Discrete-Time Replicator Equations on Parallel Neural Networks

In this paper, we are aiming to propose a novel mathematical model that studies the dynamics of synaptic damage in terms of concentrations of toxic neuropeptides (neurotransmitters) during neurotransmission processes. Our objective is to employ " Wardrop's first and second principles" within a neural network of the brain. Complete manifestations of Wardrop's first and second principles within a neural network of the brain are presented through the introduction of two novel concepts: neuropeptide's (neurotransmitter's) equilibrium and synapses optimum. In the context of a neural network within the brain, an analogue of the price of anarchy is the price of cognition which is the most unfavorable ratio between the overall impairment caused by toxic neuropeptide's (neurotransmitter's) equilibrium in comparison to the optimal state of synapses (synapses optimum). Finally, we also propose an iterative algorithm (neurodynamics) in which the synapses optimum is eventually established during the neurotransmission process. We envision that this mathematical model can serve as a source of motivation to instigate novel experimental and computational research avenues in the fields of artificial neural networks and contemporary neuroscience.