thereby resetting the voltage to a lower level.…”

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confidence: 99%

“…1 and 2 is the I+F model we study here. It is easy to simulate (1) and (2) using a digital computer (see footnote 1; Knight 1972), and methods of nonlinear oscillations, notably theories of circle mappings and rotation numbers (Carrillo and Ongay 2001;Keener et al 1981), are useful in analyzing it.…”

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confidence: 99%

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Unfolding an electronic integrate-and-fire circuit

Carrillo

Hoppensteadt

2009

Biol Cybern

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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.

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Unfolding an electronic integrate-and-fire circuit

Carrillo

Hoppensteadt

2009

Biol Cybern

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“…In these systems, most commonly used as simplified models of neuron's activity (cf. [4,6,7,10,16]), the continuous dynamics of the variable x(t), standing Downloaded by [University of Tasmania] at 10:46 30 September 2015 for the membrane voltage of the cell, is interrupted by the so-called thresholdreset behaviour (1.2), which is supposed to mimic spiking (generation of action potential). However, IF systems (and circle mappings induced by them in case of periodic forcing) can also be used in modeling of cardiac rhythms and arrhythmias [2], in some engineering applications (e.g.…”

Section: Introductionmentioning

confidence: 99%

“…Thus the natural domain of Φ is the set (compare with [6]): D Φ = t ∈ R : there exists s > t such that x(s; t, 0) = 1 .…”

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confidence: 99%

Analysis of Interspike-Intervals for the General Class of Integrate-and-Fire Models With Periodic Drive

Signerska-Rynkowska

2015

Mathematical Modelling and Analysis

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We study one-dimensional integrate-and-fire models of the general typė x = F (t, x) and analyze properties of the firing map which iterations recover consecutive spike timings. We impose very week constraints for the regularity of the function F (t, x), e.g. often it suffices to assume that F is continuous. If additionally F is periodic in t, using mathematical study of the displacement sequence of an orientation preserving circle homeomorphism, we provide a detailed description of the regularity properties of the sequence of interspike-intervals and behaviour of the interspike-interval distribution.Keywords: integrate-and-fire neuron, firing map, interspike intervals, ordinary differential equations, circle maps.

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Phase locking in integrate-and-fire models with refractory periods and modulation

Gedeon

Holzer

2004

J. Math. Biol.

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It is known [8, 11, 16, 26] that phase locking can entrain frequency information when the leaky integrate-and-fire (IF) model of a neuron is forced by a periodic function. We show that this is still the case when the IF model is made more biologically realistic. We incorporate into our model spike dependent threshold modulation and refractory periods. Consecutive firing times from this model and their respective interspike intervals are related by an annulus map. We prove a general theorem concerning orientation reversing annulus twist homeomorphisms, which shows that our map admits a unique rotation number. This implies, in particular, that chaotic behaviour is not possible in our model and phase locking is predicted.

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On the firing maps of a general class of forced integrate and fire neurons

Cited by 10 publications

References 17 publications

Unfolding an electronic integrate-and-fire circuit

Unfolding an electronic integrate-and-fire circuit

Analysis of Interspike-Intervals for the General Class of Integrate-and-Fire Models With Periodic Drive

Phase locking in integrate-and-fire models with refractory periods and modulation

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