380 research outputs found
Mean-field equations for stochastic firing-rate neural fields with delays: Derivation and noise-induced transitions
In this manuscript we analyze the collective behavior of mean-field limits of
large-scale, spatially extended stochastic neuronal networks with delays.
Rigorously, the asymptotic regime of such systems is characterized by a very
intricate stochastic delayed integro-differential McKean-Vlasov equation that
remain impenetrable, leaving the stochastic collective dynamics of such
networks poorly understood. In order to study these macroscopic dynamics, we
analyze networks of firing-rate neurons, i.e. with linear intrinsic dynamics
and sigmoidal interactions. In that case, we prove that the solution of the
mean-field equation is Gaussian, hence characterized by its two first moments,
and that these two quantities satisfy a set of coupled delayed
integro-differential equations. These equations are similar to usual neural
field equations, and incorporate noise levels as a parameter, allowing analysis
of noise-induced transitions. We identify through bifurcation analysis several
qualitative transitions due to noise in the mean-field limit. In particular,
stabilization of spatially homogeneous solutions, synchronized oscillations,
bumps, chaotic dynamics, wave or bump splitting are exhibited and arise from
static or dynamic Turing-Hopf bifurcations. These surprising phenomena allow
further exploring the role of noise in the nervous system.Comment: Updated to the latest version published, and clarified the dependence
in space of Brownian motion
The complexity of dynamics in small neural circuits
Mean-field theory is a powerful tool for studying large neural networks.
However, when the system is composed of a few neurons, macroscopic differences
between the mean-field approximation and the real behavior of the network can
arise. Here we introduce a study of the dynamics of a small firing-rate network
with excitatory and inhibitory populations, in terms of local and global
bifurcations of the neural activity. Our approach is analytically tractable in
many respects, and sheds new light on the finite-size effects of the system. In
particular, we focus on the formation of multiple branching solutions of the
neural equations through spontaneous symmetry-breaking, since this phenomenon
increases considerably the complexity of the dynamical behavior of the network.
For these reasons, branching points may reveal important mechanisms through
which neurons interact and process information, which are not accounted for by
the mean-field approximation.Comment: 34 pages, 11 figures. Supplementary materials added, colors of
figures 8 and 9 fixed, results unchange
A mathematical framework for critical transitions: normal forms, variance and applications
Critical transitions occur in a wide variety of applications including
mathematical biology, climate change, human physiology and economics. Therefore
it is highly desirable to find early-warning signs. We show that it is possible
to classify critical transitions by using bifurcation theory and normal forms
in the singular limit. Based on this elementary classification, we analyze
stochastic fluctuations and calculate scaling laws of the variance of
stochastic sample paths near critical transitions for fast subsystem
bifurcations up to codimension two. The theory is applied to several models:
the Stommel-Cessi box model for the thermohaline circulation from geoscience,
an epidemic-spreading model on an adaptive network, an activator-inhibitor
switch from systems biology, a predator-prey system from ecology and to the
Euler buckling problem from classical mechanics. For the Stommel-Cessi model we
compare different detrending techniques to calculate early-warning signs. In
the epidemics model we show that link densities could be better variables for
prediction than population densities. The activator-inhibitor switch
demonstrates effects in three time-scale systems and points out that excitable
cells and molecular units have information for subthreshold prediction. In the
predator-prey model explosive population growth near a codimension two
bifurcation is investigated and we show that early-warnings from normal forms
can be misleading in this context. In the biomechanical model we demonstrate
that early-warning signs for buckling depend crucially on the control strategy
near the instability which illustrates the effect of multiplicative noise.Comment: minor corrections to previous versio
Stability and chaos of the duopoly model of Kopel: A study based on symbolic computations
Since Kopel's duopoly model was proposed about three decades ago, there are
almost no analytical results on the equilibria and their stability in the
asymmetric case. The first objective of our study is to fill this gap. This
paper analyzes the asymmetric duopoly model of Kopel analytically by using
several tools based on symbolic computations. We discuss the possibility of the
existence of multiple positive equilibria and establish necessary and
sufficient conditions for a given number of positive equilibria to exist. The
possible positions of the equilibria in Kopel's model are also explored.
