381 research outputs found
Synchronization and oscillatory dynamics in heterogeneous mutually inhibited neurons
We study some mechanisms responsible for synchronous oscillations and loss of
synchrony at physiologically relevant frequencies (10-200 Hz) in a network of
heterogeneous inhibitory neurons. We focus on the factors that determine the
level of synchrony and frequency of the network response, as well as the
effects of mild heterogeneity on network dynamics. With mild heterogeneity,
synchrony is never perfect and is relatively fragile. In addition, the effects
of inhibition are more complex in mildly heterogeneous networks than in
homogeneous ones. In the former, synchrony is broken in two distinct ways,
depending on the ratio of the synaptic decay time to the period of repetitive
action potentials (), where can be determined either from the
network or from a single, self-inhibiting neuron. With ,
corresponding to large applied current, small synaptic strength or large
synaptic decay time, the effects of inhibition are largely tonic and
heterogeneous neurons spike relatively independently. With ,
synchrony breaks when faster cells begin to suppress their less excitable
neighbors; cells that fire remain nearly synchronous. We show numerically that
the behavior of mildly heterogeneous networks can be related to the behavior of
single, self-inhibiting cells, which can be studied analytically.Comment: 17 pages, 6 figures, Kluwer.sty. Journal of Compuational Neuroscience
(in press). Originally submitted to the neuro-sys archive which was never
publicly announced (was 9802001
Frequency control in synchronized networks of inhibitory neurons
We analyze the control of frequency for a synchronized inhibitory neuronal
network. The analysis is done for a reduced membrane model with a
biophysically-based synaptic influence. We argue that such a reduced model can
quantitatively capture the frequency behavior of a larger class of neuronal
models. We show that in different parameter regimes, the network frequency
depends in different ways on the intrinsic and synaptic time constants. Only in
one portion of the parameter space, called `phasic', is the network period
proportional to the synaptic decay time. These results are discussed in
connection with previous work of the authors, which showed that for mildly
heterogeneous networks, the synchrony breaks down, but coherence is preserved
much more for systems in the phasic regime than in the other regimes. These
results imply that for mildly heterogeneous networks, the existence of a
coherent rhythm implies a linear dependence of the network period on synaptic
decay time, and a much weaker dependence on the drive to the cells. We give
experimental evidence for this conclusion.Comment: 18 pages, 3 figures, Kluwer.sty. J. Comp. Neurosci. (in press).
Originally submitted to the neuro-sys archive which was never publicly
announced (was 9803001
Rhythms of the nervous system: mathematical themes and variations
The nervous system displays a variety of rhythms in both waking and sleep. These rhythms have been closely associated with different behavioral and cognitive states, but it is still unknown how the nervous system makes use of these rhythms to perform functionally important tasks. To address those questions, it is first useful to understood in a mechanistic way the origin of the rhythms, their interactions, the signals which create the transitions among rhythms, and the ways in which rhythms filter the signals to a network of neurons. This talk discusses how dynamical systems have been used to investigate the origin, properties and interactions of rhythms in the nervous system. It focuses on how the underlying physiology of the cells and synapses of the networks shape the dynamics of the network in different contexts, allowing the variety of dynamical behaviors to be displayed by the same network. The work is presented using a series of related case studies on different rhythms. These case studies are chosen to highlight mathematical issues, and suggest further mathematical work to be done. The topics include: different roles of excitation and inhibition in creating synchronous assemblies of cells, different kinds of building blocks for neural oscillations, and transitions among rhythms. The mathematical issues include reduction of large networks to low dimensional maps, role of noise, global bifurcations, use of probabilistic formulations.Published versio
Stability Analysis of Phase-Locked Bursting in Inhibitory Neuron Networks
