Time delays and the control of biological systems: An overview∗∗JM acknowledges support from the William R. Kenan, Jr. Charitable Trust

Observer-Based PID Control Strategy for the Stabilization of Delayed High Order Systems with up to Three Unstable Poles

In this paper, a new method to manage the stabilization and control problems of n-dimensional linear systems plus dead time, which includes one, two, or three unstable poles, is proposed. The control methodology proposed in this work is an Observer-based Proportional-Integral-Derivative (PID) strategy, where an observer and a PID controller are used to relocate the original unstable open-loop poles to stabilize the resultant closed-loop system. The observer provides an adequate estimation of the delayed-free variables and the PID uses the delay-free variables estimated by the proposed observer. Also, step-tracking is achieved in the overall control scheme. Necessary and sufficient conditions are presented to ensure closed-loop stability based on the open loop parameters of the system. The observer-based PID strategy considers five to seven constant parameters to obtain a stable closed-loop system. A general procedure to implement the proposed control strategy is presented and its performance is evaluated by means of numerical simulations.

Phase response approaches to neural activity models with distributed delay

In weakly coupled neural oscillator networks describing brain dynamics, the coupling delay is often distributed. We present a theoretical framework to calculate the phase response curve of distributed-delay induced limit cycles with infinite-dimensional phase space. Extending previous works, in which non-delayed or discrete-delay systems were investigated, we develop analytical results for phase response curves of oscillatory systems with distributed delay using Gaussian and log-normal delay distributions. We determine the scalar product and normalization condition for the linearized adjoint of the system required for the calculation of the phase response curve. As a paradigmatic example, we apply our technique to the Wilson-Cowan oscillator model of excitatory and inhibitory neuronal populations under the two delay distributions. We calculate and compare the phase response curves for the Gaussian and log-normal delay distributions. The phase response curves obtained from our adjoint calculations match those compiled by the direct perturbation method, thereby proving that the theory of weakly coupled oscillators can be applied successfully for distributed-delay-induced limit cycles. We further use the obtained phase response curves to derive phase interaction functions and determine the possible phase locked states of multiple inter-coupled populations to illuminate different synchronization scenarios. In numerical simulations, we show that the coupling delay distribution can impact the stability of the synchronization between inter-coupled gamma-oscillatory networks.

Effects of noise and time delay on E2F's expression level in a bistable Rb-E2F gene's regulatory network

The bistable Rb-E2F gene regulatory network plays a central role in regulating cellular proliferation-quiescence transition. Based on Gillespie's chemical Langevin method, the stochastic bistable Rb-E2F gene's regulatory network with time delays is proposed. It is found that under the moderate intensity of internal noise, delay in the Cyclin E synthesis rate can greatly increase the average concentration value of E2F. When the delay is considered in both E2F-related positive feedback loops, within a specific range of delay (3-13) hr , the average expression of E2F is significantly increased. Also, this range is in the scope with that experimentally given by Dong et al. [65]. By analysing the quasi-potential curves at different delay times, simulation results show that delay regulates the dynamic behaviour of the system in the following way: small delay stabilises the bistable system; the medium delay is conducive to a high steady-state, making the system fluctuate near the high steady-state; large delay induces approximately periodic transitions between high and low steady-state. Therefore, by regulating noise and time delay, the cell itself can control the expression level of E2F to respond to different situations. These findings may provide an explanation of some experimental result intricacies related to the cell cycle.

COMPARISON OF THEORIES OF REST TREMOR MECHANISM IN PARKINSON’S DISEASE: CENTRAL OSCILLATOR (SOURCE-TRIGGERED OSCILLATIONS) AND FEEDBACK-INDUCED INSTABILITY IN THE SENSORIMOTOR LOOP (SELF-SUSTAINED OSCILLATIONS)

Rest tremor is one of the most common and disabling symptoms of Parkinson’s disease (PD). The exact neural origin of rest tremor is still not clearly understood. Understanding the origin of rest tremor is important as it may aid in optimizing existing treatment strategies such as Deep Brain Stimulation or in developing new treatment strategies for rest tremor reduction. There are broadly two theories that are gaining prominence for rest tremor generation in PD. The first theory is the central oscillator theory that states that the rest tremor is triggered by an oscillatory source in the brain. The second theory is the feedback-induced instability theory that states that the rest tremor arises out of a feedback-induced instability in the sensorimotor loop. This paper analyzes validity of the two theories based on established clinical observations of Parkinsonian rest tremor by using representative simulation examples. Finally, based on our analysis, we propose two test-worthy experiments for further validation.