NEURONAL OSCILLATIONS AND STOCHASTIC LIMIT CYCLES

1996 ◽  
Vol 07 (04) ◽  
pp. 399-402 ◽  
Author(s):  
CHRISTIAN KURRER ◽  
KLAUS SCHULTEN
Keyword(s):  
Nonlinear Dynamics ◽  
Limit Cycles ◽  
Neural Activity ◽  
Oscillatory Behavior ◽  
Neuronal Oscillations ◽  
Coupled Oscillator ◽  
Stochastic Limit ◽  
Excitable Systems ◽  
The Individual ◽  
Coupled Oscillator Systems

We investigate a model for synchronous neural activity in networks of coupled neurons. The individual systems are governed by nonlinear dynamics and can continuously vary between excitable and oscillatory behavior. Analytical calculations and computer simulations show that coupled excitable systems can undergo two different phase transitions from synchronous to asynchronous firing behavior. One of the transitions is akin to the synchronization transitions in coupled oscillator systems, while the second transition can only be found in coupled excitable systems. Using the concept of Stochastic Limit Cycles, we present an analytical derivation of the two transitions and discuss implications for synchronization transitions in biological neural networks.

Delay stabilization of periodic orbits in coupled oscillator systems

2010 ◽  
Vol 368 (1911) ◽  
pp. 319-341 ◽  
Author(s):  
B. Fiedler ◽  
V. Flunkert ◽  
P. Hövel ◽  
E. Schöll
Keyword(s):  
Normal Form ◽  
Numerical Simulations ◽  
Periodic Orbits ◽  
Coupled Oscillators ◽  
Coupled Oscillator ◽  
Non Invasive ◽  
Coupling Parameters ◽  
Necessary And Sufficient ◽  
Time Delayed Feedback ◽  
Coupled Oscillator Systems

We study diffusively coupled oscillators in Hopf normal form. By introducing a non-invasive delay coupling, we are able to stabilize the inherently unstable anti-phase orbits. For the super- and subcritical cases, we state a condition on the oscillator’s nonlinearity that is necessary and sufficient to find coupling parameters for successful stabilization. We prove these conditions and review previous results on the stabilization of odd-number orbits by time-delayed feedback. Finally, we illustrate the results with numerical simulations.

Bursting in Inhibitory Interneuronal Networks: A Role for Gap-Junctional Coupling

1999 ◽  
Vol 81 (3) ◽  
pp. 1274-1283 ◽  
Author(s):  
F. K. Skinner ◽  
L. Zhang ◽  
J. L. Perez Velazquez ◽  
P. L. Carlen
Keyword(s):  
Theoretical Work ◽  
Oscillatory Behavior ◽  
Minimal Network ◽  
Junctional Coupling ◽  
Synchronization Mechanisms ◽  
The Individual ◽  
Gap Junctional ◽  
Gap Junctional Coupling ◽  
Inhibitory Networks

Bursting in inhibitory interneuronal networks: a role for gap-junctional coupling. Much work now emphasizes the concept that interneuronal networks play critical roles in generating synchronized, oscillatory behavior. Experimental work has shown that functional inhibitory networks alone can produce synchronized activity, and theoretical work has demonstrated how synchrony could occur in mutually inhibitory networks. Even though gap junctions are known to exist between interneurons, their role is far from clear. We present a mechanism by which synchronized bursting can be produced in a minimal network of mutually inhibitory and gap-junctionally coupled neurons. The bursting relies on the presence of persistent sodium and slowly inactivating potassium currents in the individual neurons. Both GABAA inhibitory currents and gap-junctional coupling are required for stable bursting behavior to be obtained. Typically, the role of gap-junctional coupling is focused on synchronization mechanisms. However, these results suggest that a possible role of gap-junctional coupling may lie in the generation and stabilization of bursting oscillatory behavior.

The unification of neural activity

1968 ◽  
Vol 171 (1024) ◽  
pp. 353-359 ◽  
Keyword(s):  
Nervous System ◽  
Neural Activity ◽  
Nerve Cells ◽  
General Nature ◽  
Central Importance ◽  
Stimulus Response ◽  
Simple Hypothesis ◽  
Neural Organization ◽  
The Individual ◽  
The Brain

In studying the brain, two levels of investigation emerge naturally. One of these concerns itself with properties of nerve cells, their numbers, patterns of firing, interconnexions, and so forth. The other considers the whole nervous system in what one may call ‘macroscopic’ terms. Thus it discusses ‘stimulus’, ‘response’, ‘decision’, etc. At this latter level, the nervous system operates with considerable unity. The individual nerve cells must therefore be linked in a well-integrated manner and the general nature of this integration has been recognized, especially by neurophysiologists such as Sherrington, to present a problem of central importance for our understanding of the brain. In previously published work, I have put forward a theory of how this unification of neural activity might be achieved and of a possible molecular biological basis of the necessary neural organization. In this talk I restrict myself to the first of these and thus give an account of what might be called the basic logic of the unification. I also indicate briefly how a simple hypothesis about the basis of memory would fit into such a theory.

PERIODIC INHIBITION OF LIVING PACEMAKER NEURONS (I): LOCKED, INTERMITTENT, MESSY, AND HOPPING BEHAVIORS

1991 ◽  
Vol 01 (03) ◽  
pp. 549-581 ◽  
Author(s):  
J. P. SEGUNDO ◽  
E. ALTSHULER ◽  
M. STIBER ◽  
A. GARFINKEL
Keyword(s):  
Nonlinear Dynamics ◽  
Point Process ◽  
Limit Cycles ◽  
Nonlinear Oscillator ◽  
Strange Attractors ◽  
Synaptic Inhibition ◽  
Pacemaker Neurons ◽  
Pacemaker Neuron ◽  
Exhaustive List

This communication is concerned with an embodiment of periodic nonlinear oscillator driving, the synaptic inhibition of one spike-producing pacemaker neuron by another. Data came from a prototypical living synapse. Analyses centered on a prolonged condition between the transients following the onset and cessation of inhibition. Evaluations were guided by point process mathematics and nonlinear dynamics. A rich and exhaustive list of discharge forms, described precisely and canonically, was observed across different inhibitory rates. Previously unrecognized at synapses, most forms were identified with several well known types from nonlinear dynamics. Ordered by decreasing regularities, they were locked, intermittent (including walk-throughs), messy (including erratic and stammerings) and hopping. Each is discussed within physiological and formal contexts. It is conjectured that (i) locked, intermittent and messy forms reflect limit cycles on 2-tori, quasiperiodic orbits and strange attractors, (ii) noise in neurons hovering around threshold contributes to certain intermittent and stammering behaviors, and (iii) hopping either reflects an attractor with several portions or is nonstationary and noise-induced.

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