281 research outputs found
Shared inputs, entrainment, and desynchrony in elliptic bursters: from slow passage to discontinuous circle maps
What input signals will lead to synchrony vs. desynchrony in a group of
biological oscillators? This question connects with both classical dynamical
systems analyses of entrainment and phase locking and with emerging studies of
stimulation patterns for controlling neural network activity. Here, we focus on
the response of a population of uncoupled, elliptically bursting neurons to a
common pulsatile input. We extend a phase reduction from the literature to
capture inputs of varied strength, leading to a circle map with discontinuities
of various orders. In a combined analytical and numerical approach, we apply
our results to both a normal form model for elliptic bursting and to a
biophysically-based neuron model from the basal ganglia. We find that,
depending on the period and amplitude of inputs, the response can either appear
chaotic (with provably positive Lyaponov exponent for the associated circle
maps), or periodic with a broad range of phase-locked periods. Throughout, we
discuss the critical underlying mechanisms, including slow-passage effects
through Hopf bifurcation, the role and origin of discontinuities, and the
impact of noiseComment: 17 figures, 40 page
Mechanism, dynamics, and biological existence of multistability in a large class of bursting neurons
Multistability, the coexistence of multiple attractors in a dynamical system,
is explored in bursting nerve cells. A modeling study is performed to show that
a large class of bursting systems, as defined by a shared topology when
represented as dynamical systems, is inherently suited to support
multistability. We derive the bifurcation structure and parametric trends
leading to multistability in these systems. Evidence for the existence of
multirhythmic behavior in neurons of the aquatic mollusc Aplysia californica
that is consistent with our proposed mechanism is presented. Although these
experimental results are preliminary, they indicate that single neurons may be
capable of dynamically storing information for longer time scales than
typically attributed to nonsynaptic mechanisms.Comment: 24 pages, 8 figure
On transition to bursting via deterministic chaos
We study statistical properties of the irregular bursting arising in a class
of neuronal models close to the transition from spiking to bursting. Prior to
the transition to bursting, the systems in this class develop chaotic
attractors, which generate irregular spiking. The chaotic spiking gives rise to
irregular bursting. The duration of bursts near the transition can be very
long. We describe the statistics of the number of spikes and the interspike
interval distributions within one burst as functions of the distance from
criticality.Comment: 8 pages, 6 figure
Border collision bifurcations of stroboscopic maps in periodically driven spiking models
In this work we consider a general non-autonomous hybrid system based on the
integrate-and-fire model, widely used as simplified version of neuronal models
and other types of excitable systems. Our unique assumption is that the system
is monotonic, possesses an attracting subthreshold equilibrium point and is
forced by means of periodic pulsatile (square wave) function.\\ In contrast to
classical methods, in our approach we use the stroboscopic map (time- return
map) instead of the so-called firing-map. It becomes a discontinuous map
potentially defined in an infinite number of partitions. By applying theory for
piecewise-smooth systems, we avoid relying on particular computations and we
develop a novel approach that can be easily extended to systems with other
topologies (expansive dynamics) and higher dimensions.\\ More precisely, we
rigorously study the bifurcation structure in the two-dimensional parameter
space formed by the amplitude and the duty cycle of the pulse. We show that it
is covered by regions of existence of periodic orbits given by period adding
structures. They do not only completely describe all the possible spiking
asymptotic dynamics but also the behavior of the firing rate, which is a
devil's staircase as a function of the parameters
Wild oscillations in a nonlinear neuron model with resets: (II) Mixed-mode oscillations
This work continues the analysis of complex dynamics in a class of
bidimensional nonlinear hybrid dynamical systems with resets modeling neuronal
voltage dynamics with adaptation and spike emission. We show that these models
can generically display a form of mixed-mode oscillations (MMOs), which are
trajectories featuring an alternation of small oscillations with spikes or
bursts (multiple consecutive spikes). The mechanism by which these are
generated relies fundamentally on the hybrid structure of the flow: invariant
manifolds of the continuous dynamics govern small oscillations, while discrete
resets govern the emission of spikes or bursts, contrasting with classical MMO
