Suppression of chaos in a partially driven recurrent neural network (2306.00900v4)

Abstract: The dynamics of recurrent neural networks (RNNs), and particularly their response to inputs, play a critical role in information processing. In many applications of RNNs, only a specific subset of the neurons generally receive inputs. However, it remains to be theoretically clarified how the restriction of the input to a specific subset of neurons affects the network dynamics. Considering RNNs with such restricted input, we investigate how the proportion, $p$, of the neurons receiving inputs (the "inputs neurons") and the strength of the input signals affect the dynamics by analytically deriving the conditional maximum Lyapunov exponent. Our results show that for sufficiently large $p$, the maximum Lyapunov exponent decreases monotonically as a function of the input strength, indicating the suppression of chaos, but if $p$ is smaller than a critical threshold, $p_c$, even significantly amplified inputs cannot suppress spontaneous chaotic dynamics. Furthermore, although the value of $p_c$ is seemingly dependent on several model parameters, such as the sparseness and strength of recurrent connections, it is proved to be intrinsically determined solely by the strength of chaos in spontaneous activity of the RNN. This is to say, despite changes in these model parameters, it is possible to represent the value of $p_c$ as a common invariant function by appropriately scaling these parameters to yield the same strength of spontaneous chaos. Our study suggests that if $p$ is above $p_c$, we can bring the neural network to the edge of chaos, thereby maximizing its information processing capacity, by amplifying inputs.

• The paper explores how feeding input to only a subset of neurons suppresses chaos in recurrent neural networks (RNNs), providing a theoretical framework.
• A key finding is that chaos can be suppressed if the proportion of neurons receiving input exceeds a critical threshold (p_c), which is robust to certain model parameters.
• The results have implications for designing more stable and computationally efficient RNNs, particularly in hardware-constrained systems like physical reservoir computers.

Suppression of Chaos in Partially Driven Recurrent Neural Networks

The paper "Suppression of chaos in a partially driven recurrent neural network" by Shotaro Takasu and Toshio Aoyagi explores the dynamics of recurrent neural networks (RNNs) when the input is only provided to a subset of neurons. The paper contributes to understanding how such configurations influence network stability and the potential for information processing, particularly through the lens of chaos suppression. It provides a theoretical framework for determining the conditions under which chaos in a randomly connected RNN can be controlled by external inputs.

Key Findings and Analytical Approach

The authors focus on calculating the conditional maximum Lyapunov exponent (MCLE), which characterizes the RNN's response to external inputs. The MCLE offers a quantitative measure of how closely two initially identical networks maintain their trajectories when subjected to the same input. A negative MCLE indicates deterministic behavior, essential for controlled information processing and reproduction of time series data—core aspects of the reservoir computing paradigm.

The central analytical result reveals that if the proportion $p$ of neurons receiving input is above a critical threshold $p_c$, chaos can be suppressed by increasing the input strength. Below this threshold, even highly amplified inputs fail to stabilize the network. Intriguingly, the threshold $p_c$ is invariant to particular model parameters like connectivity sparseness and weight strength after appropriate scaling based on spontaneous activity chaos strength. This finding significantly advances the theoretical understanding of edge-of-chaos dynamics within partially driven RNNs.

Implications and Potential Applications

The implications of these findings are notable for the design of biologically inspired and physically implemented computing systems. This research suggests that to harness and optimize RNN dynamics for computation, ensuring input to a critical mass of neurons is necessary. For physical reservoir computing, where the ability to connect input across an entire system is often constrained, strategies derived from this paper could guide the structuring of networks for maximal computational efficacy.

Future Research Directions

Given its strong theoretical foundation, future studies could expand on this work by exploring different types of RNNs, such as those with heterogeneous activation functions or synaptic plasticity mechanisms. Additionally, empirical validation of these theoretical predictions through controlled experiments would enhance understanding. The paper also opens a pathway to apply these insights into other domains such as chaotic systems control or robust sequence memory in neuromorphic computing.

Conclusion

This research enriches the comprehension of chaotic dynamics in partially driven RNNs and offers a structured approach to suppressing chaos through calculated input strategies. The insights into $p_c$ underline the systematic approach needed to design RNNs that operate optimally at the edge of chaos, maximizing their potential for applications in real-time signal processing and complex data representations.