In Pavlovian delay eyeblink conditioning, the cerebellum represents the passage-of-time (POT) between onsets of conditioned and unconditioned stimuli (CS and US, respectively). To study possible computational mechanisms of the POT representation we built a large-scale spiking network model of the cerebellum. Consistent with our previous rate-coding model, we found two conditions necessary for the present model to represent the POT with a dynamic population of active granule cells: (i) long temporal integration of input signals; and (ii) random recurrent connections between granule and Golgi cells. When these conditions were satisfied, a nonrecurrent sequence of active granule cell populations was generated in response to a CS and, conversely, the POT from the CS onset was able to be read out from the sequence. Specifically, simulated N-methyl-D-aspartate (NMDA) channels with a long decay time constant at granule and Golgi cells were responsible for the long temporal integration. Thus, blocking the NMDA channels or ablating Golgi cells impaired the POT representation. Simulated glomerulus structure made POT representation robust against noise in mossy fibre inputs. Long-term potentiation induced at mossy fibre synapses on granule cells also served to enhance the robustness. We reproduced some experimental results of Pavlovian delay eyeblink conditioning using the present model. These results suggest that the recurrent network in the granular layer and NMDA channels in granule and Golgi cells play an essential role in the timing mechanisms in the cerebellum, whereas the glomerulus serves to realize a robust representation of time.

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A neuroecological equation of the Lotka-Volterra type for mean firing rate is derived from the conventional membrane dynamics of a neural network with lateral inhibition and self-inhibition. Neural selection mechanisms employed by the competitive neural network receiving external inputs are studied with analytic and numerical calculations. A remarkable findings is that the strength of lateral inhibition relative to that of self-inhibition is crucial for determining the steady states of the network among three qualitatively different types of behavior. Equal strength of both types of inhibitory connections leads the network to the well-known winner-take-all behavior. If, however, the lateral inhibition is weaker than the self-inhibition, a certain number of neurons are activated in the steady states or the number of winners is in general more than one (the winners-share-all behavior). On the other hand, if the self-inhibition is weaker than the lateral one, only one neuron is activated, but the winner is not necessarily the neuron receiving the largest input. It is suggested that our simple network model provides a mathematical basis for understanding neural selection mechanisms.

The response of a noisy integrate-and-fire neuron with reset to periodic input is investigated. We numerically obtain the first-passage-time density of the pertaining Ornstein-Uhlenbeck process and show how the power spectral density of the resulting spike train can be determined via Fourier transform. The neuron's output clearly exhibits stochastic resonance.