Locally Contractive Dynamics in Generalized Integrate-and-Fire Neurons

Abstract

Integrate-and-fire models of biological neurons combine differential equations with discrete spike events. In the simplest case, the reset of the neuronal voltage to its resting value is the only spike event. The response of such a model to constant input injection is limited to tonic spiking. We here study a generalized model in which two simple spike-induced currents are added. We show that this neuron exhibits not only tonic spiking at various frequencies but also the commonly observed neuronal bursting. Using analytical and numerical approaches, we show that this model can be reduced to a one-dimensional map of the adaptation variable and that this map is locally contractive over a broad set of parameter values. We derive a sufficient analytical condition on the parameters for the map to be globally contractive, in which case all orbits tend to a tonic spiking state determined by the fixed point of the return map. We then show that bursting is caused by a discontinuity in the return map, in which case the map is piecewise contractive. We perform a detailed analysis of a class of piecewise contractive maps that we call bursting maps and show that they robustly generate stable bursting behavior. To the best of our knowledge, this work is the first to point out the intimate connection between bursting dynamics and piecewise contractive maps. Finally, we discuss bifurcations in this return map, which cause transitions between spiking patterns.

Event--Selected Vector Field Discontinuities Yield Piecewise--Differentiable Flows

“…In the classical setting, these constructions are obtained using the classical (alternately called Fréchet [Sch12, Section 3.1] or Jacobian [GH83, Section 1.3]) derivative of the smooth flow; our construction of the non-smooth object proceeded analogously to that of its smooth counterpart after replacing the classical derivative of the flow with our B-derivative. Thus the piecewise-differentiable dynamical systems we study bear a closer resemblance to classically differentiable dynamical systems than to discontinuous dynamical systems considered, for instance, in [PB10;Jim+13].…”

The Period Adding and Incrementing Bifurcations: From Rotation Theory to Applications

“…In the recent years, piecewise-smooth maps, both in R and in R n , have been widely investigated and very interesting bifurcation phenomena (border collisions, big bangs, period adding, period incrementing,...) have been reported by many authors. This included both works by a more applied community, like in power electronics ( [7,8,9,10,19,30,43,76,87,115]), control theory ( [46,50,51,102,135,133,134,136]), biology ( [57,106]), neuroscience ( [73,74,84,88,89,103,122,123,124,125]) or economy ( [126]) but also researchers from a more theoretical and/or computational perspective in non-smooth systems ( [12,13,14,15,17,18,20,21,22,23,24,25,26,27,28,…”

“…We show how Theorem 2.13 can be applied to describe, not only its dynamics, but relevant biological properties as the firing-rate under parameter variation. There exist in the literature other examples in neuron models from which one derives a piecewise-smooth discontinuous map such that Theorem 2.13 ii) holds; see for example [84,125]. Section 7.2 is devoted to illustrate the results revisited in Section 6 for higherdimensional piecewise-smooth maps.…”

“…Beyond its relation with homoclinic bifurcations for flows, such type of bifurcation scenarios have been observed in more applied contexts modeled by both one and n-dimensional piecewise-smooth maps. Examples of such applications where these bifurcation scenarios appear are power electronics ( [45,29,113,67,136,7,85,134]), control theory ( [46,58,51,133,135,50,51]), economics ( [126]) or neuroscience ( [57,74,84,89,103,122,116,124,125,123]). Typically, these bifurcation scenarios are numerically observed in two-dimensional parameter spaces near codimension-two bifurcation points.…”

Wild oscillations in a nonlinear neuron model with resets: (Ⅰ) Bursting, spike-adding and chaos

“…In particular, Foxall and collaborators established a relation between the adaptation map and transverse Lyapunov exponents to characterize the stability of spiking periodic orbits in nonlinear integrate-and-fire systems [19]. In [32], a generalized linear integrate-and-fire system was investigated via a similar map that is locally contractive, either globally or in a piecewise manner; conditions for spiking and bursting dynamics were established and bifurcations underlying transitions between solution patterns were studied. In the accompanying paper [57], we address the question of the dynamics of the system in the presence of two unstable equilibria of the subthreshold dynamics; in that case, the map Φ is no longer continuous, and the rotation theory for discontinuous maps is applied to characterize the dynamics.…”