Networks of neurons, which form central pattern generators (CPGs), are important for controlling animal behaviors. Of special interest are configurations or CPG motifs composed of reciprocally inhibited neurons, such as half-center oscillators (HCOs). Bursting rhythms of HCOs are shown to include stable synchrony or in-phase bursting. This in-phase bursting can co-exist with anti-phase bursting, commonly expected as the single stable state in HCOs that are connected with fast non-delayed synapses. The finding contrasts with the classical view that reciprocal inhibition has to be slow or time-delayed to synchronize such bursting neurons. Phase-locked rhythms are analyzed via Lyapunov exponents estimated with variational equations, and through the convergence rates estimated with Poincar\\u27e return maps. A new mechanism underlying multistability is proposed that is based on the spike interactions, which confer a dual property on the fast non-delayed reciprocal inhibition; this reveals the role of spikes in generating multiple co-existing phase-locked rhythms. In particular, it demonstrates that the number and temporal characteristics of spikes determine the number and stability of the multiple phase-locked states in weakly coupled HCOs. The generality of the multistability phenomenon is demonstrated by analyzing diverse models of bursting networks with various inhibitory synapses; the individual cell models include the reduced leech heart interneuron, the Sherman model for pancreatic beta cells, the Purkinje neuron model and Fitzhugh-Rinzel phenomenological model. Finally, hypothetical and experiment-based CPGs composed of HCOs are investigated. This study is relevant for various applications that use CPGs such as robotics, prosthetics, and artificial intelligence
Geometric Analysis of Synchronization in Neuronal Networks with Global Inhibition and Coupling Delays
We study synaptically coupled neuronal networks to identify the role of
coupling delays in network's synchronized behaviors. We consider a network of
excitable, relaxation oscillator neurons where two distinct populations, one
excitatory and one inhibitory, are coupled and interact with each other. The
excitatory population is uncoupled, while the inhibitory population is tightly
coupled. A geometric singular perturbation analysis yields existence and
stability conditions for synchronization states under different firing patterns
between the two populations, along with formulas for the periods of such
synchronous solutions. Our results demonstrate that the presence of coupling
delays in the network promotes synchronization. Numerical simulations are
conducted to supplement and validate analytical results. We show the results
carry over to a model for spindle sleep rhythms in thalamocortical networks,
one of the biological systems which motivated our study. The analysis helps to
explain how coupling delays in either excitatory or inhibitory synapses
contribute to producing synchronized rhythms.Comment: 43 pages, 12 figure
Rhythm generation, coordination, and initiation in the vocal pathways of male African clawed frogs
Poste
Statistical-Mechanical Measure of Stochastic Spiking Coherence in A Population of Inhibitory Subthreshold Neurons
By varying the noise intensity, we study stochastic spiking coherence (i.e.,
collective coherence between noise-induced neural spikings) in an inhibitory
population of subthreshold neurons (which cannot fire spontaneously without
noise). This stochastic spiking coherence may be well visualized in the raster
plot of neural spikes. For a coherent case, partially-occupied "stripes"
(composed of spikes and indicating collective coherence) are formed in the
raster plot. This partial occupation occurs due to "stochastic spike skipping"
which is well shown in the multi-peaked interspike interval histogram. The main
purpose of our work is to quantitatively measure the degree of stochastic
spiking coherence seen in the raster plot. We introduce a new spike-based
coherence measure by considering the occupation pattern and the pacing
pattern of spikes in the stripes. In particular, the pacing degree between
spikes is determined in a statistical-mechanical way by quantifying the average
contribution of (microscopic) individual spikes to the (macroscopic)
ensemble-averaged global potential. This "statistical-mechanical" measure
is in contrast to the conventional measures such as the "thermodynamic" order
parameter (which concerns the time-averaged fluctuations of the macroscopic
global potential), the "microscopic" correlation-based measure (based on the
cross-correlation between the microscopic individual potentials), and the
measures of precise spike timing (based on the peri-stimulus time histogram).
In terms of , we quantitatively characterize the stochastic spiking
coherence, and find that reflects the degree of collective spiking
coherence seen in the raster plot very well. Hence, the
"statistical-mechanical" spike-based measure may be used usefully to
quantify the degree of stochastic spiking coherence in a statistical-mechanical
way.Comment: 16 pages, 5 figures, to appear in the J. Comput. Neurosc
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
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