Horacio G. Rotstein

Inhibition-induced theta resonance in cortical circuits.

Both circuit and single-cell properties contribute to network rhythms. In vitro, pyramidal cells exhibit theta-band membrane potential (subthreshold) resonance, but whether and how subthreshold resonance translates into spiking resonance in freely behaving animals is unknown. Here, we used optogenetic activation to trigger spiking in pyramidal cells or parvalbumin immunoreactive interneurons (PV) in the hippocampus and neocortex of freely behaving rodents. Individual directly activated pyramidal cells exhibited narrow-band spiking centered on a wide range of frequencies. In contrast, PV photoactivation indirectly induced theta-band-limited, excess postinhibitory spiking in pyramidal cells (resonance). PV-inhibited pyramidal cells and interneurons spiked at PV-inhibition troughs, similar to CA1 cells during spontaneous theta oscillations. Pharmacological blockade of hyperpolarization-activated (I(h)) currents abolished theta resonance. Inhibition-induced theta-band spiking was replicated in a pyramidal cell-interneuron model that included I(h). Thus, PV interneurons mediate pyramidal cell spiking resonance in intact cortical networks, favoring transmission at theta frequency.

On the formation of gamma-coherent cell assemblies by oriens lacunosum-moleculare interneurons in the hippocampus

Gamma frequency (30–80 Hz) network oscillations have been observed in the hippocampus during several behavioral paradigms in which they are often modulated by a theta frequency (4–12 Hz) oscillation. Interneurons of the hippocampus have been shown to be crucially involved in rhythms generation, and several subtypes with distinct anatomy and physiology have been described. In particular, the oriens lacunosum-moleculare (O-LM) interneurons were shown to synapse on distal apical dendrites of pyramidal cells and to spike preferentially at theta frequency, even in the presence of gamma-field oscillations. O-LM cells have also recently been shown to present higher axonal ramification in the longitudinal axis of the hippocampus. By using a hippocampal network model composed of pyramidal cells and two types of interneurons (O-LM and basket cells), we show here that the O-LM interneurons lead to gamma coherence between anatomically distinct cell modules. We thus propose that this could be a mechanism for coupling longitudinally distant cells excited by entorhinal cortex inputs into gamma-coherent assemblies.

Mixed-mode oscillations in a three time-scale model for the dopaminergic neuron

Mixed-mode dynamics is a complex type of dynamical behavior that has been observed both numerically and experimentally in numerous prototypical systems in the natural sciences. The compartmental Wilson-Callaway model for the dopaminergic neuron is an example of a system that exhibits a wide variety of mixed-mode patterns upon variation of a control parameter. One characteristic feature of this system is the presence of multiple time scales. In this article, we study the Wilson-Callaway model from a geometric point of view. We show that the observed mixed-mode dynamics is caused by a slowly varying canard structure. By appropriately transforming the model equations, we reduce them to an underlying three-dimensional canonical form that can be analyzed via a slight adaptation of the approach developed by M. Krupa, N. Popovic, and N. Kopell (unpublished).

Canard Induced Mixed-Mode Oscillations in a Medial Entorhinal Cortex Layer II Stellate Cell Model ∗

Stellate cells (SCs) of the medial entorhinal cortex (layer II) display mixed-mode oscillatory activity, subthreshold oscillations (small-amplitude) interspersed with spikes (large amplitude), at theta frequencies (8–12 Hz). In this paper we study the mechanism of generation of such patterns in an SC biophysical (conductance-based) model. In particular, we show that the mechanism is based on the three-dimensional canard phenomenon and that the subthreshold oscillatory phenomenon is intrinsically nonlinear, involving the participation of both components (fast and slow) of a hyperpolarization-activated current in addition to the voltage and a persistent sodium current. We discuss some consequences of this mechanism for the SC intrinsic dynamics as well as for the interaction between SCs and external inhibitory inputs.

The dynamic structure underlying subthreshold oscillatory activity and the onset of spikes in a model of medial entorhinal cortex stellate cells

Medial entorhinal cortex layer II stellate cells display subthreshold oscillations (STOs). We study a single compartment biophysical model of such cells which qualitatively reproduces these STOs. We argue that in the subthreshold interval (STI) the seven-dimensional model can be reduced to a three-dimensional system of equations with well differentiated times scales. Using dynamical systems arguments we provide a mechanism for generations of STOs. This mechanism is based on the “canard structure,” in which relevant trajectories stay close to repelling manifolds for a significant interval of time. We also show that the transition from subthreshold oscillatory activity to spiking (“canard explosion”) is controlled in the STI by the same structure. A similar mechanism is invoked to explain why noise increases the robustness of the STO regime. Taking advantage of the reduction of the dimensionality of the full stellate cell system, we propose a nonlinear artificially spiking (NAS) model in which the STI reduced system is supplemented with a threshold for spiking and a reset voltage. We show that the synchronization properties in networks made up of the NAS cells are similar to those of networks using the full stellate cell models.

