568 research outputs found
The Utility of Phase Models in Studying Neural Synchronization
Synchronized neural spiking is associated with many cognitive functions and
thus, merits study for its own sake. The analysis of neural synchronization
naturally leads to the study of repetitive spiking and consequently to the
analysis of coupled neural oscillators. Coupled oscillator theory thus informs
the synchronization of spiking neuronal networks. A crucial aspect of coupled
oscillator theory is the phase response curve (PRC), which describes the impact
of a perturbation to the phase of an oscillator. In neural terms, the
perturbation represents an incoming synaptic potential which may either advance
or retard the timing of the next spike. The phase response curves and the form
of coupling between reciprocally coupled oscillators defines the phase
interaction function, which in turn predicts the synchronization outcome
(in-phase versus anti-phase) and the rate of convergence. We review the two
classes of PRC and demonstrate the utility of the phase model in predicting
synchronization in reciprocally coupled neural models. In addition, we compare
the rate of convergence for all combinations of reciprocally coupled Class I
and Class II oscillators. These findings predict the general synchronization
outcomes of broad classes of neurons under both inhibitory and excitatory
reciprocal coupling.Comment: 18 pages, 5 figure
Phase resetting of collective rhythm in ensembles of oscillators
Phase resetting curves characterize the way a system with a collective
periodic behavior responds to perturbations. We consider globally coupled
ensembles of Sakaguchi-Kuramoto oscillators, and use the Ott-Antonsen theory of
ensemble evolution to derive the analytical phase resetting equations. We show
the final phase reset value to be composed of two parts: an immediate phase
reset directly caused by the perturbation, and the dynamical phase reset
resulting from the relaxation of the perturbed system back to its dynamical
equilibrium. Analytical, semi-analytical and numerical approximations of the
final phase resetting curve are constructed. We support our findings with
extensive numerical evidence involving identical and non-identical oscillators.
The validity of our theory is discussed in the context of large ensembles
approximating the thermodynamic limit.Comment: submitted to Phys. Rev.
Loss of synchrony in an inhibitory network of type-I oscillators
Synchronization of excitable cells coupled by reciprocal inhibition is a topic of significant interest due to the important role that inhibitory synaptic interaction plays in the generation and regulation of coherent rhythmic activity in a variety of neural systems. While recent work revealed the synchronizing influence of inhibitory coupling on the dynamics of many networks, it is known that strong coupling can destabilize phase-locked firing. Here we examine the loss of synchrony caused by an increase in inhibitory coupling in networks of type-I Morris-Lecar model oscillators, which is characterized by a period-doubling cascade and leads to mode-locked states with alternation in the firing order of the two cells, as reported recently by Maran and Canavier (2007) for a network of Wang-Buzsáki model neurons. Although alternating- order firing has been previously reported as a near-synchronous state, we show that the stable phase difference between the spikes of the two Morris-Lecar cells can constitute as much as 70% of the unperturbed oscillation period. Further, we examine the generality of this phenomenon for a class of type-I oscillators that are close to their excitation thresholds, and provide an intuitive geometric description of such leap-frog dynamics. In the Morris-Lecar model network, the alternation in the firing order arises under the condition of fast closing of K+ channels at hyperpolarized potentials, which leads to slow dynamics of membrane potential upon synaptic inhibition, allowing the presynaptic cell to advance past the postsynaptic cell in each cycle of the oscillation. Further, we show that non-zero synaptic decay time is crucial for the existence of leap-frog firing in networks of phase oscillators. However, we demonstrate that leap-frog spiking can also be obtained in pulse-coupled inhibitory networks of one-dimensional oscillators with a multi-branched phase domain, for instance in a network of quadratic integrate-and-fire model cells. Also, we show that the entire bifurcation structure of the network can be explained by a simple scaling of the STRC (spike- time response curve) amplitude, using a simplified quadratic STRC as an example, and derive the general conditions on the shape of the STRC function that leads to leap-frog firing. Further, for the case of a homogeneous network, we establish quantitative conditions on the phase resetting properties of each cell necessary for stable alternating-order spiking, complementing the analysis of Goel and Ermentrout (2002) of the order-preserving phase transition map. We show that the extension of STRC to negative values of phase is necessary to predict the response of a model cell to several close non-weak perturbations. This allows us for instance to accurately describe the dynamics of non-weakly coupled network of three model cells. Finally, the phase return map is also extended to the heterogenous network, and is used to analyze both the order-alternating firing and the order-preserving non-zero phase locked state in this case
Timescales of spike-train correlation for neural oscillators with common drive
We examine the effect of the phase-resetting curve (PRC) on the transfer of
correlated input signals into correlated output spikes in a class of neural
models receiving noisy, super-threshold stimulation. We use linear response
theory to approximate the spike correlation coefficient in terms of moments of
the associated exit time problem, and contrast the results for Type I vs. Type
II models and across the different timescales over which spike correlations can
be assessed. We find that, on long timescales, Type I oscillators transfer
correlations much more efficiently than Type II oscillators. On short
timescales this trend reverses, with the relative efficiency switching at a
timescale that depends on the mean and standard deviation of input currents.
This switch occurs over timescales that could be exploited by downstream
circuits
Emergence and combinatorial accumulation of jittering regimes in spiking oscillators with delayed feedback
Interaction via pulses is common in many natural systems, especially
neuronal. In this article we study one of the simplest possible systems with
pulse interaction: a phase oscillator with delayed pulsatile feedback. When the
oscillator reaches a specific state, it emits a pulse, which returns after
propagating through a delay line. The impact of an incoming pulse is described
by the oscillator's phase reset curve (PRC). In such a system we discover an
unexpected phenomenon: for a sufficiently steep slope of the PRC, a periodic
regular spiking solution bifurcates with several multipliers crossing the unit
circle at the same parameter value. The number of such critical multipliers
increases linearly with the delay and thus may be arbitrary large. This
bifurcation is accompanied by the emergence of numerous "jittering" regimes
with non-equal interspike intervals (ISIs). Each of these regimes corresponds
to a periodic solution of the system with a period roughly proportional to the
delay. The number of different "jittering" solutions emerging at the
bifurcation point increases exponentially with the delay. We describe the
combinatorial mechanism that underlies the emergence of such a variety of
solutions. In particular, we show how a periodic solution exhibiting several
distinct ISIs can imply the existence of multiple other solutions obtained by
rearranging of these ISIs. We show that the theoretical results for phase
oscillators accurately predict the behavior of an experimentally implemented
electronic oscillator with pulsatile feedback
Sensitivity analysis of oscillator models in the space of phase-response curves: Oscillators as open systems
Oscillator models are central to the study of system properties such as
entrainment or synchronization. Due to their nonlinear nature, few
system-theoretic tools exist to analyze those models. The paper develops a
sensitivity analysis for phase-response curves, a fundamental one-dimensional
phase reduction of oscillator models. The proposed theoretical and numerical
analysis tools are illustrated on several system-theoretic questions and models
arising in the biology of cellular rhythms
A comparison of methods to determine neuronal phase-response curves
The phase-response curve (PRC) is an important tool to determine the
excitability type of single neurons which reveals consequences for their
synchronizing properties. We review five methods to compute the PRC from both
model data and experimental data and compare the numerically obtained results
from each method. The main difference between the methods lies in the
reliability which is influenced by the fluctuations in the spiking data and the
number of spikes available for analysis. We discuss the significance of our
results and provide guidelines to choose the best method based on the available
data.Comment: PDFLatex, 16 pages, 7 figures
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