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Determinism in the Undetermined: Deterministic Output in Charge-Conserving Continuous-Time Neuromorphic Systems with Temporal Stochasticity

Jing Yan, Kang You, Zhezhi He, Yaoyu Zhang

Abstract

Achieving deterministic computation results in asynchronous neuromorphic systems remains a fundamental challenge due to the inherent temporal stochasticity of continuous-time hardware. To address this, we develop a unified continuous-time framework for spiking neural networks (SNNs) that couples the Law of Charge Conservation with minimal neuron-level constraints. This integration ensures that the terminal state depends solely on the aggregate input charge, providing a unique cumulated output invariant to temporal stochasticity. We prove that this mapping is strictly invariant to spike timing in acyclic networks, whereas recurrent connectivity can introduce temporal sensitivity. Furthermore, we establish an exact representational correspondence between these charge-conserving SNNs and quantized artificial neural networks, bridging the gap between static deep learning and event-driven dynamics without approximation errors. These results establish a rigorous theoretical basis for designing continuous-time neuromorphic systems that harness the efficiency of asynchronous processing while maintaining algorithmic determinism.

Determinism in the Undetermined: Deterministic Output in Charge-Conserving Continuous-Time Neuromorphic Systems with Temporal Stochasticity

Abstract

Achieving deterministic computation results in asynchronous neuromorphic systems remains a fundamental challenge due to the inherent temporal stochasticity of continuous-time hardware. To address this, we develop a unified continuous-time framework for spiking neural networks (SNNs) that couples the Law of Charge Conservation with minimal neuron-level constraints. This integration ensures that the terminal state depends solely on the aggregate input charge, providing a unique cumulated output invariant to temporal stochasticity. We prove that this mapping is strictly invariant to spike timing in acyclic networks, whereas recurrent connectivity can introduce temporal sensitivity. Furthermore, we establish an exact representational correspondence between these charge-conserving SNNs and quantized artificial neural networks, bridging the gap between static deep learning and event-driven dynamics without approximation errors. These results establish a rigorous theoretical basis for designing continuous-time neuromorphic systems that harness the efficiency of asynchronous processing while maintaining algorithmic determinism.
Paper Structure (27 sections, 7 theorems, 29 equations, 2 figures)

This paper contains 27 sections, 7 theorems, 29 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related Works
  3. Asynchronous and event-driven spiking computation.
  4. Continuous-time neuron models and terminal output.
  5. Conservation- and constraint-based perspectives.
  6. QANN-SNN correspondence and terminal output computation.
  7. Network topology and determinism.
  8. Charge-Conserving Continuous-Time SNNs
  9. Single Neuron Dynamics
  10. Charge-conserving neuronal dynamics.
  11. Additional neuron-level design conditions.
  12. Network Coupling via Charge-Preserving Transmission
  13. Spike-wise transmission interface.
  14. Charge-normalized synaptic filtering and bounded delays.
  15. External episode charge.
  16. ...and 12 more sections

Key Result

Lemma 4.1

Let $V(t)$ and $Q(t)$ evolve according to the charge-based membrane dynamics Equation (eq:neuron-charge-dynamics), with spike-triggered discharge represented as an atomic measure. Then, for any $T \ge 0$, the membrane charge satisfies the identity where denotes the cumulative injected input charge.

Figures (2)

  • Figure 1: Charge-based abstraction of continuous-time spiking neurons. (A) Neuron-level dynamics. (B) Network interface. (C) Synaptic transmission.
  • Figure 2: Structural correspondence between charge-based spiking neurons and artificial neurons.

Theorems & Definitions (13)

  • Lemma 4.1: LoCC for a Single Neuron
  • Lemma 4.2: Uniqueness of discharge at silence under the decoding structure
  • Lemma 4.3: Finite spiking under strict progress toward silence
  • Theorem 4.4: Time-Independent Input--Output Map of a Single Neuron
  • Theorem 4.5: Acyclic networks admit a unique timing-invariant terminal output
  • Remark 4.6: ANN representation of the terminal computation
  • Remark 4.7: Finite-time termination without uniform bounds
  • Example 4.8: Static fixed points need not be dynamically reachable
  • Remark 5.1: Interpretation as a generalized ST-BIF neuron
  • Theorem 5.2: Terminal output admits a QANN representation
  • ...and 3 more