Fig 1.
Modeling of medium between electrically coupled FHN neurons.
The electrical signals between the neurons can pass through the inter-neuronal medium. This medium has resistive, inductive and capacitive characteristics. Therefore, it can be modeled through an RLC circuit: (a) Two neurons under external stimulation and communicating through the medium, (b) neuronal signals facing the RLC medium.
Fig 2.
The existence of a natural PID control between FHN neurons coupled through a medium.
This medium, forming an RLC circuit, can be represented as natural resistive, inductive, and capacitive properties, which are responsible for the regulation of the synchronous behavior of the neurons. The natural PID control can be understood as a mechanism for tracking the behavior of one neuron via another.
Fig 3.
Comparison of the errors under various proportional components on the synchronization of FHN neurons: kp = 0.1, kp = 0.3, kp = 0.6, kp = 1.
The synchronization error decreases with increasing kp; however, an increase in the overshoot is seen as kp gets smaller.
Fig 4.
Comparison of integral components to synchronization error: ki = 0.0001, ki = 0.3, ki = 0.6, ki = 1.
By increasing the value of ki, the synchronization error in the steady state can be reduced to zero. That is, a complete synchronization between neurons can be attained for a large value of the integral constant of the natural PID control.
Fig 5.
Comparison of derivative components to synchronization error: kd = 0.001, kd = 0.04, kd = 0.1, kd = 0.3.
By increasing the value of kd, the oscillatory effects of the synchronization error can be reduced. The derivative component of the natural PID control can improve the stability of the synchronization error system.
Fig 6.
Supervisory mechanism for PID-controlled synchronization of neurons.
By employing PSO and an additional PID control strategy, the coupling deficiency between the neurons can be improved. The proposed controller can increase the strength of the natural PID mechanism. The parameters of the new PID control can be tuned through adaptation for attainment of synchronization. PSO is employed to determine the optimal PID parameters for neuronal synchronization.
Fig 7.
Controlled synchronization by application of the proposed supervisory mechanism.
By application of this mechanism, non-synchronous neurons can be adaptively synchronized through adaptation of the proportional, integral, and derivative components: (a) Non-coherent activation potentials without the supervisory mechanism, (b) PID gains’ convergence curves for the proposed PSO approach, and (c) activation potentials using the optimal gains ,
and
of the proposed supervisory mechanism.