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Second Order systems II

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.

First Order Systems

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...

Second Order systems I

Second Order systems I

A servo system exemplifies a second-order system, featuring a proportional controller and load elements that ensure the output position aligns with the input position. The relationship between these components is described by a second-order differential equation. Applying the Laplace transform under zero initial conditions yields the transfer function, showing how inputs are converted to outputs in the system.
By reinterpreting the system, one can derive the closed-loop transfer function, which...

Excitation-Contraction Coupling in Skeletal Muscles

Excitation-Contraction Coupling in Skeletal Muscles

Excitation-contraction coupling is a series of events that occur between generating an action potential and initiating a muscle contraction. It occurs at the triad, a structure found in skeletal muscle fibers that comprise a T-tubule and terminal cisternae of the sarcoplasmic reticulum on each side. These triads are visible in longitudinally sectioned muscle fibers. They are typically located at the A-I junction — the junction between the A and I bands of the sarcomere.
When an action...

Thermodynamic Systems

Thermodynamic Systems

A thermodynamic system is a set of objects whose thermodynamic properties are of interest. The system is considered to be embedded in its surroundings or the environment. The system and its environment can exchange heat and do work on each other through a boundary that separates them. However, the immediate surroundings of the system interact with it directly and therefore have a much stronger influence on its behavior and properties.
Consider an example of tea boiling in a kettle. The...

Classification of Systems-I

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:

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Related Experiment Video

Updated: Jan 30, 2026

An Electrophysiology Protocol to Measure Reward Anticipation and Processing in Children

An Electrophysiology Protocol to Measure Reward Anticipation and Processing in Children

Published on: October 4, 2018

Anticipation via canards in excitable systems.

Elif Köksal Ersöz¹, Mathieu Desroches¹, Claudio R Mirasso²

¹MathNeuro Team, Inria Sophia Antipolis Méditerranée, 06902 Sophia Antipolis, France.

Chaos (Woodbury, N.Y.)

|February 3, 2019

Summary

Neurons can predict incoming signals using a novel mechanism involving canards. This research explains anticipation in excitable systems and proposes experimental validation for neuroscientists.

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Corticospinal Excitability Modulation During Action Observation

Corticospinal Excitability Modulation During Action Observation

Published on: December 31, 2013

Area of Science:

Computational Neuroscience
Theoretical Neuroscience
Dynamical Systems

Background:

Neuronal anticipation of incoming signals is a critical function, yet its underlying physiological mechanisms remain poorly understood.
Existing models often lack a comprehensive explanation for how receiver neurons predict sender dynamics in unidirectionally coupled systems.

Purpose of the Study:

To propose a novel theoretical framework explaining neuronal anticipation in excitable systems.
To elucidate the role of a mathematical object, known as a canard, in mediating predictive capabilities of receiver neurons.
To provide a numerical method for analyzing transient canard effects.

Main Methods:

Development of a theoretical viewpoint centered on 'canards' to explain anticipation.
Numerical simulations to quantify the transient effects of canards.
Application of the framework to established models: van der Pol, FitzHugh-Nagumo, and a Hodgkin-Huxley reduction.

Main Results:

Demonstrated that canards act as messengers enabling sender prediction in multi-scale excitable systems.
Validated the canard-mediated anticipation mechanism across diverse models, including radio-wave circuits and biophysical neuronal models.
Proposed a concrete experimental paradigm for neuroscientists to test these predictions.

Conclusions:

Canards offer a unifying theoretical explanation for anticipation in excitable systems.
The proposed framework provides a foundation for understanding predictive coding in neural networks.
Future research can extend this approach to broader classes of excitable systems and explore further implications for neural computation.