Generalized Wilson-Cowan model with short term synaptic plasticity

Abstract

A generalized version of the Wilson-Cowan (WC) model is proposed which accounts for the evolution of the synaptic resources. Adiabatic elimination of the fast variables is performed to yield a simplified framework for the coupled interaction between active excitatory and inhibitory neurons. The latter model is shown to smoothly converge to the benchmark WC model, when the appropriate limit is performed. Different dynamical regimes are identified for the reduced model and commented upon with reference to the original formulation of the generalized dynamics. This includes identifying limit cycle oscillations for population of available resources.

Figures (5)

• Figure 1: The different solutions as displayed by the generalized model (\ref{['eq_ad']}) are illustrated in the reference plane ($\omega_{EI}$, $\omega_{EE}$), for different choices of $\gamma$. More specifically $\gamma=1$ in panel (a), $\gamma=5$ in panel (b), $\gamma=20$ in panel (c). In panel (d) the landscape of possible solutions is reported for the original WC model which is recovered, for large enough values of $\gamma$. The light blue region identifies values of ($\omega_{EI}$, $\omega_{EE}$) that yields a stable fixed point solution. Working in the dark blue domain, a limit cycle is found. The other parameters are set to the following values: $a_E=a_I=1$, $h_E=-0.4$, $h_I=-0.5$, $I_E=0.5$, $I_I=0$,$\xi_E=\xi_I=1$, $\omega_{IE}=2$ and $\omega_{II}=0.1$.
• Figure 2: The landscape of possible solutions of model (\ref{['eq_ad']}) are displayed when scanning the plane ($\omega_{IE}$, $\omega_{EE}$), for a different choice of $\omega_{EI}$, as compared to Figure \ref{['Figure1']} (here, in particular, $\omega_{EI}=1.5$ while the other parameters stay unchanged). The region drawn in orange identifies a bistable regime (two stable fixed points, separated by a saddle point) while the region drawn in yellow contains a stable fixed point, a saddle and an unstable one . When the parameters are set to the values that correspond to the region colored with light blue, the system shows a stable fixed point. When $\gamma$ gets reduced, the region of bistability shrinks. Panel (a) corresponds to $\gamma=1$, panel (b) to $\gamma=5$ and panel (c) to $\gamma=20$. The WC solution is depicted in panel (c) and it is formally recovered for sufficiently large values of $\gamma$.
• Figure 3: Different dynamical regime for model (\ref{['eq_ad']}) are characterized as a function of the the baseline level of non depleted synaptic resources, namely ($\xi_I$,$\xi_E$) and for different choices of the parameter $\gamma$. Panel (a) corresponds to $\gamma=1$, panel (b) to $\gamma=2$ and panel (c) to $\gamma=7$. Panel (d) refers to the WC model: in this setting the parameters $\omega_{EE}$ and $\omega_{II}$ are modulated, while keeping the ratios $\omega_{EE}/\omega_{EI}$ and $\omega_{II}/\omega_{IE}$ frozen to the values used in generating the outputs reported in panels (a),(b) and (c). The other parameters have been here set to the values $a_E=a_I=1$, $h_E=-0.4$, $h_I=-0.5$, $I_E=0.5$, $I_I=0$,$\omega_{EE}=\omega_{IE}=1$, $\omega_{EI}=1.5$ and $\omega_{II}=0.1$. For $(\xi_I,\xi_E)$ in the light blue region, the system converges to a stable fixed point. In the dark blue region, the fixed point is unstable and the system display a limit cycle. For larger $\gamma$, two additional regions are found. The system has three fixed point two saddle nodes and one stable attractor in the yellow region. The system is instead bistable (two stable nodes and one saddle node) in the region depicted in orange.
• Figure 4: Two upper panels: the resource variables ($R_I$ in blue and $R_E$ in red), oscillate in time. The predicted limit cycle for $x$ and $y$ reverberates on the coupled variables $R_E$ and $R_I$. The oscillations fades however away for sufficiently large values of $\tau_R$ (and $\tau_D$) at constant $\gamma$. Species $x$ and $y$ keep on oscillating , independently of the chosen time scales (see lower panels). Here, $a_E=a_I=1$, $h_E=-0.4$, $h_I=-0.5$, $I_E=0.5$, $I_I=0$,$\omega_{EE}=\omega_{IE}=1$, $\omega_{EI}=1.5$, $\omega_{II}=0.1$, $\xi_E=3.5$, $\xi_I=3$ and $\gamma=\frac{\tau_d}{\tau_r}=7$.
• Figure 5: $A_E$ versus $\tau_R$ for different values of $\gamma$, respectively equal to 2 (left panel) and 7 (right panel). Here, $a_E=a_I=1$, $h_E=-0.4$, $h_I=-0.5$, $I_E=0.5$, $I_I=0$,$\omega_{EE}=\omega_{IE}=1$, $\omega_{EI}=1.5$, $\omega_{II}=0.1$, $\xi_E=4.5$, $\xi_I=4$. For that choice of the parameters for $\gamma=2$ we numerically estimate $\omega=2.3$, $\bar{R_E}=4.1$, $A_x=0.5$ and $\bar{x}=0.2$. Instead, for $\gamma=7$, we have $\Omega=2.2$, $\bar{R_E}=4.4$, $A_x=0.6$ and $\bar{x}=0.2$.