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In everyday life all of us experience a large random variability in our behaviour even when external environment does not provide any source of noise, e.g. to decide between competing opportunities or to predict the occurrence of an imminent event. Often such variability shows up in intervals between behavioural events spanning a wide range of time scales: from hundreds of milliseconds, the times needed to react to an external "go" signal, to longer periods on the scale of seconds as the durations of alternating perceptions of rival visual stimuli.
Nervous cells in the brain communicate by exchanging "digital" events, and the pooling of such "spikes", from a network of neocortical neurons, typically shows irregular temporal patterns of the instantaneous firing rate. This finite-size effect provides a natural endogenous source of noise which often has been related to behavioural variability. Furthermore, populations of synaptically coupled spiking neurons are proven to show firing rate dynamics supporting attractors such that a subset of cells may fall in "up" and "down" population states, if appropriate stimuli are provided, and there persist for arbitrary long time. Multi-stable stochastic neuronal networks has been used for these reasons to successfully model the neuronal substrate of working/short-term memory and decision making. Here we get a step further by envisaging a processing stage collecting several of these stochastic "switches", and considering upward and downward transitions driven by finite-size fluctuations, which allow to have a collective dynamics with time scales even much longer than the time constants available to the single components of the whole system. Transition probabilities are modulated both by synaptic coupling intensity and by input currents coding the external stimulation strength. Monitoring the fraction of active switches may give an accurate estimate of the "evidences" accumulated or subtracted during time and provided as stimuli. We show how and in which conditions such a stochastic multi-modular system behaves like an integrator. Because it keeps track of the fraction of active switches even in the absence of stimuli, the integrator can be seen as a quasi-continuous attractor or memory whose content may be read out by other modules. We further suggest how a bistable pattern of activity may be recognized from the analysis of single neuron inter-spike intervals (ISI) even in a strongly non-stationary condition of stimulation, predicting that suitably normalized ISI histograms of neurons in the stochastic integrator and in the read-out module should show a qualitatively different shape, even if firing rate dynamics are indistinguishable. From a wider perspective, we argue why our network of switches could be suggested as a substrate of an inner representation of a generic concept of "magnitude". In support, we compare our predictions on the dynamics driven by discrete and continuous stimulations with the recent results from experiments on counting and timing tasks, both in animals and humans. In particular, we show how the so called "scalar variability", by which fluctuations in the estimate of a magnitude are proportional to the magnitude itself, is qualitatively well expressed by our system even if a failure of the Weber-Fechner's law is predicted: the coefficient of variation of "sensation magnitude" is roughly constant only for a limited interval of magnitudes, and a U shape is what in general is expected, in agreement with measures reported by several experiments. Since the presented stochastic accumulator employs the same dynamical building blocks and mechanisms to support other kinds of cognitive processing, we further speculate about a possible relationship between working memory abilities and the "sense of magnitudes", as witnessed by an increasing amount of experimental evidence. Such a theoretical framework then suggest that the noise is a needed feature of the nervous system and not something to neglect or remove. It could even allow to make sophisticated computations as the preverbal sense of numbers and magnitudes which should account for the simple arithmetic reasoning in humans and other animals. |
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