Petri Net Description of Biological Neural Circuits for Fast Hardware Prototyping

We study the expressivity of spiking neural networks, which provide a natural framework for asynchronous, event-driven computation complementary to conventional feedforward neural networks. We consider the time-to-first-spike model in a setting for which the input-output map is continuous and piecewise linear, with affine pieces governed by causal feasibility constraints that determine which presynaptic spikes occur before a neuron fires. We first show that each neuron's firing time admits a maxout-like representation with exponentially many, highly constrained affine pieces. We then formalize causal regions as polyhedral regions with fixed causal sets and derive upper and lower bounds on the maximal number of causal regions in both shallow and multilayer feedforward spiking networks. Our theoretical and experimental results show that spiking networks can generate richer partitions of the input space than conventional feedforward ReLU networks.

Spiking neural networks (SNNs) promise low-power event-driven computation for temporally rich tasks, but commonly used neuron models often trade off gradient-based trainability, dynamical richness, and high activity sparsity. These limitations are acute in regression, where approximation error, noise and spike discretization can severely degrade continuous-valued outputs. Indeed, many state-of-the-art (SOTA) SNNs rely on simple phenomenological dynamics trained with surrogate gradients and offer limited control over spiking diversity and sparsity. To overcome such limitations, we introduce multi-timescale conductance spiking networks, a gradient-trainable framework in which neural dynamics emerge from shaping the current-voltage (I-V) curve by tuning fast, slow and ultra-slow conductances. This parametrization allows systematic control over excitability, can be implemented efficiently in analog circuits, and yields rich firing regimes including tonic, phasic and bursting responses within a single model. We derive a discrete-time formulation of these differentiable dynamics, enabling direct backpropagation through time without surrogate-gradient approximations. To probe both trainability and accuracy, we evaluate feedforward networks of these neurons at the predictability limit of Mackey-Glass time-series regression and compare them to baseline LIF and SOTA AdLIF networks. Our model outperforms LIF and AdLIF networks, while exhibiting substantially sparser activity from both communication and computational perspectives. These results highlight multi-timescale conductance spiking neurons as a promising building block for energy-aware temporal processing and neuromorphic implementation.

Short-term plasticity (STP) is fundamental to temporal information processing in biological neural systems but remains difficult to realize efficiently in neuromorphic hardware. Memristive electrochemical random-access memory (ECRAM) devices naturally exhibit non-equilibrium ionic dynamics that produce transient conductance modulation; however, these behaviors are typically treated as undesirable variability or tolerated as side effects in memory-centric computing paradigms. In this work, we instead transform these volatile dynamics from a tolerated device artifact into a computational resource through a cross-layer device-circuit-system co-design framework. We introduce a delay-feedback leaky integrate-and-fire (LIF) neuron architecture co-designed with ECRAM synapses that exploits activity-dependent conductance modulation with negligible additional circuit overhead. The architecture integrates ECRAM-based synapses with a tunable delay-feedback spike-generation path, enabling transient device dynamics to directly modulate neuron excitability and synaptic efficacy. We used experimentally characterized ECRAM devices exhibiting transient conductance modulation (1.5 KOhms per spike) to develop a compact behavioral model suitable for circuit-level simulation. Circuit simulations demonstrate two key STP behaviors -- synaptic facilitation and intrinsic excitability modulation -- while consuming 2 pJ per spike, and the same device-driven mechanisms extend across multiple neuron topologies. Network-level analysis further demonstrates frequency-selective spike processing, allowing individual synapses to act as tunable temporal filters within spiking neural networks. This work demonstrates that non-equilibrium ECRAM dynamics can serve as a native hardware substrate for STP and temporal computation in neuromorphic circuits.