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We consider the task of measuring time with probabilistic threshold gatesimplemented by bio-inspired spiking neurons. In the model of spiking neuralnetworks, network evolves in discrete rounds, where in each round, neurons firein pulses in response to a sufficiently high membrane potential. This potentialis induced by spikes from neighboring neurons that fired in the previous round,which can have either an excitatory or inhibitory effect. We first consider adeterministic implementation of a neural timer and show that $\Theta(\log t)$(deterministic) threshold gates are both sufficient and necessary. This raisedthe question of whether randomness can be leveraged to reduce the number ofneurons. We answer this question in the affirmative by considering neuraltimers with spiking neurons where the neuron $y$ is required to fire for $t$consecutive rounds with probability at least $1-\delta$, and should stop firingafter at most $2t$ rounds with probability $1-\delta$ for some input parameter$\delta \in (0,1)$. Our key result is a construction of a neural timer with$O(\log\log 1/\delta)$ spiking neurons. Interestingly, this construction usesonly one spiking neuron, while the remaining neurons can be deterministicthreshold gates. We complement this construction with a matching lower bound of$\Omega(\min\{\log\log 1/\delta, \log t\})$ neurons. This provides the firstseparation between deterministic and randomized constructions in the setting ofspiking neural networks. Finally, we demonstrate the usefulness of compressedcounting networks for synchronizing neural networks.

Counting to Ten with Two Fingers: Compressed Counting with Spiking Neurons