Learning process

Let us now consider that the primary layer is continuously stimulated by a periodic signal $\mathbf{I}^{(1)}$ (repeated every $\tau$ time steps), and the dynamics of the system is given by (2). Fig.3 presents the time evolution of the neural activity in population 2 while the system is submitted to a period-5 spatio-temporal input signal (for visual comfort, we have taken a pattern representing a frog jump).

Figure 3: Learning dynamics, between $t=1$ and $t=200$ (with $N^{(1)}=1600$, $N^{(2)}=200$). The system is continuously stimulated by a periodic spatio-temporal pattern (a frog jump). Parameters are in Tab.1. - a - Neuronal activity on secondary layer. 30 individual signals (out of 200) have been represented, and their mean activity is represented below. - b - Time evolution of input signal $\mathbf{I}^{(1)}(t)$ and feedback reinforcement $\mathbf{F}^{(12)}(t)$ (see text), between $t=51$ and $t=155$ (For readability, most of the time steps have been discarded). At each time step, the 1600 values have been represented as $40\times 40$ images, where white corresponds to 0 and black to 1 (in-between values are gray tones).

During the learning process, the synaptic weights are modified at each time step, so that the whole system continuously evolves under the constraint of the external signal $\mathbf{I}^{(1)}$. Two sorts of dynamical changes can thus be observed in the system (Fig. 3). First, the secondary layer activity, which is initially chaotic, gets closer to a periodic behavior (Fig.3-a-). The learning process tends to reduce the complexity of the initial chaotic dynamics towards a periodic dynamics (period-5 dynamics), so that the predictability between primary and secondary layers activities tends to increase. Nevertheless, changes on the weights remain very weak, and the statistics of the weight matrix remains the same as the one of the initial random matrix. Second, at each time step, a subset of neurons ($15-20\%$) is active in secondary layer. The rule strengthens the connections between the secondary layer subset which was active at time $t-1$ and the primary neurons that are active at time $t$. This reinforcement of feedback weights takes place at every time step, while the primary layer is periodically stimulated. The effective feedback signal is given by the local-field $\mathbf{h}^{(12)}$. During the first steps of the learning process, the amplitude of this signal is weak, which means that the feedback influence is almost negligible. Then, as time goes on, the amplitude of the feedback signal grows, and then some values of $\mathbf{h}^{(12)}$ reach the critical threshold value $\theta ^{(1)}$, and thus significantly increase the activation of the corresponding primary neurons. In order to estimate and represent the efficacy of this feedback signal, we consider the signal $\mathbf{F}^{(12)}=f_g(\mathbf{h}^{(12)}-\theta^{(1)})$, which corresponds to what would be the primary layer activation if no input was sent. This signal is displayed on Fig.3-b-, and compared with the current input signal $\mathbf{I}^{(1)}$. One can remark that the input signal and the feedback signal are synchronized. Knowing that the transmission delay from secondary towards primary layer is equal to 1, the feedback signal at time $t$ relies on the secondary pattern of activation at time $t-1$, so that the secondary layer anticipates the activity of primary layer. The feedback signal is thus a prediction on the forthcoming input, and corresponds to a ``top-down expectation''. The signal $\mathbf{F}^{(12)}$ also provides an objective criterion for stopping the learning process. When the value of the feedback signal is of the order of the input signal, the learning process can be stopped so that one can test the recognition properties of the system (see next section). Note that the unbounded continuation of the learning process increases too strongly the amplitude of the feedback signal and the reduction of the inner dynamics, so that the system becomes insensitive to its input signal (and thus looses its adaptivity).