2,205 research outputs found
Phase models and clustering in networks of oscillators with delayed coupling
We consider a general model for a network of oscillators with time delayed,
circulant coupling. We use the theory of weakly coupled oscillators to reduce
the system of delay differential equations to a phase model where the time
delay enters as a phase shift. We use the phase model to study the existence
and stability of cluster solutions. Cluster solutions are phase locked
solutions where the oscillators separate into groups. Oscillators within a
group are synchronized while those in different groups are phase-locked. We
give model independent existence and stability results for symmetric cluster
solutions. We show that the presence of the time delay can lead to the
coexistence of multiple stable clustering solutions. We apply our analytical
results to a network of Morris Lecar neurons and compare these results with
numerical continuation and simulation studies
Multiple firing coherence resonances in excitatory and inhibitory coupled neurons
The impact of inhibitory and excitatory synapses in delay-coupled
Hodgkin--Huxley neurons that are driven by noise is studied. If both synaptic
types are used for coupling, appropriately tuned delays in the inhibition
feedback induce multiple firing coherence resonances at sufficiently strong
coupling strengths, thus giving rise to tongues of coherency in the
corresponding delay-strength parameter plane. If only inhibitory synapses are
used, however, appropriately tuned delays also give rise to multiresonant
responses, yet the successive delays warranting an optimal coherence of
excitations obey different relations with regards to the inherent time scales
of neuronal dynamics. This leads to denser coherence resonance patterns in the
delay-strength parameter plane. The robustness of these findings to the
introduction of delay in the excitatory feedback, to noise, and to the number
of coupled neurons is determined. Mechanisms underlying our observations are
revealed, and it is suggested that the regularity of spiking across neuronal
networks can be optimized in an unexpectedly rich variety of ways, depending on
the type of coupling and the duration of delays.Comment: 7 two-column pages, 6 figures; accepted for publication in
Communications in Nonlinear Science and Numerical Simulatio
Asymptotic Stability, Orbital Stability of Hopf-Bifurcating Periodic Solution of a Simple Three-Neuron Artificial Neural Network with Distributed Delay
A distributed delay model of a class of three-neuron network has been investigated. Sufficient conditions for existence of unique equilibrium, multiple equilibria and their local stability are derived. A closed interval for a parameter of the system is identified in which Hopf-bifurcating periodic solution occurs for each point of such interval. The orbital stability of such bifurcating periodic solution at the extreme points of the interval is ascertained. Lastly global bifurcation aspect of such periodic solutions is studied. The results are illustrated by numerical simulations
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
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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