Generalization of coupled spiking models and effects of the width of an action potential on synchronization phenomena

Yasuomi Daishin Sato and Masatoshi Shiino

Phys. Rev. E 75, 011909 – Published 11 January 2007

Abstract

The integrate-and-fire (IF) neuron model and the piecewise-linear version of FitzHugh-Nagumo (PL) neuron model with a time scale parameter $μ$ are frequently being used in the study of synchronization phenomena. Although the two models are regarded to be different type, we show a certain equivalence between them by deriving the coupled IF model of an improved version with a firing duration from the recovery variable coupled system of the PL model, under taking the limit of $μ \to 0$ without a loss of any coupling properties. In the coupled IF model with the duration time, the synchronization behavior of a pair of neurons with excitatory or inhibitory synaptic coupling can be systematically explored in terms of three parameters in the model (synaptic strength, decaying relaxation rate of the synaptic coupling, and the parameter exhibiting the firing duration). We find some irregularly synchronous behavior with or without constant firing order alternations. We show that the duration of an impulse plays an important role in synchronization phenomena.

Received 8 February 2006

DOI:https://doi.org/10.1103/PhysRevE.75.011909

Authors & Affiliations

Yasuomi Daishin Sato^1,2,* and Masatoshi Shiino²

¹Frankfurt Institute for Advanced Studies, Johann Wolfgang Goethe University, Max-von-Laue-Str. 1, D-60438 Frankfurt am Main, Germany
²Department of Physics, Faculty of Science, Tokyo Institute of Technology, 2-12-1 Oh-okayama, Meguro-ku, Tokyo 152-0033, Japan

^*Electronic address: sato@fias.uni-frankfurt.de

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Vol. 75, Iss. 1 — January 2007

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Images

Figure 1
(Color) Singular perturbation approach of the PL model. (a) Phase plane of the PL model with $μ = 0.2$ . The red lines indicate dynamical flow while the green lines present the nullclines, those are $\dot{x} = 0$ and $\dot{y} = 0$ . (b) Time evolutions of membrane potential $x$ as $μ$ becomes smaller. The temporal duration between points means a velocity of $x$ . The $x$ motion on the active or inactive phases becomes gradually slower. $Ψ$ is a variable normalized by a periodicity. (c) $μ \to 0$ ; membrane potential and synaptic response are represented, respectively, as the red and blue lines. Here we set $a = 0.8$ , $b = 0.01$ , $p = 2 ∕ 3$ , $q = 0$ , $m = 2$ , $n = 1$ , $θ_{1} = - 1$ , $θ_{2} = 1$ , and $β = 1$ .Reuse & Permissions
Figure 2
$G_{syn} − β$ diagram and the bifurcation diagram for a small $γ_{1}$ and inhibitory coupling. As $γ_{1} = 1.0$ in the case of inhibitory synaptic couplings, (a) $G_{syn} − β$ phase diagram. Region I: death oscillators [as shown in Fig. 3a]. Region II: the only antiphase synchronized solution. Region III: the antiphase and quasi-in-phase synchronized solutions. Region IV: the only in-phase synchronized solution. Region V: the in-phase and quasi-antiphase synchronized solutions. Regions VI and VIII: the in-phase and antiphase synchronized solutions. Region VII: the in-phase, quasi-antiphase, and antiphase synchronized solutions. Region IX: in addition to the antiphase synchronized solution, (i) the in-phase synchronized solution with alternating of the firing order of the two neurons exists. (ii) Figs. 3b, 3c, 3dare demonstrated. (b) $ϕ − β$ diagram for the linear stability as $G_{syn} = - 0.4$ are illustrated. The solid and broken lines represent stable and unstable fixed points. $β_{2}$ and $β_{3}$ are saddle-node and inverse pitchfork bifurcation points, respectively. $β_{4}$ is a boundary point to split into type-I and -II TFPDs.Reuse & Permissions
Figure 3
(Color) Various types of synchronized solutions. (a) Death oscillators. (b) Two neurons become synchronized with a constant phase difference, alternating the firing order of the two neurons. (c) These neurons alternate their firing order, irregularly changing the phase difference. (d) In the two coupled neurons, the alternating, nonalternating of the firing order and their phase difference vary irregularly.Reuse & Permissions
Figure 4
(Color) $β − ϕ$ bifurcation diagram. (a) and (b) are $β − ϕ$ bifurcation diagrams for $γ_{1} = 10$ and $γ_{1} = 100$ . Red and green lines, respectively, represent stable and unstable fixed points.Reuse & Permissions
Figure 5
(Color) Excitatory couplings for $γ_{1} = 10$ . (a) $β − ϕ$ bifurcation diagram for $G_{syn} = 0.1$ . Red and green lines, respectively, represent stable and unstable fixed points. (b) $G_{syn} − β$ phase diagram. Region i: type-I quasi-in-phase synchronized solution. Region ii: type-II quasi-in-phase synchronized solution. Region iii: type-II quasi-antiphase synchronized solution. Region iv: the antiphase synchronized solution. $ϕ (x_{2}^{(\infty)}, s_{2}^{(\infty)}) = 0.75$ , which represents a boundary line between regions ii and iii, divides synchronized solution into the quasi-in-phase and quasi-antiphase ones.Reuse & Permissions
Figure 6
(Color) Phase reduction analysis for FHN and PL model with a finite $μ$ . (a) and (b), respectively, demonstrate $G$ plotted as a function of the phase $ϕ$ for $y$ - and $x$ -coupled system of the PL model with $μ = 0.01$ , $p = 2 ∕ 3$ , $q = 0$ , $m = n = 2 ∕ 3$ , $θ_{1} = - 1$ , and $θ_{2} = 1$ . Four different $μ$ values are shown. Stable or unstable equilibrium states correspond to 0 crossing with negative or positive slopes. (c) $β − ϕ$ bifurcation diagram for a $y$ -coupled system of FHN model [Eqs. (1, 2)] with $μ = 0.01$ . This agrees with numerical simulations for a $y$ -coupled system of the PL model.Reuse & Permissions

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