scholarly journals Sustained oscillations, irregular firing, and chaotic dynamics in hierarchical modular networks with mixtures of electrophysiological cell types

2014 ◽  
Vol 8 ◽  
Author(s):  
Petar Tomov ◽  
Rodrigo F. O. Pena ◽  
Michael A. Zaks ◽  
Antonio C. Roque
Keyword(s):  
Chaotic Dynamics ◽  
Cell Types ◽  
Sustained Oscillations ◽  
Modular Networks ◽  
Irregular Firing

scholarly journals Nonuniform Chaotic Dynamics and Effects of Noise in Biochemical Systems

1987 ◽  
Vol 42 (2) ◽  
pp. 136-142 ◽  
Author(s):  
H. Herzel ◽  
W. Ebeling ◽  
Th. Schulmeister
Keyword(s):  
Chaotic Dynamics ◽  
Chaotic Systems ◽  
Quantitative Description ◽  
Deterministic Chaos ◽  
Correlated Noise ◽  
Sustained Oscillations ◽  
Exponential Separation ◽  
Biochemical Systems ◽  
Relative Robustness

Biochemical models capable of sustained oscillations and deterministic chaos are investigated. Chaos is characterized by exponential separation of near-by trajectories in the long-term average. However, we observed rather large deviations from purely exponential separation termed "nonuniformity". A quantitative description and consequences of nonuniformity are discussed.Furthermore, the influence of short-correlated noise is treated using next-amplitude maps and Lyapunov exponents. Drastic amplification of fluctuations in non-chaotic systems and relative robustness of chaos were found.

scholarly journals Motor Cortex Microcircuit Simulation Based on Brain Activity Mapping

2014 ◽  
Vol 26 (7) ◽  
pp. 1239-1262 ◽  
Author(s):  
George L. Chadderdon ◽  
Ashutosh Mohan ◽  
Benjamin A. Suter ◽  
Samuel A. Neymotin ◽  
Cliff C. Kerr ◽  
...  
Keyword(s):  
Brain Activity ◽  
Cell Types ◽  
Population Connectivity ◽  
Dynamical Analysis ◽  
Static And Dynamic Analysis ◽  
Graph Theoretic ◽  
Sustained Oscillations ◽  
Transient Activation ◽  
Simulation Based ◽  
Sustained Responses

The deceptively simple laminar structure of neocortex belies the complexity of intra- and interlaminar connectivity. We developed a computational model based primarily on a unified set of brain activity mapping studies of mouse M1. The simulation consisted of 775 spiking neurons of 10 cell types with detailed population-to-population connectivity. Static analysis of connectivity with graph-theoretic tools revealed that the corticostriatal population showed strong centrality, suggesting that would provide a network hub. Subsequent dynamical analysis confirmed this observation, in addition to revealing network dynamics that cannot be readily predicted through analysis of the wiring diagram alone. Activation thresholds depended on the stimulated layer. Low stimulation produced transient activation, while stronger activation produced sustained oscillations where the threshold for sustained responses varied by layer: 13% in layer 2/3, 54% in layer 5A, 25% in layer 5B, and 17% in layer 6. The frequency and phase of the resulting oscillation also depended on stimulation layer. By demonstrating the effectiveness of combined static and dynamic analysis, our results show how static brain maps can be related to the results of brain activity mapping.

NULLCLINES AND NULLCLINE INTERSECTIONS

2006 ◽  
Vol 16 (10) ◽  
pp. 3023-3033 ◽  
Author(s):  
RENÉ THOMAS
Keyword(s):  
Phase Space ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Fixed Points ◽  
Ordinary Differential Equations ◽  
Chaotic Dynamics ◽  
Jacobian Matrix ◽  
Steady States ◽  
Sustained Oscillations ◽  
Sign Patterns

One purpose of this paper is to document the fact that, in dynamical systems described by ordinary differential equations, the trajectories can be organized not only around fixed points (steady states), but also around lines. In 2D, these lines are the nullclines themselves, in 3D, the intersections of the nullclines two by two, etc.We precise the concepts of "partial steady states" (i.e. steady states in a subsystem that consists of sections of phase space by planes normal to one of the axes) and of "partial multistationarity" (multistationarity in such a subsystem).Steady states, nullclines or their intersections are revisited in terms of circuits, defined from nonzero elements of the Jacobian matrix. It is shown how the mere examination of the Jacobian matrix and the sign patterns of its circuits can help interpreting (and often predicting) aspects of the dynamics of systems.The results reinforce the idea that chaotic dynamics requires both a positive circuit, to provide (if only partial) multistationarity, and a negative circuit, to provide sustained oscillations. As shown elsewhere, a single circuit may suffice if it is ambiguous (i.e. positive or negative depending on the location in phase space).The description in terms of circuits is by no means exclusive of the classical description. In many cases, a fruitful approach involves repeated feedback between the two viewpoints.

A NONLINEAR MATHEMATICAL MODEL FOR PULSATILE DISCHARGES OF LUTEINIZING HORMONE MEDIATED BY HYPOTHALAMIC AND EXTRA-HYPOTHALAMIC PATHWAYS

2002 ◽  
Vol 12 (05) ◽  
pp. 607-624 ◽  
Author(s):  
WANNAPA KUNPASURUANG ◽  
YONGWIMON LENBURY ◽  
GEERTJE HEK
Keyword(s):  
Mathematical Model ◽  
Luteinizing Hormone ◽  
Chaotic Dynamics ◽  
Hormone Secretion ◽  
Neural Pathways ◽  
Nonlinear Mathematical Model ◽  
Sustained Oscillations ◽  
Singular Perturbation Methods ◽  
Pulsatile Hormone Secretion ◽  
Lh Secretion

A mathematical model of hormone secretion in the hypothalamo-pituitary-gonadal axis in man is extended to incorporate two different neural pathways, which have been suggested by clinical data to be capable of stimulating pulsatile discharges of LH (luteinizing hormone) independently of each other. Analysis of the nonlinear model is carried out through the use of geometric singular perturbation methods. In this way, existence of a limit cycle is proved for certain ranges of the system parameters. When the LH secretion rate independent of the hypothalamus is assumed constant, dropping the hypothalamus stimulated secretion term from the model blocks the hypothalamus pathway, implying that sustained oscillations in the hormone levels may not be attainable. Therefore, a sinusoidal term is incorporated into the model so that the system can still exhibit pulsatile LH secretion independent of the hypothalamus mediation. It is shown, by a construction of a bifurcation diagram, that the pulsatile hormone secretion can develop into chaotic dynamics when the amplitude of oscillation stimulated by extra-hypothalamic structures is high enough to disturb the synchrony of hypothalamic control. The resulting numerical simulation is found to compare well with the clinically observed data.

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