Nancy Kopell

Inhibition-based rhythms: experimental and mathematical observations on network dynamics.

An increasingly large body of data exists which demonstrates that oscillations of frequency 12-80 Hz are a consequence of, or are inextricably linked to, the behaviour of inhibitory interneurons in the central nervous system. This frequency range covers the EEG bands beta 1 (12-20 Hz), beta 2 (20-30 Hz) and gamma (30-80 Hz). The pharmacological profile of both spontaneous and sensory-evoked EEG potentials reveals a very strong influence on these rhythms by drugs which have direct effects on GABA(A) receptor-mediated synaptic transmission (general anaesthetics, sedative/hypnotics) or indirect effects on inhibitory neuronal function (opiates, ketamine). In addition, a number of experimental models of, in particular, gamma-frequency oscillations, have revealed both common denominators for oscillation generation and function, and subtle differences in network dynamics between the different frequency ranges. Powerful computer and mathematical modelling techniques based around both clinical and experimental observations have recently provided invaluable insight into the behaviour of large networks of interconnected neurons. In particular, the mechanistic profile of oscillations generated as an emergent property of such networks, and the mathematical derivation of this complex phenomenon have much to contribute to our understanding of how and why neurons oscillate. This review will provide the reader with a brief outline of the basic properties of inhibition-based oscillations in the CNS by combining research from laboratory models, large-scale neuronal network simulations, and mathematical analysis.

Dynamic cross-frequency couplings of local field potential oscillations in rat striatum and hippocampus during performance of a T-maze task

Oscillatory rhythms in different frequency ranges mark different behavioral states and are thought to provide distinct temporal windows that coherently bind cooperating neuronal assemblies. However, the rhythms in different bands can also interact with each other, suggesting the possibility of higher-order representations of brain states by such rhythmic activity. To explore this possibility, we analyzed local field potential oscillations recorded simultaneously from the striatum and the hippocampus. As rats performed a task requiring active navigation and decision making, the amplitudes of multiple high-frequency oscillations were dynamically modulated in task-dependent patterns by the phase of cooccurring theta-band oscillations both within and across these structures, particularly during decision-making behavioral epochs. Moreover, the modulation patterns uncovered distinctions among both high- and low-frequency subbands. Cross-frequency coupling of multiple neuronal rhythms could be a general mechanism used by the brain to perform network-level dynamical computations underlying voluntary behavior.

Measuring Phase-Amplitude Coupling Between Neuronal Oscillations of Different Frequencies

Neuronal oscillations of different frequencies can interact in several ways. There has been particular interest in the modulation of the amplitude of high-frequency oscillations by the phase of low-frequency oscillations, since recent evidence suggests a functional role for this type of cross-frequency coupling (CFC). Phase-amplitude coupling has been reported in continuous electrophysiological signals obtained from the brain at both local and macroscopic levels. In the present work, we present a new measure for assessing phase-amplitude CFC. This measure is defined as an adaptation of the Kullback-Leibler distance-a function that is used to infer the distance between two distributions-and calculates how much an empirical amplitude distribution-like function over phase bins deviates from the uniform distribution. We show that a CFC measure defined this way is well suited for assessing the intensity of phase-amplitude coupling. We also review seven other CFC measures; we show that, by some performance benchmarks, our measure is especially attractive for this task. We also discuss some technical aspects related to the measure, such as the length of the epochs used for these analyses and the utility of surrogate control analyses. Finally, we apply the measure and a related CFC tool to actual hippocampal recordings obtained from freely moving rats and show, for the first time, that the CA3 and CA1 regions present different CFC characteristics.

Frequency Plateaus in a Chain of Weakly Coupled Oscillators, I.

A chain of

Parabolic bursting in an excitable system coupled with a slow oscillation

n + 1

Synchronization and oscillatory dynamics in heterogeneous, mutually inhibited neurons.

weakly coupled oscillators with a linear gradient in natural frequencies is shown to exhibit “frequency plateaus,” or sequences of oscillators having the same frequency, with a jump in frequency from one plateau to another. We first show that the equations for the coupled oscillators admit an invariant

Amplitude response of coupled oscillators

(n + 1)

Synchronization in networks of excitatory and inhibitory Neurons with sparse, random connectivity

-torus on which the equations have a special form, one in which an n-dimensional subsystem is approximately invariant. We then show that when the linear gradient becomes too steep to allow phaselocking, there emerges a large-scale invariant circle in this n-dimensional system which corresponds to the existence of a pair of plateaus, and whose homotopy class within the n-torus corresponds to the position of the frequency jump. Also discussed are the effects of anisotropic and nonuniform coupling.

Multiple pulse interactions and averaging in systems of coupled neural oscillators

We investigate the interaction of an excitable system with a slow oscillation. Under robust and general assumptions compatible with the more stringent assumptions usually made about excitable systems, we show that such a coupled system can display bursting, i.e. a stable solution in which some variable undergoes rapid oscillations followed by a period of quiescence, with both oscillation and quiescence continually repeated. Under a further weak condition, the bursting is “parabolic”, i.e. the local frequency of the fast oscillation increases and then decreases within a burst. The technique in this paper involves nonlinear changes of coordinates which transform the equations into ones which are closely related to Hill’s equation.

Rapid synchronization through fast threshold modulation

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.