Phase Resetting Curve Analysis of Global Synchrony, the Splay Mode and Clustering in N Neuron all to all Pulse-Coupled Networks (original) (raw)
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Journal of Neuroscience, 2009
Networks of model neurons were constructed and their activity was predicted using an iterated map based solely on the phase-resetting curves (PRCs). The predictions were quite accurate provided that the resetting to simultaneous inputs was calculated using the sum of the simultaneously active conductances, obviating the need for weak coupling assumptions. Fully synchronous activity was observed only when the slope of the PRC at a phase of zero, corresponding to spike initiation, was positive. A novel stability criterion was developed and tested for all-to-all networks of identical, identically connected neurons. When the PRC generated using N Ϫ 1 simultaneously active inputs becomes too steep, the fully synchronous mode loses stability in a network of N model neurons. Therefore, the stability of synchrony can be lost by increasing the slope of this PRC either by increasing the network size or the strength of the individual synapses. Existence and stability criteria were also developed and tested for the splay mode in which neurons fire sequentially. Finally, N/M synchronous subclusters of M neurons were predicted using the intersection of parameters that supported both between-cluster splay and within-cluster synchrony. Surprisingly, the splay mode between clusters could enforce synchrony on subclusters that were incapable of synchronizing themselves. These results can be used to gain insights into the activity of networks of biological neurons whose PRCs can be measured.
Journal of Computational Neuroscience, 2008
Our goal is to understand how nearly synchronous modes arise in heterogenous networks of neurons. In heterogenous networks, instead of exact synchrony, nearly synchronous modes arise, which include both 1:1 and 2:2 phase-locked modes. Existence and stability criteria for 2:2 phase-locked modes in reciprocally coupled two neuron circuits were derived based on the open loop phase resetting curve (PRC) without the assumption of weak coupling. The PRC for each component neuron was generated using the change in synaptic conductance produced by a presynaptic action potential as the perturbation. Separate derivations were required for modes in which the firing order is preserved and for those in which it alternates. Networks composed of two model neurons coupled by reciprocal inhibition were examined to test the predictions. The parameter regimes in which both types of nearly synchronous modes are exhibited were accurately predicted both qualitatively and quantitatively provided that the synaptic time constant is short with respect to the period and that the effect of second order resetting is considered. In contrast, PRC methods based on weak coupling could not predict 2:2 modes and did not predict the 1:1 modes with the level of accuracy achieved by the strong coupling methods. The strong coupling prediction methods provide insight into what manipulations promote near-synchrony in a two neuron network and may also have predictive value for larger networks, which can also manifest changes in firing order. We also identify a novel route by which synchrony is lost in mildly heterogenous networks.
Synchronization and stable phase-locking in a network of neurons with memory
Mathematical and Computer Modelling, 1999
consider a network of three identical neurons whose dynamics is governed by the Hopfield's model with delay to account for the finite switching speed of amplifiers (neurons). We show that in a certain region of the space of (a, p), where a and p are the normalized parameters measuring, respectively, the synaptic strength of self-connection and neighbourhood-interaction, each solution of the network is convergent to the set of synchronous states in the phase space, and this synchronization is independent of the size of the delay. We also obtain a surface, ss the graph of a continuous function of r = r(qp) (the normalized delay) in some region of (a,@, where Hopf bifurcation of periodic solutions takes place. We describe a continuous curve on such a surface where the system undergoes mode-interaction and we describe the change of patterns from stable synchronous periodic solutions to the coexistence of two stable phase-locked oscillations and several unstable mirror-reflecting waves and standing waves.
Stability of two cluster solutions in pulse coupled networks of neural oscillators
Journal of Computational Neuroscience, 2011
Phase response curves (PRCs) have been widely used to study synchronization in neural circuits comprised of pacemaking neurons. They describe how the timing of the next spike in a given spontaneously firing neuron is affected by the phase at which an input from another neuron is received. Here we study two reciprocally coupled clusters of pulse coupled oscillatory neurons. The neurons within each cluster are presumed to be identical and identically pulse coupled, but not necessarily identical to those in the other cluster. We investigate a two cluster solution in which all oscillators are synchronized within each cluster, but in which the two clusters are phase locked at nonzero phase with each other. Intuitively, one might expect this solution to be stable only when synchrony within each isolated cluster is stable, but this is not the case. We prove rigorously the stability of the two cluster solution and show how reciprocal coupling can stabilize synchrony within clusters that cannot synchronize in isolation. These stability results for the two cluster solution suggest a mechanism by which reciprocal coupling between brain regions can induce local synchronization via the network feedback loop.
