Predicting Synchronization of Three Mutually Inhibiting Groups of Oscillators with Strong Resetting (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.
Discrete Analysis of Synchronized Oscillations in Excitatory-Inhibitory Neuronal Networks
Coherent activity of two or many interconnected neurons is thought to play important role in processing information in the central nervous system. Synchronous neural activity is believed to be part of many cognitive and sensory processing tasks such as feature binding, attention and memory construction. High levels of synchronization has been implicated in brain disorders like epilepsy, schizophrenia, Alzheimer's disease and Parkinson. Therefore, studying mechanisms underlying synchrony in the brain is of fundamental importance in understanding brain processes.
Neural Computation, 2001
There are several different biophysical mechanisms for spike frequency adaptation observed in recordings from cortical neurons. The two most commonly used in modeling studies are a calcium-dependen t potassium current I ahp and a slow voltage-depe ndent potassium current, I m . We show that both of these have strong effects on the synchronization properties of excitatorily coupled neurons. Furthermore, we show that the reasons for these effects are different. We show through an analysis of some standard models, that the M-current adaptation alters the mechanism for repetitive ring, while the afterhyperpolarization adaptation works via shunting the incoming synapses. This latter mechanism applies with a network that has recurrent inhibition. The shunting behavior is captured in a simple two-variable reduced model that arises near certain types of bifurcations. A one-dimensional map is derived from the simpli ed model.
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.
Synchronization of strongly coupled excitatory neurons: relating network behavior to biophysics
Journal of computational neuroscience
Behavior of a network of neurons is closely tied to the properties of the individual neurons. We study this relationship in models of layer II stellate cells (SCs) of the medial entorhinal cortex. SCs are thought to contribute to the mammalian theta rhythm (4-12 Hz), and are notable for the slow ionic conductances that constrain them to fire at rates within this frequency range. We apply "spike time response" (STR) methods, in which the effects of synaptic perturbations on the timing of subsequent spikes are used to predict how these neurons may synchronize at theta frequencies. Predictions from STR methods are verified using network simulations. Slow conductances often make small inputs "effectively large"; we suggest that this is due to reduced attractiveness or stability of the spiking limit cycle. When inputs are (effectively) large, changes in firing times depend nonlinearly on synaptic strength. One consequence of nonlinearity is to make a periodically firi...
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.
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.
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.