Petri Net Description of Biological Neural Circuits for Fast Hardware Prototyping

Carlo daCunha; Rodrigo Pena; Marcos Turqueti

cs.NEAug 20, 2026

Petri Net Description of Biological Neural Circuits for Fast Hardware Prototyping

Authors: Carlo daCunha, Rodrigo Pena, Marcos Turqueti

Organizations: Helen and John C. Hartmann Department of Electrical Engineering, New Jersey Institute of Technology, 323 Martin Luther King Jr. Blvd., Newark, 07102, NJ, USA. · School of Applied Engineering and Technology, New Jersey Institute of Technology, 323 Martin Luther King Jr. Blvd., Newark, 07102, NJ, USA. · Department of Biological Sciences, Florida Atlantic University, 5353 Parkside Drive, Jupiter, 33458, FL, USA. · Electronic Systems Group, Lawrence Berkeley National Laboratory, 1 Cyclotron Road., Berkeley, 94720, CA, USA.

Abstract

Current approaches to simulating biological neural circuits, whether on general-purpose hardware or dedicated neuromorphic platforms, remain constrained by fixed-timestep numerical integration, hardware-imposed precision limits, and an inability to guarantee timing correctness for event-driven spiking dynamics under real-time constraints. Here, we propose a Petri net description of biological neural circuits that overcomes these limitations by modeling neurons, synapses, and spike events as a T-timed Petri net with formally verifiable timing semantics, enabling deadline-guaranteed real-time execution and analytically tractable correspondence to continuous-time leak-integrate-and-fire dynamics, independent of the underlying integration timestep. To test the model, we present the results of three simulated microcircuits: feedback inhibition, lateral inhibition, and hierarchical feature detector. The Petri neuron reproduces the expected dynamical signatures of each circuit while providing formally bounded timing guarantees throughout, with worst-case response times matching analytical predictions across all three cases.

Figures & tables

Figure 1: Phases of a biological action potential. Depolarization (rapid voltage rise past threshold), repolarization (voltage returns toward rest), hyperpolarization (transient overshoot below resting potential), and the subsequent return to rest via ionic pump activity. The refractory period spans repolarization and hyperpolarization, during which the neuron is unresponsive (absolute) or requires elevated excitation to fire (relative). This cycle motivates the input-gating, threshold-firing, and recovery transitions of the Petri neuron in Fig. 2 .

Figure 2: Petri net representation of the Petri neuron, mapping accumulation, threshold firing, axonal propagation, and refractory recovery onto places and transitions.

	$T_{input}$	$T_{spike}$	$T_{prop}$	$T_{rec}$	$T_{leak}$
$p_{acc}$	$1/0/+1$	$0/\theta/-\theta$	$0/0/0$	$0/0/0$	$0/1/-1$
$p_{rdy}$	$0/0/0$	$0/1/-1$	$0/0/0$	$1/0/+1$	$0/0/0$
$p_{pre}$	$0/0/0$	$1/0/+1$	$0/1/-1$	$0/0/0$	$0/0/0$
$p_{out}$	$0/0/0$	$0/0/0$	$1/0/+1$	$0/0/0$	$0/0/0$
$p_{rec}$	$0/0/0$	$1/0/+1$	$0/0/0$	$0/1/-1$	$0/0/0$

Table 1: Incidence matrix with entries

C^{+}/C^{-}/C

\theta

denotes the current token count of

p_{acc}

at firing time (flush arc).

Figure 3: Firing rate as a function of input excitation

I/I_{th}

for the LIF model (Eq. 3 ) and the Petri neuron (Eq. 5 ), parameterized via the Padé mapping. Agreement is close for spontaneous and moderate firing (

f<50

Hz), with relative error shrinking further as

I

increases into the high-frequency regime.

Machine	Nom. (ms)	Mean	Std	p99	$\epsilon$ (max)
AMD	$1$	$9.7$	$2.3$	$16.9$	$95.9$
	$5$	$171.4$	$221.6$	$717.2$	$901.9$
	$10$	$207.4$	$227.7$	$728.2$	$917.7$
	$50$	$198.6$	$234.7$	$728.6$	$885.0$
	$100$	$252.0$	$240.8$	$730.0$	$905.3$
RP2040	$1$	$0.03$	$0.17$	$1.00$	$1.00$

Table 2: AMD Linux POSIX timer and RP2040 time jitter. All values in

\mu

\epsilon

denotes the worst-case absolute error (

N=10{,}000

Figure 4: Feedback inhibition network. Chevron indicates an excitatory synapse, while the small black disks indicate inhibitory synapses.

Parameter	E	I
$R$ [M $\Omega$ ]	30–60	337
$C$ [ $\times 10^{-10}$ F]	1.3–4	0.44–0.74
$\tau$ [ms]	8–12	15–25
$L$ [mm]	1–3	4–8

Table 3: Neuronal parameters of excitatory (E) and inhibitory (I) neurons in a typical feedback-inhibition network.

Figure 5: Inter-event interval histogram for the firing of E and I neurons in a feedback inhibition network simulated using Petri neuron.

Figure 6: Lateral inhibition microcircuit motif with excitatory neurons

E_{A,B}

and inhibitory interneurons

I_{A,B}

Figure 7: IEIH of excitatory neurons and their firing rates as a function of time.

Figure 8: The hierarchical feature detector consisting of a

2\times 2

grid, simple neurons

S

that capture lines, and complex neurons

C

that indicate the presence of horizontal and vertical lines.

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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.

Abstract

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