Figure 4 | Scientific Reports

Figure 4

From: Bifurcation analysis of the dynamics of interacting subnetworks of a spiking network

Figure 4

Numerical integration of Eq. (5) and subnetwork firing rates for different parameters corresponding to the 6 regions. (A) For \(a=0.9\), \(b=1.3\) (region 2) and initial conditions \((0.0001,0.0001,0.02)\), the trajectory passes by the vicinity of p0,0,1, and eventually converges to p0,1,1. (B) For \(a=1.2\), \(b=0.9\), and \((0.0001,0.0001,0.02)\) as the initial condition, the trajectory follows a heteroclinic connection between p0,0,1 and p1,0,1. This parameter combination corresponds to region 3. (C) For \(a=b=1.2\), and any initial condition in the first octant, the trajectory converges to p0,0,1. (D) For \(a=b=0.9\) (region 5) and initial conditions equal to \((0.0004,0.0003,0.02)\), the trajectory passes by the points p0,0,1 and p1,1,1, and eventually converges to p1,0,1. In this case, a longer heteroclinic connection between saddle points exists. For initial conditions with \({x}_{2} > {x}_{1}\), the trajectory will converge to p0,1,1. (E) For \(a=0.90\), \(b=0.97\), and initial condition \((0.0002,0.0002,0.01)\), the trajectory represents a heteroclinic connection between p0,0,1 and p1,0,1, and will finally converge to p0,1,1. The amount of time that the trajectory spends close to the saddle points, and also how close it gets to each saddle point, depends on the initial conditions. This parameter combination corresponds to region 6. (F) For \(a=0.98\) and \(b=0.92\) which corresponds to region 4, and the initial condition \((0.001,0.001,0.01)\), the trajectory follows a heteroclinic connection between p0,0,0, p0,0,1 and p0,1,1, and eventually converges to p1,0,1. (A′F′) Firing rates of different subnetworks for each parameter combination in (AF), respectively. Mean firing rates over 20 trials for each case and each subnetwork is plotted in darker colors.

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