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  • interneurons  (2)
  • Singular perturbation  (1)
  • 1
    Electronic Resource
    Electronic Resource
    Springer
    Journal of mathematical biology 38 (1999), S. 479-533 
    ISSN: 1432-1416
    Keywords: Keywords: Central pattern generator ; Relative coordination ; Oscillation ; Singular perturbation ; Subharmonics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Biology , Mathematics
    Notes: Abstract.  We discuss a method by which the dynamics of a network of neurons, coupled by mutual inhibition, can be reduced to a one-dimensional map. This network consists of a pair of neurons, one of which is an endogenous burster, and the other excitable but not bursting in the absence of phasic input. The latter cell has more than one slow process. The reduction uses the standard separation of slow/fast processes; it also uses information about how the dynamics on the slow manifold evolve after a finite amount of slow time. From this reduction we obtain a one-dimensional map dependent on the parameters of the original biophysical equations. In some parameter regimes, one can deduce that the original equations have solutions in which the active phase of the originally excitable cell is constant from burst to burst, while in other parameter regimes it is not. The existence or absence of this kind of regulation corresponds to qualitatively different dynamics in the one-dimensional map. The computations associated with the reduction and the analysis of the dynamics includes the use of coordinates that parameterize by time along trajectories, and “singular Poincaré maps” that combine information about flows along a slow manifold with information about jumps between branches of the slow manifold.
    Type of Medium: Electronic Resource
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  • 2
    ISSN: 1573-6873
    Keywords: gamma oscillations ; hippocampus ; interneurons
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Medicine , Physics
    Notes: Abstract 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 (τs/T), where T can be determined either from the network or from a single, self-inhibiting neuron. With τs/T 〉 2, 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 τs/T 〈 1, 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.
    Type of Medium: Electronic Resource
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    Journal of computational neuroscience 5 (1998), S. 407-420 
    ISSN: 1573-6873
    Keywords: gamma oscillations ; hippocampus ; interneurons ; synchronization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Medicine , Physics
    Notes: Abstract 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.
    Type of Medium: Electronic Resource
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