Characterize Orbit Length Distributions vs Bit Width in Integer-State SNNs

Establish a formal characterization of the distribution of periodic orbit lengths as a function of the integer bit width used to represent neuron state variables in deterministic finite-state spiking neural networks modeled as bounded-integer discrete-time maps (for example, networks using integer-valued membrane potentials, synaptic weights, and thresholds with shift-based leakage updates and thresholding).

Background

The paper models hardware-oriented spiking neural networks as deterministic dynamical systems evolving on bounded integer lattices. Under finite-precision integer representations, trajectories are necessarily bounded and eventually recurrent, implying the presence of fixed points or periodic orbits.

Exploratory simulations demonstrate bounded, recurrent dynamics across various bit widths, suggesting that numerical precision acts as a dynamical design variable. However, while empirical cycle lengths are reported, a rigorous theoretical description connecting bit width to the distribution of orbit lengths is not provided, motivating the stated open problem.