class18

Causes of Oscillations

• Oscillation is a significant phenomenon in biology, affecting various physiological aspects such as body temperature, hormone levels, and gene expression.

• Primary focus:

  • Modeling stable oscillations with periodic attractors (stable limit cycles).

  • Understanding the equations that represent these models.

  • Identifying the two main causes of oscillation: steep negative feedback and time delay.

Learning Outcomes

• By the end of this section, students should be able to:

  1. Describe the two main causes of stable oscillations.

  2. Write differential equations that include time delays.

  3. Identify conditions under which differential equations exhibit time delays.

Respiratory Control of CO2

• The level of CO2 is regulated by the rate of breaths per minute, functioning through a negative feedback loop.

  • Higher CO2 levels lead to an increased breathing rate.

Mackey-Glass Model for CO2 Control

• Let X denote the concentration of CO2 in the blood. The change in concentration is influenced by both cell metabolism and ventilation:

  • Equation: X' = L - Vmax * X^n / (1 + X^n) * X

  • Factors influencing the ventilation rate include:

    • Vmax: Maximum ventilation possible.

    • n: Steepness of the response curve.

  • When X (CO2) is low, the ventilation rate is lower; as X increases, the rate approaches Vmax.

Ventilation Rate Function

• The equation for ventilation rate is given by:

  • Ventilation Rate: V = Vmax * X^n / (1 + X^n)

• Key influences:

  • Vmax: Determines the maximum value of the sigmoid function.

  • n: Controls the steepness of the function.

Time Delay in Respiration

• Recognition of time delays in physiological responses; for example, it takes time for CO2 to travel from lungs to the brain:

  • CO2 concentration in the brain is represented as a function of time delay (t).

  • This results in an adjusted respiratory control model.

Periodic Oscillations

• Periodic oscillations can be identified via time series analysis. Recognizing patterns aids in understanding oscillatory dynamics.

Investigating Ventilation Rates

• The normal ventilation rate can fluctuate, influenced by parameters such as time delay (τ) and the steepness of the feedback curve (n).

  • Different values lead to changing behaviors of the ventilation rate over time.

Causes of Oscillations

• The two identified causes of oscillations are:

  • Steep Negative Feedback: The rate at which a system corrects itself; excessive feedback can lead to oscillation.

  • Time Delay: The necessity for some processes to correct after a certain duration, contributing to oscillatory behavior.

Influence of n and τ on Ventilation Rate

• To achieve periodic oscillations, both n and τ must exceed certain thresholds.

• Different combinations of these parameters can either stabilize the system or induce oscillations.

  • Example combinations tested:

    • τ = 0.1 and n = 2

    • τ = 0.6 and n = 1

    • τ = 0.1 and n = 5

    • τ = 0.6 and n = 4

Cheyne-Stokes Breathing

• Models the condition of Cheyne-Stokes breathing:

  • Recognizes how medical conditions affect the model:

    • Heart Failure: Long circulation times due to inefficiency affects breathing patterns.

    • Stroke: Resulting reflexivity issues impact ventilation rates.

Bifurcation in Ventilation Rate

• Adjustments in parameters (τ and n) lead to a transition between steady behavior to periodic oscillations known as bifurcation.

• A bifurcation diagram displays how these parameters influence the system's long-term behavior.

Summary of Key Concepts

• Understanding of the causes of oscillations:

  • Time delay and steep negative feedback are essential for modeling respiratory control.

• The interaction of these factors highlights the complexity of physiological regulation.