mechanisms in ordinary differential equations involving more than three
dimensions and generally relying on a timescale separation. The decomposition
of mechanisms reveals the geometrical origin of MMOs, allowing a relatively
simple classification of points on the reset manifold associated to specific
numbers of small oscillations. We show that the MMO pattern can be described
through the study of orbits of a discrete adaptation map, which is singular as
it features discrete discontinuities with unbounded left- and
right-derivatives. We study orbits of the map via rotation theory for
discontinuous circle maps and elucidate in detail complex behaviors arising in
the case where MMOs display at most one small oscillation between each
consecutive pair of spikes
Conductance-Based Refractory Density Approach for a Population of Bursting Neurons
The conductance-based refractory density (CBRD) approach is a parsimonious mathematical-computational framework for modeling interact- ing populations of regular spiking neurons, which, however, has not been yet extended for a population of bursting neurons. The canonical CBRD method allows to describe the firing activity of a statistical ensemble of uncoupled Hodgkin-Huxley-like neurons (differentiated by noise) and has demonstrated its validity against experimental data. The present manuscript generalises the CBRD for a population of bursting neurons; however, in this pilot computational study we consider the simplest setting in which each individual neuron is governed by a piecewise linear bursting dynamics. The resulting popula- tion model makes use of slow-fast analysis, which leads to a novel method- ology that combines CBRD with the theory of multiple timescale dynamics. The main prospect is that it opens novel avenues for mathematical explo- rations, as well as, the derivation of more sophisticated population activity from Hodgkin-Huxley-like bursting neurons, which will allow to capture the activity of synchronised bursting activity in hyper-excitable brain states (e.g. onset of epilepsy).Russian Science Foundation grant (project 16-15- 10201)
Spanish grant MINECO-FEDER-UE MTM-2015-71509-C2-2-R
Catalan Grant number 2017SGR104
Chaotic oscillations in a map-based model of neural activity
We propose a discrete time dynamical system (a map) as phenomenological model
of excitable and spiking-bursting neurons. The model is a discontinuous
two-dimensional map. We find condition under which this map has an invariant
region on the phase plane, containing chaotic attractor. This attractor creates
chaotic spiking-bursting oscillations of the model. We also show various
regimes of other neural activities (subthreshold oscillations, phasic spiking
etc.) derived from the proposed model
Firing patterns in the adaptive exponential integrate-and-fire model
For simulations of large spiking neuron networks, an accurate, simple and versatile single-neuron modeling framework is required. Here we explore the versatility of a simple two-equation model: the adaptive exponential integrate-and-fire neuron. We show that this model generates multiple firing patterns depending on the choice of parameter values, and present a phase diagram describing the transition from one firing type to another. We give an analytical criterion to distinguish between continuous adaption, initial bursting, regular bursting and two types of tonic spiking. Also, we report that the deterministic model is capable of producing irregular spiking when stimulated with constant current, indicating low-dimensional chaos. Lastly, the simple model is fitted to real experiments of cortical neurons under step current stimulation. The results provide support for the suitability of simple models such as the adaptive exponential integrate-and-fire neuron for large network simulations
Firing patterns in the adaptive exponential integrate-and-fire model
For simulations of large spiking neuron networks, an accurate, simple and versatile single-neuron modeling framework is required. Here we explore the versatility of a simple two-equation model: the adaptive exponential integrate-and-fire neuron. We show that this model generates multiple firing patterns depending on the choice of parameter values, and present a phase diagram describing the transition from one firing type to another. We give an analytical criterion to distinguish between continuous adaption, initial bursting, regular bursting and two types of tonic spiking. Also, we report that the deterministic model is capable of producing irregular spiking when stimulated with constant current, indicating low-dimensional chaos. Lastly, the simple model is fitted to real experiments of cortical neurons under step current stimulation. The results provide support for the suitability of simple models such as the adaptive exponential integrate-and-fire neuron for large network simulation
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