Furthermore, if the duopolists adopt the best response reactions or homogeneous
adaptive expectations, we establish rigorous conditions for the existence of
distinct numbers of positive equilibria for the first time. The occurrence of
chaos in Kopel's model seems to be supported by observations through numerical
simulations, which, however, is challenging to prove rigorously. The second
objective is to prove the existence of snapback repellers in Kopel's map, which
implies the existence of chaos in the sense of Li-Yorke according to Marotto's
theorem.Comment: arXiv admin note: substantial text overlap with arXiv:2301.1262
A generalized 2D-dynamical mean-field Ising model with a rich set of bifurcations (inspired and applied to financial crises)
This is the final version of the article. Available from the publisher via the DOI in this record.We analyze an extended version of the dynamical mean-field Ising model. Instead of classical physical representation of spins and external magnetic field, the model describes traders' opinion dynamics. The external field is endogenized to represent a smoothed moving average of the past state variable. This model captures in a simple set-up the interplay between instantaneous social imitation and past trends in social coordinations. We show the existence of a rich set of bifurcations as a function of the two parameters quantifying the relative importance of instantaneous versus past social opinions on the formation of the next value of the state variable. Moreover, we present a thorough analysis of chaotic behavior, which is exhibited in certain parameter regimes. Finally, we examine several transitions through bifurcation curves and study how they could be understood as specific market scenarios. We find that the amplitude of the corrections needed to recover from a crisis and to push the system back to “normal” is often significantly larger than the strength of the causes that led to the crisis itself.This project has received funding from the European
Union’s Horizon 2020 research and innovation
programme under the Marie Sklodowska-Curie
grant agreement No. 643073
Interacting Turing-Hopf Instabilities Drive Symmetry-Breaking Transitions in a Mean-Field Model of the Cortex: A Mechanism for the Slow Oscillation
Electrical recordings of brain activity during the transition from wake to anesthetic coma show temporal and spectral alterations that are correlated with gross changes in the underlying brain state. Entry into anesthetic unconsciousness is signposted by the emergence of large, slow oscillations of electrical activity (≲1 Hz) similar to the slow waves observed in natural sleep. Here we present a two-dimensional mean-field model of the cortex in which slow spatiotemporal oscillations arise spontaneously through a Turing (spatial) symmetry-breaking bifurcation that is modulated by a Hopf (temporal) instability. In our model, populations of neurons are densely interlinked by chemical synapses, and by interneuronal gap junctions represented as an inhibitory diffusive coupling. To demonstrate cortical behavior over a wide range of distinct brain states, we explore model dynamics in the vicinity of a general-anesthetic-induced transition from “wake” to “coma.” In this region, the system is poised at a codimension-2 point where competing Turing and Hopf instabilities coexist. We model anesthesia as a moderate reduction in inhibitory diffusion, paired with an increase in inhibitory postsynaptic response, producing a coma state that is characterized by emergent low-frequency oscillations whose dynamics is chaotic in time and space. The effect of long-range axonal white-matter connectivity is probed with the inclusion of a single idealized point-to-point connection. We find that the additional excitation from the long-range connection can provoke seizurelike bursts of cortical activity when inhibitory diffusion is weak, but has little impact on an active cortex. Our proposed dynamic mechanism for the origin of anesthetic slow waves complements—and contrasts with—conventional explanations that require cyclic modulation of ion-channel conductances. We postulate that a similar bifurcation mechanism might underpin the slow waves of natural sleep and comment on the possible consequences of chaotic dynamics for memory processing and learning
Nonlocal control of pulse propagation in excitable media
We study the effects of nonlocal control of pulse propagation in excitable
media. As a generic example for an excitable medium the FitzHugh-Nagumo model
with diffusion in the activator variable is considered. Nonlocal coupling in
form of an integral term with a spatial kernel is added. We find that the
nonlocal coupling modifies the propagating pulses of the reaction-diffusion
system such that a variety of spatio-temporal patterns are generated including
acceleration, deceleration, suppression, or generation of pulses, multiple
pulses, and blinking pulse trains. It is shown that one can observe these
effects for various choices of the integral kernel and the coupling scheme,
provided that the control strength and spatial extension of the integral kernel
is appropriate. In addition, an analytical procedure is developed to describe
the stability borders of the spatially homogeneous steady state in control
parameter space in dependence on the parameters of the nonlocal coupling
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