Memory uncertainty relation and harmonic memory in random recurrent networks

Area of Science:

• Dynamical Systems Theory
• Information Theory
• Statistical Physics

Background:

• Dynamical systems possess memory capabilities crucial for information processing.
• Quantifying short-term memory and its relationship with system dynamics is an ongoing challenge.
• Understanding memory limitations is vital for designing effective computational and control systems.

Purpose of the Study:

• To derive a fundamental inequality bounding the short-term memory of dynamical systems.
• To establish an uncertainty relation between short-term memory and input-induced state fluctuations.
• To investigate the impact of input noise and state-space regularization on memory properties.

Main Methods:

• Derivation of a novel inequality for short-term memory.
• Analytical and numerical analysis of reservoir systems under input noise.
• Examination of state-space regularization effects.
• Investigation of harmonic memory and noise-induced memory.

Main Results:

• An inequality is presented that bounds short-term memory from below, interpreted as a memory-fluctuation uncertainty relation.
• The lower bound, termed harmonic memory, is achievable.
• Noise-induced memory is identified, where input noise enhances memory under regularization.
• The general memory uncertainty relation may not hold for regularized or harmonic memory.

Conclusions:

• The study establishes a fundamental limit on short-term memory in dynamical systems.
• Input noise and state-space regularization introduce complex behaviors, including noise-induced memory.
• The findings provide insights into the trade-offs between memory capacity and system dynamics, with implications for system design and analysis.