Boston University /
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Nancy Kopell

Chains of *N* FitzHugh-Nagumo oscillators with a gradient in
natural frequencies and strong diffusive coupling are analyzed in this
paper. We study the system's dynamics in the limit of infinitely
large coupling and then treat the case when the coupling is large but
finite as a perturbation of the former case. In the large coupling
limit, the 2*N* dimensional phase space has an unexpected
structure: there is a (*N*-1)-dimensional cylinder foliated by
periodic orbits with an integral that is constant on each orbit. When
the coupling is large but finite, this cylinder becomes an analog of
an inertial manifold. The phase trajectories approach the cylinder on
the fast time scale and then slowly drift along it toward a unique
limit cycle. We analyze these dynamics using the geometric theory for
singularly perturbed dynamical systems, asymptotic expansions of
solutions (rigorously justified), and the Lyapunov's method.

Experimental and modeling efforts suggest that rhythms in the CA1 region of the hippocampus that are in the beta range (12-29 Hz) have a different dynamical structure than that of gamma (30-70 Hz). We use a simplified model to show that the different rhythms employ different dynamical mechanisms to synchronize, based on different ionic currents. The beta frequency is able to synchronize over long conduction delays (corresponding to signals traveling a significant distance in the brain) that apparently cannot be tolerated by gamma rhythms. The synchronization properties are consistent with data suggesting that gamma rhythms are used for relatively local computations whereas beta rhythms are used for higher level interactions involving more distant structures.

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 neuorns, 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 process; 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 fom 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 Poincare maps" that combine information about flows along a slow manifold with information about jumps between branches of the slow manifold.

We describe a novel mechanism by which network oscillations can
arise from reciprocal inhibitory connections between two entirely
passive neurons. The model was inspired by the activation of the
gastric mill rhythm in the crab stomatogastric ganglion by the
Modulatory Comissural Ganglion Neuron 1 (MCN1), but is studied here in
general terms. One model neuron has a linear current-voltage curve
with a low (L) resting potential and the second model neuron has a
linear current-voltage curve with a high (H) resting potential. The
inhibitory connections between them are graded. There is an extrinsic
modulatory input to the L neuron and the L neuron presynaptically
inhibits the modulatory neuron. Activation of the extrinsic
modulatory neuron elicits stable network oscillations in which the L
and H neurons are active in alternation. The oscillations arise
because the graded reciprocal synapses create the equivalent of a
negative-slope conductance region in the *I-V* curves for the
cells. Geometrical methods are used to analyze the properties of and
the mechanism underlying these network oscillations.

Hippocampal networks of excitatory (E) and inhibitory (I) neurons that produce gamma-frequency rhythms display behavior in which the inhibitory cells produce spike doublets when there is strong stimulation at separated sites. It has been suggested that the doublets play a key role in the ability to synchronize over a distance. Here we analyze the mechanisms by which timing in the spike doublet can affect the synchronization process. The analysis describes two independent effects: one comes from the timing of excitation from separated local circuits to an inhibitory cell, and the other comes from the timing of inhibition from separated local circuits to an excitatory cell. We show that a network with both of these effects has different synchronization properties than a network with either excitatory or inhibitory type of coupling alone, and we give a rationale for the shorter space scales associated with inhibitory interactions.

Inhibition in oscillatory networks of neurons can have apparently paradoxical effects, sometimes creating dispersion of phases, sometimes fostering synchrony in the network. We analyze a pair of biophysically modeled neurons and show how the rates of onset and decay of inhibition interact with the time scales of the intrinsic oscillators to determine when stable synchrony is possible. We show that there are two different regimes in parameter space in which different combinations of the time constants and other parameters regulate whether the synchronous state is stable. We also discuss the construction and stability of non-synchronous solutions, and the implications of the analysis for larger networks. The analysis uses geometric techniques of singular perturbation theory that allow one to combine estimates from slow flows and fast jumps.

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.

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 where the period can be determined either from the network or from a single, self-inhibiting neuron. With that ratio larger than 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 that ratio larger than 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.

We study the periodic solutions of a two-cell network consisting of a relaxation oscillator and a bistable element. The aim is to understand how the frequency and wave form of the network depend on the intrinsic properties of the cells and on the strength of the coupling between them. The network equations constitute a fast-slow system; we show that there are four curves of saddle-node points of the fast system whose geometry in parameter space encodes information about the waveform and frequency. These curves give information about the value of the variables at which transitions are made between high and low voltage states for either of the elements, and how those transition points in phase space depend on the coupling strength. Furthermore, we develop a new geometric method to construct the curves of saddle-nodes from families of curves associated with the equations for each of the two cells. The construction allows one to see how changes in either of the elements affects the wave form of the network output. The analysis also shows that the network can produce unintuitive behavior. For example, though electric coupling may keep the network pinned longer at a higher or lower voltage level than the uncoupled oscillator, larger values of the coupling strength may be less effective at this pinning.

Send mail to nk@bu.edu.

Boston University / Mathematics Department / People / Faculty / Nancy Kopell