Stability of two cluster solutions in pulse coupled networks of neural oscillators (original) (raw)
Related papers
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.
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 Response Curves in Neuroscience, 2011
We use phase resetting curve (PRC) theory to analyze phase-locked patterns in pulse-coupled all to all network of N neurons that receive multiple inputs per cycle. The basic principles are that the phase must be updated each time an input is received, and simultaneous inputs do not sum linearly for strong coupling, but the conductances do. Therefore, the dependence of the resetting on conductance must be known. We analyze a splay mode in which no neurons fire simultaneously, global synchrony in which all neurons fire together, and clustering modes in which the firing breaks up into a small number of clusters. The key idea is to identify the appropriate perturbation in order to determine the stability of a given mode. For the splay mode, jitter is introduced into all firing times. For synchrony, a single neuron is perturbed from the rest, and for the two cluster mode, a single neuron is perturbed from one cluster. Global synchrony can be destabilized by increasing the network size or the strength of the individual synapses. At most, a small number of M clusters form because the M 1 locking points are more likely to sample destabilizing regions of the PRC as M increases. Between cluster interactions can enforce synchrony on subclusters that are incapable of synchronizing themselves. For the two cluster case, general results were obtained for clusters of any size. These results can be used to gain insights into the activity of networks of biological neurons whose PRCs can be measured.
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.
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.
Emergence of Neuronal Synchronisation in Coupled Areas
Frontiers in Computational Neuroscience
One of the most fundamental questions in the field of neuroscience is the emergence of synchronous behaviour in the brain, such as phase, anti-phase, and shift-phase synchronisation. In this work, we investigate how the connectivity between brain areas can influence the phase angle and the neuronal synchronisation. To do this, we consider brain areas connected by means of excitatory and inhibitory synapses, in which the neuron dynamics is given by the adaptive exponential integrate-and-fire model. Our simulations suggest that excitatory and inhibitory connections from one area to another play a crucial role in the emergence of these types of synchronisation. Thus, in the case of unidirectional interaction, we observe that the phase angles of the neurons in the receiver area depend on the excitatory and inhibitory synapses which arrive from the sender area. Moreover, when the neurons in the sender area are synchronised, the phase angle variability of the receiver area can be reduced ...
Neuronal synchrony: peculiarity and generality
2008
Synchronization in neuronal systems is a new and intriguing application of dynamical systems theory. Why are neuronal systems different as a subject for synchronization? ͑1͒ Neurons in themselves are multidimensional nonlinear systems that are able to exhibit a wide variety of different activity patterns. Their "dynamical repertoire" includes regular or chaotic spiking, regular or chaotic bursting, multistability, and complex transient regimes. ͑2͒ Usually, neuronal oscillations are the result of the cooperative activity of many synaptically connected neurons ͑a neuronal circuit͒. Thus, it is necessary to consider synchronization between different neuronal circuits as well. ͑3͒ The synapses that implement the coupling between neurons are also dynamical elements and their intrinsic dynamics influences the process of synchronization or entrainment significantly. In this review we will focus on four new problems: ͑i͒ the synchronization in minimal neuronal networks with plastic synapses ͑synchronization with activity dependent coupling͒, ͑ii͒ synchronization of bursts that are generated by a group of nonsymmetrically coupled inhibitory neurons ͑heteroclinic synchronization͒, ͑iii͒ the coordination of activities of two coupled neuronal networks ͑partial synchronization of small composite structures͒, and ͑iv͒ coarse grained synchronization in larger systems ͑synchronization on a mesoscopic scale͒.
Neurocomputing, 2010
Under the premise of analysis on the dynamic characteristics of the transmission mechanism among the synapses, this paper has modified the coupling term in the Tass's stochastic evolution model of neuronal oscillator population, introduced the variable higher-order coupling term. Then, we have performed the numerical simulation on the modified model. The simulation result shows that the variable coupling mechanism can induce the transition between different cluster states of the neuronal oscillator population, without the external stimulation. Another result from the numerical simulation is that, in the transient process between two different synchronization states caused by the variable coupling mechanism, it is allowed to have a full desynchronization state for a period. However, after the period of desynchronization state, the neuronal oscillator population can still reenter a new synchronization state under the action of the coupling term with the order different from initial condition.
Phase bistability between anticipated and delayed synchronization in neuronal populations
Physical Review E, 2020
Two dynamical systems unidirectionally coupled in a sender-receiver configuration can synchronize with a nonzero phase-lag. In particular, the system can exhibit anticipated synchronization (AS), which is characterized by a negative phase-lag, if the receiver (R) also receives a delayed negative self-feedback. Recently, AS was shown to occur between cortical-like neuronal populations in which the self-feedback is mediated by inhibitory synapses. In this biologically plausible scenario, a transition from the usual delayed synchronization (DS, with positive phase-lag) to AS can be mediated by the inhibitory conductances in the receiver population. Here we show that depending on the relation between excitatory and inhibitory synaptic conductances the system can also exhibit phase-bistability between anticipated and delayed synchronization. Furthermore, we show that the amount of noise at the receiver and the synaptic conductances can mediate the transition from stable phase-locking to a bistable regime and eventually to a phase-drift (PD). We suggest that our spiking neuronal populations model could be potentially useful to study phase-bistability in cortical regions related to bistable perception.