Bifurcation analysis of a general class of non-linear integrate and fire neurons.

Jonathan Touboul (1)

1 ODYSSEE - Computer and biological vision

In this paper we define a class of formal neuron models being computationally efficient and biologically plausible, i.e. able to reproduce a wide gamut of behaviors observed in in-vivo or in-vitro recordings of cortical neurons. This class includes for instance two models widely used in computational neuroscience, the Izhikevich and the Brette--Gerstner models. These models consist in a 4-parameters dynamical system. We provide the full local bifurcations diagram of the members of this class, and show that they all present the same bifurcations: an Andronov-Hopf bifurcation manifold, a saddle-node bifurcation manifold, a Bogdanov-Takens bifurcation, and possibly a Bautin bifurcation. Among other global bifurcations, this system shows a saddle homoclinic bifurcation curve. We show how this bifurcation diagram generates the most prominent cortical neuron behaviors. This study leads us to introduce a new neuron model, the \emph{quartic model}, able to reproduce among all the behaviors of the Izhikevich and Brette--Gerstner models, self-sustained subthreshold oscillations, which are of great interest in neuroscience.

Mots clés: neuron models; dynamical system analysis; nonlinear dynamics; Hopf bifurcation; saddle-node bifurcation; Bogdanov-Takens bifurcation; Bautin bifurcation; saddle homoclinic bifurcation; subthreshold neuron oscillations

Domaines: Systèmes dynamiques [math.DS]

Jonathan Touboul. Bifurcation analysis of a general class of non-linear integrate and fire neurons.. [Research Report] RR-6161, INRIA. 2008, pp.47. ⟨inria-00142987v5⟩