Distributed delays in a simple neuronal population model of decision-making

Figure 1.  Left panel: Number of steady states of System (4) depending on $ \epsilon $ and $ I $. Right panel: graphs of the function $ I-r+\epsilon f(r^2)r $ for several values of $ \epsilon $ close to $ 1 $. The change of $ I $ moves these graphs along the vertical axis

Figure 2.  Stability of the steady state depending on $ \gamma $ and the average delay. The upper greyed area is the region in which the assumptions of Theorem 2.3 are satisfied, thus the steady state is unstable independently of the delay kernel. In the lower shaded area, the assumptions of Theorem 2.5 are satisfied if the kernels are equal, and thus the steady state is stable for all kernels. In the remaing area, stability of the steady state depends on the kerenel and average delay

Figure 3.  A Hopf bifurcation in System (4) with respect to the parameter $ a $. Note that a subcritical Hopf bifurcation is observed, meaning instability of a bifurcating periodic orbit. Top panels: (LEFT) for $ a = 0.74< 0.747 $ (approx) the steady state $ S_1 $ is unstable and we observe increasing oscillations; (RIGHT) at the critical value of $ a $ a Hopf bifurcation is visible – we see almost stable oscillations. Bottom panels: (LEFT) for $ a = 0.76 > 0.747 $ (approx) $ S_1 $ is stable; (RIGHT) in the space $ \big( r_1(t), r_2(t-\tau_{av})\big) $ for the three values of the parameter $ a $ used in simulations we see: decaying oscillations (blue curve, $ a = 0.76 $), almost permanent oscillations (red curve, $ a = 0.747 $), increasing oscillations (green curve, $ a = 0.74 $); black dots denote "starting" points (for $ t = 400 $), $ t\in [400,500] $

Figure 4.  The dependence of $ \gamma_{cr} $ on $ a $ (left panel) and $ \tau_{ \rm{av}} $ (right panel) for $ n = 1 $, $ 2 $ and $ 3 $. On the right panel, the critical value of dely in the discrete delay case is plotted with dashed line for comparison. The shaded area indicates the region in which assumption of Theorem 2.5 are satisfied. The steady state is stable for $ \gamma<\gamma_{cr} $

Figure 5.  Steady states (represented by the coordinate $ \bar r $) and their stability for Erlang kernel with $ n = 1 $, reference signal $ I = 0.4 $ and different values of $ \epsilon $. The color represents the critical value $ \tau_{cr, 1} $ of the average delay $ \tau_{ \rm{av}} = \frac{1}{a} $ for which the stable steady state loses stability

Figure 6.  Destabilization of the steady state of System (4). The value of the shift $ \tau $ for which the stable steady state destabilizes depending on $ \gamma $ and $ \tau_{av, 0} = 1/a $ is presented