Metastable states and transient activity in ensembles of excitatory and inhibitory elements

Abstract

Complex activity in biological neuronal networks can be represented as a sequential transition between complicated metastable states. From a dynamical systems theory point of view sequential activity in neuronal networks is associated with the existence of stable heteroclinic contours in the phase space of the corresponding neuronal model. Previously, the conditions of existence and stability of these contours have been studied in networks consisting of only inhibitory synaptically coupled cells. In this paper the effect of excitatory neurons is studied. In contrast to early studied cases, in the considered model the heteroclinic trajectories are located on twodimensional manifolds. This leads to the existence of an infinitely large number of transitions between metastable states. The conditions for the existence of metastable states and arising sequential dynamics are presented.

Predicting the separation of time scales in a heteroclinic network

“…It is predicted in models of coupled phase oscillators [2,3], vector models [2], pulse-coupled oscillators [4] and models of winnerless competition [5] that we consider in the following. Applications are manifold and range from social [6,7] and ecological [5,8] systems, to computation [4] and neuronal [9][10][11][12][13][14][15][16] networks. As it was emphasized in the work of V. Afraimovich and his coworkers [5,9,10,12,17], heteroclinic sequences in models of winnerless competition predict transient dynamics that shares features with cognitive dynamics: being simultaneously sensitive to the input and robust against perturbations.…”

Quasiperiodic, periodic, and slowing-down states of coupled heteroclinic cycles

“…,S n } and the trajectories backward asymptotic to each saddle S i and forward asymptotic to the next one S i+1 [1]. Such a structure has been frequently found in neuronal dynamics [2][3][4][5][6][7], fluid mechanics [8][9][10], ecology [11][12][13][14], sociology [15][16][17][18][19], and other dynamical systems. A heteroclinic cycle has some significantly different characteristics from other types of dynamical states, such as limit cycles or chaotic attractors.…”

“…Recently the dynamical behavior of a small number of coupled systems that have the structure of heteroclinic cycles has attracted increasing interest [5][6][7][38][39][40]. For example, synchronization of two coupled heteroclinic cycles is of significance for understanding the microcircuit dynamics * zgzheng@bnu.edu.cn in neuronal systems [5].…”

Adaptive functional systems: Learning with chaos

“…If there is a stable heteroclinic channel in the vicinity of saddle points, then the trajectory in this channel will follow a certain heteroclinic sequence according to the principle of winnerless competition. [2][3][4][5][6][7][8][9][10] In theory, if one has a full knowledge about environment, then heteroclinic sequences for all required situations can be constructed, but only limited information is always available and the adaptive system should have some mechanisms to adjust its behavior. One of the main ideas of complexity theory is that adaptation is possible on the "edge of chaos" between regions of system's parameters with ordered and chaotic dynamics.…”