# Deep Time-Delay Reservoir Computing: Dynamics and Memory Capacity

arXiv: https://arxiv.org/abs/2006.06322 · HTML: https://mlsanity.com/paper/2006.06322/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jun 2020.

## Built on by (most cited first, 10 of 10)

- [Attractor reconstruction with reservoir computers: The effect of the reservoir's conditional Lyapunov exponents on faithful attractor reconstruction](https://mlsanity.com/paper/2401.00885/) (2023)
- [Heterogeneous Neuronal and Synaptic Dynamics for Spike-Efficient Unsupervised Learning: Theory and Design Principles](https://mlsanity.com/paper/2302.11618/) (2023)
- [Master memory function for delay-based reservoir computers with single-variable dynamics](https://mlsanity.com/paper/2108.12643/) (2021)
- [Exploiting Heterogeneity in Timescales for Sparse Recurrent Spiking Neural Networks for Energy-Efficient Edge Computing](https://mlsanity.com/paper/2407.06452/) (2024)
- [Infinite-dimensional Folded-in-time Deep Neural Networks](https://mlsanity.com/paper/2101.02966/) (2021)
- [Beyond Terabit/s Integrated Neuromorphic Photonic Processor for DSP-Free Optical Interconnects](https://mlsanity.com/paper/2504.15044/) (2025)
- [Sequential Reservoir Computing for Efficient High-Dimensional Spatiotemporal Forecasting](https://mlsanity.com/paper/2601.00172/) (2026)
- [Information Processing Capacity of Stationary Physical Systems: Theory, Data-efficient Estimation Methods, and Photonic Demonstration](https://mlsanity.com/paper/2605.19152/) (2026)
- [Organization of computation in reservoir computing](https://mlsanity.com/paper/2607.17858/) (2026)
- [Inference of Unknown Dynamical Components Using Next Generation Reservoir Computing: From Chaotic Systems to Climate Data](https://mlsanity.com/paper/2609.24754/) (2026)

## Abstract (arXiv)

The Deep Time-Delay Reservoir Computing concept utilizes unidirectionally connected systems with time-delays for supervised learning. We present how the dynamical properties of a deep Ikeda-based reservoir are related to its memory capacity (MC) and how that can be used for optimization. In particular, we analyze bifurcations of the corresponding autonomous system and compute conditional Lyapunov exponents, which measure the generalized synchronization between the input and the layer dynamics. We show how the MC is related to the systems distance to bifurcations or magnitude of the conditional Lyapunov exponent. The interplay of different dynamical regimes leads to a adjustable distribution between linear and nonlinear MC. Furthermore, numerical simulations show resonances between clock cycle and delays of the layers in all degrees of the MC. Contrary to MC losses in a single-layer reservoirs, these resonances can boost separate degrees of the MC and can be used, e.g., to design a system with maximum linear MC. Accordingly, we present two configurations that empower either high nonlinear MC or long time linear MC.