Continuous neural network with windowed Hebbian learning

Abstract

We introduce an extension of the classical neural field equation where the dynamics of the synaptic kernel satisfies the standard Hebbian type of learning (synaptic plasticity). Here, a continuous network in which changes in the weight kernel occurs in a specified time window is considered. A novelty of this model is that it admits synaptic weight decrease as well as the usual weight increase resulting from correlated activity. The resulting equation leads to a delay-type rate model for which the existence and stability of solutions such as the rest state, bumps, and traveling fronts are investigated. Some relations between the length of the time window and the bump width is derived. In addition, the effect of the delay parameter on the stability of solutions is shown. Also numerical simulations for solutions and their stability are presented.

Appendix: Proof of Theorem 1

Proof

We claim that in (9), if ${\mathrm {Re}}\lambda \ge 0$, we must have ${\mathrm {Im}}\lambda =0$. If it is not the case, assume that $\lambda =r+{\mathrm {i}}m$ and $m\ne 0$ and $r\ge 0$. Let $\theta =\alpha \gamma \tau ^{-1} f(\overline{u})^2f'(\overline{u})(W+\widehat{w}_m(\xi ))$ and considering imaginary parts of (9),

$$\begin{aligned} m-\theta \frac{r{\mathrm {e}}^{-\delta r}\sin (\delta m)-m\left( 1-{\mathrm {e}}^{-\delta r}\cos (\delta m)\right) }{r^2+m^2}=0. \end{aligned}$$

Since $m\ne 0$,

$$\begin{aligned} r^2+m^2=\theta \left( r{\mathrm {e}}^{-\delta r}\frac{\sin (\delta m)}{m}+{\mathrm {e}}^{-\delta r}\cos (\delta m)-1\right) . \end{aligned}$$

Therefore we obtain

$$\begin{aligned} r^2<\theta \left( (\delta r+1){\mathrm {e}}^{-\delta r}-1\right) \le 0, \end{aligned}$$

which is a contradiction (Note that $\theta \ge 0$).

The left-hand side of (9) is an increasing function of $\lambda \in [0,\infty ]$. Therefore, its value at $\lambda =0$ must be positive for stability, i.e.,

$$\begin{aligned}&1\!-\!(1\!-\!\alpha )f'(\overline{u})\widehat{w}_m(\xi )\!-\!\alpha \gamma \delta f(\overline{u})^2f'(\overline{u})(W+\widehat{w}_m(\xi ))>0, \\&\quad \text {for every}\,\xi \in {\mathbb {R}}. \end{aligned}$$

which concludes the results in the desired cases. (Note that $\widehat{w}_m(\xi )\le W$ when the synaptic weight kernel is positive everywhere and also in the other case, Mexican-hat kernel $w_m(x)=\frac{1}{4}(1-|x|){\mathrm {e}}^{-|x|}$, we have $W=0$ and $\widehat{w}_m(\xi )\le \frac{1}{4}$). Also in the first case, $\overline{u}$ is a root of $\big (1-\kappa {\mathrm {e}}^{-\gamma \delta f(\overline{u})^2}\big )Wf(\overline{u})-\overline{u}$ and the stability condition reads that the derivative of this function with respect to $\overline{u}$ should be negative at $\overline{u}$ and this is the case for the extreme possible values of $\overline{u}$ (which are of course positive) since the value of this function is positive at zero and negative at infinity. In fact in general, the possible constant steady states are alternatively stable and unstable. $\square$