203 research outputs found
Sensitivity to the cutoff value in the quadratic adaptive integrate-and-fire model
The quadratic adaptive integrate-and-fire model (Izhikecih 2003, 2007) is
recognized as very interesting for its computational efficiency and its ability
to reproduce many behaviors observed in cortical neurons. For this reason it is
currently widely used, in particular for large scale simulations of neural
networks. This model emulates the dynamics of the membrane potential of a
neuron together with an adaptation variable. The subthreshold dynamics is
governed by a two-parameter differential equation, and a spike is emitted when
the membrane potential variable reaches a given cutoff value. Subsequently the
membrane potential is reset, and the adaptation variable is added a fixed value
called the spike-triggered adaptation parameter. We show in this note that when
the system does not converge to an equilibrium point, both variables of the
subthreshold dynamical system blow up in finite time whatever the parameters of
the dynamics. The cutoff is therefore essential for the model to be well
defined and simulated. The divergence of the adaptation variable makes the
system very sensitive to the cutoff: changing this parameter dramatically
changes the spike patterns produced. Furthermore from a computational
viewpoint, the fact that the adaptation variable blows up and the very sharp
slope it has when the spike is emitted implies that the time step of the
numerical simulation needs to be very small (or adaptive) in order to catch an
accurate value of the adaptation at the time of the spike. It is not the case
for the similar quartic (Touboul 2008) and exponential (Brette and Gerstner
2005) models whose adaptation variable does not blow up in finite time, and
which are therefore very robust to changes in the cutoff value
Propagation of chaos in neural fields
We consider the problem of the limit of bio-inspired spatially extended
neuronal networks including an infinite number of neuronal types (space
locations), with space-dependent propagation delays modeling neural fields. The
propagation of chaos property is proved in this setting under mild assumptions
on the neuronal dynamics, valid for most models used in neuroscience, in a
mesoscopic limit, the neural-field limit, in which we can resolve the quite
fine structure of the neuron's activity in space and where averaging effects
occur. The mean-field equations obtained are of a new type: they take the form
of well-posed infinite-dimensional delayed integro-differential equations with
a nonlocal mean-field term and a singular spatio-temporal Brownian motion. We
also show how these intricate equations can be used in practice to uncover
mathematically the precise mesoscopic dynamics of the neural field in a
particular model where the mean-field equations exactly reduce to deterministic
nonlinear delayed integro-differential equations. These results have several
theoretical implications in neuroscience we review in the discussion.Comment: Updated to correct an erroneous suggestion of extension of the
results in Appendix B, and to clarify some measurability questions in the
proof of Theorem
Macroscopic equations governing noisy spiking neuronal populations
At functional scales, cortical behavior results from the complex interplay of
a large number of excitable cells operating in noisy environments. Such systems
resist to mathematical analysis, and computational neurosciences have largely
relied on heuristic partial (and partially justified) macroscopic models, which
successfully reproduced a number of relevant phenomena. The relationship
between these macroscopic models and the spiking noisy dynamics of the
underlying cells has since then been a great endeavor. Based on recent
mean-field reductions for such spiking neurons, we present here {a principled
reduction of large biologically plausible neuronal networks to firing-rate
models, providing a rigorous} relationship between the macroscopic activity of
populations of spiking neurons and popular macroscopic models, under a few
assumptions (mainly linearity of the synapses). {The reduced model we derive
consists of simple, low-dimensional ordinary differential equations with}
parameters and {nonlinearities derived from} the underlying properties of the
cells, and in particular the noise level. {These simple reduced models are
shown to reproduce accurately the dynamics of large networks in numerical
simulations}. Appropriate parameters and functions are made available {online}
for different models of neurons: McKean, Fitzhugh-Nagumo and Hodgkin-Huxley
models
Heterogeneous connections induce oscillations in large scale networks
Realistic large-scale networks display an heterogeneous distribution of
connectivity weights, that might also randomly vary in time. We show that
depending on the level of heterogeneity in the connectivity coefficients,
different qualitative macroscopic and microscopic regimes emerge. We evidence
in particular generic transitions from stationary to perfectly periodic
phase-locked regimes as the disorder parameter is increased, both in a simple
model treated analytically and in a biologically relevant network made of
excitable cells
On an explicit representation of the solution of linear stochastic partial differential equations with delays
Based on the analysis of a certain class of linear operators on a Banach
space, we provide a closed form expression for the solutions of certain linear
partial differential equations with non-autonomous input, time delays and
stochastic terms, which takes the form of an infinite series expansion
Complex oscillations in the delayed Fitzhugh-Nagumo equation
Motivated by the dynamics of neuronal responses, we analyze the dynamics of
the Fitzhugh-Nagumo slow-fast system with delayed self-coupling. This system
provides a canonical example of a canard explosion for sufficiently small
delays. Beyond this regime, delays significantly enrich the dynamics, leading
to mixed-mode oscillations, bursting and chaos. These behaviors emerge from a
delay-induced subcritical Bogdanov-Takens instability arising at the fold
points of the S-shaped critical manifold. Underlying the transition from
canard-induced to delay-induced dynamics is an abrupt switch in the nature of
the Hopf bifurcation
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