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Related Concept Videos

Second Order systems II01:18

Second Order systems II

In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.
If  ζ...
Classification of Systems-I01:26

Classification of Systems-I

Linearity is a system property characterized by a direct input-output relationship, combining homogeneity and additivity.
Homogeneity dictates that if an input x(t) is multiplied by a constant c, the output y(t) is multiplied by the same constant. Mathematically, this is expressed as:
Modeling with Differential Equations01:25

Modeling with Differential Equations

Population dynamics can be described mathematically by considering the population size P(t) as a function of time. The rate of change of the population is then represented by the derivative of P(t). A simple assumption is that the rate of growth is proportional to the size of the population itself. This leads to an exponential growth model, where the population increases rapidly without bound. While this is a useful first approximation, it does not reflect realistic long-term...
Classification of Systems-II01:31

Classification of Systems-II

Continuous-time systems have continuous input and output signals, with time measured continuously. These systems are generally defined by differential or algebraic equations. For instance, in an RC circuit, the relationship between input and output voltage is expressed through a differential equation derived from Ohm's law and the capacitor relation,
State Space Representation01:27

State Space Representation

The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
Consider an RLC circuit, a...
First Order Systems01:21

First Order Systems

First-order systems, such as RC circuits, are foundational in understanding dynamic systems due to their straightforward input-output relationship. Analyzing their responses to different input functions under zero initial conditions reveals significant insights into system behavior.
When a first-order system is subjected to a unit-step input, its response is characterized by its transfer function. By applying the Laplace transform of the unit-step input to the transfer function, expanding the...

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Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
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Stochastic hierarchical systems: excitable dynamics.

Helmar Leonhardt1, Michael A Zaks, Martin Falcke

  • 1Institute of Physics, Humboldt University at Berlin, Newtonstr. 15, D-12489, Berlin, Germany, helmar_leonhardt@web.de.

Journal of Biological Physics
|August 12, 2009
PubMed
Summary
This summary is machine-generated.

This study introduces a discrete stochastic excitability model using integral equations. The model reveals conditions for mean-field oscillations in coupled oscillators, linking to the Kuramoto model.

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Area of Science:

  • Computational Neuroscience
  • Statistical Physics
  • Dynamical Systems

Background:

  • Stochastic excitability is crucial for neural function and complex system dynamics.
  • Existing models often lack memory effects or detailed state transitions.

Purpose of the Study:

  • To develop a discrete stochastic model of excitability incorporating memory.
  • To analyze the collective behavior of coupled excitable units.
  • To investigate bifurcations and oscillations in mean-field dynamics.

Main Methods:

  • Formulated a discrete model with rest, excited, and refractory states using delayed integral equations.
  • Incorporated nonexponential waiting times via memory kernels.
  • Extended to an ensemble of globally coupled oscillators.
  • Derived and analyzed mean-field equations and their bifurcations.

Main Results:

  • Identified conditions leading to destabilization and mean-field oscillations in the stochastic ensemble.
  • Demonstrated the emergence of oscillations from nonexponential waiting times and memory effects.
  • Established a connection between the derived mean-field equations and the Kuramoto model.

Conclusions:

  • The discrete stochastic model captures essential features of excitability with memory.
  • Mean-field analysis reveals rich dynamical behaviors, including oscillations, in coupled systems.
  • The model provides a framework for understanding collective phenomena in systems with non-Markovian dynamics.