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Designing spontaneous behavioral switching via chaotic itinerancy

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Science Advances  11 Nov 2020:
Vol. 6, no. 46, eabb3989
DOI: 10.1126/sciadv.abb3989
  • Fig. 1 Experimental setups.

    (A) Schematic diagram of a high-dimensional chaotic system prepared in our experiments. The system can be divided into two parts: input echo state network (ESN) and chaotic ESN. Input ESN acts as an interface between the discrete input and the chaotic ESN, generating transient dynamics projecting onto the chaotic ESN when the symbolic input switches. To prevent the chaotic ESN from becoming nonchaotic because of the bifurcation, the connection between the input ESN and the chaotic ESN is trained to output transient dynamics converging to 0 (see the Supplementary Materials for the detailed information about the transient dynamics). (B) Two experimental schemes. In the open-loop scheme, the symbolic input is externally given. On the other hand, in the closed-loop one, the symbolic input is autonomously generated by the additional feedback loop. In our method, we change the elements represented by red arrows to embed desired CI dynamics. (C) Outline diagrams of our batch learning methods composed of a three-step procedure. In step 1, the parameters of the network and readout are trained to output the quasi-attractors and the output dynamics corresponding to the symbols. In steps 2 and 3, the symbolic sequence is autonomously yielded. We prepare periodic symbol transition patterns as the target in step 2 and stochastic symbol transition rules in step 3.

  • Fig. 2 Demonstration of step 1.

    (A) The dynamics of the reservoir before and after the innate training. In the figure, we show the RNN dynamics trained under the condition (M, Linnate) = (1,1000). The time-series data of a selected node in the input ESN are shown in the top column. Conversely, the four selected dynamics of the chaotic ESN are displayed in the bottom four columns. In each column, both the innate trajectory (black dotted) and 10 individual trajectories with different initial conditions (red) are exhibited. (B) Demonstration of open-loop dynamics. The network dynamics of the RNN trained under the condition (M, Lin, Lout) = (3,1000,1500) is used in this demonstration. Both the network dynamics and output dynamics of the trained readout are depicted. The readout is trained to output the Lissajous curve for symbol A, the “at” sign for symbol B, and the xz coordinates of the Lorenz attractor for symbol C. Note that the intervals of the symbolic input were randomly decided.

  • Fig. 3 Scalability and the validity of innate training used in step 1.

    (A) Performance of innate training over M symbols. The normalized mean square errors are calculated from the 10 trials. (B) Effect of network size Nch on the performance of innate training. (C) Evaluation of the temporal information capacity with timer task. The averaged values for 10 trials are plotted. (D) Effect of the system size on timer task capacities. Timer task capacity is defined as the integral value of the timer task function. (E) Evaluation of the LLE. The LLE is measured with the time development of the perturbation of the chaotic ESN (see the Supplementary Materials for detailed information about the calculation method of the LLE). (F) Evaluation of the system’s MLE.

  • Fig. 4 Demonstrations of closed-loop dynamics in step 2.

    (A) Three-symbol periodic transition. We prepared an RNN trained under the condition (M, Linnate) = (3,1000) and a readout trained under the condition Lout = 1500 ms to output three Lissajous curves corresponding to the symbolic input. The feedback loop fmax realizes the periodic symbol transition A-B-C switching at 2000-ms intervals. (B) Ten-symbol periodic transition. We prepared an RNN trained under the condition (M, Linnate) = (10,500) and a readout trained under the condition Lout = 500 ms to output 10 different Lissajous curves corresponding to the symbolic input. The feedback loop fmax achieves the periodic symbol transition A-B-C-D-E-F-G-H-I-J switching at 500-ms intervals. (C) Demonstration of the tasks requiring higher-order memory to be solved. The same RNN was used in the demonstration of (A). The left panel displays the periodic symbol transition pattern A-B-C-B switching at 2000-ms intervals. The right one demonstrates the periodic symbol transition pattern A-B-A-B-C switching at 2000-ms intervals. These tasks were accomplished in the same way in the demonstrations of (A) and (B), that is, only the parameters in fmax were tuned. (D) Two output dynamics: original trajectory and perturbed trajectory. A small perturbation was given to the original trajectory at t = 0 ms.

  • Fig. 5 Demonstration of step 3.

    (A) Network dynamics with fmax trained to imitate a stochastic transition rule. We used an RNN trained under the condition (M, Linnate) = (3,1000), and readout trained to output Lissajous curves under the condition Lout = 1500 ms. The feedback classifier fmax was trained to uniformly switch the symbol among the three symbols A, B, and C at 3000-ms intervals. Ten different trajectories with small perturbations are overwritten in the figure. (B) Evaluation of the embedding performance of a stochastic symbol transition. Two different stochastic symbol transition rules (patterns 1 and 2) were prepared as the target. The same RNN was used as in the demonstration of (A). The middle figures show the obtained probability density matrix, and the right ones show the average switching duration (the error bar represents SD).

  • Fig. 6 Analysis of symbolic dynamics and the final state.

    (A) Effect of a small perturbation on the terminal symbolic dynamics. We evaluated the two closed-loop setups prepared in Fig. 5B. The figures display the symbolic dynamics generated by 50 trajectories with 50 different initial values. (B) Analysis of symbolic dynamics generated by the temporal development of the initial states on a small plane and its entropy of the symbolic pattern. Two dimensions (x1 and x2) on the phase space were selected from the chaotic ESN to construct the plane. We observed the symbolic dynamics generated by the temporal development of the states on the plane. To evaluate the randomness of the obtained pattern, we calculated the entropy of the obtained symbolic pattern based on the probability distribution constructed from the 3 × 3 grid patterns. Note that the horizontal dashed line shows the maximum entropy (log23914.26).

Supplementary Materials

  • Supplementary Materials

    Designing spontaneous behavioral switching via chaotic itinerancy

    Katsuma Inoue, Kohei Nakajima, Yasuo Kuniyoshi

    Download Supplement

    The PDF file includes:

    • Algorithm of the innate training
    • Projected dynamics onto chaotic ESN
    • Distribution of eigenvalues after the innate training
    • Timer task setup
    • Maximum Lyapunov exponent
    • Local Lyapunov exponents
    • Evaluation of the embedding operability for long-period trajectory
    • Demonstration of history-dependent transition rule
    • Preferred trajectory in quasi-attractor
    • Figs. S1 to S4
    • References

    Other Supplementary Material for this manuscript includes the following:

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