Bio-Inspired Constant-Time Arithmetic Kernels in Hybrid Membrane–Neural Spiking P Systems

This work introduces Hybrid Membrane–Neural P systems (HMN P systems), a computational model that integrates principles from membrane computing and spiking neural P systems. The resulting framework offers a versatile foundation for the development of bio-inspired arithmetic architectures. Within this setting, we propose a compact family of arithmetic kernels capable of executing signed addition, subtraction, multiplication, and division in both modular and non-modular arithmetic domains. By leveraging intrinsic spike aggregation, spike–anti-spike annihilation, and exhaustive rule application, the proposed designs achieve efficient and reliable arithmetic computation in a constant number of simulation steps under exhaustive semantics and assuming synchronized input, independent of operand values. Addition and subtraction are executed intrinsically upon spike arrival, requiring no internal computation steps, while multiplication and division are completed in a single simulation step by one neuron. Furthermore, we introduce a modular-reduction kernel that operates in two simulation steps with a single neuron, and leverage its modular structure to construct modular multiplication and division through composition with non-modular arithmetic modules. Comparative evaluations against representative SNP and SNQ arithmetic designs demonstrate that HMN kernels achieve operand-independent execution time while requiring fewer neurons. Distinct from most existing approaches, the HMN framework natively supports signed operands through a dual-spike representation, thereby eliminating the need for auxiliary sign-handling mechanisms. Asynchronous spike arrivals can be managed by an optional synchronization membrane; since this mechanism is decoupled from the arithmetic kernels, its overhead is excluded from kernel performance and reported separately. Collectively, these results establish HMN systems as an efficient and modular platform for constant-time arithmetic computation, offering reusable arithmetic kernels that serve as a foundation for higher-level constructions, including those arising in elliptic-curve and modular arithmetic.