One parameter family of master equations for logistic growth and BCM theory

• Published in: Communications in nonlinear science & numerical simulation 2015-02, Vol.20 (2), p.461-468
• Main Authors: De Oliveira, L.R., Castellani, C., Turchetti, G.
• Format: Article
• Language: English
• Subjects: BCM theory; Evolution; Extinction; Gaussian; Logistic master equation; Logistics; Mathematical analysis; Mathematical models; Mean field and noise; Nonlinearity; Power law
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2014.05.026

•A family of logistic master equations with the same mean field is proposed.•Equilibrium distribution varies from maximum growth to near extinction.•Relaxation to the maximum growth occurs for large populations.•An absorbing state does not alter the picture even for a small populations.•A one parameter family of BCM models for two synapses is presented. We propose a one parameter family of master equations, for the evolution of a population, having the logistic equation as mean field limit. The parameter α determines the relative weight of linear versus nonlinear terms in the population number n⩽N entering the loss term. By varying α from 0 to 1 the equilibrium distribution changes from maximum growth to almost extinction. The former is a Gaussian centered at n=N, the latter is a power law peaked at n=1. A bimodal distribution is observed in the transition region. When N grows and tends to ∞, keeping the value of α fixed, the distribution tends to a Gaussian centered at n=N whose limit is a delta function corresponding to the stable equilibrium of the mean field equation. The choice of the master equation in this family depends on the equilibrium distribution for finite values of N. The presence of an absorbing state for n=0 does not change this picture since the extinction mean time grows exponentially fast with N. As a consequence for α close to zero extinction is not observed, whereas when α approaches 1 the relaxation to a power law is observed before extinction occurs. We extend this approach to a well known model of synaptic plasticity, the so called BCM theory in the case of a single neuron with one or two synapses.