Stochastic Synchrony and Phase Resetting Curves: Theory and Applications.

In this thesis, our goal was to study the phase synchronization between two uncoupled oscillators receiving partially correlated input. Using perturbation methods we obtain a closed-form solution for the steady-state density of phase differences between the two oscillators. Order parameters for synchrony and cross-correlation are used to quantify the degree of stochastic synchronization. We show that oscillators proscribed with Type-II phase resetting curves (PRC's) are more prone to stochastic synchronization compared to Type-I PRCs, and that the synchrony in the system can be described by a closed-form expression for the probability distribution of phase differences between the two uncoupled oscillators. We also study Morris-Lecar, leaky integrate-and-fire model and the Wang-Buzsaki interneuron model. Motivated by our theoretical developments, we study synchronization in simple neuronal network models of the olfactory bulb by applying the results from the theoretical studies to spiking neuron models with feedback to qualitatively demonstrate the emergence of self-organized synchrony. Here we again use an abstract model to obtain an expression for the averaged dynamics and compare our predicted solutions using Monte-Carlo simulations. We also show that an arbitrary mechanism that has a finite time memory of correlated inputs can cause bistability in such a system. Furthermore, we investigated the rate at which such systems approach their steady-state distribution and show that the dependence of the rate on the shape of the PRC. We obtained an expression for the rate of convergence to the steady-state density of phase differences in a two oscillators system receiving partially correlated inputs without feedback. To this end, we study the closed-form expression to obtain an approximation using a perturbation technique suited for computing large eigenvalues. It is shown that Type-II PRC's converge to their steady-state density compared to Type-I PRC's and that the rate of convergence is dependent on the input correlation. Our theoretical and numerical studies suggest a potential mechanism by which asynchronous inhibtion may promote and amplify synchronization in systems where the individual actors can be described as general oscillators at least in certain regimes of their activity with a possible source of activity dependent and partially correlated feedback

Amplification of asynchronous inhibition-mediated synchronization by feedback in recurrent networks

Synchronization of 30-80 Hz oscillatory activity of the principle neurons in the olfactory bulb (mitral cells) is believed to be important for odor discrimination. Previous theoretical studies of these fast rhythms in other brain areas have proposed that principle neuron synchrony can be mediated by short-latency, rapidly decaying inhibition. This phasic inhibition provides a narrow time window for the principle neurons to fire, thus promoting synchrony. However, in the olfactory bulb, the inhibitory granule cells produce long lasting, small amplitude, asynchronous and aperiodic inhibitory input and thus the narrow time window that is required to synchronize spiking does not exist. Instead, it has been suggested that correlated output of the granule cells could serve to synchronize uncoupled mitral cells through a mechanism called "stochastic synchronization", wherein the synchronization arises through correlation of inputs to two neural oscillators. Almost all work on synchrony due to correlations presumes that the correlation is imposed and fixed. Building on theory and experiments that we and others have developed, we show that increased synchrony in the mitral cells could produce an increase in granule cell activity for those granule cells that share a synchronous group of mitral cells. Common granule cell input increases the input correlation to the mitral cells and hence their synchrony by providing a positive feedback loop in correlation. Thus we demonstrate the emergence and temporal evolution of input correlation in recurrent networks with feedback. We explore several theoretical models of this idea, ranging from spiking models to an analytically tractable model. © 2010 Marella, Ermentrout

Dependence of the total spike count of the granule cell on the phase-difference of the two oscillators for different input strengths () and integration times of the synapse ().

<p>Higher dependence of the firing rate on the phase difference is observed for weaker and shorter synapses. The firing rate is less dependent on the phase difference for stronger and longer synapses.</p

Self-organized synchronization in a stochastic feedback network of two mitral cells and one granule cell( network).

<p>(A) Probability density of the phase-difference for different strengths of input to the granule cell. The peak at zero phase difference increases with strength of the synapse. (B) Distribution of the values of , the shared Poisson rate of the granule cell for different strengths of the synapse. (C) Plots of for and . (D) Phase difference histograms for the network. The central peak exists without decay for even larger network sizes (data not shown) suggesting that stochastic synchronization is robust against larger network sizes.</p

Evolution of in the presence of a single stable fixed point.

<p>(A) The temporal evolution of from various initial states. All initial states are attracted by the single stable fixed point. (B) Histogram of the final values of in different trials from (A). The green curve depicts the numerically calculated values of equation 7 (C) The dependence of the median probability on the amplitude of . (D) The curves depict distribution of phase differences drawn at various time points from simulations such as in (A). A slow development of synchrony on the order of hundreds of milliseconds is observed.</p

Evolution of in the bistable regime.

<p>(A) The temporal evolution of from various initial states. The initial states move randomly to either one of the stable fixed points. (B) Histogram of the final values of in different trials from (A). The green curve depicts the numerically calculated values of equation 7 for the indirect choice of . (C) The dependence of the steady state probability on the amplitude of . The taller peak of the bimodal distribution is depicted by the green curve. (D) Probability distribution of the phase difference between mitral cells for the two fixed points.</p