556 research outputs found
Hopf Bifurcation and Chaos in a Single Inertial Neuron Model with Time Delay
A delayed differential equation modelling a single neuron with inertial term
is considered in this paper. Hopf bifurcation is studied by using the normal
form theory of retarded functional differential equations. When adopting a
nonmonotonic activation function, chaotic behavior is observed. Phase plots,
waveform plots, and power spectra are presented to confirm the chaoticity.Comment: 12 pages, 7 figure
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
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
Delay-induced patterns in a two-dimensional lattice of coupled oscillators
We show how a variety of stable spatio-temporal periodic patterns can be
created in 2D-lattices of coupled oscillators with non-homogeneous coupling
delays. A "hybrid dispersion relation" is introduced, which allows studying the
stability of time-periodic patterns analytically in the limit of large delay.
The results are illustrated using the FitzHugh-Nagumo coupled neurons as well
as coupled limit cycle (Stuart-Landau) oscillators
Computational study of resting state network dynamics
Lo scopo di questa tesi è quello di mostrare, attraverso una simulazione con il software The Virtual Brain, le più importanti proprietà della dinamica cerebrale durante il resting state, ovvero quando non si è coinvolti in nessun compito preciso e non si è sottoposti a nessuno stimolo particolare. Si comincia con lo spiegare cos’è il resting state attraverso una breve revisione storica della sua scoperta, quindi si passano in rassegna alcuni metodi sperimentali utilizzati nell’analisi dell’attività cerebrale, per poi evidenziare la differenza tra connettività strutturale e funzionale. In seguito, si riassumono brevemente i concetti dei sistemi dinamici, teoria indispensabile per capire un sistema complesso come il cervello. Nel capitolo successivo, attraverso un approccio ‘bottom-up’, si illustrano sotto il profilo biologico le principali strutture del sistema nervoso, dal neurone alla corteccia cerebrale. Tutto ciò viene spiegato anche dal punto di vista dei sistemi dinamici, illustrando il pionieristico modello di Hodgkin-Huxley e poi il concetto di dinamica di popolazione. Dopo questa prima parte preliminare si entra nel dettaglio della simulazione. Prima di tutto si danno maggiori informazioni sul software The Virtual Brain, si definisce il modello di network del resting state utilizzato nella simulazione e si descrive il ‘connettoma’ adoperato. Successivamente vengono mostrati i risultati dell’analisi svolta sui dati ricavati, dai quali si mostra come la criticità e il rumore svolgano un ruolo chiave nell'emergenza di questa attività di fondo del cervello. Questi risultati vengono poi confrontati con le più importanti e recenti ricerche in questo ambito, le quali confermano i risultati del nostro lavoro. Infine, si riportano brevemente le conseguenze che porterebbe in campo medico e clinico una piena comprensione del fenomeno del resting state e la possibilità di virtualizzare l’attività cerebrale
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