Phase resetting and phase locking in hybrid circuits of one model and one biological neuron
Biophysical journal, 2004
To determine why elements of central pattern generators phase lock in a particular pattern under some conditions but not others, we tested a theoretical pattern prediction method. The method is based on the tabulated open loop pulsatile interactions of bursting neurons on a cycle-by-cycle basis and was tested in closed loop hybrid circuits composed of one bursting biological neuron and one bursting model neuron coupled using the dynamic clamp. A total of 164 hybrid networks were formed by varying the synaptic conductances. The prediction of 1:1 phase locking agreed qualitatively with the experimental observations, except in three hybrid circuits in which 1:1 locking was predicted but not observed. Correct predictions sometimes required consideration of the second order phase resetting, which measures the change in the timing of the second burst after the perturbation. The method was robust to offsets between the initiation of bursting in the presynaptic neuron and the activation of the synaptic coupling with the postsynaptic neuron. The quantitative accuracy of the predictions fell within the variability (10%) in the experimentally observed intrinsic period and phase resetting curve (PRC), despite changes in the burst duration of the neurons between open and closed loop conditions.
Phase synchronization of bursting neurons in clustered small-world networks
Physical Review E, 2012
We investigate the collective dynamics of bursting neurons on clustered networks. The clustered network model is composed of subnetworks, each of them presenting the so-called small-world property. This model can also be regarded as a network of networks. In each subnetwork a neuron is connected to other ones with regular as well as random connections, the latter with a given intracluster probability. Moreover, in a given subnetwork each neuron has an intercluster probability to be connected to the other subnetworks. The local neuron dynamics has two time scales (fast and slow) and is modeled by a two-dimensional map. In such small-world network the neuron parameters are chosen to be slightly different such that, if the coupling strength is large enough, there may be synchronization of the bursting (slow) activity. We give bounds for the critical coupling strength to obtain global burst synchronization in terms of the network structure, that is, the probabilities of intracluster and intercluster connections. We find that, as the heterogeneity in the network is reduced, the network global synchronizability is improved. We show that the transitions to global synchrony may be abrupt or smooth depending on the intercluster probability.
Synchronization with an Arbitrary Phase Shift in a Pair of Synaptically Coupled Neural Oscillators
The phase dynamics of a pair of spiking neural oscillators coupled by a unidirectional nonlinear connection has been studied. The synchronization effect with the controlled relative phase of spikes has been obtained for various coupling strengths and depolarization parameters. It has been found that the phase value is deter mined by the difference between the depolarization levels of neurons and is independent of the synaptic cou pling strength. The synchronization mechanism has been studied by means of the construction and analysis of one dimensional phase maps. The phase locking effect for spikes has been interpreted in application to the synaptic plasticity in neurobiology.
Phase-locked oscillations in a neuronal network model
Neurocomputing, 2002
We analyzed the oscillatory activities in a neuronal network model as the basis of synchrony of the activities in the brain. The model consists of two groups of neurons that are interconnected. One group is composed of an excitatory and an inhibitory neuron which are expressed by Hodgkin-Huxley equations. The network shows di erent phase-locked oscillations depending on the structure and intensity of interconnection between groups or coupling of neurons in the group, or the value of synaptic latency. The oscillations include various periodic solutions in which the two groups oscillate not only in in-phase or anti-phase but also in continuously changing phase di erence with the parameters of coupling and latency.
Stability of the splay state in networks of pulse-coupled neurons
The Journal of Mathematical Neuroscience, 2012
We analytically investigate the stability of splay states in networks of N pulse-coupled phase-like models of neurons. By developing a perturbative technique, we find that, in the limit of large N , the Floquet spectrum scales as 1/N 2 for generic discontinuous velocity fields. Moreover, the stability of the socalled short-wavelength component is determined by the sign of the jump at the discontinuity. Altogether, the form of the spectrum depends on the pulse shape but is independent of the velocity field.