Introduction to Focus Issue: Mixed Mode Oscillations: Experiment, Computation, and Analysis

Mixed mode oscillations (MMOs) occur when a dynamical system switches between fast and slow motion and small and large amplitude. MMOs appear in a variety of systems in nature, and may be simple or complex. This focus issue presents a series of articles on theoretical, numerical, and experimental aspects of MMOs. The applications cover physical, chemical, and biological systems.

Low-Dimensional Maps Encoding Dynamics in Entorhinal Cortex and Hippocampus

Cells that produce intrinsic theta oscillations often contain the hyperpolarization-activated current Ih. In this article, we use models and dynamic clamp experiments to investigate the synchronization properties of two such cells (stellate cells of the entorhinal cortex and O-LM cells of the hippocampus) in networks with fast-spiking (FS) interneurons. The model we use for stellate cells and O-LM cells is the same, but the stellate cells are excitatory and the O-LM cells are inhibitory, with inhibitory postsynaptic potential considerably longer than those from FS interneurons. We use spike time response curve methods (STRC), expanding that technique to three-cell networks and giving two different ways in which the analysis of the three-cell network reduces to that of a two-cell network. We show that adding FS cells to a network of stellate cells can desynchronize the stellate cells, while adding them to a network of O-LM cells can synchronize the O-LM cells. These synchronization and desynchronization properties critically depend on Ih. The analysis of the deterministic system allows us to understand some effects of noise on the phase relationships in the stellate networks. The dynamic clamp experiments use biophysical stellate cells and in silico FS cells, with connections that mimic excitation or inhibition, the latter with decay times associated with FS cells or O-LM cells. The results obtained in the dynamic clamp experiments are in a good agreement with the analytical framework.

Canard-Like Explosion of Limit Cycles in Two-Dimensional Piecewise-Linear Models of FitzHugh–Nagumo Type

We investigate the mechanism of abrupt transition between small- and large amplitude oscillations in fast-slow piecewise-linear (PWL) models of FitzHugh–Nagumo (FHN) type. In the context of neuroscience, these oscillatory regimes correspond to subthreshold oscillations and action potentials (spikes), respectively. The minimal model that shows such phenomena has a cubic-like nullcline (for the fast equation) with two or more linear pieces in the middle branch and one piece in the left and right branches. Simpler models with only one linear piece in the middle branch or a discontinuity between the left and right branches (McKean model) show a single oscillatory mode. As the number of linear pieces increases, PWL models of FHN type approach smooth FHN-type models. For the minimal model we investigate the bifurcation structure; we describe the mechanism that leads to the abrupt, canard-like transition between subthreshold oscillations and spikes; and we provide an analytical way of predicting the amplitude re...

A CANARD MECHANISM FOR LOCALIZATION IN SYSTEMS OF GLOBALLY COUPLED OSCILLATORS

Localization in a discrete system of oscillators refers to the partition of the population into a subset that oscillates at high amplitudes and anotherthat oscillates at much loweramplitudes. Motivated by experimental results on the Belousov-Zhabotinsky reaction, which oscillates in the relaxation regime, we study a mechanism of localization in a discrete system of relaxation oscillators globally coupled via inhibition. The mechanism is based on the canard phenomenon for a single relaxation oscillator: a rapid explosion in the amplitude of the limit cycle as a parameter governing the relative position of the nullclines is varied. Starting from a parameter regime in which each uncoupled oscillatorhas a lar ge amplitude and no otherper iodic orotherstable solutions, we show that the canard phenomenon can be induced by increasing a global negative feedback parameter γ, with the network then partitioned into low and high amplitude oscillators. For the case in which the oscillators are synchronous within each of the two such populations, we can assign a canard-inducing critical value of γ separately to each of the two clusters; localization occurs when the value for the system is between the critical values of the two clusters. We show that the larger the cluster size, the smaller is the corresponding critical value of γ, implying that it is the smallerclusterthat oscillates at large amplitude. The theory shows that the above results come from a kind of self-inhibition of each cluster induced by the local feedback. In the full system, there are also effects of interactions between the clusters, and we present simulations showing that these nonlocal interactions do not destroy the localization created by the self-inhibition.

PHASE FIELD EQUATIONS WITH MEMORY: THE HYPERBOLIC CASE*

We present a phenomenological theory for phase transition dynamics with memory which yields a hyperbolic generalization of the classical phase field model when the relaxation kernels are assumed to be exponential. Thereafter, we focus on the implications of our theory in the hyperbolic case, and we derive asymptotically an equation of motion in two dimensions for the interface between two different phases. This equation can be considered as a hyperbolic generalization of the classical flow by mean curvature equation, as well as a generalization of the Born--Infeld equation. We use a crystalline algorithm to study the motion of closed curves for the generalized hyperbolic flow by mean curvature equation our hyperbolic generalization of flow by mean curvature and present some numerical results which indicate that a certain type of two-dimensional relaxation damped oscillation may occur.