898 research outputs found
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
Synchronous Behavior of Two Coupled Electronic Neurons
We report on experimental studies of synchronization phenomena in a pair of
analog electronic neurons (ENs). The ENs were designed to reproduce the
observed membrane voltage oscillations of isolated biological neurons from the
stomatogastric ganglion of the California spiny lobster Panulirus interruptus.
The ENs are simple analog circuits which integrate four dimensional
differential equations representing fast and slow subcellular mechanisms that
produce the characteristic regular/chaotic spiking-bursting behavior of these
cells. In this paper we study their dynamical behavior as we couple them in the
same configurations as we have done for their counterpart biological neurons.
The interconnections we use for these neural oscillators are both direct
electrical connections and excitatory and inhibitory chemical connections: each
realized by analog circuitry and suggested by biological examples. We provide
here quantitative evidence that the ENs and the biological neurons behave
similarly when coupled in the same manner. They each display well defined
bifurcations in their mutual synchronization and regularization. We report
briefly on an experiment on coupled biological neurons and four dimensional ENs
which provides further ground for testing the validity of our numerical and
electronic models of individual neural behavior. Our experiments as a whole
present interesting new examples of regularization and synchronization in
coupled nonlinear oscillators.Comment: 26 pages, 10 figure
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
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
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