scholarly journals Iterative maps with multiple fixed points for excitation of two level systems

1987 ◽  
Vol 86 (6) ◽  
pp. 3089-3106 ◽  
Author(s):  
H. Cho ◽  
J. Baum ◽  
A. Pines
Keyword(s):  
Fixed Points ◽  
Iterative Maps ◽  
Two Level Systems ◽  
Multiple Fixed Points

A new criterion for the existence of multiple solutions in cones

2012 ◽  
Vol 142 (5) ◽  
pp. 1043-1050 ◽  
Author(s):  
Daniel Franco ◽  
Gennaro Infante ◽  
Juan Perán
Keyword(s):  
Boundary Value Problem ◽  
Fixed Points ◽  
Banach Spaces ◽  
Multiple Solutions ◽  
Nonlinear Boundary ◽  
Sufficient Conditions ◽  
Extra Information ◽  
New Criterion ◽  
Multiple Fixed Points ◽  
Ordered Banach Spaces

We provide new sufficient conditions for the existence of multiple fixed points for a map between ordered Banach spaces. An interesting feature of this approach is that we require conditions not on two boundaries, but rather on one boundary and a point with some extra information on the monotonicity of the nonlinearity on a certain set. We apply our results to prove the existence of at least two positive solutions for a nonlinear boundary-value problem that models a thermostat.

scholarly journals Attractor-state itinerancy in neural circuits with synaptic depression

2019 ◽  
Author(s):  
Bolun Chen ◽  
Paul Miller
Keyword(s):  
Fixed Points ◽  
Synaptic Depression ◽  
Basins Of Attraction ◽  
Dependent Manner ◽  
Neural Populations ◽  
History Of ◽  
Point Attractor ◽  
Dynamical Features ◽  
The Stability ◽  
Multiple Fixed Points

AbstractNeural populations with strong excitatory recurrent connections can support bistable states in their mean firing rates. Multiple fixed points in a network of such bistable units can be used to model memory retrieval and pattern separation. The stability of fixed points may change on a slower timescale than that of the dynamics due to short-term synaptic depression, leading to transitions between quasi-stable point attractor states in a sequence that depends on the history of stimuli. To better understand these behaviors, we study a minimal model, which characterizes multiple fixed points and transitions between them in response to stimuli with diverse time- and amplitude-dependences. The interplay between the fast dynamics of firing rate and synaptic responses and the slower timescale of synaptic depression makes the neural activity sensitive to the amplitude and duration of square-pulse stimuli in a non-trivial, history-dependent manner. Weak cross-couplings further deform the basins of attraction for different fixed points into intricate shapes. Our analysis provides a natural explanation for the system’s rich responses to stimuli of different durations and amplitudes while demonstrating the encoding capability of bistable neural populations for dynamical features of incoming stimuli.

scholarly journals Attractor-state itinerancy in neural circuits with synaptic depression

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Bolun Chen ◽  
Paul Miller
Keyword(s):  
Fixed Points ◽  
Synaptic Depression ◽  
Basins Of Attraction ◽  
Dependent Manner ◽  
Short Term ◽  
Neural Populations ◽  
History Of ◽  
Point Attractor ◽  
The Stability ◽  
Multiple Fixed Points

Abstract Neural populations with strong excitatory recurrent connections can support bistable states in their mean firing rates. Multiple fixed points in a network of such bistable units can be used to model memory retrieval and pattern separation. The stability of fixed points may change on a slower timescale than that of the dynamics due to short-term synaptic depression, leading to transitions between quasi-stable point attractor states in a sequence that depends on the history of stimuli. To better understand these behaviors, we study a minimal model, which characterizes multiple fixed points and transitions between them in response to stimuli with diverse time- and amplitude-dependencies. The interplay between the fast dynamics of firing rate and synaptic responses and the slower timescale of synaptic depression makes the neural activity sensitive to the amplitude and duration of square-pulse stimuli in a nontrivial, history-dependent manner. Weak cross-couplings further deform the basins of attraction for different fixed points into intricate shapes. We find that while short-term synaptic depression can reduce the total number of stable fixed points in a network, it tends to strongly increase the number of fixed points visited upon repetitions of fixed stimuli. Our analysis provides a natural explanation for the system’s rich responses to stimuli of different durations and amplitudes while demonstrating the encoding capability of bistable neural populations for dynamical features of incoming stimuli.

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